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Do complementary factors lead to economic fluctuations?
*
Kirsten Ralf
The American University of Paris and Paris Graduate School of Management, 23 Rue de la Roquette, 75011 Paris,
France
Received 2 March 2000; received in revised form 12 October 2000; accepted 7 November 2000
Abstract
In a two-sector overlapping generations model with fixed coefficients the steady state may be indeterminate, if production in the consumption goods sector is more capital intensive than in the investment goods sector. Fluctuations are the consequence of capital-labour substitutions between the two sectors caused by changes in relative input prices. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Fixed coefficients; Two sectors; Indeterminacy
JEL classification: E32
1. Introduction
It is frequently stated that the observed fluctuations of macroeconomic aggregates are too large to be explained by the change of economic fundamentals only. Other sources for cyclical behaviour have to be looked for. The present paper concentrates on the substitution possibilities in a two-sector overlapping generations model with fixed coefficients. In doing this it draws a parallel with the two-sector model of optimal growth with infinite horizon by Nishimura and Yano (1995). They show that the optimal time path of capital accumulation may be chaotic for any rate of time preference, if production in the consumption goods sector takes place with a higher capital intensity than in the investment goods sector.
This capital intensity condition is also necessary for the steady state to be indeterminate in the overlapping generations model. Here, endogenous cycles may exist or the change of tastes and beliefs may cause economic fluctuations. Yet, we do not observe ergodically chaotic behaviour as in the Nishimura / Yano case. This is due to the fact that in optimal growth models with infinite horizon the
*Tel.:133-1-4805-4438.
E-mail address: ralfkirsten@aol.com (K. Ralf).
representative consumer takes into account the production functions of both, the consumption good and the investment good. There is no producer of investment goods maximizing profits and the price of the consumption good in terms of the investment good, the real wage, and the interest rate therefore do not enter into the analysis. In the overlapping generations model, on the contrary, prices are derived explicitly from a free entry condition on the market for the consumption good and the market for the investment good. The fact that prices, interest rates, and wages have to be positive excludes chaotic dynamics. The driving force for economic fluctuations, however, is in both cases the optimal change from the production of consumption goods to the production of investment goods and vice versa.
The rest of the paper is organized as follows: Section 2 describes the decision problem of the agents and the production of consumption and investment goods, respectively. In Section 3 market equilibrium is defined and the dynamic system is derived. In Section 4 the stability of the steady state and the dynamic behaviour of the dynamic system are analysed. Section 5 concludes the paper.
2. The model
2.1. Consumption and savings
Overlapping generations models provide a framework in which the hypothesis of complete markets is abandoned since the agents can trade only with other agents which live at the same period of time. It is assumed that individuals live for two periods. In every period there are thus two kinds of individuals, namely old individuals and young individuals. We assume L young individuals to plant
their lifetime consumption in an optimal way. They work only in the first period of their lives supplying one unit of labour inelastically. For the representative young individual, labour income, w ,t
t
has to be split between consumption expenditures, c , and savings, s , for next period’s consumption.t t With these savings the individual buys the capital stock from which he gains interest payments in the next period. The income of the second period of his lives is given as interest income only. Here, we choose a utility function with constant elasticity of substitution between current and future consumption. The young individual has to solve the following problem:
g g ( 1 /g)
where c denotes consumption of the individual in the first part of his life and ct t11 consumption of the individual in the second part of his life. s are the savings, w wage income, and rt t t11 interest payments on savings. p is the price of the consumption good in terms of the investment good. The investmentt
wt
]]]]]]]
st5 g/ (g21 ) (1)
11(rt11p /pt t11)
Savings are thus increasing with increasing wages and future prices and decreasing with current prices and future interest rates.
2.2. Production
We assume two output goods, a consumption good and an investment good, which are produced with different technologies. Production takes place with labour and capital at fixed coefficients. The production functions in the consumption goods sector and in the investment goods sector are given as:
c c c c
Ct5F (K , L )c t t 5minhK ,t aLtj
i i i i
It5F (K , L )i t t 5mminhK ,t bLtj, m.1.
