1. By: Brian Murphy

2.  Given a function for cost with respect to quantity produced by a firm and market demand with respect to price set by the firm, find the price for a manufactured good that will optimize profits for the firm.  Key variables:  p = price of manufactured good  Q = quantity manufactured  Q(p) = market demand function  C(Q) = cost function for manufacturing process  Π (Q) = profit function = R(Q) – C(Q)

3.  Given cost and demand function:  Take market demand function and solve for p in terms of Q to get inverse market demand (p(Q)).  Calculate Revenue function (R(Q) = p(Q)*Q  Find marginal revenue function MR(Q) = dR(Q)/dQ  Find marginal cost function MC(Q) = dC(Q)/dQ  Set MR = MC and solve for optimal quantity Q*.  Plug Q* into p(Q) to get profit maximizing price p*.  Plug Q* into Π (Q) to calculate profit for p*.

4. A firm faces the following market demand: Q(p) = 27.5 -0.5p and the following costs: C(Q) = 100 – 5Q + Q 2 What price should the firm set to maximize profits?

5. Find inverse market demand: Take Q(p) = 27.5 – 0.5p 0.5p = 27.5 –Q p(Q) = 55 – 2Q Find revenue function: R(Q) = p(Q) * Q = 55Q – 2Q 2 Find marginal revenue function: MR(Q) = dR(Q)/dQ = 55 – 4Q

6. Find marginal cost function: C(Q) = 100 – 5Q + Q 2 MC(Q) = -5 + 2Q Set marginal revenue equal to marginal cost: MC(Q) = MR(Q) -> 55 – 4Q = -5+2Q 6Q = 60 -> Q* = 10. Plug Q* into p(Q): p(Q) = 55 – 2Q, p(Q*) = 35 = p*.

7. Calculate profit function: Π (Q) = R(Q) – C(Q) = 55Q – 2Q 2 -100 +5Q – Q 2 = 60Q – 3Q 2 -100 With Q* = 10 Π (Q*) = $200 = maximized profit.