Su¢ ciency Conditions and the Existence of Equilibria in the
Two-Sector Endogenous Growth Model with Leisure
Michael Ben-Gad
University of Haifa
February 2007
Preliminary and Very Incomplete
Abstract
This paper examines the existence and nature of equilibria and balanced growth paths in the
Uzawa-Lucas two-sector endogenous growth model with elastic labor supply. We demonstrate
that it is not possible to merely insert a labor-leisure choice into the model, as has occasionally
been done in in the past— the resulting …rst order conditions for the representative agent violate
the Pontyagrin maximum principle. We also explore di¤erent alterations to the model that
introduce enough concavity to the initial optimization problem to satisfy the Arrow second
order su¢ ciency conditions for optimization. Finally, we consider under what circumstances the
altered models generate multiple balanced growth paths, and whether the equilibria are unique
or indeterminate.
JEL Classi…cation Numbers: E23, J24, O41.
Key Words: Endogenous Growth, Leisure, Su¢ cient Conditions.
1
Introduction
For most of the last two decades, macroeconomists have treated long-run aggregate output growth,
and the cyclical component of aggregate output, as disinct phenomena; building models that analyze
each seperately. What uni…es these two parts of the literature is their reliance on di¤erent extensions
of the same basic underlying model— the optimal growth model …rst developed by Ramsey (1928)
and later by Cass (1965) and Koopmans (1965).
By introducing technonology shocks and a labor-leisure choice into the decision process of the
representative agent, Kydland and Prescott (1982) demonstrated that the Ramsey growth model
Department of Economics, Mt. Carmel, Haifa 31905, Israel, [email protected]. I thank Russell Cooper
for comments and advice.
1
could be a useful tool for understanding business cycles. Subsequant re…nements have produced a
large set of models designed to produce arti…cial time series that mimic the cyclical component of
GDP.
Similarly work by Romer (1986) and Lucas (1988) adapted the same growth model to include
human capital, endogenizing the rate of technological change. Lucas’model of endogenous growth
is essentially the result of combining the Ramsey growth model with the Uzawa (1961) two sector
model of growth, where the second sector produces human capital as an input in the production of
consumer goods and physical capital.
The practice of modeling cyclical output within models that explicitly abstract from longrun growth has generated some criticism. Cogley and Nason (1995) demonstrate that because
traditional real business cycle models have weak endogenous propagation mechanisms, they have
trouble replicating observed autocorrelations, or impulse responses not incorporated within the
dynamics of the impulses themselves. These models also fail to mimic the shape of the spectrum
of output— they neither capture its low frequency properties or generate a peak in its spectrum at
business cycle frequencies. Introducing endogenous growth and human capital accumulation into
a one sector model, Collard (1999) is able to generate a non-zero-valued spectrum for output at
the zero frequency. The model produces output series with positive serial correlation, and unlike
models with exogenous growth, Collard’s model can produce the hump-shaped pattern of impulse
responses to transitory shocks also generated by VAR estimates of the U.S. economy.
These results have motivated e¤orts to combine some of the features of the RBC models, such
as elastic labor supply, with those of the two sector endogenous growth literature, the endogeneity
of human capital. The …rst e¤orts by Benhabib and Perli (1994), Ladron et. al. (1997, 1999) and
Ben-Gad (2004) were largely concerned with analyzing the dynamic structure of such models. More
recent work by Jones et. al. (2005a) demonstrate that in an endogenous growth model with elastic
labor supply, long-run growth is highly dependent on the variability of shocks to fundamentals.
Similarly Jones et. al. (2005b) demonstrate that the same type of model generates arti…cial times
series that better mimic U.S. data, than standard real business cycle models that treat technological
change and trend growth as exogenous.
