Macroeconomic Theory and Policy
David Andolfatto
January 2009
ii
Contents
1 The Employment of Nations
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
1.2 OECD Data . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Employment Rates . . . . . . . . . . . . . . .
1.2.2 Hours of Work per Employed Person . . . . .
1.2.3 The Aggregate Labor Input . . . . . . . . . .
1.3 A Basic Model Economy . . . . . . . . . . . . . . . .
1.3.1 Preferences . . . . . . . . . . . . . . . . . . .
1.3.2 Constraints . . . . . . . . . . . . . . . . . . .
1.4 Determining Individual Behavior . . . . . . . . . . .
1.4.1 Formalizing a Person’s Choice Problem . . .
1.4.2 Characterizing the Solution . . . . . . . . . .
1.4.3 Properties of the Solution . . . . . . . . . . .
1.5 Determining Aggregate Behavior . . . . . . . . . . .
1.5.1 Distribution of Personal Characteristics . . .
1.5.2 Computing Aggregates . . . . . . . . . . . . .
1.5.3 Measuring Economic Welfare . . . . . . . . .
1.6 Redistributive Policies and Employment . . . . . . .
1.6.1 Tax Rates on Labor Income Across Countries
1.6.2 Modeling Redistribution Policy . . . . . . . .
1.7 Taking Theory to Data . . . . . . . . . . . . . . . . .
1.7.1 Calibrating the Model . . . . . . . . . . . . .
1.7.2 Quantifying the Welfare Effects of Tax Policy
1.8 A Case for Redistribution? . . . . . . . . . . . . . .
1.8.1 The Distribution of After-Tax Income . . . .
1.8.2 Socially Optimal Tax Policy . . . . . . . . .
1.8.3 The Political Economy of Redistribution . . .
1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Problem Set . . . . . . . . . . . . . . . . . . . . . .
1.11 References . . . . . . . . . . . . . . . . . . . . . . . .
iii
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1
1
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31
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iv
CONTENTS
Chapter 1
The Employment of Nations
1.1
Introduction
Our material living standards depend on a variety of goods and services that
are somehow made available to us. A roof over our heads, a comfortable bed,
food and drink, a bus ride from here to there, and so on. Humans naturally
attach value to these things.
But the goods and services available to us do not just fall out of the sky—
someone must be employed toward their production. If this is correct, then the
“wealth of nations” will depend at least in part on the “employment of nations.”
To put it another way, the general availability of goods and services within an
economy will depend in some manner on the amount of time that is collectively
allocated within that economy toward the production of goods and services.
Macroeconomists label the collective amount of time allocated to production
the aggregate labor input (referred to loosely at times as employment and more
accurately at times as hours worked ). I begin this chapter by presenting data
that reveals some rather large and persistent differences in the aggregate labor
input across countries. I then present a basic theoretical framework that can
help us explain and interpret these differences.
The underlying premise of my approach—which is applied consistently throughout the text—is that aggregate behavior is best understood as the outcome associated with the sum of individual choices. That is, to understand aggregate
behavior, we must first understand the motivation behind the individual decisions that generate it.
1
2
CHAPTER 1. THE EMPLOYMENT OF NATIONS
.
Table 2.1
Aggregate Employment Rates
Selected OECD Countries
Canada
United States
Japan
Australia
New Zealand
Belgium
Denmark
France
Germany
Ireland
Italy
Netherlands
Norway
Portugal
Spain
Sweden
Switzerland
United Kingdom
1965
60.9
62.9
71.5
66.6
63.7
60.5
72.2
66.4
69.6
64.4
58.8
59.9
63.0
59.8
59.8
71.9
78.7
71.4
1970
61.3
64.5
71.1
68.8
64.1
61.3
74.3
65.8
68.6
62.0
56.3
57.3
64.1
62.2
60.1
72.7
77.5
70.4
1975
63.4
64.4
69.7
67.5
64.8
61.1
73.3
65.4
65.7
58.0
55.6
54.5
68.9
64.5
58.1
76.3
75.3
70.8
1980
65.5
66.8
70.3
65.5
64.2
58.4
73.1
63.5
63.6
57.0
55.7
53.6
73.8
63.8
50.2
78.7
73.7
68.7
1985
66.1
68.5
70.6
64.7
62.3
54.5
75.3
59.4
62.5
51.7
54.0
53.3
76.0
64.6
45.4
79.3
74.6
65.7
1990
68.9
72.4
72.5
67.6
66.8
56.6
76.8
60.1
65.8
52.7
55.0
61.5
74.3
68.9
48.9
79.8
81.7
70.7
1995
67.3
73.4
74.4
67.4
69.2
56.3
74.1
58.9
65.2
55.2
52.5
65.3
74.2
67.3
46.4
70.4
79.9
68.6
Mean
Standard Deviation
65.7
5.71
65.7
5.91
65.4
6.38
64.8
7.67
63.8
9.45
66.7
9.30
65.9
8.82
1.2
OECD Data
The acronym OECD stands for the Organization for Economic Cooperation and
Development; a collection of about 30 countries that follow (in varying degrees)
the principles of representative democracy and free-market economy. In what
follows, I present some aggregate data on measures of the labor input across
several OECD countries, based on the numbers reported in Rogerson (2001).
The available data measure two different dimensions of the labor input. The
first dimension is properly labeled employment; which is commonly defined as
the number of people who have performed any amount of “paid work” over a
given interval of time. The second dimension is the hours worked per employed
person. The aggregate labor input, or total hours worked, is constructed as the
product of these two measures.
1.2. OECD DATA
1.2.1
3
Employment Rates
The employment rate is constructed by dividing employment by some measure of
the population; which below is taken to be those aged between 15-64 years. The
numbers reported in Table 2.1 for each year correspond to five-year averages.1
From Table 2.1, it appears that the average employment rate across this
selection of countries has remained remarkably stable over time (roughly 66%).
The averages within countries have in some cases changed significantly over time.
But the most striking impression relates to the large and persistent differences
in employment rates across these countries.
1.2.2
Hours of Work per Employed Person
Table 2.2 reports the annual hours worked per employed person across a subset
of the countries considered in Table 2.1.2 This table reveals a secular decline in
the average work week (from about 36.7 hours per week in 1970, to about 32
hours per week in 1996). The information also shows that it may be misleading
in some cases to use the employment rate as a measure of the labor input.
Norway, for example, has a relatively high employment rate; but Norwegian
workers appear to work relatively few hours. The opposite holds true for Spain.
Table 2.2
Annual Hours Worked per Employed Person
Selected OECD Countries
Canada
United States
Japan
France
Germany
Italy
Norway
Spain
Sweden
1970
1890
1889
2201
1962
1949
1969
1766
n.a.
1641
1975
1837
1832
2112
1865
1801
1841
1653
n.a.
1516
1979
1832
1845
2126
1806
1696
1722
1514
2022
1516
1983
1780
1808
2095
1712
1657
1699
1485
1912
1518
1990
1788
1819
2031
1657
1598
1674
1432
1824
1546
1996
1784
1839
1892
1608
1511
1636
1407
1810
1623
Mean
Standard Deviation
1908
163.1
1807
172.2
1757
197.8
1719
189.9
1693
185.0
1663
165.6
1 See
Table 1 in Rogerson (2001).
Table 10 in Rogerson (2001); who is careful to provide the caveats associated with
making strict cross-country comparisons with these measurements.
2 See
4
CHAPTER 1. THE EMPLOYMENT OF NATIONS
1.2.3
The Aggregate Labor Input
To compute a measure of the aggregate labor input, I take the product of the
employment rate in Table 2.1 and hours worked per employed person in Table
2.2.3 I then divide this product by 8,760 (the number of hours per year). The
result is hours worked as a fraction of total time.4 Figure 2.1 plots this data for
the nine countries in Table 2.2. The countries ranked from highest to lowest in
1995 are: [1] Japan, [2] U.S., [3] Canada, [4] Sweden, [5] Norway, [7] France, [8]
Italy, and [9] Spain.
Figure 2.1
Hours Worked as Fraction of Total Time
Selected OECD Countries
0.25
0.2
0.15
0.1
0.05
0
1965
1970
1975
1980
1985
1990
1995
2000
Figure 2.1 reveals that most countries in this sample appear to have experienced at least a moderate secular decline in hours worked since 1970. The
major exception appears to be the United States (highlighted in bold). But
once again, the most striking impression is the large and persistent differences
in hours worked across countries. These differences, when measured in terms
of percent, are enormous. The difference between Japan and Spain in 1995, for
example, is almost 68%.
3 The
4 The
years do not correspond exactly, but they are close enough for the purpose at hand.
measure of hours worked H is computed here by the formula,
H=
Employment
Annual Hours Worked per Employed Person
×
Population
8,760
1.3. A BASIC MODEL ECONOMY
5
Understanding why these differences exist across countries and across time
constitutes an important line of enquiry in applied macroeconomic theory. The
answer to this question is not obvious. In reality, there are likely to be many
complicated forces that influence the behavior of individuals and economies.
