1. Macroeconomics Chapter 9
Capital Utilization and Unemployment
Macroeconomics Chapter 9

2. Macroeconomics Chapter 9
Capital Input
Capital utilization rate
The fraction of the capital stock used in production.
κ (the Greek letter kappa) represent the utilization rate for the capital stock, K.
Y= A· F(κK, L)
Macroeconomics Chapter 9

3. Macroeconomics Chapter 9
Capital Input
κK, rises with the utilization rate, κ.
Therefore, an increase in κ raises real GDP, Y, for a given technology level, A, capital stock, K, and labor input, L.
Macroeconomics Chapter 9

4. Macroeconomics Chapter 9
Capital Input
The Demand for Capital Services
Firms maximize real profit
π/P = A·F[(κK)d, Ld)]
− (w/P)·Ld − (R/P)·(κK)d
Macroeconomics Chapter 9

5. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

6. Macroeconomics Chapter 9
Capital Input
The Demand for Capital Services
Assume that the technology level rises from A to A’. This change raises the MPK at a given quantity.
Macroeconomics Chapter 9

7. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

8. Macroeconomics Chapter 9
Capital Input
The Supply of Capital Services
For a given stock of capital, K, owners can supply more or less capital services per year by varying κ.
One reason to set the utilization rate, κ, below its maximum is that increases in κ tend to raise the depreciation rate, δ.
δ = δ(κ)
Other reasons include utility cost of capital.
Macroeconomics Chapter 9

9. Macroeconomics Chapter 9
Capital Input
The Supply of Capital Services
Net real income from supplying capital services
= real rental payments − depreciation
= (R/P) · κK − δ(κ) · K
= K·[(R/P)·κ − δ(κ)]
Rate of return from owning capital
= (R/P)·κ − δ(κ)
Macroeconomics Chapter 9

10. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

11. Macroeconomics Chapter 9
Capital Input
Owners of capital (households) select the utilization rate, κ, that maximizes
(R/P)·κ−δ(κ)
The solution is κ = κ∗
Macroeconomics Chapter 9

12. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

13. Macroeconomics Chapter 9
Capital Input
Thus, an increase in the real rental price raises the capital utilization rate; the higher real rental price makes it worthwhile to raise κ despite the resulting increase in the depreciation rate, δ(κ).
Macroeconomics Chapter 9

14. Macroeconomics Chapter 9
Capital Input
Market Clearing and Capital Utilization
The supply curve slopes up because an increase in the real rental price, R/P, motivates a higher capital utilization rate, κ.
The technology level, A, does not affect the choice of the capital utilization rate, κ
Macroeconomics Chapter 9

15. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

16. Macroeconomics Chapter 9
Capital Input
Market Clearing and Capital Utilization
When the technology level rises to A, the demand curve for capital services shifts to the right, and the supply curve does not shift.
Therefore the market clears at the higher real rental price, [(R/P)∗], and the larger quantity of capital services, [(κK)∗].
Macroeconomics Chapter 9

17. Macroeconomics Chapter 9
Capital Input
We now have three reasons why real GDP rises in a boom and falls in a recession.
First a high or low technology level, A, causes real GDP to be correspondingly high or low.
A high or low A causes L to be correspondingly high or low
A high or low A causes the capital utilization rate, κ, and, thereby, the quantity of capital services, κK, to be correspondingly high or low.
Macroeconomics Chapter 9

18. Macroeconomics Chapter 9
Capital Input
Market Clearing and Capital Utilization
i = (R/ P) · κ − δ(κ)
Rate of return on bonds
= rate of return on ownership of capital
Increase in the technology level, A, raises the rate of return from owning capital, the interest rate, i, increases. The interest rate is still procyclical in the model.
Macroeconomics Chapter 9

19. Macroeconomics Chapter 9
Capital Input
The Cyclical Behavior of Capacity Utilization
The Federal Reserve computes capacity utilization by expressing a sector’s output of goods as a percentage of the estimated “normal capacity” of each sector to produce goods
Macroeconomics Chapter 9

20. Macroeconomics Chapter 9
Capital Input
Macroeconomics Chapter 9

21. The Labor Force, Employment, and Unemployment
Basic Concepts and Empirical Patterns
u = number unemployed/labor force
= ( labor force− number employed) /labor force
= 1 − (number employed/labor force)
= 1 − employment rate
employment rate= 1 − u
Macroeconomics Chapter 9

22. The Labor Force, Employment, and Unemployment
Employment =（1 − u）· Labor force
Total working hours
= （1 − u）· Labor force · working hours per worker
Macroeconomics Chapter 9

23. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

24. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

25. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

26. The Labor Force, Employment, and Unemployment
The equilibrium business-cycle model is probably satisfactory for understanding fluctuations in the labor force and hours worked per worker.
The real wage rate, w/P adjusts to equate the quantity of labor supplied,Ls, to the quantity demanded, Ld.
However, this approach leaves unexplained the most important factor—the fluctuations in the employment rate or, equivalently, in the unemployment rate.
Macroeconomics Chapter 9

