The mathematical equivalence of Marshallian analysis and "General Equilibrium" theory
BY PHILIP GEORGE

Summary

Attacks on New Classical Economics, Dynamic Stochastic General Equilibrium (DSGE) or Real Business Cycle theory usually focus on their poor record in forecasting or on issues like identification and parameterization. Here, we take a different tack, choosing instead to study the root of New Classical Economics which is General Equilibrium (GE) theory.

We show that (a) Marshallian demand analysis is not any less general than GE theory, and (b) that the unstated assumption of GE theory is that aggregate demand is constant. Together, these two results amount to saying that, shorn of the complicated math, GE theory is equivalent to Marshallian demand analysis. It also explains why the two arrive at identical results on subjects like involuntary unemployment.

Introduction

Of the battle between Ricardian and Malthusian ideas John Maynard Keynes wrote in the General Theory: "... Ricardo conquered England as completely as the Holy Inquisition conquered Spain. Not only was his theory accepted by the city, by statesmen and by the academic world. But controversy ceased; the other point of view completely disappeared; it ceased to be discussed." Not long after Keynes wrote those words his own ideas conquered the world of academic economics as completely as Ricardo's had in his time. In turn it too has been thrust aside by General Equilibrium (GE) theory so that now only isolated pockets of resistance remain.

Unlike its predecessors GE shows no sign of going away any time soon. It has now ruled academia longer than classic Keynesianism did. And although it has come in for attack on various counts few have challenged its basic mathematical foundations. That is what we do in this paper. We question some of the basic claims of GE, choosing to focus mainly on the exposition by Kenneth Arrow. The reason is that Arrow was more conscious of the assumptions that GE was making and did not make the grandiose claims that some of the latter-day proponents of GE are wont to make.

A. The Generality of General Equilibrium Theory

One of the principal claims of General Equilibrium theory is that a) it takes all markets into simultaneous consideration and thus more realistically reflects the interconnectedness of all markets in the economy than b) Marshallian economics which deals only with one isolated market at a time and is thus a case of partial equilibrium.

It is the second part of the above statement that we shall contest, showing that Marshallian demand analysis too deals with the economy as a whole and is thus no less general than General Equilibrium theory.

Kenneth Arrow's Nobel Memorial Lecture sets out the GE argument: "The consumer starts with the possession of some quantities of economically valuable goods, such as labor of particular types, land, or other possessions. Let us imagine there are n commodities altogether, and let $\bar x_{hi}$ be the amount of commodity i owned initially by individual h (this may well be zero for most commodities). If $p_i$ is the price of the $i^{th}$ commodity, then his total income available for expenditure is

$$\sum_{i=1}^n p_i \bar x_{hi} \tag{1}$$

"Hence, he can choose for consumption any bundle of goods, $x_{h1}, ..., x_{hn}$, which cost no more than his income,

$$\sum_{i=1}^n p_i x_{hi} \le \sum_{i=1}^n p_i \bar x_{hi} \tag{2}$$

"Within this budget set of possible consumption bundles, the individual is presumed to choose his most preferred bundle... The most preferred bundle then is a function, $$x_{hi}(p_1,...,p_n) \tag{3}$$ of all prices. Notice that, from this viewpoint, all prices clearly enter into the determination of the demand for any one commodity. For one thing, the rise in any one price clearly diminishes the residual income available for all other commodities. More specifically, however, the demands for some commodities are closely interrelated with others; thus, the demand for gasoline is perhaps more influenced by the use of automobiles and therefore by their price than it is by its own price. The interrelation of all demands is clearly displayed here."