2
A generalized endogenous growth model with leisure
Consider an economy populated by a continuum of individuals indexed by i 2 [0; 1], who choose in
every period t the amounts of consumption c (i; t) and leisure l (i; t) that maximize a utility function
U : R2+ ! R:
max
c;l
Z
1
e
t
U [c (i; t) ; l (i; t)] dt
0
2
(P.1)
subject to the constraints:
_ t) = F [k(i; t); u(i; t)h (i; t)]
k(i;
_ t) = G [(1
h(i;
u (i; t)
[ (t); h (t)]
k(i; t)
c(i; t)
(1)
l (i; t)) h(i; t)] [ (t)]
(2)
We assume throughout that U1 ; U2 > 0, and U11 ; U22 < 0. Equation (1) represents the budget
constraint for individual i. Individual income is produced by the constant returns to scale production function F : R2+ ! R+ that combines individual stocks of physical capital k(i; t) with e¤ective
labor. E¤ective labor is that portion u(i; t) of non-leisure time devoted to market work multiplied
by the individual’s stock of human capital h(i; t). Individual productivity may also depend on
an external e¤ect
: R+ ! R+ that depends on either the level of per-capita human capital
R1
R1
h(t) = 0 h(i; t)di as in Lucas (1988), or e¤ective labor (t) = 0 u(i; t)h(i; t)di as in Ben-Gad
(2003). Both functions F and
are increasing and concave in all their elements; physical capital
depreciates at the constant rate : Equation (2) describes individual human capital accumulation
as an increasing function G : R3+ ! R+ of the amount of time each devotes to its aquisition, multiplied by its current stock. Here as well there is a possible external e¤ect— the economy-wide level
R1
of per-capita e¤ort devoted to human capital accumulation (t) = 0 (1 u (i; t) l (i; t)) h(i; t)di
reinforces accumulation at the individual level through the function
: R+ ! R+ . Finally, we
de…ne the per-capita values of consumption, market work, leisure, physical and human capital
R1
R1
R1
R1
as, respectively: c(t) = 0 c(i; t)di, u(t) = 0 u(i; t)di; l(t) = 0 l(i; t)di, k(t) = 0 k(i; t)di, and
R1
h(t) = 0 h(i; t)di.
For each individual i, the current value Hamiltonian associated with the optimization problem
P.1 is:
H (i)0 = U [c (i; t) ; l (i; t)] + (i; t) [F [k(i; t); u(i; t)h (i; t)]
+ (i; t)G [(1
u (i; t)
[ (t); h (t)]
k(i; t)
c(i; t)]
(H.1)
l (i; t)) h(i; t)] [ (t)] :
where (i; t) and (i; t) are the co-state variables that correspond to the constraints (1) and (2),
respectively. The …rst order necessary conditions for an interior optimal solution are:
Uc [c (i; t) ; l (i; t)] = (i; t)
Ul [c (i; t) ; l (i; t)] = (i; t)G0 [(1
(i; t)F2 [k(i; t); u(i; t)h (i; t)]
l (i; t)
u (i; t)) h(i; t)] h(i; t) [ (t]
[ (t); h (t)] = (i; t)G0 [(1
(i; t)F1 [k(i; t); u(i; t)h (i; t)]
(i; t) (1
l (i; t)
u (i; t)) G0 [(1
[ (t); h (t)]
l (i; t)
+ (i; t)F2 [k(i; t); u(i; t)h (i; t)]
3
(3)
l (i; t)
=
(4)
u (i; t)) h(i; t)] [ (t)]
(i; t)
(i; t)
(6)
u (i; t)) h(i; t)] [ (t)]
[ (t); h (t)] u (i; t) =
(5)
(7)
(i; t)
(i; t)
along with the two transversality conditions:
lim (i; t)k(i; t) = 0
(8)
lim (i; t)h(i; t) = 0:
(9)
t!1
t!1
Each individual i, treats the per-capita economy-wide values h(t);
(t) and
(t) as given. In
equilibrium, marginal products for physical and human capital equal each inputs’ economy-wide
rental rate. Hence, equilibrium marginal products across the di¤erent individuals are equal:
Fi [k(i; t); u(i; t)h (i; t)] = Fi [k(t); (t)] ; 8i 2 f1; 2g ; i 2 [0; 1]
(10)
Integrating (6) with respect to t; and combining the result with (3), yields the intertemporal
condition:
Uc [c (i; t) ; l (i; t)]
=e
Uc [c (i; s) ; l (i; s)]
Rt
s (F1 [k(t);
(t)] [ (t);h(t)]
)dv
:
(11)
Combining (3)-(5) yields the intratemporal condition:
Ul [c (i; t) ; l (i; t)]
= F2 [k(t); (t)]
Uc [c (i; t) ; l (i; t)]
[ (t)] h(i; t)
(12)
where the right-hand side is the observed wage of individual i.
