Presentation from the MGTA event in Rome
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Jun 25, 2024
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About This Presentation
Presentation from the MGTA event in Rome
Slide Content
Optimal Redistribution
via Income Taxation and Market Design
Mohammad Akbarpour (Stanford University)
Paweł Doligalski(University of Bristol)
Piotr Dworczak
?
(Northwestern University; GRAPE)
Scott Duke Kominers(Harvard University; a16z)
June 19, 2024
Summer School, Markets and Governments: a Theoretical Appraisal
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the European Research Council.
via Income Taxation and Market Design
Mohammad Akbarpour (Stanford University)
Paweł Doligalski(University of Bristol)
Piotr Dworczak
?
(Northwestern University; GRAPE)
Scott Duke Kominers(Harvard University; a16z)
June 19, 2024
Summer School, Markets and Governments: a Theoretical Appraisal
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the European Research Council.
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
žHousing(rent control; public housing; LIHTC in the US)žHealth care(public health care in Europe, Medicare and Medicaid in the US)žFood(food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)
žEnergy(electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )
žCovid-19 vaccines(“communist” solution within countries versus “wild west
capitalism” across countries)
žRoad accessžLegal services
ž...Policymakers didn’t take Econ 101...?
Inequality-aware Market Design
žRunning assumptions throughout:
žI Welfare Theorem holds.
žSocial planner is benevolent and competent.
žSocial planner has redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
žRunning assumptions throughout:
žI Welfare Theorem holds.
žSocial planner is benevolent and competent.
žSocial planner has redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
Inequality-aware Market Design
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
Inequality-aware Market Design
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
Inequality-aware Market Design
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
Inequality-aware Market Design
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
žConventional wisdom #1:
II Welfare Theoremimplies that markets are a superior
allocation mechanism even under redistributive preferences.
žCaveat: II WT does not account forprivate information!žInequality-aware Market Design (IMD):
How to designindividualmarkets in the presence of
socioeconomicinequalityandprivate information?
žMain takeaway: Marketdistortions(taxes, subsidies, inefficient
rationing) can be part ofoptimalmechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
žThere are 3 agents: Ann, Bob, and Claire.
žThere are 2 homogeneous goods (e.g., houses) to allocate.žAgents have the following utilitiesufor a house:
Ann: 1/4, Bob: 3/4, Claire: 1.
žHowever, these agents also differ in wealthw:
Ann: 8, Bob: 1, Claire: 1.
žAll agents have the same utility function
log(w) +u˙x;
wherexis the probability of having a house.
žSocial planner aims to maximize the sum of utilities.
Inequality-aware Market Design—Example
Inequality-aware Market Design—Example
Inequality-aware Market Design—Example
Inequality-aware Market Design—Example
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8
Bob 3/4 1
Claire 1 1
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8
Bob 3/4 1
Claire 1 1
Inequality-aware Market Design—Example
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Alternativeinterpretation of the problem:
žNormalize the values for money in agent’s utilities to 1 (the usual
quasi-linear model);
žThink of values for money as (unobserved) welfare weights.
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Alternativeinterpretation of the problem:
žNormalize the values for money in agent’s utilities to 1 (the usual
quasi-linear model);
žThink of values for money as (unobserved) welfare weights.
Inequality-aware Market Design—Example
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Alternativeinterpretation of the problem:
žNormalize the values for money in agent’s utilities to 1 (the usual
quasi-linear model);
žThink of values for money as (unobserved) welfare weights.
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Alternativeinterpretation of the problem:
žNormalize the values for money in agent’s utilities to 1 (the usual
quasi-linear model);
žThink of values for money as (unobserved) welfare weights.
Inequality-aware Market Design—Example
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
value for goodvalue for moneywillingness to pay
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙1=3
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
0+1˙2=3
Claire 1 1 1
1Γ1˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
2=3
Claire 1 1 1
11˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
2=3
Claire 1 1 1
11˙1=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
2=3
Claire 1 1 1
2=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
5=24
Bob 3/4 1 3/4
2=3
Claire 1 1 1
2=3
Mechanism #1:Market equilibrium(+ lump-sum transfers)Market-clearing price: 1
House owners: Ann & Claire
Lump-sum transfer: 2=3Total utility: 37=24
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
2=3˙1=4
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
2=3˙3=4
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3˙1
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Total utility: 32=24 (<37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Among all mechanisms charging a single pricep,p=1 is optimal.