To begin to make sense of this complicated reality, it will prove useful to
begin by constructing simple model economies that emphasize a limited number
of the potential (and presumably important) forces at work. The next section
provides the building blocks of a basic theory; which will subsequently be used
to interpret the data.
1.3
A Basic Model Economy
I begin this section with what I hope are two self-evident facts. The first is that
people are endowed with a limited amount of time. The second is that people
generally have competing uses for this time. These two facts present people with
a basic and fundamental problem: How should people allocate their limited time
across its competing uses?
In what follows, I describe a simple framework that economists regularly use
to organize their thinking on this matter. The basic idea is to build a model
economy, populated by model people that share some (though certainly not all)
of the characteristics of real people. Our model people will have preferences
for different goods and they will be motivated to acquire these goods to satisfy
their preferences. At the same time, our model people are endowed with limited resources and they will have to deal with certain trade-offs when making
decisions.
The decisions that are ultimately made by our model people will generally
depend on the trade-offs they face. Hence, our theory can be used as a device to
make predictions about the way behavior will change in response to any given
change in the trade-offs people face. As a predictive device, the theory can be
tested by comparing its predictions against what is observed in the data. A
theory that is judged (by some measure) to be successful in this regard can then
be used to interpret observed behavior and to make predictions over hypothetical events (including the likely effects of hypothetical changes in government
policies).
1.3.1
Preferences
Consider an economy consisting of a set of people. What do these people care
about? While people care about all sorts of things in reality, I assume here that
their personal welfare depends in a direct manner on only a limited set of goods
and services.
To keep things simple, assume that there are only two types of goods; labeled
6
CHAPTER 1. THE EMPLOYMENT OF NATIONS
consumer goods and home goods. Let c ≥ 0 denote a quantity of consumer goods
and let h ≥ 0 denote a quantity of home goods. Then the pair (c, h) is called
a commodity bundle; and the collection of all such commodity bundles is called
the commodity space; see Figure 2.2.
Figure 2.2
Commodity Space
c
commodity bundles
A
convex combination
of bundles A and B
C
B
0
h
The points depicted in Figure 2.2 are examples of commodity bundles. For
example, bundle A might be something like (c, h) = (10, 5), while bundle B
might be something like (c0 , h0 ) = (3, 10). The bundles lying on the straight
line connecting A and B represent bundles that constitute linear combinations
(weighted averages) of the bundles A and B. A bundle lying on this line, like
bundle C, is a convex combination of A and B.5
Next, assume that people have preferences defined over the commodity space.
That is, assume that people are able to rank every commodity bundle (c, h) in
the commodity space from best to worst. Let U (c, h) denote the numerical rank
attached to bundle (c, h). The function U that performs this ranking is called
a utility function. You will be better served, however, to just think of it as a
ranking function.6
In what follows, I assume that personal preferences can be represented by
5 The line segment connecting A and B is constructed as follows. Consider any fraction 0 ≤
α ≤ 1. Then define a new bundle by constructing cα = αc + (1 − α)c0 and hα = αh + (1 − α)h0 .
By varying α, we can vary the position of point C along the straight line connecting A and B
in Figure 2.2.
6 That is, the actual level of U is, in itself, a meaningless number. The only relevant aspect
of U is in its ability to rank alternatives; i.e., to make relative comparisons.
1.3. A BASIC MODEL ECONOMY
7
an “additively separable” utility function in the form,
U (c, h) = u(c) + v(h)
(1.1)
where u and v are increasing and concave functions. Let me explain what this
means.
First, the fact that u and v are increasing implies that the utility function
U is increasing in both c and h. In words, this expresses the plausible idea that
people prefer more to less. Mathematically, this can be expressed as follows,
u0 (c) > 0 and v 0 (h) > 0
(1.2)
where u0 (c) = ∂U (c, h)/∂c and v 0 (h) = ∂U (c, h)/∂h.
Note that u0 measures the slope of U in the c direction, while v 0 measures the
slope of U in the h direction. These slopes are referred to as marginal utilities.
Intuitively, marginal utility measures the change in utility U brought about by
a (small) change in either c or h. Hence, the restriction that u0 > 0 and v 0 > 0
implies that an increase in either c or h will necessarily generate a higher utility
payoff (i.e., the new commodity bundle will be accorded a higher rank).
In what follows, I also impose the following two conditions (called Inada
conditions): u0 (0) = ∞ and v 0 (0) = ∞. In words, this expresses the idea that
people attach an infinite value (marginal utility) to consumer and home goods
when these goods are close to zero levels. A starving man will do anything for
a crumb of bread, so to speak.
Second, the fact that u and v are concave implies that the utility function
U is concave in c and h. In words, this expresses the plausible idea that while
people may prefer more to less, the value they attach to (say) one extra beer
(the marginal utility) is decreasing in the amount of beer they plan to consume.
Formally, this concept can be expressed as follows,
u00 (c) < 0 and v 00 (h) < 0
(1.3)
where u00 (c) = ∂ 2 U (c, h)/∂c2 and v 00 (h) = ∂ 2 U (c, h)/∂h2 .
Concavity of the utility function implies the property of diminishing marginal
utility. That is, the slopes u0 and v 0 are decreasing in c and h, respectively.
Finally, note that because U is additively separable in c and h, it follows
that ∂ 2 U (c, h)/∂c∂h = 0. In words, this means that the marginal utility of c
does not depend on the level of h; and vice-versa. This may not be the most
natural restriction to impose for all applications, but I impose it here primarily
because it will let me express the most important ideas with the least amount
of clutter.
Exercise 2.1 Let u(c) = ln(c) and v(h) = λ ln(h) with λ > 0; so that U (c, h) =
ln(c)+ λ ln(h). Demonstrate that this utility function satisfies the properties
described above. Provide an economic interpretation for the parameter λ.
8
CHAPTER 1. THE EMPLOYMENT OF NATIONS
For the purpose of describing some ideas diagrammatically, it will be useful
to make reference to indifference curves. An indifference curve is simply a
combination of commodity bundles (c, h) that deliver the same utility payoff
(so that the person is indifferent between any such bundle). Indifference curves
are a useful analytical device because they demonstrate the important idea that
people are generally willing to substitute across different commodities.
Consider, for example, an arbitrary rank (utility) number R. Then for the
utility function (1.1), an indifference curve constitutes all bundles (c, h) that
satisfy,
R = u(c) + v(h)
(1.4)
Since u0 > 0 and v 0 > 0, it follows that higher levels of h are associated with
lower levels of c on any given indifference curve.
It is useful to think of condition (1.4) as implicitly defining the level of
consumption that would deliver a given rank R for a given level of h. In other
words, the indifference curve is a function c = I(h, R) that implicitly satisfies,
u (I(h, R)) + v(h) − R ≡ 0
(1.5)
Exercise 2.2 Derive the indifference curve c = I(h, R) for the utility function
U (c, h) = ln(c) + λ ln(h). Demonstrate that ∂I/∂h < 0 and ∂ 2 I/∂h2 > 0.
Exercise 2.2 reveals that the indifference curves for a particular utility function are decreasing and convex. In fact, this can be shown to be true for any
utility function where u and v satisfy the restrictions imposed above. In particular, note that condition (1.5) implies,
∙ 0
¸
∂I
v (h)
=− 0
< 0.
∂h
u (c)
(1.6)
The derivative ∂I/∂h measures the slope of an indifference curve. It is
useful to remember that the slope of an indifference curve is simply the ratio of
marginal utilities in (1.2). The absolute value of this slope is called the marginal
rate of substitution; or MRS for short. The economic interpretation of the MRS
is that it measures the relative value that a person attaches to goods c and h at
a particular point in the commodity space. Alternatively, one can think of the
MRS as the rate at which a person is willing to substitute one good for another
(in a manner that leaves them with roughly the same utility).
1.3. A BASIC MODEL ECONOMY
9
Figure 2.3
Indifference Curves
c
Note:
[1] RH > RL
[2] U(C) > U(A) = U(B)
[3] MRS(A) > MRS(C) > MRS(B)
A
convex combination
of bundles A and B
C
B
I(h,RH)
I(h,RL)
h
0
Figure 2.4
Homotheticity
c
0
h
10
CHAPTER 1. THE EMPLOYMENT OF NATIONS
The indifference curves associated with a concave utility function are convex
functions; that is, ∂ 2 I/∂h2 > 0. Hence, their general shape is as depicted in
Figure 2.3. Point A depicts a commodity bundle that contains a high level
of consumer goods and a low level of home goods; while point B depicts the
opposite case. Intuitively, one might expect that at point A, the value a person
attaches to home goods will be high relative to consumer goods (reflecting the
relative scarcity of home goods). This same intuition would suggest that the
opposite holds true at point B. Indeed, these intuitive statements are reflected
by the fact that MRS(A) > MRS(B).
