Wednesday, May 11, 2016

The great economic equations

A few days ago, Unlearning Economics twitted a link to an article on "The 17 equations that changed the world." Only one was an economic equation, The Black-Scholes one, and in all fairness it did not change the world, and is not even a central one in economics. First of all, Nassim Taleb has argued convincingly (for example, here) that Black, Scholes and Merton did not invent the formula, and what they really did was to provide a theoretical justification that was compatible with Arrow-Debreu general equilibrium (GE) views. Haug and Taleb say it clearly:
Indeed what Black, Scholes and Merton did was “marketing”, finding a way to make a well-known formula palatable to the economics establishment of the time, little else, and in fact distorting its essence.
So market participants already had formulas to price options, and the idea that their version of the formula "helped create the now multi-trillion dollar derivatives market," as suggested by Andy Kiersz, is clearly incorrect. Unregulated financial markets didn't need the Black-Scholes formula, economists did. And we know how well that ended. Deregulation and the nature of competition in financial markets would have led to the expansion of derivative markets anyway. But GE would not look like it could provide practical answers to real economic problems. Which turns out it couldn't. Besides, as discussed here before, the Arrow-Debreu general equilibrium model is not devoid of problems.

So what are, if any, the great economic equations, you ask. If I had to say one it would be either Keynes' multiplier formula, Y = I/(1 – c), or Sraffa's demonstration of the inverse relation between wages and profits, r = R(1  – w).* The first clearly shows that spending determines the level of activity and provides formal justification for counter-cyclical policies, which have indeed reduced the effects of recessions on the economy, even if, as Kalecki had noted it would happen, austerity is often imposed for political reasons. The second resolved an issue first clearly posed by Ricardo, is part of the clear understanding of what determines long-term prices, and shows the conflictive nature of the capitalist system. Both are central to understanding the way capitalist economies work.

* Where all variables have the standard meaning, Y is output, I investment, c the propensity to consume, r is the rate of profit, R is the maximum rate of profit, and w the wage share.


  1. A bit nitpicky, but shouldn't the multiplier be written:

    dY/dG,etc.=I/(1-c) since it tells you the responsiveness of output to a change in spending rather than the overall level of output?

    1. The numerator is an I for investment. In levels is correct, and I assumed the simplest version with no government or foreign sector, and with no autonomous consumption. In Twitter you must economize

  2. Ah that clears it up, thanks. Also, how about the Fisher Equation? Accurate or not, it seems to have inspired tons of movements, like monetarism, rational expectations, etc.

    Or, going even farther back, to Jevons and the basis of economics, where "every mind is inscrutable to every other mind," and the law of diminishing marginal utility: d^2U/dg^2<0

    1. Hi Harold:
      I do not believe in he law of diminishing returns. Outside of issues of scale, replicability in manufacturing is always possible. Classical authors only used it in the context of agriculture to explain differential rents. I don't think Fisher's equation to be comparable to effective demand and conflictive distribution.

  3. Andres Carbacho-BurgosMay 13, 2016 at 5:37 PM

    Hmm. The two above are a start, but I would add a few to round out the heterodox world view:

    P = U/(1-pi)

    The necessary complement of letting output be determined by aggregate demand is to have the price level P set as a markup (1/(1-pi), with pi the profit share of income) on average unit cost U. Prices become tools to validate production costs rather than to clear markets, which is now done by output adjustment.

    M/P = L(r,Y)

    The interest rate r moves to set the supply and demand for money equal, not to set savings equal to investment. It took economists until 1936 to think this one up...

    M --> C --> (P) --> C' --> M'

    Ok, not an equation, but Marx's Arrow (as I like to call it) is a shorthand for a completely different view of the economic process, where the object is not to redistribute commodities via market price signals (C --> M --> C') but to have the labor purchase and production process act as a engine of accumulation.

    One more:

    Y = A*L^a*K^b

    Ok, just kidding ;-)

    1. The Sraffian equation is related to long term prices, and there is a relation between those prices and the cost based mark up stories you suggest. And Keynes effective demand (multiplier) works in what he called a monetary economy of production, which is, according to Keynes, equivalent to Marx's capitalism (M - C - M').

  4. May I suggest that you explain Sraffa's work in a few posts?

    1. It's has been done. There is actually a post that provides the links to a possible reading of the Sraffa posts called "Sraffa's contributions to economics: a crash course"


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