Chapter 13: Problem 8

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 week will not keep until the next week. A small caviar producer has a cost function given by \\[ T C=\sim q^{2}+5 q+\mathbf{1 0 0} \\] 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 }=\mathrm{V}_{\mathrm{TT}} \\] where $T T$ is weekly profits. What is the expected utility associated with the output strategy denned 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 prof its be?

Short Answer

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a. Optimal production quantity for expected profit maximization Step 1: Calculate profit function in good and bad weather scenarios Good weather profit: $\pi_G = 30q - (q^2 + 5q + 100)$ Bad weather profit: $\pi_B = 20q - (q^2 + 5q + 100)$ Step 2: Calculate expected profit function Expected Profit (EP): $EP = 0.5 (\pi_G) + 0.5 (\pi_B) = 0.5(30q - (q^2 + 5q + 100)) + 0.5(20q - (q^2 + 5q + 100))$ Step 3: Differentiate expected profit function $\frac{d(EP)}{dq} = 0$ Step 4: Solve for optimal production quantity _optimal_output_ b. Expected utility associated with optimal output strategy Step 1: Calculate profit with optimal output in each scenario $\pi_G = 30q^* - (q^{*2} + 5q^* + 100)$ $\pi_B = 20q^* - (q^{*2} + 5q^* + 100)$ Step 2: Calculate expected utility $EU = 0.5 (\sqrt{\pi_G}) + 0.5 (\sqrt{\pi_B})$ _expected_utility_ c. Possibility of attaining higher utility with different output Step 1: Analyze the utility function _utility_analysis_ d. Optimal strategy when next week's price can be predicted Step 1: Determine profit function when price can be predicted $\pi = PQ - (q^2 + 5q + 100)$, where $P$ can be either $30$ or $20$. Step 2: Calculate optimal output for each price level $\frac{d(\pi)}{dq} = 0$ _optimal_output_predicted_ Step 3: Calculate expected profits $EP = 0.5(\pi_{30}) + 0.5(\pi_{20})$ _expected_profit_predicted_

Step by step solution

Calculate profit function in good and bad weather scenarios

First, we need to determine the profit function in both good and bad weather scenarios. Profit is defined as revenue minus total cost (TC). In good weather, the price per pound is $30 and revenue can be calculated as $(30q)$. In bad weather, the price per pound is $20, so the revenue is $(20q)$. The total cost (TC) function is given: $TC= q^2 + 5q + 100$. We can calculate the profit in both scenarios: Good weather profit: $\pi_G = 30q - (q^2 + 5q + 100)$ Bad weather profit: $\pi_B = 20q - (q^2 + 5q + 100)$

Calculate expected profit function

Since there is a 0.5 probability of good weather and a 0.5 probability of bad weather, we can calculate the expected profit function as the weighted sum of profits in each scenario: Expected Profit (EP): $EP = 0.5 (\pi_G) + 0.5 (\pi_B) = 0.5(30q - (q^2 + 5q + 100)) + 0.5(20q - (q^2 + 5q + 100))$

Differentiate expected profit function

To find the optimal production quantity, we need to find the critical points by differentiating the expected profit function with respect to $q$ and setting it equal to 0: $\frac{d(EP)}{dq} = 0$

Solve for optimal production quantity

Solve the equation obtained in Step 3 to find the optimal production quantity $q$: _optimal_output_ #b. Expected utility associated with optimal output strategy#

Calculate profit with optimal output in each scenario

Using the optimal output quantity obtained in part (a), calculate the profit in both good and bad weather scenarios: $\pi_G = 30q^* - (q^{*2} + 5q^* + 100)$ $\pi_B = 20q^* - (q^{*2} + 5q^* + 100)$

Calculate expected utility

Now, we can calculate the expected utility using the owner's utility function, which is defined as $Utility = \sqrt{TT}$, where $TT$ is weekly profits: $EU = 0.5 (\sqrt{\pi_G}) + 0.5 (\sqrt{\pi_B})$ _expected_utility_ #c. Possibility of attaining higher utility with different output#

Analyze the utility function

To determine if the owner can achieve higher utility by producing a different output, inspect the shape and convexity of the utility function. If the utility function is concave, the owner can produce an output quantity other than the one in parts (a) and (b) that yields higher utility. _utility_analysis_ #d. Optimal strategy when next week's price can be predicted#

Determine profit function when price can be predicted

If the firm can predict next week's price, it can set the same marginal cost and marginal revenue. The profit function will be: $\pi = PQ - (q^2 + 5q + 100)$, where $P$ can be either $30$ or $20$.

Calculate optimal output for each price level

Differentiate the profit function with respect to $q$, set equal to 0, and solve for optimal production quantity $q^*_P$ for each price level $P$: $\frac{d(\pi)}{dq} = 0$ _optimal_output_predicted_

Calculate expected profits

Using the optimal output quantities for both price levels, calculate the expected profit for each scenario: $EP = 0.5(\pi_{30}) + 0.5(\pi_{20})$ _expected_profit_predicted_

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Function

Understanding the cost function is crucial for any business aiming to maximize profit. It represents the total cost of producing a certain quantity of goods, typically involving variable costs that change with the level of output, and fixed costs that remain constant regardless of production levels.

In our caviar producer's scenario, the cost function is defined as $ TC = q^2 + 5q + 100 $. Here, $ q^2 + 5q $ represents the variable costs that increase with the quantity of caviar produced, while $ 100 $ symbolizes the fixed costs that do not fluctuate with production. Being aware of this function helps the firm in planning and decision-making, particularly when predicting and calculating expected profits under different weather scenarios, which in turn affect demand and selling price.

Utility Function

A utility function expresses a consumer's or firm owner's preferences, translating these into a quantifiable measure. In economics, it is used to model satisfaction or happiness with the consumption of goods or, in the case of our caviar producer, the satisfaction from generating profit.

The provided utility function $ Utility = \sqrt{TT} $, where $ TT $ is the weekly profits, is a square root function, indicating diminishing marginal utility of income. This signifies that the satisfaction from each additional unit of profit decreases as the profit increases. When assessing expected utility associated with different production levels, this function becomes vital, as it affects the producer's decision making, especially when weighing risks involved with uncertain weather conditions.

Marginal Analysis

Marginal analysis is an examination of the additional benefits of an activity compared to the additional costs incurred by that same activity. Companies use marginal analysis as a decision-making tool to help determine the ideal level of production.

For our caviar producer, this involves differentiating the expected profit function with respect to the production quantity $ q $ to find where the additional revenue from producing one more pound of caviar equals the additional cost of producing that pound. This critical point is where profit is maximized, as expanding production beyond this point would reduce the overall profit due to the increasing marginal cost outweighing the marginal revenue.

Optimal Production Quantity

The optimal production quantity is the output that maximizes a firm's profits given its cost structure and market conditions. It is found where marginal revenue equals marginal cost, which in the case of the caviar producer, is complicated by uncertain weather affecting the selling price.

Through proper application of marginal analysis, the caviar producer can calculate the expected profit for both good and bad weather scenarios. The optimal point is where the derivative of the expected profit function is zero, indicating that the firm has balanced additional revenue against additional costs efficiently. This balance ensures that the producer does not create excess supply, leading to spoiled caviar, or underproduce, missing out on potential profits.