1. Compensating Differentials

2. Reading List
S. Rosen (1986). “The Theory of Equalizing Differences” in the Handbook of Labor Economics, vol. 1:
J. Roback (1982) “Wages, Rent and the Quality of Life”. Journal of Political Economy, vol. 90.6:
Borjas, Chapter 5
Cahuc & Zylberberg, Chapter 5.1.
Ehrenberg & Smith, Chapter 8.

3. Compensating Differentials
(or equalizing differences)
In the Roy model people cared only about income, but differed in skills.
In the simplest version of this model, people:
Have identical skills.
Heterogeneity in tastes for jobs.
The idea is that an employer must pay a premium to get you to do some job you don’t want to do.

4. Disamenities
Let D represent a disamenity of work, like how dangerous it is.
Suppose
D = 0 represents safe jobs that pay W0.
D = 1 represents dangerous jobs that pay W1.
All safe jobs will pay the same because workers are identical and labor market is competitive (and frictionless).

5. Preferences People have preferences for consumption and disamenities.
Ui(C, D)
C0 = Y + W0
C1 = Y + W1
where Y is nonlabor income.
You choose the job with D = 0 if
U(C0,0) > U(C1, 1).

6. Wages
If C0 = C1 = C, then everyone prefers D = 0: U(C,0) > U(C, 1).
Then, in equilibrium, if we see people in both jobs, it must be that W1 > W0.
Thus in equilibrium a wage differential arises.

7. Linear Utilities
Take a really simple case with linear utility so that:
Ui(C,D) = C – diD
Then individual i chooses to work in the dangerous sector if:
Ui(C0,0) < Ui(C1, 1)
Y + W0 < Y + W1 – di
di < W1 – W0 = DW

8. Heterogeneity
Now suppose that di varies over the population with measure G.
The supply of people to dangerous jobs can be written as:
Where 1(.) is the indicator function.

9. Notice that:
This is just the cdf of di.
As DW increases ,more people do the dangerous job.
Elasticity of supply.
So the elasticity depends on the density of people who are indifferent.

10. Examples of Supply Curves
Case 1: No heterogeneity
Case 2: Several types of workers
Case 3: Continuous case

11. Firm Side Now let’s think about the firm side of the market.
If costs money to make the workplace safe.
The cost varies across jobs.
It is easier for a university than a coal mine.
Each firm (job) hires one worker and there are as many firms as workers.

12. Profits Production for firm j is Fj.
The cost of making the work environment safe is bj.
So profits as a function of working environment are:
Fj – bj(1-D) –WD

13. Demand Thus the workplace is dangerous if W1 < W0 + bj bj > DW
Let F be the distribution of bj, then the demand for workers in dangerous jobs is
So demand also looks like a cdf.

14. Hedonic Price Model More generally, suppose that danger is continuous.
Let W(D) be the wage paid in a job with mortality rate D.
The workers chooses D to maximize:
Ui(Y + W(D), D), so UicW’ = -UiD
The firm minimizes costs of production.
W(D) + bj(D), so W’ = -bj’
(see Rosen for details)

15. Applications Occupational choice Migration Environment
Local public finance
Industry wage differentials
Human capital / signaling
Labor supply

16. More applications
Felfe, C. (2012) “The Willingness to Pay for Job Amenities: Evidence from Mothers' Return to Work.” Industrial and Labor Relations Review, 65(2),
   
Felfe, C. (2012) “The Motherhood Wage Gap - What about Job Amenities?” Labour Economics, 19(1),

17. Roback (1982) Some cities are more “liveable” than others.
Compensating wage differential to attract workers to “worse” cities.
Problem: why would a firm locate in a less liveable city?
There must be some productivity advantage.

18. Hedonic Model of Intercity Location
Problem: both the land and the labor markets must clear.
Example: all firms and workers are identical.
Only one type of job will be offered.
In the spatial allocation problem, people cannot all occupy the same space!
The scarcity of land gives rise to an additional constraint.

19. Roback’s paper General equilibrium model.
Mobile factors (labor) and site-specific factors (land).
Possibility that the amenities may influence productivity.
Calculate implicit prices of amenities.
Compute quality of life rankings.
Regional differences in wages are explained by differences in amenities.

20. Price Determination Cities vary in the quantity of amenity s.
Residents consume and produce a consumption commodity X.
Capital and labor are completely mobile across cities.
Costs of moving are zero.
Land is fixed among cities.

