G. Duranton J. of Economic Behavior Org. 44 2001 295–314 301
In this section, we assume that the income elasticity of leisure demand is zero a
0 Inelasticity of leisure demand : ∂l
t
Z
t
∂Z
t + 1
= ∀
Z
t + 1
. Under assumption a0, the solution to 5 is such that the income effect exactly offsets the
substitution effect, which implies that the choice of the effort is constant and independent of the expected returns of the savings. The dynamic behavior of the capital stock is then
given by K
t + 1
K
t
= 1 − βA1 − L
1−β
. Consider for instance U = lnl
t
+ α lnc
t + 1
. One finds l
t
= L
t
= 11 + α and K
t + 1
K
t
= 1 − βAα1 + α
1−β
. Thus, depending on the value of α, the economy experiences steady negative growth low α, a no-growth
steady-state or steady positive growth high α. See Fig. 1 for an illustration. As a conclusion for this section, note that the inelasticity-of-labor-supply assumption is
very convenient because it amounts to neglecting future events and thus implies a constant demand for leisure. However, this simplification is obtained with an important loss of
generality. Nonetheless, except for Eriksson 1996 in a continuous time framework but in his case labor supply does not affect accumulation or Hahn 1989 for neoclassical
growth models, papers dealing with growth and endogenous labor supply usually adopt this assumption. The fact is that labor supply in these papers is just an ingredient not their
specific object of study, which is either the endogenous fluctuations of the economy or the impact of taxation see Benhabib and Perli, 1994; Jones et al., 1993 or King et al., 1988 for
examples of such treatments. The rest of the paper explores the behavior of our prototype endogenous growth model with endogenous labor supply when the assumption of zero wage
elasticity is relaxed.
3. Substitutability between leisure and consumption
A first justification for this section is that substitutability between leisure and consumption should be considered as a theoretical possibility. It is also empirically relevant for low levels
of income as assumed by traditional labor supply curves. Assume the following: a1 Continuity: lZ
t + 1
is continuous and twice differentiable. a2 Substitutability: ∂lZ
t + 1
∂Z
t + 1
0. a3 Convexity: ∂
2
lZ
t + 1
∂Z
2 t +
1
0. a4 Boundary conditions: lim
Z
t + 1
→
lZ
t + 1
= 1 and lim
Z
t + 1
→∞
lZ
t + 1
= 0.
Assumption a1 requires that the demand for leisure is well defined. Assumption a2 is the main assumption of substitutability between leisure and consumption. Assumption a3
ensures that the labor supply curve must be concave or convex demand for leisure. Note that the boundary conditions in a4 could be relaxed. We just need labor supply to be able
to create positive and negative growth. The use of 0 and 1 will just make the proofs easier. These assumptions are summarized on Fig. 2. They lead to the following lemma.
Lemma 1. Assumptions
a1–a4 define a map L
t + 1
L
t
, K
t
with the following properties: P1 ∂L
t + 1
L
t
, K
t
∂K
t
≥ 0 and ∂L
t + 1
L
t
, K
t
∂L
t
≥ 0.
302 G. Duranton J. of Economic Behavior Org. 44 2001 295–314
Fig. 2. The leisure demand function under substitutability.
P2 ∂
2
L
t + 1
L
t
, K
t
∂L
2 t
≤ 0.
P3 For any K
t
0, L
t + 1
1, K
t
= 1 and ∂L
t + 1
L
t
, K
t
∂L
t
|
L
t
= 1
= 0.
P4 lim
L
t
→
L
t + 1
L
t
, K
t
= −∞ and ∀ε ∈
0, 1, ∃K
t
such as L
t + 1
ε, K
t
= 0.
Proof. From Eq. 7, we obtain directly lZ
t + 1
− L
t
= 0 with Z
t + 1
= β
1 − βA
2
K
t
1 − L
t + 1
1−β
. A direct application of the implicit function theorem yields ∂L
t + 1
∂K
t
= −
∂Z
t + 1
∂K
t
∂Z
t + 1
∂L
t + 1
≥ 0. Another application of the same theorem implies
∂L
t + 1
∂L
t
= ∂lZ
t + 1
∂Z
t + 1
∂Z
t + 1
∂L
t + 1
− 1
≥ 0, which ends the proof of P1. A
further application of the same theorem yields P2. Note that if L
t
= 1, then K
t + 1
= which implies K
t + 2
= 0 and thus Z
t + 2
+ 0. Due to the disutility of labor, this implies
L
t + 1
= 1. Since ∂L
t
∂L
t + 1
= − β
1 − β
2
A
2
K
t
1 − L
t + 1
− β
∂lZ
t + 1
∂Z
t + 1
and since L
t
= 1 implies L
t + 1
= 1 then ∂L
t + 1
L
t
, K
t
∂L
t
|
L
t
= 1
= 0. This proves P3. By a2 and
a4, L
t
can be zero only when Z
t + 1
is infinite. This is the case for any K
t
0 only when L
t + 1
is negative and infinite. By a2 and a4, for any ε between 0 and 1 there is a value of Z
t + 1
such that L
t
Z
t + 1
ε = ε . Note then that K
t
= Z
t + 1
εβ 1 − βA
2
corresponds to L
t + 1
= 0, which proves P4.
