Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

uncertain. If we assume utility is additive across the two periods, we have \(E\left[U\left(c_{1}, h_{1}, c_{2}, h_{2}\right)\right]=U\left(c_{1}, h_{1}\right)+E\left[U\left(c_{2}, h_{2}\right)\right]\) Show that the first-order conditions for utility maximization in period 1 are the same as those shown in Chapter \(16 ;\) in particular, show \(M R S\left(c_{1} \text { for } h_{1}\right)=w_{1} .\) Explain how changes in \(W_{0}\) will affect the actual choices of \(c_{1}\) and \(h_{1}\) b. Explain why the indirect utility function for the second period can be written as \(V\left(w_{2}, W^{*}\right),\) where \(W^{*}=W_{0}+w_{1}\left(1-h_{1}\right)-\) \(c_{1}\). (Note that because \(w_{2}\) is a random variable, \(V\) is also random.) c. Use the envelope theorem to show that optimal choice of \(W^{*}\) requires that the Lagrange multipliers for the wealth constraint in the two periods obey the condition \(\lambda_{1}=E\left(\lambda_{2}\right)\) (where \(\lambda_{1}\) is the Lagrange multiplier for the original problem and \(\lambda_{2}\) is the implied Lagrange multiplier for the period 2 utility- maximization problem). That is, the expected marginal utility of wealth should be the same in the two periods. Explain this result intuitively. d. Although the comparative statics of this model will depend on the specific form of the utility function, discuss in general terms how a governmental policy that added \(k\) dollars to all period 2 wages might be expected to affect choices in both periods.

Short Answer

Expert verified
Answer: The first-order conditions for utility maximization in period 1 imply that the marginal rate of substitution (MRS) between consumption and leisure is equal to the wage in the first period. Changes in initial wealth will affect the optimal choice of consumption and leisure in the first period, with an increase in wealth leading to higher consumption and more leisure, and a decrease in wealth leading to lower consumption and reduced leisure.

Step by step solution

01

Finding the First-Order Conditions for Utility Maximization in Period 1

To maximize the expected utility in period 1, the first step is to take the partial derivatives of the objective function with respect to its arguments, \(c_1\) and \(h_1\), and set them equal to zero. Denote \(U_1 \equiv U(c_1, h_1)\) and \(U_2 \equiv E[U(c_2, h_2)]\), then \(E[U(c_1, h_1, c_2, h_2)] = U_1 + U_2\). The first-order conditions are: \begin{align*} \frac{\partial E[U]}{\partial c_1} = \frac{\partial U_1}{\partial c_1} - \lambda = 0 \\ \frac{\partial E[U]}{\partial h_1} = \frac{\partial U_1}{\partial h_1} - w_1\lambda = 0 \end{align*} where \(\lambda\) is the Lagrange multiplier. The first-order conditions imply that the marginal rate of substitution (MRS) between \(c_1\) and \(h_1\) is equal to \(w_1\): \(MRS(c_1 \text{ for } h_1) = w_1\).
02

Impact of Changes in Initial Wealth on Choices

Changes in initial wealth (\(W_0\)) will affect the optimal choice of consumption and leisure in the first period. An increase in \(W_0\) would relax the wealth constraint and lead to higher consumption and more leisure in period 1. Conversely, a lower initial wealth would reduce consumption and leisure in period 1.
03

Indirect Utility Function for Period 2

The indirect utility function for period 2 can be written as \(V(w_2, W^*)\), where \(W^* = W_0 + w_1(1-h_1) - c_1\). Given that period 2 wage (\(w_2\)) is a random variable, the indirect utility function \(V\) is also random.
04

Envelope Theorem and Optimal Choice of Wealth

Using the envelope theorem, we can show that optimal choice of \(W^*\) requires that the Lagrange multipliers obey the following condition: \(\lambda_1 = E(\lambda_2)\), where \(\lambda_1\) is the Lagrange multiplier for the original problem, and \(\lambda_2\) is the implied Lagrange multiplier for the period 2 utility-maximization problem. This implies that the expected marginal utility of wealth should be the same in the two periods. Intuitively, this result suggests that agents should distribute their wealth between the two periods such that the additional utility derived from additional wealth is the same across both periods.
05

