# Profit and contemporary neoclassical models

Contemporary neoclassical theory unfolds around at least two lines of research: the theory of economic growth and the theory of output fluctuations. Both areas of research have been developed upon the neoclassical theory of distribution; that is, upon hypotheses concerning the subdivision of output into different income categories. For full studies of contemporary neoclassical economics see, for instance, Solow (2000), Romer (2001), and Mankiw (2015), from which we draw some causes for reflection in the analysis that follows. It shall be noted that the assumptions of neoclassical models have not backed out of criticism in the past. As profit is concerned, no significant progress has been made in neoclassical models since the end of the 19th century.

**Economic growth and output fluctuations**

Economic growth pertains to one of the most important and fascinating areas of economic theory since the 1950s. Classical authors had already sought to study those conditions that would allow output to grow over time. Yet, from the end of the 19th century through the first half of the 20th century, economic analysis developed on the basis of static or short-run considerations. The neoclassical theory of general economic equilibrium and Keynes’s research represent the most important examples of this tendency. The focus of neoclassical authors during the early days was the determination of relative prices by means of equation-based systems. The early versions of the theory of marginal utility and marginal productivity also left no space for considerations of a dynamic type. As it shall be observed further on in the volume, Keynes’s work was frequently criticized by modern authors of economic growth, inasmuch as Keynes studied employment trends in the short-run, rather than analyzing its evolution in the mid- to long-term.

Harrod’s and Domar’s studies of economic activity over the long-run (see Harrod 1939, 1948; Domar 1946) were, instead, a starting point for two different lines of thinking: the neoclassical theory of distribution and growth - including the neoclassical interpretations of Keynes’s *General Theory* (1936) - and the post-Keynesian theory of distribution and growth. The neoclassical theory of distribution evolved to satisfy formal requirements, so as to facilitate the study of production functions (analogous with utility functions) through mathematical analysis. The subdivision of income into wages and profit, an idea inherited from classical times by neoclassical economists, is dictated essentially by formal requirements taken for granted: the term ‘profit’ has even fallen into disuse in traditional economic theory; in its place, neoclassical economists favor the use of the word ‘interest’.

Nowadays, research on income distribution is often carried out on econometric, rather than on theoretical, grounds. Also, it must be observed that income distribution and growth is the object of study of welfare economists. As Mishan (1981: 3) puts it, welfare theory ‘can be defined as the study of criteria for ranking alternative economic situations on the scale of better or worse’, and it ‘implies “ought” propositions that derive ultimately from the ethics of society’, its purposes being, for instance, those of reaching ‘high employment levels’, guaranteeing ‘the free movement of goods and factors’, and pursuing ‘economic and social stability’. Since the times of Edgeworth and Pareto, and particularly over the last forty years, welfare economics has greatly improved. Notable contributions in the field have in fact tried to enhance the discourse by adding the human factor. Examples of such improvements are Sen (1970, 1971) on the ‘impossibility of a Paretian liberal’, Peacock and Rowley’s 1972 article against Pareto optimality, Sen’s 1977 ‘Rational Fools’, and Hirschman’s 1984 work on ‘human endowments’ and ‘tensions’. See also, for instance, Atkinson (1975, 1980, 2011, 2015), Sen (1992, 1994, 1997), Cohen, Piketty, and Saint-Paul (2002), Atkinson and Piketty (2010), and Galbraith (2012, 2014). Interestingly, empirical research has been carried out so as to understand increasing inequality and decreasing growth rates in a number of countries and regions (see, for instance, Gordon’s 2016 account of the US economy).

Neoclassical growth theory is therefore more recent than Walrasian price theory and the theories of marginal utility and marginal productivity. The entire theory of growth was developed with the aim of studying long-run trends in national output, isolating short-run fluctuations from such trends. Therefore, the kernel of modern neoclassical theory is the growth of potential output. The reasons why economists distinguish between short-run phenomena and long-run trends are also important. According to mainstream economics, short-run output fluctuations are thought to be determined by aggregate demand, which also affects the price of output in the long-term.

Output and prices in the short-run are thought to depend on aggregate supply. On the one hand, aggregate demand shifts, for instance, following changes in the money supply, in government spending, in taxes, in net exports, and in the economic expectations of consumers and companies; on the other hand, aggregate supply is thought to be affected, for example, by changes in the price of productive factors, in supply shocks, and in government and fiscal regulations (see e.g. Mishkin 2016).