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=6
Bob 3/4 1 3/4
1=2
Claire 1 1 1
2=3
Mechanism #2:Random allocation
Price: 0
House owners: 66% Ann & 66% Bob & 66% Claire
Lump-sum transfer: 0Among all mechanisms charging a single pricep,p=1 is optimal.
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
1=4Γ1=8˙2=3
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
1=2˙3=4+1˙1=3
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
1=2˙1Γ1˙1=3
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
20=24
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
20=24
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3Total utility: 41=24(>37=24)
Inequality-aware Market Design—Example
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
20=24
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3This is in fact the optimal mechanism! (subject to IC constraints)
val. goodval. moneyWTP utility
Ann 1/4 1/8 2
4=24
Bob 3/4 1 3/4
17=24
Claire 1 1 1
20=24
Mechanism #3: Allocate 1st unit bylottery, sell 2nd unit inmarketMarket-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4Γ1=8˙1=1=2˙1=4)
House owners: Ann & 50% Claire & 50% Bob
Lump-sum transfer: 1=3This is in fact the optimal mechanism! (subject to IC constraints)
Inequality-aware Market Design—Example
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
Inequality-aware Market Design—Example
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
Inequality-aware Market Design—Example
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
Inequality-aware Market Design—Example
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
val. moneyWTPutility—marketutility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:žValues for money (welfare weights) are negatively correlated with
willingness to pay.
žMarket behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
žThe designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.
Inequality-aware Market Design—Papers
žDKA (2021):
žTwo-sided market with buyers and sellers for an indivisible good
žRationing used if significant inequality within a side of the market
žTaxes/ subsidies used if inequality across the two sides of the market
žADK (2023):
žOne-sided allocation problem with goods differing in quality, observable
information, and flexible preferences over revenue
žA Budish DK (2023):
žApplication to allocation of Covid-19 vaccines (interaction of redistributive
preferences with allocative externalities)
žTokarski KAD (2023):Application to energy pricing in Europe
žD (2023), Yang DA (2024):Optimality of using costly screening (e.g., queuing)
žDKA (2021):
žTwo-sided market with buyers and sellers for an indivisible good
žRationing used if significant inequality within a side of the market
žTaxes/ subsidies used if inequality across the two sides of the market
žADK (2023):
žOne-sided allocation problem with goods differing in quality, observable
information, and flexible preferences over revenue
žA Budish DK (2023):
žApplication to allocation of Covid-19 vaccines (interaction of redistributive
preferences with allocative externalities)
žTokarski KAD (2023):Application to energy pricing in Europe
žD (2023), Yang DA (2024):Optimality of using costly screening (e.g., queuing)
IMD and Income Taxation
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
IMD and Income Taxation
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
IMD and Income Taxation
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
IMD and Income Taxation
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
IMD and Income Taxation
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
žConventional wisdom #2:
The Atkinson-Stiglitz theoremimplies that redistribution should
be accomplished entirely via income taxation.
žCaveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.
žMain questions for this project:
žShould markets be optimally distorted when there is also
heterogeneity in tastes for goods?
žWhat is the interaction between IMD and income taxation?
IMD and Income Taxation
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
IMD and Income Taxation
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
IMD and Income Taxation
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
IMD and Income Taxation
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
IMD and Income Taxation
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Result #1:Assuming separable utility functions and multidimensional
heterogeneity, if
1.
2.
3.
then the conclusion of the Atkinson-Stiglitz theorem holds.
Result #2:If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.
žWe provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
žDistortions in markets are “generically” optimal.
Literature Review
žSuboptimalityofgoodsmarketdistortions:
Diamond and Mirrlees (1971), Atkinson and Stiglitz (1976), Gauthier and
Laroque (2009), Doligalski et al. (2023), ...
žTasteheterogeneityandfailureofAtkinson-Stiglitztheorem:
Cremer and Gavhari (1995, 1998, 2002), Saez (2002), Kaplow (2008),
Golosov et al. (2013), Gauthier and Henriet (2018), Scheuer and Slemrod
(2020), Hellwig and Werquin (2023), Ferey et al. (2023), ...
