Q1. Suppose a firm has the Cobb-Douglas production function Q = f(K, L) = AKαLβ, where α, β > 0, α + β = 1; A is a positive parameter that represents technology, K is capital, and L is labor. Using this function, show the following.
(1) Does this production function exhibit increasing, constant, or decreasing returns to scale? Why?
(2) Please draw the set of three isoquants that support your answer in part (1). What are the characteristics of these isoquants?
(3) Please find MPL and MPK.
(4) Please show diminishing marginal productivities for both K and L.
(5) Given the total cost outlay such as C = wL + rK where w is the wage of labor and r is the rental price of capital. Please determine the amount of labor (L*) and capital (K*) that the firm should use in order to maximize output.
(6) Suppose that Q = 5√KL, w = 5, r = 20, C = 1000. Please find the optimal quantities of labor and capital that this firm should hire. How much output can the firm produce?
Q2. A widget manufacturer has an infinitely substitutable production function of the form:
q = 4K + 3L
(1) Graph the isoquant maps for q=24, q=48, and q=60. What is the RTS along these isoquants?
(2) If the wage rate (w) is $1 and the rental rate on capital (r) is $1, what costminimizing combination of K and L will the manufacturer employ for the three different production levels in part (1)? What is the manufacturer's expansion path?
(3) How would your answer to part (2) change if r rose to $2 with w remaining at $1?
Q3. Suppose that the Acme Gumball Company has a fixed proportions production function that requires it to use three gumball presses and two workers to produce 1000 gumballs per hours.
(1) Please find the cost per hour of producing 1000 gumballs. (Suppose r is the hourly rent for gumball presses and w is the hourly wage.)
(2) Assume Acme can produce any number of gumballs they want using this technology. Please find the total cost function in this case. (Suppose q is output of gumballs per hour, measured in thousands of gumballs.)
(3) Please find the average and marginal cost of gumball production (again, measure output in thousands of gumballs).
(4) Please graph the average and marginal cost curves for gumballs assuming r=2, w=3.
(5) Now graph these curves for r = 3, w = 3. Have these curves in part (4) shifted?
Q4. The short-run total cost function for a firm producing skateboards is
TC = 18q2 - 9q + 72
where q is the number of skateboards per week.
(1) What is the fixed cost? What is the variable cost?
(2) Please calculate the average cost function for skateboards.
(3) Please calculate the marginal cost function for skateboards.
(4) Please show that when average cost reaches a minimum, marginal cost intersects average cost. Also, at what level of skateboard output does average cost reach a minimum? What is the average cost at this level of output?