Question

A firm has the following production function:

Q = K^0.8L^0.1

1. Show if this is a constant, decreasing, or increasing returns to scale production function? Explain.
2. Find the marginal rate of technical substitution (MRTS) and discuss how MRTS changes as the firm uses more L, holding output constant.
3. Suppose that the wage rate is $10 per hour and the rental rate of capital is $10 per hour. If the firm wants to produce 500 units of output, what is the cost-minimizing bundle of capital and labor?

Answer 1

Firm's production function

Returns to scale of this production function will help to check what will happen to returns to firm in long run when scale of production increases by say . is greater than 1 here.

If both and increase by then firms output function will be :

Since the power of is lesser than 1 production function exhibits decreasing returns to scale. Decreasing returns to scale imply as inputs K and L in production function increases, then output increases less than proportionately.

Marginal rate of technical substitution is obtained by dividing marginal product of Labor by marginal product of capital.

Marginal product of Labor

Marginal product of Capital

Therefore , Marginal rate of technical substitution (MRTS)

In order to check what happen to MRTS when L changes holding output constant, take partial derivative of MRTS with respect to L i.e. . It is a negative sign , so when L changes MRTS decline.

Wage rate given is $10 per hour and rental rate of capital given is $10 per hour.

Cost minimizing bundle of K and L is obtained at a point where ratio of wage rate to rental rate of capital equalizes MRTS.

At output 500 units, production function is of form

Put in to get cost minimizing bundle of K and L--------------------

Put L=156.23 in to get K

Cost minimizing bundle of K is 1249.84 and L is 156.23.