The features of the neoclassical model were developed with the aim of solving the Ramsey (1928) problem. ‘The first problem I propose to tackle is this: how much of its income should a nation save?’ (Ramsey 1928: 543). Ramsey’s model represented a theoretical economy managed by an immortal planner who maximizes utility over an infinite period of time, given a particular technology. A large number of companies faithfully follow the plan of a single planner. There are two well-known versions of Ramsey’s problem: the version described above, i.e. with a centralized or planned economy, and the version with a decentralized economy, in which companies act freely in perfect competition. Be that as it may, in both cases the economy follows an equilibrium path.

Growth models may be divided in various categories, each one delineated by an elevated level of related research. The main groups of models are the following.

The *Solow growth model* (1955-6), or basic neoclassical model of growth. One of the main features of this model is technical progress, which is considered as exogenous to the economic system. Individuals’ choices and the behavior do not affect the stock of capital in the hands of companies (see also Swan 1956).

*Infinite-horizon and overlapping-generations models*

*See, for* instance, Allais (1947), Cass (1965), Koopmans (1965), Diamond (1965), Modigliani (1966), and Blanchard and Fischer (1989). These models usually assume the existence of a representative household, a representative firm, and a social planner. The stock of capital is endogenously determined by utility functions. This means that households, seeking to maximize their utility, shall determine the quantity of savings to be supplied to companies.

*New growth theory*

One of the main features of new growth theory is the inclusion of human capital and technical knowledge. ‘There are two kinds of capital, [. . .] in the system: physical capital that is accumulated and utilized in production under a familiar neoclassical technology, and human capital that enhances the productivity of both labor and physical capital, and that is accumulated’

(Lucas 1988: 39). Another important feature is that technological progress is taken as endogenous. See, for instance, Romer (1986) and Lucas (1988) See also Vercelli (1991).

Money *and growth*

Generally, little is said concerning money in neoclassical models of growth. Neoclassical models are based on the concept of commodity-money. The literature is full of examples of this tendency. For instance, Lucas (1988: 6) states: ‘I will [. . .] be abstracting from all monetary matters, treating all exchange as though it involved goods-for-goods’. However, a small number of neoclassical authors must be acknowledged as having included money in their models (for an in-depth view, see, for instance, Tobin 1955, 1965; Orphanides and Solow 1990).

If, on the one hand, economists pay great attention to the concept of economic growth, on the other hand, research into the short-run has in no sense been abandoned. It is in fact true that, particularly from the 1980s onwards, the so-called fluctuation theory has achieved great success. The theory of output fluctuations subdivides into two lines of research, though sharing the same methodology. On the one hand, real-business-cycle theory is based on the idea that output levels vary cyclically, following real shocks. On the other hand, new Keynesian theory represents the most significant attempt to formalize Keynes’s *General Theory* and, therefore, to study the possibility that output fluctuations are the result of nominal (monetary) shocks. Although the basic hypotheses are diametrically opposed, both theories resort to the use of so-called models of general economic equilibrium, these being of a stochastic and dynamic type (DSGE models). The use of such models in studying output fluctuations has been criticized by many non-mainstream authors, but - and this is not without significance - these models have also met the opposition of notable neoclassical authors. For example, Solow’s criticism of DSGE models (2010: 12) is well known:

I do not think that the currently popular DSGE models pass the smell test. They take it for granted that the whole economy can be thought about as if it were a single, consistent person or dynasty carrying out a rationally designed, long-run plan, occasionally disturbed by unexpected shocks, but adapting to them in a rational, consistent way [.. .]. The protagonists of this idea make a claim to respectability by asserting that it is founded on what we know about microeconomic behavior, but I think that this claim is generally phony. The advocates no doubt believe what they say, but they seem to have stopped sniffing or to have lost their sense of smell altogether.

(Solow 2010: 12)

As noted by David Romer (2001: 174), the subdivision into the two types of fluctuation theories is an ‘oversimplification’, because it omits the possibility that *‘real non-Walrasian* theories’ exist. According to the American author, under the hypothesis that nominal shocks have no effect on output fluctuation, output variations might potentially be explained by taking a distance (‘departures’) from the traditional Walrasian model. Among aspects to be considered, he includes ‘imperfect competition, externalities, asymmetric information’, and so on. Interesting is his opinion (Romer 2001: 174) that there are many more ways of approaching macroeconomics than those that exist at present.

