# profit functions

#### arslan

1. A firm can price discriminate between two markets. Assume that its TR functions are

Market 1 TR_1 = 25Q_1 - 2Q_1^2

Market 2 TR_2 = 17Q_2 - Q_2^2

and its TC function is TC = 2 + 5Q_1 + 5Q_2

a. Find the firm’s profit function.

b. Find the output levels needed to maximise profit.
Use the second-order derivatives to test for a maximum.
What is the maximum profit?

What price will the firm charge in each market to maximise its profit?

#### SpringFan25

Profit = TR - TC

$$\displaystyle \Pi(Q_1,Q_2) = (25Q_1+17Q_2 -2Q_1^2 -Q_2^2) - (2 + 5Q_1 + 5Q_2)$$

b.
Find the marginal costs/revenues by differentiating the total costs/revenues with respect to the output level of the relevant good.

Find maximum profit by equating marginal cost = marginal revienue in each market

Market 1:
$$\displaystyle MR_1 = 25 - 4Q_1$$
$$\displaystyle MC_1 = 5$$

$$\displaystyle 25-4Q_1 = 5$$
$$\displaystyle Q_1 = 5$$

market 2
$$\displaystyle MR_2 = 17 - 2Q_2$$
$$\displaystyle MC_2 = 5$$

$$\displaystyle 17-2Q_2 = 5$$
$$\displaystyle Q_2 = 6$$

I assume you know what a second derivative is so you should be able to do the rest yourself.

To get the price, use the fact that (Total Revenue / Quantity) = Price.