Distribution Theories: Neoclassical

Abstract

Whenever a theory becomes involved in controversy the question of what constitutes that theory itself becomes a contentious issue, and the neoclassical theory of distribution is no exception to that general rule. Some have seen marginal productivity as an essential feature of neoclassical theory. Others have regarded the aggregation of capital or an aggregate production function (even a function of the Cobb–Douglas form) as essential. Neoclassical distribution theory is viewed as general equilibrium theory by many but Friedman has defended the ‘Marshallian’ or partial equilibrium approach.

Whenever a theory becomes involved in controversy the question of what constitutes that theory itself becomes a contentious issue, and the neoclassical theory of distribution is no exception to that general rule. Some have seen marginal productivity as an essential feature of neoclassical theory. Others have regarded the aggregation of capital or an aggregate production function (even a function of the Cobb–Douglas form) as essential. Neoclassical distribution theory is viewed as general equilibrium theory by many but Friedman has defended the ‘Marshallian’ or partial equilibrium approach.

The truth is that any body of ideas widely maintained for a long time inevitably develops and transforms itself, absorbs some ideas, discards others, and fathers traditions and sub-traditions. As the neoclassical theory of distribution has been the predominant view in the leading countries for the development of economics for over 100 years, it is not surprising that it conformed to this pattern and expressed itself in diverse even contradictory voices. Many, whether or not they like neoclassical theory, hold that one voice represents the true message, but neoclassical theory, like christian doctrine, may stand on certain fundamentals but is not and could not be monolithic.

It is important to distinguish between ‘neoclassical theory’ on the one hand and the history of the development of that theory on the other. Both are valid subjects for study but a scientific assessment of the theory should address itself to the best modern statements. This principle has not always been respected, particularly in the heat of controversy, and some maintain that the theory went wrong from the start, and that if one could only go back to where the vital mistakes were made everything would become clearer. (For an extensive development and discussion of this line of argument, see Baranzini and Scazzieri 1986.) However, the development of economic theory is not like a complicated calculation in which every step is supported by every earlier step. As with any other discipline, the logical standing of a theory and the history of the development of that theory are distinct entities.

By way of illustration of the last point, consider the way in which the theory developed in its early stages. The ‘neoclassical’ movement, whose leading members may be taken to include Böhm-Bawerk, Edgeworth, Gossen, Jevons, Marshall, Menger, Walras, Wieser and Wicksell, did not begin with a theory of distribution but quite neglected that side of the economic problem. By focusing on marginal utility and the demand for given resources in a barter economy, the neoclassical economists were able to develop a powerful and flexible method, the marginal principle, so impressive that it has often been taken to define their approach. The so-called ‘psychic’ notion of marginal utility represented the refinement, no more, of the old idea of ‘value in use’. However with its help the neoclassicals eventually succeeded in clarifying, as Smith, Ricardo and Mill had all failed to clarify, how value in use, value in exchange and cost of production could coexist. Only the Austrians with their concept of ‘imputation’ hung on to the idea that utilities were in some sense primary and other values derived.

Put in unashamedly modern terms, the central neoclassical idea is that the pricing of goods and the pricing of factors of production are governed by common principles, mainly the forces of supply and demand generated by agents who maximize their objectives. From the perspective of the history of the development of the theory the definition is anachronistic. Economics did not develop and refine the notion of a factor of production or the concept of maximizing an objective and later arrive at the neoclassical theory of distribution. Rather the two processes took place in tandem. Despite the lip service to classical ideas paid by some members of the neoclassical school, notably Marshall, neoclassical is a misnomer. The neoclassicals were not revivalists of classical economic ideas, an Oxford Movement of classical political economy. They were revolutionaries.

The Distribution of Income

The theories with which we are concerned are designed to explain the levels of payment to the various factors of production – rents, wage rates, and rates of profit – and by extension the shares of the various factors in the total product. That is to say that they are concerned with the functional distribution of income.

