Question

A cost-minimizing firm has the following production function: Q=LK+2M. Where L denotes Labor, K denotes Capital, and M denotes Materials. The prices for the inputs are as follows: w=$4, r=$8, and m=$1. The firm is operating in the long run. Answer the following questions as you solve for the total cost of producing 400 units of output. Assume an interior solution (i.e. positive values of all inputs).

a) Set up constrained optimization problem of the firm:

b) Write out the Lagrangian:

?c) First Order Conditions

?d) Using FOCs for labor and capital, solve for capital as a function of labor: K=f(L)

?e) Using FOCs for capital and materials, solve for L explicitly.

f) Using d and e answers, solve for K explicitly.

?g) Using e and f answers plug in to the constraint and solve explicitly for M.

?h) Finally, solve for the (minimum) total cost of producing 400 units.

Answer 1

a) The optimization problem is to minimize the cost subject to the constraint 400= KL+2M.

b) Setting up the Lagrange for minimizing the cost of production:

Lagrange = 8K + 4L + M + lambda (400-LK-2M)

c) Taking first order conditions:

dLagrange/dK = 8 + lamda(-L) = 0 ........(1)

dLagrange/dL = 4 + lambda(-K) = 0.......(2)

dLagrange/dM = 1+ lambda(-2).............(3)

d) Solving for capital as a function of labour:

4/K = 8/L or K = L/2.

e) Solving for L, L = 8/lambda

since lambda = 1/2 (from (3)), we get:

L = 16.

f) K = 8.

g) M = 136

h) Total cost, C = 8K + 4L + M

= 64+64+136 = 264