Intermediate Microeconomics Cheat Sheets

University of Notre Dame, Fall 2018

1. a>b
.
perfectCompte
-
.
MRTX
.
,×z=maxamt
Strictlypreferred
merits
:
fixedratiosof
good
Zthatcan
aIb
U=mih(
ax
,,bXz)
purchase
WII unit
weaklypreferred
.
perfect
subs
: reduction in
good
't
,
aisatleastasgood
total amount
negslopeofbudget
asb
U=aX
,tbxz line
,
Px
.
/p×
a
a~b
.
MRS
,
,
,×z=
the
.
Optimal
bundle
:
indifferent Max amount of
MRSAPEMRTAB
.
Or
.
Complete
:either Good2tobe
given
at kink
,
or corner
xdyoryhorbothupfor1moreof.HU/p,7V2/pz-s
.
Symmetric
:
similar
good
1.the
neg
.
of
Spend
more onX
,
to indifference the derivativeof
-
diminishing
MRS
:
-
transitive:ifXdy
indifference curve
strictly
convex
,
more
E.
yd2.thenXd2 Xz=fCX ,
)
,
'/MRS×a,×
,
Of
goodcompared
to
.
locally
nonsatiated
:
=
U
,102 other
,
less
"
Valuable
"
noblisspoint
.
Utility
Function
:
.
Lagrangian
:
.
Monotonicity
:
moreX1yiffUNRULY
)1.L=UH
.
,xzHXLgCx
))
ispreferred DUIDHLO
→
decreasing
2942×942×2,24×3=0
.
Strict
convexity
: dUldH70→
increasing
3.Solve forX
X12,412
,
then monotonic 4.X.
=fC
Xz)
XX
tLI
-
a)
y
>
2
.
Monotonic Trans
-
5.
Plug
intoMx
.
rational
:
complete
formations
:
same 6.SolveforXz
and transitive preferencesLderma
-
Complements
.
indifference curve
:fivemustbepositive
)
equally
desirable
.
Cobb
-
douglas
:
(w/ rational
,
mono
-
U=X.
a
Xz
'
-
a
tonic
,
E convex pref
-
MRSx.x.FM/u.asxzerencesmustbethin.Asa-
1.x.
→
important
Substitutes
non
-
increasing,
non
-
.
Budget
Line
:
setof
crossing
,preferred allbundles thatcauses
fartherfrom
origin
, an individual to
convexE.everywhere
)spendallincome

2. -
Individual demand curve
:
the
graphical
3.Tofind EV
,
.
Steps
to determine optimal decision under
relationship btw
p
,
EX
,
when
optimizing
i.Calculate
Utility
of
new
bundle Wnew)
uncertainty
When U (X)=XFXzLl
-
a)
Over
budget

set
x.
= a.

¥
.
×z=u
-as
.
¥
ii.Solve forX.

equiv
and
Xzequiv
I.Determine thepossible Statesof

theworld
L= ULXITXCI
-
pix
,
-
pzxz
)
(Mlk
,
1MHz )
=
Pxilpxz 2.Under each state
,
determine the
-
Law

of demand
:
If
p.lv
,
then X.
T Uncw
=
ULX
,,
Xz)
t
Use
Oldprices wealth the individual would have
If

¥170
,
Law of demand holds
iii.Solve for

compensated income 3.Setup

expected utility
.
Engle
Curve
:
function that

shows an
I
equiv
=p
,
orig
X.

