Question

Problem 2: A firm has the following production function: ?(?, ?) = ? + ?

A) Show whether this firm’s technology exhibits constant, increasing, or decreasing returns to scale.

B) Suppose the firm wants to produce exactly ? units and that input ? costs $?? per unit and input ? costs $?? per unit. What are the firm’s input demand functions?

C) Write down the formula for the firm’s total cost function as a function of ?? , ??, and y

Answer 1

The production function of the firm is given as

f(L, M) = L + M

This is a perfect substitute type production function.

(A) If we increase the inputs L and M by proportion "k", then we get

f(k.L, k.M) = k.L + k.M

or, f(k.L, k.M ) = k.(L + M)

or, f(k.L, k.M) = k.f(L, M)

We can see that, the output also increases by proportion "k". Hence, output and inputs increases in a same proportion. Hence, the production function exibits constant returns to scale.

The firm's technology exibits Constant Returns to Scale (CRS).

(B) The firm wants to produce y units of output.

Hence,

f(L, M) = y

Input L costs $wL per unit and input M costs $wM per unit.

Now, the production function is perfect substitute type. The firm will use the input which has lower cost per unit.

• When, $wL < $wM, the firm uses only L as input. Hence, M = 0.

We get,

f(L, M) = L + M

or, y = L + 0

or, L = y, when $wL < $wM

• When, $wL > $wM, the firm uses only M as input. Hence, L = 0.

We get,

f(L, M) = L + M

or, y = 0 + M

or, M = y, when $wL > $wM

The firm's input demand functions are,

L = y, when $wL<$wM

M = y, when $wL>$wM

(C) Hence, the firm's total cost function is

C = L.wL + M.wM

We put L = y and M = y from part (B) and get,

C = y.wL + y.wM

or, C = (wL + wM).y

This is the total cost function of the firm.

Hope the solutions are clear to you my friend