The Harrod–Domar Model

The year 1939 was marked by the appearance of Harrod (1939) which gave a major impetus to the development of growth theory. Harrod was concerned with the problem of probable inconsistency between the conditions of full employment and a steady state of economic growth. The conditions under which full employment is secured are necessarily of a short-run nature, while a steady state of growth requires certain fundamental dynamic equations to be satisfied.

One of the fundamental equations introduced by Harrod expresses the equilibrium of a steady state of growth:

$$ {g}_w{c}_r=s, $$

where gw is the warranted rate of growth, cr is the required capital coefficient, and s is the saving coefficient. Harrod’s warranted rate of growth gw is defined as

the rate of growth, if it occurs, will have satisfied all members of the economy, while the required capital coefficient cr is defined as the requirement for new capital divided by the increment of total output to sustain which the new capital is required.

Harrod assumed that the saving ratio s is a constant, to be dependent upon the psychological and social characteristics of the economy. Under the assumptions of the neutrality of inventions and of the constancy of the rate of interest, the required capital coefficient cr is also a constant. If the rate of growth g is higher than the warranted rate of growth gw, then the capital coefficient c is lower than the required capital coefficient cr. The accumulation of capital then would be insufficient to sustain a steady state of growth. On the other hand, if g is lower than gw, c is higher than cr. Some portion of capital would be necessarily left unutilised at a steady state of growth. Harrod thus gave a simple proof for the instability of processes of economic growth in a capitalist economy.

The analysis of the instability of the process of economic growth in a capitalist economy, as discussed by Harrod, was one of the major attempts to extend Keynes’s General Theory and regarded as one of basic pillars upon which the modern growth theory has been built.

In the General Theory, Keynes attempted to formulate institutional arrangements of the modern capitalist economy in terms of a coherent macroeconomic analytical framework and showed that the allocative mechanism in a decentralized, private-enterprise market economy resulted in a state of involuntary unemployment, unless stabilizing fiscal and monetary policies are effectively utilised. Harrod’s dynamic analysis may be regarded as an extension of the Keynesian analysis to cover the economy at a steady state of growth. However, it was after the end of World War II that the Keynes–Harrod analysis received fuller attention. Indeed, the problems of economic growth and full employment were at the centre of attention of the political and social planners in major capitalist countries, both developed and less developed, and the period of approximately twenty-five years up to the end of the 1960s may be regarded as one of stable economic growth, largely due to the adaption of what may be properly termed a Keynesian policy.

Harrod’s analysis was further elaborated by Evsey Domar (1946), where some of the underlying assumptions in the Harrod’s model were more explicitly brought out and the long-run implications were discussed in more detail. While Keynes was primarily concerned with the role of investment as an instrument for generating income, both Harrod and Domar focused their attention upon the effect of investment to increase productive capacity.

  1. (a)

    The amount of capital and labour required to produce a unit of output are both technologically given.

  2. (b)

    A constant fraction of income is saved.

  3. (c)

    The rate of increase in labour forces is exogenously given.

  4. (d)

    Inventions are neutral in the sense of Harrod and the rate of increase in labour efficiency is exogenously given.

Under these assumptions, the Harrod knife-edge instability of a steady state of economic growth was rigorously proved by Domar (1946). The equality of the natural rate of growth with the warranted rate of growth, on which the existence of a steady state of growth crucially hinges, occurs only for an economy for which the saving ratio, the capital coefficient, and the rate of increase in labour forces satisfy particular relationships. Any path of capital accumulation in such an economy generally exhibits an unstable feature; either involuntary unemployment tends to be increased without limit or capital continues to be accumulated in such a manner that the stock of unutilized capital piles up indefinitely.

Neoclassical Growth Models

The instability property of the process of capital accumulation inherent in the Harrod–Domar model may crucially hinge upon the nature of the basic assumptions concerning technical and social structure of the capitalist economy in question.

In particular, the assumption of a constant capital coefficient seems to be a pivotal one in the Harrod–Domar analysis. The neoclassical growth models, developed by Tobin (1955), Solow (1956), Swan (1956), Ara (1958) and Meade (1961), among others, take an explicit note of the possibility of the substitution between capital and labour and conclude that growth paths in a capitalist economy have a trend to converge to a steady state.

