Question

A firm has the following production function: ?(?, ?) = ? 3/4?1/4 .

A) What is the firm’s Technical Rate of Substitution?

B) What is the optimality condition that determines the firm’s optimal level of inputs?

C) Suppose the firm wants to produce exactly ? units and that input ? costs $?? per unit and input ? costs $?? per unit. What are the firm’s conditional input demand functions? D) Using the information from part C, write down the firm’s total cost function as a function of ?? , ??, and ?.

Answer 1

a)

f(L,M)=y=L^3/4M^1/4

MP_L=dy/dL=0.75L^-1/4M^1/4

MP_M=dy/dM=0.25L^3/4M^-3/4

MRTS=MPL/MPM=(0.75L^-1/4M^1/4)/(0.25L^3/4M^-3/4)=3M/L

b)

Optimal level of inputs requires that MRTS should be equal to ratio of input prices i.e.

MRTS=w_L/w_M

_c)

MRTS=w_L/w_M

3M/L=w_L/w_M

M=(w_L/3w_M)*L

We know that

y=L^3/4M^1/4

y=L^3/4*[(w_L/3w_M)*L]^1/4

y=(w_L/3w_M)^1/4*L

L=y*(3w_M/w_L)^1/4 -------conditional factor demand for L

M=(wL/3wM)*L=(wL/3wM)*y*(3w_M/w_L)^1/4

M=y*(w_L/3w_M)^3/4  -------conditional factor demand for M

d)

TC=L*w_L+M*w_M

TC=y*(3w_M/w_L)^1/4*w_L+w_M*y*(w_L/3w_M)^3/4

TC=y*3^1/4*w_M^1/4*w_L^3/4+y*(1/3)^3/4w_M^1/4*w_L^3/4

TC=0.57735*y*w_M^1/4*w_L^3/4