As I explained in my last post, the cost curve in the Treasury’s model of the RSPT is based on a fantasy. The situation is even worse when we turn to the other side of the model, the demand curve: it is based not merely on a fantasy, but an outright fallacy.
Legions of economists believe that the demand curve for a competitive firm is horizontal, but their belief is based on the most fundamental of mathematical errors: confusing a very small amount (an infinitesimal) with zero. The Treasury repeats this fallacy—without knowing it is one—in its explanation of why a royalty reduces output levels but the RSPT won’t.
Figure 1: Treasury’s drawing of the demand and supply curves facing a mining venture
In the absence of taxes and royalties, the full?production rate of output would be determined by the intersection of the marginal revenue curve (line P) and the marginal cost curve… With the RSPT, however, the selected output would remain at QA, since the price line P describes the relevant marginal revenues. (The Resource Super Profits Tax: a fair return to the nation, p. 39)
Treasury describes the line P as “the marginal revenue curve”, where marginal revenue means the increase in revenue that the firm gets for selling an extra unit of output (say a ton). Treasury argues that this equals the market price, and is independent of the amount the firm produces: if the market price is $100 a ton, then the firm gets $100 for every extra ton it produces, from zero out to a million tons.
Here’s Treasury is simply parroting First Year Microeconomics propositions that the market price falls as the quantity supplied to the market increases, but that at the same time, any individual firm in a competitive industry has no impact on market price (see Figure 2).
Figure 2: Neoclassical micro “marginal revenue equals price for competitive firms”
You won’t pass an exam in Microeconomics unless you accept that these two propositions are compatible—that an increased supply to the market will cause the market price to fall, but individual firms have no impact on the market—and yet mathematically they are incompatible.
Amazingly, this was first proven back in 1957 by one of the staunchest defenders of Neoclassical economics ever, George Stigler (Stigler (1957)), using one of the simplest rules of mathematics, the “Chain Rule”. Stigler decomposed the allegedly horizontal individual firm demand curve faced into two components: the market demand curve, and how much the individual firm’s change in output changes market output.
The first factor is unambiguously negative: according to Neoclassical theory, the market demand curve slopes downwards, so if supply increases, the market price must fall. So for the firm’s demand curve to be horizontal—in mathematical terms, to have a slope of zero—the second factor would have to be zero: an increase in supply by an individual firm would have to make no change to market supply.
This would only happen if the rest of the industry reduced output as much as the single firm increased it; but that implies collusion when the theory assumes competition! Stigler instead gave this second factor a value of 1: if a single firm increases output by (say) 1000 tons, the amount supplied to the market rises by 1000 tons, so dQ/dqi equals 1 (Figure 3).
Figure 3: Stigler uses the chain rule to show that dP/dq = dP/dQ
The logic behind Stigler’s proposition that simply applies the “atomism” assumption that is part of the theory of competitive markets: that firms in a competitive industry do not react strategically to the hypothetical actions of other firms. Spelling this out in more detail, the result follows as shown in equation in the Appendix. Spelling that equation out verbally:
The change in market price (dP) caused by a change in the output of a single firm (dqi) equals
- The change in market price (dP) caused by a change in market output (dQ)
- The change in market output (dQ) caused by a change in the output of a single firm (dqi)
- The first component is negative since the market demand curve slopes downwards
The second component is 1 because
- Market output (Q) is the sum of the output of the n firms in the industry;
- If a single firm (qi) changes its output (by dqi)
- the other firms in the industry don’t change theirs in response
- so the increase in total market output equals the increase in output by the single firm
- So since the second component is 1, the slope of the individual firm’s demand curve is the same as the slope of the market demand curve.
The upshot of Stigler’s very straightforward mathematics was that the demand curve facing the individual firm cannot be horizontal, but has exactly the same negative slope as the market demand curve: if the firm increases its supply, market price (and the price the firm gets) will fall.
Stigler’s maths puts the lie to the standard drawings via which economists learn microeconomics—like the one in Figure 4.
Stigler’s mathematics is both impeccable and simple, and it’s easily illustrated in diagrams too. Firstly, if the market demand curve slopes down, then any small portion of it also slopes down—even the bit seen by a single competitive firm. Secondly, for that firm to have no impact on market price, regardless of the quantity it sells, then the market price must be the same for the quantity P(Q+qi)—if the firm produces some output—as it is for P(Q)—if the firm produces nothing. But that contradicts the proposition that the market demand curve slopes downwards (Figure 5).
Figure 5: A small segment of a downward sloping line is still downward sloping
Equally, firms are described as “rational profit maximizers” in neoclassical theory—but what’s rational about believing that an increase in your output won’t affect market price, if you also believe that market price will fall if the quantity supplied to market rises (Figure 6)?
