Question

Consider a competitive firm with initial profits given by π0 = 15, and total cost function c(q) = 10 + q^2. The market price is stochastically distributed as follows-
P˜ =(
8 with prob 1/2
2 with prob 1/2
)

If the firm manager’s utility function is given by-
u(π) = E(π) −1/9Var(π).
derive firm’s optimal output level and associated profit level.

Answer 1

Solution:

We find the price of the goods on the market as the mathematical expectation of a random variable described in the condition of the problem.

$$M(p) =\displaystyle\sum_{i=1}^n price_i probability_i$$

So, price will be

$$P=8\times \frac{1}{2}+2\times \frac{1}{2}=5$$

$$MC=P$$

$$MC=C^/(q)=2q$$

$$2q=5$$

$$q=2.5$$

$$\pi=15+(TR-TC)$$

$$\pi=15+5q-10-q^2$$

$$\pi=5+5q-q^2$$

$$\pi=5+5\times2.5-2.5^2=17.5-6.25=11.25$$

Answer: q=2.5, $\pi$ =11.25