This note shows that indeterminacy arising from an economy exhibiting production with social constant returns to scale may be related to the instability of the consumption goods market equilibrium. Furthermore, trade does not contribute to indeterminacy; indeterminacy arises becasue each country’s equilibrium path is already indeterminate before trade.Indeterminacy, Market Instability

Kong-Weng Ho

Nicholas C.S. Sim

English

Research Papers in Economics

 
 
 
Economic Growth Centre 
Working Paper Series 
 
 
 
 
 
 
 
Indeterminacy and Market Instability 
 
 
 
by 
 
  
Nicholas C.S. SIM and HO Kong Weng 
 
 
 
 
 
 
 
 
Economic Growth Centre 
Division of Economics 
School of Humanities and Social Sciences 
Nanyang Technological University 
Nanyang Avenue 
SINGAPORE 639798 
 
Website:  http://www.ntu.edu.sg/hss/egc/ 
Working Paper No:  2007/05 Copies of the working papers are available from the World Wide Web at: 
 
Website:  http://www.ntu.edu.sg/hss/egc/ 
 
 
 
 
 
 
 
 
 
 
 
 
The author bears sole responsibility for this paper.  Views expressed in this 
paper are those of the author(s) and not necessarily those of the Economic 
Growth Centre. INDETERMINACY AND MARKET INSTABILITY
Nicholas C.S. Sim1 and Kong-Weng Ho2∗
1 Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02452 USA.
2 Department of Economics, School of Humanities and Social Sciences, Nanyang Technological University,
Nanyang Avenue, Singapore 639798. Tel: (65)67904496, Fax: (65)679466303, Email: KWHo@ntu.edu.sg.
∗ To whom correspondence should be addressed.
Abstract. This note shows that indeterminacy arising from an economy exhibiting production
with social constant returns to scale may be related to the instability of the consumption goods
market equilibrium. Furthermore, trade does not contribute to indeterminacy; indeterminacy arises
becasue each country’s equilibrium path is already indeterminate before trade.
JEL Classiﬁcation Numbers: E32, F00, F11, F43.
Key Words: Indeterminacy, Market Instability.
1I. Introduction
Seminal works by Benhabib and Farmer (1994, 1996) and Benhabib and Nishimura (1998) on
equilibrium path indeterminacy have attracted considerable attention in macroeconomic dynamics
analysis. Studies along this direction have strived to elucidate why and how indeterminacy can
arise even for parameterization consistent with the data. However, existing works in the literature
mainly employ the autarkic framework as the modelling workhorse. The ﬁrst exception is the
two-country Heckscher-Ohlin framework of Nishimura and Shimomura (2002). The authors show
that the equilibrium path in the world market is possible when private and social factor intensity
rankings are the reverse of the other and intertemporal elasticity of substitution in consumption
is small enough. Consequently, the indeterminacy of the realized path implies that the long-run
Heckscher-Ohlin prediction of trade is uncertain.
Our objective is to highlight a counter-intuitive result relating to market instability and indeter-
minacy by employing an autarkic variant of Nishimura and Shimomura (2002). We believe that this
result may have potential implications on other works in indeterminacy especially those employing
a similar production setup. In particular, we show that indeterminacy may entail a downward
sloping supply schedule of consumption goods, which we view as evidence against the plausibility
of indeterminacy in Nishimura and Shimomura. Moreover, it is possible for the consumption goods
market to be unstable when the demand schedule cuts the downward sloping supply schedule from
above.
The rest of the paper is as follows. Section II outlines the autarkic variant of the Nishimura
and Shimomura model. Discussions about the stability properties of the steady state as well as our
main result are presented in Section III. Section IV concludes.
II. The Model
The outline of the model is as follows. The economy is populated by an inﬁnitely-lived pseudo-
planner who solves the following dynamic problem:
2Max
C
Z ∞
0
C1−η
1 − η
e−ρtdt (1)
subject to
ˆ I = L
aI
I K
bI
I L
αI
I K
βI
I and ˆ C = L
aC
C K
bC
C L
αC
C K
βC
C (2)
K = KI + KC and L = LI + LC (3)
˙ K = ˆ I + p ˆ C − δK − pC (4)
K0 given
where C and I denote the level of demand for consumption and investment, ρ ∈ (0,∞) is the