In Figure 2.3, the commodity bundle C is a weighted average (convex combination) of bundles A and B. Hence, concavity of the utility function (convexity
of the indifference curves) implies that people generally prefer “average” bundles to “extreme” bundles. In particular, note that bundle C sits on a higher
indifference curve.
Macroeconomists frequently impose an additional restriction on the utility
function called homotheticity. In terms of an indifference curve diagram, homotheticity implies that slopes of indifference curves are constant along any given
ray from the origin; i.e., see Figure 2.4. Homotheticity has an important implication for how individual behavior translates into aggregate behavior; but I will
reserve discussion on this matter for later.
1.3.2
Constraints
Preferences determine personal objectives and imply a general willingness to
substitute across different commodities to meet these objectives. Unfortunately
(except possibly for the science of economics), the environment people live in
generally places limitations on what can be achieved. Among other things, these
limitations determine the ability to substitute across different commodities. In
short, life is full of constraints. Let me describe now the constraints placed on
our model people.
Each person is endowed with a limited quantity of available time, labeled
a > 0. I do not necessarily assume that this time endowment is the same for all
people. For example, from a life-time perspective, the young have more time
remaining than the old. Similarly, if mortality rates differ across countries, then
even people of the same age may differ in their (expected) amounted of time
remaining. Alternatively, even over shorter time intervals, people may differ in
their physiological need for sleep; so that some people may have more available
time than others.
What can people do with their time? Assume, for simplicity, that there are
only two uses of time, labeled work and leisure. Let n ≥ 0 and l ≥ 0 denote the
time devoted to work and leisure, respectively. Then a person with available
time a faces the time-constraint,
n+l =a
(1.7)
1.3. A BASIC MODEL ECONOMY
11
Condition (1.7) describes the limitations placed on the competing uses of
time. Next, I describe what these time-use categories are good for. In what
follows, I assume that work time is used to produce consumers goods and that
leisure time is used to produce home goods. While there are more general
ways in which to describe exactly how these time-inputs are transformed into
commodities, here I adopt the following simple specification,
c ≤ ωn
h ≤ l
(1.8)
(1.9)
where ω > 0 is a person-specific parameter.
The constraints (1.8) and (1.9) are linear technologies. The parameter ω can
be interpreted as measuring person’s skill in producing consumer goods.7 The
specification in (1.9) assumes that people are equally skilled in producing home
goods. In words, these constraints describe the levels of production (c, h) that
are technologically possible for any given input (n, l) of time.
In what follows, I assume that the home good (leisure) is not transferable
across agents.8 In this case, while h ≤ l is feasible, we can anticipate that
it will not be optimal. That is, technological efficiency will imply that h =
l. Combining this with the time-constraint (1.7) then implies n = a − h. If
we substitute this into (1.8), then the constraints on (c, h) are conveniently
summarized by,
c ≤ ω (a − h) .
(1.10)
Once again, note that while c < ω (a − h) is feasible, we can anticipate
that it will not be optimal. That is, technological efficiency will imply that
c = ω (a − h) . I will refer to the (c, h) bundles that satisfy (1.10) with equality
as the feasible line; see Figure 2.5. In this figure, the feasible set is defined as
the commodity bundles (c, h) that satisfy (1.10). The figure also depicts how
the feasible set depends on the parameters ω and a.
The parameter ω determines a person’s ability to substitute across commodities c and h. An increase in ω is seen to result in an upward rotation of the
feasible line; i.e., a person is now better able to substitute home goods (leisure)
for consumer goods (work). This parameter also has the interpretation of a
relative price; i.e., an increase in ω makes consumer goods (work) cheaper relative to home goods (leisure). Another useful way of thinking about ω is that it
reflects an exchange rate; i.e., the rate at which a person is able (although not
necessarily willing) to exchange home goods for consumer goods.
7 In a competitive market economy, ω would correspond to the real wage paid to labor of
skill-level ω. In this case, (1.8) can also be interpreted as a budget constraint.
8 The plausibility of this restriction is open to question for many types of home-produced
goods. On the other hand, think of leisure time spent in the production of a Caribbean
vacation. How is one to enjoy a vacation without employing one’s own leisure time?
12
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Figure 2.5
Feasible Commodity Bundles
c
An increase in ω
ωa
Feasible Line: c = ω(a - h)
A decrease in a
Feasible Set
a
0
h
The parameter a can be usefully interpreted as measuring a person’s wealth.
In Figure 2.5, a reduction in a is seen to result in a parallel leftward shift
in the feasible line, leading to a reduction in the set of feasible commodity
bundles. Note that because the shift is parallel, the ability to substitute remains
unchanged.
1.4
Determining Individual Behavior
1.4.1
Formalizing a Person’s Choice Problem
To an economist, it seems natural to suppose that human behavior is governed
by the following basic principle: People try to do the best they can in fulfilling
their objectives, given their circumstances. If we apply this principle to our
model economy, it implies that a person will try to make choices (c, h) in a
manner that generates the highest utility rank U (c, h) possible, subject to the
constraints that are imposed on these choices by condition (1.10). Formally, this
choice problem can be expressed mathematically as the following constrained
optimization problem,
W (θ) ≡ max {u(c) + v(h) : c ≤ ω(a − h)}
c,h
(1.11)
1.4. DETERMINING INDIVIDUAL BEHAVIOR
13
where θ denotes a list of parameters that describe an individual’s personal characteristics (like ω, a and any parameters describing the functions u and v).
When you see something like (1.11), you should read it as follows. Consider
a person with attributes θ. The problem they wish to solve entails a choice c
and h that maximizes utility u(c) + v(h), subject to the constraint c ≤ ω(a − h).
The optimal choice of c and h (the solution) generates a maximum utility rank
W (θ).
The function W (θ) is called an indirect utility function.9 That is, while
utility U (c, h) depends directly on (c, h); the maximum utility attainable will
depend indirectly on the parameters θ that describe an individuals’ personal
circumstances. I use W to denote the indirect utility function because it is
usefully interpreted as an individual welfare function.
1.4.2
Characterizing the Solution
Having formulated the choice problem, we need some way to describe the properties of the solution. One way to do this is to employ simple calculus. Let me
now describe the procedure involved in this approach.
An educated guess tells us that an optimal choice will entail c = ω(a − h);
that is, the solution will be technologically efficient (why waste time producing
consumer goods, if they are not to be consumed?).10 Alternatively, we can write
c = ωn and h = a − n. In this case, utility may be expressed as a function of a
single variable; i.e.,
V (n) ≡ u(ωn) + v(a − n)
(1.12)
and the choice problem can be recast as,
W (θ) ≡ max V (n)
n
(1.13)
In doing so, we have transformed the original constrained optimization problem
into an unconstrained maximization problem.11
Hence, the choice problem basically boils down to an appropriate choice of
n. Once n is determined, the remaining choice variables are determined by the
constraints c = ω and h = a − n (with l = h).
So how is the choice of n determined? The utility function (1.12) gives us a
hint. In particular, note that an increase in n confers a utility benefit in that
more work time translates into a higher income ωn; which can be used to acquire
more consumer goods. On the other hand, there is a utility cost associated with
9 In
mathematics, it is called a maximum value function.
that in this model, consumer goods cannot be used to purchase home goods
(home goods are non-transferable). Nor is there any scope in this model for saving; a subject
that I reserve for a later chapter.
1 1 Of course, I am assuming here that the corner constraints do not bind; so that the solution
falls in the range 0 < n < a.
1 0 Remember
14
CHAPTER 1. THE EMPLOYMENT OF NATIONS
increasing work time. In particular, more work time necessarily implies less
time available for the production of home goods. How might a person deal with
this apparent trade-off?
To answer this question, it is useful to differentiate the utility function (1.12)
twice with respect to n; i.e.,
V 0 (n) = ωu0 (ωn) − v 0 (a − n) ≷ 0
V 00 (n) = ω 2 u00 (ωn) + v 00 (a − n) < 0
(1.14)
(1.15)
We can infer from the Inada conditions I imposed earlier that V 0 (n) > 0 when
n = a (since v 0 (0) = ∞) and that V 0 (n) < 0 when n = 0 (since u0 (0) = ∞).
Condition (1.15) reveals that V is concave in n. Together then, this information
suggests that (1.12) is in the shape of a “hill;” see Figure 2.6.
A person’s choice problem then can be thought of choosing actions that
get him or her to the top of their “utility hill.” The solution to this problem,
for a person endowed with attributes θ, is denoted in Figure 2.6 as n(θ).12 The
solution is written in this manner to emphasize the fact that the optimal amount
work time will generally depend on the parameters θ. Or, to put things another
way, the height and positioning of the peak of the “utility hill” in Figure 2.6
will in general depend on θ.
