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Solution Manual For Modern Labor Economics: Theory and Public Policy, 13th Edition

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    Answers to Chapter 1 • Review Questions 1. Answer d. In product markets, firms sell and individuals buy. Also, human beings are not as intimately involved in the transactions, so nonpecuniary factors are less important. 2. Answer d. If people strive to make themselves as happy as they can (i.e., maximize their utility), and a human being is “attached” when labor market transactions take place, it seems likely that feelings, emotions, personalities, and working conditions will play a role in determining the nature of these transactions. All these factors seem to play an important role in determining a human being’s happiness. 3. Answer b. This is the only statement that refers to “what is” true about the labor market. All the others, either explicitly or implicitly, make reference to “what should be” true about labor market outcomes. 4. Answer a. This statement makes reference to “what should be,” while all the others refer to how labor market variables are actually linked. 5. Answer a. A model is useful precisely because it is abstract. If it accurately captured all of the complexity of the real world or people’s actual thought processes, it would be so intricate that no patterns or principles could be discerned. Models can be used not only to predict behavior but also to illustrate the kind of behavior that should take place to reach certain goals. 6. Answer d. Individuals or households are assumed to maximize utility, while firms are assumed to maximize profits. 7. Answer d. The two most fundamental assumptions of positive economics are scarcity (people cannot get everything they want) and rationality (people consistently weigh costs and benefits). 8. Answer c. Utility maximization means people strive to be as happy as possible, and that happiness may depend in part on how other people are doing, not solely on an individual’s own circumstances. The concept of mutually beneficial transactions is a normative one, unrelated to what actually happens in some instances. Positive models refer to actual links between economic variables, and these links can only be exposed by holding other factors constant. 9. Answer b. The test of a positive economic model is how well it predicts average tendencies. It does not have to capture everyone’s idiosyncrasies all the time. Normative models, on the other hand, relate to standards that someone feels should be attained.

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    236 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition 10. Answer b. Normative statements are frequently connected to mandated transactions, not just voluntary transactions. These mandates can be driven by the quest for mutually beneficial gain, but they need not be. Other ethical principles or goals like fairness may be the driving force behind normative statements. 11. Answer d. If one person gains and no one else loses, that is a transaction that seemingly could be supported by unanimous consent. Therefore, if such a transaction can still be made, a Pareto efficient outcome has not been achieved. 12. Answer b. In most situations there are many different Pareto efficient outcomes, but many of these outcomes would probably be considered unfair. The outcome that is finally achieved may be arrived at voluntarily or by government mandate. 13. Answer d. Moving from one Pareto efficient outcome, even if it is to another Pareto efficient outcome, will make one person worse off at the same time it makes someone else better off. Deciding among Pareto efficient outcomes, therefore, involves some assessment of the fairness or equity of the outcomes. These judgments have traditionally been made in the political arena and are often influenced by each party’s initial income or wealth position. Most political systems have been willing to support at least some redistribution of wealth from the rich to the poor. 14. Answer d. Other reasons include price distortions caused by government taxes or the lack of a market where the transaction can be worked out. 15. Answer d. Also, in some instances, government action is the impediment to the Pareto efficient outcome. In these instances the government must get out of the way before the transaction can be completed. 16. Answer c. Least squares regression makes no assumptions with respect to the relationship between independent variables (those on the right side of the equation) or the size of the coefficients attached to these variables. For the estimated parameters to be unbiased estimates of the true values, however, the error term must be random. 17. Answer b. One can reject the hypothesis that the true value of a particular coefficient is zero only if the estimated parameter is more than twice the size of its standard error (the t -statistic will then be greater than 2). A small standard error is no guarantee of a non-zero parameter if the estimated parameter is also very small. A random error term helps to ensure that the estimates are as precise as possible, but the existence of a random error term does not have any bearing on the existence of a relationship between the independent and dependent variables. 18. Answer b. A coefficient in a multiple regression indicates the effect on the dependent variable, on average, of a one-unit change in the independent variable, assuming the other variables are held constant. A non-zero estimate for the coefficient does not ensure that the true value of the parameter is different from zero. In this case, however, a parameter estimate four times as large as its standard error allows us to reject the hypothesis of a “true” value of zero.

