Indeterminacy in a small open economy with endogenous labor supply

Indeterminacy in a small open economy with
endogenous labor supply¤
Qinglai Meng1 and Andrés Velasco2
1Department of Economics
Chinese University of Hong Kong, Shatin, Hong Kong
(email: meng2000@cuhk.edu.hk)
2Kennedy School of Government
Harvard University, Cambridge, MA 02138, USA, and NBER.
(email: andres_velasco@harvard.edu)
April 3, 2002
Summary.
We establish conditions under which indeterminacy can occur in a small
open economy business cycle model with endogenous labor supply. Indeterminacy requires
small externalities in technologies with social constant returns to scale, independently of the
intertemporal elasticities in both consumption and labor.
Keywords and Phrases: Indeterminacy, Small Open Economy, Business Cycles
JEL Classi…cation Numbers: E32, F12, F4
¤The paper has bene…ted from discussions with Jess Benhabib and Mark Weder, as well as from the
comments of an anonymous referee. Send correspondence to Q. Meng
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1
Introduction
It is well understood by now that under some conditions closed-economy real business cy-
cle models can be subject to indeterminacy, in the sense that there exist a continuum of
equilibrium trajectories converging to a steady state.1
The literature on indeterminacy
underscores that equilibria need not be uniquely determined by the fundamentals of the
economy, and that the existence of indeterminate equilibria is associated with the possibility
of self-ful…lling prophecies. The remaining question is how plausible are the requirements
to generate such indeterminacy.
Early models relied on relatively large increasing returns to scale to generate indetermi-
nacy.2 The estimates by Hall ([10], [11]) and others made this plausible. But over time, the
empirical evidence has mounted against large increasing returns.3 More recently, Benhabib
and Farmer [3] showed that in a two-sector model the size of increasing returns need not be
large. Benhabib and Nishimura [5] went further, showing that decreasing marginal costs are
not necessary to render the steady state indeterminate.
In their model, small production
externalities with social constant returns are su¢cient to generate multiple equilibria.
But these closed-economy models also place restrictions on preferences. They work best,
in the sense of requiring only small distortions to generate indeterminacy, either when the
intertemporal elasticity of substitution in consumption or/and the elasticity on labor supply
is high.
The intuition for why these conditions are needed is straightforward. Take for
instance, the two-sector model of Benhabib and Farmer [3]. They provide the intuition for
indeterminacy succinctly (pp 423):
“Consider starting with an arbitrary equilibrium trajectory of investment or con-
sumption, and inquire whether a faster rate of accumulation and growth can also
1 For an excellent survey of the literature, see Benhabib and Farmer [4].
2 See, e.g., Benhabib and Farmer [2].
3 On recent empirical estimates, see Basu and Fernald [1] and Burnside, Eichenbaum and Rebelo [7].
These papers …nd little evidence of increasing marginal returns. Indeed, they conclude that returns to scale
are roughly constant and that market imperfections are small.
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be justi…ed as an equilibrium.
This would require a higher return on invest-
ment. If higher anticipated stocks of future capital raise the marginal product
of capital by drawing labor out of leisure, or by reallocating labor across sectors,
the expected higher rate of return may be self-ful…lling... If... there are su¢-
cient increasing returns that are consistent with optimization, either because of
externalities or because of imperfect competition that generate markups, these
increasing returns may amplify the movement of labor into production and pro-
vide a su¢cient boost to private rates of returns to justify multiple equilibria.
The critical parameters are the magnitudes of increasing returns or externalities,
and the ease with which labor can be drawn into employment – that is – the
elasticity of labor supply.”
Benhabib and Farmer [3] assume a utility function that is logarithmic in consumption.
But with that formulation it is relatively costly to reallocate labor out of the consumption
sector. Therefore they need high labor elasticity in order to draw labor out of leisure and to
raise the marginal product of capital. If one assumes larger values of intertemporal elasticity
of substitution in consumption, then high labor elasticity is not required for indeterminacy.
In particular, if one uses linear consumption (¾ = 0), then indeterminacy can arise in a
broader range of values for labor elasticity.
Obviously, with linear consumption (so that
intertemporal elasticity in consumption is in…nite), labor can be freely reallocated from the
consumption sector (in order to increase the marginal product of capital in the capital goods
sector) with little e¤ect on leisure. In that case, indeterminacy can arise even when labor
supply is …xed.4
In this paper we extend research on indeterminacy to a small open economy real busi-
ness cycle model, in a way that addresses some of the limitations of earlier theorizing. We
combine the preferences proposed by Greenwood, Hercowitz and Hu¤man [9] with the tech-
4 The same intuition applies to the model in Benhabib and Nisimura [5], and indeed it may be more pro-
nounced there. With linear consumption, they show that indeterminacy conditions are in fact independent
of labor elasticity.
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nologies of social constant returns to scale introduced by Benhabib and Nishimura [5]. In
that setup, indeterminacy can occur regardless of the elasticities on both consumption and
labor, for technologies with very small or even negligible external e¤ects.