subjective rate of time discount, 1/η ∈ (1,∞) is the intertemporal elasticity of substitution in
consumption, and K0 is the initial capital stock. Production in each sector follows Cobb-Douglas
technology, where the production frontier for each good is described by Equation (2) with consump-
tion and investment output denoted by ˆ C and ˆ I respectively. Labor and capital are represented
by Li and Ki, and in each sector, technology exhibits constant returns to scale from the social
perspective, i.e. ai + αi + bi + βi = 1, but decreasing returns to scale from the private perspective,
i.e. ai +bi < 1. Equation (3) describes the resource constraints for capital and labor and Equation
(4) is capital accumulation process where δ ∈ (0,1) is the capital depreciation factor. Using the
market clearing conditions, that is C = ˆ C and I = ˆ I, the current value Hamiltonian is
H =
C1−η
1 − η
+ λ(I − δK) + PI(L
aI
I K
bI
I L
αI
I K
βI
I − I)
+ PC(L
aC
C K
bC
C L
αC
C K
βC
C − C) + r(K − KI − KC) + w(L − LI − LC)
where PI, PC, r and w are the shadow price of investment, consumption, capital and labor re-
spectively. Let p be the price of consumption in terms of investment and normalize r = r/PI and
w = w/PI. The necessary conditions are
C−η = λp (5)
3w = aIL
aI−1
I K
bI
I L
αI
I K
βI
I = paCL
aC−1
C K
bC
C L
αC
C K
βC
C (6)
r = bIL
aI
I K
bI−1
I L
αI
I K
βI
I = pbCL
aC
C K
bC−1
C L
αC
C K
βC
C (7)
˙ λ = λ(ρ + δ − r) (8)
lim
t→∞
K(t)λ(t)e−ρt = 0 (9)
Equation (5) states that the marginal utility and the marginal cost of consumption are equalized
under optimality, while (6) and (7) state that factor returns are equalized across sectors due to
frictionless mobility. Equation (8) is the familiar intertemporal arbitrage condition and (9) is
the transversality condition. Denote θi = ai + αi as the labor share in sector i from the social
perspective.
Lemma 1. From (6) and (7), we have
pr0(p)
r(p)
=
θI
θI − θC
(10)
pw0(p)
w(p)
=
1 − θI
θC − θI
(11)
where θC − θI measures the social factor intensity rankings.
Proof. See Nishimura and Shimomura (2002). 
Equations (10) and (11) can be rationalized as the Stolper-Samuelson theorem. When labor
is socially more intensive in the consumption sector, one has θC − θI > 0. Therefore, as (10) and
(11) indicate, an increase in the relative price of consumption will increase the returns to labor and
decrease the returns to capital.
Lemma 2. Goods market clearing implies
I =
bCw(p)L − aCr(p)K
∆
(12)
C =
aIr(p)K − bIw(p)L
p∆
(13)
4where ∆ = aIbC − aCbI.
Proof. See Nishimura and Shimomura (2002). 
Using (12), the capital accumulation process can be expressed as
˙ K =
bCw(p)L − aCr(p)K
∆
− δK (14)
The transitional path of the economy is determined by (8) and (14). In the steady state, ˙ λ = 0
implies ρ + δ − r(p) = 0, from which the steady state relative price, pe, is determined. Before
taking linear approximations around the steady state, note that K and λ are one-to-one and onto
functions in p, which can be determined by substituting consumption demand from (5) into (13)
and totally diﬀerentiating the resulting expression:
dp
dK
= −
ΣaI(ρ + δ)(λp)1/η
∆
(15)
dp
dλ
= −
pΣ
ηλ
(16)
where
1
Σ
=
1
η
+
θCaIbC(ρ + δ) + bI(1 − θC)(ρaC + δaIbC + δaC(1 − bI))
(θI − θC)∆(ρ + δ(1 − bI))
(17)
As it is usual in the literature, we take a ﬁrst order approximation of the dynamic equations. The
dynamic property of the economy is then determined by the signs of the determinant and trace of
the Jacobian matrix derived from linearly approximating the dynamic system. In particular, for
indeterminacy to take place, the determinant must be positive and the trace must be negative, so
that there are two stable (but possibly complex) eigenvalues and hence two stable solutions that
solve the dynamic equations. Since any linear combination of the two (stable) solutions is also a
(stable) solution,1 there are now inﬁnitely many possible stable solutions giving rise to the notion
of indeterminacy of the equilibrium path. However, if the determinant is negative, then there must
be one unstable and one stable eigenvalue, implying that the stable solution is unique. This is the
case that corresponds to saddle-path stability.
1This is the well-known principle of superposition in diﬀerential equations.
5III. Discussion
The determinant and trace of the Jacobian matrix are:
Det(J) = −