equivtpzxzequivEU-p.ULX.lt/2zULXzlti..tpnULXn
)
individual'sdemand foragood
atdiff
.
iv.EV=I
equiv
-
long
4.Take first

order conditions)ofEU
Levelsof

income
3.TOfindCV
,
function

wrtthevariablels )the individual
-
ElasticityA.B
:
EA
, ,,
elasticity
ofAWrt
i.Calc
.
Origutility
(
Uorig
) can

choose
B
,
B
elasticity
ofA.thef.
change
in
ii.SolveforXsubsdaysubs
5.Solvefor

variables)to determine
the
Athat

results from
a
I't

change
inB t.MU//MUy)=px1py
choice that

would maximize EU
Uorig
=
ULX
,
y
)
-
Risk Adverse
:
avoid
an even
bet
,
dislike
EA,B=fF3
.Ben = -
BF
iii.Solvefor

compensated income risk'swould
pay
to

avoid it
,
ULX)is

strictly
.
magnitude
OfEa
, B
→
whether effect

Of
lump
=
phew Xsubtpyysub
concave
,
U'
'
LN LO
B becomes
"
magnified
"
(IEA
,
13171)or
iv.C✓=/
orig
-
lamp
.
Risk Neutral
:
indifferent to
even
bet
.
"
dampened
"
(IEA
,
BILL)asit
Changes
A
CVEEV canbe used to evaluate policies
.
indifferent to

riskda wouldn't
pay
to
.
sign
ofEA.rs
→
A moves in same l
t
)Or
Ex
:
subsidy
costSIXE.est
.
sum of

society
'savoidit
.
UCX)is linear
,
U'
'
CXI=0
opposite
(
-
)direction asB
CVIEV isSly
→
inefficient

if$4>SIX
.
Risk
Loving
:
would take
an even
bet
,
like
-
Demand Elasticity
:
Ex
, ,
.
Marginal
Rate ofTime Preference
:
risk'swould
pay
for
a
gamble
,
ULX)is
-
Own Price Demand Elasticity
:E
x.
p
1.
The
rate at
which
a consumer
is

willing
to
concave
,
U'
'
LX)>O
Iflawof demand holds
,
Ex,pL0
substitute

current

consumption for future
.
Risk Premium
:
the amount Less than EV
IfIEx.pl
>I
→
elastic
,
<I
→
inelastic
,
=l→ consumption a consumer would
accept

in

exchange
unit elastic(where rev
.
Maximized )
2.MRS
x. ixz
When
X
,
is
con
.
today
and forthelottery
Cobb
-
Douglas
:
-
I Xz
is
con
.
tomorrow 2.the
' '
fee
"
paid
to
avoid risk

inherent
.
Cross Price
Elasticity
of Demand
:
Ex
;
,pj
Usually MRSX
.,xz
>I(impatient
)
tothe
lottery
Exi
,
pj
LO
→
complements
↳U=X9
'
XYZ Where
a
,
>Az 3.The value
,r
,
such that ULEV
-
r
)=EU
Exi
,
pj
>O
→
substitutes
*
Need Unitsof

money insametimeper
.
where EU isthe expected utilityofthe
-
Income Demand Elasticity
:E
x.
I
lntom
.
's$
,
budget
constraint
:
lottery
E
x. I
>O
→
normal
(Iti )IitIz= LIti)XitX2
.
To solvefor

risk
premium
:
Ex
,
ISO
→
inferior
'
Marginal
Rate

of Intertemporal Trans
:
I.solveforEv
MRT×
,
,×z=
Iti(
"
luz
=
Iti)
2.Solve forEU
.
Substitution Effect
:
the
Change
in

an
.
L=UL×
,
,×z)t×uHilI
,
+
Iz
-
Lltilx
,
-
Xz)
3.
plug
into
equation
,
ULEV
-
r)=EU
.
E
individual'S
consumption Of
a
good
due
.
Human capital production Function
: a
solvefor
r
tothefactthe relative
prices
have
Changed
math functiondescribing
how

consumption
U
,
Uz
)
U
,
s
-
s
-
-
=
-
good
I
pm
,
)
¥22
Purchase
more
today
affects future

income
.
1. Find optimal bundle before
.z Draw new
budget
line
p
,
pz
becomes of
good