Thus, in open
economies facing a perfect bond market, indeterminacy can obtain under empirically plau-
sible conditions.
This paper is also a realistic extension of Weder [15] and related work in the literature,
in that we incorporate endogenous labor supply to an otherwise standard Ramsey model
of a small open economy.
Weder [15] uses the Benhabib and Farmer [3] technology, with
inelastic labor supply, in such a model, and shows that indeterminacy can obtain more easily
than a closed-economy variant of Benhabib and Farmer [3].5
But with …xed labor supply,
unemployment (or employment) ‡uctuations –a key element in business cycle ‡uctuations–
simply can not be explained. In this paper we bring this feature back into the picture. An
additional advantage of our analysis is that we can derive an explicit closed-form condition
for indeterminacy, something that is not possible in models with endogenous labor supply
and other parameter speci…cations.6
2
The Two-Sector Open Economy with Endogenous
Labor Supply
Consider a small open economy inhabited by an in…nite-lived representative agent who max-
imizes the intertemporal utility function
5 With inelastic labor supply, Meng and Velasco [14] obtain similar results in an open economy version
of Benhabib and Nishimura [5].
In a small open economy endogenous growth model, again with …xed
labor, Lahiri [12] shows also that it is easier to obtain multiple transitional growth paths than in the closed
economy.
6 Therefore, with endogenous labor supply there are situations under which indeterminacy can occur in
the closed economy, but it is unclear whether indeterminacy can also happen in the open economy.
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Z 1u(ct;lt)e¡½tdt
(1)
0
where ct is consumption of traded goods, lt labor supply and ½ the parameter of time pref-
erence.
Assume the economy is open to full international capital mobility, so that the
domestic representative agent can borrow from and lend to the outside world freely. This
agent has access to net foreign bonds dt, denominated in units of the tradable good, that
pay a world interest rate r, which is exogenously given to the small open economy. Assume
that consumption goods are tradable and capital goods non-tradeable, as in Weder [15].
On the production side there are two sectors: one producing a consumption tradable good
(y1t) and the other and investment non-tradeable good (y2t). The production functions are
assumed to be Cobb-Douglas with externality components
y1t = l®0k®1la0ka1; where ®
1t
1t 1t
1t
0 + ®1 + a0 + a1 = 1
(2)
y2t = l¯0k¯1lb0kb1; where ¯ + ¯ + b
2t
2t 2t 2t
0
1
0 + b1 = 1
(3)
where
l1t + l2t = lt;
k1t + k2t = kt
(4)
Here l1t and k1t denote the capital and labor services used by the individual …rm in the
consumption good producing sector, and l2t and k2t for the investment good producing
sector.
The components la0ka1 and lb0kb1 of the production functions represent external
1t
1t
2t 2t
e¤ects that are viewed as functions of time by the agent.
Constant returns coupled with small external e¤ects imply that some sectors must display
a small degree of decreasing returns at the private level.
This is in contrast to models of
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indeterminacy with social increasing, but private constant returns to scale.7
The agent’s budget constraint is
_
dt = rdt + y1t + pty2t ¡ ct ¡ ptit
(5)
where pt is the relative price of the investment or non-tradeable good to the traded good.
Sometimes this price is referred to as the real exchange rate. Note that in (5) the traded good
is taken to be the numeraire. Note also that pt is taken as exogenously given by the agent,
but is determined by market-clearing conditions. The variable it denotes gross investment,
so that the law of motion for capital is
_kt = it ¡ ±kt
(6)
Equations (5) and (6) can be consolidated into
_at = rat + y1t + pty2t ¡ ct + kt( _pt ¡ rpt ¡ ±pt)
(7)
where at = dt + ptkt. The agent’s problem is to choose ct, l1t, l2t, it, k1t, k2t and dt to
maximize (1), subject to (2), (3), (4) and (7), and given k0 and d0.
The Hamiltonian is
H = u(ct; lt) + ¸t(rat + y1t + pty2t ¡ ct + kt( _pt ¡ rpt ¡ ±pt)) +
¹ (k
t
¡ k1t ¡ k2t) + !t(lt ¡ l1t ¡ l2t)
where ¸t is a costate; ¹ and !
t
t are the rental rate of capital goods and the wage rate of
labor, all in terms of the consumption good. First-order conditions are
7 Although we adopt the production functions with social constant returns to scale as in Benhabib and
Nishimura [5], similar results to those obtained below carry over to the case of increasing returns to scale
that is speci…ed in Benhabib and Farmer [3] and used in Weder [15]. We use the Benhabib and Nishimura
[5] setup for expositional simplicity.
