The Fatal Flaw in Cost-Benefit Analysis

In Part I of this commentary on the Sarkozy-Stiglitz Commission on the Measurement of Economic Performance and Social Progress, the focus was on the social engineering perspective underlying the search for such an index. But at the end of that commentary, I noted that the Commission’s discussion of different indices was rather “academic” since there is one dominant index used in governmental decision-making: the monetized gains minus the monetized losses of cost-benefit analysis. A proponent of cost-benefit (CB) analysis would roll up all the Commission’s discussion into the question of the better “costing out” of all the direct and indirect impacts of a social decision.

The principle behind CB analysis is the Kaldor-Hicks Principle, but the core ideas figure prominently in much of conventional economics. John Hicks has outlined the tradition in economics that descends from Smith and Ricardo through Marshall and Pigou to Hicks among others as the tradition that focused on the “production and distribution of the national product” [The Scope and Status of Welfare Economics, Oxford Economic Papers, 1975]. In contrast to this rather mainstream tradition, Hicks juxtaposed the tradition of the Lausanne School (Walras and Pareto) and the Austrian School of economics which focused on the mutually beneficial transactions of market exchanges.

In more narrow terms, the contrast between the two traditions is embodied in the contrast between the aforementioned notion of Kaldor-Hicks efficiency and the notion of Pareto efficiency. A change in the social state of affairs is a Pareto improvement if someone becomes better off (in terms of their own welfare or utility) and no one becomes worse off. The criterion for a Pareto improvement is often criticized as being too stringent, so the notion of a Kaldor-Hicks improvement is intended as a more practical alternative.

Given a possible change in the social state, monetize the gain of the people who become better off and monetize the losses of those who are made worse off, and then since the gains and losses are now in the same units, one can consider if the gains outweigh the losses to give a positive net benefit—a Kaldor-Hicks improvement. The basic idea is that the monetized gain is the maximum amount of money the gainers would be willing to pay to have the change, and the monetized losses is the minimum amount of compensation the losers would have to receive so they would not be worse off by the change. If the gains exceeded the losses, then the winners could in theory compensate the losers so the whole “change + compensation” would be a Pareto improvement.

The Kaldor-Hicks analysis parses the whole “change + compensation” into the original change seen as the production of social wealth plus the compensation seen as a redistribution of social wealth. That is the miniature version of the “production and distribution of the national product” in the mainstream tradition outlined by Hicks. Often the “pie metaphor” is used. The increase in social wealth is a change in the size of the pie while the redistribution is only a change in the slices of the pie (not its total size). The question of the size of the social-wealth pie is a question of efficiency that is the professional concern of economists while the question of slices is a question of equity that is the concern of politicians, philosophers, poets, or the like. Hence economists qua economists can “scientifically” recommend a Kaldor-Hicks improvement even without the compensatory redistribution that would be necessary so the change + compensation would be a Pareto improvement.

Richard Posner, the main founder of the economic analysis of law, makes this point in the context of law and economics as well as cost-benefit analysis. He notes that “Kaldor-Hicks efficiency” leaves distributive considerations to one side.

But to the extent that distributive justice can be shown to be the proper business of some other branch of government or policy instrument…, it is possible to set distributive considerations to one side and use the Kaldor-Hicks approach with a good conscience. This assumes, …, that efficiency in the Kaldor-Hicks sense—making the pie larger without worrying about how the relative size of the slices changes—is a social value. [“Cost-Benefit Analysis: Definition, Justification, and Comment on Conference Papers,” In Cost-Benefit Analysis: Legal, Economic, and Philosophical Perspectives. 2001]

Hence the KH principle can be used “with a good conscience” to recommend a certain law, project, or policy in the name of “efficiency” even though it, by itself, without the compensation would not be a Pareto improvement. That in a nutshell is the basis for cost-benefit analysis and for the economic analysis of law known as “law and economics.”

Recently it has become clear that there is a basic logical-methodological error behind the Kaldor-Hicks Principle, an error that vitiates the practice of cost-benefit analysis as well as the theory behind the “economic analysis of law.” The standard criticism of the KH principle is that it, by itself, ignores distributional questions which should be taken into account in social decisions. But the proponents of CB analysis (including law and economics) perfectly agree with this point as indicated by the Posner quote; they are only making recommendations from the viewpoint of efficiency.