The idea that general equilibrium theory is superior to alternative methods can also be seen in Arrow and General Equilibrium Theory by Darrel Duffie and Hugo Sonnenschein. "The Walrasian [general equilibrium] theory," they wrote, "has the capacity to explain the influence of taste, technology, and the distribution of wealth and resources on the determination of value. Nothing that came before the Walrasian theory had this capacity. Neither partial equilibrium theory nor theories that depend on technology and resources alone provide as strong an explanation of value. Although, for certain markets, it is possible to explain how price responds to small parameter changes with partial equilibrium reasoning, few economists would contend that this method is adequate when economies are disturbed in a major way." Or again, a little later: "The essence of general equilibrium does not preclude aggregation; what is essential is an emphasis on intermarket relations and the requirement that variables are not held fixed in an ad hoc manner."

We turn next to Marshallian demand curves to see if this accusation is true. Fig 1 below shows a linear demand curve for the fish market.

The original demand curve is AB. The demand curve is drawn assuming that people's incomes are constant as are their tastes. If people's incomes increase then at every price they can buy more fish and the demand curve moves to CD. Similarly, if people develop an increased taste for fish the demand curve moves up.

T is the midpoint of the demand curve. The segment AT is the elastic zone. The segment TB is the inelastic zone.

Assume that the initial equilibrium is at point R (in the elastic zone) where the price is P1 and the quantity sold is Q1. Assume also that at this point individuals spend all their income and do not save anything. Next suppose that because of a movement of the supply curve the price falls to P2, also in the elastic zone, as a result of which the equilibrium moves to S. The quantity of fish bought increases to Q2. We can also see from the graph that the money spent on fish rises; the initial amount spent is the area of OP1RQ1 and the final amount spent is the area of OP2SQ2.

But here we run into a problem. We had assumed that individuals spent all their income at the first equilibrium point R and now we find them spending a larger amount on fish at S. This can happen only if they spend a smaller amount on some other good or goods so as to maintain their spending constant.

To summarize, along the elastic portion of a linear demand curve, when the price of fish falls not only does the quantity of fish bought increase but the money spent on fish also increases. So the money spent in other markets has to fall so as to maintain our initial assumption of a constant income. In general, the money spent at any point on the demand curve is different from that spent at any other point. To compensate for this difference the money spent in other markets, and therefore the demand and price in those markets, have to change.

This feature of linear demand curves also applies to demand curves of other shapes, with a solitary exception: the rectangular hyperbola PQ = constant. An analysis involving the rectangular hyperbola is too complex to be gone into here but can be read at Why is there involuntary unemployment?

What is true of the fish market is also true of every other Marshallian market. When the price and demand for any good changes it affects demands and prices in other markets.

So it is clearly not true that the Marshallian demand curve is drawn on the assumption that demand and prices in all other markets is constant. The Marshallian analysis is therefore no less general than General Equilibrium analysis. The charge is that in the Marshallian analysis, the individual's demand for the ith commodity is $x_{hi}(p_i)$ whereas in reality it is $x_{hi}(p_1,...,p_n)$ which happens to be the demand for good i in GE theory.

B. General Equilibrium Theory and Involuntary Unemployment

In his Nobel lecture mentioned earlier Arrow describes the various stages by which general equilibrium theory arrives at an equilibrium that is also Pareto efficient. But after the proof is done he mentions a caveat: "There is one loose end that should now be picked up. It has been assumed that the demand functions of the individual are continuous. But one of the surprising discoveries that Debreu and I made in the course of our study was that even under all the usual strong assumptions about the behavior of individuals, this cannot be true everywhere in the price simplex except under very artificial conditions. The trouble is that the individual's income also depends upon prices, and if the prices of those commodities which the individual owns originally fall to zero, his income falls to zero. When some prices and income are zero, however, the demand for the now-free goods may jump discontinuously. To illustrate, suppose an individual owned initially only one good, say, labor. So long as the price of that good was positive, he might retain some for his own use, but in any case could never consume more than he had initially. But when the price fell to zero, he could demand the same labor from others and in any amount he chooses. The existence of competitive equilibrium then does depend on assumptions which insure that for each individual there is at least one commodity he owns initially which is bound to have positive value."

