Problems for Carlos
1. Afirmproducesoutput, y, byusingcapital,k,andlabor,l,accordingtotheproduction function:
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)andxy2(w,y).
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. 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 thatThe monopolist’s quantityanddemand pricearethe following:
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:
d) Show that thetaxrevenue and deadweight loss correspondingto thesituation in part (c)are
Attachment:- Problems for MicroTheory.docx