21. Workers Workers are assumed to be identical in tastes and skills.
Each person supplies a unit of labor.
The worker’s problem: choose X and l to maximize utility and satisfy the budget constraint.
Market equilibrium condition:
Wages and rents must adjust to equalize utility in all occupied locations.

22. Firms X is produced according to a CRS production function f.
The firm minimizes costs subject to the production function.
Equilibrium condition: unit cost must equal product price (assumed to be 1).
Otherwise the firm would have an incentive to move to more profitable cities.

23. Productivity of Amenities
Amenities can be:
productive (“lack of earthquakes”, “ocean”)
unproductive (“clean air”)
neither (“sunny days”, “good schools”).

24. Equilibrium
The equilibrium levels of wages and rents are determined by the interaction of the equilibrium conditions of workers and firms.
Effect of different quantities of s on wages and rents?
If s is unproductive, in more amenable places the wages should be lower (rents?).
If s is productive, in more amenable cities rents would be higher (wages?).

25. Two Cities We can think of 4 different cases: VS > 0, CS = 0

26. Large Number of Cities
Let’s think about equilibrium over a large number of cities.
We know that C(w,r; s) = 1
V(w,r; s) = k
So, dV/ds =
dC/ds =

27. We can substitute for ∂r/∂s and solve for ∂w/∂ s.
Thus, if
Cs = 0 ∂w/∂ s ∂r/∂s
Cs > 0 ∂w/∂ s ∂r/∂s
Cs < 0 ∂w/∂ s ∂r/∂s

28. Empirical Implementation
How can we implement this model empirically?
We want to measure how tastes for cities vary.
We can observe how wages and rental rates vary across cities.
We can use them to measure “revealed preference” for amenities.

29. We know that:
Cw = N/X
Cr = lp/X
lc = -Vr/Vw (Roy’s identity)

30. Value of the Amenity The shadow price of the amenity: ps* = Vs/Vw
How much consumers value the amenity.
Income required to compensate for a small change in s.
The worth to the firm: CsX.
We have 2 equations (∂w/∂s, ∂r/∂s) where everything is observable except Cs and Vs/Vw
Once we substitute Cr, Cw and use Roy’s identity.

31. Value of the Amenity (ii)
We solve for Cs and Vs/Vw.
The total value of the amenity: ps*N – CsX.
Roback implements this to estimate the value of life in each city.

32. Worker and Housing Quality
One problem is that we’re assuming that worker quality and housing quality is the same across regions.
Roback relaxes the first part by assuming that workers are perfect substitutes.
As for housing, Roback adds “housing services” to the model.

33. Housing Services Problem: we don’t really observe land prices.
Houses are different within a city.
Roback adds “housing services”.
As well as other possible “nontraded goods”.

34. Workers The vector of non-traded goods enters the utility function.
Now the consumption of land is considered an input into the production of housing.
Land increases utility indirectly through the consumption of housing.
The indirect utility function now depends on the price of nontraded relative to traded goods.
But not directly on the rental rate of land.

35. Production Non-traded goods sector. Unit cost function.
These 3 equations are sufficient to determine w, r and p.

36. Earnings
Worker i in city j earns:
Log earnings

37. Empirical Results CPS data. 98 largest US cities.
Coefficients of city characteristics on earnings regressions.
All the coefficients go in the expected direction.

39. Regional Earnings Differences
What is the influence of the city attributes on the well-known regional differences in earnings?
Earnings regressions with region dummies.
Adding the amenities should reduce the effect of region.

41. Quality of Life R. also runs land price equations.
Computes the “implicit price” of each attribute.
These prices can be used to compute quality of life indices.
Los Angeles and San Francisco are # 1 and 3.
New York is #11, Chicago #16.

42. Conclusions
City amenities are reflected in both wages and land prices.
Sign and size depend on:
the influence of the amenity on production, and
the strenght of consumer preferences.
Recent applications to quality of life estimates across countries.

43. Problems with hedonic wage regressions
Need to account for workers’ labor market productivity.
Otherwise, estimates of workers’ willingness to pay for amenities will be biased towards 0.
Example: Bonhomme & Jolivet (Journal of Applied Econometrics, 2009).
Recent work tries to overcome this issue.
Use of panel data, etc.