žMulti-dimensionalheterogeneityinpublicfinance:
Cremer et al. (2003), Blomquist and Christiansen (2008), Diamond and
Spinnewijn (2011), Piketty and Saez (2013), Saez and Stantcheva (2018),
Golosov and Krasikov (2023), Bierbrauer et al. (2023), ...
žIMDprecursors:Spence (1977), Weitzman (1977), Nichols and
Zeckhauser (1982), Besley and Coate(1991), Condorelli (2013), ...
žRelatedmechanism-designpapers:Jullien (2000), Che et al. (2013),
Fiat et al. (2016), Li (2021), Dworczak and Muir (2024), ...
žSuboptimalityofgoodsmarketdistortions:
Diamond and Mirrlees (1971), Atkinson and Stiglitz (1976), Gauthier and
Laroque (2009), Doligalski et al. (2023), ...
žTasteheterogeneityandfailureofAtkinson-Stiglitztheorem:
Cremer and Gavhari (1995, 1998, 2002), Saez (2002), Kaplow (2008),
Golosov et al. (2013), Gauthier and Henriet (2018), Scheuer and Slemrod
(2020), Hellwig and Werquin (2023), Ferey et al. (2023), ...
žMulti-dimensionalheterogeneityinpublicfinance:
Cremer et al. (2003), Blomquist and Christiansen (2008), Diamond and
Spinnewijn (2011), Piketty and Saez (2013), Saez and Stantcheva (2018),
Golosov and Krasikov (2023), Bierbrauer et al. (2023), ...
žIMDprecursors:Spence (1977), Weitzman (1977), Nichols and
Zeckhauser (1982), Besley and Coate(1991), Condorelli (2013), ...
žRelatedmechanism-designpapers:Jullien (2000), Che et al. (2013),
Fiat et al. (2016), Li (2021), Dworczak and Muir (2024), ...
Literature Review
žSuboptimalityofgoodsmarketdistortions:
Diamond and Mirrlees (1971),Atkinson and Stiglitz (1976), Gauthier and
Laroque (2009), Doligalski et al. (2023), ...
žTasteheterogeneityandfailureofAtkinson-Stiglitztheorem:
Cremer and Gavhari (1995, 1998, 2002), Saez (2002), Kaplow (2008),
Golosov et al. (2013), Gauthier and Henriet (2018), Scheuer and Slemrod
(2020), Hellwig and Werquin (2023), Ferey et al. (2023), ...
žMulti-dimensionalheterogeneityinpublicfinance:
Cremer et al. (2003), Blomquist and Christiansen (2008), Diamond and
Spinnewijn (2011), Piketty and Saez (2013), Saez and Stantcheva (2018),
Golosov and Krasikov (2023), Bierbrauer et al. (2023), ...
žIMDprecursors:Spence (1977), Weitzman (1977), Nichols and
Zeckhauser (1982), Besley and Coate(1991), Condorelli (2013), ...
žRelatedmechanism-designpapers:Jullien (2000), Che et al. (2013),
Fiat et al. (2016), Li (2021), Dworczak and Muir (2024), ...
žSuboptimalityofgoodsmarketdistortions:
Diamond and Mirrlees (1971),Atkinson and Stiglitz (1976), Gauthier and
Laroque (2009), Doligalski et al. (2023), ...
žTasteheterogeneityandfailureofAtkinson-Stiglitztheorem:
Cremer and Gavhari (1995, 1998, 2002), Saez (2002), Kaplow (2008),
Golosov et al. (2013), Gauthier and Henriet (2018), Scheuer and Slemrod
(2020), Hellwig and Werquin (2023), Ferey et al. (2023), ...
žMulti-dimensionalheterogeneityinpublicfinance:
Cremer et al. (2003), Blomquist and Christiansen (2008), Diamond and
Spinnewijn (2011), Piketty and Saez (2013), Saez and Stantcheva (2018),
Golosov and Krasikov (2023), Bierbrauer et al. (2023), ...
žIMDprecursors:Spence (1977), Weitzman (1977), Nichols and
Zeckhauser (1982), Besley and Coate(1991), Condorelli (2013), ...
