Time to build in dynamics of economic models II: models of economic growth

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Abstract

In this paper we investigate the relationship between the time delay for building investment goods and cycles under the conditions of general equilibrium. We introduce a delay parameter in the spirit of Kalecki into the simplest class of models of growth theory. They encompass a wide range of models of neoclassical economics with a production function which shows constant return to scale, and also multi-sector models. Kalecki’s delay parameter is introduced into the equation for the accumulation of capital, assuming that the growth of the amount of capital at time t is a function of the total output of capital at time tT.

We consider the simplest case, when the time of building investment goods is constant, but we also suggest a more realistic economic situation, when the building time is represented by a distribution function. We show that the limit cycle is indeed the optimal time path in a model of economic growth.

Introduction

Since Jevons economists have been aware that a delay in production could cause cycles in the economy, however it was Kalecki who first formally and systematically considered the problem [8]. In the 30s the subject was also studied in two important works by Frisch and Holme [4] and James and Beltz [7]. They considered the question was whether the time delay could induce cyclic behaviour in the model. Later Larson [11] used a model with a time delay to explain the observed cycles in the price of pork. Kalecki’s approach was revived in the work of Zak [1], [16], [17], who argue that the delay in the production of investment capital influences the fluctuations in the model of economic growth. He made an thorough review of possible application of functional differential equation in economics.

Although Kalecki’s idea of the time for building investment capital appeared in the context of explaining changes in the market cycle, in this work we will apply them to the theory of growth, using advanced methods from the theories of dynamical systems and bifurcation which were unavailable to Kalecki. In theories of growth the time delay takes the form of a delay in the production of capital, that is the time which passes before new capital is produced and available.

The idea of introducing a time delay into models of growth theory is not new. A production delay was used in models of optimal growth by Cass and Yaari in the expanded version of Solow’s growth theory [3]. In a more complicated situation Rustichini [14] shows that a pair of delays, the capital building time and the production time generate cyclic behaviour in the theory of optimal growth in an economy with a single commodity.

The aim of this work is to analyse the simplest situation of the influence of time to build on the dynamics of economic growth in the spirit of Kalecki. We want to show in the simplest possible way that in standard models of economic growth, the mechanism of generating cyclic behaviour through the time delay parameter is universal (generic). We will show this fact in models with and without optimization. We will thus prove that Kalecki’s ideas continue to drive contemporary research in economics more than 100 years after his birth.

Expanding Kalecki’s approach we will introduce the delay parameter into the simplest and most general class of models possible in the theory of growth. The dynamics of these models is governed by (or can be reduced to) one equation of the form k̇=Lm−1h(k), where k=K/L (K––capital, L––labour; quantities written in lower case are evaluated per capita), m––the level of homogeneity of the production function Y=F(K,L)=Lmf(k). When m=1 then the governing function (the right-hand side of the equation) takes the form of the dynamical system k̇=g(k)−nk, where g(k) (examples are given in Table 1).

Such systems encompass a wide range of models of neoclassical economies with a production function which exhibits constant returns to scale (i.e. a function homogeneous of degree one).

We introduce Kalecki’s delay parameter into the equation for the accumulation of capital, assuming that the growth of capital at time t is a function of the amount of capital held at time tT, since newly-produced capital is available for use by time T. We consider the simplest case, where the time for building capital is constant, but we also suggest a generalization of our model, aiming for a more realistic description of the economy. We emphasise that our primary goal is not so much a confrontation of the model with empirical data (that will be the subject of our future research), as showing on the simplest possible model the effects of cyclic economic changes induced by the delay parameter in growth theory models. We will show that Kalecki’s idea in introducing the delay parameter into the dynamics of economic processes is still attractive and viable, allow simple explanation of the sources of inherent cyclic behaviour of the economy, which makes this mechanism obvious and ubiquitous.