Figure 2.6
Maximizing Utility
V
W(θ)
V(n0 )
V’(n(θ)) = 0
V’(n0 ) > 0
V(n)
0
n0
n(θ)
n
Given the concavity of the function V (n), there is one—and only one—peak;
1 2 This
is bad notation as I am now using h to denote both a real number and a function;
but I do this to economize on excessive notation.
1.4. DETERMINING INDIVIDUAL BEHAVIOR
15
and it occurs precisely at n(θ). The slope of the utility hill at its peak is zero;
i.e., V 0 = 0. This, together with (1.14), implies that the solution to this problem
is given by a value of n that satisfies,
ωu0 (ωn) = v 0 (a − n)
(1.16)
The left-hand-side of (1.16) measures the utility benefit of allocating one
(small) unit of time to work; hence, this measures the marginal utility benefit of
work. To put things another way, increasing work time by one unit generates ω
units of additional consumer goods; and these goods are valued by the marginal
utility u0 (c). Similarly, the right-hand-side of (1.16) measures the marginal utility
cost of work. At an optimum, these two margins must be equal.
Figure 2.7
Utility Maximization
c
D
C
A
c(θ)
B
I(h,W )
0
a
h(θ)
h
n(θ)
There is another useful way to characterize this solution to the choice problem. To see this, note that condition (1.16) may also be expressed as,
v 0 (h)
= ω and c = ω(a − h)
u0 (c)
(1.17)
These two conditions represent two equations in the two unknowns (c, h). The
solution to these two equations is also a solution to the choice problem (and
once h is known, n can be determined from the time-constraint).
It should not have escaped your attention that v 0 (h)/u0 (c) is the MRS. Hence,
the first condition in (1.17) states that the solution must be such that the slope
16
CHAPTER 1. THE EMPLOYMENT OF NATIONS
of the indifference curve is equal to the slope of the feasible line. The second
condition requires that the solution lie on the feasible line. These two conditions
are satisfied simultaneously only at a point like A in Figure 2.7.
Exercise 2.3 What assumption accounts for the fact that the tangency points A,
B, and C in Figure 2.7 lie on a straight line from the origin?
Exercise 2.4 The commodity bundles described by points B, C and D in Figure 2.7
each satisfy only one of the two conditions in (1.17). For each bundle, explain
which condition is violated.
1.4.3
Properties of the Solution
At this point, you might be forgiven for feeling a modest sense of accomplishment. In particular, out of the infinite points in the commodity space, you have
(with the help of our theory) narrowed things down to a specific prediction; i.e.,
point A in Figure 2.7. But this is not all. The theory also predicts how individual behavior will depend on a person’s personal characteristics (ω, a). Moreover,
it provides us with a framework that helps us explain why this should be the
case.
Experiment 1: An Increase in the Time Endowment
There are two ways to think of this experiment. One way is to imagine that
a person has an endowment of time a and that it has suddenly (for reasons
beyond his control) changed to a0 > a. As I alluded to earlier, this might be
interpreted as the arrival of a new drug that lowers mortality rates or the time
necessary for sleep. The second is to imagine that we are comparing two people;
one of which has a and the other a0 (and who are otherwise identical). Again,
we might interpret this as two people with identical skills living in two separate
economies with different mortality rates.13
To begin, consider condition (1.16), which characterizes the work choice
n(ω, a). How does this choice depend on the parameter a? A simple way to find
out how a “small” change in a influences the work choice is by using the implicit
function theorem to derive expression,
¸
∙
∂n
v 00
>0
(1.18)
= − 2 00
∂a
ω u + v 00
This is perhaps not a surprising result, as it tells us that a person with
more time on their hands is likely to spend more time working. On the other
hand, the result is not something that should have been considered a foregone
1 3 Alternatively,
think of two people with identical skills but different in age.
1.4. DETERMINING INDIVIDUAL BEHAVIOR
17
conclusion. Who is to say, for example, that the additional time might not have
been spent entirely in the form of additional leisure?14
In fact, the result in (1.18) reveals that 0 < ∂n/∂a < 1; which is to say that
additional time will be divided in some manner between additional work and
and additional leisure. By appealing to the constraints c = ωn and h = (a − n),
it then follows that δc/∂a = ω∂n/∂a > 0 and ∂h/∂a = 1 − ∂n/∂a > 0. That is,
the additional time ultimately manifests itself in the form of additional consumer
and home goods. And while it seems obvious that this must entail an increase in
welfare, the result can be demonstrated formally by appealing to (1.13), together
with an application of the envelope theorem,
∂W
= v0 > 0
∂a
(1.19)
The intuition here can be developed further by examining the effect of an
increase in a in terms of a diagram like Figure 2.7. That is, imagine that point
A in Figure 2.7 describes some initial condition; and that a is then increased to
a0 . The result is depicted in Figure 2.8.
Figure 2.8
Pure Wealth Effect
c
Δc
cB
cA
0
Note:
Δh < Δa
implies
Δn > 0
B
A
hB
Δh
a
a’
h
Δa
I mentioned earlier that a might be interpreted as measuring a person’s
wealth (measured in units of time). Note that in this experiment, the exchange
rate between consumer and home goods (ω) remains unchanged. That is, the
incentive to substitute across commodities remains the same. Therefore, any
1 4 This
is something that would have been true, in particular, if v 00 = 0.
18
CHAPTER 1. THE EMPLOYMENT OF NATIONS
predicted change in behavior must be entirely the product of a change in wealth.
For this reason, the movement from A to B in Figure 2.8 is labeled a pure wealth
effect. I will leave it to the student to confirm that the predicted changes in
behavior in Figure 2.8 are consistent with what has been derived mathematically
above.
There is in fact a simpler way in which to think about the impact of a change
in wealth when relative prices remain unchanged. In particular, just remember
that an increase in wealth increases the demand for all normal goods (this is
the definition of a normal good ). Since the goods (c, h) are normal here (a byproduct of the separability and concavity of the utility function), it follows that
an increase in wealth leads to an increase in the demand for both c and h.
In the experiment just considered, the level of a is best interpreted as the
amount of time remaining in one’s life (say, measured in years). An increase in
a is seen to lead to an expansion in both work and leisure time in one’s lifetime.
But does the analysis here have anything to say about how work time per year
is affected?
To answer this question, note that homotheticity implies that the ratio c/h
remains invariant to changes in a (see Figure 2.8). In turn, this implies that
the ratio ωn/(a − n) remains invariant to changes in a. If we define the variable
e ≡ n/a (work time per year), then this in turn implies that,
ωea
ωe
=
= constant
(1 − e)a
(1 − e)
In other words, e is also invariant to a.
Experiment 2: An Increase in Skill
To begin, consider again condition (1.16), which characterizes the work choice
n(θ). How does this choice depend on the parameter ω? To find the effect of a
“small” change in ω, apply the implicit function theorem once more to derive,
¸
∙ 0
∂n
u + ω 2 u00
≷0
(1.20)
= − 2 00
∂ω
ω u + v 00
We know that the denominator of this expression is negative. However, the
numerator may be either positive or negative, depending on the curvature properties of u. For example, note that if u is sufficiently linear (so that u00 is close
to zero), then ∂n/∂ω > 0.
The way to interpret this apparent ambiguity in the predicted response of
work to the “real wage rate” ω is that it reflects the effect of two economic
forces operating in opposite directions. These two effects are recorded by the
two terms in the numerator of (1.20).
The first term u0 > 0 reflects a substitution effect brought about by a change
in the relative price of consumer goods vis-à-vis home goods. That is, consumer
1.4. DETERMINING INDIVIDUAL BEHAVIOR
19
goods are now relatively cheaper; and our model person is both willing and
able to substitute out of home goods (leisure) into consumer goods (work). The
second term u00 < 0 reflects a wealth effect; which has already been discussed
in the previous experiment. Note that while while wealth measured in unit of
time (a) has not changed, wealth measured in units of consumer goods (ωa) has
increased. Higher wealth translates into an increase in the demand for all normal
goods. Since h is a normal good, the time constraint implies that n = a − h
must fall (when a is held fixed).
The effect of an increase in ω is depicted in Figure 2.9 for the case in which
the substitution and wealth effects on work time exactly cancel (a movement
from point A to B). Conceptually, the movement from A to B in Figure 2.9
might be decomposed into a movement from A to C (wealth effect) and then
from C to B (substitution effect). The relative strength of these two effects
determine whether B ends up to the left or the right of A.
Exercise 2.5 Prove that regardless of whether an increase in skill level increases
or decreases desired work time, that the desired level of consumer goods rises
unambiguously. Explain why this is so.