žRelatedmechanism-designpapers:Jullien (2000), Che et al. (2013),
Fiat et al. (2016), Li (2021), Dworczak and Muir (2024), ...
Talk outline
1.
2.
3.
žOptimal design with a simple income effect
žOptimal design with welfare weights depending on taste
žOptimal design under correlation of ability and taste
1.
2.
3.
žOptimal design with a simple income effect
žOptimal design with welfare weights depending on taste
žOptimal design under correlation of ability and taste
Model
Model
Model
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
žWe introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.
žThere is a unit mass of agents characterized by two-dimensional
types:(t; ȷ), wheretis “taste” andȷis “ability.”
žSocial planner allocates:
žnumeraire consumption goodc2R,
žvector of goodsx2X∂R
L
+,žearnings (pre-tax)y2R.
žEach agent’s utility function is
U((c;x;y);(t; ȷ)) =u(c) +v(x;t)Γw(y;ȷ);
where the functionu;v;ware non-decreasing.
Model
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
Model
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
Model
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
Model
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
žTypes(t; ȷ)have a joint distributionF(t; ȷ)in the population.
žGoods inxcan be produced at fixed marginal costsk2R
L
+
(in terms of the numeraire).
žThe planner must respect the resource constraint:
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB;
whereBcould be positive (government expenditure) or negative
(outside source of revenue).
žThe planner uses social welfare weightsffl(t; ȷ)ffl0, with
average weight normalized to 1.
Planner’s problem
Choose an allocationa= (c;x;y)to maximize
Z
ffl(t; ȷ)U(a(t; ȷ);(t; ȷ))dF(t; ȷ)
subject to incentive-compatibility constraints
U(a(t; ȷ);(t; ȷ))fflU(a(t
0
; ȷ
0
);(t; ȷ));8(t; ȷ);(t
0
; ȷ
0
);
and the resource constraint
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB:
Choose an allocationa= (c;x;y)to maximize
Z
ffl(t; ȷ)U(a(t; ȷ);(t; ȷ))dF(t; ȷ)
subject to incentive-compatibility constraints
U(a(t; ȷ);(t; ȷ))fflU(a(t
0
; ȷ
0
);(t; ȷ));8(t; ȷ);(t
0
; ȷ
0
);
and the resource constraint
Z
[y(t; ȷ)Γc(t; ȷ)Γk˙x(t; ȷ)]dF(t; ȷ)fflB:
Atkinson-Stiglitz result
Atkinson-Stiglitz result
Atkinson-Stiglitz result
Atkinson-Stiglitz result
Given the additive separability, the efficient allocation of goods is
x
?
(t; ȷ) =argmax
x
fu(˜c(t; ȷ)Γk˙x) +v(x;t)g, where˜cdenotes
post-tax earnings (or total consumption expenditures).
Theorem
Suppose that:
1.There are no income effects: u(c) =c;c2R;
2.Welfare weights depend only on the ability type:
ffl(t; ȷ) =
¯
ffl(ȷ);8(t; ȷ);
3.Ability and taste types are statistically independent:
F(t; ȷ) =Ft(t)Fȷ(ȷ);8(t; ȷ), where Ftand Fȷdenote the marginal
distributions.
Then, the optimal mechanism induces the efficient choice of x, and
can be implemented with an income tax alone.
Given the additive separability, the efficient allocation of goods is
x
?
(t; ȷ) =argmax
x
fu(˜c(t; ȷ)Γk˙x) +v(x;t)g, where˜cdenotes
post-tax earnings (or total consumption expenditures).
Theorem
Suppose that:
1.There are no income effects: u(c) =c;c2R;
2.Welfare weights depend only on the ability type:
ffl(t; ȷ) =
¯
ffl(ȷ);8(t; ȷ);
3.Ability and taste types are statistically independent:
F(t; ȷ) =Ft(t)Fȷ(ȷ);8(t; ȷ), where Ftand Fȷdenote the marginal
distributions.
Then, the optimal mechanism induces the efficient choice of x, and
can be implemented with an income tax alone.
Atkinson-Stiglitz result
Given the additive separability, the efficient allocation of goods is
x
?