Impact of a Governmental Policy Adding to All Period 2 Wages

The comparative statics of this model depend on the specific form of the utility function. However, in general terms, a governmental policy that adds \(k\) dollars to all period 2 wages could encourage agents to reallocate their choices towards more consumption and leisure in the second period, given the increase in income. In anticipation of higher income in period 2, agents may choose to save more in period 1 or potentially even reduce their consumption and leisure in period 1 to enjoy more benefits in period 2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Utility Maximization
At its core, utility maximization is the economic premise that individuals choose to allocate their resources in a way that maximizes their satisfaction or utility. In the two-period model presented in the exercise, individuals aim to distribute consumption (\(c_1, c_2\)) and leisure (\(h_1, h_2\)) between both periods to achieve the highest utility. The first-order conditions for maximizing utility reflect the trade-offs between these goods. Essentially, to maximize utility, the marginal utility per dollar spent on each good must be equalized across all goods. For instance, if an individual values an extra hour of leisure more than the goods that could be purchased with an hour's wage, they will choose more leisure until their marginal rate of substitution between consumption and leisure equals their wage rate, balancing the utility gained from each additional unit of good or leisure time.
Marginal Rate of Substitution
The marginal rate of substitution (MRS) is a key concept in understanding consumer choice. In the context of the exercise, MRS quantifies how much additional consumption (\(c_1\)) an individual requires to compensate for a reduction in leisure (\(h_1\)), while maintaining the same level of utility. Mathematically, it is the negative ratio of the marginal utilities: \[ MRS(c_1 \text{ for } h_1) = -\frac{\partial U/\text{\partial }c_1}{\partial U/\text{\partial }h_1} \.\] When the MRS equals the wage rate (\(w_1\)), it implies that the individual is optimizing their utility given their budget constraint. At this point, the extra utility from an additional unit of consumption is exactly offset by the loss of utility from less leisure. Understanding the MRS helps to explain how consumers make decisions at the margin.
Lagrange Multiplier
The Lagrange multiplier technique is a powerful tool for solving optimization problems subject to constraints. In our exercise, the Lagrange multiplier (\(\lambda\)) represents the shadow price of wealth, or the additional utility a consumer would obtain by relaxing their budget constraint by one unit. The condition \[ \lambda_{1} = E(\lambda_{2}) \.\] suggests that the optimal wealth allocation between the two periods should be such that the expected marginal utility of wealth is equal in both periods. If the marginal utility of wealth were higher today than in the future, the consumer would shift resources (consumption and leisure time) into the first period until the balance is restored. This is where the concept of intertemporal utility maximization comes into play, and the Lagrange multiplier is a pivotal factor in ensuring that maximization occurs in the presence of constraints.
Envelope Theorem
The envelope theorem plays a crucial role in simplifying the analysis of how optimal decisions change in response to changes in the constraints. Imagine an envelope smoothly covering the surface of all possible utility levels as wealth changes — the theorem states that at the optimal level of wealth, the slope of this 'envelope' gives us the additional utility from an incremental increase in wealth, which is the Lagrange multiplier in our case. When applied to our two-period model, the envelope theorem leads to an intuitive result: the increase in utility from an extra unit of wealth would be the same today as its expected increase would be in the future. This demonstrates a key principle in economics — that individuals equate the value of resources across different time periods.
Comparative Statics
Finally, comparative statics is a technique used to analyze how changes in exogenous variables affect the equilibrium state of a model. In our scenario, if a government policy introduced a subsidy to period 2 wages (\(k\) dollars), how would it alter the consumer's choices in both periods? Comparative statics can predict that the subsidy would likely shift consumption and leisure preferences towards the future period when the subsidy is received. This shift occurs because the policy changes the relative advantages of consuming or taking leisure in the second period versus the first. Individuals might save more in period 1 in anticipation of greater period 2 income, or they could adjust their labor supply in period 1, all revealed through the comparative statics analysis of the model post-policy change.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose demand for labor is given by \\[ l=-50 w+450 \\] and supply is given by \\[ l=100 w \\] where \(l\) represents the number of people employed and \(w\) is the real wage rate per hour. a. What will be the equilibrium levels for \(w\) and \(l\) in this market? b. Suppose the government wishes to increase the equilibrium wage to \(\$ 4\) per hour by offering a subsidy to employers for each person hired. How much will this subsidy have to be? What will the new equilibrium level of employment be? How much total subsidy will be paid? c. Suppose instead that the government declared a minimum wage of \(\$ 4\) per hour. How much labor would be demanded at this price? How much unemployment would there be? d. Graph your results.