Figure 2.9
Substitution and Wealth Effects
c
Δc
Note:
Δh = 0
implies
Δn = 0
B
cB
C
cA
A
a
0
h
An Example
Applied economists frequently impose additional structure on their model economies
by assuming that preferences take on a specific functional form; for example,
U (c, h) = ln(c) + λ ln(h)
(1.21)
20
CHAPTER 1. THE EMPLOYMENT OF NATIONS
where λ > 0 is a preference parameter. An individual’s personal characteristics
are then summarized by the vector θ = (ω, a, λ). The MRS for this functional
form is given by λc/h. In this case, we can use the conditions in (1.17) to derive
an explicit solution; i.e.,
µ
¶
1
c(ω, a, λ) =
ωa
(1.22)
1+λ
µ
¶
λ
h(ω, a, λ) =
a
1+λ
µ
¶
1
n(ω, a, λ) =
a
1+λ
As n and h do not depend on ω here, this is clearly an example of exactly
offsetting substitution and wealth effects; see Figure 2.9.
Exercise 2.6 Let U (c, h) = c1/2 + λh1/2 . Derive the solution analytically and
explain its properties.
1.5
Determining Aggregate Behavior
Macroeconomists are concerned primarily with understanding the behavior of
economic aggregates (total or average quantities). To construct these aggregates, we need to add up individual behavior. Since individual behavior depends on personal characteristics, aggregate behavior will depend on how these
personal characteristics are distributed across the population.
1.5.1
Distribution of Personal Characteristics
The people of our model economy potentially differ along many dimensions. We
need some way to describe these differences. This can be done by specifying a
distribution function π(θ); which identifies the fraction of the population with
attributes θ. This distribution satisfies the following two properties,
X
0 ≤ π(θ) ≤ 1 and
π(θ) = 1
(1.23)
θ
1.5.2
Computing Aggregates
Once the distribution π has been specified, computing aggregates is easy—one
just add things up. That is, if type θ people choose an action x(θ), then the
aggregate (or average) value of these actions together is given by,
X
X(π) =
x(θ)π(θ)
θ
1.5. DETERMINING AGGREGATE BEHAVIOR
21
The notation X(π) emphasizes that aggregate behavior will generally depend
on the properties of the distribution function π.
Let me consider a specific example. Assume, for example, that people have
the log-linear utility functions specified in (1.21). In this case, people potentially differ along three dimensions θ = (ω, a, λ) and their predicted behavior
is described by (1.22). Letting capital letters denote aggregates (or per capita
quantities), we have
C(π) =
Xµ
ω,a,λ
1
1+λ
¶
ωaπ(ω, a, λ)
Xµ λ ¶
aπ(ω, a, λ)
H(π) =
1+λ
ω,a,λ
Xµ 1 ¶
aπ(ω, a, λ)
N (π) =
1+λ
ω,a,λ
For some purposes, additional restrictions might be employed. For example, if we were to take the view that people share identical (and homothetic)
utility functions and an identical time endowment, then aggregate behavior is
independent of the manner in which ω is distributed across the population; i.e.,
µ
¶
1
C(ω) =
aω
1+λ
µ
¶
λ
H(ω) =
a
1+λ
µ
¶
1
N (ω) =
a
1+λ
P
where ω ≡
ωπ(ω) denotes the average skill level.15 Hence, there are cases
for which distributional aspects (beyond the mean of a distribution) might be
ignored when studying aggregate behavior.16
In the very special case where π(ω, a, λ) = 1 for a particular (ω, a, λ), people
are exactly identical. When a model economy has this property, it is referred
to as a representative agent economy (that is, the behavior of one person will
be representative of all others). This assumption, while extreme, is frequently
employed for teaching purposes or in applications where distributional aspects
are reasonably expected not to play a major role in understanding a specific
phenomenon.
1 5 H and N here do not depend on ω only because of the offsetting substitution and wealth
effects associated with these log-linear preferences.
1 6 Polemarchakis (1983) provides a more general analysis of this topic.
22
1.5.3
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Measuring Economic Welfare
Ranking Outcomes
In some applications, economists are interested in measuring the welfare impact
of different government policies on individuals and the economy. I will talk
more about government policies later on; but for now, imagine that policies
are indexed by a parameter p and that people differ along the one dimension
ω.Then conditional on a given p, theory predicts behavior c(ω, p) and h(ω, p);
which generates a maximum utility W (ω, p) for a person of type ω.
Now, consider a person faced with an hypothetical choice of living in a world
governed by one of two policy regimes, p or p0 . If W (ω, p) > W (ω, p0 ), then he
prefers the former to the latter; and vice-versa, if W (ω, p) < W (ω, p0 ). This
constitutes a simple way to rank policy regimes, from the perspective of each
person.
While ranking different policy regimes for each person is relatively straightforward, the same cannot be said for ranking different regimes from the perspective of “society.” It is generally the case that any change in policy regime
will result in both winners and losers. How “society” might rank policies when
some gain at the expense of others is not immediately clear. It is not even clear
whether it is in anyway meaningful to think of “society” as having preferences
that are distinct from the preferences of any given person (in particular, the
person in charge of choosing policies).
Despite these philosophical difficulties, economists have devised various ways
to construct what they label social welfare functions; which can be used to rank
different policies from a “social” perspective. There is, however, no unique or
obvious way to construct such functions; so that the choice of any particular
social welfare function leaves the chooser open to the criticism that the choice
simply reflects their own underlying preferences.
With this caveat in mind, let me describe one popular method called the
utilitarian approach. This approach asserts that the different outcomes that
arise from any given policy p can be ranked by a weighted sum of individual
utilities; e.g.,
X
W(p) ≡
W (ω, p)π(ω)
(1.24)
ω
One way to interpret the function W(p) above is that it might possibly reflect
how each person individually would rank a policy p if they did not know their
type ω a priori. That is, imagine that we are all asked to operate under a “veil
of ignorance” in the sense that we do not know a prior i whether we will be born
with high or low skill sets. Imagine, however, that we do know W (ω, p) and
π(ω); that is, the utility payoff associated with skill set ω, and the probability
of being born with this skill set. In this case, (1.24) can be interpreted as each
person’s expected utility as a function of p.
1.5. DETERMINING AGGREGATE BEHAVIOR
23
Quantitative Measures
One drawback with using W (ω, p) as a measure of personal welfare is that it
can only be used to rank outcomes. That is, it can tell us whether a person
prefers p or p0 , but it cannot tell us by how much. For example, it would be
invalid to conclude that if W (ω, p) = 2W (ω, p0 ), then person ω values policy p
twice as much as policy p0 .
The reason for this is because any given set of preferences (a ranking system) can be represented by different utility functions. For example, the utility functions U1 (c, h) = chλ and U2 (c, h) = ln(c) + λ ln(h) each represent
identical preference orderings in the sense that if U1 (c, h) ≷ U1 (c0 , h0 ), then
U2 (c, h) ≷ U2 (c0 , h0 ).
Exercise 2.6 Demonstrate that the utility function U1 (c, h) = chλ implies the
behavior predicted in (1.22).
In this sense, we are free to choose either utility function U1 or U2 for the
purpose of predicting behavior. But it is not generally true that U1 (c, h) =
xU1 (c0 , h0 ) implies U2 (c, h) = xU2 (c0 , h0 ) for any x > 0. So, if we find for example
that U1 (c, h) = 2U1 (c0 , h0 ) and U2 (c, h) = 3U2 (c0 , h0 ), we would have to question
the conclusion that the person values the bundle (c, h) twice as much as (c0 , h0 )
with utility function U1 and three times as much with utility function U2 . Such a
statement is nonsense, as both utility functions represent the same preferences.
There are, however, ways in which meaningful quantitative welfare measures
can be constructed. To show how this might be done, consider three hypothetical
policies p = A, B and C. Let c(p) and h(p) denote a person’s optimal choice
under policy regime p; and let W (p) denote the utility payoff associated with
policy p. Assume that W (C) > W (A) > W (B); so that the situation is as
depicted in Figure 2.10.
Now, let us use policy A as the reference policy. Clearly, the person is made
better off under policy C and worse off under policy B. But by how much?