(t; ȷ) =argmax
x
fu(˜c(t; ȷ)Γk˙x) +v(x;t)g, where˜cdenotes
post-tax earnings (or total consumption expenditures).
Theorem
Suppose that:
1.There are no income effects: u(c) =c;c2R;
2.Welfare weights depend only on the ability type:
ffl(t; ȷ) =
¯
ffl(ȷ);8(t; ȷ);
3.Ability and taste types are statistically independent:
F(t; ȷ) =Ft(t)Fȷ(ȷ);8(t; ȷ), where Ftand Fȷdenote the marginal
distributions.
Then, the optimal mechanism induces the efficient choice of x, and
can be implemented with an income tax alone.
Given the additive separability, the efficient allocation of goods is
x
?
(t; ȷ) =argmax
x
fu(˜c(t; ȷ)Γk˙x) +v(x;t)g, where˜cdenotes
post-tax earnings (or total consumption expenditures).
Theorem
Suppose that:
1.There are no income effects: u(c) =c;c2R;
2.Welfare weights depend only on the ability type:
ffl(t; ȷ) =
¯
ffl(ȷ);8(t; ȷ);
3.Ability and taste types are statistically independent:
F(t; ȷ) =Ft(t)Fȷ(ȷ);8(t; ȷ), where Ftand Fȷdenote the marginal
distributions.
Then, the optimal mechanism induces the efficient choice of x, and
can be implemented with an income tax alone.
Atkinson-Stiglitz result
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Atkinson-Stiglitz result
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Atkinson-Stiglitz result
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Atkinson-Stiglitz result
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Atkinson-Stiglitz result
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Proof outline:
1.
2. t, dropping the IC
constraints of reportingttruthfully.
3.
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).
4.
˜c(t; ȷ) =c(t; ȷ) +k˙x
?
(t)(note:x
?
does not depend onȷ).
5. (˜c;y)that solves the relaxed problem for fixedtdoes
not depend ont.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Proof outline continued:
5.(˜c;y)does not depend ontbecause:žBy assumption 2, the distribution ofȷis the same for allt;
žBy assumption 3, the welfare weights do not vary witht;žBy assumption 1,tonly affects the “split” of total
consumption between numeraire and other goods.
6. (˜c(ȷ);y(ȷ);x
?
(t))is IC for the full problem:
žTotal expenditure and labor choices are incentive-compatible
by construction (they do not depend ont)
žExpenditure on goodsxis incentive-compatible because
each typetobtains the efficient amount.
Atkinson-Stiglitz result
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Atkinson-Stiglitz result
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Atkinson-Stiglitz result
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Comments:
žAssumption 1 is crucial—in particular, we have used the fact that
ccan be potentially negative.
žAssumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
žIf we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.
Optimality of market distortions
Optimality of market distortions
Optimality of market distortions
Framework
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
Framework
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
Framework
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
Framework
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
Framework
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
žTo solve the model without Assumptions 1–3, we make further
simplifying assumptions.
žThere is a single indivisible good,x2[0;1](can be interpreted
as bounded quality); earningsy2[0;¯y].
žAbility typeȷis binary:ȷ2 fȷ
L; ȷ
Hg, whereȷ
L≥0, andȷ
H>1.žLet≠
ibe the mass of ability typesi, andF
i(t)the CDF of tastet
conditional on ability typei, supported on[0;¯t], fori2 fL;Hg.
žUtility function is piece-wise linear:
(
c+t˙xΓ
1
ȷ
˙y cfflc;
Γ1 c<c;
wherecis a “subsistence” level.
Framework
Framework
Framework
Framework
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Framework
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Framework
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Commentsabout the framework:
žWillingness to pay is
WTP=
t c >c;
2[0;t]c=c;
0 c<c:
žSo efficiency requires
x
?
(t;c) =
1 c>candt>k;
2[0;1]c=candt>k;
0 c<cort<k:
žThe framework is a “best-case” scenario for income taxation.
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Results
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
We show that in the optimal mechanism:
žProvision of labor is always efficient.
žThe allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
žRough idea of the proof:
žLow-ability agents cannot mimic high-ability agents;
žHigh-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
žSuch constraints can be handled by adapting Myerson’s
ironing technique.
