Mathematical Economics

This is the fourth entry in a series of loosely connected posts on economics. Previous entries are here, here, and here.

Economics is a study of human behavior, not an exercise in mathematical modeling or statistical analysis, though both endeavors may augment an understanding of human behavior. Economics is about four things:

  • wants, as they are perceived by the persons who have those wants
  • how people try to satisfy their wants through mutually beneficial, cooperative action, which includes but is far from limited to market-based exchanges
  • how exogenous forces, including government interventions, enable or thwart the satisfaction of wants
  • the relationships between private action, government interventions, and changes in the composition, rate, and direction of economic activity

In sum, economics is about the behavior of human beings, which is why it’s called a social science. Well, economics used to be called a social science, but it’s been a long time (perhaps fifty years) since I’ve heard or read an economist refer to it as a social science. The term is too reminiscent of “soft and fuzzy” disciplines such as history, social psychology, sociology, political science, and civics or social studies (names for the amalgam of sociology and government that was taught in high schools way back when). No “soft and fuzzy” stuff for physics-envying economists.

However, the behavior of human beings — their thoughts and emotions, how those things affect their actions, and how they interact — is fuzzy, to say the least. Which explains why mathematical economics is largely an exercise in mental masturbation.

In my disdain for mathematical economics, I am in league with Arnold Kling, who is the most insightful economist I have yet encountered in more than fifty years of studying and reading about economics. I especially recommend Kling’s Specialization and Trade: A Reintroduction to Economics. It’s a short book, but chock-full of wisdom and straight thinking about what makes the economy tick. Here’s the blurb from

Since the end of the second World War, economics professors and classroom textbooks have been telling us that the economy is one big machine that can be effectively regulated by economic experts and tuned by government agencies like the Federal Reserve Board. It turns out they were wrong. Their equations do not hold up. Their policies have not produced the promised results. Their interpretations of economic events — as reported by the media — are often of-the-mark, and unconvincing.

A key alternative to the one big machine mindset is to recognize how the economy is instead an evolutionary system, with constantly-changing patterns of specialization and trade. This book introduces you to this powerful approach for understanding economic performance. By putting specialization at the center of economic analysis, Arnold Kling provides you with new ways to think about issues like sustainability, financial instability, job creation, and inflation. In short, he removes stiff, narrow perspectives and instead provides a full, multi-dimensional perspective on a continually evolving system.

And he does, without using a single graph. He uses only a few simple equations to illustrate the bankruptcy of macroeconomic theory.

Those economists who rely heavily on mathematics like to say (and perhaps even believe) that mathematical expression is more precise than mere words. But, as Kling points out in “An Important Emerging Economic Paradigm,” mathematical economics is a language of “faux precision,” which is useful only when applied to well defined, narrow problems. It can’t address the big issues — such as economic growth — which depend on variables such as the rule of law and social norms which defy mathematical expression and quantification.

I would go a step further and argue that mathematical economics borders on obscurantism. It’s a cult whose followers speak an arcane language not only to communicate among themselves but to obscure the essentially bankrupt nature of their craft from others. Mathematical expression actually hides the assumptions that underlie it. It’s far easier to identify and challenge the assumptions of “literary” economics than it is to identify and challenge the assumptions of mathematical economics.

I daresay that this is true even for persons who are conversant in mathematics. They may be able to manipulate easily the equations of mathematical economics, but they are able to do so without grasping the deeper meanings — the assumptions and complexities — hidden by those equations. In fact, the ease of manipulating the equations gives them a false sense of mastery of the underlying, real concepts.

Much of the economics profession is nevertheless dedicated to the protection and preservation of the essential incompetence of mathematical economists. This is from “An Important Emerging Economic Paradigm”:

One of the best incumbent-protection rackets going today is for mathematical theorists in economics departments. The top departments will not certify someone as being qualified to have an advanced degree without first subjecting the student to the most rigorous mathematical economic theory. The rationale for this is reminiscent of fraternity hazing. “We went through it, so should they.”

Mathematical hazing persists even though there are signs that the prestige of math is on the decline within the profession. The important Clark Medal, awarded to the most accomplished American economist under the age of 40, has not gone to a mathematical theorist since 1989.

These hazing rituals can have real consequences. In medicine, the controversial tradition of long work hours for medical residents has come under scrutiny over the last few years. In economics, mathematical hazing is not causing immediate harm to medical patients. But it probably is working to the long-term detriment of the profession.

