Exercise Set 2
Partial Derivatives

Problem 1

Consider the production function Q = F(K,L) = 6K^0.75L^0.25, where Q represents output, K capital and L labor.

1. Find the first-order partial derivatives of the function F(K,L). Are they positive or negative?

2. Find the second-order partial derivatives of F(K,L). Show that the cross-partials are equal to each other.

3. Suppose the firm is currently using 200 workers and 50 machines. What is the firm's marginal product of capital? Marginal product of labor?

4. As the firm hires additional workers, ceteris paribus, does the marginal product of labor rise--or fall? Does the production function exhibit the Law of Diminishing Returns?

5. Suppose the firm has a capital stock of 16 machines.

   1. Plot Q vs. L.
   2. What is the slope of the curve at L = 10?
   3. What does it represent?
      
      
6. Does the given production function exhibit constant returns to scale?

Answers to Problem 1

a. The first-order partial derivatives are positive, implying that as K (or L) increases, so does output.

b. Note that F₁₂ = F₂₁.

c. Suppose the firm is currently using 200 workers and 50 machines.

MP_K = 6.36

MP_L = 0.53

d. Note that F₂₂ < 0, which means that F₂ decreases as L increases. Thus, as the firm hires additional workers, ceteris paribus , the marginal product of labor falls.

e. Since F₂₂ < 0, yes: the production function does exhibit the Law of Diminishing Returns. Also applies to F₁₁ < 0: As more capital is added, the marginal product of capital falls.

f. Q = 48L^0.25.

Slope of the production function at L = 10 is 2.13.

This means that an infinitesimal increase in L, ceteris paribus , will cause output to rise by 2.13 units.

g. Yes. If you double the values of K and L, output will double too. This CRTS feature is associated with Cobb-Douglas functions .

Problem 2

Consider the production function Q = F(K,L) = 40KL^2/3, where Q represents output, K capital and L labor.

a. Find the first-order partial derivatives of the function F(K,L). Are they positive or negative?

b. Find the second-order partial derivatives of F(K,L). Show that the cross-partials are equal to each other.

c. Suppose the firm is currently using 700 workers and 60 machines. What is the firm's marginal product of labor?

d. As the firm hires additional workers, ceteris paribus, does the marginal product of labor rise--or fall?

e. Does the production function exhibit the Law of Diminishing Returns with respect to capital?

f. Suppose the firm has a capital stock of 10 machines.

• Plot Q vs. L.
• What is the slope of the curve at L = 5?
• What does it represent?

g. Does the given production function exhibit CRTS?(Hint: Try doubling the values of K and L. What is the corresponding change in output?)