A 75th Anniversary Present for Sraffa
Steve Keen (UWS), John Legge (La Trobe) and Geoff Fishburn (UNSW)
In 1926, Sraffa argued that the 'Marshallian' theory of the firm was untenable. Though his paper sparked intense debate at the time, it is not too harsh to conclude that his arguments were ignored. We believe that a major—if largely unstated—reason for this was the belief that the mathematics behind the marginal analysis of profit maximising behaviour was incontrovertible. In this paper, we show that this confidence was and is misplaced. The neoclassical model of perfect competition, the welfare implications of different market structures, and even the general rule for profit maximisation, are all mathematically flawed. We conclude that economics must, as Sraffa argued, abandon its a priori theory of the firm, and instead develop a theory grounded in empirical research.
Sraffa's critique
The gist of Sraffa's critique was that two key precepts to supply and demand theory are mutually contradictory. The model of a single commodity market in the short run requires that supply and demand curves are independent, and that supply is constrained by a fixed resource. Given these two conditions, independent supply and demand curves can be drawn in which the slope of the demand curve is negative with respect to quantity, and the slope of the supply curve positive (Figure 1).
Figure 1: Supply and demand analysis
However, Sraffa argued that these two precepts are mutually inconsistent: in circumstances where the former applies, the latter will not, and vice versa.
The assumption of independence will be invalid if a broad definition of a market is used—say, defining the commodity as 'agriculture', and the factors of production as 'land' and 'labour'. In this case, production of agriculture will undoubtedly be subject to diminishing returns, since the amount of the fixed resource (land) devoted to agriculture can only be increased with difficulty. However, since agriculture employs a large percentage of the factors 'land' and 'labour', its increasing demand for the variable factor 'labour' will alter the demand for 'labour' on an economy-wide scale, with resultant effects upon income distribution and demand for agriculture (and other equally broadly defined 'products'). The movement from one point on a supply curve for 'agriculture' to another will thererfore alter the conditions of demand, requiring a separate demand curve. The idea of a unique equilibrium price is therefore unsustainable.
Sraffa put the argument the following way:
“If in the production of a particular commodity a considerable part of a factor is employed, the total amount of which is fixed or can be increased only at a more than proportional cost, a small increase in the production of the commodity will necessitate a more intense utilisation of that factor, and this will affect in the same manner the cost of the commodity in question and the cost of the other commodities into the production of which that factor enters; and since commodities into the production of which a common special factor enters are frequently, to a certain extent, substitutes for one another ... the modification in their price will not be without appreciable effects upon demand in the industry concerned.” (Sraffa 1926)
Graphically, Sraffa’s argument is that, rather than the traditional picture shown in Figure 1, the situation will be as illustrated by Figure 2. In this case, the analysis of a single market in isolation from others is meaningless. Clearly, a narrow definition of industry should be used.
Figure 2: Multiple equilibria with a broad definition of industry
However, if we use a narrow definition, then Sraffa argued that the concept of a fixed resource becomes questionable. Each industry will use such a small quantity of the allegedly fixed resource that, in general, extra units of it are easily acquired by either bringing fallow resources into use, or by converting resources from one related usage to another. As Sraffa put it,
“If we next take an industry which employs only a small part of the 'constant factor' (which appears more appropriate for the study of the particular equilibrium of a single industry), we find that a (small) increase in its production is generally met much more by drawing 'marginal doses' of the constant factor from other industries than by intensifying its own utilisation of it; thus the increase in cost will be practically negligible…” (Sraffa 1926)
The logical import of Sraffa’s argument was that, if it can be drawn at all, the short run supply curve for an individual market should be horizontal (or even downward-sloping), which would return economics to the classical analysis of price determination. Rather than the Marshallian vision of price being set by the scissors of supply and demand, supply sets (short-run price), while demand determines the quantity sold at that price, as in Figure 3. If Sraffa was right, then the neoclassical theory of the firm, and especially the model of perfect competition, was untenable.