c c i i
where K and L denote capital input and labour input in the consumption goods sector, K and Lt t t t
capital input and labour input in the investment goods sector. In per capita terms we denote
c c c i i i
kt 5K /L capital per employed worker in the consumption goods sector and kt t t5K /L capital pert t
employed worker in the investment goods sector. With total labour force L divided between the twot
c i c c i i
sectors Lt5Lt1L and relative employment shares denoted by lt t5L /L and lt t t5L /L , we havet t
c i
lt1lt51. (2)
The capital stock per head in the economy is
c c i i
kt5l kt t1l k .t t (3)
Efficient production in both sectors requires:
c c c
Kt5aLt→kt5a (4)
i i i
Kt5bLt→kt5b. (5)
The capital stock per worker has to be constant in both sectors. Production in the consumption goods sector takes place with a higher capital intensity than in the investment goods sector, if a.b. The input factors are assumed to be mobile across sectors and they are homogeneous. In economic equilibrium wages and interest rates are the same in both sectors. We assume a free entry condition in both sectors, i.e., profits in both sectors are zero. Then:
wt
]
pt5 1rt (6)
a
wt5bm2br .t (7)
i
it5bml ,t (8)
with i output in the investment goods sector per head, I /L .t t t
3. Equilibrium and the dynamical system
The capital stock is assumed to depreciate fully after one period. Output of the investment goods sector can be used in the following period, i.e., the capital stock in period t11 equals the output of the investment good in period t. Equilibrium on the market for the investment good therefore requires:
i
kt115it5bml .t (9)
On the other hand, investment in period t has to equal savings in period t such that in per capita terms we have:
st5i ,t (10)
with st5S /L .t t
Since the capital stock per worker in each sector is a constant fluctuations can occur only because of relative factor price changes and because of intertemporal substitution of consumption. The dynamical system is derived in k and p . It can be written ast t
mb ]]
kt115 b2a (kt2a) (11)
which follows from equations (2), (8) and (9) and
mb
which follows from Eq. (10), rearranging the savings function (1) using Eqs. (2)–(8). For prices, wages, interest rates, and the capital stock to remain positive, certain restrictions on the variables depending on the values of the parameters of the system can be found. If the investment goods sector is more capital intensive than the consumption goods sector, b.a, we have the following conditions:
b.kt.a
kt b
] ]
m ,pt#m ;t.
a a
b,kt,a
The steady state solution for the capital stock per head can be derived directly as:
mab
¯ ]]]]]
k5 .
(m21)b1a
¯
The steady state k is positive since m.1 and it is always contained in the range of definition. For the price of the consumption good the steady state solution cannot be derived in such a simple
¯
way. The price of the consumption good in the steady state p has to fulfill the following condition:
g21
For given parameter values a, b, m, and g satisfying the above assumptions, there exists a unique steady state solution, i.e., a zero of the functionj within the range of definition. This can be seen by calculating the value of the function j at the boundaries of the range of definition and applying the mean value theorem.
4. Stability and indeterminacy
The steady state is locally asymptotically stable, if both eigenvalues of the Jacobian of the dynamical system derived at the steady state are less than unity in absolute values. These eigenvalues are:
For a,b, the perfect foresight dynamics is always determinate, since the first eigenvalue l1 is always greater than 1, the steady state is a source if l2, 21 and it is a saddle point if l2. 21. Fora.b, the steady state may be indeterminate. As an example takeg5 20.5,b50.3,a50.7,
¯ ¯
and m51.1. Then the steady state values are k50.3146 and p50.4715. Both eigenvalues are negative, l15 20.8265 and l25 20.0054 and the steady state is indeterminate. Decreasing the capital intensity in the consumption goods sector decreases the first eigenvalue and for a50.63 we have l15 21, taking the other parameter values as before. Here regular cycles of order 2 exist depending on the initial condition. On such a cycle the capital stock per head is low whenever the price of the consumption good in terms of the investment good is high and vice versa. We have a low capital stock per head because with a high price of the consumption good today and the expectation of a low price of the consumption good tomorrow, the expected interest rate is high and the current wage is low. This makes a shift of production from the consumption goods sector to the investment goods sector reasonable since the consumption good is produced with a higher capital intensity. We have thus a high investment today which increases the capital stock tomorrow. Decreasing the parameter value further, the steady state becomes a saddle point and is therefore determinate.
5. Conclusion
The analysis of the present paper was dedicated to the dynamical properties of a two-sector overlapping generations model. Substitution of labour and capital between the sectors were seen as relevant for indeterminacy of the steady state. Intertemporal substitution of consumption does not only change the relative price of current consumption with respect to future consumption, but also the interest rate on capital. With a higher capital intensity in the consumption goods sector this effect is larger. This capital intensity condition is also important in the two-sector OLG-model derived by Galor (1992). Assuming substitutability of input factors within the sector, the production functions are different and Eqs. (5) and (6) are replaced by the condition that factor prices equal marginal productivity. Then, the production technologies have to be relatively dissimilar and the income effect to be sufficiently stronger than the substitution effect in order to lead to indeterminacy. With the above specified utility function indeterminacy is not possible.
The result of indeterminacy deserves particular attention in the light of the findings of Chiappori et al. (1992). Their main result is that whenever the steady state is indeterminate, there is a continuum of stationary sunspot equilibria of finite order in any neighbourhood of the steady state. For the model on hand this means that self-fulfilling economic fluctuations are possible in the case of a capital intensive consumption goods sector.
Acknowledgements
References
Chiappori, P.-A., Geoffard, P.-Y., Guesnerie, R., 1992. Sunspot fluctuations around a steady state: the case of multi-dimensional, one-step forward looking economic models. Econometrica 60, 1097–1126.
Galor, O., 1992. Two-sector overlapping-generations model: A global characterization of the dynamical system. Econo-metrica 60, 1351–1386.