The hazing ritual in economics has as least two real and damaging consequences. First, it discourages entry into the economics profession by persons who aren’t high-IQ freaks, and who, like Kling, can discuss economic behavior without resorting to the sterile language of mathematics. Second, it leads to economics that’s irrelevant to the real world — and dead wrong.

Reaching back into my archives, I found a good example of irrelevance and wrongness in Thomas Schelling‘s game-theoretic analysis of segregation. Eleven years ago, Tyler Cowen (Marginal Revolution), who was mentored by Schelling at Harvard, praised Schelling’s Nobel prize by noting, among other things, Schelling’s analysis of the economics of segregation:

Tom showed how communities can end up segregated even when no single individual cares to live in a segregated neighborhood. Under the right conditions, it only need be the case that the person does not want to live as a minority in the neighborhood, and will move to a neighborhood where the family can be in the majority. Try playing this game with white and black chess pieces, I bet you will get to segregation pretty quickly.

Like many game-theoretic tricks, Schelling’s segregation gambit omits much important detail. It’s artificial to treat segregation as a game in which all whites are willing to live with black neighbors as long as they (the whites) aren’t in the minority. Most whites (including most liberals) do not want to live anywhere near any “black rednecks” if they can help it. Living in relatively safe, quiet, and attractive surroundings comes far ahead of whatever value there might be in “diversity.”

“Diversity” for its own sake is nevertheless a “good thing” in the liberal lexicon. The Houston Chronicle noted Schelling’s Nobel by saying that Schelling’s work

helps explain why housing segregation continues to be a problem, even in areas where residents say they have no extreme prejudice to another group.

Segregation isn’t a “problem,” it’s the solution to a potential problem. Segregation today is mainly a social phenomenon, not a legal one. It reflects a rational aversion on the part of whites to having neighbors whose culture breeds crime and other types of undesirable behavior.

As for what people say about their racial attitudes: Believe what they do, not what they say. Most well-to-do liberals — including black one like the Obamas — choose to segregate themselves and their children from black rednecks. That kind of voluntary segregation, aside from demonstrating liberal hypocrisy about black redneck culture, also demonstrates the rationality of choosing to live in safer and more decorous surroundings.

Dave Patterson of the defunct Order from Chaos put it this way:

[G]ame theory has one major flaw inherent in it: The arbitrary assignment of expected outcomes and the assumption that the values of both parties are equally reflected in these external outcomes. By this I mean a matrix is filled out by [a conductor, and] it is up to that conductor’s discretion to assign outcome values to that grid. This means that there is an inherent bias towards the expected outcomes of conductor.

Or: Garbage in, garbage out.

Game theory points to the essential flaw in mathematical economics, which is reductionism: “An attempt or tendency to explain a complex set of facts, entities, phenomena, or structures by another, simpler set.”

Reductionism is invaluable in many settings. To take an example from everyday life, children are warned — in appropriate stern language — not to touch a hot stove or poke a metal object into an electrical outlet. The reasons given are simple ones: “You’ll burn yourself” and “You’ll get a shock and it will hurt you.” It would be futile (in almost all cases) to try to explain to a small child the physical and physiological bases for the warnings. The child wouldn’t understand the explanations, and the barrage of words might cause him to forget the warnings.

The details matter in economics. It’s easy enough to say, for example, that a market equilibrium exists where the relevant supply and demand curves cross (in a graphical representation) or where the supply and demand functions yield equal values of price and quantity (in a mathematical representation). But those are gross abstractions from reality, as any economist knows — or should know. Expressing economic relationships in mathematical terms lends them an unwarranted air of precision.

Further, all mathematical expressions, no matter how complex, can be expressed in plain language, though it may be hard to do so when the words become too many and their relationships too convoluted. But until one tries to do so, one is at the mercy of the mathematical economist whose equation has no counterpart in the real world of economic activity. In other words, an equation represents nothing more than the manipulation of mathematical relationships until it’s brought to earth by plain language and empirical testing. Short of that, it’s as meaningful as Urdu is to a Cockney.

Finally, mathematical economics lends aid and comfort to proponents of economic control. Whether or not they understand the mathematics or the economics, the expression of congenial ideas in mathematical form lends unearned — and dangerous — credibility to the controller’s agenda. The relatively simple multiplier is a case in point. As I explain in “The Keynesian Multiplier: Phony Math,”

the Keynesian investment/government-spending multiplier simply tells us that if ∆Y = $5 trillion, and if b = 0.8, then it is a matter of mathematical necessity that ∆C = $4 trillion and ∆I + ∆G = $1 trillion. In other words, a rise in I + G of $1 trillion doesn’t cause a rise in Y of $5 trillion; rather, Y must rise by $5 trillion for C to rise by $4 trillion and I + G to rise by $1 trillion. If there’s a causal relationship between ∆G and ∆Y, the multiplier doesn’t portray it.