A neoclassical growth model typically is formulated in terms of a one-commodity economy, where output is produced by two factors of production, capital and labour. Total output Y is given by the aggregate production function

$$ Y=F\left(K,L\right) $$

where K and L represent inputs of capital and labour, respectively.

The possibility of substitution between capital and labour then is expressed by the assumption that the aggregate production function F(·,·) is continuously differentiable so that the marginal rate of substitution between capital and labour is well defined. Constant returns to scale is also assumed so that the aggregate production function F(·,·) is linear and homogeneous.

At each time t, real output Y(t) is produced by using the stock of capital K(t) and labour services L(t). A constant portion of real income Y(t) is assumed to be consumed and the rest is saved. Net investment is assumed to be equal to savings. If we denote by s the saving ratio, then the rate of accumulation of capital is given by

$$ \frac{\mathrm{d}K(t)}{\mathrm{d}t}= sY(t)= sF\left[K(t),L(t)\right], $$

while the available labour is assumed to grow at a constant rate n:

$$ \frac{\mathrm{d}L(t)}{\mathrm{d}t}= nL(t). $$

Growth paths in a neoclassical model then are completely described by these two differential equations, which may be, due to the constant returns to scale assumption, reduced to the following:

$$ \frac{\mathrm{d}k(t)}{\mathrm{d}t}= sf\left[k(t)\right]- nk(t), $$
(1)

where k(t) = K(t)/L(t) is the capital–labour ratio at time t and f [k(t)] = F [k(t),l] is real output per capita at time t.

The assumption of diminishing marginal rates of substitution between capital and labour may be expressed by the concavity of the per capital output function f (k); namely,

$$ {f}^{{\prime\prime} }(k)<0,\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ \ \ k>0. $$

Hence, the solution paths to the differential equation (1) tend to converge to the stationary state k* of the system (1):

$$ sf\left({k}^{\ast}\right)={nk}^{\ast }. $$

The existence of the stationary state k* is generally guaranteed, particularly when f′(0) = ∞ and f′(∞) = 0.

The neoclassical growth models have many variants, in particular concerning the saving ratio assumption. While the constancy of the saving ratio s has been adapted in most of the neoclassical models, some have taken an explicit cognizance of the fact that it may depend upon the level of per capita real income and the rate of interest. The assumption that the rate of population growth is exogenously given has been cricially examined, particularly by Buttrick (1960). The stability property, however, has been verified in most of the neoclassical models.

Kaldor’s ‘Stylized’ Facts

The perspective of the growth theory may be best illustrated by the six ‘styled’ facts put forward by Nicholas Kaldor (1961), which have been obtained by observing the process of economic growth in capital economies. They are (1) the continued growth in the aggregate output and in the per capita output at an ever-increasing rate; (2) the capital-labour ratio has continuously increased; (3) the rate of profit on capital has been steady, significantly higher than the real rate of interest, at least for most of the more advanced capitalist economies; (4) the steady capital coefficient has been maintained; (5) the share of investment in output has been highly correlated with the share of profits in income; (6) the divergence of the long-run rate of increase in labour productivity and of the aggregate output in different economies.

Some of Kaldor’s ‘stylized’ facts may not be necessarily borne out by the observed statistical data, particularly in the latter half of the 20th century. However, they may be taken as a convenient starting point for the construction of theoretical growth models. In the light of Kaldor’s ‘stylized’ facts, both the Harrod–Domar model and neoclassical growth models may need a re-examination of the basic assumptions. Particular attention was paid to the apparent inconsistency between the continued increase in the capital–labour ratio and a constant capital-coefficient. This indeed was one of the problems Harrod addressed himself in Harrod (1937), and later elaborated in Joan Robinson (1937–8). It is related to the role which inventions have played in the process of economic growth. A technical invention was defined by Harrod to be neutral if the optimum capital coefficient remains constant when the rate of interest is kept constant. It was shown by Joan Robinson that if a technical invention is neutral in Harrod’s sense, then the increase in the efficiency of labour is determined independently of the stock of capital being utilized. In terms of the aggregate production function, the characterization of the Harrod neutrality was explicitly brought out in Uzawa (1961a). Let technological conditions change over time, so that the aggregate production function may be represented by

$$ Y=F\left(K,L,t\right). $$

Then it was proved that technical inventions are neutral in the sense of Harrod if and only if the aggregate production function Y = F(K, L, t) is written as

$$ Y=G\left[K,A(t)L\right], $$

where A(t) indicates the efficiency measure for labour at time t, to be determined independently of K and L.

