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Exam review and Compensating wage dierentials

Econ 152

Fall 2008

Raymundo M. Campos-Vazquez

October 27, 2008

Contents

1 Exam review 1

1.1 Income Eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Labor Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Compensating wage dierentials 2

1 Exam review

1.1 Income Eects

Consider the general case of a utility-maximizing worker with some positive non-labor income

V0 > 0 who faces the choice between consumption and leisure.

1. Suppose this worker faces an increase in non-labor income to V1, V1 > V0, holding the

wage constant. What is the eect of this increase on her hours of work (support your

argument with a graph):

(a) If leisure is a normal good.

(b) If leisure is an inferior good.

2. Suppose leisure is indeed a normal good. Moreover, suppose that the worker retains

some positive non-labor income V0 > 0. What happens to the workers hours of work

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when she faces an increase in her wage, holding non-labor income constant? Illustrate

the two possibilities in two separate graphs. In both cases, include the decomposition

that leads from the workers initial optimum to the new optimum. Give a verbal

explanation of your results, in particular the reason for the ambiguity in the relationship

between hours of work and the wage.

1.2 Labor Demand

1. What is the dierence between the rms employment decision in the short run and

in the long run? What do we want to express with the elasticity of labor demand and

how is this elasticity dened? Would you expect the rms short-run demand curve or

its long run demand curve to be more elastic and why?

2. Now suppose the rms technology distinguishes between native and immigrant labor.

The production function is q = f(K;N ;I) where K refers to capital, N to native

workers and I to immigrant workers. Suppose the initial prot maximizing input mix

is given by N=180, I=20, K=1000. The initial wage for immigrant labor is 10/hr.

Suppose that new immigrants arrive to the labor market and they reduce the wage of

immigrants to 8/hr. As a consequence, the rm decreases K to 800 and reduces its

level of native labor to N=150.

(a) Are the inputs "immigrant" and "native" complements or substitutes? Why?

(b) Are the inputs "immigrant" and "capital" complements or substitutes? Why?

2 Compensating wage dierentials

Suppose all surgeons utility functions are given by:

U = w1=3 4i;

where w is the wage and i is the probability of pinching your nger with a contaminated

needle. Assume there are two types of jobs for surgeons: (1) the safe academic jobs wherei = 0; and (2) the risky hospital jobs where i = 0:25: Let wsafe be the wage in the safe jobs,

and wrisk be the wage in the risky jobs. Suppose the safe academic jobs pay $64 per hour.

1. (a) How much should the hospital physicians be paid per hour?

Answer: Recall that the compensating wage dierential is the equilibrium dif-

ferential in wages between a risky and a safe job; that is, w = wrisk wsafe.

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Figure 1: Compensating wage dierential

Figure 2: Indierence curve of marginal worker

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The fact that it is the equilibrium dierential means that it is the dierential in

wages needed to attract the marginal worker (although all workers are going to

be paid this dierential in a competitive market) (See Figure 1). So the rst

thing we need to nd is the utility that the marginal worker gets in the safe job.

Substituting the wage and the risk from the safe job, the utility in the safe job is:

Usafe = (64)1=3 4 0

= 4

In order to bribe the marginal worker to accept the risky job we need to pay her

enough so that she would get at least as much utility as in the safe job (that is,

we need to compensate her for incurring the risk) (See Figure 2). Then we need

to solve for wrisk in the following equation:

Urisk = 4 = w1=3risk 4 0:25

) 5 = w1=3risk

) wrisk = 53

) wrisk = 125

The compensating wage dierential is thus:

w = wrisk

wsafe= 125 64

= 61

(b) Suppose the demand for risky jobs is given by the following function:

w = 1009 12E

What is the equilibrium level of employment of doctors in the hospitals?

Answer: Recall I told you that all doctors had exactly the same preferences,

this means that all doctors will be enticed to work if the compensating wage

dierential is equal to 61 dollars. So at w = 61, the hospitals are going to be

able to attract any amount of doctors.

Question for you: how does the supply curve looks like?

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In equilibrium, supply of risky workers equals demand for risky workers, hence:

61 = 1009 12E

) E = 79

(c) Suppose that instead of having only one type of workers you have two types of

workers. The rst types preferences are as before:

U1 = w1=3 4i

The second types preferences are given by:

U2 = w1=2 10i:

Also, assume we have wrisk = 121; wsafe = 64; isafe = 0; and irisk = 0:25:

i. Among these two types, whos the less risk averse?

Answer: Recall the more risk-averse worker will have a steeper indierence

curve. In this case, the slope of the indierence curve is given by:

MRS = M Urisk

M Uw

Given that risk is a bad (i.e. M Urisk < 0), we know that MRS > 0: Now

we need to estimate the marginal utilities of risk and wages for each type of

workers. The following table presents the marginal utilities:

M U Risk w

Type-one -4 13

w2=3

Type-two -10 12

w1=2

Thus the M RS for each type is:

M RS1 = 4

1

3w2=3

= 12w2=3

M RS2 = 10

1

2w1=2

= 20w1=2

Evaluating at an arbitrary wage level, say the safe jobs wage, we have:

M RS1 = 12 (64)2=3 = 192

M RS2 = 20 (64)1=2 = 160

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Hence, type-two worker is less risk averse than type-one worker (i.e. he needs

a lower bribe to be enticed to work in the risky job).

ii. Where is each worker going to work?

Answer: In order to answer this question we need to know what is the utility

level of each doctor in each job. The doctor will work in the job that gives her

the higher utility level. But even before estimating the utility levels, we know

that the rst type will work in the safe job, because the wage dierential is not

big enough to bribe type-one into the risky job. The following table presents the

utility levels for both workers in both type of jobs.

Safe Risky

U1 4 3.946

U2 8 8.5

So type-one is going to work in the safe job, as we predicted, and type-two isgoing to work in the risky-job.

(d) What is the reservation price of type-two?

The utility from the safe job is equal to 8. Hence we need wrisk such that the

worker is indierent between the two types of jobs:

8 = w1=2risk 10 0:25

) wrisk = (8 + 2:5)2

) wrisk = 110:25

So type-two only needs w = 46:25 to take the risky job.

Note that ifwrisk < 110:25; then both workers will take the safe job. So it is not

necessarily true that the less risk averse will always take the risky job, there is

just a higher probability the the less risk averse will end up in the riskier jobs.

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