We shall not discuss the distribution of personal or household incomes, sometimes called the size distribution. The size distribution of household incomes takes the form of a function relating the level of income and the number of units receiving that income. It is true that given the distribution of the ownership of factors among units, strictly the quantities supplied to the various markets, and given also the rates at which those factors are remunerated, the size distribution may be derived. However, except in the short run, the interrelationship between the functional and size distributions is more complicated. This is mainly so because the quantities of factors which may be accumulated by individual units, land and capital, and even the quantity of labour, respond to rates of return to the various factors. Pasinetti (1962) presents a model which unusually takes this inter-relationship into account. For a discussion of the Pasinetti model and some of the criticisms which it has attracted, see Marglin (1984, pp. 324–8). On the distribution of personal income and wealth, see Atkinson (1975).

Factors of Production

It is not surprising that the concept of a factor of production plays a leading role in neoclassical theory because it lends itself to the view that the inputs used in production stand to each other in a relation of symmetry, governed by common principles. This is not to say that no differences between the conditions applying to factors are admitted. The symmetry is most marked in the treatment of the demand for factors, while on the supply side important differences are recognized.

The membership of the trinity of land, labour and capital, which have always been taken to be factors of production, goes back to the classical writers, and an additional factor called ‘entrepreneurship’ is widely recognized by neoclassical and classical writers alike. The development of the theory along formal lines has tended until recently to suppress the role of the entrepreneur and to make the firm into a rather lifeless object. However lately the increasing employment of economic theory in industrial economics has given rise to some richer treatments of the firm.

The employment of the concept of a factor of production has been criticized. It has been argued that labour in particular does not submit itself to the laws of supply and demand like any other input. The introduction of distinctive features of the various factors and their markets tends to undermine the simple symmetry of pure theory. Some have detected apologetics in the designation of capital as a factor of production. On this see the discussion of ideology below.

Marginal Productivity and the Determination of Factor Prices

Do marginal productivities determine factor rewards? This apparently straightforward question conceals conceptual complications and, depending on the context to which the question is applied, either ‘no’ or ‘yes’ may be defended as reasonable answers. Robertson (1931) argued that the wage rate ‘measures’ the marginal productivity of labour. The reference is to the demand curve for labour, which is the schedule of the marginal productivities of various quantities of labour. Robertson was reminding his readers that the wage rate in a competitive market is determined by the intersection of the demand curve and the supply curve – both blades of the scissors cut the paper. If marginal productivities are values determined by the equilibrium solution as much as are wages and prices, talk of one determining the other is misplaced. The same point applies when the marginal product of capital and the return to capital are under consideration.

In certain contexts however it is reasonable to see marginal productivity as the determinant and the payment to the factor as determined. Consider the claim that managers of large enterprises are paid very large salaries because the marginal value productivity of a good manager amounts to a great deal of money. Supposing this argument correct, the high marginal productivity is a general feature which does not depend upon solving out the whole equilibrium. Contrast this with the case of a micro unit, say a farm, facing a given wage rate for labour and able to vary the quantity of labour employed. For that exercise the wage rate is given and the marginal product is determined by it.

A Simple Neoclassical Model

In this section we examine a static model. Growth and capital will be considered below. The idea is to construct a model in which factor prices will drive everything else, including goods prices through cost functions. This requires special assumptions but makes for a model which can be easily presented and which suffices to illustrate some points about the neoclassical model of distribution. For a much more thorough review of neoclassical models, see Ferguson (1969).

We assume factors and goods to be distinct and that factors are not directly consumed. Let there be F factors available in given quantities, and G goods producible from those factors, F and G need not be equal and there may be more goods than factors, or less or the same number. The production function for the ith good is:

$$ {v}^i={f}^i\left({x}^i\right)\, \left(i=1,\dots, G\right); $$

(1)

where vⁱ is the output of the i th good and xⁱ is a vector of factor inputs to the production of the ith good. fⁱ( ) is a concave constant returns production function. The cost function shows the unit cost of producing good i given factor prices. Factor prices are a vector w and the unit cost of the i th good is Cⁱ(w), where Cⁱ(w) is the solution to the programme:

$$ \min \limits_{x^i}\;w{x}^i; $$

(2)

subject to:

$$ {f}^{\mathrm{i}}\left({w}^{\mathrm{i}}\right)\ge 1. $$

(3)

