CHAPTER 13. Duration of Spell (in months) Exit Rate

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1 CHAPTER Suppose there are 25,000 unemployed persons in the economy. You are given the following data about the length of unemployment spells: Duration of Spell (in months) Exit Rate where the exit rate for month t gives the fraction of unemployed persons who have been unemployed t months and who escape unemployment at the end of the month. The data can be used in the problem to calculate the number of workers who have 1 month of unemployment, the number who have 2 months of unemployment, and so on, and how many months of unemployment are associated with workers who get a job after a given duration. Duration (Months) # Unemp: Start of Month # Months For Duration Exit Rate # of Exiters # of Stayers ,000 15,000 10,000 15, ,000 2,000 8,000 4, ,000 1,600 6,400 4, ,400 1,280 5,120 5, ,120 1,024 4,096 5, ,096 4, ,576 (a) How many unemployment-months will the 25,000 unemployed workers experience? The 25,000 workers will experience 58,616 months of unemployment, an average of 2.34 months per worker. (b) What fraction of persons who are unemployed are long-term unemployed (that is, are in unemployment spells which last 5 or more months)? Only 5,120 of the 25,000 workers (20.5 percent) are in spells lasting 5 or more months. (c) What fraction of unemployment months can be attributed to persons who are long-term unemployed? Although only 20.5 percent of workers are unemployed for 5 or more months, they account for 29,696 of the 58,616 (50.7 percent) of months of unemployment. 89

2 (d) What is the nature of the unemployment problem in this example: too many workers losing their jobs or too many long spells? Most spells are short-lived, but workers in long spells account for most of the unemployment observed in this economy. Thus, the main problem is too many long spells Consider Table 599 of the 2002 U.S. Statistical Abstract. (a) How many workers aged 20 or older were unemployed in the United States during 2001? How many of these were unemployed less than 5 weeks, 5 to 14 weeks, 15 to 26 weeks, and 27 or more weeks? In total, 5,554,000 workers aged 20 years or older were unemployed in the U.S. in Of these, 40 percent, or 2,221,600 workers, were unemployed for less than 5 weeks; 32.2 percent, or 1,788,388 workers, were unemployed between 5 and 14 weeks; 15.5 percent, or 860,870 workers, were unemployed 15 to 26 weeks, and 12.9 percent, or 716,466 workers were unemployed for 27 or more weeks. (b) Assume that the average spell of unemployment is 2.5 weeks for anyone unemployed for less than 5 weeks. Similarly, assume the average spell is 10 weeks, 20 weeks, and 35 weeks for the remaining categories. How many weeks did the average unemployed worker remain unemployed? What percent of total months of unemployment are attributable to the workers that remained unemployed for at least 15 weeks? The total number of weeks of unemployment is calculated as: 2,221,600(2.5) + 1,788,388(10) + 860,870(20) + 716,466(35) = 65,731,590 months of unemployment spread over 5,554,000 unemployed workers, implies that the average unemployed worker remained unemployed for weeks. The percent of total months of unemployment attributable to the workers that remained unemployed for at least 15 weeks is [ 860,870(20) + 716,466(35) ] / 65,731,590 = percent Suppose the marginal revenue from search is MR = w, where w is the wage offer at hand. The marginal cost of search is MC = 5 + w. (a) Why is the marginal revenue from search a negative function of the wage offer at hand? If the offer-at-hand is relatively low, it pays to keep on searching as the next offer is likely higher than the offer-at-hand. If the offer-at-hand is very high, however, it does not pay to keep on searching since it is unlikely that the next search will generate a higher wage offer. 90