1
price
change
cheaper -
Intertemporal Utility
Maximization

with
"
-
"
-
.
Income Effect
:
change
in consumption
-
-
human

capital development 's
access
to
s
-
s
-
Ofa
good
due to
change
in
budget
set
,
n n
controlling
for substitution effect
financial markets
:
go, go,
-1€
,
-
Compensated budget
line
:
shows all
1.Maximize Lifetime Income subject to
,
:
y
,.
,
:
y
↳
bundles an individual can affordat new
human

cap
.
constraint
. ,.
.
2.Max lifetime
Utilitysubject
to
making
. i . . i . . .orig
. .
. i . . i . .
loris
. .
I 2 3 4 S
I 2 3 4 S
pricesassumingthey
were compensated
SO
rig good, rig Good'
Max

amount

Of lifetime income
s
-
3. Find optimal bundle after s
- 4. To calc
.W
. shift new
they can afford
Orig
.
levelOfhappiness
Ex
:
constraint
:
zoo-2×12×22
,
i10%0=+1213×2113
-
price
change
-
budget
line until it is
Anychanges
inoptimal bundle must be
4
-
4
-
tangent
to
original
L
=
(1.1)X.

t
Xztx(300-2×12
-X
})
-
- indifference curve
due to
change
inprices
CSub
.
effect)
s
-
s
-
X.
=
7.52×2=13.67
N
N
.
GiffenGood
:
a
good
that

isinferior's
o
-
o
-
f-X
,
"3Xz"3tX( (1.174.523+13.67
-
I.IX
,
-
Xz
)
802!802
-\•
ewtiescincmmag
.EE#Es9Fwdo:tYumsan3nx.=issoxz=isi
.
.
Lottery
:
Bundleof
goods
(X
,
,Xz
,
...
,
Xn)
-
I
- •
-
Bhim
.
I I I I l
-
l l
toric
I I
, , , ,
I
, ,
I
, orig, ,
-
Compensating
Variation
:

amount

Of
w/
a
lottery
(
p
,,
pz
,
...
,
pm)
rig
' Z
gig
?
" s
n
.ge#ebcti3eIIgIEIew3
4 s
income someone would bewilling
to
give
s
-
s
- Good'
I
.
p
,
t
pzt
... t
pm

=/
- 5. Label intersection
.
4. To calc EV
, shift
Blorig
Up(need)after aprice
reduction(increase
,
z
.
pizo
4
-
Of
BLUEY
-axis
,
4
.
until
itistangenttolnew
.
"
Xznew
"
G
Blcompda
to maintain
Utility
before
change
y-axis as
"
xzcomp
"
-
3.
pi

istheprob
.
of

receiving
bundle Xi
,
-
6.
CV=p{
Xznew
-
Xzcomp)
3
-
new
price
,
Oldutility
Level
.
Degenerate lottery
:
all
prob
. on
I outcome
o
-
512-1.25=3.75
-
-
.
equivalent
variation
:
amount of

income
.
expected
value
:p ,×,+pz×z+
... +
pn×
,
I÷÷u¥•§↳;§o
,
.§,
a consumer would need (be
willing
to
give
.
Mixing
:
Given 2 lotteriesLp
,,
Pz
,
....
pm
)E
=
i
-
a
- •
I
Blcoymp
-
IBL
equiv
Inew
Up)beforeaprice
reduction (increase)to (t
,
,tz
,
..
.tn)and OLXLI
,
a
MIX

Of
. . . . . .
Kris
. .
, , .
Kori
. ,
iorig
.
?
rig
I
Zxcompxnew
} 4 S
I 2
=
3 4 S
give
the same levelof
utility
as afterthe these 2 lotteriesisthe
new
lottery
:
good,
rig
yoga
,
S
-
5. Label intersection of
>
rice
change
(
AP .tl/-X)ti,XPztll-X)t2
,
...
,
April
'
-
d)th)
n
-
Blom
.gg
y-axis
"
xzorig
"
's
Sub
.
Effect
:X sub
-
Xorig/
Old
prices
,
new
utility
.
Independence
:
If
lotteryPIlotteryq
E I
"
-
Bleauivday
-axis as
Income

