### Chapter One

**The Labor Wedge**

Throughout this book, I study the interaction of optimizing households
and firms in a closed economy. I begin in this chapter by developing
a competitive, representative-agent version of the model. The chapter
has two objectives. First, I introduce much of the notation that I rely on
throughout the book. Because of this, I include details in this chapter
that are not really necessary for the second, more substantive objective:
I use the model to measure and analyze the behavior of the *labor wedge*,
the wedge between the marginal rate of substitution of consumption for
leisure and the marginal product of labor. I confirm the well-known result
that the labor wedge tends to rise during recessions, so the economy
behaves as if there is a countercyclical tax on labor. The remainder of the
book explores whether extending the model to incorporate labor market
search frictions can explain the behavior of the labor wedge.

I start the chapter by laying out the essential features of the model: optimizing households, optimizing firms, a government that sets taxes and spending, and equilibrium conditions that link the various agents. In section 1.2, I use pieces of the model to derive a static equation that relates hours worked, the consumption-output ratio, and the labor wedge. Section 1.3 discusses how I measure the first two concepts and uses these measures to calculate the implied behavior of the labor wedge in the United States. I establish the main substantive result: that the labor wedge rose strongly during every recession since 1970. I show the robustness of my results to alternative specifications of preferences in section 1.4 and discuss the possibility that the results are driven by preference shocks in section 1.5. I finish the chapter with a brief discussion in section 1.6 on the empirical relationship between the fluctuations in hours, which I analyze here, and fluctuations in employment and unemployment, which are the main topic of subsequent chapters.

**1.1 A Representative-Agent Model**

I denote time by *t* = 0, 1, 2, ... and the state of the economy at time *t* by
[*s*.sub.*t*]. Let [*s*.sup.*t*] = {[*s*.sub.0], [*s*.sub.1], ..., [*s*.sub.t]} denote the history of the economy and [PHI]([*s*.sup.*t*])
denote the time-0 belief about the probability of observing an arbitrary
history [*s*.sup.*t*] through time *t*. Exogenous variables like aggregate productivity,
government spending, and distortionary tax rates may depend on
the history [*s*.sup.*t*]. At date 0, there is an initial capital stock [*s*.sub.0] = *k*([*s*.sup.0]) and an
initial stock of government debt [*b*.sub.0] = *b*([*s*.sup.0]). The capital stock is owned
by firms, while households hold the debt and own the firms.

**Households**

A representative household is infinitely lived and has preferences over
history-[*s*.sup.*t*] consumption *c*([*s*.sup.*t*]) and history-[*s*.sup.*t*] hours of work *h*([*s*.sup.*t*]). To
start, I assume that preferences are ordered by the utility function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where ? [member of] (0, 1) is the discount factor, [gamma] > 0 measures the disutility of working, and, as I show below, [epsilon] > 0 is the Frisch (constant marginal utility of wealth) elasticity of labor supply.

This formulation implies that preferences are additively separable over time and across states of the world. It also implies that preferences are consistent with balanced growth-doubling a household's initial assets and its income in every state of the world doubles its consumption but does not affect its labor supply. This is consistent with the absence of a secular trend in hours worked per household, at least in the United States (Aguiar and Hurst 2007; Ramey and Francis 2009). I maintain both of these assumptions throughout this book. The formulation also imposes that the marginal utility of consumption is independent of the worker's leisure. This restriction is more questionable and so I relax it in section 1.4 below.