The Ajax Coal Company is the only hirer of labor in its area. It can hire any number of female workers or male workers it wishes. The supply curve for women is given by \\[ l_{f}=100 w_{f} \\] and for men by \\[ l_{m}=9 w_{m}^{2} \\] where \(w_{f}\) and \(w_{m}\) are the hourly wage rates paid to female and male workers, respectively. Assume that Ajax sells its coal in a perfectly competitive market at \(\$ 5\) per ton and that each worker hired (both men and women) can mine 2 tons per hour. If the firm wishes to maximize profits, how many female and male workers should be hired, and what will the wage rates be for these two groups? How much will Ajax earn in profits per hour on its mine machinery? How will that result compare to one in which Ajax was constrained (say, by market forces) to pay all workers the same wage based on the value of their marginal products?

Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by \\[ l=80 w \\] where \(l\) is the number of workers hired and \(w\) is their hourly wage. Assume also that Carl's labor demand (marginal revenue product \()\) curve is given by \\[ l=400-40 M R P_{1} \\] a. How many workers will Carl hire to maximize his profits, and what wage will he pay? b. Assume now that the government implements a minimum wage law covering all garment workers. How many workers will Carl now hire, and how much unemployment will there be if the minimum wage is set at \(\$ 4\) per hour? c. Graph your results. under perfect competition? (Assume the minimum wage is above the market- determined wage.)

A welfare program for low-income people offers a family a basic grant of \(\$ 6,000\) per year. This grant is reduced by \(\$ 0.75\) for each \(\$ 1\) of other income the family has. a. How much in welfare benefits does the family receive if it has no other income? If the head of the family earns \(\$ 2,000\) per year? How about \(\$ 4,000\) per year? b. At what level of earnings does the welfare grant become zero? c. Assume the head of this family can earn \(\$ 4\) per hour and that the family has no other income. What is the annual budget constraint for this family if it does not participate in the welfare program? That is, how are consumption ( \(c\) ) and hours of leisure ( \(h\) ) related? What is the budget constraint if the family opts to participate in the welfare program? (Remember, the welfare grant can only be positive. e. Graph your results from parts (c) and (d). f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn. How would this change your answers to parts (d) and (e)? g. Using your results from part (f), can you predict whether the head of this family will work more or less under the new rules described in part (f)?

Suppose there are 8,000 hours in a year (actually there are 8,760 ) and that an individual has a potential market wage of \(\$ 5\) per hour. a. What is the individual's full income? If he or she chooses to devote 75 percent of this income to leisure, how many hours will be worked? b. Suppose a rich uncle dies and leaves the individual an annual income of \(\$ 4,000\) per year. If he or she continues to devote 75 percent of full income to leisure, how many hours will be worked? c. How would your answer to part (b) change if the market wage were \(\$ 10\) per hour instead of \(\$ 5\) per hour? d. Graph the individual's supply of labor curve implied by parts (b) and (c).

See all solutions

Recommended explanations on Economics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free