Duffie and Sonnenschein's paper has more details about this assumption. They quote Arrow: "Debreu and I sent our manuscripts to each other and so discovered our common purpose. We also detected the same flaw in each other's work; we had ignored the possibility of discontinuity when prices vary in such a way that some consumers' incomes approach zero. We then collaborated, mostly by correspondence, until we had come to some resolution of this problem." D&S go on to explain: "This resolution was to require, in theorem 1 of their paper, that the initial endowment of each household be interior to its consumption set. (Arrow had faced a difficulty much related to the demand discontinuity problem in his earlier work on the second welfare theorem.)"

But if the existence of general equilibrium requires ruling out a situation when a consumer's initial endowment has a market price of zero it also requires ruling out a situation when the consumer possesses only a single commodity and is unable to sell it because this would practically mean that he had no initial endowment. When the commodity in question is labor the situation is of course what we call involuntary unemployment. That is to say, labor has a market price but not all consumers who possess it can find a buyer.

So, contrary to what at least some general equilibrium theorists would have us believe, general equilibrium theory does not disprove the existence of involuntary unemployment. Rather, for general equilibrium to exist, involuntary unemployment must first be assumed out of existence. General equilibrium can exist only if its assumptions guarantee that the labour market has no disequilibrium. Marshallian demand analysis logically arrives at the conclusion that involuntary unemployment is impossible but GE theory needs to assume it in the first place to ensure general equilibrium.

C. General Equilibrium Theory and Unemployment

Apart from consumption, General Equilibrium theory also takes production into consideration.

To quote from Arrow's Nobel lecture again: "A productive unit or firm is characterized by a relation between possible outputs and inputs. A firm may have, of course, more than one output. Then firm f may be characterized by its transformation surface, defined by an equation, $T(y_{f1}, ..., y_{fn}) = 0$, where $y_{fi}$ is taken to be an output if positive and input if negative; the surface is taken to define the efficient possible input-output vectors for the firm, that is, those which yield maximum output of one commodity for given inputs and given outputs of other commodities. The optimizing behavior of the firm is taken to be the maximization of profit among the points on its transformation surface. Because of the sign conventions for inputs and outputs, the firm is seeking to maximize,

$$\sum_{i=1}^n p_i y_{fi} \tag{4}$$

And a little later: "For any commodity i, there will be some demands and some supplies at any given set of prices. Following Hicks, we will speak of the excess demand for commodity i as the sum over all individuals and firms of demands and supplies, the latter being taken as negative. The demand by individual h is $x_{hi}(p_1,...,p_n )$, so that the total demand by all households is

$$\sum_h x_{hi} (p_1,...,p_n) \tag{5}$$

"The supply by households is the aggregate amount they have to begin with, i.e.,

$$\sum_h \bar x_{hi} \tag{6}$$

"Finally, the aggregate demand by firms is

$$\sum_f y_{fi}(p_1,...,p_n); \tag{7}$$

some firms may be demanders rather than suppliers, but the sign convention assures that the above sum gives the aggregate net supply by firms, i.e., after cancelling out demands by one firm which are supplied by another.

"... Further, the satisfaction of the budget constraint for each individual also restricts the excess demand functions. Since for each individual, the monetary value of expenditure planned at any set of prices equals the monetary value of his initial endowments plus his share of the profits, we have in the aggregate that the money value of planned expenditure by all households equals the money value of total endowments plus total profits, or

$$\sum_h \sum_{i=1}^n p_i x_{hi}(p_1,..., p_n) = \sum_h \sum_{i=1}^n p_i \bar x_{hi} + \sum_f \sum_{i=1}^n p_i y_i(p_1,...,p_n), \tag{8}$$

or, from the definition of excess demand,

$$\sum p_i z_i (p_1,...,p_n) \equiv 0 \tag{9}$$

[$z_i$ is the market excess demand for commodity i] where the identity symbol reminds that this relation, called by Lange [1942] Walras' Law, holds for all values of the prices."