This might be answered by noting that if we were to augment consumption by
z(C) − c(A) in policy regime A, the person would be indifferent between policy
A and C. Since the value z(C)−c(A) is positive, this constitutes a “consumption
equivalent” measure of the welfare benefit associated with moving from policy
A to policy C. Expressed in terms of a fraction of consumption, we can measure
this welfare benefit by,
x(C) =
∙
¸
z(C) − c(A)
>0
c(A)
24
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Figure 2.10
Welfare Benefit and Cost
Measured in Consumption Equivalent Units
c
x(p) = (z(p) - c(A))/c(A)
C
z(C)
A
c(A)
z(B)
B
h(A)
0
h
We can apply the same procedure for evaluating the welfare benefit of moving
instead to policy B. In this case, the value z(B) − c(A) is negative; i.e., it
constitutes a “consumption equivalent” measure of the welfare cost associated
with moving from policy A to B. Expressed in terms of a fraction, we can
measure this welfare benefit by,
x(B) =
∙
¸
z(B) − c(A)
<0
c(A)
In general, one can calculate the x(p) that solves,
W (p) = u ((1 + x(p))c(A)) + v(h(A))
1.6
(1.25)
Redistributive Policies and Employment
Having described a basic framework for understanding the forces involved in
determining time allocation, I return now to the original question that motivated
the analysis. That is, what accounts for the large discrepancy in employment
patterns observed across countries and across time; see Figure 2.1.
1.6. REDISTRIBUTIVE POLICIES AND EMPLOYMENT
1.6.1
25
Tax Rates on Labor Income Across Countries
Beginning with Prescott (2004), a recent body of literature argues that much of
these observed differences might largely be explained by the observed differences
in tax rates on labor income across countries.
McDaniel (2007, Table 7) provides measures of the average tax rate on labor
income for a subset of the countries in Figure 2.1 (no figures are provided for
Norway). Ranked in terms of their 1970 tax rate, the countries are: [1] Sweden,
[2] Germany, [3] France, [4] Italy, [5] U.S., [6] Canada, [7] Spain, and [8] Japan;
see Figure 2.11.
Figure 2.11
Average Tax Rate on Labor Income
Selected OECD Countries
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1965
1970
1975
1980
1985
1990
1995
2000
In comparing Figures 2.1 and 2.11, it is hard not to suspect tax policies playing at least some role in generating the observed differences in hours worked. In
particular, note the large and persistent differences in tax rates across countries
and across time. Moreover, there appears to be an upward secular trend in tax
rates corresponding to the downward secular trend in hours worked. The tax
rate in some countries appear to have risen much more sharply than in the U.S.
(bold line).
In fact, the argument is a little more subtle than this. In particular, the
policy differences analyzed in the literature entail not just the taxation of labor, but also an associated policy of redistributing the resulting tax revenue
in a particular manner. The effect of these tax-transfer (redistributive) poli-
26
CHAPTER 1. THE EMPLOYMENT OF NATIONS
cies is examined by Rogerson (2008) using a framework very similar to the one
developed in this chapter. I turn now to explaining how this can be done.
1.6.2
Modeling Redistribution Policy
The particular fiscal policy studied here is one in which the government taxes
all labor income at a flat tax rate 0 < τ < 0 and then uses the the resulting
tax revenue to finance a lump-sum transfer of income T to all individuals. This
type of policy is called a negative income tax, or NIT.17 While tax policies in
reality are not nearly so simple, the NIT is a simple way to model the progressive
nature of most tax systems.
I begin by considering the choice problem of a person. I assume that people
are identical in every dimension, except possibly in their skill level ω. Because
many of the steps involved are similar to what I have described before, I will be
more economical in my explanations this time around.
From the perspective of an individual, the policy variables τ and T are
parameters.18 A person living in this world now faces the following budget and
time constraints,
c ≤ (1 − τ )ωn + T and h = a − n
(1.26)
That is, (1 −τ )ωn represents “after-tax labor income” and T represents transfer
income. The important distinction to be made here is that while a person’s tax
bill depends on their income, the transfer they receive does not (it is lump-sum).
Anticipating that the budget constraint will bind, we can substitute the
constraints (1.26) into the utility function as before; so that once again, the
choice problem boils down to choosing an appropriate level of n,
W ≡ max {u((1 − τ )ωn + T ) + v(a − n)}
n
(1.27)
This yields the following first-order condition,
(1 − τ )ωu0 ((1 − τ )ωn + T ) = v 0 (a − n)
(1.28)
which characterizes the solution n(ω, τ , T ). From (1.28), we can derive the following properties of the solution,
¸
∙ 0
∂n
ωu + (1 − τ )ω 2 nu00 )
≷0
= −
∂τ
(1 − τ )2 ω 2 u00 + v 00
¸
∙
∂n
(1 − τ )ωu00
<0
= −
∂T
(1 − τ )2 ω 2 u00 + v 00
1 7 See:
http://en.wikipedia.org/wiki/Negative_income_tax
is, an individual perceives that these variables are beyond his or her personal control,
even if they may, in the end be determined by collective behavior.
1 8 That
1.6. REDISTRIBUTIVE POLICIES AND EMPLOYMENT
27
A change in the tax rate operates much in the same way as a change in
skill (the real wage) ω studied earlier; i.e., there are substitution and wealth
effects that operate in opposite directions in determining optimal work time.
An increase in the size of the lump-sum transfer T is seen to reduce optimal
work time. The force at work here is a pure wealth effect.
Exercise 2.7 Let U (c, h) = ln(c)+λ ln(h). Use condition (1.28) to solve for the
optimal work time n(ω, τ , T ). Describe and explain its properties.
Note that instead of (1.28), one might alternatively characterize the solution
by,
v 0 (h)
= (1 − τ )ω and c = (1 − τ )ω(a − h) + T
u0 (c)
(1.29)
Writing things this way is helpful because we can use it to depict the solution
in a diagram like Figure 2.12.
Figure 2.12
Negative Income Tax
c
Budget Line:
c = (1 - τ)ω(a - h) + T
C
B
A
T
0
a
h
Figure 2.12 presents just one example of how a NIT might affect individual
decision-making. Point A depicts the solution absent any government policy.
Point C depicts what would happen in the case of τ = 0 and T > 0 (a pure
wealth effect). Point B depicts the case for which both τ > 0 and T > 0. In this
hypothetical experiment, work time is seen to decrease (leading to an increase
in the consumption of home goods) and this person enjoys roughly the same
28
CHAPTER 1. THE EMPLOYMENT OF NATIONS
level of consumer goods. Moreover, individual welfare is improved (point B sits
on a higher indifference curve).
There is something “fishy” about the analysis so far. In particular, it suggests
the possibility that everyone might be made better off by simply redistributing
income. Simple intuition suggests that this is unlikely to be the case. So we are
led to ask what is missing from the analysis developed to this point.
The answer is simple. What we have done to this point is consider an arbitrary NIT policy (τ , T ) without bothering to ask whether such a policy is budget
feasible for the government. The next step in the analysis requires that we impose a government budget constraint (GBC). If the government cannot borrow
(a topic we will explore later on), then government budget balance requires,
X
τ
n(ω, τ , T )π(ω) = T
(1.30)
ω
The left-hand-side of (1.30) represents total (per capita) tax revenue; and the
right-hand-side denotes total (per capita) value of transfers.
The government budget constraint reveals that the program parameters τ
and T cannot be chosen independently of each other (this appears to be something that politicians frequently try to ignore). The government must either
pick a T and then determine a budget-balancing τ ; or the government can pick
a τ and use the resulting tax revenue to finance an implied level of T. If we identify τ as an arbitrary policy parameter, then condition (1.30) can be thought of
as one equation in the one unknown, T.
It is worthwhile noting that the introduction of the government budget constraint (1.30) into our model introduces a certain “feedback effect.” In particular, if we fix τ , then an increase in individual work time leads to an increase in
taxes payable, τ n. From an individual’s perspective, the contribution to total
tax revenue is negligible (and the individual is assumed to ignore this effect
in formulating his choice problem). On the other hand, if others are similarly
motivated to increase their work time, then the aggregate impact is not negligible; the government will have more tax revenue T to distribute as a transfer.
The feedback effect I mentioned earlier works through the fact that individual
(and hence aggregate) work time depends on T. In short, aggregate employment depends on T ; and T depends on aggregate employment. In a general
equilibrium, these forces must be consistent with individual maximization and
the government budget constraint.
This then completes the description of our model economy; we are now in a
position to study how individual (and aggregate) behavior responds to different
tax rates τ .
To avoid overwhelming (at least some) students, I begin the investigation by
assuming a representative agent. While this may sound strange when analyzing
the effects of redistributive policies, it turns out that the predictions for aggregate behavior are equivalent to a model with different skill levels and identical
1.6. REDISTRIBUTIVE POLICIES AND EMPLOYMENT
29
homothetic preferences. I will discuss the added implications of different skill
levels later on; but for now, note that the government budget constraint (1.30)
reduces to,
τ n(ω, τ , T ) = T
(1.31)
What we are left with then is one equation (1.28) that describes individual
behavior; and one equation (1.31) that describes government behavior. By combining these two restrictions, we are left with one equation in the one unknown
n(τ ); i.e.,
(1 − τ )ωu0 (ωn) = v 0 (a − n)
(1.32)
This restriction forms our basis for predicting how work time n(τ ) varies with
the tax rate on labor income; i.e.,
∙
¸
ωu0
n (τ ) =
<0
(1 − τ )ω 2 u00 + v 00
0
(1.33)
It follows as a simple corollary that c0 (τ ) < 0 and h0 (τ ) > 0.
