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Does Benefit-Cost Analysis Attempt the Impossible?

Before tackling some of the practical problems involved in the use of Benefit-Cost Analysis (BCA), I want to take a look at two foundational questions.  How do we incorporate the interests of affected individuals into a BCA model?  And how do we assemble those individual interests into a social welfare function that allows us to make a collective decision?

The fact that BCA is used to make collective decisions is what distinguishes it from many similar methodologies (such as discounted cash-flow analysis) that are used by private individuals or businesses.  BCA purports to evaluate a decision from the perspective of multiple affected parties, whose views and interests are not aligned.  One will often see a simplistic description of the BCA procedure:  assume that individuals are utility maximizers when they make their own decisions, and then choose government policies that maximize the sum of the affected individuals’ utility.  Up until about 75 years ago this was a pretty accurate description of how it worked.  But during the 20th century (along with most of the rest of microeconomics) BCA made the transition from neoclassical welfare economics, in which individuals were presumed to have quantitative “cardinal” utility functions, to modern welfare theory, in which individuals are presumed only to have ordered preferences.  This is not the place to try to explain that transition; suffice it to say that, if cardinal utility functions exist, they don’t matter because we cannot observe them.  All we can observe are the choices people make in the marketplace, and from those we can infer an ordinal set of preferences.

But that inference requires an assumption:  that individuals’ preferences are transitive, so that they can be put in an unambiguous order.  Sometimes this is called internal consistency; more often the transitivity assumption is what economists mean when they say “rational” preferences.  If I prefer A to B, and B to C, you can assume that I prefer A to C.  I am certainly free to change my mind, depending on my circumstances and mood.  But if I am persistently irrational (i.e., intransitive), my preferences won’t fit very well into any economic model.

How big a problem is that?  Well, irrational behavior is not exactly rare.  But that doesn’t mean that it is economically important.  If I have consistently irrational preferences you can turn me into a money pump – you could charge me a penny to exchange B for A, another penny to exchange A for C, and a third penny to exchange C for B.  We are back where we started, except that you have some of my money; and soon, unless I learn to be more rational, I will be penniless.  For this reason, economists are comfortable assuming that intransitive individual preferences do not play a major role in shaping the economy.

Transitivity of preferences can be even more important on a large scale.  A nation that displayed intransitive preferences in its trade patterns, for example, would soon find that it had nothing left to trade.  But there is a problem:  rationality does not automatically scale up.  This brings us to the Condorcet Paradox:

Suppose we have three options, A, B, & C, and a committee of three to decide.

Alice’s ranking is  A > B > C; Bob’s ranking is  B > C > A; Chris’s ranking is  C > A > B.

It is plain to see that, for each option, there exists another option that is preferred by a majority of members.  And the remarkable thing is that not only do a majority agree that a better option exists, but a majority agree on a specific option that would be preferable.  And yet going there does not solve the problem.  No matter what option we choose, a majority will agree that it is inferior to a particular available alternative.  It should be obvious why this paradox has been much on my mind in recent weeks.

The Marquis de Condorcet published in 1785.  Many mathematicians since then have tackled the problem, including Charles Dodgson, better known as Lewis Carroll, the author of Alice’s Adventures in Wonderland and Though the Looking Glass.  But a major advance (or perhaps retreat) was made by Kenneth Arrow, winner of the 1972 Nobel Prize in Economics, while a graduate student at Stanford.  He proved what we know as the Arrow Impossibility Theorem.  In mathematical form it can be complex, but here is how the Stanford Encyclopedia of Philosophy describes it:

“Which procedures are there for deriving, from what is known or can be found out about [people’s] preferences, a collective or ‘social’ ordering of the alternatives from better to worse?  The answer is startling. Arrow’s theorem says there are no such procedures whatsoever.”

Well, that seems discouraging.  Is it really impossible for rational individuals to come up with a rational way to decide things as a group?  There are some escapes from the theorem’s logic.  One is the dictatorship option.  If Alice is able to impose her own preferences on everyone else in society, then policy choices will be clear; there is nothing to debate.  Many households work this way, but it is not attractive on a larger scale.

Another exception is the set of easy problems:  the Pareto improvements that everyone can simply agree on.  This is the domain of market transactions.  Since dictatorship is so unpleasant, we should try to use markets as much as possible – establishing property rights in fisheries, for example, so that markets can work their magic, and make it unnecessary to come up with some kind of groupthink fishery policy or a dictator of fish.

