Question

A firm has a Cobb-Douglas production function of Q = K^(0.25) L^(0.75)
(a) Does this production technology exhibit increasing, constant, or decreasing returns to scale?

2. (b) Suppose that the rental rate of capital is r = 1, the wage rate is w = 1, and the ?rm wants to produce Q = 3. In the long-run, what combination of L and K should they use? (It would be good to practice doing this with the Lagrangian, even if you can potentially solve it more rapidly with your knowledge of Cobb-Douglas.)

3. (c) Suppose now that the wage rate increases to w = 6. In the long run, what is the new combination of L and K that would be cost-minimizing if the ?rm wishes to continue producing Q = 3?

Answer 1

o cokol Lars Increase buth K und t by d. de = (d) 0.25 (12) 0.75 10.25 0.25 10.15 10.15 = 10:25 1015 KORT 75 21 (028 L75) =de Increase in inputs by I will lead to increase in wuthut (a) by only, it implic production to exhibit constant return to scale 2K + ابيا :C c=1L TIK C=L+K I=K0.25, 0.75 from wants to produce 0= 3. - 3 = K.25 L from wants to minimize cest c= WLtok subject to 3= kolo Set up the Lagrangin for I= WLtok - d (80 K.25 0.25 -3) de. wood (0.15 4025 [02) =

و عده دید (4) پارسی و ( 10- 15 (2 ) = (3 - 15. کد Now egante equation (1) and (2). بیا / 10 - 25 . / ۱۰ - 75 6.5 (R 15 هه )) : " *

prt w-1, r=1 in ey (4) 1 x 4 – (5) put cg (5) in ej (3) 10.25. 27 = (3) (3) 0.25 L = 3.948 12=3.95 and K = 3.95 K K = 1.316 = 1.32 In long awn from will use L = 3.95 Preduce co-3. and k = 1.32 in order to now w: 6 and r=1 put it in eg (4) k=6L 3 1 K = 2L - (6)

I put eg (6) in eg (3) 40.25 Lon=3 (2270:25 (0.15 = 3 (2) 0.25 0.25 10.15 = 3 L= 3 (2)0:25 11 -2:52 L = 1.52 and K = 22 K= 2 (2.52) K = 5.04 New combination of L and K is L= 2-52 and K= 5.04