But the new criticism shows that the whole parsing of the “change + compensation” into the efficiency + equity part is itself flawed. One can redescribe exactly that same situation in such a way that the two parts reverse themselves. The so-called “equity” part becomes the pie-increasing efficiency part, and the so-called “efficiency” part becomes the merely redistributive equity part—so the whole analysis breaks down in incoherence and arbitrariness.

To illustrate the reversal, we use the example of the Kaldor-Hicks principle from David (son of Milton) Friedman’s widely-used textbook in law and economics, Law’s Order: What Economics has to do with Law and why it matters (2000). Mary has an apple which she values at fifty cents while John values an apple at one dollar. There might be a voluntary exchange where Mary sold the apple to John for, say, seventy-five cents. There are two changes in that Pareto improvement: the transfer of the apple from Mary to John and the transfer of seventy-five cents from John to Mary.

Let us apply social wealth maximization reasoning to the transfer of the apple using money as the numeraire or unit of account. Since the apple was worth fifty cents to Mary and a dollar to John, social wealth would be increased by fifty cents by the apple transfer from Mary to John. That is an increase in efficiency. The other change, the transfer of seventy-five cents from John to Mary, is a question of distribution or equity. Social wealth (measured in dollars and cents) would be unchanged by the mere transfer of seventy-five cents from one person to another.

It would still be an improvement, and by the same amount, if John stole the apple—price zero—or if Mary lost it and John found it. Mary is fifty cents worse off, John is a dollar better off, net gain fifty cents. All of these represent the same efficient allocation of the apple: to John, who values it more than Mary. They differ in the associated distribution of income: how much money John and Mary each end up with.

Since we are measuring value in dollars it is easy to confuse “gaining value” with “getting money.” But consider our example. The total amount of money never changes; we are simply shifting it from one person to another. The total quantity of goods never changes either, since we are cutting off our analysis after John gets the apple but before he eats it. Yet total value increases by fifty cents. It increases because the same apple is worth more to John than to Mary. Shifting money around does not change total value. One dollar is worth the same number of dollars to everyone: one. [p. 20]

Now comes the reversal. Describe exactly the same situation but take apples as the numeraire or unit of account. For John the (marginal) rate of substitution of dollars for apples was one so John would value the loss of the three-quarters of a dollar at three-quarters of an apple. The acquired apple would be worth one apple so the total change is worth one-quarter apple to John. Mary’s rate of substitution of dollars for apples was one-half dollar per apple so the reciprocal rate of substitution of apples for dollars is two apples per dollar. Hence she values the acquired three-quarters of a dollar at (3/4) x 2 = 1.5 apples. Since the loss of her apple is worth one apple to her, she also has a positive net gain (half an apple) from the exchange.

But now the total “pie” of social wealth is an apple pie. When we evaluate the social wealth consequences of the money transfer and the apple transfer, we find that the money transfer of three-quarters of a dollar increased the size of the social apple pie by (3/2) – (3/4) = 3/4 of an apple. The apple transfer had no effect on the size of the apple pie—an apple’s an apple for all that—so the apple transfer was merely redistributive in this redescription. One apple is worth the same number of apples to everyone: one.

Hence the whole attempt to take a shortcut around the Pareto principle using the Kaldor-Hicks parsing into efficiency plus equity parts of the total change (apple plus money transfers) breaks down. A simple redescription of the same transfers with the other good as numeraire reverses the conclusions. In the name of efficiency, economists should recommend making the $0.75 payment, but whether the apple is also transferred is only a matter of equity outside the bailiwick of the “Science of Economics.”

This is only a simple example used by Friedman to illustrate the Kaldor-Hicks principle that is the foundation for the economic analysis of law and cost-benefit analysis. The logical-methodological error, that is so obvious in this textbook example, also applies to the more complicated real-world examples based on the same principle.

This new analysis of the Kaldor-Hicks principle is developed at length in my just-published 2009 paper: “Numeraire Illusion: The Final Demise of the Kaldor-Hicks Principle,” In Theoretical Foundations of Law and Economics. Mark D. White ed., New York: Cambridge University Press: 96-118, which can also be downloaded from my website here.