Mathematica Problem 2
Use Mathematica to solve the following problems:
A firm produces output Q using labor, L, and capital, K. Its production
function is:
.6 .4
Q = 10 L K
1. Plot the production function in 3 dimensions with Q on the vertical axis.
Set the range on L from 0 to 15 and the range on K from 0 to 1000.
2. Hold K constant at 800, and graph the MP and AP of labor. Set the range
on L from 0 to 15.
3. Hold L constant at 6 and graph the MP and AP of capital. Set the range on
K from 0 to 1000.
4. Suppose the wage, W, equals 10 and the price of capital, r, equals .05,
graph the equal-cost curves for total costs, TC, of $50, $100, and $200.
5. Graph the equal-product curves for output equal to 225, 425, 625.
6. For Q = 425, W = 10, r = .05, and TC = $100, graph the equal-product and
equal cost-curves on the same diagram. What is the cost-minimizing
combination of capital and labor that should be used to produce 425 units of
Q?
7. If W = 10 and r = .05, TC = 10 L + .05 K. Solve for TC in terms of Q
(Hint: use the least cost rule) and graph the TC curve. Q should be on the
horizontal axis. Limit the range of Q to 600.
8. Graph the MC and AC curves for W = 10 and r = .05. Limit the range of Q
to 600.
9. Suppose this is a monopoly firm. It faces the following demand curve:
P = 1 - .0009 Q
where P is the price of its output. Graph the demand and marginal revenue,
MR, curve for this firm.
10. Add the MC and AC curve to your diagram in 9 and determine the profit
maximizing output for the firm. What are the monopoly profits?
11. Determine the price and output if this were a competitive industry.
12. Determine the deadweight loss of the monopoly.