While the neoclassical theory of growth sets out to study the long-run evolution of output, the neoclassical theory of the medium-run is based on the idea that output fluctuates regularly, following cycles with high and low points that alternate with each other over very brief intervals of time. More specifically, real-business-cycle (RBC) theory studies year-to-year increases and decreases in employment and output (see, for instance, Long and Plosser 1983). The analysis undertaken focuses exclusively on real economic activity: in this theoretical context, money in no way influences the real economy - in other words, the hypothesis is that economic disturbances are exclusively of a real type. Fluctuation models are an elaborated version of Ramsey’s model, given that they include shocks and disturbances inserted into a long-run context of growth. It should be noted that all RBC models are no more than a single version, among many, of the basic Walrasian model, for these are based on the concept of equilibrium between labor supply and labor demand: employment levels are made to depend upon households’ utility or satisfaction, which is a function of consumption and other variables, among which is leisure time. It should also be mentioned that fluctuation models exist in two forms: one with an analytical solution and a version that, on the contrary, gives no analytical solution. See, for instance, Kydland and Prescott (1982), Hansen (1985), Christiano and Eichenbaum (1992), and Baxter and King (1993).

Suffice to mention here some of the main features of neoclassical models. Although, in the neoclassical theory, production is made to depend on an unspecified number of inputs, the factors considered are usually reduced to two or three: ‘the factors of production are the inputs used to produce goods and services. Labor, land, and capital are the three most important factors of production’ (Mankiw 2015: 374). Labor is understood as the number of individuals engaged in a given productive activity or the number of working hours. Capital is meant as the machines used in this same activity. It should be said that the theory is confined to the limited case in which a homogeneous commodity is produced, being destined for sale and accumulation. Namely, labor and capital serve to produce a single typology of good that is partly consumed by purchasers and partly used as capital. Moreover, labor is taken as exogenous, while capital depreciates. It is useful to note that ‘[t]here is no special connection between the neoclassical model of growth and the determination of factor prices. The usual practice is to appeal to the same view of factor pricing that characterizes static neoclassical equilibrium theory’ (Solow 2000: 378). This means that ‘factor’ prices, i.e. unitary wages and unitary interest (profit), are made to depend upon a certain number of equations and a number of unknowns inserted into a mathematical system, as is the case when determining the relative prices of goods. We previously noted that, according to some authors (see Cencini 1982, 2001, 2015; Schmitt 1984, 1996a; Schmitt and De Gottardi 2003), in the neoclassical theory, the relative prices of goods and services cannot be determined, since the system of equations proves to be over-determined. Analogy therefore leads one to conclude that, were this the case, the prices of productive factors could not be determined, either. However, hereafter it shall be assumed that factor prices are determinable, so as to go on in the investigation.

Labor is supplied by households and, just as if it were a commodity, it is given a price; i.e. the wage rate. One should ask, however, whether wages may truly be considered the ‘price’ of labor, as if labor were a commodity. Households consume output with the aim of maximizing their utility or satisfaction. In itself, the concept of utility is abstract, and its measurement cannot be made objectively. Yet, for the last few decades, growth theorists have introduced inter-temporal utility functions into their models (see Cass 1965; Koopmans 1965). ‘The social welfare to be maximized is [. . .] represented by the total of the discounted utility of consumption per capita, where a general concave utility index is employed’ (Cass 1965: 233). Furthermore, households purchase goods subject to a budget constraint. It is assumed that disposable income is fully spent on the basis of well-defined preferences. On the other hand, company investment is finalized with the aim of maximizing profit, according to the technology available on the market. Profit or benefit, in this theoretical context, corresponds to the difference between earnings and costs (defined via functions with their own properties). Remarkably, ‘the assumption of profit maximization by firms can be replaced by some other systematic criterion of behavior’ (Solow 2000: 352), a criterion chosen arbitrarily. Moreover, traditional neoclassical models assume there is a regime of perfect competition for labor and goods, in which companies are price-takers. The observation that large interest groups in the real world are instead pricemakers has led neoclassical theoreticians to a new formulation of the basic model, thus assuming imperfect competition (e.g. see Romer 1990).