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uc(ct; lt) = ¸t
(8)
ul(ct; lt) = ¯ ¸
k¯1+b1
(9)
0 tptl¯0+b0¡1
2t
2t
!t = ¸t®0l®0+a0¡1k®1+a1 = ¸
p
k¯1+b1
(10)
1t
1t
t¯0 tl¯0+b0¡1
2t
2t
¹ = ¸
k®1+a1¡1 = ¸
p
k¯1+b1¡1
(11)
t
t®1l®0+a0
1t
1t
t¯1 tl¯0+b0
2t
2t
_¸t = ¸t(½ ¡ r)
(12)
_
pt = pt(r + ± ¡ ¯ l¯0+b0k¯1+b1¡1);
(13)
1 2t
2t
together with the transversality conditions
t ! 1lim¸tdte¡½t = t ! 1lim¸tptkte¡½t = 0:
(14)
Preference Structure
We adopt the following utility function popularized by Greenwood, Hercowitz and Hu¤-
man [9]:
1
1
[(c
l1+Â)1¡¾
1
t
¡ ¾
¡ 1 + Â t
¡ 1]
(15)
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where ¾ > 0 is the inverse of the intertemporal elasticity of substitution in consumption, and
 corresponds to the intertemporal elasticity of substitution in labor supply.
The utility
function form in (15) implies that the marginal rate of substitution between consumption
and labor e¤ort depends on the latter only8
¡ul(ct;lt)=uc(ct; lt) = lÂ;
(16)
t
so that labor e¤ort is determined independently of the intertemporal consumption-savings
choice. Such a property is essential in obtaining the indeterminacy results below.9
As is standard in international macroeconomics, we impose ½ = r; a condition that ensures
a well-de…ned steady-state with constant bond-holdings. This assumption will also imply,
by (12), that marginal utility remains constant over all time –that is, ¸t = ¹
¸. Substituting
¸t = ¹
¸ into other …rst-order conditions, by (8), (9) and (16) we have
1
ct ¡
l1+Â = ¹
¸¡ 1¾
(17)
1 + Â t
lÂ
t = ¯ p
k¯1+b1
(18)
0 tl¯0+b0¡1
2t
2t
Note from (17) that when marginal utility remains at a constant level, so does instant
utility (15). Dividing (11) by (10) yields
®1 l1t
¯ l
=
1
2t
(19)
®0 k1t
¯ k
0
2t
8 We ignore the extreme case when  = 0, which implies that consumption and leisure are perfect
substitutes.
9 While Greenwood, Hercowitz and Hu¤man [9] …rst proposed a utility function of the form in (15) for
the closed economy, it was later used for small open economy real business cycle models by a number of
authors. See, e.g., Mendoza [13] and Correia, et al. [8].
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Using (11) and (19) to solve for l2t , we have
k2t
l
1
1
2t = ´p®0+a0¡¯0¡b0 = ´p(®0+a0)(¯1+b1)¡(®1+a1)(¯0+b0)
k
t
t
´ g(pt)
(20)
2t
where ´ = ¯1
® ³®1¯0´®0+a0. Thus, from (18) we have
1
®0¯1
lt = £¯ p0tg(pt)¯0+b0¡1¤1 =l(pt)
(21)
which is the equilibrium labor supply equation. Substituting (4), (19) and (20) into (21),
we can solve for k2t
®
k
0¯1
2t =
k
®
t + h(pt)
(22)
0¯1 ¡ ®1¯0
where
®
ptg(pt)¯0+b0¡1
h(p
1¯0
£¯0
(23)
g(pt)
¤1Â
t) = ¡®0¯1 ¡ ®1¯0
In addition, the market clearing conditions for the investment (non-tradeable) good and
the economy’s current account are, respectively
_kt = l¯0+b0k¯1+b1
2t
2t
¡ ±kt
(24)
_
dt = rdt + l®0+a0k®1+a1
1t
1t
¡ ct
(25)
Substituting (20) and (22) into (13) and (24), we obtain the following di¤erential equations
for kt and pt:
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_
pt = pt £r+±¡¯ g(p
1
t)¯0+b0 ¤
(26)
_
®
k
0¯1
t = [
g(p
®
t)¯0+b0 ¡ ±]kt + h(pt)g(pt)¯0+b0
(27)
0¯1 ¡ ®1¯0
These two equations describe the dynamics of the economy. The solution to this system
can then be used, in conjunction with the other conditions laid out above, to solve for all
variables of interest. In particular, the current account equation (25) and the transversality
(14) combined determine the equilibrium consumption pro…le.
To see this, integrate over
(25), by using (17) and (21), to obtain
Z 1
1
(de¡rt)0dt = Z 1y1t(pt;kt)e¡rtdt¡Z 1(
l1+Â
t
(pt) + ¹
¸¡ 1¾ )e¡rtdt
(28)
0
0
0
1 + Â
Using the transversality condition we have
¹
1
¸ = [r Z 1(y1t(pt;kt)¡
l1+Â
t
(pt))e¡rtdt ¡ rd0]¡¾
(29)
0
1 + Â
Consumption can be obtained from ((17), i.e.
1
ct =
l1+Â
1 + Â t
(pt) + ¹
¸¡ 1¾
(30)
Therefore, once the solution path for (kt; pt) is determined, other variables including con-
sumption can all be uniquely determined, and at the same time the transversality condition
is satis…ed. That is, indeterminacy in the two-by-two system in kt and pt implies that the
overall economy is indeterminate.
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