Question

Please show all your work step by step

1. Suppose the production function is Cobb-Douglas and f(x1,22) = x122. (a) Write an expression for the marginal product of xy at the point (21,22). (b) Holding a fixed, for small increases in 31, will the marginal product of increase, decrease or remain constant? (c) What's the marginal product of factor 2? Will it increase, remain constant or decrease for small increases in x2? (d) Does an increase in the amount of c, increase, leave unchanged or decrease the marginal product of xn? (e) What's the technical rate of substitution TRS (21,22)? (f) Does this technology have diminishing technical rate of substitution? (g) Does this technology demonstrate increasing, constant or decreasing returns to scale?

Answer 1

la) f(n) an 2/2 - af (2,22) Marginal product of no = 20 A = 1/22 - 11/22 222 an 72 I holding a fixed for small increase in ni Por = - 1/2 X /22, -3/2 3/2 --Yux,-9/2 % Lo I so, Mor decoreans for sineruare in a o MP = 28 (213 X 2) = 2/2 2/2 - 1 222, 2202 12 - 3,127,1277-72 23 x, -/2 > 0 for small increan in de Mine increases 4 MPa = /2-1/2 12 / 2 I anpa - 10% 2-1/2 12 / 2 2 MPN 202 I so, for a increase in m2, MPa, increas e MRPS = MPM - 1/2, -1/2 2 2 3/2 MPnr ₂2, 1/222/2 = 1/3. 1 972 12

f AMATS - -Y 2 20. E so, the technology follaus diminishing rate of rechnical substitution, Eg increase both the intents inpub by amant Plano, A₂) = (2x)/2 A 22 ) % = 1/2+² , 1/2, 3/2 So, if both inputs increase by x amamb, then, fcn , n) increases more than & amamb. So, technology fallaus increasing returns to scale.