That leaves a set of problems for which markets are not working well – the public goods, externalities, and other familiar market imperfections.  People will have different opinions about how large or how important this set is, and whether collective decision making will be able to do any better than imperfect market outcomes.  I think we can agree, however, that there exists a very large set of government programs who purport to be occupying this space, busily pursuing what they call the public’s business.  And we can get pretty wide agreement that we should cut back on those programs which are unable to demonstrate that they are doing more good than harm.

To test this, BCA applies the Kaldor-Hicks test, in which those who support a particular policy outcome are imagined to be able to compensate those who oppose it, and thereby to change their minds.  A decision passes the Kaldor-Hicks criterion if, when such compensating side payments are made, the decision becomes unanimous.

Unanimous sounds good!  Such decisions are called potential Pareto improvements – they would be Pareto improvements, and would be accomplished by the market instead of the government, if the market were better able to find these bargains.  (And, thanks to advances in technology, the market is getting better at finding bargains all the time!)  But with a few exceptions, like the ingenious Clarke tax, the compensating payments are not actually made.  Thus the Kaldor-Hicks methodology does not actually constitute an exception to Arrow’s theorem, because the choices that it evaluates are not identical to the choices actually made.  Over a large number of transactions we can hope that, for any individual, the pluses and minuses of the unpaid compensating variations will even out.  But perhaps not.

The Kaldor-Hicks criterion has another noteworthy strength, and another weakness.  The strength is that it produces a transitive ranking of options, and so appears to be “rational,” in that narrow but important sense.  The weakness is that the rankings depend upon the initial distribution of income or wealth, which is taken as a given.  The initial endowment affects how much people are willing to pay for their preferred policy option, and a different income distribution will result in different policy rankings.  I’ll revisit that property in another blog.  Meanwhile, my answer to the title question is:  Yes, BCA attempts to do the impossible.  And of course it does not fully succeed, but I give it partial credit for what it is able to accomplish.

Reader Discussion

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on November 04, 2016 at 09:16:38 am

"The initial endowment affects how much people are willing to pay for their preferred policy option,"

And therein lies the real problem. Those crafting and implementing THEIR preferred policy options ARE NOT the ones paying for it.

How, I wonder, does one factor that into the computations?

Oh, I get it - that's why we are having an election this year!!!

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gabe
on November 04, 2016 at 10:41:36 am

What is going on with this site:

Where the tallies for Facebook and Twitter usually show a "tally" now there is a series of Facebook and Twitter logos representing EACH response.

Annoying!

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gabe
on November 04, 2016 at 13:01:35 pm

You talk about the impossibility of getting a rational ordering in groups but dont seem to apply that to the Kaldor-Hicks criterion. If it is so impossible to get a ranked ordering, how do you know what the benefits or costs are in a group? Instead you just state that "The strength is that it produces a transitive ranking of options, and so appears to be 'rational,' in that narrow but important sense." How do you get that ranked ordering if it is "impossible" to do so in a group even for the single issue? You cant add up benefits A & B and subtract costs of C & D if you cant rank their order. Individuals might be able to rank them and be rational, but not groups.

Of course there are other ways around Arrow Impossibility Theorem than just by having a dictator (which is just one way to reject the premises). For instance another way to get around the problem is to say that every new majority to override an old majority must be larger % of the population. So if 50% agree something is good, then you need at least 51% to overturn that for something better. This solves the theorem (although it adds a slight path dependence), and probably would be a better way to decide solutions to these problems than some unaccountable bureaucrats in the executive branch trying to hypothesize what they personally think are the benefits or costs of a given policy rather than based off the larger community.

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Devin Watkins
on November 04, 2016 at 14:58:44 pm

And remember the government drones ARE NOT the ones upon whom the costs will fall, further lessening the strength of the "rational / transitive ordering" thesis.

BTW: Devin: check out Originalsim Blog - fellow commented on your essay.

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gabe
on November 04, 2016 at 18:04:27 pm

Government BCA comes in two basic forms. There's the classic cost-effectiveness analysis, used mainly in the Department of Defense, where the question is how to get the most bang for the buck. Assuming a given defense budget (which is a reasonable assumption at the margin), this is an unexceptionable kind of BCA. The other kind of BCA tries to weigh private costs against so-called social benefits; e.g., whether the costs of building a dam are less or greater than the benefits derived from it (as in the electrification of a region that has no electric power). This kind of BCA relies on an impossible act of omniscience: weighing the costs (born by taxpayers and those persons who lose their homes because of the creation of an upstream lake) against the benefits (the value to the beneficiaries of electric power, most of whom didn't lose their homes and many of whom don't pay their "fair share" of the cost of building the dam).

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Thomas
on November 06, 2016 at 05:27:32 am

I'm aware. I was in all-day meetings Wend.-Fri. of last week. I should have time early next week to write a response.

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Devin Watkins

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