Figure 3: Constant returns with a narrow definition of industry
Sraffa later made this explicit when commenting upon Robertsons' attempt to counter his critique of the instability of the neoclassical model of the firm in the face of increasing returns. Sraffa concluded that
I am trying to find what are the assumptions implicit in Marshall’s theory; if Mr Robertson regards them as extremely unreal, I sympathise with him. We seem to be agreed that the theory cannot be interpreted in a way which makes it logically self-consistent and, at the same time, reconciles it with the facts it sets out to explain. Mr Robertson’s remedy is to discard mathematics, and he suggests that my remedy is to discard the facts; perhaps I ought to have explained that, in the circumstances, I think it is Marshall’s theory that should be discarded. (Sraffa 1930: 93)
It is obvious that Sraffa’s call has had no impact on mainstream economics. Subsequent papers (Pigou 1922, 1927, 1928; Robertson 1924, 1930; Robbins 1932, Harrod 1934; etc.) accepted that diminishing marginal productivity in the short run was indisputable, and that perfect competition was theoretically watertight. Today’s introductory to advanced microeconomics textbooks propound the same belief, while even advanced neoclassical theory implicitly treats perfect competition as the ideal (with a research agenda of deriving its customary conclusions from less restrictive market models).
Why was Sraffa's critique ignored? There are undoubtedly many reasons,but one which we believe played a major role was the apparently conclusive mathematics behind the marginalist theory of the firm: logic, apparently, was overruled by mathematics. It is our aim in this paper to show that,far from being conclusive,this mathematics is internally contradictory. To counter erroneous assertions (made during previous presentations) that our results involve mathematical errors, all symbolic calculations in this paper have been done using symbolic mathematics programs (Mathematica and Mathcad).
Mathematical Flaws
The neoclassical theory of the firm begin with the definition that profit is the gap between revenue and cost (equation 1):
1
In keeping with the Marshallian concept of the short run, price and cost are treated as functions of quantity only (rather than, for example, time and quantity):
2
The standard assumptions are that the demand and cost curves are continuous in the relevant range and at least twice differentiable, with the first differential of price negative and the first differential of cost positive (to simplify our argument we assume, without loss of generality, linear demand and marginal cost curves). Then the zero of the differential of profit indicates where profit is maximised:
3
The conclusion from this reasoning is that firms maximise profits by equating marginal revenue to marginal cost. In the case of a small firm in a competitive industry, the profit maximisation exercise is to maximise profits derived from its output q:
4
The model of perfect competition arises from the assumption
that, since the small firm is such a small part of the industry output 
.
This reduces its profit maximising problem to the perfectly competitive
condition that price equals marginal cost: 
.
This result means that the marginal cost curve becomes the supply curve for the firm, and the industry supply curve is then derived by summing the marginal cost curves of all firms in it. In this respect, the perfectly competitive industry differs from all other market forms in that a supply curve can be derived. As is illustrated by Figure 4, with a horizontal demand curve facing the competitive firm, its point of supply coincides with its marginal cost curve for all levels of output (above the minimum of average variable cost). For a monopoly, the point of supply never coincides with the marginal cost curve, and an apparent increase in demand can actually lead to a fall in the supply price.
Figure 4: Supply curve can only be derived for perfect competition
These differences in the nature of the demand curve perceived by the individual firms is also a foundation for guidance about welfare, and for the one, peculiar radicalism of mainstream economics: its advocacy of small, competitive firms in preference to large, monopolistic ones. The following indicative diagram and explanation are taken from a recent edition of an influential textbook (Lipsey & Chrystal 1998, Figure 10.6):
Figure 5: Welfare arguments in favour of perfect competition over monopoly
Figure 10.6 The deadweight loss of monopoly. Monopoly is allocatively inefficient because it does not maximise the sum of consumers' and producers' surpluses. At the perfectly competitive equilibrium Ec consumers surplus is the sum of the yellow shaded areas 1, 5 and 6. When the industry is monopolized, price rises to pm and consumers' surplus falls to area 5. Consumers lose area 1 because that output is not produced; they lose area 6 because the price rise has transferred it to the monopolist. Producers' surplus in a competitive equilibrium is the sum of the dark blue areas 7 and 2. When the market is monopolized and price rises to pm the surplus area 2 is lost because the output is not produced. However, the monopolist gains area 6 from consumers. Area 6 is known to be greater than 2 because pm maximises profits. Thus areas 1 and 2 are lost to society. They represent the deadweight loss resulting from monopoly and account for its allocative inefficiency. (Lipsey & Chrystal 1998: 161)
In what follows, we show that the 'perfectly competitive' condition of price equal to marginal cost cannot be reconciled with a market demand curve where demand rises as price falls. Our method is to consider whether changing output from the allegedly profit maximising level of output in fact reduces profit. To illustrate the applicability of the logic, we first demonstrate it with the uncontested case of profit maximisation under monopoly.