I followed that post with “The True Multiplier“:

Math trickery aside, there is evidence that the Keynesian multiplier is less than 1. Robert J. Barro of Harvard University opens an article in The Wall Street Journal with the statement that “economists have not come up with explanations … for multipliers above one.”

Barro continues:

A much more plausible starting point is a multiplier of zero. In this case, the GDP is given, and a rise in government purchases requires an equal fall in the total of other parts of GDP — consumption, investment and net export. . . .

What do the data show about multipliers? Because it is not easy to separate movements in government purchases from overall business fluctuations, the best evidence comes from large changes in military purchases that are driven by shifts in war and peace. A particularly good experiment is the massive expansion of U.S. defense expenditures during World War II. The usual Keynesian view is that the World War II fiscal expansion provided the stimulus that finally got us out of the Great Depression. Thus, I think that most macroeconomists would regard this case as a fair one for seeing whether a large multiplier ever exists.

I have estimated that World War II raised U.S. defense expenditures by $540 billion (1996 dollars) per year at the peak in 1943-44, amounting to 44% of real GDP. I also estimated that the war raised real GDP by $430 billion per year in 1943-44. Thus, the multiplier was 0.8 (430/540). The other way to put this is that the war lowered components of GDP aside from military purchases. The main declines were in private investment, nonmilitary parts of government purchases, and net exports — personal consumer expenditure changed little. Wartime production siphoned off resources from other economic uses — there was a dampener, rather than a multiplier. . . .

There are reasons to believe that the war-based multiplier of 0.8 substantially overstates the multiplier that applies to peacetime government purchases. For one thing, people would expect the added wartime outlays to be partly temporary (so that consumer demand would not fall a lot). Second, the use of the military draft in wartime has a direct, coercive effect on total employment. Finally, the U.S. economy was already growing rapidly after 1933 (aside from the 1938 recession), and it is probably unfair to ascribe all of the rapid GDP growth from 1941 to 1945 to the added military outlays. [“Government Spending Is No Free Lunch,” The Wall Street Journal (, January 22, 2009]

This is from Valerie A. Ramsey of  the University of California-San Diego and the National Bureau of Economic Research:

. . . [I]t appears that a rise in government spending does not stimulate private spending; most estimates suggest that it significantly lowers private spending. These results imply that the government spending multiplier is below unity. Adjusting the implied multiplier for increases in tax rates has only a small effect. The results imply a multiplier on total GDP of around 0.5. [“Government Spending and Private Activity,” January 2012]

In fact,

for the period 1947-2012 I estimated the year-over-year percentage change in GDP (denoted as Y%) as a function of G/GDP (denoted as G/Y):

Y% = 0.09 – 0.17(G/Y)

Solving for Y% = 0 yields G/Y = 0.53; that is, Y% will drop to zero if G/Y rises to 0.53 (or thereabouts). At the present level of G/Y (about 0.4), Y% will hover just above 2 percent, as it has done in recent years. (See the graph immediately above.)

If G/Y had remained at 0.234, its value in 1947:

  • Real growth would have been about 5 percent a year, instead of 3.2 percent (the actual value for 1947-2012).
  • The total value of Y for 1947-2012 would have been higher by $500 trillion (98 percent).
  • The total value of G would have been lower by $61 trillion (34 percent).

The last two points, taken together, imply a cumulative government-spending multiplier (K) for 1947-2012 of about -8. That is, aggregate output in 1947-2012 declined by 8 dollars for every dollar of government spending above the amount represented by G/Y = 0.234.

But -8 is only an average value for 1947-2012. It gets worse. The reduction in Y is cumulative; that is, every extra dollar of G reduces the amount of Y that is available for growth-producing investment, which leads to a further reduction in Y, which leads to a further reduction in growth-producing investment, and on and on. (Think of the phenomenon as negative compounding; take a dollar from your savings account today, and the value of the savings account years from now will be lower than it would have been by a multiple of that dollar: [1 + interest rate] raised to nth power, where n = number of years.) Because of this cumulative effect, the effective value of K in 2012 was about -14.

The multiplier is a seductive and easy-to-grasp mathematical construct. But in the hands of politicians and their economist-enablers, it has been an instrument of economic destruction.

Perhaps “higher” mathematical economics is potentially less destructive because it’s inside game played by economists for the benefit of economists. I devoutly hope that’s true.