Effect
:X new
-
Xsubs
or -
"
Xzequiv
''
CV=pz(Xznew
-
Xzcomp
)
both
are
Mixed with
lottery
titherlottery
xzeausi-6.eu-pzlxzeauiu-xzon.gl
s I
2178
p

is

stillpreferred
to
lotteryq
- Scs-23
.
-s
EV
=
Pz
(
Xzequiv
-
Xzorig
)
xzorig
-
Find CVEEV
mathematically
.
ExpectedUtility
Theorem
:
VNM Can be
2
,
;
¥
,
Yg
,
1.
Use
Lagrangian
tofindoptimal bundle interpreted
as
the cardinalUtility
received
.
Iaea
" s
>
i i ,
Phon
, ,
Iorio
, ,(
giffen
before
pricechange
→
X
orig
from
good
Isuch that(
p
,,
Pz
,
...
,
Pm
)I
rig
, z
Inns
4 s
I
ICO
2.Use
Lagrangian
tofind optimal bundle (
q
,
,qz
,
...
,
qn
)Iff
p
.
U
,
tpzuzt
.. .
tpnun
Good'
III
>
151
after
pricechange
→
Knew Iq
,
U
,
tqzuzt
.
. .
tqnvn

3. Production Fundamentals
'
Cost
minimizing input
blend
:
f
'
Az
=
W'
lwz
'
Production Function
:
y
=
fCX
.,
Xz
,
...
,
Xn)isthe (if
using
both
goods
)
amount of
output
,
y
,
that can be
efficiently
.tk/Wk=f4WL L
marginal productivity per
producedusingX
..
Xz
,
...
,
Xn dollarspent
)
.
Efficient Production
:
given inputs
,
firm produces
Cost Minimization
largest
amount of output possible
.
L
=
W
,
X
,
t
WzXz
t
X(
q
-
f-LX
,,
Xz)
.
NO free lunch
:
impossible
toproduce output
-
Factor Demand Function
:
specifies
w/o
usinginputs relationship btw
prices
of inputgoods
,
'
Possibility
of inaction
:
Xi20
quantity
of output produced
.
E amount
'
Free disposal
:
inputs
can be
disposed
of at no Of an inputgood
a firm will select
cost
; dfldxi O for
every input
.
Total Cost

=
w
,
X
,
t Wzxz
.
Decreasing
Returns to scale
:
a production set
.
Cost Function
:
Clq
,
w
,,
wz)
=
displaysdecreasing
returns to scale if fctxlctfcx) W
,
X
,
(
q
.
W
,,
Wz)
t
Wz Xz(
q
,
w
,,
Wz)
forallt71 Where Xn(q
.
W
.
,
Wz) are the firm'S
.
Increasing
Returns to Scale
:
fCtx)>tfCx) factor demand functions
.
Constant Returns to Scale
:
fLtX)
=
tfCx)
.
Market Demand Function
:
sum of
'
Cobb
-
Douglas
Production Function
:
FCK
,
L)
=
akaL
's
individual demand functions (be
↳
Ifa
t
BCI
→
decreasing
careful of corner solutions
!)
↳
Ifa
t
B>I
→
increasing
Profit Maximization
-
Fixed Proportions
Production Function
:
.
ITL
p
)
=pDcp)
-
CLDcp
))
fLk
,
L)
=
minLak
,
BL)
.
IT(
q
)
=p
L
q
)
q
-
C
Cq)
.
Linear Production Function
:
perfect subs
,
'
IT

=p
(
q
)
q
-
CCq
) Marginal
f-CK
,
L)
=
aK
t
BL
=
Req
)
-
CC
q
) Revenue
=
.
No monotonic transformations for production R
'
(
q
)
-
C
'
(91=0 Marginal
functions! R
'
(
q)
=
C
'
(
q
) Cost
.
15090
ants
:
graphical
setof bundles that allow
.
DIT
1dg=p
C
q
)t
p
'
(
q
)
q
-
C
'
(
q
)
=
O
afirmto
produce
the same levelof output
.
Marginal
Rate of Technical substitution
'
lE÷pI=
-
¥
-
To
=
=L
(MRTS
×
,,
xz)
:
Max amount of