Di¤erentiating (3) and (5) with respect to t, and combining with (7) and (12) yields an implicit
equation in the amount of individual lesiure l (i; t):
l (i; t) = 1
where the term
1 (t)
1 (t)
and the term
2 (i; t)
(1
0[
(t)]
(t)
[ (t)]
2 (t)
(13)
includes only per-capita economy-wide terms:
F1 [k(t); (t)]
+
1
1 (t)
l (i; t) u (i; t)) h(i; t)] [ (t)]
G00 [(1 l (i; t) u (i; t)) h(i; t)]
2 (i; t)
G0 [(1 l (i; t) u (i; t)) h(i; t)]2 [ (t)]
G0 [(1
F21 [k(t); (t)]
(F [k(t); (t)]
F2 [k(t); (t)]
0 [ (t)]
F22 [k(t); (t)]
+
(t);
[ (t)]
F2 [k(t); (t)]
[ (t); h (t)]
[ (t)]
k(t)
c(t))
includes terms that are particular to individual i :
l (i; t)
u (i; t)) G [(1
l (i; t)
u (i; t)) h(i; t)] [ (t)]
l (i; t) + u (i; t) h(i; t):
Proposition 1 First order conditions (3)-(7) cannot correspond to an optimization of P.1 or support competetive equilibrium if G is linear.
4
Proof. If function G is linear, then term multiplying
2 (i; t)
in (13) is zero, G0 [(1
l (i; t)
u (i; t)) h(i; t)]
reduces to a constant scalar, and the value of leisure l (i; t) in (13) is only a function of economywide variables. Hence the value of leisure at time t is identical across the di¤erent individuals
in the economy, and from (12) the value of individual consumption c (i; t) is directly proportional
to the contemporaneous stock of the individual’s human capital h (i; t). Finally, if individuals i
and j each possess identical levels of human capital at time s, h (i; s) = h (j; s) then from (12)
consumption at time s is also identical, c (i; s) = c (j; s). From (11), for all time t in the future,
t > s, c (i; t) = c (j; t) as well. This holds true even if at time s, the two individuals own di¤erent
amounts of physical capital, for example, k (i; s) > k (j; s), and implying (i; t) = 0, a violation of
the Pontyagrin Maximum Principle necessary conditions for an optimum.
Proposition 1 has far-reaching implications for how the Lucas (1988) model of endogenous
growth model may, and may not be extended. It is not possible to merely insert a labor-leisure
choice into the model without some other alteration to the model, for example adding an external
e¤ect to the human capital production. This also means that the validity of the examples of multiple
interior Ladron et. al., (1997,1999) calculated for the two sector model with human capital functions
that are linear, must be reconsidered.
De…nition 1 A balanced growth path for this economy is an optimal solution fc(i; t),l(i; t),u(i; t),
k(i; t),h(i; t)gi2[0;1] , to problem (P.1) for some initial conditions fk(i; 0); h(i; 0)gi2[0;1] , such that
fl(i; t); u(i; t)gi2[0;1] remain constant, the growth rates of c(i; t), k(i; t), and h(i; t) are all constant
and equal to the per-capita growth rates for c(t), k(t), and h(t). Furthermore, the output-capital
ratio F [k(i; t); u(i; t)h (i; t)]
[ (t); h (t)] =k(i; t) and capital’s share in output for all i 2 [0; 1] are
constant.