Monopoly
In the case of monopoly, the firm faces the entire market
demand curve. If it is true that the profit maximising level of output qm
is the level at which marginal revenue equals marginal cost, then it should be
provable that a small deviation dq from
this level will cause a reduction in profit. Consider the output level (qm+dq) where 
.
Profit at (qm+dq) is
5
To compare this to profit at qm, we can expand terms using a Taylor series. The symbolic engine returns:
converts to the series
Some explanation of notation is in order here. The
expression 
means "the derivative of the function
P(t) evaluated at the point where t=qm". In standard notation,
this is 
.
Similarly, 
is "the derivative of the function TC(t)
evaluated at the point where t=qm" is 
,
or marginal cost evaluated at qm (we emphasise this because
this has been a point of contention in explaining our analysis to neoclassical
economists). In more conventional notation, we therefore have that profit at qm+dq is:
6
Since we have assumed, for the sake of simplicity, that the second and higher order derivatives of P and TC are zero, this reduces to:
7
The term 
is clearly 
.
Also, we have defined qm as the output level at which marginal cost
equals marginal revenue. The term 
in this equation is therefore zero.
Rearranging, we get the result that
8
Since, by assumption, the market demand curve is downward
sloping, 
.
Since 
,
we have that 
.
Therefore this firm-level analysis confirms the standard argument, that a
monopoly will maximise profits by equating marginal revenue and marginal cost.
Now we apply the same analysis to perfect competition.
Perfect competition
Profit for a perfectly competitive firm is
Neoclassical theory argues that the profit maximising level
of output for a perfectly competitive firm is where price equals marginal cost.
Call this output level qm: then 
.
The theory asserts that price is independent of the output level of the single
firm, so this can also be written 
.
However, this is also equivalent to 
where 
.
Since we wish to investigate what this condition actually means in the context
of a market demand curve where 
,
we will continue to write the price condition in the full form. Profit at
output level qm will therefore be:
Profit at (qm+dq) will be
Expanding this using the symbolic engine:
converts to the series
Putting this into the same standard form as for monopoly, we have
Making the same simplification as for monopoly, we set the second and higher order differentials to zero to yield
As before, 
.
Therefore:
Since by definition, qm is the level of output at which price equals marginal cost, we can cancel only the first and third of the three terms in the first expression (unlike monopoly, where the entire expression can be cancelled). This leaves us with
9
Whereas the sign for monopoly was unequivocally negative,
the sign for this expression is at best zero (if 
),
or negative (if 
).
The only condition under which qm can be profit maximising is if 
;
if it is even slightly negative, then a perfectly competitive firm can increase
its profit by reducing its output. Under what conditions can 
?
We can get some general guidance by expanding 
:
10
Since we know that 
<0,
can equal zero only if 
.
This may look acceptable to neoclassical economists, on the basis that the
output of the single firm is so much smaller than the output of the industry
that 
could
quite feasibly be zero. This attitude can be seen in any standard textbook
explanation of perfect competition. Lipsey & Chrystal, for example, argue
that:
Because all products have negatively sloped market demand curves, any increase in the industry’s output will cause some fall in the market price. The calculations given below show, however, that any conceivable increase that one wheat farm could make in its output has such a negligible effect on the industry’s price that the farmer correctly ignores it… The calculations given below … [show that] the elasticity of demand for the farm’s product is very high… It is not surprising, therefore, that the farmer regards the price of wheat as unaffected by any change in output that his one farm could conceivably make. For all intents and purposes, the wheat-producing farm faces a perfectly elastic demand curve for its product… (Lipsey & Chrystal 1998: 141)
This attitude—that 
is effectively zero while 
is negative—is an incorrect deduction from
the fact that the elasticity of demand for a single firm (
)
is very large compared to the elasticity of demand for the market as a whole (
).
This latter true proposition is simply due to the fact that Q is much larger
than q: it tells us nothing about the relative size of 
and 
.
