Question

A firm has two independently operated plants that make the same good using the same two inputs. They differ, however, in the technology of production. One plant produces output according to the production function ~a22- while the other plant produces output according to the product on What is the firm's cost function c(w, w2,q)? b,1-b

Answer 1

Let's assume that plant 1 produces quantity $q_1$ and plant 2 produces quantity $q_2$ .

Plant 1's objective is to minimize cost subject to the production function:

At the point of equilibrium, marginal rate of technical substitution is equal to the ratio of factor prices:

W1Z1(1 - a) az2 (1 - a)zz (1 - a)z1W2 aw2

Substituting the value of $z_2$ in the production function, we get:

a21-a W1(1-a) aw2

and,

aw2

The cost function for plant 1 is:

aw2 W1(1

Similarly, for plant 2, the cost function is:

aw2 -b W1(1

The total cost is given by:

$C=C_1+C_2$

The firm's objective is to minimize the total cost subject to the constraint $q_1+q_2=Q$

The 2 cost functions are identical except for a and b. If a>b, plant 1's isoquant will be steeper than that of plant 2 at all points and hence, the marginal cost of plant 2 will be lower than plant 1 at all point. Therefore, firm will only use plant 1 for production. Similarly, if b>a, firm will only use plant 1 for production. Therefore, the firm's cost function is given by:

bW2 Уь, if a > b W1(1 -b)

$C=w_1q(\frac{aw_2}{w_1(1-a)})^(1-a)+w_2q(\frac{aw_2}{w_1(1-a)})^{-a} , \ if \ b>a$