Question

(d) A representative consumer has a utility function U (x, y) = xy. A representative firm makes

good x and has a production function x = f(k, l) = (kl)^0.25and an unavoidable fixed cost

equal to A. There are 100 consumers and, initially, 100 firms. Prices are w = v = P_y = 1 and P_x is determined in a competitive market. Representative consumer income is I = 2.

(d) Solve for the equilibrium price and quantity of good x?

Answer 1

OCX, Y 2 = xy Du = mox ox = y ou = muy = x - Oy ų MRS = mux = muy BL: Pxx+ Pyy = I - At optimal choice mRs = put y = (Px) x in eque - Pxx Ź Pyy = 1 PxX + 841 px x = I Px x + px x 2 px x = = I. 1 2px I = 2 There are 100 consumers ther, Market demand of goodx * DCPX? = 10011) DLPx) = 100 PX

x = = fck, D x (ke) 0.25 x0.25 0.25 Ox = mpe = 0.25%0-25 20.25-1 o e mpe= 0.25k 0.25 e 0.75 pxmpx = 0.25 x 0.25-) 20-25 ok MPx = 0.25k-0.75 0.25 MRTS = mle = 0.25 +0.25 075 0.25 1-0.75 10-2 K At optimal choice MRTS = w put x= x= x= k =e k=1 in prodh fh ko. 25 e 0.25 10.25 e 0.25 10.50 d = x2 K = x2 c= wet vk

c= 0x2 + 12x2 c= 2x2 de = nec = 4x O suppey curve for an firm is given by individual p= mc Рx = Tмск 4x Px = x = Px since Thurs, there are 100 market supply firms B S[P = 100 S[Px) = 25 px competitive Equilibrium S(Px) = D(Px) 25px = 100 Px 25 P² = 100 P² i 4 Px = 2 and eque librium quantity of x = 25 (22 = 50