With those definitions in place and a long discursion through more math we come to the section "The Existence of Competitive Equilibrium"

"A set of prices defines a competitive equilibrium if supply and demand balance on each market, including the possibility of corners, with some choice of the profit-maximizing input-output vector for each firm. Formally, we will say that a price vector $p^*$, an input-output vector $y^*_f$ for each firm, and a consumption vector, $x^*_h$ = $x_h(p^*)$, for each individual together constitute a competitive equilibrium if the following [four] conditions hold."

We are concerned here with the second condition.

"(b) for each commodity i,

$$\sum_h \bar x_{hi} + \sum_f y^*_{fi} \geqslant \sum_h x^*_{hi}" \tag{10}$$

This is a straightforward conservation law. In plain English it says that the total demand of a commodity by households cannot be greater than the sum of the initial holding of that commodity by households together with the output of that commodity by firms.

This applies to any commodity; therefore it must also apply to labour. Since households are suppliers of labour but not consumers of labour the right hand side of the above equation is equal to zero and the first term on the left hand side is positive. Similarly, since firms are consumers but not suppliers of labour the second term on the left hand side is negative.

So the equation above tells us that the supply of labour will always be equal to or greater than the demand for labour. This curious result derived from the equations of General Equilibrium theory is in complete accord with reality (with the sign in practice being one of inequality) because we know that even in economies that are not undergoing a recession, there is a certain amount of unemployment, which is sought to be explained under the rubrics of frictional unemployment, non-accelerating inflation rate of unemployment, search-match delays and so on.

But the equation tells us that there is a simpler, more logical explanation for such unemployment. Most households have only the single commodity called labour to sell. On the other hand nearly all firms in all industries have to buy greater or lesser quantities of labour, so labour is the largest market in the economy. Compensation paid to employees in the US amounts to about 44% of GDP at present. So even when every market is only slightly displaced from equilibrium, to an extent that the disequilibrium is barely perceptible, the small displacements from perfect equilibrium in every market add up in the case of the input, labour, so that unemployment amounts to a relatively large proportion of the labour market, say, of the order of 5% in the US. The constant presence of unemployment is actually proof that the economy is always in the process of attaining equilibrium but never quite there.

Unfortunately, this explanation is not permitted by the rules of General Equilibrium theory.

To quote from Arrow's Nobel lecture again, "(c) for any commodity for which the strict inequality holds in (b) [eqn 10], we must have $p^*_i = 0$"

Now this condition has been used earlier to account for the case of free goods. For example, the supply of air is greater than the demand for air, so its price must be zero. By that token, if the supply of labour is greater than the demand for air, its price must be zero.

We come away with the impression that General Equilibrium theorists do not wish to listen to what their own equations are telling them but prefer to torture them so that the equations say what the theorists want to hear, viz. that the economy is in equilibrium.

D. The relationship between supply and demand

For this section an extensive extract from Arrow's Nobel lecture will be required. To get a proper grasp of the context it is advisable to read the complete speech, though it is not needed for our purpose.

"We begin to see that a Pareto efficient allocation is an equilibrium of supply and demand in the generalized sense which includes corners. We also see that,

$$\sum_{i=1}^n p_i(z_i - z_i^0) \geqslant 0 \ for \ z \ in \ Z. \tag{11}$$

[$z_i$ is the excess demand vector for commodity i, Z is the set of all excess demand vectors and the superscript 0 refers to the Pareto-optimal case.]