Details
Case # 1: Income effect
Case # 1: Income effect
Case # 1: Income effect
Case # 1: Income effect
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
Case # 1: Income effect
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
Case # 1: Income effect
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
Case # 1: Income effect
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
Case # 1: Income effect
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
žSuppose that Assumptions 2 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žffl
H(t) =0 and≠
Lffl
L(t) =1, for allt.
žSuppose that buyer and seller virtual surpluses
tΓ(1ΓF(t))=f(t)andt+F(t)=f(t)are non-decreasing.
žFinally, assume that
≠
H
ȷ
1Γ
1
ȷ
H
ff
<B+c+k;
=)income effect has bite.
Case # 1: Income effect
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Case # 1: Income effect
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Case # 1: Income effect
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Case # 1: Income effect
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Case # 1: Income effect
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Theorem
There exists an optimal mechanism that can be implemented as
follows:
žIncome is taxed at a rate1Γ1=ȷ
H(high-ability agents work);
žAnyone can purchase eitherž¯x
Lof the good at a unit price p
L<k; or
ž¯x
Hunits of the good at a unit price p
H(1Γ¯x
L=¯x
H) +p
L;
where p
Hlies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if¯x
H=1)
žEveryone receives a lump-sum transfer equal to c+p
L
¯x
L.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Discussion:
žIncome tax extracts all surplus but is not distortionary.žLow-ability agents with sufficiently hightpurchase the good at a
subsidized price subject to rationing.
žBasic intuition:žSuppose the good is priced atk. Low-ability agents are
constrained and buyx<1 despite having high tastet.
žConsider perturbing the price top=kΓž, for smallž >0.žThere is a second-order loss in efficiency (marginal effect)...ž... but there is a first-order gain in welfare since all types
t>kcan now buyžmore of the good (inframarginal effect)
žHigh-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.
Case # 1: Income effect
Goods allocation in optimal mechanism
Goods allocation in optimal mechanism
Case # 1: Income effect
Goods allocation in optimal mechanism
Goods allocation in optimal mechanism
Case # 1: Income effect
Numeraire consumption in optimal mechanism
Numeraire consumption in optimal mechanism
Case # 1: Income effect
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Case # 1: Income effect
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Case # 1: Income effect
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Case # 1: Income effect
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Case # 1: Income effect
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Is the result driven by our stylized “income effect”?
žModel with income effect very difficult to solve analytically.
žWe are working on two extensions:
žConcave utilityu(c)in a 2fi2 model;
žOptimality of distortion in the “same direction” in a model
with concave utilityu(c).
žNumerically, results appear robust.
Case # 1: Income effect
Goods allocation in optimal mechanism
Goods allocation in optimal mechanism
Case # 1: Income effect
Goods allocation in optimal mechanism (u(c) =
p
c)
Goods allocation in optimal mechanism (u(c) =
p
c)
Case # 1: Income effect
Goods allocation in optimal mechanism (u(c) =
p
c)
Goods allocation in optimal mechanism (u(c) =
p
c)
Case # 1: Income effect
Numeraire consumption in optimal mechanism
Numeraire consumption in optimal mechanism
Case # 1: Income effect
Numeraire consumption in optimal mechanism (u(c) =
p
c)
Numeraire consumption in optimal mechanism (u(c) =
p
c)
Case # 1: Income effect
Goods allocation in optimal mechanism (multiple ability types)
Goods allocation in optimal mechanism (multiple ability types)
Case # 1: Income effect
Numeraire consumption in optimal mechanism (multiple ability types)
Numeraire consumption in optimal mechanism (multiple ability types)
Case # 1: Income effect
Labor supply in optimal mechanism (multiple ability types)
Labor supply in optimal mechanism (multiple ability types)
Case # 1: Income effect
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Case # 1: Income effect
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Case # 1: Income effect
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Case # 1: Income effect
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Intuitionfor the result under concaveu:
žPurchasing the good implies higher marginal utility for money
u
0
(c)relative to agents who do not buy.žThus, the planner endogenously values giving more money to
agents who buy the good (higher typet).
žWelfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
žHelpful thought exercise: Think of the good being treatment for a
serious illness.
