Question

Castro’s daily production function for producing custom-printed mugs is q = 15K^0.5*L^0.5. The price of labor is $40 and the price of capital is $10.

a. Find and sketch the expansion path.

b. How much capital and labor should Castro employ to produce 150 mugs a day?

c. What is the lowest possible cost of producing 150 mugs a day?

d. Repeat parts (a) and (b), for an output level of 450 mugs.

e. What is the firm’s LRAC at q = 150? At q = 450? What does the firm’s LRAC look like over this range of output? Are there economies of scale? Explain.

Answer 1

(a) q= 15k 0.5 10.5 09 = MP2 = 1510.5)ko .5 20.5-1 O2 MPL = 7.56 0.5,-0.5 Ok - MPx = 1510967 * 0.5-4,0.5 mPx = 7.5*0.5, 0.5 MRTS = MPL = zu ko.5 -0.5 MPK 7.5 K™0-5,0.5 MRTS = K L is given by Expansion path MRTS = w { : Yo = = 4 YL Expansion path K=4L

b) eh put - 9 = K=4L in proan 15kous 20.5 - 14.12.0-5 10-5 a = a 2005 Lois - a= 22 = 20 K= 4L = 4/2) = 4a 2=150 L= 150=5 L=5 30 K = 4x150 5 K=20 c= olt ak = 40x5 + 10 x 20 = 200 + 200 = 400 q= 450 Expansion path MRTS= w - Ķ = yo => ķ = 4 « = 42 Again, plet k = UL in prodh fh 2 = 15L0-5 X0.5 : = 2, k = 42 = 450 K= 46 450) L = 15, k=60

e) we have ha a x = 42 с 4) + К = yoq + 14422 - 402 4oz 802 30 30 LRATC = C = 84 | 1) LRATC - cet q = 150 LRATC= 8 LRATC= 700 at (450) LRATC is constant Since LRATC is constant for value of q. Thus, there economies of scale. a is given constant

Thus,there is constant economies of scale.