ρaC + δaIbC + δaC(1 − bI)
∆

ΣθI(ρ + δ)
η(θI − θC)

(18)
Trace(J) = −
ΣaI(ρ + δ)
∆2

(1 − θC)(1 − θI)(δ∆ + aC(ρ + δ)) + θCθIbC(ρ + δ)
(θC − θI)(ρ + (1 − bI)δ)

−
ρaI + δaIbC + δaC(1 − bI)
∆
+
ΣθI(ρ + δ)
η(θI − θC)
(19)
We now examine the nature of the equilibrium under three cases. First, we consider the case
where private and social factor intensities have the same ranking. Next, we examine the dynamic
properties of the economy under two cases that result in factor intensity reversal between private
and social factor intensities.
Case 1. Let θI − θC < 0 and ∆ < 0.
These restrictions imply that 1/Σ > 0. Therefore, the determinant is negative, indicating that
the system only has one stable eigenvalue. Consequently, the steady state is saddle-path stable.
Case 2. Let θI − θC < 0 and ∆ > 0.
Clearly, the sign of the determinant depends on the sign of 1/Σ. In particular, if
η ∈

0,
(θC − θI)∆(ρ + δ(1 − bI))
θCaIbC(ρ + δ) + bI(1 − θC)(ρaC + δaIbC + δaC(1 − bI))

