University of California Department of Economics

Field Exam August 2010

Labor Economics

There are three parts in the exam. Each part will be weighted equally and

should take approximately one hour. Make sure you allocate your time

carefully, answering all parts as fully as possible given the time constraint. Use

equations and graphs whenever possible to clarify your reasoning.

WRITE YOUR ANSWERS FOR EACH QUESTION IN A SEPARATE BOOK

PART I Answer both questions in this part. Each question should take about 30

minutes.

Question 1: LABOR SUPPLY

a) Derive the Slutsky equation for hours of work in a standard two-good model with

leisure and consumption. Be sure to define your notation, and carefully state any

assumptions that you have used in your derivation.

b) The recent financial crisis led to a dramatic decline in the net worth of many American

households. Suppose that for a representative household the elasticity of hours worked

per year with respect to income is −0.1 and that the budget share of earned income in

total consumption is 0.8. How much of an increase in hours worked should we expect if

the crisis destroyed ten percent of lifetime wealth and wages remained constant? Using a

reasonable estimate of the compensated elasticity of labor supply, how much would

wages need to fall for total hours worked to be unaffected by the crisis?

Question 2: LABOR DEMAND

A pair of well known economists proposed cutting payroll taxes during the recent

economic downturn in order to boost employment. Note: Payroll taxes are federal taxes

levied on wages paid by firms to workers (e.g. firms pay an after tax wage of w(1+τ)).

a) Suppose firms produce output using a constant returns to scale technology F(K,L)

taking capital (K) and labor (L) as inputs. Assume the rate of return on capital r is fixed

exogenously by international capital markets. Derive an expression for the employment

effects of a 10% decrease in the federal payroll tax τ. Be sure to state any additional

assumptions needed to derive your answer, and discuss their plausibility.

2

b) Now consider a representative firm facing the following dynamic objective:

Π(A,K,L)= max AF(K′,L′) − c1 1[L′<L] − c2(K′-(1-δ)K) − wL′ + β E Π (A′,K′,L′)

K′,L′

where A is the firm's total factor productivity, which follows some stochastic process, F

is a production function, c1 represents the fixed cost of firing workers, 1[.] is an indicator

function for the expression in brackets being true, c2 is the cost of new capital, w is the

wage rate, and β<1 is a discount factor.

Using this model, discuss the dynamic effects of a small, temporary cut in the payroll tax

(e.g., a reduction in the tax rate that will last for 1 year).

c) How does your answer to b) depend on the firm's level of uncertainty regarding next

period's level of productivity A′?

PART II Answer all parts of this question. The question is designed to take 1 hour. Use

equations and graphs whenever possible to clarify your reasoning.

In this question you will develop a variant of the Roback (JPE, 1982) model of the joint

determination of wages and cost of living. In particular, consider the case where there are

2 cities (A and B) and 2 skill groups: skilled workers and unskilled workers. Skilled and

unskilled workers are imperfect substitutes. Variation in the cost of living depends only

on variation in cost of land which is assumed to be the same for all workers in the same

city, irrespective skill.

1) State all the assumptions of the Roback model. (for example: what are you assuming

about workers' and firms' mobility? Which goods are traded and which goods are local?

What are you assuming about firms' profits?) . Full credit will be awarded for a full

description of all the assumptions.

2) Assume for now that the two cities are identical in terms of amenities and production

technology, and there are no externalities. Describe the equilibrium in words. Now,

describe the equilibrium graphically. (Hint: Draw two graphs side by side. The left

graph is for the skilled workers. The right graph is for the unskilled workers. Label the

axis and all the curves and explain in detail why each curve looks the way it does.)

3) Now suppose that city A is less attractive than city B because schools have lower

quality. For simplicity, assume that school quality directly enters workers' utility

functions. Assume also that schools are not financed locally and that skilled and

unskilled workers value schools equally. In a graph, show what happens to wages and

rents in equilibrium. Label all the curves. Explain what is happening in words.

4) In equilibrium, both skill groups are present in both cities. Since workers are free to