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    Answers to Chapter 1 237 19. Answer d. The relatively small value of the standard error means the coefficient on the W variable has been measured with a relatively small margin of error. Also, since the coefficient is 2.5 times its standard error, the hypothesis that the true value of the coefficient is zero can be rejected. However, a relationship that is significant in a statistical sense is not necessarily significant in an economic sense. Holding all else constant, in this equation, every $1 increase in the wage is accompanied by just a one-half hour reduction in total annual hours, a very insignificant amount. 20. Answer a. Since W and V are positively correlated, the coefficient on W would pick up some of the negative effect V exerts on H , making the coefficient on W smaller. • Problems 21a. The drawing is a model because it is a deliberate abstraction designed to illustrate certain key characteristics of human beings. 21b. The model highlights that human beings walk upright on two legs. It also highlights the relative size of the head and limbs. It pushes all the details of our appearance, particularly things like the shape of our hands and feet, into the background. It also says nothing about the mental capacities that distinguish us from other animals. There are, of course, other characteristics that it doesn’t capture either. 21c. A simple drawing was easy to make. Anything more involved would require some artistic skill. Also, such a simple drawing highlights the idea that models are abstract. A more realistic drawing would not have gotten across the idea that “humans walk upright on two legs” any better than this simple drawing. In fact, a more realistic drawing might have obscured this point by focusing attention on other features. 21d. This is a positive model since it attempts to show “what is” true about human beings. It does not attempt to assess our bodily design or illustrate “what should be.” 22a. The person is being inconsistent. If it is worth taking 30 minutes to save $5 in the first situation, it should also be worth it in the second situation. However, he may be acting like this because the percentage savings is much smaller in the second case. Studies of perception have revealed that the larger or more intense the stimulus (in this case the original expenditure on insurance), the larger any change (in this case the cost savings) has to be before it will be recognized. 22b. No, a model with poor predictive power may still be useful in a normative sense, in this case as a guide to more rational decision making. In this example, if confronted with his inconsistency, the worker may realize that his choice in the second instance was obscured by the large dollar amounts. As a result he may be able to avoid similar mistakes in the future. 23a. The outcome is Pareto efficient because there is no way any of the remaining firms or workers can reach an agreement that is mutually beneficial. For example, firm 9 will only benefit from a wage below $360, but worker 9 will not benefit from anything below $400. Since all the mutually beneficial deals have already been made, the outcome is Pareto efficient.

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    238 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition 23b. With deals below $480 prohibited, only firms 1, 2, and 3 will be interested in hiring. On the other hand, all 10 workers would be willing to work. Suppose for simplicity that workers 1, 2, and 3 are hired. Worker 4, who has been shut out of the market, could easily strike a mutually beneficial deal with firm 4 by offering to work for, say, $400, but the law prohibits this. Since mutually beneficial deals can still be made, the outcome under the legislated wage is not Pareto efficient. 23c. The three workers who continue to be hired each gain $100 (the difference between the legislated wage of $480 and the wage of $380 that would have emerged if all mutually beneficial trades were allowed). The three firms that hire them each lose $100 relative to what they would have gained under the lower wage. The five workers who would have been hired under the lower wage also lose. Assuming workers 4 through 8 have been shut out, the total loss to these workers is $200 ($80 + $60 + $40 + $20). Combining the worker and firm losses yields a total loss of $500. 23d. The winners do not gain enough ($300) to compensate the losers ( − $500) and still come out ahead. Such a law can only be justified using principles developed outside of economics. For example, those who gain might have been very poor to begin with, while those who lost might have been well off. In such a situation, the redistribution might be justified by appealing to notions of equity or fairness. 23e. Eight workers would be hired, the same outcome as in 23a . 23f. This is a Pareto efficient outcome since no mutually beneficial deals still exist. Even if person 9 were paid his minimum acceptable wage of $400, the firm would not gain by hiring him. While the outcome is Pareto efficient, it is not necessarily equitable since the firm has reaped all the benefits. 24a. See the scatter of points in Figure 1-2. 24b. See the straight line in Figure 1-2. Figure 1-2

