# Firm's Optimal Quantity, Price and Profit

1. The original revenue function for the microchip producer is R=170Q-20Q2. Derive the expression for marginal revenue, and use it to find the output level at which revenue is maximized. Confirm that this is greater than the firm's profit-maximizing output, and explain why.

2. Suppose a firm's inverse demand curve is given by P=120- .5Q, and its cost equation is C=420+60Q+ Q2

(1) Find the firm's optimal quantity, price, and profit (1) by using the profit and marginal profit equations and (2) by setting MR equal to MC. Also provide a graph of MR and MC.

(2) Suppose instead that the firm can sell any and all of its output at the fixed market price P=120. Find the firm's optimal output.

3. Suppose a firm assesses its profit function as

profit =-10-48Q+15Q2-Q3

(1) Compute the firm's profit for the following levels of output:

Q=2, 8, and 14.

(2) Derive an expression for marginal profit. Compute marginal profit at Q=2, 8, and 14. Confirm that profit is maximized at Q=8. (Why is profit not maximized at Q=2?)

#### Solution Preview

1. The original revenue function for the microchip producer is R=170Q-20Q2. Derive the expression for marginal revenue, and use it to find the output level at which revenue is maximized. Confirm that this is greater than the firm's profit-maximizing output, and explain why.

MR = 170 - 40Q

Setting this to zero gives us the point where revenue is maximized:

170 -40Q = 0

40Q = 170

Q = 4.25

However, profit is maximized when MR = MC. We do not have information to find MC in this question, however, we can assume that it is greater than zero (it should cost something to produce the microchips). If we set any positive number equal to MR we find that 4.25 is too large:

170 - 40Q < MC when Q = 4.25

When we decrease the value of Q, the total ...

#### Solution Summary

A firm's optimal quantity, price, and profit is determined in this solution. Calculations are plainly laid out and often explained in words.