Question

You are the manager of a firm that sells a “commodity” in a market that resembles perfect competition, and your cost function is$\mathit{C}\mathbf{\left(}\mathit{Q}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathit{Q}\mathbf{+}\mathbf{3}{\mathit{Q}}^{\mathbf{2}}$. Unfortunately, due to production lags, you must make your output decision prior to knowing for certain the price that will prevail in the market. You believe that there is a$\mathbf{70}$percent chance the market price will be$\mathit{\backslash (}\mathbf{200}$and a$\mathbf{30}$percent chance it will be$\mathit{\backslash )}\mathbf{600}$.

a. Calculate the expected market price.

b. What output should you produce in order to maximize expected profits?

c. What are your expected profits?

Short answer

a. The expected market price is $\$320$.

b. The output that be produced in order to maximize expected profits is 11 units.

c. The expected profits are $\mathit{\$}\mathbf{3}\mathbf{,}\mathbf{746.07}$.

Step-by-step solution

Concept Introduction

Marginal Cost: In economics, marginal cost depicts the change in the total cost that occurs due to the change in the output produced by the firm. If the cost of producing additional quantity increases the cost of the firm, it signifies that the marginal cost of the firm has increased.

Profit Maximization: The profit of the firm is maximized when the firm chooses the optimal quantity and price that increases its revenue exceeding the total cost at the maximum level.

Expected Market Price 

a.

The expected market price of the commodity can be calculated as the sum of the different probabilities that a result willoccur times the resulting payoffs. This can be represented in the following equation –

$E\left(X\right)=q_1{x}_1+q_2{x}_2+\dots q_n{x}_n\mathrm{......}\left(1\right)$

Taking into account that there is a $70\%$probability that the market price will be $\$200$and a$30\%$ probability that the price will be $\$600$, it can be substituted in equation$1$to obtain the expected market price.

Thus,

role="math" localid="1658729504294" $\begin{array}{c}E\left(X\right)=\frac{70}{100}\times 200+\frac{30}{100}\times 600\\ E\left(X\right)=0.7\times 200+0.3\times 600\\ E\left(X\right)=140+180\\ E\left(X\right)=320\end{array}$

Therefore, the value for price is obtained as$\$320$ .

Maximizing the Profits

b.

Since the real price is not known, the firm's revenues and profits cannot be determined either, therefore in acompetitive market, the marginal cost (MC) will equal the price. Then it can be assumed that –

$\begin{array}{c}E\left(p\right)=\$320\\ E\left(p\right)=\mathrm{Cos}t\mathrm{Pr}oduction\end{array}$

Therefore, replace the previous equation with the production function and solve for$Q$ to find that the output that maximizes expected profits. Thus,$\$320=2Q+3Q^2$

Equating and rearranging the equation to zero to solve the quadratic equation –

$\begin{array}{c}2Q+3Q^2-320=0\\ Q=\frac{-2+\sqrt{2^2-4\left(3\right(-320\left)\right)}}{2\left(3\right)}\\ Q=\frac{-2+\sqrt{3844}}{6}\\ Q=10.7\end{array}$

Therefore, the value for units produced is obtained as 11 units.

The Expected Profits 

c.

The expected profits can be calculated with the following equation –

$\begin{array}{c}E\left(\pi \right)=\{E\left(p\right)Q-C\left(Q\right)\}\\ E\left(\pi \right)=320\times 10.7-2\times 10.7+3{\left(10.7\right)}^2\\ E\left(\pi \right)=3,746.07\end{array}$

The value for expected profit is obtained as $\$3,746.07$.