input
2firm
.
p
=
MC (Markup
=
¥
)
would be
willing
to
giveup
to
get
one more of
input
Iwhile
keeping
total output
the same ;
.
Profit w/ Fixed Prices
:
(
negative
of) derivative of
isoquant Xz=fCx
,
)
;
ITL
q
)
=
pq
-
C
Cq
)
MRTS
a,
B
=
fa
HB Lfa
:
marginal productivity
Wrt
Al p
=
MR
=
MC
.
ISOcost Line
:
graphical
set of
inputgood
'
Market
Supply
Function Scp)
:
sum of
bundles that cost the same amount individual
supply
functions
.
Factor Price Ratio btw InputIda Input
2
:
.
Market PriceinPC
amount of
input
2 the firm must
giveup
1
.
Solve for each consumer 's demand
to
get
one more of
input
II maintain function forthe specifiedgood
the same cost level;L
negative
of)the
slope
2
.
Find market demand
ofthe boost line w/ input
Ion x
-
axis
;
if 3.Solve for
q
each firm willproduce
at
prices
are fixedda
nothing
is
being given
agiven price
away
for free
=
w
'
lwz 4
.
Find market
supply
5
.
Find p where Qs
=
QD

4. .
Derfeet Price Discrimination
:
firm
1.
Determine firm'sprofit function
sells each unitat

maximum amount as function ofquantity
Cor
price
)
.
Cournot
game
ex
each customer is
willingtopay
IT

=
q
,
p
,
t
qzpz
-
C(
q
,
t
ga
)
-
producers get
all surplus
,
more units 2.Take first

order condition Wrt
MC for each firm
:
$2
sold than

w/o
price
discrim
.,
more
both
quantity
Lprice
) variables
p
(Q)
=
10
-
Q
total surplus generated
,
MR re
pre
-
3
.
Solvefor
profitmaximizing prices
sent

edby
demand curve
L
quantities
) IT
,
=
(10
-
q
,
-
qz
)
q
,
-
29
,
.
Non
-
linearprice discrimination
:
price
4
.
Plug
into

market demand to
IT
z
=
(10
-
q
,
-
qz
)
qz
-
Zqz
varies w/quantitypurchased
butall
determine profit

maximizing prices
consumers purchasing
the same low

antities) DIT
,
1dg
,
=
8
-
ofz
-
2g
,
=
O
quantitypay
the same price
5
.
Plug
into profit
function to find
q
,
*
=
(q
.
q
y,2
-
used when it

is difficult toidentify
profitLevel
2
orillegal
to
charge
different
prices
'
Dominant Strategy
: an
action that ditzldqz
=
8
-
q
,
-
292=0
to
groups
w/
highest
WTP provides
ahigher payoff regardless
qz*
,
Lq
-
q
,
),z
-
BlockPricing
:
firms

choose one
price
Ofhow an opponent plays
→
if
any
forthefirstfew unitsE another
price
player
has
a
dominant strategy
,
a
q
,
=
(8
-
L8
-
q
,
)12)12
for subsequent

units
dominant

strategy
solution exists
zq
,
=
q
-
4 +I
q
,
=
4+
I
q
,
1. Determine firm'sprofitfunction as
.
l
Strictly
)DominatedStrategy
:
an
function of
quantity
2 block a
action that

always provides a lower 3/2
q
,
T
4
IT
plof
,
)
q
,
t
plqz)(
q
z
-
q
,
)
-
C(
q
z
)payoff
than another
possible
action
q
,
=
4
.
I
=
8/3
2.Take first

order condition Wrt both regardless
ofhow the
player
's
quantity
variables opponents play qz
=
8/3
3.Solve for
quantity
variables
.
Nash Equilibrium
:
set

of
strategies
.
SubscriptionGunit
price
EX
4.
Plug
into market demand tofind Such that
no
player
has an incentive
profit