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    Answers to Chapter 1 239 *24c. See Table 1-5 and the points plotted as squares in Figure 1-3. Table 1-5 UR Actual % ∆ W Predicted % ∆ W Residual 5.4 3.4 3.3 0.1 6.5 3.0 2.1 0.9 5.4 3.4 3.3 0.1 5.5 2.8 3.2 − 0.4 5.0 2.8 3.7 − 0.9 4.4 3.6 4.4 − 0.8 3.7 4.3 5.2 − 0.9 3.7 5.0 5.2 − 0.2 3.5 6.1 5.4 0.7 3.4 6.7 5.5 1.2 *24d. The residuals do not seem random. They tend to be positive when the unemployment rate is relatively high or relatively low, and negative otherwise. From the scatter of points in Figure 1-2, it appears a convex curve would fit the data better. *24e. See the solid curve in Figure 1-2. Figure 1-3

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    240 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition *24f. See Table 1-6 and the points plotted as triangles in Figure 1-3. For six of the ten years the fit improved. There were two years where it worsened slightly and two years where it was essentially unchanged. The fit improved most at the lowest and highest values of the unemployment rate. For both values, the residual was 0.3 units less when the nonlinear function was used. Table 1-6. UR Actual % ∆ W Predicted % ∆ W Residual 5.4 3.4 3.1 0.3 6.5 3.0 2.4 0.6 5.4 3.4 3.1 0.3 5.5 2.8 3.0 − 0.2 5.0 2.8 3.5 − 0.7 4.4 3.6 4.2 − 0.6 3.7 4.3 5.2 − 0.9 3.7 5.0 5.2 − 0.2 3.5 6.1 5.6 0.5 3.4 6.7 5.8 0.9 24g. See the scatter of points in Figure 1-4. Figure 1-4 24h. See the lines labeled “1960s” and “1970s” in Figure 1-4. The 1970s line lies 3.7 units above the 1960s line at every value of the unemployment rate. 24i. See the line labeled “Combined” in Figure 1-4. The omission of the D 70 variable severely biases the coefficient on the unemployment variable in an upward direction. The D 70 variable normally would exert a positive effect on the dependent variable. However, since D 70 and the unemployment rate are positively correlated (notice that both unemployment and inflation were higher in the 1970s), when the D 70 variable is omitted, most of its positive effect on the dependent variable will be attributed to the unemployment rate variable. Notice that omitting a variable is a problem here because the omitted variable and the remaining variable are correlated. Omitting a variable that is not correlated with the independent variable would not create bias in the estimated coefficient.