In fact, given the assumption of atomism which is an
integral part of the model of perfect competition--so that firms do not react
to the behaviour of other firms--then if a single firm changes its output by dq, total industry output should also change
by dq. The assumed value of 
in perfect competition is therefore 1. As a
result, 
and 
are equal and negative.
11
Feeding this result back into our previous equation, we find that qm--the level of output at which market price equals marginal cost--cannot possibly be a profit maximising equilibrium for the firm. If dq is negative then
12
The individual competitive firm can increase its profit by reducing its output below the level at which price equals marginal cost.
The behaviour of other firms
As alluded to above, the only circumstance in which price
equals marginal cost can be profit maximising for the individual firm is where 
.
We can examine what this means in terms of the behaviour of the rest of an industry
by breaking total output Q down into output due to a single firm q, and output
by the rest of the industry QR.
13
Thus for 
to be zero, it must be that 
:
te behaviour of other firms in the industry must counteract the behaviour of
any single firm. Collusion certainly cannot be appealed to to reach this
result: if this behaviour occurs, then it must be due to atomistic profit
maximising behaviour by the other firms.
But this is simply not feasible. If all firms are producing a quantity where price equals marginal cost, and it is true of any one firm that it can increase its profit by reducing its output, then it is true of all other firms. Also, all other firms benefit from the reduction in output by a single firm, even if they themselves do not alter their output. The new market price level will be that resulting from an industry level output of Q+dq, where dq<0. Profit for any firm that does not also change its output will be:
The series expansion of this is:
Using our linear simplifications, this reduces to
14
This is greater than the profit earned when the industry
output level was Q, since 
if dq<0.
So all other firms in the industry will indirectly benefit from the reduction
in output by a single firm, and they can of course directly benefit if they
also reduce their output.
Where will this process of output reduction stop? With the standard assumptions of a downward sloping market demand curve and diminishing marginal productivity, it will stop at the same level of output as for a monopoly: where the marginal revenue for each firm equals its marginal cost.
The theory of perfect competition is therefore fundamentally flawed. Far from being an ideal market structure by which all other market types should be judged, it is an illusion built upon fundamental mathematical fallacies. As a careful examination of standard economic models emphasises, if a competitive firm set output at a level where marginal cost equalled price (rather than a level at which marginal cost equalled marginal revenue), all output beyond the point where marginal cost equalled marginal revenue would be produced at a loss. Price equals marginal cost is incompatible with profit maximising behaviour.
As a corollary, the welfare analysis based upon maximising producer and consumer surplus is also flawed. Not only will competitive markets and monopolies have similar welfare profiles (in that both will produce where marginal cost equals marginal revenue), but the welfare ideal of maximising the sum of consumer and producer surplus can only be achieved if producers (voluntarily or otherwise) forego surplus by individually and collectively producing some output at a loss. This is evident even in Lipsey & Chrystal's Figure 10.6 above, and comparable diagrams like it. When price equals marginal cost, producer surplus is equal to the sum of the areas 7 and 2; when marginal revenue equals marginal cost, it is the sum of areas 7 and 6, which is substantially larger. Producers must therefore forego producer surplus to produce beyond the point at which marginal cost equals marginal revenue. While this 'net loss of consumer and producer surplus' has been portrayed by economists as part of the deadweight loss due to monopoly, it is in fact part of the inevitable 'deadweight loss' of profit-maximising behaviour.
The analysis also points out an inherent fallacy in the proposition that profit-maximising behaviour is consistent with market output being determined by the intersection of demand and supply. This proposition cannot be reconciled with profit maximising behaviour, since no the number of firms in an industry has no impact on the existence or otherwise of a marginal revenue curve. If the demand curve is downward sloping, then there is a market marginal revenue curve which is below it and more steeply sloped. This curve, and not the demand curve, determines the behaviour of profit maximisers--whether directly at the level of the market, or indirectly via the behaviour of myriad individual firms. The model of perfect competition was based on the erroneous belief that the marginal revenue curve could somehow be made to disappear.