Given De…nition 1 there are a number of necessary restrictions to the functional forms in the
optimization problem (2). We start with the functions that produce human capital in (2).
Proposition 2 If human capital and the time devoted to its accumulation are the only inputs used
in the production of human capital, along a balanced growth path the aggregate production function
for human capital must be homogenous of degree one.
Proof. From De…nition 1, along the balanced growth path, the growth rate of human capital is
_
constant, i.e.: @ h(s;t)
h(s;t) =@t=0.
Di¤erentiating
_
h(s;t)
h(s;t) with
@
respect to t:
_ t)
h(i;
• t)=h(i; t)
=@t = h(i;
h(i; t)
5
_ t)=h(i; t)
h(i;
2
:
Di¤erentiating (??) with respect to t, and inserting yields:
@
where
_ t)
h(i;
=@t =
h(i; t)
(t) =
R1h
0
(1
[ (t)] G0 [(1
h
(1 l (i; t)
l (i; t)
u (i; t)) h(i; t)]
i
l_ (i; t) + u_ (i; t) h(i; t)
)
_ t)2
h(i;
+ 0 [ (t)] G [(1 l (i; t) u (i; t)) h(i; t)] (t)
=h(i; t):
h(i; t)
i
_ t)di
l (i; t) u (i; t)) h(i;
l_ (i; t) + u_ (i; t) h(i; t) di. From De…nition 1,
_ t)
u (i; t)) h(i;
along the balanced growth path, l_ (t) = u_ (t) = l_ (i; t) = u_ (i; t) = 0; 8i. Therefore along the
balanced growth path
(t) = (1
l (t)
_
u (t)) h(t).
Furthermore from De…nition 1, along the
_
_ t)h(t)=h(i; t); 8i and therefore (t) = (t)h(t)=h(t).
_
balanced growth path: h(t)
= h(i;
Together
these conditions imply that along the balanced growth path:
_ t) =
h(i;
[ (t)] G0 [(1
+
0
[ (t)] G [(1
l (i; t)
l (i; t)
u (i; t)) h(i; t)] (1
l (i; t)
u (i; t)) h(i; t)
(14)
u (i; t)) h(i; t)] (t)
Inserting (??), the left-hand side of expression (14) is equal to
[ (t)] G [(1
l (i; t)
u (i; t)) h(i; t)]
and the result corresponds to Euler’s Theorem for functions that are homogenous of degree one.
Proposition 2 is a generalization of the result in Mulligan and Sala-i-Martin (1993) that found
the conditions for balanced growth when the aggregate production functions for both sectors take
the Cobb-Douglas form. In order to ensure the existence of a balanced growth path we henceforth
impose the condition that the aggregate production function for human capital is homogeneous of
degree one. Combining this condition with the non-linearity of the function G imples:
Proposition 3 To ensure the existence of a balance growth path in the two sector model with
leisure, the external e¤ ect
both G and
in the human capital sector must be non-degenerate. Furthermore,
are strictly concave.
Proof. Follows directly from the homogeneity of degree one of the production function for human
capital and Proposition 1.
Finally, to ensure the existence of a balanced growth path, we must also place some restricition
on preferences.
Proposition 4 For the balanced growth path to exist, the utility function must display constant
intertemporal elasticity of substitution and be multiplicatively seperable in labor.
Proof. Di¤erentiating (3) with respect to t, combining the result with (6), and setting l (s; t) = 0
yields the equation:
Uc [c (i; t) ; l (i; t)]
c_ (i; t)
=
[ +
c (i; t)
Ucc [c (i; t) ; l (i; t)] c (i; t)
6
F1 [k(t); (t)]
[ (t); h (t)]]
(15)
Along the balanced growth the left-hand side of (15) is constant and so is the term in square brackets
on the right (otherwise the share of capital in output will not be constant). Di¤erentiating (12)
with respect to t, setting l (s; t) = 0, and assuming that along the balanced growth path payments
to labor grow at the same rate as consumption yields:
Ucl [c (i; t) ; l (i; t)] = (1
where
=
)
Ul [c (i; t) ; l (i; t)]
c (i; t)
(16)
Ucc [c (i; t) ; l (i; t)] c (i; t) =Uc [c (i; t) ; l (i; t)] . The solution to the di¤erential equation
(in the variable c (i; t)) can be expressed as U [c (i; t) ; l (i; t)] = c (i; t)(1
)
(l (i; t)) = (1
), where
(l (i; t)) is any increasing concave function.