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    Answers to Chapter 1 241 0 0 1 *24j. To allow both the position and slope of the curve to be different, estimate the equation % ∆ W t = a 0 + a 1 UR t + a 2 D 70 t + a 3 D 70 UR t + e i . For the 1960s, the vertical intercept would be given by the estimate of a , while for the 1970s it would be given by a + a 2 . Similarly, for the 1960s, the slope would be given by the estimate of a , while for the 1970s it would be given by a + a 3 . • Applications 25a. The notion of exercising a preferential option for the poor is very different from promoting the norm of Pareto efficiency. Under Pareto efficiency, only mutually beneficial changes should be undertaken. But when exercising a preferential option for the poor, some parties may be called upon to sacrifice so that the poor can advance. 25b. When the goal is to promote Pareto efficient outcomes, the government mainly plays a facilitating role. Its task is to promote voluntary exchange based on accurate and complete information. When barriers to such transactions exist, however, the government sometimes can act in a way to overcome these barriers. Some examples of these actions include providing for the production of public goods, eliminating capital market imperfections, or establishing substitutes for the market. Pope Leo XIII sees the task of government as promoting and protecting the interests of the poor and disadvantaged. Government need not act on behalf of all members of a society. Rather, government actions must be judged by their impact on the poor, even when these actions require sacrifice by others. 26a. The survey suggests people care about where their income stands relative to others in the population. 26b. The norm of Pareto efficiency says that all mutually beneficial changes should be undertaken. In this case, many people disagree that the program should be enacted, even though everyone gains. Therefore the responses are inconsistent with the norm of Pareto efficiency. 26c. It appears that the premise of the question would have to have everyone gaining by roughly the same amount for the reform to be supported unanimously. 27a. Estimating the parameters α 0 and α 1 would require cross-sectional data on earnings and employee- generated revenues for a number of workers at one or more firms. Estimates of the parameters would be derived by choosing values for α 0 and α 1 such that the squared differences between the actual and predicted values of employee earnings were minimized. Most statistical and spreadsheet programs have commands that will use this criterion to find the parameter estimates. 27b. The precision of an estimate is indicated by its standard error. If the estimated parameter is approximately twice the value of its standard error, it is possible to reject with reasonable confidence the hypothesis that the true value of the parameter equals zero. *27c. If the hypothesis is that parameter α 1 equals one, rather than zero, then the value of the estimated parameter, minus one, must be approximately twice the standard error of estimated parameter. 1

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    242 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition *27d. Frank’s hypothesis implies that the relationship between earnings and productivity is much flatter than the standard theory predicts. For a graphical depiction of this effect, see Figure 1-5. Line ab represents the relationship between E and R implicit in the standard model, while line cd would be consistent with Frank’s hypothesis. Consequently, the prediction would be that α 0 would be much greater than it is in the standard model, while the value of α 1 would be much less (and definitely less than one). Figure 1-5 *27e. The equation to be estimated is E i = α 0 + α 1 R i + α 2 D i + α 3 D i R i + e i . This equation allows both the intercept and slope to change as the degree of employee interaction increases. Since the hypothesis is that relationship between earnings and productivity flattens out as the degree of interaction increases, the expectation would be that the estimate of the parameter α 2 would be greater than zero and the estimate of the parameter α 3 would be less than zero. The effect of increased interaction is also shown in Figure 1-5. Line cd represents a possible relationship between E and R when there is modest worker interaction, while line ef represents a situation of more extensive interaction. *27f. It seems unlikely that D is correlated with the level of employee productivity ( R ) since high- productivity people can be found in all types of work environments. Therefore, eliminating D should not bias the estimates of the remaining parameters α 0 and α 1 .

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    6 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition • Review Questions Choose the letter that represents the BEST response. Labor Economics: Some Basic Concepts 1. A unique feature of the labor market relative to standard product markets is a. individuals are the sellers, firms are the buyers. b. nonpecuniary characteristics play a larger role in labor market transactions. c. intense competition exists among the sellers. d. both a and b . 2. Feelings, emotions, personalities, and working conditions play an important role in labor market transactions because a. labor services cannot be separated from workers. b. people strive to make themselves as happy as they can. c. people are sometimes irrational. d. both a and b . 3. Which of the following is a statement associated with positive economic reasoning? a. Workers are entitled to a just wage. b. An increase in the payroll tax paid by firms will reduce wages and employment opportunities. c. Employers should be required to pay for health insurance for their employees. d. Professional athletes are overpaid. 4. Which of the following is a statement associated with normative economic reasoning? a. It is important that everyone who wants to attend college be able to do so. b. Holding all else constant, workers in more dangerous occupations earn more than those employed under safer conditions. c. Welfare programs can lead to reductions in work incentives. d. Subsidies given to firms that purchase new capital equipment will lead to an increase in employment. 5. Which of the following is a true statement about economic models? a. A model is deliberately abstract. b. A model accurately captures the all of the complexity of the real world. c. A model shows the thought process people go through when making decisions. d. A model must accurately predict behavior to be useful. The Models and Predictions of Positive Economics 6. The goal often pursued in positive economic models of the labor market is a. utility maximization. b. profit maximization. c. income maximization. d. either a or b .

