Numeraire Illusion: The Final Demise of the Kaldor-Hicks Principle David Ellerman

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Numeraire Illusion: The Final Demise of the Kaldor-Hicks Principle

  • David Ellerman

  • Visiting Scholar

  • University of California/Riverside


A “Proof” that inflation is impossible!

  • What’s wrong with this “argument”?

  • How much would a dollar buy in 1900?

    • Answer: A dollar’s worth of goods.
  • How much would a dollar buy in 2000?

    • Answer: A dollar’s worth of goods.
  • Since a dollar buys the same amount of goods in 1900 and 2000 (or any two times), inflation is impossible. Q.E.D.

A “proof” that all transfers are useless!

  • What’s wrong with this “argument”?

  • What’s an apple worth to John?

    • In terms of apples as numeraire, the value of an apple to John is exactly one.
  • What’s an apple worth to Mary?

    • Similarly, the value of an apple to Mary is exactly one.
  • Since an apple has exactly the same value to any John and Mary, any transfer of apples between them is pointless and useless!

  • Repeat the argument with any other commodity substituted for apples. Q.E.D.

  • Corollary: Since all transfers of commodities between people are useless!

A “Proof” that yardsticks cannot expand or contract.

  • What’s wrong with this argument?

  • We have a yardstick that we suspect has expanded or contracted. How to check it?

  • We will mark off the length of the yardstick on the edge of a table, and then we will measure the distance to see if it is a yard long.

  • We mark off the distance on the table, and then we check it with our handy yardstick.

  • We find exactly one yard marked off on the table so we conclude that the yardstick has not expanded or contracted. Q.E.D.

The Church Vindicated: “Proof” the Earth does not move!

  • Let X(t) = (x1(t), x2(t), x3(t)) be the trajectory of a body through 3-space, e.g., E(t) for Earth and S(t) for the Sun.

  • A body does not move if X(t) is constant.

  • Since the choice of origin is arbitrary, we use the geocentric coordinates: X(t) E(t).

  • The trajectory of the Earth is E(t) E(t) = (0,0,0) which is constant! Eppur non muove!

“Proof” the MU of income is constant

  • Max U(X) such that PX = I. MUI is rate of change of U w.r.t. I along income-consumption path.

  • Any monotonic transform of U is also a utility function.

  • Let Um(X) = Min expenditure E = PX* such that U(X*) = U(X). Um is a utility function (money- metric utility function).

  • Rate of change of Um w.r.t. I along income-consumption path is constant (= 1) so MU of income is constant.

Samuelson on MUmI = 1

  • “[T]he money-metric marginal utility of income is constant at unity. For how could it be otherwise? If you are measuring utility by money, it must remain constant with respect to money: a yardstick cannot change in terms of itself.”

  • [Complementarity: An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory. JEL, Vol. 12, No. 4, Dec. 1979, p. 1264]

What the “proofs” really show

  • Just as the eye has a “blind spot” (where the optical nerve is connected), so

  • Every system of measurement-relative-to-a-base has an informational “blind spot” at the base:

    • Location of the origin relative to the origin is always 0
    • Value of the numeraire relative to the numeraire is always 1.
  • All the “proofs” derive that tautology and then erroneously generalize as if it were true for any base.

  • “If the Earth does not move relative to a geocentric coordinate system, then the Earth does not move.”

  • I call this fallacy “numeraire illusion.”

Marshall-Pigou Framework

  • Pigou makes basic distinction between “production” and “distribution” of national dividend or product, i.e., size and distribution of pie.

  • Marshall’s notion of consumer & producer surplus: max pie at comp. equilibrium.

  • “Pie” measured in money, not welfare. After Keynes, GNP roughly identified as “pie”

  • Size of "social product" pie = “efficiency” question

  • Distribution of "social product" pie = “equity” question.

Kaldor-Hicks Counter-Reformation

  • Paretian definitions “impractical”

  • Kaldor-Hicks criterion:

    • = $gains exceeds $losses
    • = net increase in pie
    • = potential Pareto improvement.
    • Change is “increase in efficiency”
  • There is compensation $C such that $gains > $C > $losses so “change + compensation” = actual Pareto improvement.

    • Change = “efficiency” part
    • Compensation = “equity” part.
  • As economists, the change can be recommended on efficiency grounds while the actual payment of compensation is a separate question of equity.

K-H rehab of M-P

  • Kaldor rehabilitates Pigou:

  • “This argument lends justification to the procedure, adopted by Professor Pigou in The Economics of Welfare, of dividing ‘welfare economics’ into two parts: the first relating to production, and the second to distribution.”

  • [Welfare Propositions of Economics and Interpersonal Comparisons of Utility. EJ, 1939, p. 551]

  • Hicks sort-of rehabilitates Marshall: The Rehabilitation of Consumers' Surplus. RES. 1941.

  • Willig really rehabilitates Marshall: Consumer's Surplus Without Apology. AER. 1976.

Economics based on MPKH Methodology

  • Cost-Benefit Analysis:

    • Change = “Project” (e.g., “project evaluation”)
    • Might do if project $gains exceed $losses
    • Actual compensation is controversial separate question.
  • Wealth Maximization (“Chicago”) School of Law and Economics:

    • Change = “legal change”
    • Increase in efficiency if $gains – $losses = net change in social wealth is positive.
    • Compensation is again a separate question usually considered not feasible.