Figure 6: Irrational versus rational beliefs for a competitive firm
Why don’t economists accept this mathematical logic and devise a better theory of the firm? Because, if it is accepted, the whole theory of competitive markets fails: market price is not set by the intersection of supply and demand curves, but by a markup on the cost of production. The vision of perfect competition, which frames how economists think about the real world, is both a fantasy and a fallacy.
This is so destructive of the overall Neoclassical vision of how a market economy works that this very accurate mathematics is suppressed—in the teaching of a discipline that prides itself on being mathematical. Instead, economists continue teaching—and 99% of them continue believing—that the slope of the demand curve facing the individual demand curve is horizontal (or “The price line is horizontal because the operator is a price taker” as Treasury puts it, or some other euphemism for a mathematical fallacy).
I rediscovered this when writing Debunking Economics in 2000—and subsequently found that Stigler had beaten me to it, before I’d even gone to school. So how did neoclassical economists justify suppressing such a simple demolition of a key plank of their story?
Because Stigler argued that he’d found a way out of the conundrum in that same article: yes, individual firms face downward sloping demand curves, with exactly the same slope as for the market itself; but if they all set marginal cost equal to marginal revenue, then in the limit marginal revenue for the individual firm will converge to price, and the original unblemished story can be restored (See Figure 7).
Figure 7: Stigler’s way out
This is mathematically true: if firms equate marginal cost and marginal revenue, then marginal revenue will converge to market price for the individual firm. Given this, and the belief that equating marginal revenue and marginal cost will maximize profits, economists are content to ignore Stigler’s analysis and just stick with the old but false proposition that the demand curve facing the individual competitive firm is horizontal.
In itself this is mendacious: why teach a fallacy, when there’s a more legitimate way to reach the same result? But there’s a more important problem that I discovered after writing Debunking Economics: what neoclassical economists call “profit maximizing behavior” doesn’t actually maximize profits—even if their model of the firm was otherwise correct. Instead, a firm following their advice would produce about twice the level that actually would maximize profits, and lose money on the deal as well.
Equating marginal cost and marginal revenue doesn’t maximize profits
One of the basic mantras of neoclassical economics is that all firms, from a monopoly to a tiny competitive firm, maximize profits by equating marginal cost and marginal revenue. It’s the basis of Stigler’s “rescue” of perfect competition back in 1957: that even though the demand curve facing the individual firm can’t be horizontal, as more firms enter an industry, marginal revenue converges to price if they all maximize profits by equating marginal revenue and marginal cost.
My personal contribution to this literature was to show that this mantra isn’t true: equating marginal revenue and marginal cost only maximizes profits if there is just one firm in the industry—a monopoly. With more than one firm, the correct profit-maximizing formula is to have marginal revenue exceed marginal cost.
Here n represents the number of firms in the industry, lowercase letters represent an individual firm, and uppercase letters represent the market.
I’ve proven this is many different ways, but the starting point of them all is that—ironically—the neoclassical mantra ignores the market: even if an individual firm can’t affect what other firms in the market do, what other firms in the market do does affect the individual firm. Therefore to maximize profits, the firm needs to behave as if it’s in a market where it is only one of the factors affecting its profitability. The profit-maximization problem is therefore one of total differentials, rather than the ordinary differential used by neoclassical economists.
The solution, as explained in these papers, is the one shown in equation later on. And it doesn’t involve collusion—just the firm altering its production levels in response to changes in its profits.
Why does this matter?
To bring this back to practical issues, what neoclassical economists describe as profit-maximizing behavior actually involves selling about 50% of output at a loss.
Normally this doesn’t have any practical impact—beyond making economists singularly useless in advising a firm on output levels. But it has a real impact when competition regulators decide that a breakup of a monopoly will result in lower costs for the consumer because of competition, or that a utility should sell its output at its marginal cost—because according to neoclassical theory, that is both socially optimal AND profit-maximizing for the firm.
In fact, as consumers of privatized services often find, costs don’t fall, and competition fragments what was more effective as a monopoly-delivered product; and as some utilities find out to their great cost, marginal cost pricing involves cheap services to the public for a while (cheap electricity, water, etc.) but a low or even negative rate of profit that means the utility can’t be maintained. Breakdown of the system—as with electricity supplies in California earlier this century—is often the outcome.
This theory is also the justification for the policies of competition watchdogs like the Australian Competition and Consumer Commission (ACCC). While monopolies can certainly have deleterious consequences in some contexts, the automatic bias that this theory leads to in favour of competition are not justified when the theory itself is erroneous. Bad theory can lead to good policy only by sheer accident, and that’s an accident that I wouldn’t expect all that often.
