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Chapter 11: Problem 4

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

Expert verified
**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

01

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: \(\pi_B(q) = R_B(q) - C(q) = 20q - (0.5 q^{2} + 5q + 100)\)
02

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: \(\pi_E(q) = 0.5 \pi_G(q) + 0.5 \pi_B(q)\)
03

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 \(q\): \(\frac{d \pi_E(q)}{d q} = 0\)
04

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: \(U_E = 0.5\sqrt{\pi_G(q^*)} + 0.5\sqrt{\pi_B(q^*)}\)
05

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.
06

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.

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Key Concepts

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

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.

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Most popular questions from this chapter

The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry's demand for inputs. To do so, we assume that an industry produces a homogencous good, \(Q,\) under constant returns to scale using only capital and labor. The demand function for \(Q\) is given by \(Q=D(P)\), where \(P\) is the market price of the good being produced. Because of the constant returns-to- scale assumption, \(P=M C=A C\). Throughout this problem let \(C(v, w, 1)\) be the firm's unit cost function. a. Explain why the total industry demands for capital and labor are given by \(K=Q C_{v}\) and \(L=Q C_{w}\) b. Show that \\[ \frac{\partial K}{\partial v}=Q C_{v v}+D^{\prime} C_{v}^{2} \quad \text { and } \quad \frac{\partial L}{\partial w}=Q C_{w w}+D^{\prime} C_{w}^{2} \\] c. Prove that \\[ C_{w v}=\frac{-w}{v} C_{v w} \quad \text { and } \quad C_{w w}=\frac{-v}{w} C_{N w} \\] d. Use the results from parts (b) and (c) together with the elasticity of substitution defined as \(\sigma=C C_{v n} / C_{\nu} C_{w}\) to show that \\[ \frac{\partial K}{\partial v}=\frac{w L}{Q} \cdot \frac{\sigma K}{v C}+\frac{D^{\prime} K^{2}}{Q^{2}} \text { and } \frac{\partial L}{\partial w}=\frac{v K}{Q} \cdot \frac{\sigma L}{w C}+\frac{D^{\prime} L^{2}}{Q^{2}} \\] e. Convert the derivatives in part (d) into elasticities to show that \\[ e_{K, v}=-s_{L} \sigma+s_{K} e_{Q, p} \quad \text { and } \quad e_{L, w}=-s_{K} \sigma+s_{L} e_{Q, P} \\] where \(e_{Q, P}\) is the price elasticity of demand for the product being produced. f. Discuss the importance of the results in part (e) using the notions of substitution and output effects from Chapter 11 Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall. The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993).

Because firms have greater flexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the first economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Châtelier's Principle. The basic idea of the principle is that any disturbance to an equilibrium (such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking firm that chooses its inputs to maximize a profit function of the form \(\Pi(P, v, w)=P f(k, 1)-w l-v k .\) This maximization process will yield optimal solutions of the general form \(q^{*}(P, v, w), I^{*}(P, v, w),\) and \(k^{*}(P, v, w) .\) If we constrain capital input to be fixed at \(\bar{k}\) in the short run, this firm's short-run responses can be represented by \(q^{s}(P, w, \bar{k})\) and \(I^{*}(P, w, \bar{k})\) a. Using the definitional relation \(q^{*}(P, v, w)=q^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that $$\frac{\partial q^{*}}{\partial P}=\frac{\partial q^{s}}{\partial P}+\frac{-\left(\frac{\partial k^{*}}{\partial P}\right)^{2}}{\frac{\partial k^{*}}{\partial v}}$$ Do this in three steps. First, differentiate the definitional relation with respect to \(P\) using the chain rule. Next, differentiate the definitional relation with respect to \(v\) (again using the chain rule), and use the result to substitute for \(\partial q^{3} / \partial k\) in the initial derivative. Finally, substitute a result analogous to part (c) of Problem 11.10 to give the displayed equation. b. Use the result from part (a) to argue that \(\partial q^{*} / \partial P \geq \partial q^{s} / \partial P\). This establishes Le Châtelier's Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses. c. Using similar methods as in parts (a) and (b), prove that Le Châtelier's Principle applies to the effect of the wage on labor demand. That is, starting from the definitional relation \(l^{*}(P, v, w)=l^{s}\left(P, w, k^{*}(P, v, w)\right),\) show that \(\partial l^{*} / \partial w \leq \partial l^{s} / \partial w\) implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative). d. Develop your own analysis of the difference between the short- and long-run responses of the firm's cost function \([C(v, w, q)]\) to a change in the wage \((w)\)

This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is \(-1 / b \cdot q\) where \(b(<0)\) is the slope of the demand curve. c. Show that, for a constant elasticity demand curve of the form \(q=a P^{b}\), the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)-(d) of this problem.

Suppose that a firm's production function exhibits technical improvements over time and that the form of the function is \(q=f(k, l, t) .\) In this case, we can measure the proportional rate of technical change as \\[ \frac{\partial \ln q}{\partial t}=\frac{f_{t}}{f} \\] (compare this with the treatment in Chapter 9 ). Show that this rate of change can also be measured using the profit function as \\[ \frac{\partial \ln q}{\partial t}=\frac{\Pi(P, v, w, t)}{P q} \cdot \frac{\partial \ln \Pi}{\partial t} \\] That is, rather than using the production function directly, technical change can be measured by knowing the share of profits in total revenue and the proportionate change in profits over time (holding all prices constant). This approach to measuring technical change may be preferable when data on actual input levels do not exist.

With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12 a. Use steps (b), (d), and (e) from Problem 11.12 to show that \\[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text { and } \quad e_{L, v}=s_{K}\left(\sigma+e_{Q, P}\right) \\] b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part (e) of Problem 11.12 than they do in part (a) of this problem. c. The expression computed in part (a) can be easily generalized to the many- input case as \(e_{x_{i}, w_{i}}=s_{j}\left(A_{i j}+e_{Q, P}\right),\) where \(A_{i j}\) is the Allen elasticity of substitution defined in Problem 10.12 . For reasons described in Problems 10.11 and 10.12 , this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case \(i=j\) this expression seems to say that \(e_{L, w}=s_{L}\left(A_{L L}+e_{Q . P}\right),\) and if we jumped to the conclusion that \(A_{L L}=\sigma\) in the two-input case, then this would contradict the result from Problem \(11.12 .\) You can resolve this paradox by using the definitions from Problem 10.12 to show that, with two inputs, \(A_{L L}=\left(-s_{K} / s_{L}\right) \cdot A_{K L}=\left(-s_{K} / s_{L}\right) \cdot \sigma\) and so there is no disagreement.
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