maximizing prices
to unilaterally
deviate 9
,
=
6
-
p
S
.
Plug
intoprofit
function to find
.
Prisoner's Dilemma
:
players
have a
qz
=3
-
O
.
5
p
no MC
profitlevel dominant
Strategy
to cheat
,
preventing
-
firms must be able to prevent beneficial cooperation
lower demander
:
qz
resaleG must have market
power
,
.
Coordination Games
:
multiple Nash p
PS increases
,
CS decreases
.
more
Equilibria
exist
,
each
corresponding 6
-
units sold than w/o price
discrim
,
to each
playerdoing
what the other
more total surplus
,
W/ more blocks
players
are
doing p
-
p
can approach efficient outcome
- "
pushing
"
a
coordination
game
into
D
.
2 Part Tariff
:
charge
lump sum
a
' '
good
"
equilibrium
can be achieved
3-
.
's
p
Q
,
subscription fee
,
S
,
for the
right
to
through
costless expectations
;
however
,
buy
all
;
unitprices
,u
,
are uniform
players
must believe
you
are

willing
to
CS
z
=
I(6
-
p
)(3
-
.
5
p
)
1. Calculate consumer surplus
as
a
pay
ifthe
equilibrium
doesn't

occur
function of
u
for the lower
-
HawkIDove Games
:
multiple
Nash IT

=
25
t
pg
.
t
pqz
-
C
Cq
,
tqz
)
demanding
customer
.
Set this
Equilibria
exist
,
each
corresponding
to
equaltos
.
one
playerbeing
"
strong
"
and the
=
L6
-
p
)(3-

.
Sp
)
t
p
(6
-
D)
2. Determine profit function as
a
other
being
' '
weak
"
(
opposite
actions)
t
p
(3-
.
Sp
)
-
O
function

of U
-
Tobe
a
hawk
,
commit

irreversibly
to
IT

=
25
t
09
,
t
Uqz
-
C
Cq
,
t
q
z
)
the
strongposition
=
I8
t
3
p
-
P2
3.Take the first

order condition
-
ifthe
game
isrepeated L both dITIdp
=3
-
213=0
Wrt U
players
are stubborn
,
consider a
4.Solve for
profit
-
maximizing
u p
=
3/2
5.Plug
U
into demand functions to
compromise
.
Steps tosolve fortheNE of
a
2
q
,
=
6
-
I
.
5
=
4
.
5
determine profit
-
maximizingquantities
player
continuous
game
6.
Plug
intoprofit
function to find
qz
=3
-
.
5(I
.
S)
=
2
.
25
I
.
For
player
I
,
derive
player
I'sbest
profitlevel
response
LBR)for

each possible S
=
.
5(6
-
I
.
S)L3-
.
SLI
.
S))
=
5
-
firms must be able to
prevent
resale
.
strategy
of
player
2
must have market power Eidentify 1000ofeach
type
→
customer WTP
types
,
firms
charge
2
.
For
player
2
.
derive
player
2'sBR for
user fee equal
to consumer surplus
each possiblestrategy
of
player
I Max fixed cost?
of lower demander
,
When demand
3
.
Findtheset

of
Strategies
thatSimUlta
-
1000(g
+
1.g(4.s))
+
types
are similar firms
charge
low
newly
solves the BR functions
Uand
high
S
,
when demand
types
-
Cournot Game
:
firms
produce

identical 1000(S
t
I
.
SL2
.
25))
=
20125
are different firms
charge high
u goods
,
firms commit

irreversibly
to
a
and lowS
,
producer doesn't
certain quantity
level
,
when
qtpt
,
extract

all
surplus
each firmisprofitmaximizing
.
Tiein Sales
:
in order tobuy
one
.
Bertrand
game
:
each firm sets prices;
item
,
customer must
buy
another
the firm that has the lowest pricegets
.
Group Price Discrimination
:
price
all customers
;
if
they
have identical
variesby group
,
used when difficult
prices
,
they
share customers 50150
toprice
on individual WTP but can
determine
avg
WTP fora
group