27. The Labor Force, Employment, and Unemployment
A Model of Job Finding
Macroeconomics Chapter 9

28. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

29. The Labor Force, Employment, and Unemployment
A Model of Job Finding
Distribution of wage offers.
For each real wage rate, w/P, on the horizontal axis, the value on the vertical axis shows the probability of receiving that offer.
The value ω, which is the effective real income received while unemployed.
Key decision is whether to accept a real wage, w/P, when it is greater than ω.
Macroeconomics Chapter 9

30. The Labor Force, Employment, and Unemployment
A Model of Job Finding
Reservation real wage, denoted by (w/P)’. Offers below (w/P)’ are rejected, and those above (w/P)’ are accepted.
Macroeconomics Chapter 9

31. The Labor Force, Employment, and Unemployment
For a given distribution of wage offers an increase in ω makes it more likely that w/P < ω will apply, because ω increases.
Job offers will be rejected more often. It follows that job searchers tend to take longer to find a position when ω increases. For a group of persons, we therefore predict that a rise in ω reduces the job-finding rate, which is the rate at which job seekers find positions.
Correspondingly, a rise in ω raises the expected duration of unemployment, which is the amount of time that the typical unemployed person stays unemployed.
Macroeconomics Chapter 9

32. The Labor Force, Employment, and Unemployment
The effect of the increase in A
The real offer rises
Reservation real wage increases, but not so large as the increase in the real offer
Short run effect of A
The effect of ω
Macroeconomics Chapter 9

33. The Labor Force, Employment, and Unemployment
Search by Firms is important in a well-functioning labor market, the allowance for this search does not change our major conclusions.
It still takes time for workers to match with jobs, so that the expected durations of unemployment and vacancies are greater than zero.
An increase in workers’ effective real incomes while unemployed, ω, lowers the job-finding rate and raises the expected duration of unemployment.
A favorable shock to productivity raises the job-finding rate and lowers the expected duration of unemployment.
Macroeconomics Chapter 9

34. The Labor Force, Employment, and Unemployment
Job Separations
Job-separation rate, the rate at which job matches dissolve.
This rate is high among inexperienced workers who are hard to evaluate or among young persons who are likely to experience changes in family size or job preferences.
The separation rate is also high in industries that are subject to frequent shocks to technology or product demand.
Macroeconomics Chapter 9

35. The Labor Force, Employment, and Unemployment
Job Separations, Job Finding, and the Natural Unemployment Rate
Macroeconomics Chapter 9

36. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

37. The Labor Force, Employment, and Unemployment
Job Separations, Job Finding, and the Natural Unemployment Rate
Job−finding rate = (number of hires per month) /U
Macroeconomics Chapter 9

38. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

39. The Labor Force, Employment, and Unemployment
Job Separations, Job Finding, and the Natural Unemployment Rate
The natural unemployment rate is the unemployment rate that economy tends towards, given the rates at which people lose and find jobs
Macroeconomics Chapter 9

40. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

41. The Labor Force, Employment, and Unemployment
Job Separations, Job Finding, and the Natural Unemployment Rate
The change in the number of persons employed over a month, ∆L, is given by
∆ L = ϕ U − σ L
= job findings− job separations
The long-run levels of L and U are constant.
∆L=0
ϕ U = σ L
job findings= job separations
Macroeconomics Chapter 9

42. The Labor Force, Employment, and Unemployment
Job Separations, Job Finding, and the Natural Unemployment Rate
Key equation (natural unemployment rate):
un = σ/(ϕ + σ)
A higher job-separation, σ , raises the natural unemployment rate, un, whereas a higher job-finding rate, ϕ, lowers un.
Macroeconomics Chapter 9

43. The Labor Force, Employment, and Unemployment
Economic Fluctuations, Employment, and Unemployment
An adverse shock to the technology level, A, reduces the marginal product of labor for the typical worker and job. The job-finding rate, ϕ, falls because market opportunities became poorer—probably temporarily—relative to the real income received while unemployed, ω.
Macroeconomics Chapter 9

44. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

45. The Labor Force, Employment, and Unemployment
Vacancies
An increase in the value of prospective marginal products, MPL, raises the number of job openings
An increase in the real wage rate, w/P, required to get workers to accept jobs lowers the number of job openings.
A reduction in costs of posting jobs and processing applications raises the number of job openings.
Macroeconomics Chapter 9

46. The Labor Force, Employment, and Unemployment
Vacancies
An increase in the technology level, A, raises the MPL. For given real wage rates, w/P, firms post more job openings. Therefore, job vacancies increase.
An unfavorable shock reduces vacancies.
Thus, our prediction is that vacancies are procyclical — high in booms and low in recessions.
Macroeconomics Chapter 9

47. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9

48. The Labor Force, Employment, and Unemployment
Macroeconomics Chapter 9