Profit and time
This critique strengthens part of Sraffa's attack on the Marshallian theory of the firm and short run price determination. The model of perfect competition is crucial to the argument that price is set by intersecting demand and supply curves, since only in that model can a market supply curve be derived. We have shown that this model is not feasible, and therefore the problems with Marshall's vision are even greater than those put by Sraffa. Whereas Sraffa argued that the supply curve was probably horizontal, in the absence of perfect competition, a supply curve cannot even be derived.
Sraffa reached his result by logically deconstructing the concept of diminishing marginal productivity. We have reached ours by accepting that concept, but deconstructing the concept of the supply curve. This would appear to leave intact the core proposition that profit is maximised by equating marginal cost to marginal revenue (even if it is no longer tenable that marginal revenue can be equal to price). However, Sraffa's critique of diminishing marginal returns is strengthened if it is appreciated that, at the core of Sraffa's objection to Marshall's theory, was a rejection of the Marshallian--and neoclassical--treatment of time.
Time in neoclassical analysis is divided into three segments, on the basis of the variability of factors of production. The market period is defined as that period of time in which no factors can be varied; the short run is the period of time in which (at least) one factor cannot be varied; while the long run is when all factors can be varied. Sraffa effectively argued that the short run does not exist for the vast majority of products in an industrial economy, so that constant returns are the rule. However, the concept of diminishing marginal productivity is so much a part of the economic vision that Sraffa's verbal critique received short shrift from the economics profession.
It is feasible to see Sraffa's critique as simply an attempt to take seriously the limitations which Jevons, Walras, Marshall and Clarke all acknowledged were endemic in using static methods to analyse what are clearly dynamic problems. Their defence for the use of static tools was the inherent difficulty of dynamic analysis, and the absence of suitable tools. No such defence is available to modern economics, since dynamics is now a far more developed field of analysis (in sciences other than economics), and so many tools exist to analyse dynamic systems dynamically. We can begin this process of recasting economics as a dynamic science by taking Sraffa's critique to heart and drop altogether the neoclassical treatment of time. Our first step here should be to take time seriously by treating revenue and costs as functions of not just quantity, but also time. At any point in time, a firm can decide to produce from zero to 100% of its installed capacity, so that quantity itself is a function of time. Similarly, demand and costs will vary with time (via changing tastes, changing technology, etc.). We then have in general that price, costs and quantity itself are all functions of time. We can now specify the profit function of a single firm as:
15
In this dynamic format, the objective of the firm is no
longer the static one of maximising profit at any point in time. Instead, it is
more sensible to see the key variable in the firm's objective function as being
the rate of change of profit over time: 
.
We can easily define this using the above definition. Using Mathematica for the
symbolic derivation, and taking advantage of the fact that 
,
we find that 
is equal to:
16
Some explanation of Mathematica's notation is in order here.
F(1,0)(x,y) means 
while F(0,1)(x,y) means 
.
Converting the above into more familiar partial differential notation (and
dropping the explicit statement of dependence of P and TC on q and t, and of q
on t), we find that
Grouping terms, we find that:
17
This simple exercise establishes a point which is well-known
to practitioners of dynamic analysis, but not well understood by economists:
static and dynamic analysis can lead to contrary conclusions. Note that the
first expression in parentheses is marginal revenue minus marginal cost: 
.
Therefore we can write:
18
Static profit maximisation analysis argues that 
should be set to zero to maximise profits.
But from the above equation, the only conditions under which equating marginal
cost and marginal revenue will maximise the rate of growth of profits is if 
.
If 
--so
that the firm's sales grow with time--then maximising the rate of growth profit
requires that 
.
A firm that took conventional static economic advice literally would find that
its profit grew more slowly.
We anticipate that neoclassical economists will find this result, if anything, harder to accept than our previous result that perfect competition is impossible. But it is just as soundly based. The difficulty comes, not from any difficulty in the analysis--which is itself very simple--but from the difficulty of escaping from the "habitual modes of expression and thought" of which Keynes once spoke.
Economists have so absorbed the image of intersecting supply and demand curves, and intersecting marginal revenue and marginal product curves, that it is very difficult to appreciate that they are constructs of the imagination rather than real entities. As Clarke remarked over a century ago, "A static state is imaginary. All actual societies are dynamic; and those that we have principally to study are highly so. Heroically theoretical is the study that creates, in the imagination, a static society" (Clark 1898). The vision of intersecting static marginal revenue and marginal cost curves is just such a heroic abstraction, when in the real world, prices, quantities and costs are forever varying.
