Question

The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for \(\$ 30\) per pound. In bad weather it sells for only \(\$ 20\) per pound. Caviar produced one weck will not keep until the next week. A small caviar producer has a cost function given by $$C=0.5 q^{2}+5 q+100$$ where \(q\) is weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5 a. How much caviar should this firm produce if it wishes to maximize the expected value of its profits? b. Suppose the owner of this firm has a utility function of the form \\[ \text { utility }=\sqrt{\pi} \\] where \(\pi\) is weekly profits. What is the expected utility associated with the output strategy defined in part (a)? c. Can this firm owner obtain a higher utility of profits by producing some output other than that specified in parts (a) and (b)? Explain. d. Suppose this firm could predict next week's price but could not influence that price. What strategy would maximize expected profits in this case? What would expected profits be?

Short answer

**Answer:** 1. Calculate revenue and profit functions for good and bad weather scenarios. 2. Calculate the expected profit function considering the probabilities of good and bad weather. 3. Maximize the expected profit function by taking its first derivative and setting it to zero. 4. Calculate the expected utility using the optimal production level and given utility function. 5. Investigate alternative production strategies using the second derivative test. 6. Consider a scenario where the firm can predict next week's price but not influence it and calculate the expected profits for each weather scenario.

Step-by-step solution

Calculate Revenue and Profit Functions

First, we need to calculate the revenue and profit functions for both good and bad weather scenarios. In good weather, the price per pound of caviar is \(30, and in bad weather, it's \)20. The revenue and profit functions are as follows: Good weather revenue: \(R_G(q) = 30q\) Bad weather revenue: \(R_B(q) = 20q\) To find the profit function (\(\pi\)) for both scenarios, subtract the cost function (\(C(q)\)) from the respective revenue functions: Good weather profit: \(\pi_G(q) = R_G(q) - C(q) = 30q - (0.5 q^{2} + 5q + 100)\) Bad weather profit: \(\pi_B(q) = R_B(q) - C(q) = 20q - (0.5 q^{2} + 5q + 100)\)

Calculate the Expected Profit Function

Since each weather scenario has an equal probability of occurring (0.5), the expected profit function can be calculated as the weighted average of the good and bad weather profit functions: Expected Profit: \(\pi_E(q) = 0.5 \pi_G(q) + 0.5 \pi_B(q)\)

Maximize Expected Profit

To find the optimal production level that maximizes expected profit, we need to take the first derivative of the expected profit function \(\pi_E(q)\) with respect to \(q\) and set it equal to zero. Solve for \(q\): \(\frac{d \pi_E(q)}{d q} = 0\)

Calculate the Expected Utility

Using the optimal production level from Step 3 (denoted as \(q^*\)), calculate the expected utility for the owner with the utility function provided: \(\text { utility }=\sqrt{\pi}\) Expected Utility: \(U_E = 0.5\sqrt{\pi_G(q^*)} + 0.5\sqrt{\pi_B(q^*)}\)

Investigating Alternative Production Strategies

To answer whether a different production strategy could lead to a higher utility in this case, we need to show that any alternative production level would yield a lower expected utility, or at least, no higher expected utility. The second derivative test can help establish this condition.

Predicting Next Week's Price

Assuming the firm can predict next week's price but not influence it, we need to find the production strategy that maximizes expected profits in this case. We'll consider both good and bad weather scenarios and calculate the profits for each, and then calculate the expected profits.

Key concepts

Expected Value of Profits

When firms face uncertainties in market conditions, such as fluctuating prices due to weather, they often aim to maximize the expected value of their profits. The expected value of profits is a statistical concept that calculates the average of all possible profits, weighted by the likelihood of each scenario occurring.

In function terms, the expected profit \(\pi_E(q)\) for our caviar producer is obtained by calculating the weighted average of profits for both good (\(\pi_G(q)\)) and bad (\(\pi_B(q)\)) weather, considering that both weather conditions have equal chances of occurring, which is expressed mathematically as:
\[\pi_E(q) = 0.5 \pi_G(q) + 0.5 \pi_B(q)\].

The producer must decide on the quantity (\(q\)) prior to knowing whether conditions will be favorable, aiming to choose the production level that will, on average, provide the highest profit. To determine this, the firm calculates the derivative of the expected profit function with respect to quantity and sets it equal to zero to find the production level that maximizes expected profit. This concept is crucial because it guides the firm in making the most informed decision under uncertainty, thereby enhancing its profitability over the long term.

Utility Function

A utility function represents the satisfaction or preference an individual or firm gains from consuming goods or services, or in this case, achieving a certain level of profits. For the caviar producer, the given utility function is expressed as a function of weekly profits: \(\sqrt{\pi}\).

The square root utility function signifies a risk-averse behavior, as each additional unit of profit yields a progressively smaller increase in utility. The expected utility of a certain production strategy is evaluated by taking the average of the utility obtained from profits under different scenarios, weighted by the probability of each scenario. In our producer's case, we used the following formula:
\[U_E = 0.5\sqrt{\pi_G(q^*)} + 0.5\sqrt{\pi_B(q^*)}\]

Where \(q^*\) is the production level that maximizes expected profit, obtained from the previous step of maximizing the expected value of profits. The utility function plays a pivotal role in decision-making under uncertainty, especially when the owner's risk preferences are taken into account.

Profit Maximization

Profit maximization is the process by which a firm determines the price and output level that returns the highest profit. Under conditions of certainty, firms produce where marginal costs equal marginal revenues. However, in uncertain environments, like our caviar market affected by weather variations, firms need to take into account the probabilities of different scenarios.

In the scenario where the firm can predict the upcoming price, it can directly calculate the profit-maximizing output level for that price, considering its cost function. This contrasts with the earlier scenario where the firm needs to consider both possible prices and their probabilities. Both methods focus on finding the point at which the production level leads to the optimal outcome under the given circumstances, whether they include uncertainty or not.

In this case, whether the owner should use the strategy from part (a) depends on their utility function and attitude towards risk. If the utility from expected profits under the strategy from part (a) does not exceed that of all other possible strategies, then the owner might consider alternative production levels. Indeed, these alternative strategies should be evaluated using the utility function to ensure decisions are made that align with the owner's risk preferences and maximize their satisfaction from profits.