The reigning mathematical tool in the neoclassical theory of growth is the production function. Its features, always true for the most widely used production function, the Cobb-Douglas (1928) production function, are highly restrictive. After all, ‘it should be made clear that we do not claim to have actually solved the law of production’ (Cobb and Douglas 1928: 164). The concept of returns to scale is well known and underlies the entire basic neoclassical model. Namely, the assumption is that the physical product or output varies by a multiplicative factor *t* whenever each productive input is multiplied by such factor *t.* Moreover, it is assumed that the sum of the distributive quotas is always equal to the unit: if one quota diminishes, then the other increases, and vice versa, in such a way that equality is always respected and, therefore, one productive factor may be substituted for another (see, for instance, Sylos Labini’s 1995 inquiry on the Cobb-Douglas exponents). The hypothesis of diminishing returns to scale of individual factors is also important. It in fact allows for productive inputs to be aggregated, as if they were a unitary and homogeneous productive factor. However, it defies understanding to imagine how factors such as labor and capital may possibly be aggregated *as if* they formed a single homogeneous physical factor. Therefore, the concept of returns to scale of individual factors appears to belong exclusively to a world that, according to Solow, does not correspond to the real one. One should consider the ‘[neoclassical growth model as being a story about an imaginary economy that has only one produced good that can be consumed directly or stockpiled for use as a capital good’ (Solow 2000: 351).

The neoclassical theory of production is based on a mathematical model whose aim is to minimize production costs, these being subject to an output constraint (see, for instance, Mas-Colell, Whinston, and Green 1995). Dealing with the problem of optimization would allow a solution in equilibrium to be reached for the consumer. The final goal is in fact to derive an equilibrium cost function that provides the minimum production cost, given the prices of productive factors (usually, labor and capital). It should be observed that the neoclassical problem of optimization in the productive context was developed in perfect symmetry with the neoclassical theory of utility. In fact, the study of individual utility involves an exercise in optimization in which a consumer spending constraint must be minimized given a given utility function.

In both growth models and RBC models, output is subdivided into two parts: one that is consumed and one that is accumulated and used as capital. So it is that the part not destined for consumption is invested. Otherwise said, investment always amounts to the sum of households’ savings. As Solow (2000: 356) states, the neoclassical theory assumes that consumption and investment take place on the basis of ‘mechanical rules’ or ‘mechanical optimization procedures’, in such a way that the sums for consumption and investment ‘add up to total output’. This is a delicate matter: if the distribution and the allocation of income are not sufficiently investigated, it is likely that neoclassical models do not explain either the existence of profit or its relation with investment.

# The neoclassical theory of distribution

The neoclassical theory of distribution is often an object of debate, as happened, for instance, in 2000, when Nobel Prize winner Robert Solow and

Keynesian economist Luigi Pasinetti produced two articles concerning the neoclassical theory of growth and distribution, on the one hand, and its critique, on the other hand. Hereafter, a short assessment shall follow of the neoclassical theory of distribution, drawing on the exposition made by Pasinetti. As it shall be observed, this theory has met serious criticism over the last century.

Although the problem of income distribution was already tackled during classical times, it remains a controversial subject. The established marginal principle in the neoclassical theory goes back to the Ricardian principle of diminishing returns from land, which neoclassical authors extend to those elements that can be defined as productive ‘inputs’ or ‘factors’. Therefore, from the neoclassical viewpoint, an extension of the principle of diminishing returns to all presumed productive factors, such as labor and capital, renders the principle of diminishing returns from land a particular case within a wider theory. There is an enormous, fundamental difference between classical and neoclassical approaches: while classical authors believe that the value of goods and services derives exclusively from labor, neoclassical economists limit labor to being one among a few productive factors, together with land and capital. From Walras onwards, labor and, then, capital, were inserted into the theory of production in the same way in which Ricardo considered land.

