Question

Consider a textile manufacturing firm that uses labor and capital inputs and has the production technology given by the equation Q = 8K0.25L 0.5 , where Q is output, K is capital and L is labor. Each unit of capital costs 10 TL while each unit of labor costs 5 TL.

a) Does this firm have increasing, decreasing or constant returns to scale? (1)

b) Define the cost minimization problem faced by firm. What is the objective function, what is the constraint? (1)

c) Write down the Lagrangean of the cost minimization problem. (1)

d) Write down the first order conditions of the Lagrangean. (3)

e) What are the conditional input demand functions for capital and labor of this firm? (2)

f) What is the total cost function of the firm? (2)

g) What is the supply function of the firm? (2)

Answer 1

0000000 1 M T W T F S Su to scale, we will 2. олі Poodmetion function: 22.86%4's :) To determine the returns increase both K & L by m. Then new production function & We will compare Black) "" ( mb 1/2 tt ha keto la create ' I to . 0 8 m - K 8 ktu 1 1/2 am 21% og m Blu m) 1 m <m Now, because Therefore, we scale. have deeree creasing returns to -6) The cost constraint is given C = P L + Pk K Where eis total cost P = 5TL P = 10TL 15= SL + 104 . Production / antput function 08 ALAK

00 M T W T F S Su Now, for cost minimization, we will have to minize minimize minimize cire. 52+ lok = 15 m, L+2k 23 subject to the output fine 8=8kt 2 C) Lagrangian for cost minimization Z L+ 2k + Ba(6-8kku [2) Where a Lagrange multiplies (d) F.o.ch sa SL 1 8 2 / 2 - - - 2 klu zo SK sk 2 824 2 4 L k La 0 sa s Sa 8k uL tull so