The Employment of Nations
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The Employment of Nations
Introduction
• Material living standards depend on the availability of goods and services
• Availability depends on production; production depends on employment
(more precisely: the aggregate labor input)
• What determines how much time (a scarce resource) is allocated to production in an economy?
• In short: what determines the employment of nations?
The Aggregate Labor Input
• Extensive Margin: Number of people employed (employment)
• Intensive Margin: Hours worked per employed person
• A commonly-used measure of the aggregate labor input:
Employment
Hours Worked per Employed Person
Hours per capita = Adult
×
Population
Available Time per Employed Person
Table 2.1
Aggregate Employment Rates
Selected OECD Countries
Canada
United States
Japan
Australia
Denmark
France
Germany
Ireland
Italy
Netherlands
Spain
Sweden
United Kingdom
1965 1970 1975 1980 1985 1990 1995
60.9 61.3 63.4 65.5 66.1 68.9 67.3
62.9 64.5 64.4 66.8 68.5 72.4 73.4
71.5 71.1 69.7 70.3 70.6 72.5 74.4
66.6 68.8 67.5 65.5 64.7 67.6 67.4
72.2 74.3 73.3 73.1 75.3 76.8 74.1
66.4 65.8 65.4 63.5 59.4 60.1 58.9
69.6 68.6 65.7 63.6 62.5 65.8 65.2
64.4 62.0 58.0 57.0 51.7 52.7 55.2
58.8 56.3 55.6 55.7 54.0 55.0 52.5
59.9 57.3 54.5 53.6 53.3 61.5 65.3
59.8 60.1 58.1 50.2 45.4 48.9 46.4
71.9 72.7 76.3 78.7 79.3 79.8 70.4
71.4 70.4 70.8 68.7 65.7 70.7 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
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
1908 1807 1757 1719 1693 1663
Standard Deviation 163.1 172.2 197.8 189.9 185.0 165.6
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
Approach
• Observation: large and persistent differences in employment patterns across
nations
• Question: what accounts for these differences?
• Aggregate behavior is simply the sum of individual behaviors
• Implication: to understand aggregate behavior, we need to understand
what motivates the individual behavior that generates it
• We need a theory of individual decision-making, applicable to the question
at hand
• Let us begin with two self-evident facts:
1. People are endowed with a limited amount of time;
2. People generally have competing uses for this time.
• Implication: people must decide how to allocate scarce time across competing uses (this is called a time-allocation problem)
• Approach: build a model economy, populated by model people, similar
(but by no means identical) to real people along a few selected dimensions
• We need to address a few questions:
1. What motivates our model people? (Specify an objective)
2. What actions are under their control? (Endogenous variables)
3. What objects are beyond their control? (Exogenous variables)
4. How do objectives interact with constraints to determine behavior?
(Prediction)
A Basic Model Economy
• There are two commodities: consumer goods () and home goods ()
• The pair ( ) ∈ R2+ is called a commodity bundle (where R2+ constitutes
the commodity space)
• What motivates a model person? Assume that they derive pleasure from
consuming ( )
• Moreover, assume that they can attach a number ( ) to each ( ) ∈
R2+ that ranks each commodity bundle in the commodity space
• Assume that ( ) is strictly increasing in and ; and strictly concave
in and
0 and
0
2
2
0 and
0
2
2
• Common example: ( ) = ln() + ln(); ≥ 0
• An indifference set is defined to be all combinations of ( ) that are
equally ranked; i.e., that satisfy
( ) = 0
for some 0 ∈ R (note: 0 could be negative)
• Note: given the assumptions made on this indifference set will be an
“indifference curve” that slopes downward in a convex manner from left
to right in the commodity space
• The marginal rate of substitution or MRS is (the absolute value of) the
slope of the indifference curve
• Denote by ( ); measures the relative valuation attached to and
by an individual at the point ( )
• Also: measures an individual’s willingness to substitute across different
commodities
• Additional assumption: homotheticity
Example
( ) = ln() + ln()
• Now set ( ) = 0 so that
ln() + ln() = 0
• This implies an indifference curve
= 0 −
• Slope is
• So MRS is given by
µ ¶
= −0 −−1 = −
µ ¶
( ) ≡
• Note: these preferences are homothetic
Constraints
• Let denote time endowment
• Let denote skill level in producing consumer goods
• Let denote time devoted to producing consumer goods (employment)
• Let denote time devoted to producing home goods (leisure)
• Assume that = (can only consume home goods produced by self)
• Then constraints are given by
+ =
=
• Or, combining...