Case # 2: Welfare weights depend on taste type
Case # 2: Welfare weights depend on taste type
Case # 2: Welfare weights depend on taste type
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
žSuppose that Assumptions 1 and 3 from the AS result hold:
žF
L(t) =F
H(t) =F(t), for allt;
žu(c) =c;c2R.žSuppose that welfare weights can depend on the taste type.žLetΛ
i(t)be the average welfare weight on all types higher
thant, conditional oni2 fL;Hg.
žLetΛ(t) =≠
LΛ
L(t) +≠
HΛ
H(t):žLeth(t)be the inverse hazard rate, andJ(t) =tΓh(t)be
the virtual surplus function associated with distributionF.
žAssume thatž(Λ
i(t)h(t) +J(t)Γk)f(t)is non-decreasing whenever it is
negative, fori2 fL;Hg.
žΛ
L(t)fflΛ
H(t);for allt(weaker thanffl
L(t)fflffl
H(t)for allt).
Case # 2: Welfare weights depend on taste type
Theorem
There are two candidate optimal mechanism:
1.High-ability agents work efficiently at net wage of1=ȷ
H(with the
remaining surplus taxed away), and the good is provided at a
single price p
?
given by
p
?
=k+ (1ΓΛ(p
?
))h(p
?
):
2.High-ability agents work efficiently at net wage w>1=ȷ
H;and
the good is sold at a lower price to low-ability agents than to
high-ability agents.
Theorem
There are two candidate optimal mechanism:
1.High-ability agents work efficiently at net wage of1=ȷ
H(with the
remaining surplus taxed away), and the good is provided at a
single price p
?
given by
p
?
=k+ (1ΓΛ(p
?
))h(p
?
):
2.High-ability agents work efficiently at net wage w>1=ȷ
H;and
the good is sold at a lower price to low-ability agents than to
high-ability agents.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 2: Welfare weights depend on taste type
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Discussion:
žIn the first case, there is:
žSubsidy for the good if the average welfare weight on agents
buying it,Λ(p
?
), exceeds the average welfare weight 1.
žTax on the good if the average welfare weight on agents
buying it,Λ(p
?
), is below the average welfare weight 1.
žIn the second case:žThe reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
žIncentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.
Case # 3: Correlation of taste and ability types
Case # 3: Correlation of taste and ability types
Case # 3: Correlation of taste and ability types
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
žSuppose that Assumptions 1 and 2 from the AS result hold:
žffl
i(t) =¯ffl
i, for allt;
žu(c) =c;c2R.
žSuppose thatF
L(t)6=F
H(t).žLetΛ(t)be the average welfare weight on all types higher thant:
Λ(p) =
¯ffl
L≠
L(1ΓF
L(p)) +¯ffl
H≠
H(1ΓF
H(p))
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
:
žAssume thatž
Γ
tΓ(1Γ¯ffl
H)h
H(t)Γk
˙
f
H(t)is increasing when negative;
ž
Γ
tΓ(1Γ¯ffl
L)h
L(t)Γk
˙
f
L(t)crosses zero once from below;ž(
¯
ffl
LΓ1)h
L(r)ffl(
¯
ffl
HΓ1)h
H(r);for allt:
Case # 3: Correlation of taste and ability types
Theorem
There are two candidate optimal mechanism:
1.High-ability agents work efficiently at net wage of1=ȷ
H(with the
remaining surplus taxed away), and the good is provided at a
single price p
?
given by
p
?
=k+ (1ΓΛ(p
?
))h(p
?
):
2.High-ability agents work efficiently at net wage w>1=ȷ
H;and
the good is sold at a lower price to low-ability agents than to
high-ability agents.
Theorem
There are two candidate optimal mechanism:
1.High-ability agents work efficiently at net wage of1=ȷ
H(with the
remaining surplus taxed away), and the good is provided at a
single price p
?
given by
p
?
=k+ (1ΓΛ(p
?
))h(p
?
):
2.High-ability agents work efficiently at net wage w>1=ȷ
H;and
the good is sold at a lower price to low-ability agents than to
high-ability agents.
Case # 3: Correlation of taste and ability types
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Case # 3: Correlation of taste and ability types
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Case # 3: Correlation of taste and ability types
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Case # 3: Correlation of taste and ability types
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Case # 3: Correlation of taste and ability types
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Discussion:(recall thatp
?