The household chooses a sequence for consumption and hours of work to maximize utility subject to a single lifetime budget constraint,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

The household has initial assets [*a*.sub.0] = *a*([*s*.sup.0]). In addition, [tau]([*s*.sup.*t*]) is the labor
income tax rate, *w*([*s*.sup.*t*]) is the hourly wage rate, and *T*([*s*.sup.*t*]) is a lump-sum
transfer in history [*s*.sup.*t*], all denominated in contemporaneous units of consumption.
Thus *c*-(1-[tau])*wh* - *T* represents consumption in excess of
after-tax labor income and transfers, which is discounted back to time 0
according to the intertemporal price [*q*.sub.0]([*s*.sup.*t*]). That is, [*q*.sub.0]([*s*.sup.*t*]) represents the
cost in history [*s*.sup.0] of purchasing one unit of consumption in history [*s*.sup.*t*],
denominated in units of history-[*s*.sup.0] consumption. Put differently, [*q*.sub.0]( [*s*.sup.*t*])
is the history-[*s*.sup.0] price of an Arrow-Debreu security that pays one unit of
consumption in history [*s*.sup.*t*] and nothing otherwise. Equation (1.2) states
that the household's net purchase of Arrow-Debreu securities in history
[*s*.sup.0] must be equal to its initial assets [*a*.sub.0].

It will be useful to define the assets of the household, following history
[*s*.sup.*t*], as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the notation [*s*.sup.*t'*] | [*s*.sup.*t*] indicates that the summation is taken over histories
[*s*.sup.*t'*] that are continuation histories of [*s*.sup.*t*], i.e., [*s*.sup.*t'*]
= {[*s*.sup.*t*], [*s*.sub.*t*+1], [*s*.sub.*t*+2],
..., [*s*.sub.*t'*]} for some states {[*s*.sub.*t*+1], [*s*.sub.*t*+2], ..., [*s*.sub.*t'*]}.
Then [*q*.sub.*t*]( [*s*.sub.*t'*+1]) is the price
of a unit of consumption in history [*s*.sub.*t'*+1] = {[*s*.sup.*t*], [*s*.sub.*t*+1],
[*s*.sub.*t*+2], ..., [*s*.sub.*t'*]} paid
in units of history-[*s*.sup.*t*] consumption. The absence of arbitrage opportunities
requires that q0(st)qt(st+1) = q0(st+1) for all st and for all
[*s*.sup.*t*+1] = {[*s*.sup.*t*], [*s*.sub.*t*+1]}.
Equivalently, the lifetime budget constraint implies a sequence of intertemporal budget
constraints,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

so assets plus labor income plus transfers in history [*s*.sup.*t*] is equal to
consumption plus purchases of assets in continuation histories [*s*.sup.*t*+1].

**Firms**

The representative firm owns the capital stock [*k*.sub.0] = *k*([*s*.sup.0]) and has
access to a Cobb-Douglas production function, producing gross output *z*([*s*.sup.*t*])
*k*([*s*.sup.*t*])[sup.*a*]*h] sup.d]
[([s.sup.t]).sup.1-a] in history [s.sup.t], where
z([s.sup.t]) is history-contingent
total factor productivity, k([s.sup.t]) is its capital stock, [h.sup.d]([s.sup.t]) is the labor it
demands, and [alpha] [member of] [0, 1) is the capital share of income. A fraction [delta] of
the capital depreciates in production each period, while at the end of
period t, the firm purchases any capital that it plans to employ in period
t + 1. That is, history-[s.sup.t+1] = {[s.sup.t],
[s.sub.t+1]} capital k([s.sup.t+1]) is purchased in history
[s.sup.t] and so must be measurable with respect to [s.sup.t]. The present value
of the firm's profits is then given by
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
*

*
Note that this expression presumes that the firm does not pay any taxes. I
do this for notational simplicity alone. In particular, any payroll taxes are
rolled into the labor income tax rate [tau]. The firm chooses the sequences
[ h.sup.d]([s.sup.t]) and k([s.sup.t+1]) to maximize J.
*

*
I can also write the value of the firm's profits from history [ s.sup.t] on as
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
*

