Sample Midterm Ans

7/24/2019 Sample Midterm Ans

1/7

Economics 402: Business Cycles Prof. Yamin Ahmad

SampleMidtermSolutions

Instructions:Please answerbothquestions.You should show yourworking and calculations foreach

applicable

problem.

Correct

answers

without

working

will

get

you

relatively

few

points.

Make

sure

to

putyournameand idnumberonthetopofeverysinglepieceofpaperthatyouturn in.Also,please

write in apagenumber at thebottomof everypieceofpaper that you turn in and assemble them

sequentiallybeforesubmitting.

Question1: MathandOptimization(50%)

Consider the Primalproblem facedby aproducer.Namely,aproducer (or firm) canbe thought to

maximize profits, subject to producing their goods based on a particular kind of technology, or

productionfunction.

Supposethatproductionofgoodyrequirestwoinputs:K(capital)andL(Labor),whichhaveinputprices

r and w respectively. Furthermore, assume that the firms price is normalized to 1, and that the

productionfunctionisgivenbyaconstantelasticityofsubstitution(CES)productionfunction:

1

,y F K L K L

[Quickaside:

Justasanaside,andasapointof interest (sincesomeofthepaperswemayreadmightusetheCES

function),herearesomeofthepropertiesofthefunctionabove.Asitsnamesuggests,theelasticityof

substitution (whichinthiscasewouldmeasurethecurvatureoftheproductionisoquant)isconstant

for the CES production function. More specifically, the elasticity of substitution measures the

percentage change in the factor ratio divided by the percentage change in the technical rate of

substitution,whilstholdingoutputfixed.Thatis:

/ln / ln //

/ ln / ln

/

K L

d K L d K LK LdK dL d dK dL d TRS

dK dL


2/7


The usefulness of the CES production function in replicating different types of production functions

becomesapparentaswevarytheparameter.Forexample:

CaseI:Linearproductionfunction:

Whenweset 1 ,theproductionfunctionabovebecomes: y K L ,wherethetwoinputs,capital

andlaborareperfectsubstitutes.

Case2:CobbDouglasproductionfunction:

When tends to0, i.e.0lim y , the isoquantsof theCESproduction function lookverymuch like

those of the CobbDouglas production function. This can be shown a variety of different ways

mathematically,buttheeasiestistocomputethetechnicalrateofsubstitution.Assuch,thetwoinputs

inthiscaseareimperfectsubstitutes,wheretheproductionisoquantsaredownwardsloping.

Case3:LeontieffProductionfunction:

When tends to , limi.e. y , the production isoquants become Lshaped, which we

associatewiththeperfectcomplementscaseforinputs.

EndAside]

a. [5pts]

Calculate

the

partial

derivatives

of

ywith

respect

to

Kand

with

respect

to

L.

Answer:

1

1 11

1 1

1 11

1 1

1,

1

where

y K L

Fz K z K z K L

K

Fz L z L

L

b. [5pts]

Consider

that

the

total

derivative

of

the

production

function

above

would

be

calculated

as:

F Fdy dk dL

K L

.Thetechnicalrateofsubstitutionmeasureshowoneoftheinputsmust

adjustinordertokeepoutputconstant(i.e.when 0dy )whentheotherchanges,andcanbe


3/7


calculated from the totalderivativeaboveas:/

/

dK F LTRS

dL F K

.Usingyouranswer to

part(a),calculatethetechnicalrateofsubstitutionfortheCESproductionfunctionabove.

Answer:

11 11

1

1

dK z L L K TRS

dL K Lz K

c. [4 pts] Using your answer from part (b), rewrite the equation so that you have (K/L) as a

functionoftheTRS.[Hint:taketheabsolutevaluesoftheTRSandthenrearrangefor(K/L)]].

Answer:

1

1

1

KTRS

L

KTRS

L

d. [4pts]Take logsofbothsidesanddifferentiatetoshowthattheelasticityofsubstitution isa

constant

Answer:

1ln / ln

1

1ln / ln1

ln / 1

ln 1

K L TRS

d K L d TRS

d K L

d TRS

Analternativewaytothinkabouttheproducersproblem istoview itasafirmschoicetoproducea

certainamountofoutput, y,andthentheywishtominimizetheamountofcostsneededtoproduce

thatamountofoutput. ThisistypicallytheDualproblemtothefirmsdecisions.

e. [6pts]

Given

input

prices

r

andw

for

Kand

Lrespectively,

write

out

the

Lagrangean

for

the

firmscostminimizationproblemforproducinganamountofoutput y.

Answer:Thefirmsproblemisto,

minK L

rK wL subjectto:1

K L y .Itisactuallyeasier

totransformtheconstraintabit,sothattheproductionfunctionisnow: K L y .