3 (b) Can you give an economic interpretation of the intercept in the marginal cost equation; in other words, what does it mean to say that the intercept equals $5? Similarly, what does it mean to say that the slope in the marginal cost equation equals one dollar? The $5 indicates the out-of-pocket search costs. Even if the offer-at-hand is zero (so that there is no opportunity cost to search), it still costs money to get to the firm and learn about the details of the potential job offer. The slope equals $1, because the costs of search also vary directly with the opportunity cost of search which is the wage offer at hand. If the wage offer at hand is $10, the opportunity cost from one more search equal $10; if the wage offer at hand is $11, the opportunity cost would be $11, and so on. (c) What is the worker s asking wage? Will a worker accept a job offer of $15? The asking wage is obtained by equating the marginal revenue of search to the marginal cost of search, or w = 5 + w. Solving for w implies that the asking wage is $18. The worker, therefore, would not accept a job offer of $15. (d) Suppose UI benefits are reduced, causing the marginal cost of search to increase to MC = 20 + w. What is the new asking wage? Will the worker accept a job offer of $15? If we equate the new marginal cost equation to the marginal revenue equation we find that the asking wage drops to $12. The worker will now accept a wage offer of $ (a) How does the exclusion of non-working welfare recipients affect the calculation of the unemployment rate? Use the 2002 U.S. Statistical Abstract to estimate what the 2000 unemployment rate would have been if welfare recipients had been included in the calculation. Excluding non-working welfare recipients from the unemployment rate biases the unemployment rate downward. Table 560 of the Statistical Abstract shows that the labor force in 2000 totaled 140,863,000, while the number employed totaled 135,208,000. Thus, the unemployment rate was 4.0 percent. Table 513 of the Statistical Abstract shows that 2,253,000 people received public assistance in Assuming all of these were non-working and out of the labor force, their inclusion in the calculation of the unemployment rate would increase the unemployment rate to (140,863, ,253, ,208,000) / (140,863, ,253,000) = 5.5%. (b) How does the exclusion of black market workers affect the calculation of the unemployment rate? Estimate, the best you can, what the 2000 unemployment rate would have been if workers in the underground economy had been included in the calculation. Excluding black market workers from the calculation of the unemployment rate keeps the unemployment rate artificially high. At the best, black market workers are not counted in the labor force, but at the worst, the black market workers claim to be in the labor force and without a job. The problem is that there is no (or very little) data on the black market by definition. Some researchers have estimated the underground economy to be on the order of 10 to 20 percent of activity in the U.S. Suppose half of underground economy workers have regular market jobs as well. Suppose further that of the remaining half, one half claim to be unemployed while the other half is out of the labor force. Thus, 10 percent of the unemployed workers are the number of underground economy workers in the labor force without a job: 10 percent of 91

4 (140,863, ,208,000) = 565,500. An equal number say they are not in the labor force. (And twice this number is already reporting having a legitimate job.) Using these estimates, therefore, the labor force should be 140,863, ,500 = 141,428,500, while the number employed should be 135,208, , ,500 = 136,339,000. Thus, the unemployment rate would be (141,428, ,339,000) / 141,428,500 = 3.6% Compare two unemployed workers; the first is 25 years old and the second is 55 years old. Both workers have similar skills and face the same wage offer distribution. Suppose that both workers also incur similar search costs. Which worker will have a higher asking wage? Why? Can search theory explain why the unemployment rate of young workers differs from that of older workers? The marginal revenue of search depends on the length of the payoff period. Younger workers have the most to gain from obtaining higher paying jobs, since they can then collect the returns from their search investment over a longer expected work-life. As a result, it pays for younger workers to set their asking wage at a relatively high level. This implies that younger workers will tend to have higher unemployment rates and longer spells of unemployment than older workers Suppose the government proposes to increase the level of UI benefits for unemployed workers. A particular industry is now paying efficiency wages to its workers in order to discourage them from shirking. What is the effect of the proposed legislation on the wage and on the unemployment rate for workers in that industry? The introduction of UI benefits shifts the no-shirking supply curve upwards (from NS to NS ), because a higher wage would have to be paid in order to attract the same number of workers who do not shirk. As a result, the new equilibrium (point Q ) entails a higher efficiency wage and leads to a larger number of unemployed workers. Dollars S Q NS Q NS P D E Employment 92

5 13-7. It is well known that more-educated workers are less likely to be unemployed and have shorter unemployment spells than less-educated workers. Which theory(ies), the job search model, the sectoral shifts hypothesis, or the efficiency wage model, can explain this empirical correlation? Two of the theories in question can be formulated in such a way that they each predict that more-educated workers will have less unemployment and shorter spells of unemployment than less-educated workers. The job search model suggests that highly educated workers would have a lower unemployment rate either if they have relatively higher search costs or relatively lower gains from search. It could then be argued, for example, that search costs are (relatively) higher for highly-educated workers than for lesseducated workers, perhaps because the nature of the job match between a highly educated worker and a firm is much more complex than the type of job match required between the firm and a worker who does repetitive tasks. Similarly, it is plausible to argue that it is easier to retool a highly-educated worker than a less-educated worker. The decline in demand in particular industries, then, leads to sectoral shift that can be better weathered by highly-educated workers. The efficiency wage model, however, has a harder time explaining the correlation. Presumably, the output of highly-educated workers is more difficult to measure than the output of less-educated workers. As a result, efficiency wages are more likely to arise in industries that employ highly-educated workers, and these industries (which pay above-market wages) would have to maintain a pool of unemployed workers in order to keep the employed workers in line Suppose a country has 100 million inhabitants. The population can be divided into the employed, the unemployed, and the persons who are out of the labor force (OLF). In any given year, the transition probabilities among the various categories are given by: Moving From: Moving Into: Employed Unemployed OLF Employed Unemployed OLF These transition probabilities are interpreted as follows. In any given year, 2 percent of the workers who are employed become unemployed; 20 percent of the workers who are unemployed find jobs, and so on. What will be the steady-state unemployment rate? Use E for the number of employed people, U for the number of unemployed, and N for the number not participating. The flows of people among the three categories can be found by multiplying these numbers with respective probabilities in the table. In the steady-state, the flows into each category must exactly balance the outflows. This produces the following equations: Employed: E =.94E +.20U +.05N Unemployed: U =.02E +.65U +.03N OLF: N =.04E +.15U +.92N 93

