Assignment 6

Consider a monopolist's profit maximization problem.

1. The firm faces a demand curve given by P = 1000 - 2Q. The firm's cost function is C = aQ² - 6Q + 2.
   Q is output, P is price, and a is a parameter.
   
   
   1. Select a suitable value for the parameter: a = _________.
   2. Obtain the firm's profit function (in terms of Q).
   3. Obtain the first-order necessary condition for a maximum.
   4. Obtain the critical value of output. Call it Q*.
   5. Does the second-order sufficient condition for a maximum hold? Explain.
   6. What is the firm's optimal price?
   7. What is the firm's maximum profit?
   8. Obtain the firm's MR and MC equations. Sketch the curves.
   9. Sketch the following on a graph (with Q on the horizontal axis): Demand, MR, MC.
      Indicate the optimal price and output on the graph.
      
      
2. Suppose the government imposes a tax on the monopolist. The tax t is charged per unit of output. Let t = $2.
   
   
   1. Obtain the firm's profit function.
   2. Obtain the optimal price and output combination for the firm.
   3. After the imposition of the tax, will the firm produce more (or less) output? Will it charge a higher (or lower) price? Will the firm make bigger (or smaller) profits? Confirm all this.
   4. Indicate the results on the graph.