"Let us go back to the definition of excess demand, as a sum of individual and firm demands and supplies.

$$z_i = \sum_h x_{hi} - \sum_h \bar x_{hi} - \sum_f y_{fi} \tag{12}$$

where $y_f = (y_{f1},...,y_{fn})$ is a technologically possible vector of inputs and outputs for firm f and $x_h = (x_{h1},...,x_{hn})$ is a possible vector of consumptions for individual h. In particular, the excess demands defined by the Pareto efficient allocation can be written in this form,

$$z_i^0 = \sum_h x_{hi}^0 - \sum_h \bar x_{hi} - \sum_f y_{fi}^0, \tag{13}$$

and then, if z belongs to Z, we must have, for each h, that the consumption vector of individual h, $(x_{h1},...,x_{hn})$ is preferred to that under the Pareto efficient allocation $(x_{h1}^0, ...,x_{hn}^0)$. Then,

$$\sum_h(\sum_{i=1}^n p_i x_{hi} - \sum_{i=1}^n p_i x_{hi}^0 ) - \sum_f(\sum_{i=1}^n p_i y_{fi} - \sum_{i=1}^n p_i y_{fi}^0) \geqslant 0 \tag{14}$$

if, for each h, $x_h$ is preferred by individual h to $x_h^0$

"Now the elementary point about this inequality is that the variable vectors $x_h$, $y_f$ are independent of each other. It is not hard to see that this inequality can hold only if it holds for each individual and each firm separately. For a firm f, this means that,

$$\sum_{i=1}^n p_i y_{fi}^0 \geqslant \sum_{i=1}^n p_i y_{fi} \ for \ all \ possible \ y_f \tag{15}$$

that is, if we interpret the $p_i$'s as prices, each firm is maximizing its profits. The corresponding interpretation for individuals is somewhat less simple; it is that the consumption vector prescribed by the given Pareto efficient allocation is the cheapest way of deriving that much satisfaction."

The details of the above derivation are not so important as the assumption that makes it possible: "Now the elementary point about this inequality is that the variable vectors $x_h, y_f$ are independent of each other."

In Section A we showed that General Equilibrium theory does not represent an advance over Marshallian demand analysis. But the assumption above, that consumption and production are independent of each other, represents a giant step backwards compared with Keynesian theory. Indeed, I think I exaggerate only a little if I say that the central purpose of writing The General Theory was to show that consumption and production are not independent of each other as classical economics had assumed. This would of course have been clearer if Keynes had not gone on a wild goose chase after investment.

It requires quite some flexing of the imagination to believe that the demand for bread is dependent on the price of steel, but it requires no imagination at all to perceive that the output of steel affects the demand for bread, by affecting the incomes paid out to labour involved in the production of steel. The demand for bread is a weak function of the price of steel. The demand for bread is also a weak function of the output of steel, but it is probably a stronger function than the first.

So the problems of General Equilibrium theory can be traced to its fundamental assumptions. It assumes that the individual's demand for commodity i is $x_{hi}(p_1,...,p_n)$ whereas in reality it is $x_{hi}(p_1,...,p_n,y_{f1},...,y_{fn})$. When aggregate income is constant the cross effects cancel out so that the first expression can be taken as a good approximation. This is because we are primarily interested in how demand changes in response to changes in prices, and vice versa. Similarly the firm's demand for commodity i is not $y_{fi}(p_1,...,p_n)$ but $y_{fi}(p_1,...,p_n,x_{h1},...,x_{hn})$.

In order to solve for general equilibrium, using the simpler but erroneous expressions for consumer and firm demand, GE theory had to make a number of assumptions that all but removed any resemblance to reality. This suggests that using the correct but far more complex expressions for consumer and firm demand would make the system of equations unamenable to a solution.

The fact that the consumer demand for a commodity is only a weak function of the output of other commodities means that it is only in the aggregate, and during recessions, that such effects are felt. And, as noted earlier, the effect of a fall in aggregate demand is first felt in the largest market of all, the labour market.

It must be here pointed out that the assumption that firms' profits are paid out to consumers does not integrate production and consumption. What matters is not profit but income. In other words, what matters is not

$$\sum_f \sum_{i=1}^n p_i y_{fi} \tag{4}$$

where outputs are positive and inputs are negative, but the same quantity with no negative signs attached to inputs.

What Keynes recognized was that it was possible to solve this problem in the aggregate using income as an intermediate variable. We show below a simple numerical example, with intertemporal substitution, using aggregate demand as the intermediating quantity.