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    Chapter 1 Introduction 7 7. Positive economic models typically assume a. people cannot get everything they want and so must make choices. b. people undertake a particular course of action if the marginal benefits of that action outweigh the marginal costs. c. people are often inconsistent and inflexible when making their choices. d. both a and b . 8. Predictions of positive economic models are usually conditional on a. individuals caring only about themselves. b. the transactions being mutually beneficial. c. everything else being held constant. d. both a and b . 9. For a positive economic model to be useful, it must a. explain how people should behave. b. tend to predict how people behave on average. c. explain how people actually behave. d. apply to everyone. Normative Economics 10. Normative economic statements a. must be based on the principle of mutually beneficial gain. b. are based on ethical principles or goals. c. relate to voluntary transactions only. d. both b and c . 11. A Pareto efficient outcome has been achieved when a. all mutually beneficial transactions have been made. b. it is impossible to make someone better off without making someone else worse off. c. at least one person can gain and no one else will lose. d. both a and b . 12. Which of the following is true about a Pareto efficient outcome? a. the outcome must have been purely voluntary. b. the outcome may not satisfy society’s sense of fairness. c. the outcome is unique. d. Both a and b . 13. Policies that move two parties from one Pareto efficient outcome to another a. will make one of the parties worse off. b. are often influenced by the parties’ initial endowment of resources. c. can be designed scientifically outside of the political system. d. both a and b .

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    8 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition 14. Pareto efficient transactions may not occur because a. parties may not have all the relevant information. b. they may be against the law. c. the cost associated with completing the transaction may deter one or more of the parties. d. all of the above. 15. Government may help to bring about Pareto efficient outcomes by a. producing public goods. b. eliminating capital market imperfections. c. providing a way to substitute for market transactions. d. all of the above. Statistical Testing of Labor Market Hypotheses (Appendix 1A) 16. Least squares regression assumes that a. the relationship between the independent variables is linear. b. the coefficients associated with the independent variables are greater than zero. c. a random error term accounts for any differences between the actual and predicted values of the dependent variable. d. all of the above. 17. One can be reasonably confident a relationship between an independent and a dependent variable has been found if a. the standard error of the estimated parameter is small. b. the standard error of the estimated parameter is less than half the value of the estimated parameter. c. the error term of the least squares regression is random. d. the t -statistic for the estimated parameter is about 0.5. In answering Questions 18 through 20 , please refer to the following information. Suppose that for a sample of 10,000 individuals, the relationship between annual hours supplied ( H ), the hourly wage rate in dollars ( W ), and the level of nonlabor income in dollars ( V ) (e.g., interest income, gifts, public assistance) was estimated by multiple regression to be H i = 1200 + 0.5 W i − 0.04 V i . (225) (0.2) (0.01) The subscript i refers to the i th household where i = 1 to 10,000. Standard errors of the estimated parameters are in parentheses. 18. The coefficient of − 0.04 associated with the independent variable V means that a. on average, when a person’s wage and nonlabor income both increase by $1, he wishes to work 0.04 hours less. b. on average, when a person’s nonlabor income increases by $1, he wishes to work 0.04 hours less, holding all else constant. c. the true value of the parameter associated with the V variable is equal to zero. d. both b and c .