*
This implies the recursive equation
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
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*
The value of a firm that starts history [ s.sup.t] with capital k([s.sup.t]) comes from current production z([s.sup.t])k
([s.sup.t])[sup.a]h]sup.d]
[([s.sup.t]).sup.1-a] minus the cost of investment
k([s.sup.t+1]) - (1 - [delta])k([s.sup.t]) minus labor
costs w([s.sup.t])[h.sup.d]
([s.sup.t]) plus the value of
starting the following period in history [s.sup.t+1] =
{[s.sup.t], [s.sub.t+1]} with k
({[s.sup.t+1]})
units of capital.
*

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Government
*

*
A government sets the path of taxes, transfers, and government debt to
fund some spending g([s.sup.t]). I assume government spending is wasteful or
at least is separable from consumption and leisure in preferences. The
government faces a budget constraint in any history [s.sup.t],
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)
*

*
so debt b([s.sup.t]) is equal to the present value of future tax receipts in
excess of spending and lump-sum transfers. Again, this is equivalent
to a sequence of budget constraints of the form
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.7)
*

*
so initial debt plus current spending and transfers is equal to current
tax revenue plus new debt issues.
*

*
*

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Market Clearing
*

*
There are three markets in this economy: the labor market, the capital
market, and the goods market. All of them must clear in equilibrium.
Labor market clearing dictates that labor supply equals labor demand
in all histories, h([s.sup.t]) = [h.sup.d]
([s.sup.t]). Capital market clearing dictates that
household assets are equal to firms' valuation plus government debt,
a([s.sup.t]) = J([s.sup.t],
k([s.sup.t])) + b([s.sup.t]).
Goods market clearing dictates that output
plus undepreciated capital is equal to consumption plus government
spending plus next period's capital stock:
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
*

*
One can confirm that goods market clearing is implied by the household
budget constraint (equation (1.3)), the firm's value function (equation
(1.5)), the government budget constraint (equation (1.7)), and capital
and labor market clearing. This is an application of Walras's law.
*

*
*

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Equilibrium
*

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Given arbitrary paths for government spending g([s.sup.t]),
taxes t([s.sup.t]), and
government debt b([s.sup.t]), an equilibrium consists of paths for consumption
c([s.sup.t]), labor supply h([s.sup.t]),
labor demand [h.sup.d]([s.sup.t]),
capital k([s.sup.t]), assets
a([s.sup.t]), transfers T([s.sup.t]),
intertemporal prices [q.sub.0]([s.sup.t]), and the wage rate
w([s.sup.t]) such that:
*

*
{ c([s.sup.t])}, {h([s.sup.t])},
and {a([s.sup.t])} solve the household's utility-maximization
problem, maximizing equation (1.1) subject to the budget
constraint (1.2) given {q([s.sup.t])}, {w([s.sup.t])},
{[tau]([s.sup.t])}, and {T([s.sup.t])};
*

*
{[ h.sup.d]([s.sup.t])} and {k([s.sup.t])}
maximize firms' profits in (1.4) given {[q.sub.0]( [s.sup.t])}
and {w([s.sup.t])};
*

*
the government budget is balanced, so equation (1.6) holds; and
*

*
the labor, capital, and goods markets clear.
*

*
*

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1.2 Deriving the Labor Wedge
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*
To see the implications of this model for the labor wedge, I focus on
a subset of the equilibrium conditions. First, consider the household's
choice of history-[ s.sup.t] consumption and labor supply. These must satisfy
the first-order conditions
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
*

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and
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
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where [lambda] is the Lagrange multiplier on the budget constraint: equation
(1.2). Note from the second equation that a one percent increase
in the after-tax wage (1 - [tau]) w raises labor supply h by [epsilon] percent, holding
fixed the Lagrange multiplier [lambda] and the intertemporal price [q.sub.0]([s.sup.t]).
Thus [epsilon] is the Frisch elasticity of labor supply, a key parameter in this
chapter.
*

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*

*
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(Continues...)
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Excerpted from "Labor Markets and Business Cycles (CREI Lectures in Macroeconomics)" by Robert Shimer. Copyright © 0 by Robert Shimer. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher. Excerpts are provided solely for the personal use of visitors to this web site.
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