Conclusion
The above analysis supports Sraffa's call in 1926, by showing that the superficially robust edifice of the microeconomics of production is in fact devoid of content. Its canonical model, of perfect competition, is untenable, while its canonical insight, that equating marginal cost and marginal revenue maximizes profit, is invalid in a dynamic setting.[1] What then should economics do?
We would argue that microeconomics has to become an empirical discipline. Rather than trying to prove results about firms and market structures from a priori reasoning, microeconomics should instead be based in robust empirical research into how firms actually behave, and how actual markets function over time. Economics has a tradition of ignoring empirical work, on the grounds that, whatever firms might say they are doing, their behaviour can be interpreted "as if" they are following neoclassical precepts. We have shown that this cannot possibly be so, since the standard neoclassical a prioris are in fact erroneous.
The very good empirical work that has been done (see Lee 1998 for an authoritative survey) has generally found that the neoclassical concepts perfect competition, monopoly, marginal cost and marginal revenue are regarded as meaningless concepts by factory managers--a result that should no longer be surprising after the above. Instead, firms normally exist in markets where there is a range of firms of different sizes, complex inter-relations between them, price normally far exceeds marginal cost, and variable costs tend to fall as production rises, rather than being U-shaped.
The apparently neat, self-contained visage of conventional neoclassical microeconomics hides an empty soul. An empirical economics is bound not to have as tidy a visage, but it will at least have content.
References
Bishop, R.L., (1948). 'Cost discontinuities, declining costs and marginal analysis', American Economic Review 38: 607-617.
Clapham, J.H., (1922a). 'Of empty economic boxes', Economic Journal, 32: 303-314.
_ (1922b). 'The economic boxes - a rejoinder', Economic Journal, 32: 560-563.
Clark, J.B. 1898). 'The future of economic theory', Quarterly Journal of Economics, 13: 1-14).
Eiteman, W.J., (1947). 'Factors determining the location of the least cost point', American Economic Review 37: 910-918.
_ (1948). 'The least cost point, capacity and marginal analysis: a rejoinder', American Economic Review 38: 899-904.
Eiteman, W.J. And Guthrie, G.E., (1952). 'The shape of the average cost curve', American Economic Review 42: 832-838.
Haines, W.W., (1948). 'Capacity production and the least cost point', American Economic Review 38: 617-624.
Keen, S., (2001). Debunking Economics, Pluto Press & Zed Books, Sydney.
Langlois, C., (1989). 'Markup pricing versus marginalism: a controversy revisited', Journal of Post Keynesian Economics 12: 127-151.
Lee, F., (1996). 'Pricing and the business enterprise', in C.W. Whalen (ed.), Political Economy for the 21st century, ME Sharpe, Armonk.
_ Lee, F., (1998). Post Keynesian Price Theory, Cambridge University Press, Cambridge.
Lipsey, R.G., and Chrystal, K.A., (1998). Principles of Economics, Oxford University Press, Oxford.
Means, G.C, (1972). 'The administered-price thesis reconsidered', American Economic Review 62: 292-306.
Pigou, A.C., (1927). 'The law of diminishing and increasing cost', Economic Journal, 41: 188-197
_ (1928). 'An analysis of supply', Economic Journal, 42: 238-257.
Robertson D.H., (1924). 'Those empty boxes', Economic Journal, 34: 16-31.
_ (1930). 'The trees of the forest', Economic Journal, 44: 80-89.
_ (1930). “Increasing returns and the representative firm”, Economic Journal 44: 79-89.
Shepherd, W.G., (1984). '‘Contestability’ vs. competition', American Economic Review, 74: 572-587.
Shove, G.F., (1930). 'The representative firm and increasing returns', Economic Journal, 44: 93-116.
Sraffa,P., (1925). “On the relation between cost and quantity produced”, Annali di Economia 2: 277-328. (translation available at http://iml.umkc.edu/econ/economics/faculty/Lee/ReadingsLee/ReadingsLee.html)
_ (1926). “The law of returns under competitive conditions”, Economic Journal, 36: 535-550.
(1930). “Increasing returns and the representative firm: a criticism”, Economic Journal 40: 89-93., 44: 93-116.