It was then shown in Uzawa (1961a) that, if technical inventions are neutral in the sense of Harrod, the steady state of the neoclassical growth model is characterized by the conditions that the capital coefficient remains constant while the capital–labour ratio continues to increase at the rate equal to that of labour efficiency, and paths of economic growth necessarily converge to the steady state.

In the neoclassical growth models, the rate of profit has been largely identified with the rate of interest. At the same time, savings were regarded as determined by total income, largely independently of the way total income is divided between the factors of production. These properties are related to the way the working of an economic system is viewed. In the traditional neoclassical economic theory, the working of economic activities is described by the representative homo economicus who behaves himself in accordance with the subjective value judgement he possesses independently of the economic environments and historical and social circumstances. The representative homo economicus acts as a producer and a consumer at the same time, and he is the owner of all the factors of production, including labour. He divides his income between consumption and savings in such a manner that his intertemporal preference ordering is satisfied. The aggregate savings then are channelled into investment. Full employment necessarily results in such a neoclassical economy, and investment is automatically determined by the amount of savings.

Marxian and Kaldorian Growth Models

Unlike the neoclassical growth models, the Marxian and Keynesian growth models have been built upon the basic premises that a capitalist economy is composed of different, occasionally conflicting, classes, and patterns of economic growth would reflect the interaction of classes in the process of resource allocation and income distribution. A typical Marxian growth model is the one where the neoclassical production function is assumed to summarize the production processes, but the amount of savings depends upon the way total product is divided between wages and profits. The accumulation of capital then is given by dK dt

$$ \frac{\mathrm{d}K}{\mathrm{d}t}={s}_PP+{s}_WW, $$

where P and W stand for profits and wages, respectively, and sP and sW are the average propensities to save out of profits and wages, respectively.

The simplest Marxian case may be represented by the conditions: sP = 1 and sW = 0. The stability of growth paths has been shown for the general case when

$$ 0\leqq {s}_w<{s}_p\leqq 1, $$

and profits P and wages W are determined by marginal products capital and labour, respectively.

Kaldor (1956) introduced a slightly different model, in which the distribution of income Y between profits P and wages W is so determined as to equate the forthcoming savings S with investment I, the latter being independently determined by entrepreneurs. Namely, profits P and wages W are determined by the following two equations:

$$ {\displaystyle \begin{array}{rcl}Y& =& P+W\\ {}I& =& {s}_PP+{s}_WW,\end{array}} $$

where Y and I are exogenously given.

The stability of growth processes in a model where distribution is determined by what Kaldor has termed the Keynesian theory of distribution is related to the way entrepreneurs decide total investment I.

The Marx–Kaldor theory of economic growth was further elaborated by Pasinetti (1962).

Two-Sector Growth Models

The neoclassical growth models have been based upon the concept of the aggregate production function which relates the total output measured in terms of a certain homogeneous quantity to the inputs of labour and capital. A number of attempts were made to extend the analysis to cover the situation where there exist various types of goods which are produced by different technologies. Particular attention was paid to the two-sector growth models, where there are two types of goods, investment goods and consumption goods, to be produced by two factors of production, labour and capital.

The simple case where both goods are produced with constant coefficient technologies was discussed by Shinkai (1960). Shinkai’s main conclusion was to relate the stability of the growth process in such a two-sector model to the relative intensities of two goods. Shinkai’s model was extended to the case in which substitution between capital and labour is possible in the production of both investment goods and consumption goods (Meade 1961; Uzawa 1961b, 1963).

The basic premises upon which the two-sector growth models have been built may be briefly summarized. Two-sector growth models consider an economy in which there are two sectors, one producing consumption goods and the other investment goods, to be labelled C and I respectively. Both consumption goods and investment goods are composed of homogeneous quantities and produced by two homogeneous factors of production, labour and capital. Consumption goods are instantaneously consumed, while capital goods, being the accumulation of investment goods, depreciate at a fixed rate. In each sector, production is subject to constant returns to scale and diminishing marginal rates of substitution between capital and labour. Joint products are excluded and neither external economies nor dis-economies exist. The quantity of each good to be produced is related to the quantities of capital and labour to be allocated in each sector. The production function in each sector then is assumed to be a linear homogeneous, continuously differentiable function of two variables, capital and labour.