4/7


Assuch,thelagrangeanfortheproblemis:

rK wL y K L

f. [6pts]Derivethefirstorderconditions(FOC)fortheproblemabove.

Answer:

1

1

0

0

(I)

(II)

(III)

r KK

w LL

K L y

g. [10pts]Solvefortheoptimalvaluesof* *andK L .

Answer:Usingequations (I)and (II)above,wecaneliminate toget theexpression that the

absolutevalueofthemarginalrateoftechnicalsubstitutionequalsthe(input)priceratio:

11

1* * (IV)

K r rK L

L w w

WenowhaveanexpressionforKintermsofL.Finally,byplugging(IV)into(III)(whichwehave

notuseduptothispoint),wecansolveforL*firstandthenK*asfollows:

1

1*

* 1 1 1

11

* 1 1 1

11

* 1 1 1

1

rL L y

w

rL y

w

L w w r y

L w w r y

K r w r y

h. [10pts]CalculatetheValuefunctionforthecostfunction,whichisobtainedbyplugginginthe

optimalvaluesofK*andL*thatyouobtainedinpart(g)aboveintothecostfunctionitself.


5/7


Answer:Byplugging in theoptimalvaluesforK*andL* into thecostfunction,wecanobtain

totalcostsasafunctionoftheinputpricesandthelevelofoutputthefirmwishestoachieve.

Thatis:

* *

1 11 1

1 11 1 1 1 1 1

1

1 1 1 1

1

1 1

1

, ,C r w y rK wL

r w r y w w r y

y r w w r

y r w

y r w

[Notice that theformof thecostfunction is the sameas theoriginalCESproductionfunction

exceptwithreplacedwithinstead,andtheinputsreplacedbytheirfactorprices.]

Question2: ISLMquestion(50%)

ConsiderthefollowingISLMmodel:

1200 ( )

4

1150 10004

250

200

, 2 8000

1600

C Y T

I Y r

G

T

L r Y Y r

M

P

a. [8pts]DerivetheISequation

Answer:TogettheISequation,justaddupC+I+Gasfollows:

1 1

: 200 200 150 1000 2504 4

1550 1000

2

1100 2000 (1)

IS Y Y Y r

Y Y r

Y r


6/7


b. [8pts]Derivetheequation fortheLMcurve. [Hint,youwanttowrite itas interestratesasa

functionofeverythingelse].

Answer:TogettheLMequation,justsetdemandequaltosupplyforrealmoneybalances:

,

1600 2 8000

8000 1600 2

0.2 0.00025 (2)

M

L r YP

Y r

r Y

r Y

c. [8pts]Solvefortheequilibriumoutput.[Hint:substitutetheLMequation intotheISequation

andthensolveforoutput]

Answer:

*

1100 2000( 0.2 0.00025 )

1500 0.5

1000

Y Y

Y

Y

d. [8pts]Solvefortheequilibriuminterestrate.[Hint:substitutethevalueyouobtainedforoutput

inpart (c) intoeither the ISorLMequations,andsolve for interest rates.You shouldget the

samenumberfrombothifyoudidthemathcorrectly]

Answer:PluggingintotheLMequation,weget:

*

0.2 0.00025 1000

0.2 0.25 0.05

r

r

Thusinterestratesare5%inequilibrium.

e. [8pts]Calculatethevaluesofconsumptionandinvestmentinequilibrium.Verifythatthesumof

C,IandGadduptooutput.

Answer:

*

*

* *

200 0.25 (1000 200) 400

150 0.25 1000 1000 0.05 350

250

400 350 250 100

C

I

G

C I G

f. [10pts]Suppose that themoney supply increases toM/P=1840.Solve forY, r,Cand Iand

describeinwordstheimpactofamonetaryexpansion.

Answer:Intuitively,amonetaryexpansionshouldimpacttheLMcurve.Itshouldcreateasurplus

ofmoney inthemoneymarketandcause interestratestogodown,shiftingtheLMcurveout.


7/7


Thedecreaseininterestratesshouldspurinvestmentandoutputandwewouldmovealongthe

IScurveintheshortrun.Sinceoutputgoesup,consumptionshouldtoo.

1840 2 8000

0.23 0.00025

Y r

r Y

This

is

the

new

LM

curve.

Plugging

into

the

IS

curve

yields:

* *

*

*

*

*

*

1100 2000( 0.23 0.00025 )

1.5 1560

1040

0.23 0.00025 1040 0.03

200 0.25 1040 200 410

150 0.25 1040 1000 0.03 380

Y Y

Y

Y

r

C

I

Thus

our

observations

are

verified.

Documents

Sample Midterm Ans