The restrictions on the model imposed by Propositions 1- 4 are important to bear in mind
because Proposition 1 merely states a necessary, not a su¢ cient condition for (3)-(9) to correspond
to a maximum value of P.1. One su¢ cient condition guaranteeing solutions satisfying (3)-(9) are
h
i
0 (c; l; u; h; k)
of the current value
indeed optimal, is that the Hessian matrix H = Hij
i=1;5;j=1;5
Hamiltonian (H.1) is jointly concave in all the states and controls (Mangassarian (1966)).
Proposition 5 The Hamiltonian (H.1) does not satisfy the Mangasarian su¢ ciency conditions for
optimization.
Proof. We calculate H by di¤erentiating H0 with respect to the individual levels of consumption,
work, leisure, physical and human capital. The determinant of the Hessian matrix is:
jHj =
u (i; t) G00 [(1 l (i; t) u (i; t))h (i; t)] F11 [k(t); (t)]
k (i; t) (l (i; t))G0 [(1 l (i; t) u (i; t))h (i; t)] [ (t); h (t)]2
3
0 (l (i; t))2
(1
(1
)h (i; t)
)2 h (i; t) F2 [k(t); (t)]
(17)
[ (t); h (t)] (l (i; t))F2 [k(t); (t)]
0 (l (i; t))
5(1
The determinant (17) is the …fth principal minor and is strictly positive— the matrix is not negative
semi-de…nite.
Fortunately the Arrow su¢ ciency conditions o¤er a weaker set of restrictions, and under certain
circumstances, the model does ful…ll. First we must establish one more necessary property of the
model.
Proposition 6 The functional form of the functions G and
take the form: G [(1
(t)1
, where
G,
u (i; t)
, and
l (i; t)) h(i; t)] =
G [(1
are constants and 0 <
u (i; t)
must be exponential; i.e. must
l (i; t)) h(i; t)]
and
[ (t)] =
< 1:
Proof. From Proposition 1:
G0 [ ]
=
G[ ]
0[
[ ]
[ ]
7
]
(18)
)
where we surpress the terms for individuals and time and
respect to
u
l. Integrating both sides with
:
0[
ln G [ ] = ln
where
=1
ln G [ ] =
ln +
[ ]
0[ ]
[ ]
is a constant of integration. From 1
]
(19)
G0 [ ]
G[ ]
=1
G0 [ ]
ln +
G[ ]
:
(20)
G
Solving this di¤erential equation:
G[ ] =
where
(21)
G
is a constant. The same argument yields:
[ ]=
(22)
Finally from the property of homogeneity of degree one:
Henceforth we de…ne
=
= 1.
, and rede…ne (2) as:
G
_ t) =
h(i;
3
+
[(1
u (i; t)
l (i; t)) h(i; t)]
(t)1
(23)
The model with Cobb Douglas production and CRRA Utility
Propositions 1- 6 determine most of the functional forms represented in the model. We are left to
assume the forms of the functions F ,
; and
. The production function is homogenous of degree
one, we assume it takes the Cobb Douglas form: F [k; uh] = k (uh)1
e¤ect is sector-speci…c and
, and the associated external
[ (t); h (t)] = (t) . We assume the functional form
1
that the utility function is U [c; l] = cl
= (1
(l) = l
(1
)
so
).
