Chapter 5. Comparative statistics.pdf

Eco

1. C
HAPTER
3.C
OMPARATIVE
S
TATISTICS
1

2. T
HENATUREOFCOMPARATIVESTATISTICS

Comparativestatics,asthenamesuggests,isconcernedwiththe
comparisonofdifferentequilibriumpositionsassociatedwiththe
differentvaluesoftheexogenousvariablesandtheparametersin
themodel.

Comparativestaticsanswersthequestion“howwillthe
equilibriumvalueofanendogenousvariablechangewhenthere
isachangeinanyoftheexogenousvariablesorparameters.”isachangeinanyoftheexogenousvariablesorparameters.”

Comparativestaticsallowseconomiststoestimatesuchthings
as:

theresponsivenessofconsumerdemandtoaprojectedexcisetax,tariff,
orsubsidy;

theeffectonnationalincomeofachangeininvestment,government
spending,ortheinterestrate;and

thelikelypriceofacommoditygivensomechangeinweatherconditions,
priceofinputs,oravailabilityoftransportation.
2

3. E
XAMPLE
: T
HE
M
ARKET
M
ODEL

Consider the simple one-commodity market model:

At equilibrium Q
d
=Q
s
, using this the market clearing
solutions to the problem will be:
3

4. E
XAMPLE
: T
HE
M
ARKET
M
ODEL

Thesesolutionsarereferredtoasareducedform
equation

thetwoendogenousvariableshavebeenreducedtoexplicit
expressionsofthefourmutuallyindependentparameters
a,b,c,andd.

Tofindhowaninfinitesimalchangeinoneofthe
parameterswillaffectthevalueofP*,wesimplyparameterswillaffectthevalueofP*,wesimply
differentiate(1)w.r.teachoftheparameters.

Similarly,wecandrawqualitativeorquantitative
conclusionsfromthepartialderivativesofQ*w.r.t
eachparameter,suchas:

Toavoidmisunderstanding,however,aclear
distinctionshouldbemadebetweenthetwo
derivativesand
4
*
Q



*
Q



Q




5. C
ONT
’
D

Thelaterderivativeisaconceptappropriatetothe
demandfunctiontakenalone,andwithoutregardto
thesupplyfunction.

Thederivativepertains,ontheotherhand,tothe
equilibriumquantityin(2),whichtakesintoaccount
theinteractionofdemandandsupplytogether.
ConcentratingonP*forthetimebeing,wecangetthe
*
Q




ConcentratingonP*forthetimebeing,wecangetthe
followingfourpartialderivativefromequation1given
as:
5

6. E
XAMPLE
: T
HE
M
ARKET
M
ODEL
6

7. E
XAMPLE
: T
HE
M
ARKET
M
ODEL

Sincealltheparametersarerestrictedtobeing
positiveinthepresentmodel,wecanconcludethat:
Forafullerappreciationoftheresultsin(3),letus

Forafullerappreciationoftheresultsin(3),letus
lookatFigure1,whereeachdiagramshowsachange
inoneoftheparameters.

NoticethatweareplottingQ(ratherthanP)onthe
verticalaxis.
7

8. E
XAMPLE
: T
HE
M
ARKET
M
ODEL
8

9. E
XAMPLE
: T
HE
N
ATIONAL
-I
NCOME
M
ODEL

Considerasimplenational-incomemodelwithtwo
exogenousvariables,investment(I
0
)andgovernment
expenditure(G
0
):

ThismodelcanbesolvedforYbysubstitutingthe
thirdequationof(4)intothesecondandthen
substitutingtheresultingequationintothefirst.

Theequilibriumincome(inreducedform)is
9

10. E
XAMPLE
: T
HE
N
ATIONAL
-I
NCOME
M
ODEL

Similarequilibriumvaluescanalsobefoundforthe
endogenousvariablesCandT,butweshall
concentrateontheequilibriumincome.

From(5),therecanbeobtainedsixcomparative-
equilibriumderivatives.

Amongthese,thefollowingthreehavespecialpolicy
significance:significance:
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11. C
OMPARATIVE
S
TATICSOF
G
ENERALFUNCTIONMODELS

Inthecomparativestaticproblemsconsideredbefore,
equilibriumvaluesofendogenousvariablesofthe
modelcouldbeexplicitlyexpressedintermsofthe
exogenousvariables,

Accordingly,thetechniqueofsimplepartial
differentiationwasallweneedtoobtainthedesireddifferentiationwasallweneedtoobtainthedesired
comparativestaticinformation.

However,whenamodelcontainsfunctionsexpressed
ingeneralform,explicitsolutionsarenotavailable.

Insuchcases,anewtechniquemustbeemployedthat
makesuseofsuchconceptsas implicitfunctionruleto
findthecomparativestaticderivativesdirectlyfrom
thegivengeneralfunctionmodel.
11

12. C
ONT
’
D

Example:-Consideramarketmodel:
Atequilibrium


0,
0,0,
0
01











p
s
psQs
y
D
p
D
ypDQd

Atequilibrium

Where,p=endogenous
Y
0=
exogenous

Weknowthateveryequationpriceisafunctionof
income.i.e..

Therefore,theequilibriumconditioncanbetakentobe
anidentifyintheequilibriumsolution.
0)(),()(),(
00  PSyPDPSyPD

0
*
ypp
12

13. C
ONT
’
D

Thecomparativestaticanalysisofthismodelwill
thereforebeconcernedwithhowachangeiny
0
will
affecttheequilibriumpositionofthemodel,i.e.
i.whatistheeffectofachangeinY
0
onp
*
?
 

0,
0,
0
*
*
0
*


ypF
psypD
i.whatistheeffectofachangeinY
0
onp?
0
)()(
*
*
0
*
0
0
*













p
s
p
D
yD
pF
y
F
dy
dp
13

14. C
ONT
’
D

Thus, increase y P
*
or the other way.

What is the effect of a change in Y
0
on Q
*
?

At equation , Q
*
=Qd=Qs

We can write ,


0
***
., yPPandPSQ 

Thus, the comparative static results convey the proposition
that an up –ward shift of the demand curve ( due to a rise in
income) will result in a higher equilibrium price as well as a
higher equilibrium quantity.
0.
0
*
0
*

dy
dp
dp
ds
dy
dQ
14

15. L
IMITATIONSOFCOMPARATIVESTATICANALYSIS

By its very nature, comparative statics has the
following limitations.

Ignores the process of adjustment from the old equilibrium
to the new one.

Neglects the time element (length of time ) involved in the
adjustment process from one to another equilibrium.

Assumes that a new equilibrium can be defined and

Assumes that a new equilibrium can be defined and
attained after a disequilibrating change in a parameter,

i.e. disregards the possibility that the new
equilibrium may not be attained ever because of
the inherent instability of the model all these
limitation are addressed by dynamic analysis
which will be dealt in the next chapter.
15

16. 
Reading assignment

Differentiating systems of equations

The Jacobianand hessian determinants
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