The difference between the two approaches is substantial: although both are based on a physical concept of inputs and output, the causal relationship changes according to the theory under study. There is a clear symmetry between productive factors in the neoclassical theory - impossible, instead, in Ricardo’s theory - since they are placed on the same conceptual level. Translated into mathematical terms, symmetry is evident in the production function, developed initially by Wicksell (1901 [1934]). The properties of this function are well known, as are productive function typologies. What matters in terms of the discourse here is the way in which income is distributed within this theoretical environment. The first neoclassical models were based on the idea that physical production depends on quantities of land and on the ‘quantity’ of labor employed in the productive activity. Given a unitary income from land, *r,* and the unitary ‘price’ of labor, *w,* a typical budgetary constraint for this model was: Y = *rT + wL,* where Y stands for output, *T* for land, and L for labor. Output is thus distributed to landlords and laborers in a complementary manner. Unitary wages and unitary income from land therefore complement each other. This means that, for the neoclassical theory, rents and wages are two incomes placed on the same conceptual level: indeed, were labor to be absent, rents could still exist. The way in which unitary income from land and unitary wages are determined remains unspecified - as previously observed, their determination is entrusted to an explanation analogous to that used in explaining relative prices.

There is even more: the hypotheses that the function is homogeneous and linear means that output distribution among the ‘social classes’ (laborers and landlords in the example) in no way depends on the class that receives the residual income - i.e. the class that undertakes productive activity. The introduction of the two hypotheses allows for the equality *Y = (ôY/ôT) T + (ôY/ôL) L.* In the absence of any asymmetry, unitary rent is made equal to the marginal product from land, and unitary wages are made equal to the marginal product from labor. ‘The temptation that naturally comes is to assume that F is precisely a function of this type, so as to make equality satisfied and coincidental with’ the equation *Y = rT + wL* (Pasinetti 2000: 391). There is no logical-factual motivation justifying the use of this type of production function. Yet, merely formal considerations have induced generations of neoclassical economists to adopt the production function.

In order to create a production function that may be studied using mathematical analysis, other properties are attributed to this function. The hypotheses that the first derivatives of the function are positive and the second derivatives are negative guarantee that the function is convex. ‘It turns out to imply constant return to scale and diminishing returns to the variation of the proportions between the two factors of production (a process which can be interpreted as a phenomenon of substitution between the two factors)’ (Pasinetti 2000: 391).

Whoever be the entrepreneur (who maximizes his own earnings), and whatever be the initial endowment of the production factors, a market with free competition will lead to such factors prices (i.e. to a wage rate and to a unit rent) which not only maximize the product, but also distribute to each factor of production precisely its physical marginal product, without leaving any (positive or negative) residual to anyone.

(Pasinetti 2000: 391)

The reasoning behind land and labor factors may be extended to the case in which production is a function of an *n* number of factors. While developing their models, neoclassical authors from the last few decades have assumed that production is a function of just two factors: labor and capital. The introduction of capital, *K,* into the production function is interesting for this inquiry, since it has led to the introduction of profit into the neoclassical theory of growth. More particularly, profit or interest is identified with the part of income destined to the owners of capital. If households received no wages, profit would still exist. The basic relationship thus becomes the following, analogous to what has been previously stated about the income from land:

*Y = (dY / dL)L + (dY / dK)K*

where the marginal product of labor is equal to the wage rate, and the marginal product of capital is equal to the profit rate, 7t:

Y = *wL + *ttK

The majority of neoclassical models of growth are based on this distribution theory. Yet, this theory has often been the object of criticism, in particular following the works of Wicksell more than a century ago.

The Swedish economist Knut Wicksell (1901) [. . .] realized that [. . .] the marginal product of capital always turns out to be smaller than the rate of profits, because the variation of capital leads to changes in its price, *i.e.* to changes of the (current) unit in terms of which capital itself is measured. This divergence of the marginal product of capital from the rate of profits came to be known as the *Wicksell effect.* [. . .] [I]n a subsequent discussion with Dr Akerman, Wicksell (1923) did confirm his previous finding of a divergence between the rate of profits and the marginal product of capital, but at the same time also realized, to his surprise, that not always did the divergence occur in the same direction. It could in certain cases - as he had always thought - be negative (the Wicksell effect), but it could also in other cases turn out to be positive, generating a Wicksell effect in reverse.

(Pasinetti 2000: 404)

Criticism of the neoclassical theory became animated soon after Wicksell’s findings. John Maynard Keynes’s theory came to the fore in the 1930s as an alternative to neoclassical economics. The following chapter is devoted to the analysis of profit that was carried out by Keynes in two of his most influential volumes: *A Treatise on Money* (1930a [1971]) and *The General Theory of Employment, Interest and Money* (1936 [1964]).