Through looking at Kalecki’s achievements through contemporary mathematical methods we can discover his true genius and discern his deep economic intuition, which always preceded a mathematical analysis of the model—in other words, the mathematics was always only the tool of proof, not an end in itself. The laws of macroeconomic dynamics which Kalecki demonstrates can, just as in physics or biology, be formulated in the form of differential equations. In my opinion, M. Kalecki was deeply aware of this fact, and looked for the source of complex economic behaviour in a simple model.

The beginnings of the modern theory of growth can be traced to the moment when Ramsey constructed his theory of the optimisation of consumer choice with time [13], while Pontriagin expanded it and formulated it in strictly mathematical form [12]. It is interesting to carry out an experiment to see how viable Kalecki’s idea is if we add dynamic optimisation to the model. To this end we consider an economy with infinitely long-lived individuals. The preferences of representative persons are defined by a continuous and strictly rising, upwardly convex utility function U(c(t)), which meets the standard Inada conditions and a subjective discount rate ρ>0. In this economy the period T>0 is the period required for the production and installation of new capital equipment. Consumers save in relation to their earned income, the interest rate and their wealth.

The problem of planning for an infinite temporal horizon in such an economy is given by the functionalmaxc(t)0U(c(t))e−ρtdtwith the constraintk̇(t)=e−nTg(k(t−T))−nk(t)−c(t),where k(t)=φ(t) for every t∈[−T,0], consumption 0<c(t)⩽g(k(tT))−nk(t) and n∈[0,1]. If we include the depreciation rate of capital δ∈[0,1], then a term δ(t) appears on the right-hand side of the constraint equation.

For the problem formulated in this way we carry out a generalized analysis of the optimal conditions based on Hamilton’s principle and Pontriagin’s maximum principle. We obtain a nonlinear equation of two equations with a delayed argument, for which we prove the existence of a limit cycle which represents the cyclic behaviour of the amount of capital k(t) and level of consumption c(t). In this way we explicitly show the presence of Hopf’s cycle as the optimal solution––the optimal evolutionary path––for the problem of the maximum utility function, since such a solution fulfils the appropriate Euler equations of motion.

The theorem explicitly shows that Kalecki discovered a real and universal mechanism leading to cyclic behaviour inherent in models of growth; a mechanism which clearly stands the test of time. His model is firstly simple, and secondly extracts from a real process the most important factor, which determines the evolution of the phenomenon. What is more, numerical analyses of the parameters of the cycle (period, amplitude) give reasonable values for the fitted real constants of the model. This is the subject of our current research.

Section snippets

Growth theories as one-dimensional dynamical systems

We consider the dynamics of fundamental theories of growth in the phase plane (capital K, labour L). Initially it is more convenient to consider the problem of dynamics very generally, not concentrating on the economic meaning of the equations (K=x, L=y). It takes the form of a two-dimensional autonomous dynamical system, whose right-hand sides are homogeneous functions of degree mẋ=F(x,y),ẏ=G(x,y),where (x,y)∈R2 and F(·) and G(·) are functions of class C1 on R2⧹{0,0}. The homogeneity of the

Models of optimal growth with Kalecki’s delay time

We now pose the question of optimal economic growth and consider a growth model in which society maximises the utility of consumption in an infinite temporal horizon.

Households, receiving reward for labour, save by allocating their assets, depending on their earnings, wealth and the interest rate. We treat the household as a continuum of individuals living for ever (the assumption is that the family is immortal, and made up of succeeding generations) so that the utility is maximised on the

Conclusion

The aim of this paper was to show the vitality of Kalecki’s ideas using the example of his proposition of the time for building investment goods as the source of cyclic behaviour in economic systems. For proof modern methods of analysing bifurcations in functional equations (unknown in the nineteen-thirties) were used. The author earlier used these methods to describe changes in market conditions [10], [15]. Thanks to this way of looking at Kalecki’s idea we have been able to follow how it has

Acknowledgements

I would like to thank Paul J. Zak, Sue Ann Campbell for the very interesting papers and Adam Krawiec for discussions, without which this work would not have come about.

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