Question

**Cobb-Douglas Production Function & Cost of
Production**

- A firm’s production function is given as –

q =
2K^{0.4}N^{0.6}

- What kind of returns to scale does this production technology exhibit? Justify your answer.
- Find out the expression for the marginal product of labor.
- Find out the expression for the marginal product of capital.
- Find out the expression for MRTS.

Answer #1

a)

Given production function is Cobb-Douglas. Let us check sum exponents.

Sum of exponents=0.40+0.60=1.0

**Since sum is equal to 1, we can say that production
function exhibits constant returns to scale (CTS)**

b)

Marginal product of labor can be determined by differentiating q with respect to N, we get

**MP _{N}=dq/dN=0.6*2K^{0.4}N^{0.6-1}=1.2K^{0.4}N^{-0.4}**

c)

Marginal product of capital can be determined by differentiating q with respect to K, we get

**MP _{K}=dq/dK=0.4*2K^{0.4-1}N^{0.6}=0.8K^{-0.6}N^{0.6}**

d)

A firm’s production is represented by the following Cobb-Douglas
function: ? = ?^2/3?^1/3. The rental rate, r, of capital is given
by $200 and the price of labor is $100.
a) For a given level of output, what should be the ratio of
capital to labor in order to minimize costs?
b) How much capital and labor should be used to produce those
300 units?
c) What is the minimum cost of producing 300 units?
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capital to labor in order to minimize costs?
b) How much capital and labor should be used to produce those
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