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    Chapter 1 Introduction 9 19. Comparing the size of the coefficient on the W variable and its standard error, one can conclude that a. the relationship between H and W has been estimated with reasonable precision. b. statistically, one can be confident that the true relationship between H and W is not zero. c. from an economic standpoint, the relationship between H and W is insensitive to changes in W . d. all of the above. 20. Suppose W and V are positively correlated (e.g., people with high wages save more and thus earn more interest income). What would happen if V were omitted from the multiple regression? a. The coefficient on W would become smaller. b. The coefficient on W would become larger. c. The standard error associated with the coefficient on W would become smaller. d. Both b and c . • Problems Labor Economics: Some Basic Concepts 21. The concept of a model is one of the most important concepts in economics. Helping the reader to understand and apply labor market models is the primary purpose of this book. Figure 1-1 is an example of a model, but in this case, it is a model of a human being. Figure 1-1 21a. What does it mean to call this drawing a model? 21b. What essential characteristics of a human being does it capture? What characteristics does it push into the background? 21c. Why do you think such a simple drawing was used to illustrate the concept of a model? Would a more realistic drawing make for a better model? 21d. Is this model a positive or normative model? Explain your reasoning. 22. Models of positive economics typically assert that people weigh costs and benefits in a consistent manner. That is, if the benefits of following a particular course of action exceed the costs, then that action will be taken, otherwise it will not.

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    10 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition Consider a worker who is told he can save $5 on his dental insurance premium of $50 if he goes to the company’s personnel office and fills out a form. The whole process will take 30 minutes. Suppose the worker fills out the form. A month later, the same worker is told that he can reduce his health insurance premium from $1,000 to $995 by taking 30 minutes to fill out a form in the personnel office. The worker declines the offer. 22a. Does the model of rational choice based on costs and benefits serve as a positive economic model for this person? If not, can you explain why the person might have acted as he did? 22b. When an economic model fails to predict behavior accurately, does that mean it is useless? What other purpose could the model serve? Pareto Efficiency 23. Consider a labor market in which there are 10 firms and 10 workers. Each firm can hire only one of these workers. Each firm has a maximum it would be willing to pay the worker. It would like to hire its worker for as far below this maximum as possible, but if no other options exist, it will hire a worker at a wage equal to its maximum willingness to pay value. The maximum wage each firm is willing to pay is shown in Table 1-2. Suppose that each worker in this market can work for only one firm. Each worker has a minimum wage that will just be acceptable. Any offer below this minimum will be rejected and the worker will not participate in the market. Each worker would like to be hired at a wage as much above the minimum acceptable wage as possible, but if no other options exist, will work at a wage just equal to this minimum. The minimum wage acceptable to each worker is also shown in Table 1-2. Table 1-2 Maximum Acceptable Minimum Acceptable Firm # Wage ($) Person # Wage ($) 1 520 1 240 2 500 2 260 3 480 3 280 4 460 4 300 5 440 5 320 6 420 6 340 7 400 7 360 8 380 8 380 9 360 9 400 10 340 10 420 23a. Suppose that the first 8 workers are hired by the first 8 firms and each worker is paid a wage of $380. Show why this is a Pareto efficient outcome. 23b. Concerned that workers will be exploited by firms, suppose that the government passes a law that prohibits deals below $480; that is, individuals and firms must contract for work at $480 or above. What will be the consequences of this law? 23c. Assuming the Pareto efficient outcome from 23a would have emerged without government interference, which workers gain because of the law in 23b ? How much do they gain? Who loses because of the law? How much do they lose?