=k+ (1ΓΛ(p
?
))h(p
?
))
žConditional on buying the good,tfflp
?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.
žThus, the goods market can be used to redistribute more utility
from high- to low-ability agents.
žAssuming that
¯
ffl
H=0, we have that
Λ(p)ı
1ΓF
L(p)
≠
L(1ΓF
L(p)) +≠
H(1ΓF
H(p))
<1()F
L(p)>F
H(p):
žIf ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
žIf ability and taste are negatively correlated, the good is
subsidized.
Concluding Remarks
Concluding Remarks
Concluding Remarks
Concluding Remarks
žWe investigated the optimaljoint designof anincome taxand
amarketfor goods.
žIncome taxation is used as the only redistributive instrument
under narrow assumptions.
žWhenever these assumptions fail,redistribution through
marketscan also be useful.
žThis isnotto say that all market interventions motivated by
redistributive motives are justified...
ž... but this does mean that traditionaltheoretical foundations
opposing goods market distortions are insufficient.
žWe investigated the optimaljoint designof anincome taxand
amarketfor goods.
žIncome taxation is used as the only redistributive instrument
under narrow assumptions.
žWhenever these assumptions fail,redistribution through
marketscan also be useful.
žThis isnotto say that all market interventions motivated by
redistributive motives are justified...
ž... but this does mean that traditionaltheoretical foundations
opposing goods market distortions are insufficient.
Concluding Remarks
Future research directions:
žRicher models of how Inequality-aware Market Design
interacts with optimal taxation.
žEmpirical applications.
žPolitical economy.
žAlternative approaches to measuring welfare and inequality.
Future research directions:
žRicher models of how Inequality-aware Market Design
interacts with optimal taxation.
žEmpirical applications.
žPolitical economy.
žAlternative approaches to measuring welfare and inequality.
Appendix
Appendix
Appendix
Results
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Results
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Results
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Results
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Results
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Lemma
There exists an optimal mechanism that features
žefficient provision of labor (high-ability agents choose y=¯y,
low-ability agents choose y=0);
žone-step allocation rule x
L(possibly with rationing) for low-ability
agents;
žat most a three-step allocation rule x
Hfor high-ability agents.
If there is no income effect (c! Γ1or B! Γ1), then both
allocation rules reach1at the highest step, and x
Hhas at most two
steps.
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Proof overview:
žIf high-ability agents do not work full time, increase their labor
supplyyand consumptioncto keep them indifferent:
žleaves utilities unchanged;
žrelaxes the incentive-compatibility constraints;žrelaxes the resource constraint=)strict improvement.žUse the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
žLump-sum payment pinned down by the resource constraint.žThe “subsistence constraint” (c
i(t)fflc) binds only at the highest
type¯tfor eachi2 fL;Hg, and hence depends onx
i(¯t).
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
žParameterize¯x
L=x
L(¯t),¯x
H=x
H(¯t), and assign Lagrange
multipliersı
Lffl0; ı
Hffl0 to the two subsistence constraints.
žIncentive-compatibility constraints:
žNo misreport oftconditional on true report ofȷ=ȷ
L:
=)x
L(t)is non-decreasing;
žNo misreport oftconditional on true report ofȷ=ȷ
H:
=)x
H(t)is non-decreasing;
žNo misreport oftand misreport ofȷ=ȷ
L
()ruled out by the assumption thatȷ
L≥0.
žNo misreport oftand misreport ofȷ=ȷ
H
()no misreport ofȷ=ȷ
Hbut truthful report oft
()an outside option constraint.
Back
Results
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
Results
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
Results
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
Results
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
Results
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
žAn outside option constraint:
∆ +
Z
t
0
x
H(∏)d∏fflU
L(t) :=
Z
t
0
x
L(∏)d∏;8t2[0;¯t]:
žIf we fixx
L, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
žDworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=)The optimal(∆;x
H)depend linearly onx
L.
žThe problem of optimizing overx
Lis linear with no constraints:
=)The optimalx
Ltakes the formx
L(t) =¯x
L˙1
ftfflt
?
L
g.
žThis implies a bound on the number of steps ofx
H(t). ⁄
Back
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