Too low a rate of profit
The final flaw of the RSPT is probably the most important of all: the rate of profit that it regards as indicating super-profits is too low. The proposition that “super profits” occur when the rate of profit exceeds the long term bond rate is nonsense: any venture that appeared to offer a rate of return that low wouldn’t be undertaken by any sensible capitalist.
If the best estimate of a project is that it would return the long run bond rate, then why not buy long term bonds? At least they will exist in ten or so years time (well, maybe before the GFC anyway…) whereas your project may be brought undone by something you can’t foresee—like a BP disaster in the Gulf of Mexico bringing your plans for offshore drilling in East Timor unstuck.
The brilliant Australian Professor of Applied Mathematics John Blatt showed decades ago that the best estimate of whether a project should be undertaken was a concept that neoclassical economists deride: the “payback period” (Blatt (1980)). Blatt showed that the payback period considers both the discount rate that should be applied to expected future cash streams—where the long term bond rate is a reasonable guide—and what Donald Rumsfeld latter famously called “known unknowns” and “unknown unknowns”. The upshot is that the long term bond rate seriously underestimates the minimum expected rate of return that any prudent investor should consider. A rate that low isn’t super at all—and nor is the logic behind the RSPT, even though the concept itself is a valid one.
Appendix: Some mathematics
As I detail in Debunking Economics, the neoclassical theory of the firm is almost totally irrelevant to how actual firms and markets behave. Worse, it distracts attention from the real-world processes that actually matter, in particular the technological and other forms of product competition that firms undertake that both introduce new products over time and reduce the costs of existing ones.
It also encourages economists to be “revolutionaries”: restructuring industries from the pattern they have evolved into, to the one that the textbook says is the best.
We’d be far better off abandoning the theory completely, and building an entirely different model that suits real-world data—which shows that most industries have a range of firm sizes rather than the economic fantasy of monopoly at one end and perfect competition at the other—and for which Schumpeter’s model of competition is far more suitable (Schumpeter (1934)).
So why waste time pulling it apart mathematically? Because the last time economics attempted to break out of the neoclassical straightjacket, the escape was subverted by economists reworking macroeconomics until it was consistent with neoclassical microeconomics. The so-called “microfoundations debate” undermined macroeconomics because economists believed that micro had sound foundations while macro did not.
The reality is that micro itself is unsound—but even now when I present to an audience that includes neoclassical economists, I often hear them arguing that macro might be a mess (something they are forced to concede because of the total failure of neoclassical macroeconomics to anticipate the GFC, and the success of policies to counter the GFC that neoclassical theory said could never work), but micro is OK.
No it’s not. Every aspect of neoclassical microeconomics is a pseudo-science, including the theory of the firm. But given the seductiveness of the vision it offers of a perfect, competitive world in harmonious equilibrium, a proper exposure of its flaws is necessary.
With that caveat, I’ll turn to the proof of why “perfect” competition can’t exist. This proof that equating marginal cost and marginal revenue doesn’t maximize profits inevitably involves mathematics, but I know that for most readers that leads to what is known as the “MEGO” effect: “My Eyes Glaze Over”. For those who can cope with mathematics, here’s the proof. For everyone else, it’s time to read a different blog!
Firstly, the proof that the slope of the individual firm’s demand curve is the same as that of the market demand curve. Applying the chain rule, the slope of the individual firm’s demand curve can be broken into the market demand curve, times how much market output changes if a single firm alters its output. Since market output is the sum of the output of all individual firms, and in the Marshallian model firms don’t react strategically to each other (in contrast to the Cournot-Nash version), this last term is 1, as shown by a simple expansion of market output:
There are many ways to prove that the actual profit-maximizing output level is given by equation rather than by equating marginal cost and marginal revenue:
The technically most correct proof starts from the correct initial definition that the maximum of the profit function for a single firm isn’t simply a function of its own behavior, but also of what all other firms do. So the true profit-maximizing problem is to solve the following equation:
That’s do-able (and it’s done below), but it’s normally a stretch for economists to follow, so a simpler proof can start from working from the starting point of assuming that firms do what economists advise them to do, and see what the outcome of that is when we correct for the mistake of putting rather than the correct expansion that . If you can handle the more complicated maths, skip the relatively messy explanation in equations to , and jump straight to equation .
We start from assuming that the condition in equation applies:
Then we expand what mr and mc are for the ith firm:
The first term in the expansion for marginal revenue gives us n copies of P; in the second term can be replaced by , and the expression can be rearranged to . So the expansion of marginal revenue is now complete:
The expansion of marginal cost simply gives us n copies of industry level marginal cost:
So the total expression in equation is as shown in equation :
So if all firms follow neoclassical profit-maximizing advice, the aggregate situation for the industry will be shown by the right hand side of equation :
What does that apply for the industry level of output? A bit of rearranging explains it: split off one term for P and one for MC and we get equation :
Rearranging this (and omitting the dependence on Q to simplify the notation) gives us:
Only for a monopoly—where n=1—will equation equal zero. For all other industry structures, what neoclassical theory calls profit maximizing behavior actually results in part of market output being produced at a loss—since then marginal revenue will be less than marginal cost. But the industry is just the sum of the firms in it; therefore if each firm equates its marginal revenue and marginal cost, each firm must producing more than the profit-maximizing level.