Since L
t + 1
is monotonically increasing in L
t
and can take negative values and any positive values below 1, for any K
t
, there is a unique the fixed point LK
t
of L
t + 1
L
t
, K
t
such that LK
t
= L
t + 1
LK
t
, K
t
and L
t
6= 1. This temporary fixed point is the demand
for leisure in t such that the same demand is expected in t + 1. Define also the unique demand for leisure in t if zero leisure demand is expected in t + 1, L
K
t
, which is such that L
t + 1
L K
t
, K
t
= 0. Fig. 3 gives a simple diagrammatic illustration of some typical
situations. For example the CES utility function U
t
= l
1−σ t
1 − σ + γ c
1−σ t +
1
1 − σ with γ
0 and 0 σ 1 satisfies assumptions a1–a4 and consequently P1–P4. Note that this forward-looking dynamic system is strongly non-linear. Hence, there is
no hope of generating a complete analytical solution of our problem. Nonetheless, some results can be derived.
Result 1. Under
a1–a4, there exist two no-growth steady-states.
Proof.
If the economy expects L
t + 1
= 1, P3 implies L
t
= 1. Then, K
t + 1
= 0 and the
economy is trapped in this equilibrium forever. We call this equilibrium with no output
G. Duranton J. of Economic Behavior Org. 44 2001 295–314 303
Fig. 3. Substitutability between leisure and consumption: a a typical case; b growth; c inevitable recession.
the trivial steady-state. We can also define K
∗
such that L
t + 1
L
∗
, K
∗
= L
∗
. The exis- tence and uniqueness of K
∗
directly stem from the intermediate value theorem with a1, a2 and a4. We can check easily that L
∗
, K
∗
is a no-growth steady-state with strictly positive output since i K
t + 1
L
∗
, K
∗
= K
∗
and ii demanding L
∗
is self-fulfilling. From Eq. 4, no other level of leisure demand except for unity can imply a steady-state.
Any level of capital, except for K
∗
, makes the demand L
∗
inconsistent with rational expectations.
304 G. Duranton J. of Economic Behavior Org. 44 2001 295–314
Result 2. Under
a1–a4, there exists at least one equilibrium path with sustained positive growth for a large enough initial K. It is such that the demand for leisure tends to zero
asymptotically .
Proof.
Proof in Appendix A. The argument runs as follows. If L
t
≥ LK
t
, this implies for some t
′
t , L
t
′
L
∗
and thus negative growth until K K
∗
. There is also a neighborhood left of LK
t
, for which the demand for leisure ends up being superior to L
∗
. This implies the same outcome as previously. Note also that L
t
L K
t
implies L
t + 1
0, which is inconsistent. There is also a neighborhood right of L
K
t
such that for some t
′
t , L
t
′
L K
t
′
and thus L
t
′
+ 1
0 which is again inconsistent. Since no point can belong to both neighborhoods, there is at least one equilibrium path with sustained growth between L
and L. Moreover, since L
t
L K
t
, L is driven to zero when K becomes very large. Any equilibrium with growth then implies that leisure time should converge to zero.
Result 3. Under
a1–a4, the equilibrium path with growth is unique.
Proof.
From Lemma 1, consider L
t + 2
L
t + 1
L
t
, K
t
, K
t + 1
L
t
, K
t
. Full differentiation leads to dL
t + 2
dL
t
= ∂L
t + 2
∂L
t + 1
× ∂L
t + 1
∂L
t
+ ∂L
t + 2
∂K
t + 1
× ∂K
t + 1
∂L
t
. Since ∂L
τ + 1
∂L
τ
≥ 0, then dL
t + 2
dL
t
≥ ∂L
t + 2
∂K
t + 1
× ∂K
t + 1
∂L
t
. After replacement, this implies dL
t + 2
dL
t
≥ 1 − L
t + 2
1 − β × K
t + 1
K
t
0. Note further that for K large enough, L
t
β and K
t + 1
K
t
1. This implies dL
t + 2
dL
t
1. Thus, if for any level of capital, two different levels of leisure demand were in equilibrium, the two equilibrium
paths would be diverging. This is impossible since L tends towards 0. Hence, the growth path is unique.
Result 4.
Under a1–a4, there exists at least one equilibrium path with negative growth.
Negative growth is the only outcome when K is less than K
∗
.
Proof. The first part of the result is straightforward: the trivial steady-state is always possible
see the proof of Result 1. Sustained positive growth requires L
t
L
∗
for any t and {L
t
} decreasing. Then, if we reverse the time scale {L
− t
} is increasing and bounded from above.
Then it must converge toward a level lower than L
∗
if it was converging towards L L
∗
, then growth would be impossible in the first stages. From Result 3, the lowest level of
capital consistent with that assertion i.e., LK
t
= L
∗
is K
∗
. By contraposition, K K
∗
implies negative growth.
Corollary 1. The non-trivial steady-state with no growth is unstable
.