= −
• Think of this as a budget constraint; where denotes the price of
measured in units of (the real wage)
• Alternatively + = ; where denotes the value of your time
endowment measured in units of
Summing Up
• A person has preferences defined over ( ) ranked by a function ( )
• Assume that ( ) are choice variables
• Therefore, choose the ( ) that delivers the highest rank ( )
• But without violating the budget constraint ≤ −
• In short, the choice problem can be stated as
( ) ≡ max { ( ) : ≤ − }
( ) ≡ max { ( ) : ≤ − }
• The choice variables ( ) are endogenous variables (objects that the theory is designed to predict/explain)
• The parameters ( ) are exogenous variables (objects that are not explained, but which potentially influence behavior)
• The solution to the choice problem is an optimal ( ); this solution will
be a function of parameters ( )
• ( ) is a maximum value function; or indirect utility function (think
of it as a welfare function)
Review
• Model economy consists of people with preferences defined over commodity
bundles ( )
• Preferences are represented by a ranking (or utility) function ( ); e.g.,
( ) = ln() + ln()
• ( ) measures a person’s willingess to substitute across and
• Model people endowed with time and skill
• There are two uses of time: + = (time constraint)
• Assume that = ; so that ≤
• Constraint boils down to ≤ −
• measures a person’s ability to substitute across and (interpret as
real wage; it is the price of leisure measured in units of consumption)
• Now, assume that people try to do the best they can, subject to their
constraints
• In this model, choose ( ) to maximize ( ) subject to ≤ −
• Solution is a pair of functions ∗( ) and ∗( ); which implies
∗( ) = − ∗( )
• Note: solution is a conditional forecast
• That is, this theory has the structure
( ) ⇒ (∗ ∗ ∗)
• We can use the theory to predict and interpret how any exogenous change
in ( ) will affect individual behavior
• We can also use theory to evaluate welfare consequences; ( ) ≡
(∗ ∗)
• Mathematically, solution is characterized by
(∗ ∗) =
∗ = − ∗
• Example 1: ( ) = ln() + ln() implies ( ) = (); so
solution is
µ
¶
1
∗ =
1 + ¶
µ
∗ =
1 + ¶
µ
1
∗
=
1+
• Note: ∗ is increasing in ; while time allocation is invariant to (interpretation: SE and WE stemming from a change in exactly cancel)
• Example 2: ( ) = 212 + 212 implies ( ) = ()12;
so solution is
∗ =
Ã
2
!
1 + 2
Ã
!
1
∗
=
2
1 +
Ã
!
2
∗ =
2
1 +
• Note: ∗ is increasing in ; and now ∗ is increasing in (so that ∗ is
decreasing in )
• For these preferences, SE dominates WE
From Individual to Aggregate Behavior
• Let Π(X
) denote fraction of population with characteristics ( );
Π( ) = 1
so that
• Special Case [Representative Agent Model] is when Π( ) = 1 for
some ( )
• Aggregate labor input is given by
∗ =
X
∗( )Π( )
• For representative agent model, aggregate labor input is simply ∗( )
Interpreting Cross-Country Differences in Employment
• Fact: over long periods of time, aggregate labor input (per capita) is
approximately constant (in relatively advanced economies)
• One interpretation: for permanent wage changes, SE and WE roughly
cancel
• This suggests that cross-country wage/productivity differentials is not likely
the primary source of observed differences in employment patterns
• Another possibility is cross-country differences in ; but not generally considered a plausible explanation
Redistributive Policies
• Ed Prescott argues that cross-country differences in tax/transfer policies
may be important
• In fact, there appear to be large and persistent differences in tax rates
across countries
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
• Simple explanation is that higher labor income tax reduces the return to
working; leading to a substitution away from work into leisure
• In fact, the argument is a little more subtle than this
• In particular, a higher labor income tax is like a reduction in the real wage
• But if SE and WE roughly cancel, not likely to have a big impact on time
allocation choices
• Key lies here: what does the government do with tax revenue?
• Assume that they redistribute it in lump-sum manner (not a bad approximation)
Modeling a Redistribution Policy
• Let 0 ≤ ≤ 1 denote labor income tax rate; and let ≥ 0 denote
lump-sum transfer
• Then an individual’s budget constraint is
≤ (1 − )( − ) +
• The slope of the budget line is now −(1 − )
• That is, an increase in makes the budget line “flatter” (distorts the
relative price of consumption and leisure)
• So an increase in increases the price of consumption relative to leisure
(leisure is now relatively cheaper)
• On the other hand, an increase in is like an increase in wealth
• Choice problem is
max { ( ) : ≤ (1 − )( − ) + }
• Solution is characterized by
(̂ ̂) = (1 − )
̂ = (1 − )( − ̂) +
• Assume that people only differ in ; then solution is a pair of functions
̂( ) and ̂( )
• Example: ( ) = ()
µ
¶
µ
¶µ
¶
1
̂ =
−
+
1+
1 + Ã 1 − !
