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 the wi’s are the pricesoffactors (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 x p1(w,p).
f) Useyour results from parts (b), (d), and(e) toconfirm that:
xy1 w,y(w,p)
x p1(w,p)
3. Let us return to question (1).This time assume that capital is held fixed at levelk, and for
simplicity assume that
.
a) Find the short-run (capital fixed) cost function
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
Wherep is 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:
a t
2b
a t
p(t)
2
y(t)
b) Show that the correspondingtaxrevenueand deadweight loss are
(a t)t
2b
(a t)2
DW(t)
8b
R(t)
c) Supposethat instead of a per unit tax, an ad valorem tax with rate is imposed (tax
revenue now is py , and tax per unit sold is p ). Show that now the monopolist’s
quantityand demand price arethe following:
a
2b
a
p( )
2
y( )
d) Show that thetaxrevenue and deadweight loss correspondingto thesituation in part (c)
are
a 2
R( )
4b
a2
DW( )
8b
e) For a given level of tax revenue, which tax has the smaller dead weight loss? Why is this
so? Illustrate your 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
ac
. Assume thata pUS
.
cy2, where y y K
c b
yUS
a) Find
b) Whyis
pK,yK And yUS.Illustrateyouranswer with adiagram.
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
i
xi
Where y is another good whose price is normalized to one throughout this problem. Assume
that the are equal number so the two types of 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 1)2
1
, x2
1x
4c 2
4c 2
b) Findthepriceof each package.
9. A firm has the following two-factor production function:
y
f(x1,x2)
x1 2 x2
Where y is 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.
40
x for its product. Its
x* and p *.Illustrateyouranswerwith
a) Findthemonopolist’soutputandprice.Callthese
adiagram withp on the vertical axis andxon the horizontal axis.
b) Suppose that a price ceiling of p p*isimposed.Onadiagramliketheoneinpart(a),
Show themonopolist’seffectivedemandandeffectivemarginalrevenuecurveswitha price
ceiling .Find the monopolist s optimal price, p, and quantity, x .In your answer
p 0.
Distinguish between priceceilings wherep* p 20, and20
c) Supposethatthemonopolistsellsintwoseparatemarkets.Inonemarket,denotedby
subscript1, theinversedemandcurveisasbefore, i.e., p1
40 x1, while in the second
it is p2 32 x2. Total cost are now given by
0.5(x1 x2)2. Assuming thatthe
Monopoly is tis free to charge whatever prices it wishes (no price ceiling), find the
*1 2
2 }.Illustrateyouranswerwith
monopolist’spricesandthequantitiesitsells,{p*, p*1,x*,x
adiagram. Explain brieflywhy p* 1
p2*.
d) Supposethat under theconditions of part (c) thegovernment imposes a priceceiling,p.
Find themonopolist’s prices andthe quantities itsells,{p*1, p*2,x*1 ,x*2}for each of the
Following three price ceilings:{p1 27,p2
25,p3
23}.Illustrateyouranswers.