At each moment, total quantity of capital available to the economy is determined as the result of past investment, while the available labour forces are assumed to be exogenously given, to grow at a certain fixed rate. Then the quantities of capital and labour allocated to the two sectors are constrained by the quantities of capital and labour available at that moment.

In two sector growth models, both outputs and factors of production are allocated in perfectly competitive markets, so that in each sector the wage is equal to the marginal product of labour and the rentals of capital to the marginal product of capital. In each sector, the optimum capital–labour ratio then is uniquely determined by the wage–rental ratio. Consumption goods are defined to be always relatively more (or less) capital intensive than investment goods if the optimum capital–labour ratio is higher (or lower) in the C-sector than in the I-sector for all possible wage–rentals ratios.

The allocation of capital and labour between two sectors is uniquely determined if the relative price of two goods is given. If consumption goods are more capital intensive than investment goods, then the relative price of consumption goods in terms of investment goods will be increased when the wage–rentals ratio is decreased.

The relative price of two goods will be determined once the demand conditions are specified.

In a Marxian situation where labourers consume all their wages and capitalists save all their profits, the short-run equilibrium will be uniquely determined if consumption goods are always more capital intensive than investment goods. Paths of growth equilibrium have been shown to be stable under the same capital-intensity condition.

On the other hand, in a neoclassical economy where a fixed proportion of total income is spent on consumption and the rest is saved, the short-run equilibrium has been shown to be uniquely determined, regardless of the capital-intensity condition. However, the stability of growth of growth equilibrium is established only for the case where consumption goods are always more capital intensive than investment goods.

The concept of capital utilized in the two-sector growth models has been based upon the neoclassical theory in the sense its use can be shifted from one sector to another without incurring any additional cost or any time lag. The model developed in Inada (1966) recognizes that most of capital embodied in modern technologies cannot be freely shifted from one sector to another and has to stay in the sector where it has been invested.

This is related to the line of model construction, which may be traced back to Fel’dman (1928), and paves a way to the development of the Keynesian theory of economic growth.

As for growth models with heterogeneous capital goods, a number of what may be termed vintage models have been constructed and their implications on the pattern of growth equilibrium have been analysed in detail. A particular interest is with the one built by Solow (1960). Solow’s model considers an economy which is composed of homogeneous labour and capital equipment of various vintages, each of which embodies the technologies of the time when it is built. If the technical progress embodied in vintage capital in Solow’s model is neutral in the sense of Harrod, then it has been proved that any path of growth equilibrium asymptotically approaches the steady state where the vintage distribution of capital equipment remains stationary (Uzawa 1964b).

Further contributions to the theory of vintage capital and the related topic of induced investment have been made by Atkinson and Stiglitz (1969), Phelps (1966), Kennedy (1964), and Leif Johansen (1959).

Optimum Economic Growth

One of the basic problems in economic planning, in particular in underdeveloped countries, is concerned with the rate at which society should save out of current income to achieve an optimum growth. It is closely related to the problem of how to allocate scarce resources at each moment of time between the production of consumption goods and investment goods. It was analysed within the context of the two-sector growth models, as introduced in Meade (1961), Srinivasan (1964), and Uzawa (1964a). The Srinivasan–Uzawa analysis focused its attention on evaluating the impact of roundabout methods of production upon the welfare of the society, as expressed by a discounted sum of per capita consumption over time. It abstracted from the complications that would arise by taking into account those factors such as changing technology and structure of demand, the role of foreign trade (in particular of capital movements) and tax policy that may be generally regarded as decisive in the course of economic development. It is postulated that a certain quantity of consumption goods per capita is required to sustain a given rate of population growth. The constraint will become effective for an economy with relative shortage of capital, and it results in the phenomenon of ‘the vicious circle of poverty’.