6 Only two of the three equations can be used, however, as E + U + N = 100 million. The steady-state solutions for E, U, and N can now be found with brute force algebra. The solution is that million are employed, million are unemployed, and million are not in the labor force. The steadystate unemployment rate is then u U = 100 = 100 = % U + E Consider an economy with 250,000 adults, of which, 40,000 are retired senior citizens, 20,000 are college students, 120,000 are employed, 8,000 are looking for work, and 62,000 stay at home. What is the labor force participation rate? What is the unemployment rate? The labor force participation rate = ( 120, ,000 ) / 250,000 = 51.2 percent. The unemployment rate = 8,000 / ( 120, ,000 ) = 6.25 percent Consider an economy with 3 types of jobs. The table below shows the jobs, the frequency with which vacancies open up on a yearly basis, and the income associated with each job. Searching for a job costs $C per year and generates at most 1 job offer. There is a 20 percent chance of not generating any offer in a year. (Note: the expected search duration for a job with probability p of appearing is 1/p years.) Job Type Frequency Income A 30 percent $60,000 B 20 percent $100,000 C 30 percent $80,000 As a function of C, specify the optimal job search strategy if the worker maximizes her expected income net of search costs. There are four possible strategies for the worker: accept any job paying at least $100,000, accept any job paying at least $80,000, accept any job paying at least $60,000, or do not search for a job. Under the first strategy, the worker will only accept a B job. The expected search time for a B job is 5 years. Thus, the expected value of this strategy is $100,000 5C. Under the second strategy, the worker will accept a B or a C job. The expected search time, therefore, is 2 years (i.e., there is a 50 percent chance of receiving a B or a C job in a year, so the search time is 1/0.5 = 2). Thus, the expected value of this strategy is ($100,000) + ($80,000) 2C = $88,000 2C

7 Under the third strategy, the worker will accept any of the three jobs. The expected search time, therefore, is 1.25 years, and the expected value of this strategy is 0.2 ($100,000) ($80,000) + ($60,000) 1.25C = $77, C Under the fourth strategy, the worker does not search, and therefore does not find a job or incur search costs, for an expected payoff of $0. One can now compare the four expected payoffs to determine that the optimal search strategy is: Accept the first B job if C < $4,000. Accept the first B or C job if $4,000 # C < $14,000. Accept the first job of any kind if $14,000 # C < $62,000. Don t search if $62,000 # C (a) A country is debating whether to fund a national database of job openings and giving all unemployed workers free access to it. What effect would this plan have on the long-run unemployment rate? What effect would this plan have on the average duration of unemployment? Why? Lower search costs would probably increase the unemployment rate. Lower search costs (similar to extended UI benefits) will cause workers to keep their asking wage higher for a longer time. Thus, unemployed workers will be more selective in the jobs they accept. The result is that workers remain unemployed for longer periods of time. It is also possible that lower search costs will entice workers to quit their jobs in order to look for a better job. This will also increase the unemployment rate. (b) A country is debating whether to impose a $10,000 tax on employers for every worker they layoff. What effect would this plan have on the long-run unemployment rate? What effect would this plan have on the average duration of unemployment? This policy would cause firms, in the long-run, to be very cautious in hiring new workers. (This cautious behavior is evident in Japan and some countries in Europe. The UI system in the US may also cause firms to be a bit cautious in hiring new workers.) Thus, the unemployed will remain unemployed for longer durations. The effect on the unemployment rate is a little ambiguous. The unemployment rate would likely increase in the long-run, but one could also argue that the end result will be less labor turnover and, therefore, a lower rate of unemployment, especially if unemployed workers under the proposed rules are more easily discouraged and exit the labor force all together. 95