Problems for Carlos

1. Afirmproducesoutput, y, byusingcapital,k,andlabor,l,accordingtotheproduction function:

y=kαlβ

The firm can purchase allthe capital and labor it wants atfactor pricesrandw, respectively. a) UsethemethodofLagrangemultiplierstofindthecostfunction,c(r,w,y).Findaverage

and marginal costfunctions.

b) Whatis theinterpretation of theLagrangemultiplier in part (a)?

c) Whatis the importanceof theterm (α+β) beingless than, equal to, orgreaterthan one?

2. A firm has a cost function given bythefollowing:

c(w1,w2,y) =w1w2y2/(w1+w2)

Where thewi’s are thepricesoffactors (inputs)x1and x2, respectively,andyis output. a) Is this a legitimate cost function?

b) Fromthecostfunctionderivethefirm’sconditional(outputconstant)demandfunctions

xy1 (w,y)and

xy2(w,y).

c) Show that the firm’s production function is yf(x1,x2)

x1 

x2 .

d) Foroutputpricep, usethecostfunctiontoderivethefirm’ssupplyfunction,

and the firm'sprofitfunction,(w,p).

y(w,p),

e) UseHotelling'slemmatoderivethefirm’sunconditional(MarshallianorWalrasianor

price constant) demand function

xp1(w,p).

f)Useyour results from parts (b), (d), and(e) toconfirm that:

xy1w,y(w,p)xp1(w,p)

3. Letusreturnto question (1).Thistimeassumethatcapitalisheldfixedatlevelk,andfor simplicityassumethat.

a) Findtheshort-run(capitalfixed) costfunction total cost.

ck(w,r,y,k).Identifyfixedandvariable

b) Whatis the relationship betweenc (w, r,y)

andck(w,r,y,k(w,r,y)), wherek(w,r,y)is

Theconditional demand function from question (1)?(Remember that).

6. A monopolist faces the followingdemandcurve forits product:

p aby

Wherepis thedemand price andyis quantitydemanded. Themonopolisthaszero marginal cost (c0).

a) Supposethat a specifictaxof t dollars per unit is imposed on the monopolist. Show that

The monopolist’s quantityanddemand pricearethe following:

y(t)at

2b

p(t)at

2

b) Show that the correspondingtaxrevenueand deadweight loss are

R(t)(at)t

2b

(at)2

DW(t)

8b

c) Supposethat instead of a per unit tax,an ad valoremtaxwith rateis imposed (tax revenue now is py, and taxper unit sold is p). Show that now the monopolist’s quantityand demand price arethe following:

y()a

2b

p()a

2

d) Show that thetaxrevenue and deadweight loss correspondingto thesituation in part (c)

are

R()

4b

a2

DW()

8b

e) Foragiven level of taxrevenue, which taxhas the smallerdeadweight loss?Whyis this so?Illustrateyour explanation with a diagram.

7. Supposethat dueto tariffprotection, the South Korean automobile maker,Hyundai, is the onlyseller of automobiles in the country.Its demand curveis

pK abyK

wherethe subscriptKstands forKorea. Hyundaican also sellits cars in theU.S. whereit

Acts as aperfect competitor. Thepriceof cars in theU.S. is

pUS. Hyundai’s costfunction is

cy2, where

yy

K yUS

. Assume thatapUS

ac .

cb

a) Find

pK,yK

And

yUS.Illustrateyouranswer with adiagram.

b) Whyis

pK pUS?Explain.

8. Suppose that Cubatours, the official travelagencyofthe Cubangovernment, offers visitors to theIslandtwotouristpackages.Thetwopackagesdifferintermsofthelengthofstay,

xi,i1,2,andintermsoftheirprices,

ri,i1,2.TherearetwotypesofvisitorstoCuba,

each having a utilityfunction given bythe following:

ui(y,x)yixi

Where y isanothergoodwhosepriceisnormalizedtoone throughout thisproblem.Assume thattheareequalnumbersothetwotypesof visitors, and that 1 2 1/2.Cubatours has a constant marginal cost of length of stay,c.

a) Show that each packagehas length of staygiven by:

2(2

)2

x 1 ,

14c2

x2 1

24c2

b) Findthepriceof each package.

9. A firm has the following two-factor production function:

yf(x1,x2)

x12 x2

Whereyis output and(x1,x2)are the factors. The factor prices are(w1, w2).

a) Derivethefirm’s cost function and the output constant factor demand functions.

b) Does this firm’s profitfunction for output price exist?Explain.

11. Consideramonopolistthatfacestheinversedemandcurve p

cost function is0.5x2.

40x forits product.Its

a) Findthemonopolist’soutputandprice.Callthese

x* and

p*.Illustrateyouranswerwith

adiagram withp on the vertical axis andxon the horizontal axis.

b) Supposethatapriceceilingof

pp*isimposed.Onadiagramliketheoneinpart(a),

Showthemonopolist’seffectivedemandandeffectivemarginalrevenuecurveswitha priceceiling.Findthemonopolistsoptimal price,p, andquantity, x.Inyouranswer

Distinguish between priceceilings wherep*

p20, and20

p0.

c) Supposethatthemonopolistsellsintwoseparatemarkets.Inonemarket,denotedby

subscript1, theinversedemandcurveisasbefore, i.e., p1

40x1, whileinthesecond

2

it isp2

32x2. Total cost are now given by

0.5(x1

x2)

. Assuming thatthe

Monopoly istis free to charge whatever prices it wishes (no price ceiling), find the

monopolist’spricesandthequantitiesitsells,{p*, p*,x*,x*}.Illustrateyouranswerwith

adiagram. Explain brieflywhy p*

p*.

12

d) Supposethat under theconditions of part (c) thegovernment imposes a priceceiling,p.

Find themonopolist’s prices andthe quantities itsells,{p*, p*,x*,x*}for each of the

Following threeprice ceilings:{p1

27,p2

25,p3

23}.Illustrateyouranswers.