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    Chapter 1 Introduction 11 23d. Is it possible for the winners in 23c to fully compensate the losers and still come out ahead? If not, how could such a law be justified? 23e. Now suppose that the maximum willingness to pay values in Table 1-2 represent values for a single firm that is the only buyer of labor. Also suppose that this firm is aware of each worker’s minimum acceptable wage. In this situation, the firm could offer the first worker a wage of $240, and since the only other option would be not to work at all, the worker would accept. Similarly, the second worker could be offered a wage of $260 (the first would still earn $240) and he would accept. How many workers would be hired under this pay scheme? 23f. Is the answer to 23e a Pareto efficient outcome? Is it an equitable outcome? Statistical Testing of Labor Market Hypotheses (Appendix 1A) 24. Up until the 1970s economists believed there was a stable tradeoff between the unemployment rate and the rate of wage inflation. This relationship is known as the Phillips curve. To summarize briefly, the logic of the Phillips curve rests on two assertions. The first is that the general rate of change in wages is proportional to the amount by which the number of jobs exceeds the number of people looking for work on a percentage basis. For example, when the number of jobs exceeds the number of people willing to work by 10 percent, the wage is presumed to rise at a faster pace than when the number of jobs exceeds the number of people willing to work by only 5 percent. The second assertion is that the unemployment rate is inversely related (probably in a nonlinear way) to the percentage difference between the number of jobs and the number of people willing to work. As Chapter 15 will point out, even when the number of jobs and the number of people willing to work are equal, there will still be a small positive rate of unemployment. This rate should diminish (probably at a decreasing rate) as jobs become more plentiful relative to the supply of workers. On the other hand, as the number of jobs falls below the number of workers, the unemployment rate will rise. The result of these two assertions is that the rate of wage inflation will be inversely related (probably in a nonlinear way) to the unemployment rate. By the 1970s economists began to realize that there were also a number of other factors that could change over time and cause the terms of the entire tradeoff to either worsen or improve. In other words, a stable tradeoff exists only at a particular point in time. The difficulties a shifting tradeoff can cause for economists wishing to empirically measure the Phillips curve are explored in the problems below. 24a. Plot the 1960 − 69 data in Table 1-3 with the unemployment rate on the horizontal axis and wage inflation on the vertical axis. 24b. Fitting a linear equation to the data for 1960–69 using the least squares regression technique yields % ∆ W t = 9.2 − 1.1 UR t , (1.21) (0.25) where % ∆ W t refers to the wage inflation for any year t , and UR t refers to the unemployment rate for any year t . The standard errors of the estimated parameters (the vertical intercept and the slope) are in parentheses. Add this line to the graph produced in Question 24a .

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    12 Ehrenberg/Smith • Modern Labor Economics: Theory and Public Policy, T hirte enth Edition t Table 1-3 contains the annual unemployment rate and rate of wage inflation experienced by the U.S. economy each year over the period 1960–79. Table 1-3 Year Unemployment Rate (%) Rate of Wage Inflation (%) 1960 5. 4 3. 4 1961 6. 5 3. 0 1962 5. 4 3. 4 1963 5. 5 2. 8 1964 5. 0 2. 8 1965 4. 4 3. 6 1966 3. 7 4. 3 1967 3. 7 5. 0 1968 3. 5 6. 1 1969 3. 4 6. 7 1970 4. 8 6. 6 1971 5. 8 7. 2 1972 5. 5 6. 2 1973 4. 8 6. 2 1974 5. 5 8. 0 1975 8. 3 8. 4 1976 7. 6 7. 2 1977 6. 9 7. 6 1978 6. 0 8. 1 1979 5. 8 8. 0 Source : Economic Report of the President: 1989 , Table B-39 (p. 352) and Table B-44 (p. 358). *24c. Assess how well this line fits the data. For each of the years 1960–69, find the difference between the actual level of wage inflation and the level predicted by the regression line (these differences are called residuals). Make a plot where the unemployment rate is on the horizontal axis and the corresponding residual value is on the vertical axis. *24d. Do the residuals plotted in 24c seem to be the result of just random errors? Is there a curve other than a straight line that could fit this data a little better? *24e. To capture the nonlinearities suggested by the plot of the data and the residuals from the linear regression line, the rate of wage inflation was regressed on the reciprocal of the unemployment rate (since the reciprocal function decreases at a decreasing rate). The least squares technique yielded the following relationship. % ∆ W = − 1.43 + 24.6 1 . UR t (0.98) (4.25) Add a plot of this line to the graph created in 24a and 24b . *24f. Compute the residual errors using the nonlinear relationship estimated above. Plot the new residuals (using different symbols) on your previous residual plot. Where has the fit of the line improved most?

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