Exercise Set 6
Profit Maximization

A firm's demand and cost functions are given by (1) and (2) respectively:
P = 60 - 2Q,                                         (1)
C = (1/3)Q3 -14Q2 + 140Q + 100,     (2)
where Q is output, P is price, and C represents total cost.

1. Optimal output and price
  1. What is the firm's fixed cost?
  2. What is the firm's variable cost?
  3. Obtain the firms' revenue function R(Q).
  4. Obtain the firm's MR and MC functions.
  5. Obtain the firm's profit in terms of output.
  6. Obtain the first-order derivative of the profit function.
  7. Obtain the second-order derivative of the profit function.
  8. Plot the profit function over Q = 0 to Q = 30.
  9. Using the first-order condition, obtain the critical values of Q.
  10. Using the second-order condition, establish the maximum and minimum.
  11. Obtain the (local) maximum and minimum values of the profit function. Indicate the values on the graph.
  12. Is MR = MC at Q = 20? At Q = 4?
  13. Confirm the solution to the firm's problem: Optimal output = _____ units. Optimal price = __________. Maximum profit = ____________.

2. Suppose the firm's fixed costs fall. Does the firm change its price and output? Confirm using a new value for the fixed cost. Explain why the firm's optimal price-output combination remains the same.

3. Suppose the demand for the firm's product rises. Using a new demand equation, obtain the optimal price and output.

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