Exercise 2.8 Prove that economic welfare W (τ ) is strictly decreasing in τ .
In terms of a diagram, the result is depicted in Figure 2.13 as a movement
from point A to B. Note that point B lies both on the original (dashed) budget
line and on the new (solid) budget line. How do we know this? Well, we know
that optimal behavior implies that c = (1 − τ )τ ωn + T. We also know that
government budget balance requires τ n = T. Combining these two equations
implies that c = ωn; which is the equation describing the original budget line.
How do we know that point B lies to the right of point A and not to the left
of it; for example, as in point C? To prove that this cannot be the case, draw a
budget line through point C and depict the optimal choice. You should be able
to see that this optimal choice cannot be a point like C. Therefore, the only
possibility remaining is that the new point lies to the right of A. Hence, the
tax-transfer policy necessarily reduces work effort and results in a lower level of
welfare.
What is going on here? The way to understand these results is as follows.
The tax on labor operates like a decline in the real wage. This implies two
effects: [1] a substitution away from labor (consumer goods) into leisure (home
goods) and [2] a decrease in the demand for all normal goods, including leisure
(home goods). On the other hand, the lump-sum transfer generates a pure
wealth effect; leading to [3] an increase in the demand for all normal goods,
including leisure (home goods).
30
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Figure 2.13
Negative Income Tax: Representative Agent
c
Budget Line:
c = (1 - τ)ω(a - h) + T
C
A
B
T
0
a
h
What is happening here is that the negative wealth effect implied by [2]
essentially cancels the positive wealth effect implied by [3]; leaving only the
substitution effect implied by [1]. In short, the income subsidy to people (which
is independent of their work effort) and the tax on labor income together combine
to reduce the incentive to work. The model’s basic prediction then is that
we should expect to see relatively low levels of employment in societies with
relatively generous transfer programs (and relatively high tax rates to finance
such programs).
Let me say something briefly about how the results described above change
when we extend the representative agent model to allow for some form of heterogeneity. A simple way to do this is to imagine that the population is divided
in some manner between “high-skill” and “low-skill” people. What we would
discover in this case is that the high-skill people would be net tax contributors
(the subsidy they receive is not enough to compensate for the taxes they pay)
and that the low-skill people would be net tax recipients (the subsidy they receive more than compensates for the taxes they pay).19 The welfare benefits
in this case clearly depend on a person’s type. The high-skill people are made
worse off, and the low-skill are made better off. This outcome is presumably
the goal of a redistributive policy.
In terms of predicting the response of aggregate employment, however, the
results remain unchanged. High skill people end up working about the same, or
1 9 Figure 2.12 depicts what things would look like for low-skill people. The situation would
be reversed for high-skill people.
1.7. TAKING THEORY TO DATA
31
a little less; while low-skill people reduce their work time proportionately more
(the positive wealth effect works much more strongly for them, as they are net
recipients). Overall, the aggregate level of employment declines, just as in the
representative agent model (and precisely for the same economic reasons).
1.7
Taking Theory to Data
In comparing the data in Figure 2.1 and Figure 2.11, the visual impression
is that of a negative correlation between hours worked and the tax rate on
labor income. As this negative relation forms the basic prediction of our model,
this theory can be used to interpret the general pattern displayed in the data.
Countries that have, at a point in time, relatively high labor tax rates (and
correspondingly generous transfer programs) are likely to have relatively low
levels of employment. The explanation offered by our theory is that a high tax
rate and generous transfer program simultaneously exert forces that discourage
people from allocating time to work (encouraging them in home production).
1.7.1
Calibrating the Model
The theory developed here can also be used to make a quantitative assessment
regarding the likely importance of tax policy on employment. To do so, we will
have to assume a functional form for the utility function. If u(c) = ln(c) and
v(h) = λ ln(h), then we can use condition (1.32) to solve for,
∙
¸
1−τ
n(τ ) =
(1.34)
λ+1−τ
The next step is to identify a plausible numerical value for the preference
parameter λ. The approach I take here is to “calibrate” condition (1.34) so
that it exactly matches the behavior of a benchmark economy in a benchmark
year. As a benchmark economy, I choose the United States. As a benchmark
year, I choose 1985 (roughly the mid-point of the sample in Figure 2.1). For
this benchmark, we have n = 0.1414 and τ = 0.209. I use these observations,
together with (1.34), to estimate λ; i.e.,
∙
¸
1 − (0.209)
0.1414 =
λ + 1 − (0.209)
so that λ = 4.803.
In this manner, our model’s quantitative prediction is given by,
∙
¸
1−τ
n(τ ) =
4.803 + 1 − τ
I plot this prediction in Figure 2.14 (solid line).
32
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Figure 2.14
Actual and Predicted Hours Worked
Hours Worked / Total Time
0.2
0.18
0.16
0.14
0.12
Predicted
0.1
0.08
0.06
Actual
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
Tax Rate
Next, I construct a sample of the data as follows. Consider the eight countries
plotted in Figure 2.11 (Canada, France, Germany, Italy, Japan, Spain, and
Sweden). Consider two years, one at each extreme of the sample (1970 and
1995). For each of these two years, match the hours worked observation with
the corresponding tax rate for each country in both years.20 The result is the
scatterplot of points depicted in Figure 2.14.
This simple exercise that our simple calibrated model does a reasonably good
job of matching the data quantitatively. Of course, we should not to get too
carried away with this apparent success. In particular, note that the model does
not match the data perfectly. Evidently, there is a good deal of variation in the
data that is left unexplained by our hypothesis.
Although it is likely that many other factors help determine employment,
our calibrated model can be used to quantify the effect of redistributive tax
policy, taking as fixed these other factors. Our model roughly predicts a two
percentage point decline in hours worked (about 3.3 weeks per year) for every
ten percentage point increase in the labor tax rate.
2 0 The observation for Spain in 1970 is omitted as the hours worked for that year are not
available in the data set used here.
1.7. TAKING THEORY TO DATA
1.7.2
33
Quantifying the Welfare Effects of Tax Policy
We already know from Exercise 2.8 and Figure 2.13 that welfare is decreasing
in the tax rate for a representative agent. And while a representative agent
framework ignores the potential benefits of a redistributive policy (something
that I will address in short order), it is nevertheless instructive to see how
we might go about quantifying the welfare cost using the analysis surrounding
Figure 2.10.
An allocation in this economy is described by the pair c(τ ) and h(τ ); in
particular,
µ
¶
1−τ
c(τ ) = ω
4.803 + 1 − τ
µ
¶
4.803
h(τ ) =
4.803 + 1 − τ
As the value of ω will play no role in what is to follow, I simply normalize the
value to ω = 1. This generates the indirect utility function,
¶
µ
¶
µ
4.803
1−τ
+ 4.803 ln
W (τ ) = ln
4.803 + 1 − τ
4.803 + 1 − τ
Now, let me take as a benchmark the “laissez-faire” regime τ = 0 (this is
also the optimal policy, from the perspective of a representative agent). I now
use (1.25) to compute the x(τ ) that solves,
¶¶
µ
¶
µ
µ
4.803
1
+ 4.803 ln
W (τ ) = ln (1 + x)
4.803 + 1
4.803 + 1
The resulting value x(τ ) measures the fraction by which we would have to
increase the representative agent’s consumption in the laissez-faire regime to
make him indifferent between remaining there or moving to a world with policy
regime τ . The result of this exercise is reported in Table 2.3 for five different
tax rates.
Table 2.3
Welfare Benefit
Measured as a Fraction of Consumption
τ
0.10
0.20
0.30
0.40
0.50
x(τ ) −0.0045 −0.0194 −0.0475 −0.0910 −0.1566
Not surprisingly, the values for x(τ ) in Table 2.3 are negative; which is
to say that the welfare benefits associated with a progressively generous taxtransfer program are negative for our representative agent. For a tax of 10%,
the welfare cost is 0.45% measured in terms of consumption. For a tax of 50%,
the welfare cost is 15.66% measured in terms of consumption. The calibrated
model suggests that the welfare cost associated for a typical tax rate of 30% is
almost 5%; which is fairly significant.
34
CHAPTER 1. THE EMPLOYMENT OF NATIONS
1.8
A Case for Redistribution?
Those of you who live on the “left-wing” of the planet are not likely to be pleased
with the conclusion associated with Table 2.3. There is, however, a method by
which to satisfy your interventionist inclinations. Let me describe how.