Consider a simple, closed economy operating at full employment. It produces \$90 worth of consumption goods. An equal amount is therefore paid out in income flows: wages, rent, interest, profit and the like. The saving rate is 10%, so \$9 of these income flows is saved and \$81 is spent on consumption goods. That means \$9 worth of consumption goods is unsold. But then the economy also produces \$10 worth of investment goods. An equal amount is therefore paid out as income. Since the saving rate is 10%, \$1 of this is saved and \$9 spent on consumption goods. Financial institutions of course turn the \$10 saving into loans for investment. So all consumption goods are sold and savings are sufficient to pay for investment goods, as in the table below.

 Consumption goods Investment goods Income $90$10 Consumption $81$9 Saving $9$1

Assume also that individuals have, on average, accumulated 20 years worth of saving, which at present is valued at \$200. Then thanks to a housing and stock market crash, an average of 10 years worth of accumulated saving is lost. (This is not an outlandish figure; in the 2008 housing and stock market crash the median US household lost 18 years of real net worth. In an attempt to recover the lost net worth, households increase their saving rate from 10% to 15%. At this rate, they reckon they would take 20 years to recover their lost net worth. As a result of the higher saving rate, the money spent on consumption goods falls from \$90 to \$85. When manufacturers of consumption goods see their output remaining unsold they cut production to \$85 and also cut purchases of investment goods to \$8. Manufacturers of investment goods then cut production to \$8. So total income now paid out is \$93, a fall from \$100 earlier. The economy now looks as in the table below.

 Consumption goods Investment goods Income $85$8 Consumption $72.25$6.8 Saving $12.75$1.2

Now total consumption is \$79.05, so \$5.95 worth of consumption goods remain unsold. As a result manufacturers of consumption goods cut their production further and curtail purchases of investment goods even more. The downward spiral continues unabated. It is of course possible to see that there is a damping factor. The saving in the first year is \$13.95, so consumers might lower their saving rate a little as they begin to recover their lost net worth. But this must be balanced against the fact that consumers had expected a saving of$15, which did not materialize because the higher saving rate reduced the aggregate income. Whatever the details, the purpose of our example was to show the interconnection of production and consumption, which Keynes of course recognized and which General Equilibrium theory assumes out of existence.

In the final extract from Arrow it can be seen that the independence of production and consumption is a crucial assumption in proving profit maximization by firms and utility maximization by individuals. But if the assumption is wrong, as we have shown it is, we can question both profit maximization and utility maximization. During a recession following a large asset market crash the minimization of consumption takes precedence over the maximization of utility. We suggest therefore that utility maximization is an idea applicable only to the special case of an economy in equilibrium, ie. where aggregate demand is not falling. Similarly for firms during a recession, survival is more important than profit maximization.

Conclusion

We have shown that the problems of New Classical Economics, DGSE and Real Business Cycle theory can be traced back to its parent, GE Theory. Marshallian demand analysis has the same order of generality as GE theory. Moreover, both make the same assumption of constant aggregate demand, explicitly in the case of Marshallian demand analysis and implicitly in the case of GE theory. Mathematically, the two are equivalent, which is why they arrive at identical results, as on the issue of involuntary unemployment.

References

1. Arrow, Kenneth; General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice, Nobel Memorial Lecture, December 12, 1972, http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1972/arrow-lecture.pdf

2. Duffie, Darrell and Sonnenschein, Hugo: Arrow and General Equilibrium Theory, Journal of Economic Literature, June 1989, http://www.darrellduffie.com/uploads/pubs/DuffieSonnenschein1989.pdf

3. St Louis Federal Reserve Bank web site; Graph of ratio of labour compensation to GDP in the US, https://fred.stlouisfed.org/graph/?g=75e6

4. Changes in U.S. Family Finances from 2007 to 2010: Evidence from the Survey of Consumer Finances; http://www.federalreserve.gov/pubs/bulletin/2012/pdf/scf12.pdf