Question

Exercise 1. Short-Run Industry Supply Curve

In a perfectly competitive market there are n firms with identical technology: yi=Li½Ki½. Each firm’s cost function is Ci=wLi+rKi where w=r=1.

a) In the short run all firms have a fixed level of Ki=100, so that yi=10Li½ and Ci=Li+100. What is the cost function Ci(yi)? What is the short-run average cost function ACi(yi)?

b) What is each firm’s marginal cost function MCi(yi)? What is each firm’s short-run supply function si(p)? Find the inverse of the marginal cost function.

c) The industry’s supply function is the summation of each firm’s supply curve: S(p)=Σisi(p). There are n=60 identical firms, so that S(p)=n∙si(p). What is S(p)?

d) The market demand curve is D(p)=20,000-5,000p. What is the market equilibrium p? At this market price, how much profit does each firm make? How many workers does each firm employ?

e) In the long-run each firm can choose the optimal level of capital Ki as well. From cost minimization find the conditional factor demands Li(yi) and Ki(yI)? Plug these factor demands into the cost function to find the long-run cost function Ci(yi)?

f) What is the long-run marginal cost function MCi(yi)? What is the long-run supply function si(p)? How much would each firm supply if price does not adjust?

g) Suppose that new firms will enter the market as long as there are profits to be made. The zero profit condition for each firm is p=ACi, which must also equal MCi if a firm is profit maximizing. What is the zero-profit price level p* where ACi=MCi?

h) How many firms are in the market in the long run? Find total output in the long run by using D(p*) from the market demand in part d. This will equal S(p*), which is n*∙si(p*). Find n*.

Answer 1

- Answer : yo = L 12K 12 I = Whitrki W =r=t la - C = Lit Ki Ki=100 yo = 102112 C= 100+ Li Substiluting the value of Litrom Ci=100+ you Voo get eq ♡ we a Cily) = 100+ ye? - Aclui) - lily) = (oo + y Aclyid = 100 + you yo

DATE: 1 eby Milyi) ality, JOU dyni ( 160 24 +0050 - Melyi) . y. firm's short sun suapply function : Silp); p = Milyi) Inverse of Mailyi) - 50 supa n silb) - Gor you com by® Los Dip) = 20,000 botolp] at equilibrium D(p)= S(p) 20.000 - 5000(p)' = Gyi 20,000 - 5ood ( ) Gyi 20000 - sy' + 500 20,000 - Gyi tesoo

10 0000 s Ye - 100000 506 506 yo - 197:63. I p. = 197.63 = 3.95 ~ 4 Market equilibrium p = 4 - beopt = px y 100+ y 2 ) 4x 19 7063-100 - (197.6312 100 = 790.52-100 - 390.58 > hept = 299.94 z 500 Lia 42 = 390.58 ~ 391 workers.