1.8.1
The Distribution of After-Tax Income
In reality, people are born with different skill sets. We can model this heterogeneity in a simple way by assuming that there are three different skill levels
ω ∈ {ω 1 , ω 2 , ω 3 } , with ω 1 < ω 2 < ω 3 . Imagine that each person has an equal
probability of being born with any one of these skill levels; so that π(ω) = 1/3
for each ω. I assume that ω 1 = 75, ω 2 = 100, ω 3 = 150. That is, I normalize
the median skill level to 100 and assume that low-skill people are 25% less productive than the median, while high-skill people are 50% more productive than
the median.
Exercise 2.9 For the parameters used above, demonstrate that the average skill
level is lower than the median skill level.
Now, consider a given redistribution policy (τ , T ). From Exercise 2.7, we
know that this induces behavior,
n(ω, τ , T ) =
∙
(1 − τ )ω − λT
(1 − τ )ω(1 + λ)
¸
(1.35)
In what follows, I assume that λ = 4.803. From this information, we can calculate each person’s consumption (after-tax income), c(ω, τ , T ) = (1−τ )ωn(ω, τ , T )+
T.
For a given value of τ , the general equilibrium allocation must also satisfy
the government budget constraint (1.30); i.e.,
(1/3)τ ω 1 n(ω 1 , τ , T ) + (1/3)τ ω 2 n(ω 2 , τ , T ) + (1/3)τ ω 3 n(ω 3 , τ , T ) = T (1.36)
If we substitute (1.35) into (1.36), we are left with one equation in the one
unknown, T ; which we can solve for any arbitrary tax rate τ . Having solved
for this T (τ ), we can plug it back into (1.35) to calculate the hours work for a
person of type ω. Similarly, we can compute each person’s after-tax income for
any given tax policy τ .
In Figure 2.15, I plot the after-tax income for each type (low, medium, and
high skill) and average income relative to the average income associated with
the laissez-faire regime (τ = 0).
1.8. A CASE FOR REDISTRIBUTION?
35
Figure 2.15
After-tax Income as a Function of the Tax Rate
1.6
After-tax Income
1.4
1.2
Low
Med
1
0.8
High
Ave
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
Tax Rate
Figure 2.15 shows that average income (employment) is a decreasing function
of the tax rate. This occurs for precisely the same reason highlighted in our
representative agent model. On the other hand, note that in this heterogeneous
agent model, the dispersion in after-tax incomes (consumption) is decreasing
in the tax rate. That is, a progressively more generous redistribution policy
has the expected effect of making after-tax incomes more equal (although all
incomes are at lower levels).
1.8.2
Socially Optimal Tax Policy
Any given tax policy τ results in a particular allocation of resources and an
associated utility payoff for each type of person, W (ω, τ ). If people do not know
beforehand which skill set they will be born into, then one might reasonably
argue that the appropriate social welfare function is given by (1.24); i.e,
W(τ ) = (1/3)W (ω 1 , τ ) + (1/3)W (ω 2 , τ ) + (1/3)W (ω 3 , τ )
By a “socially optimal” tax policy, I mean a tax policy that maximizes W(τ ).
Note that it is by no means clear that the optimal policy here will be one in
which τ = 0. In particular, the “risk-aversion” implied by the strict concavity
of each person’s utility function suggests that people may very well like (from
an ex ante perspective) to have their consumption “smoothed” across different
36
CHAPTER 1. THE EMPLOYMENT OF NATIONS
states of the world.21 Of course, any such benefit must be weighed against
the welfare-reducing distortions caused by the resulting tax (the point that was
highlighted in our representative agent model).
Figure 2.16
Social Welfare as a Function of Tax Rate
Heterogeneous Agents Model
2
1.98
Social Welfare
1.96
1.94
1.92
1.9
1.88
1.86
1.84
0
0.1
0.2
0.3
0.4
0.5
Tax Rate
Figure 2.16 plots the social welfare function W(τ ) for various tax rates. For
this calibrated model, it appears that social welfare achieves a maximum at
τ = 0.09. In other words, according to this model, the “socially optimal” tax
rate is not zero. On the other hand, it does not appear to be very large either;
at least, in comparison to the tax rates observed in reality.
1.8.3
The Political Economy of Redistribution
If the “socially desirable” tax rate is in the order of 10% as our model suggests,
then how do we explain tax rates that commonly much larger than this?
One possible explanation is that our model abstracts from other forms of
government expenditure that may be deemed desirable from a social perspective
(public infrastructure, military defense spending, etc.). On the other hand, our
model also abstracts from other sources of tax revenue (consumption taxes,
capital income taxes, etc.).
2 1 That is, ex ante, people would like to insure themselves against the possibility of being
born with a low skill set. Of course, what people might hypothetically agree on ex ante need
not correspond with what they would like ex post.
1.9. SUMMARY
37
Utility Payoff
Figure 2.17
Optimal Tax Rates Across Types
2.5
2.4
2.3
2.2
2.1
2
1.9
Low
Med
High
1.8
1.7
1.6
1.5
0
0.1
0.2
0.3
0.4
0.5
Tax Rate
But another possible explanation is to be found in the relative political
influence of different groups in society. In Figure 2.17, I plot the indirect utility
function W (ω, τ ) for each skill type across different tax rates. Each group in
this society has their own preferred tax rate. For the high-skill people, the
optimal tax rate is (not surprisingly) τ = 0. For the medium-skill people, the
optimal tax rate is τ = 0.09; and for the low-skill people, the optimal tax rate
is τ = 0.33. This suggests that different political structures are likely to result
in different “optimal” tax rates.
1.9
Summary
People must decide how to allocate their time across competing uses. An important time-use category is time devoted toward the production of consumer
goods and services; the general availability of which helps determine our material living standards.
Economists interpret observed time-use patterns as the outcome of individuallyrational choices. That is, behavior is interpreted as the outcome of people doing
the best they can (according to their own preferences) and subject to the constraints that are placed on their decision-making. Economic theory predicts
that individual (and hence, aggregate) behavior will generally respond to any
change in incentives or constraints.
38
CHAPTER 1. THE EMPLOYMENT OF NATIONS
Employment (or hours worked) patterns differ substantially across nations.
One hypothesis explored in this chapter is that these differences might be largely
explained by cross-country differences in tax (redistribution) policies. Theory
predicts that progressively more generous redistribution policies are likely to
result in lower levels of employment. The reason is that transfers of income and
taxes on labor both serve to reduce the private incentive to work.
While redistributive policies are likely to reduce the average level of employment and income in an economy, the theory developed above does not imply that
the optimal tax rate is necessarily equal to zero. That is, a redistributive policy
can be thought of generating benefits similar to that of an insurance policy.
As private insurance markets are not available to the unborn, social insurance
in the form of a redistributive policy can be thought of replacing this missing
market. The level of distribution chosen in any society, however, is likely to be
influenced greatly by the relative political power of its constituent groups.
1.10. PROBLEM SET
1.10
39
Problem Set
1. Consider the representative agent model described in this chapter and assume that preferences are given by the utility function U (c, h) = ln(c) +
λ ln(h). Explain why cross-country differences in labor productivity (ω) is
not likely to account for the observed cross-country differences in employment.
2. Using the same preferences as in Question 1, explain how differences in
the parameter λ may, in principle, be used to explain the observed crosscountry differences in employment. Would such an explanation be plausible in your view? Explain why or why not.
3. Based on what you learned in this chapter, is it legitimate to conclude that
any government policy that reduces per capita income will necessarily be
welfare-reducing? What lessons are to be drawn from this in associating
measures of income with welfare?
4. Figure 2.17 demonstrates that if it was left up to the poor, they would
legislate a tax rate much less than 100%. Do you find this result surprising,
given that the poor are net recipients of tax revenue? Why do they not
prefer a much higher tax on labor earnings? Explain the economic factors
at work here that limit the political decision to tax.
5. Perform a Google search on “The Median Voter Theorem.” Can you identify the “median voter” in Figure 2.17? Which of the three optimal tax
rates in Figure 2.17 would you expect to emerge in a representative democracy?
40
CHAPTER 1. THE EMPLOYMENT OF NATIONS
1.11
References
1. McDaniel, Cara (2007). “Average Tax Rates on Consumption, Investment, Labor and Capital in the OECD 1950-2003,” Manuscript: Arizona
State University.
2. Polemarchakis, Heraklis M. (1983). “Homotheticity and the Aggregation
of Consumer Demands,” The Quarterly Journal of Economics, 98 (2):
363—369.
3. Prescott, Edward (2004). “Why Do Americans Work So Much More than
Europeans?” Federal Reserve Bank of Minneapolis Review, July, 2—13.
4. Rogerson, Richard (2001). “The Employment of Nations—A Primer,” Federal Reserve Bank of Cleveland Economic Review, Q4: 27—50.
5. Rogerson, Richard (2008). “Market Work, Home Work, and Taxes: A
Cross-Country Analysis,” NBER Working Paper 14400.