We denote the prices of goods by vector c(w), where the i th element of c(w) is Cⁱ(w). There are H households. Let the h th household own factors x^h, in which case its income will be w · x^h. All the household’s income is assumed spent on goods and the vector of goods demanded by household h is denoted z^h and is given by the h th household’s demand function:

$$ {z}^{\mathrm{h}}={z}^{\mathrm{h}}\left[c(w),w\cdot {x}^{\mathrm{h}}\right]\quad \left(h=1,\dots, H\right). $$

(4)

Now note that factor prices w imply demands for factors as may be seen by the following line of reasoning. Given w, we have household incomes w · x^h and goods prices c(w). Hence we have total demands for goods:

$$ \sum \limits_{\mathrm{h}}{z}^{\mathrm{h}}\left[c(w),w\cdot {x}^{\mathrm{h}}\right]=z. $$

(5)

The amount of factor j used in the production of good i is the partial derivative of Cⁱ(w) with respect to w^j, denoted ${c}_{\mathrm{j}}^{\mathrm{i}}$. The matrix of these coefficients, denoted C, depends on w only. Hence demand for factors is C ⋅ z, supply is ∑x^h, and we have shown that excess demands for factors are a function of factor prices.

To prove the existence of factor prices such that factor demands and supplies are equal (strictly such that there is excess demand for no factor), one has to establish the continuity of the relationship between factor prices and excess demands for factors, and then employ a fixed point theorem (see Arrow and Hahn, 1971, ch. 5).

We note some salient features of this model. First, prices of factors are determined by the supply and demand for those factors although demands for factors are derived demands depending on their employment to produce goods. Secondly, both the technology of production and tastes influence the solution for factor prices. Thirdly, factor prices measure the marginal products of factors, a property which is ensured by the process of cost minimization. However there is clearly no sense in which marginal products are prior to prices.

More and Less General Models

The model of the previous section is designed to illustrate the manner in which the determination of the distribution of income may be viewed as the outcome of a general equilibrium of supplies and demands for factors of production. The model is less general than the standard general equilibrium model. It exhibits, for example, constant returns to scale production functions, no joint production and no direct consumption of factor services. Also, goods are not used as inputs to the production of goods. The introduction of those features would undermine the model’s neatness without introducing fundamentally new principles.

More striking results are produced when the model is made still more specialised. The factor input coefficients may be treated as constants independent of w. In this fixed coefficient case the marginal product of a factor in producing a good is undefined. In an extreme case there is only one factor, usually labour, with the result that relative goods prices are independent of demand. (For a discussion of this non-substitution result and its extension to an economy which uses fixed capital, see Bliss, 1975, ch. 11.) Models of this kind typically introduce the use of goods as intermediate inputs to the production of goods. However so long as there is no genuine joint production (the term genuine joint production is used to distinguish the production of final demands jointly from the notional joint production that arises when fixed capital goods are treated as one of the products of the productive process.), the inputs used to produce final output may all be reduced to the quantities of the factor incorporated in them.

The model of Sraffa (1960), sometimes known as the neo-Ricardian model, will be seen to be a version of this model, but including an elegant extension to fixed capital goods. Hahn (1982) has argued against the claim that the neo-Ricardian approach leads to new insights by pointing out that the model is a special case of the general equilibrium model.

The Problem with Capital

The introduction of capital into the theory of distribution raises two issues which should be distinguished, even though they are not entirely unrelated. One is the aggregation of capital, the other is the nature of the supply of capital in the long run.

Although many expositions of the theory have been expressed in terms of an aggregate called capital, and there have even been attempts to formally underwrite this approach, it is now generally recognized that there is no rigorous method of aggregating a heterogeneous collection of capital goods. (The most famous attack on the use of aggregate capital is Robinson, 1953–4); see also Champernowne, 1953–4; Harcourt, 1972 and Marglin, 1984, ch. 12.) In this respect capital stands on a par with other types of input, labour for example. Highly aggregated models should therefore be seen as simple devices for illustrating how a type of model functions and not as descriptions of the world. Unfortunately, some writers who emphasize the problems of aggregating capital are quite cavalier when it comes to discussing the aggregation of labour or output. Formally however there is little difference between the cases.