The structure of optimum paths of capital accumulation differs significantly according to whether consumption goods are relatively more capital-intensive or less capital-intensive than investment goods. If consumption goods are always more capital-intensive than investment goods then there exist two critical–labour ratios, k*I and k*C such that, if the initial capital–labour ratio of the economy is less than k*I, then, along the optimum path, the economy produces just enough consumption goods to meet the minimum requirements and devotes the rest of scarce resources in the production of investment goods until the time when the economy’s capital–labour ratio reaches the capital ratio k*I, and from then on it proceeds to produce both investment goods and consumption goods, keeping the imputed price of the two outputs constant and approaching the stationary state. On the other hand, if the initial capital–labour ratio of the economy is larger than the critical ratio k*C, then, along the optimum path, the economy is specialized to the production of consumption goods until the capital–labour ratio is reduced to the critical ratio k*C. When the critical ratio k*C is reached, then both consumption goods and investment goods are produced, asymptotically approaching the stationary state.

Two-Class Models of Economic Growth

Most of the growth models described above have been built upon premises directly involving aggregate variables, without specifying the postulates which govern the behaviour of individual units comprising the national economy. In particular, the specifications of aggregate savings have seldom been based upon analysis of rational behaviour concerning savings and consumption. Similarly, the aggregate behaviour of investment has not been derived from the rational behaviour of business firms; instead, it has been postulated in terms of ad hoc relations involving market rate of interest, rate of profit, and other variables. In Uzawa (1969), an attempt was made to build a formal model of economic growth for which the aggregate variables such as consumption, savings and investment are described in terms of individual units’ rational behaviour. A private-enterprise economy is divided into two sectors; the household sector and the corporate sector. Households decide how to consume goods and services produced in the corporate sector; they are endowed with labour and possess the securities issued by the corporate sector. A business firm in the corporate sector consists of a complex of fixed factors of production, such as factories, machinery, and others, including managerial abilities and technological skills. Real capital is regarded as an index to measure the productive capacity of such a complex of capital goods endowed within the firm at each moment of time; it is increased as the stock of fixed factors of production is accumulated as the result of investment activities. The relationships between real investment and the resulting increase in real capital may be characterised by what may be called the Penrose curve, which incorporates the basic tenure of the analysis expounded by Edith Penrose (1959). Each business firm plans the levels of employment and investment in such a way that the discounted present value of expected future net cash flows is maximized. The desired level of investment then depends upon the expected rate of profit and the market rate of interest.

The behaviour of an individual household may be analysed in terms of Irving Fisher’s theory of time preference, as formulated by Koopmans (1960). The marginal rate of substitution between current and future consumption is represented by the Fisherian schedule, which relates the rate of time preference to the current level of consumption and to the utility level for all future consumption. The optimum propensities to consume and save are then derived as functions of the expected market rate of interest and permanent income.

These analyses are put together to formulate a two-class model of economic growth, where the stability of the short-run and long-run equilibrium is discussed in terms of the Penrose curve and the Fisherian schedule.

Keynesian Models of Economic Growth

Whether or not the dynamic allocation of scarce resources through the market mechanism may achieve stable economic growth is not simply a matter of theoretical interest, but is indispensable in the discussion of the effect of public policy. There are two opposing approaches to the problem of dynamic stability of the market mechanism. One approach is based upon the analytical framework of neoclassical economic theory, and the other is discussed in terms of the Keynes–Harrod theory of economic dynamics. The neoclassical approach derives the conclusion that the process of equilibrium growth in a market economy is dynamically stable and the conditions of full employment generally prevail. The Keynes–Harrod approach, on the other hand, concludes that the market allocation of scarce resources is inherently instable in a modern capitalist economy and that maintaining stable economic growth, together with full employment and price stability, is akin to walking on the edge of a knife. It may be worthwhile to examine the reason why these two opposing conclusions concerning the stability of the growth process in a market economy are obtained.

One of the crucial elements which distinguish one approach from the other is concerned with the concept of capital. In the neoclassical approach, capital refers to the various factors of production which have been accumulated through refraining from consumption in the past. Production is carried out by utilizing capital together with labour and other variable factors of production which are obtained through markets. In the neoclassical theory, the phenomenon of the fixity of capital has not been handled explicitly, so that the market price of the stock of capital is the same for newly produced capital goods and for existing capital goods which are the result of investment in the past. Any member of the economy may engage in productive activity either by purchasing capital goods or by renting the services of capital goods, and at the same time may engage in consumption activity, resulting in the disappearance of the essential difference between consumers and producers. Accordingly, various members of the economy may hold either physical or financial assets in whatever manner they prefer. Investment, as an accumulation of fixed capital, loses its essential meaning, and the difference between rate of interest and rate of profit disappears. Markets for outputs and factors of production are assumed to be perfectly competitive.