The current value Hamiltonian associated with the optimization problem P.1 now becomes:
H (i)0 =
c(i; t)l(i; t)
1
+ (i; t) [(1
h
+ (i; t) k(i; t) (u(i; t)h (i; t))1
1
u (i; t)
;
i
c(i; t) (H.2)
= (i; t)
(24)
(t)1
l (i; t)) h(i; t)]
(t)
k(i; t)
and the …rst order conditions become:
c(i; t)
c(i; t)1
(i; t) (1
l(i; t)
(1
l(i; t)
) 1
(1
= v (1
) k (i; t) (u (i; t) h (i; t))
(i; t) k (i; t)
1
(u (i; t) h (i; t))1
8
)
) (i; t)h(i; t) (t)1
(t) = v (1
(t)
=
(25)
) (i; t) (t)1
(i; t)
(i; t)
(26)
(27)
(i; t)v (1
) (1
u (i; t)) (t)1
l (i; t)
) k (i; t) u (i; t)1
+ (i; t) (1
(28)
h (i; t)
(t) =
(i; t)
(i; t)
along with the transversality conditions (8) and (9).
Four aggregate laws of motion describe the dynamic behavior of the economy:
1
(1
c_ (t) =
u_ (t) =
[r (t)
)
(v (1
(1
+
) (1
l (t)) + (1
(
(t)1
1
where r (t) = k(t)
+
)v(1
v (u (t)
) +v(
)k (t)
(t)1
_
k(t)
= k(t)
_ (t) = ((1
)(r (t)
+
k(t)
u(t)c(t)
)k(t) (t)1
(1
+
l (t))))] c (t)
) u (t)) k (t)
c (t)
u (t)
c(t)
(29)
(30)
(31)
l(t)) + (1
) ) k(t)
(
)k(t)
and l (t) =
(1
c(t)
(t)
(32)
.
Of the four variables in (29)-(32), consumption, hours worked, physical capital, and e¤ective
labor, only hours worked is stationary. In order to make this system stationary, we de…ne stationary
consumption and physical capital: c~(t) = c(t) (t)
1
+
1
~
, k(t)
= k(t) (t)
1
+
1
. The dynamic
system reduces to three stationary laws of motion:
:
1
(1
c~ (t) =
1
)
+
1
"
:
~
k~ (t) = k(t)
u_ (t) =
(r (t)
#
_ (t)
c~ (t) ;
(t)
c~(t)
~
k(t)
1
(v (1
+
1
+
(1
((1
)(r (t)
)v(1
1
) (1
l (t)) + (1
(
v (u (t)
(1
) u (t)) k~ (t)
(33)
#
c~(t) ~
k (t) ; (34)
~
l(t)) + (1
) ) k(t)
~
(
)k(t)
) +v(
~
)k (t)
l (t)))))
c~ (t)
u (t) ;
(35)
where the interest rates and leisure are now de…ned in terms of the stationary variables: r (t) =
~
k(t)
1
and l (t) =
(1
u(t)~
c(t)
~
)k(t)
.
Setting (33)-(35) equal to zero, the rates of return to physical capital r are constant along the
balanced growth path, and solve the quadratic equation:
(1
+((2
( + (1
where
= [(1
)(
)
+ (
)
) )
+
) )
(1
(1
+ ) (1
) ( ( + )
(1
)
2
+ (1
+ )((1
)(
)v + ) +
=0
(1
) ) + ((1
+ )(1
9
)
(1
)
) )] .
r2
(36)
)r
Proposition 7 There can be no more than two positive valued levels of physical capital k~ along a
balanced growth path for the economy that correspond the optimization problem H.2.
Proof. Follows directly form (36) and k~ =
1
r
1
:
In contrast to the …ndings of Ladron et. al., (1997,1999) that numerically calculate speci…c
examples from H.2 with as many as three di¤erent levels of capital, according to Proposition 7
there can be no more than two.
References
[1] Ben-Gad, Michael, “Fiscal Policy and Indeterminacy in Models of Endogenous Growth,”Journal of Economic Theory, 108, (2003), 322-344.
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