(20)
is holds, then Σ > 0 emerges, resulting in a positive determinant. This is the necessary condition
for indeterminacy, in the sense that a positive determinant is necessary to rule out saddle-path
stability. Furthermore, since the determinant is positive and the trace is negative, both eigenvalues
are stable. Hence, any equilibrium path as diﬀerentiated by the choice of λ0 given K0 will always
be convergent.
Note that indeterminacy arises in this system when two conditions are met: i) θI − θC < 0
and ∆ > 0, ii) 1/Σ > 0. Although it is diﬃcult to reconcile these conditions intuitively, some
6observations related to the consumption market equilibrium can be made. If condition i) is satisﬁed,
then consumption supply becomes negatively sloping. This can be seen by diﬀerentiating (13) to
obtain
dCS
dp
=
aIr0(p)K − bIw0(p)L
p∆
+
aIr(p)
p∆
dK
dp
−
C
p
< 0 (21)
using the fact that r0(p) < 0, w0(p) > 0, dC/dK > 0 and dC/dL < 0 given θI − θC < 0 and
∆ > 0. These conditions imply that the dual relationship between the Stolper-Samuelson and the
Rybzcynski eﬀects is broken. Suppose it is not. Then with a labor intensive consumption good,
two things must be true: the Stolper-Samuelson theorem asserts that wage should rise given an
appreciation in the relative price of consumption, i.e. w0(p) > 0; the Rybczynski theorem asserts
that consumption goods supply should increase alongside an increase in the relative supply of labor,
i.e. dC/dL > 0. However, while w0(p) > 0 holds, we have dC/dL < 0 instead. Therefore, it must
be true that the dual relationship is broken. The phenomenon is ﬁrst discovered by Benhabib and
Nishimura (1998).
Now, what is interesting is that with a negatively sloping consumption supply schedule, the
consumption market equilibrium is unstable if the demand schedule cuts the supply schedule from
above. This can indeed occur with condition ii), which states that η must be small enough so that
1/Σ > 0. For instance, for η suﬃciently close to 0 by which the second condition is satisﬁed, the
slope of the consumption demand schedule, which is
dCD
dp
= −
1
η
λ
1
ηp
− 1
η−1 (22)
can always be made steeper than the slope of the consumption supply schedule with an appropriate
choice of η that is small.
Nishimura and Shimomura demonstrate that indeterminacy may arise in the Heckscher-Ohlin
world in the presence of sector-speciﬁc externalities whenever θI − θC < 0, ∆ > 0 and (20) hold.
However, these are exactly the same set of restrictions giving rise to indeterminacy in the autarkic
economy. Therefore, whenever indeterminacy holds under autarky, indeterminacy will hold in the
7Heckscher-Ohlin model once two of these autarkic economies are opened to trade.
The intuition behind this outcome is straightforward. Incomplete specialization in each country
under trade implies that the countries are functioning together as one integrated world economy.
In the trade literature, this feature is known as the integrated world equilibrium (IWE). With the
IWE, one may pool the endowments of all countries and treat all the countries as a single entity with
an enlarged endowment set. Clearly, as long as indeterminacy occurs in any one of the countries
under autarky, indeterminacy will occur for the IWE as well. That is, as long as indeterminacy
occurs in any one of the countries under autarky, it must occur for all countries under autarky since
each country is only diﬀerentiated by its factor endowments. Since indeterminacy exists in each
country under autarky to begin with, it is not surprising that IWE preserves indeterminacy under
trade ` a l` a Heckscher-Ohlin with incomplete specialization.
Case 3. Let θI − θC > 0 and ∆ < 0.
Case 3 states that the consumption good is capital intensive from the social perspective but
labor intensive from the private perspective. Interestingly, an unstable steady state may arise under
this scenario. The determinant of the Jacobian remains positive just as in Case 2. As mentioned
in the above, this implies that there are either two eigenvalues with positive real parts or two
eigenvalues with negative real parts, the outcome of which lies in the sign of the trace. A cursory
observation of the trace in (19) reveals that it may be positive. For instance, the trace is necessarily
positive if δ∆ + aC(ρ + δ) > 0, which is easily satisﬁed for a large enough ρ or aC, and a small
enough ∆.
Note that Case 3 is not examined in Nishimura and Shimomura (2002). A point that can be
made from examining Case 3 is that factor intensity reversal may break the saddle-point property of
the steady state. However, the steady state is not necessarily a sink. From our analysis, it is a sink
if the investment good is intensive in the factor that the investment good is used to accumulate. It
would be important to further investigate whether this statement is generalizable in a multisector
economy.
The downward sloping supply schedule for the consumption good remains. The condition
θI − θC > 0 implies r0(p) > 0 and w0(p) < 0. Therefore, the term aIr0(p)K − bIw0(p)L in (21) is
8positive. However, given that ∆ < 0, we have (aIr0(p)K − bIw0(p)L)/p∆ < 0. Since dp/dK > 0, it
is clear that dCS/dp < 0, implying that the supply schedule for the consumption good is downward
sloping yet again. As before, with an appropriate choice of η, the market clearing condition for the
consumption good becomes unstable.
IV. Conclusion
Using the autarkic variant of Nishimura and Shimomura, we show that non-saddle-path stability is
related to an unstable consumption goods market equilibrium, where the supply curve of consump-
tion goods is downward sloping and is intersected by the demand curve of consumption goods from
above. In addition, we point out that trade on its very own in the Heckscher-Ohlin framework does
not contribute to indeterminacy in Nishimura and Shimomura. Instead, indeterminacy arises in
the Heckscher-Ohlin framework because each country’s equilibrium path is already indeterminate
before trade.
Reference
Benhabib, J., Farmer, R. (1994) Indeterminacy and increasing returns to scale, Journal of Eco-
nomic Theory 63, 19-41.
Benhabib J., Farmer R. (1996) Indeterminacy and sector-speciﬁc externalities, Journal of Mon-
etary Economics 37, 421-443.
Benhabib J., Nishimura K. (1998) Indeterminacy and sunspots with constant returns, Journal
of Economic Theory 81 (1998), 58-96.
Nishimura K., Shimomura K. (2002) Trade and indeterminacy in a dynamic general equilibrium
model, Journal of Economic Theory 105 (2002), 244-260.
9

Indeterminacy and Market Instability

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics