Q1. 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 v 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 cost function in this case. (Suppose q is output of gumballs per hour, measured in thousands of gumballs.)
(3) What is the average and marginal cost of gumball production (again, measure output in thousands of gumballs)?
(4) Graph the average and marginal cost curves for gumballs assuming v=4, w=5.
(5) Now graph these curves for v=5, w =5. Have these curves shifted? Why?
You need to explain what make you lead to this answer in order to receive full credit.
(2) Please find MPL. ?
(3) Please find MPK. ?
(4) Please show diminishing marginal productivity for K. ?
(5) Please show diminishing marginal productivity for L.
Q2. where 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?
In Q3, part (3) and part (5) count 8% each, whereas the rest of question counts 4% each.
Q3. A widget manufacturer has an infinitely substitutable production function of the form:
q = 12K + 9L
(1) Graph the isoquant maps for q=72 and q=180. ?
(2) What is the RTS along these isoquants q=72 and q=180? ?
(3) If the wage rate (w) is $3 and the rental rate on capital (v) is $3, what cost- ?minimizing combination of K and L will the manufacturer employ for the two ?different production levels at q=72 and q=180? ?
(4) What is the manufacturer's expansion path? ?
(5) How would your answer to part (3) change if v rose to $6 with w remaining at ?$3?
Suppose a firm has the Cobb-Douglas production function Q = f(K, L) = 2K0.7L0.8,
Q4. Given a Cobb-Douglas production function ?? = 100K0.4L0.6, the price of labor per unit is $60, and the price of capital per unit is $40. Please use the Lagrangian method to answer this question. You need to show all the steps including how to take partial derivatives of the Lagrangian function with respect to three endogenous variables K, L, and λ.Please make total cost to $2,400.
(1) Please find the quantity of labor and capital that this firm should use to maximize output.
(2) What is this level of output??In Q4, part (1) counts 8% and part (2) counts 4%.
Q5. A firm producing hockey sticks has a production function given by q = 4K0.5L0.5?In the short run, the firm's amount of capital equipment is fixed at K = 64. The rental rate for K is v=$1, and the wage rate for L is w=$4.
(1) Calculate the firm's short-run total cost function (STC). ?
(2) Calculate the short-run average cost function (SAC). ?
(3) Calculate the short-run marginal cost function (SMC). ?
(4) What are the SAC and the SMC for the firm if it produces 32 hockey sticks? 64 hockey ?sticks? 128 hockey sticks? 256 hockey sticks??You may use SMC= qto work for part (4) if you cannot answer how to derive the ?128 ?SMC. ?
(5) Please graph the SAC and SMC curves for the firm based on your answer in part (4). Please also indicate at which point the SAC and SMC intersects. To find the intersection point, you can use your findings in the graph you drew, or take derivative of the SAC with respect to q.
Bonus Credit:
You can choose to answer this question and earn bonus credit to bring up your grade. Please remember that you need to show the steps for part (3) and part (4) to get full credit. If you just show the final answers in part (3) and part (4) without showing how you derive them, you will not receive any credit.
The long-run total cost function for a firm is
TC = q3 -18q2 + 60q + 50 where q is the produced quantity.
(1) What is the fixed cost?
(2) What is variable cost?
(3) Please find the average cost.
(4) Please find the marginal cost.