Question

Calculate the profit-maximizing output and price in an environment of uncertainty.

Short answer

The presence of uncertainty directly impacts the consumers and their choice of commodities.

At this point, the firm must consider consumer behavior to understand their demand, depending on the output decisions.

The profit-maximizing output is $1300$units and the price is $\$1.3$respectively.

Step-by-step solution

Profit Maximisation

It is the condition where a firm earns profits where its expected marginal revenue exceeds marginal cost, thus, expanding its profits.

Profit Maximisation under Risk Neutral Environment

Let us say the manager is risk-neutral; that is, he is indifferent to risk. His goal is to maximize its expected profit.

The manager determines the output before being sure about the demand for the commodity.

Since demand is uncertain hence, revenues thus expected are uncertain as well. Thus, the manager equates its expected marginal revenue with$E\left(MR\right)=MC$.

If expected marginal revenue increases, the firm enjoys profits from expanding output and vice versa.

Mathematical Explanation

Suppose Pop Hut produces popsicles and sells them in the competitive market. Since the manager is risk-neutral, he determines the output before any knowledge about the market demand or price in the market.

However, economists suggest a $30\%$chance that the market price would be $\$2$per popsicle and a $70\%$chance it will be $\$1$when it hits the market.

Let the cost function be, $C=200+0.0005Q^2$. Let’s find out the output that needs to be produced to maximize the expected profit.

The profit function can be written as:

$\pi =pQ-200-0.0005Q^2$

For a competitive firm:

$$\begin{gathered} MR=p \\ \mathrm{MC}=\frac{\partial C}{\partial Q} \\ =0.001Q \end{gathered}$$

Expected price:

$$\begin{gathered} E\left(MR\right)=E\left(p\right) \\ =\left\{\frac{30}{100}\times 2\right\}+\left\{\frac{70}{100}\times 1\right\} \\ =(0.3*2)+(0.7*1) \\ =1.3 \end{gathered}$$

Hence the expected price is $\$1.3$

We know,

$$\begin{gathered} E\left(MR\right)=MC \\ 1.3=0.001Q \\ Q=1300 \end{gathered}$$

Hence, the firm produces$1300$ popsicles.

The expected profit can be written as:

$$\begin{gathered} E\left(\pi \right)=E\left(p\right)Q-200-0.0005Q^2 \\ =1.3*1300-200-0.0005{\left(1300\right)}^2 \\ =645 \end{gathered}$$

Thus, the expected profit is $\$645$

The profit-maximizing output and price for Pop Hut are $1300$units and $\$1.3$respectively.