Proof.
This corollary stems directly from Results 1 and 4. The most important result of this section is that growth, although possible, is not auto-
matically given. Our assumptions imply multiple equilibria that can be Pareto ranked. If the economy starts at a sufficiently low level of capital, growth will not occur and the trivial
G. Duranton J. of Economic Behavior Org. 44 2001 295–314 305
steady-state is eventually reached. Even above the threshold, the trivial steady-state cannot be ruled out. The trivial steady-state can be interpreted as a poverty trap.
4
The only equilibrium path that involves sustained growth also implies a sustained decrease of leisure demand towards zero. The intuition of the result is the following. The current
generation works hard because it expects the next generation to work very hard as well and, in so doing, to offer the future retired people high yields for their savings. The next
generation has the same beliefs. Since, the amount of accumulated capital is higher, wages are higher. The incentive to substitute leisure for work is even stronger. Thus, labor supply
has to increase over time and it tends to unity.
5
By contrast, when individuals expect low returns for their savings, their demand for leisure is high, so that capital at the next period is scarce and wages are low. This makes
work even less attractive for the next generation, which confirms previous expectations. Labor supply decreases until the trivial steady-state is reached.
6
This coordination failure arises because of the OLG structure where no contract between generations all committing
to a high supply of labor is feasible. The externality in the aggregate production function which enables growth is usually at the root of the inefficiencies in endogenous growth
models. Here, it only plays a very minor role. Even if savers were to receive the full marginal product of their investments, it would only affect Z
t + 1
multiplicatively by a factor 11−β. This would modify K
∗
but would leave Results 1–4 unchanged. When the economy is in the low equilibrium, there is no possibility to escape it in under
laisser-faire. This argument may act as a rationale for government intervention justified only by a Utilitarian and not a Pareto criterion in low income countries, for which the
substitutability assumption might be valid. This intervention may not create too many dis- tortions, since it suffices for the government to intervene for a finite number of periods.
Once the high growth equilibrium path is triggered, no more intervention is required.
Results 1–4 hold under well-behaved utility functions exhibiting substitutability between leisure and consumption.
7
If individuals consume when young as well as when old, results get more complicated. However, it seems natural to focus on the assumption of comple-
4
When the depreciation rate is strictly below 1 or when minimum labor supply is restricted to be strictly positive, the low steady-state implies a strictly positive level of capital. Note also that this poverty trap does not stem from
restrictions on the production function as in Azariadis and Drazen 1990 i.e., an exogenous non-convexity in the returns of the production function. but it results from the structure of tastes, initial conditions and expectations.
Of course, the model can also be “growth-constrained” if the minimum labor supply is high enough to generate growth whatever the expectations or if the depreciation rate is zero.
5
Note that upper-corner solutions for labor supply are impossible because of the infinite marginal utility of leisure as assumed implicitly with the demand function, when leisure demand is close to zero.
6
An interpretation in terms of product multiplicity can be also developed with l figuring some index of con- sumption of goods produced with constant returns such as agricultural products or manufactured goods produced
with a traditional technology.
7
They are also robust to the formation of expectations. The rational expectation assumption might look at bit strong. If one thinks of simple possible rules of thumb, one may supply L
t
such that L
t + 1
= L
t
where one is unable to forecast the changes of economic conditions due to the accumulation of capital. Under this rule of thumb, we
can see that L
t
= LK
t
. Then, we have three different possible situations. If K = K
∗
, then the economy stays at the unstable steady-state. If K K
∗
, then L
t
L
∗
. The economy keeps growing: K
t + 1
K
t
and L
t + 1
L
t
. Captial grows unbounded, whereas leisure tends to zero. If K K
∗
, then L
t
L
∗
. The economy experiences negative growth with labor supply and capital going to zero. Our rule of thumb then yields results with the same
flavor as the model with rational expectations.
306 G. Duranton J. of Economic Behavior Org. 44 2001 295–314
mentarity or multiplicative separability of the two consumptions. As a consequence, in- tertemporal consumption smoothing leads to both consumption levels to be “tied” in the
sense that pessimistic expectations about future returns do not induce people to substitute future consumption for present consumption but consumption for leisure. The essence of
the results then should not be modified by this refinement of the model.
Finally, note that sociological theories of growth often emphasize the importance of culture in the process of development. Cultural values are supposed to govern the attitude
towards work. Here, we simply assume, that the economic returns are higher when one works more. If we accept here the idea that consumption and leisure are substitutes when
people are poor, there exist multiple equilibria. Consequently, poverty may not be the result of cultural values through a direct causality, but a matter of self-fulfilling expectations and
initial conditions. Nonetheless, sociological explanations may still have a part to play since expectations matter. Furthermore, to escape the trivial or low steady-state, Weber 1930
emphasizes some form of irrationality with respect to our specification for utility. In his introduction, he underlines that With Protestantism man is dominated by the making of
money, by acquisition as the ultimate purpose of his life
. [. . . ] Economic acquisition is no longer subordinated to man as the means for the satisfaction of material needs
.
4. Complementarity between leisure and consumption