So if following neoclassical advice causes firms to produce more than the profit-maximizing level, can the advice be modified to work out the true profit-maximizing level? If each firm actually does produce the profit maximizing level for itself and no more, then industry level marginal revenue will also equal industry level marginal cost. Using the expression for this in equation , the real profit-maximizing level can be worked out using equation :
Taking P and MC inside the summation sign yields:
The profit-maximizing level for the single firm is thus to set equation to zero:
That’s the messy way to derive the result. A cleaner method is to start from the total differential that actually does maximize profits:
We can drop since this is 1. Expanding out yields:
Expanding the first expression gives us one copy of P (since if and on the one occasion that )
The second expression in is how much the jth firm’s costs change for a change in output by the ith firm. This is zero n‑1 times and marginal cost for the ith firm once. So the complete expression is that the jth firm will maximize its profits if it produces at the level at which its marginal cost equals industry-level marginal revenue:
This is operationally equivalent to the expression in .
There’s much more to the critique than this, but that’s all I have time for here. Other aspects include that the conditions for the aggregation of marginal cost curves from competitive firms to equal that for a monopoly require either a fluke or constant marginal costs (see Appendix A.1 to Keen (2004)), that Cournot-Nash behavior is irrational (Keen and Standish (2006, pp. 82–83)), and that a population of instrumentally rational profit maximizers will converge to the Keen solution, not the Cournot-Nash solution (Keen and Standish (2006, pp. 83–85)).
Again I’m doing this too quickly because I want to get back to the important stuff of developing an alternative credit-based macroeconomics from Minsky’s foundations, but one useful additional way that I undercut the conventional model was to use computer simulation.
The theory tells us that a market populated by profit maximizing competitive firms would produce where marginal revenue equals marginal cost. I instead predicted that such an industry would produce the amount predicted by equation . I tested this by building a model of a hypothetical market with a downward-sloping demand curve:
Figure 8: Price, marginal cost, marginal revenue and neoclassical predictions
Identical firms were defined in such a way that the marginal cost curve for a monopoly (with a single factory) was identical to the sum of the marginal cost curves of a 100 firm competitive industry (with one factory per firm):
Figure 9: Firm costs defined to ensure that different scales of output have the same aggregate marginal cost curve
Neoclassical theory predicts that the amount produced will be a function of the number of firms in the industry: the more firms, the higher the output and the lower the market price:
Figure 10: Neoclassical predictions
The program then runs through simulations of the behavior of “instrumental profit maximizers”, who simply choose an output level, choose an amount to vary output by, and vary output in search of higher profit. It starts with a simulation of a monopoly, and ends with a simulation of a 100 firm industry:
Figure 11: The program in Mathcad
The aggregate outcome of the model was very different to the neoclassical predictions: rather than a competitive 100-firm industry producing much more than a monopoly, they both produced much the same amount.
Figure 12: Outcome of the model for monopoly and 100 firm industry
A neoclassical referee who rejected this paper for a mainstream journal claimed that this was because all the firms were doing the same thing and hence effectively colluding. A close look at the behavior of three firms chosen at random from the 100-firm simulation shows that this was not the case:
Figure 13: Output of 3 firms chosen at random and industry average output over 1000 iterations
What’s more, the instrumentally rational profit maximizers made higher profits than neoclassical theory predicted:
In general, the number of firms in the industry had no impact on the amount produced by the industry:
And market price was also unaffected by the number of firms:
In summary, the neoclassical theory of the firm and competition is “much ado about nothing”.
Blatt, J. M. 1980, ‘The Utility of Being Hanged on the Gallows’, Journal of Post Keynesian Economics, vol. 2, no. 2, pp 231–239.
Keen, S. 2004, ‘Deregulator: Judgment Day for Microeconomics’, Utilities Policy, vol. 12, pp 109–125.
Keen, S. and Standish, R. 2006, ‘Profit maximization, industry structure, and competition: A critique of neoclassical theory’, Physica A: Statistical Mechanics and its Applications, vol. 370, no. 1, pp 81–85.
Schumpeter, J. A. 1934, The theory of economic development : an inquiry into profits, capital, credit, interest and the business cycle, Harvard University Press, Cambridge, Massachusetts.
Stigler, G. J. 1957, ‘Perfect Competition, Historically Contemplated’, The Journal of Political Economy, vol. 65, no. 1, pp 1–17.