Under these neoclassical assumptions, Say’s Law is shown to hold true and full employment necessarily results. Growth equilibrium is generally shown to be dynamically stable.

The stability of monetary growth in neoclassical theory has been similarly handled, particularly by Tobin (1965), Sidrauski (1967), Harry Johnson (1966), Levhari and Patinkin (1968) and Uzawa (1974). The neoclassical theory of monetary growth typically ignores the institutional details of the mechanism by which money is supplied by the central bank, and money is assumed to be distributed to the economic units of the economy through transfer payments. In neoclassical theory, money performs the function of a consumer good, contributing to the increase in the level of utility for each individual, and it may also serve the role of a factor of production, increasing the marginal products of real factors of production. The demand for money thus may be assumed to depend upon the market rate of interest and the level of income.

The aggregate demands for real capital and money are related to the price level, and the equilibrium price level then is determined as the level at which the demand for the holding of money balances is equated to the supply of money. The rate of interest then becomes the real rate of interest plus the expected rate of price increase. The stability of monetary growth in the neoclassical setting is thus related to the way expectations concerning future price changes are formed. If expectations are adjusted according to adaptive expectations of the Cagan–Nerlove type, then equilibrium growth in the neoclassical model of monetary growth is dynamically stable provided the speed of adjustment in expectations is relatively small, as one would expect from similar analyses such as Cagan (1956).

The Keynesian model of monetary growth, on the other hand, may be formulated in terms of the two-class economy as briefly outlined above. The private sector consists of households and business firms, and the government sector provides various public goods and services which are financed through taxes or the issue of money. Money supply is increased to meet fiscal deficits, but money supplied through open market operations changes the pattern of portfolio balances in the economy.

The market system is divided into three types; the goods and services market, the labour market, and the financial market. The goods and services market and the financial market both are assumed to be instantaneously adjusted to the equilibrium positions. In the labour market, however, when the demand for labour exceeds the supply, the money wage rate is instantaneously adjusted to the equilibrium level, but when the demand for labour is less than the supply, the money wage rate remains at the current level, resulting in involuntary unemployment. Money and short-term securities are dealt with in efficiently organized markets, but price adjustments for long-term securities are not necessarily efficient and there is a time lag in the adjustment of long-term securities prices.

The schedule of the aggregate supply price is then defined; it relates the aggregate amount of goods and services measured in wage-units to total employment in such a manner that entrepreneurs’ profits are maximized subject to the constraints imposed by the conditions prevailing in the economy. On the other hand, aggregate demand is determined by the behaviour of households, business firms and government concerning consumption and investment.

Equilibrium in the goods and services market is obtained when aggregate supply is equal to aggregate demand. The level of total employment at the equilibrium does not correspond to the full employment level, resulting in the situation of involuntary unemployment. The effective demand is closely related to the level of investment, which in turn is influenced by the market rate of interest in the long-term securities market.

The market rate of interest is determined by the equilibrium conditions in the markets where money and short-term securities are transacted. Thus, in order for the economy to sustain the conditions of full employment and continuous economic growth, certain conditions have to be satisfied between the long-term rate of interest, the rate of increase in money supply, and the rate of increase in productive capacity of the economy. It is then possible to prove that equilibrium growth paths in such a Keynesian model tend to exhibit an instability of the Harrod knife-edge type; along any growth path, either the level of employment remains at the level below full employment or the price level tends to increase with an accelerating rate. To stabilize the process of economic growth, it becomes necessary to adopt a flexible policy concerning the supply of money which is directed toward stabilization of the market rate of interest or the rate of increase in the price level.

The phenomenon of economic growth exhibits a quite different picture, according to whether we take the neoclassical approach or the Keynesian approach. In the neoclassical growth model, the path of economic growth is dynamically stable under fairly general conditions, while in the Keynesian model there is an intrinsic tendency for the process of growth to be dynamically unstable unless stabilizing monetary and fiscal policies are adopted.

See Also