With many distinct capital goods, demands for inputs are demands for the services of particular capital goods. However the supply of capital in the long run is the supply of saving, which may translate itself as required into particular capital services. Hence a long-run neoclassical theory of distribution depends on a model of long-run saving, a point which deserves emphasis.

We show how the solution for the quantities of capital goods and equilibrium prices may be obtained in a simple constant returns growth model. Let there be N goods, and let the quantities of them which make up the capital stock used by one unit of labour be represented by the elements of a vector x. Let consumption be proportional to a vector c_o, and γ the rate of growth of the labour force. Let y be the total stock of goods available next period for consumption and as inputs to next period’s production. The production function corresponding to a unit labour input is:

$$ F\left(y,x\right)=0. $$

(6)

In steady state growth with a per capita consumption of αc_o, y will be αc_o + (1 + γ)x. Hence:

$$ F\left[\alpha {c}_o+\left(1+\gamma \right)x,x\right]=0 $$

(7)

Given a particular per capita consumption αc_o, (7) may be satisfied by various values of x, but only one of these will be the efficient and equilibrium value. To see this let V(x¹) be the maximum value of β such that βc_o is a sustainable per capita consumption starting with a capital stock x¹. If x¹ is the steady state composition of the capital stock for consumption βc_o then x²=x¹ must solve:

$$ \max \limits_{x^2}\alpha $$

(8)

subject to:

$$ F\left[\left(1+\lambda \right){x}^2+\alpha {c}_o,{x}^1\right]\ge 0; $$

(9)

and

$$ \mathrm{V}\left({x}^2\right)\ge \beta {c}_o. $$

(10)

Let the Lagrange multipliers attaching to the constraints (9) and (10) be respectively μ and η and let F_i(i = y, x) denote the vector of partial derivatives of F with respect to the output and input vectors. The necessary conditions for a solution to (8)–(10) are:

$$ 1+\mu {c}_o\cdot {F}_{\mathrm{y}}=0; $$

(11)

and

$$ \mu {F}_{\mathrm{y}}\left(1+\lambda \right)+\eta {V}_{\mathrm{x}}=0. $$

(12)

Equation (12) states that the marginal rates of substitution between outputs of the various goods shall be equal to the marginal rates of substitution between those same goods as inputs to the long-term provision of future consumption; compare Dorfman et al. (1958, ch. 12). This condition reduces the degrees of freedom enjoyed by the steady state capital stock to one. That last degree of freedom depends on the level of steady state consumption the determination of which requires a saving condition.

Theory and Ideology

According to its Marxist critics, neoclassical distribution theory is irredeemably apologetic in character, and it is indeed the case that some economists in the past saw the theory, and in particular the concept of marginal productivity, as throwing a relatively favourable light on capitalism. When the justification for the earnings of capital owning rentiers was being questioned, the notion that capital earns no more than its ‘contribution’ to production was not unwelcome in the salons. The idea that the rich are rewarded according to the marginal productivity of their ‘waiting’ sounded better still.

It can need a positive effort to see that all this is strictly irrelevant to the scientific standing of neoclassical theory. No one supposes that Newton’s mechanics should be dismissed because its author saw in it the justification of a hierarchical organization of social life. A play on the overtones of words such as ‘earning’ or ‘waiting’ to justify the distribution of income should be similarly disregarded. Of course the neoclassical theory of distribution can be used to analyse the effects of policy, including policies to redistribute income. In a perfect world the conclusions which emerged from such investigations would be independent of the political stance of the investigator. We do not live in a perfect world, but the fact that the scientific ideal is never fully attainable should not lead us to conclude that economics can know nothing but self-serving apologetics.

See Also

• Adding-up Problem

• Clark, John Bates (1847–1938)

• Marginal Productivity Theory

• Wicksteed, Philip Henry (1844–1927)