µ
¶
µ
¶
+
̂ =
1+
1+
(1 − )
Ã
!
µ
¶
µ
¶
1
̂ =
−
1+
1+
(1 − )
• Notice: reduces to what we had before if = = 0
• Theory predicts that increase in generally
— decreases increases decreases
• and that increase in generally
— increases increases decreases
• So an increase in both ( ) appears to have ambiguous impact on ; but
clearly leads to a decrease in
• Think about it this way:
— increase in implies ↓ (SE) and ↑ (negative WE)
— increase in implies ↑ (positive WE)
• Negative and positive WE basically cancel; leaving the SE
• But why should policy increase both and ? Why not just set tax rate
to zero and give everyone a transfer?
• Answer: Because such a policy is not budget feasible for the government
The Government Budget Constraint (GBC)
• In this model economy, the GBC is given by
=
X
̂( )Π()
• So, for exogenous policy this can be solved for the budget-balancing
tax rate ̂
• Or, for exogenous policy this can be solved for the budget-balancing
lump-sum transfer ̂
A Representative Agent Economy
• For a given theory delivers the following restrictions
(̂ ̂) = (1 − )
̂ = (1 − )( − ̂) + ̂
̂ = ( − ̂)
(1)
(2)
(3)
• Hence, 3 equations in the 3 unknowns (̂ ̂ ̂ ) as a function of parameters
(in particular, )
• And, of course, once we know ̂ we can deduce ̂
• Condition (2) asserts that (̂ ̂) lies on the budget line ̂ = (1 − )( −
̂) + ̂
• If we combine (2) and (3), we get ̂ = ( − ̂)
— this implies that (̂ ̂) also lies on the feasible line
• Now, since (̂ ̂) = (1 − ) = (∗ ∗) it follows that
̂ ∗ and ̂ ∗ for any 0
• It follows that ̂ is decreasing in ; which is broadly consistent with the
data
• For a RA economy, economic welfare is also decreasing in
A Heterogeneous Agent Economy
• Assume that people differ according to only; distributed according to
Π()
• Now, for any given we can compute an equilibrium allocation ̂( ) ̂( )
for each
— we can also compute individual welfare ( ) ≡ (̂( ) ̂( ))
• Generally speaking, low types welfare is increasing in ; while high
types welfare is decreasing in (redistribution)
• The redistribution policy modeled here is called a negative income tax
(NIT)
• People pay tax and receive transfer ; so that their net tax obligation
is −
• For high types, − 0 (net contributors)
• For low types, − 0 (net recipients)
• Increasing (and hence ) has the effect of increasing inequality in gross
earnings
• Why?
• Because ↑ leads low-skill to reduce labor supply more than high-skill
• Why?
• Because high-skill are net contributors (negative wealth effect implies greater
willingness to work)
• Because low-skill are net recipients (positive wealth effect implies they can
afford more leisure)
• On the other hand, increasing (and hence ) has the effect of reducing
inequality in after-tax earnings (1 − ) +
Socially Optimal Tax Policy
• The answer to this question depends on what one assumes is a reasonable
measure of social welfare
• A case could be made for the Rawlsian veil of ignorance
W=
X
( )Π()
• If government objective is to maximize W then generally speaking, optimal
tax rate is positive
• Essentially, redistributive program insures risk-averse agents against the
“bad luck” of being born with low skill
Taking the Theory to the Data
• Assume ( ) = ln() + ln() and identical across people and
countries (representative agent model)
• This implies
∙
1−
̂( ) =
+1−
¸
• Calibrate this to the U.S. in 1985; where = 01414 and = 0209
• This implies
∙
1 − 0209
01414 =
+ 1 − 0209
¸
• Solve for = 4803
• Now, we can use our “estimated” model’s prediction
∙
1−
̂( ) =
4803 + 1 −
¸
• Data is for Canada, France, Germany, Italy, Japan, Spain and Sweden;
each in the years 1970 and 1995
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
Tax Rate
0.4
0.5
Conclusions
• In the context of the theory developed here, cross-country differences in
employment not likely the consequence of cross-country differences in productivity
• Cross-country differences in the “generosity” of redistributive programs
may be quantitatively important (but clearly not the entire story)
• Higher labor-income tax rates may induce lower levels of employment and
per capita income (GDP), but this does not necessarily imply that lower
GDP is associated with lower “social” welfare
⇒ do not confuse GDP or employment with social welfare
