13-1. 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
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.
# Unemp: # Months
Duration Exit Start of # of # of For
(Months) Rate Month Exiters Stayers Duration
3 0.20 8,000 1,6006,4004,800
4 0.20 6,400 1,2805,1205,120
5 0.20 5,120 1,0244,0965,120
6 1.00 4,096 4,096 0 24,576
(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
(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
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 (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.
13-2. 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
In total, 5,554,000 workers aged 20 years or older were unemployed in the U.S. in 2001. 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 11.835 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 = 64.34 percent.
13-3. Suppose the marginal revenue from search is
MR = 50 - 1.5w,
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 (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
50 – 1.5w = 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 $15.
13-4. (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 2000.
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
(140,863,000 + 2,253,000 – 135,208,000) / (140,863,000 + 2,253,000) = 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 (140,863,000 – 135,208,000) = 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 140,863,000 + 565,500 = 141,428,500, while the number employed should be 135,208,000 +
565,500 + 565,500 = 136,339,000. Thus, the unemployment rate would be
(141,428,500 – 136,339,000) / 141,428,500 = 3.6%.
13-5. 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
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 young