Two Schools

  • Hicks sees two basic approaches to economics [The Scope and Status of Welfare Economics, OEP, 1975]:

  • The production (Smith) and distribution (Ricardo) school developed by Marshall & Pigou (and modernized by Kaldor and Hicks—and Keynes);

  • The exchange (catallactics) school of the marginalist revolution in its Lausanne (Pareto and Walras), Austrian (Menger) and English (Jevons) varieties.

  • Is the economy conceptualized as:

  • A Social Product to be maximized and (fairly) distributed, or

  • A mechanism to allocate resources to make some better off without making others worse off?

The Basic Argument

  • The production-distribution school (MPKH+) is based on parsing a resource reallocation—into two parts:

    • “Production” or efficiency part that changes the size of the social pie, and
    • “Distribution” or equity part that does not change the size of the social pie.
  • But the judgment that the distribution-equity part does not change the size of the social pie is pure numeraire illusion—the resources reallocated in the “compensation” and the size of “social pie” are both measured in the same numeraire units.

  • "It should be emphasized that pure transfers of purchasing power from one household or firm to another per se should be typically attributed no value." [Boadway, Robin. The Economic Evaluation of Projects. 2000, 30] Or again, "pure transfers of funds among households, firms and governments should themselves have no effect on project benefits and costs." [Boadway 2000, 35]

  • Reverse the numeraire, and the efficiency-equity parts reverse themselves—just as moving from geocentric to heliocentric coordinates with reverse the results about which one—the sun or the earth—moves.

Only apple transfer increases $pie

  • “It would still be an improvement, and by the same amount, if John stole the apple-price zero-or it 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.” [Friedman 2000, 20]

From Marginals to Totals

From “Quid pro Quo” to “Quo pro Quid”

  • Every description of a market by a supply & demand curve has an inverted description.

  • Interpret the “demand for apples” as the “supply of $-spent-on-apples”

  • Interpret the “supply of apples” as the “demand for $-spent-on-apples”

  • Prices are in “apples per $-spent-on-apples”

  • Equilibrium quantity is R* (= P*A*) and eq. price is P’* (= 1/P*) so payment is P’*R* = P*A*/P* = A* apples.

  • Thus exactly same transfers as before, A* one way and R* the other way but with roles of goods transfer and payment transfer reversed.

  • Then consumer + seller surpluses attach to R* transfer while A* transfer gives no change in apple pie.

Inverted Description Graphed

Math of Inverted Description of Market

  • Demand for apples = Ad = D(P)

  • Supply of apples = As = S(P)

  • Equilibrium: A* = D(P*) = S(P*)

  • Inverted Description: R = $-spent-on-apples

  • P = R per apple so P’ = 1/P = apples per R

  • Demand for R = Rd(P’) = S(1/P’)/P’

  • Supply of R = Rs(P’) = D(1/P’)/P’

  • Equilibrium: R* = Rd(P’*) = Rs(P’*) so multiply thru by P’* to get D(1/P’*) = S(1/P’*) which holds at 1/P’* = P*. Thus R* = Rd(P’*) = S(P*)P* = P*A* and payment is: P’*R* = P*A*/P* = A*.

Summing Up I

  • Theorem: If Project + Compensation is a Pareto improvement, then:

  • $ = change in dollar pie > 0 ($ = numeraire)

    • But d$ = “Compensation” makes no change in $ pie;
  • A = change in apple pie > 0 (apples = numeraire)

  • N = change in nut pie > 0 (nuts = numeraire)

    • Where both dA and d$ contribute to the nut pie.
  • MPKH-methodology infers from “$ = numeraire” description that there is something “productive” about the dA = “Project” while the d$ is merely “redistributive.”

  • But this is not numeraire-invariant. In the “apples = numeraire” description, the same d$ is “productive” and the same dA is merely “redistributive.” To avoid numeraire illusion, use a third non-involved numeraire in which case both dA and d$ are “productive.”

Summing Up II

  • Kaldor-Hicks criterion is not numeraire invariant.

  • KH is based on an incidental feature of the particular description, not a numeraire-invariant property of the underlying resource allocations being described.

  • Therefore KH criterion cannot be sustained.

  • The MPKH methodology dissolves into a kind of “money mysticism”—where attributes of a description with money as numeraire are taken as revealing “basic properties” of the underlying resource allocations being described, properties that disappear under a mere change of numeraire.

  • Like the Church taking “The earth does not move” and “The sun moves” as basic underlying properties rather than just features of the choice of geocentric coordinates.

Fallout of “KH-efficiency” Failure I

  • Failure of KH-efficiency in Welfare Economics and Cost-Benefit Analysis:

  • “The purpose of considering hypothetical redistributions is to try and separate the efficiency and equity aspects of the policy change under consideration. It is argued that whether or not the redistribution is actually carried out is an important but separate decision. The mere fact that is it possible to create potential Pareto improving redistribution possibilities is enough to rank one state above another on efficiency grounds.”

  • [Boadway and Bruce, Welfare Economics, 1984, p. 97]

Fallout of “KH-efficiency” Failure II

  • Failure of KH efficiency in Wealth-Max (“Chicago”) School of Law & Economics.

  • “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.”

  • [Posner, Richard. Cost-Benefit Analysis, Journal of Legal Studies. 2000, pp. 1154-5]

The End


Algebraic appendix: 3rd Uninvolved Numeraire

  • Nuts = (MRSJN/A MRSMN/A)dA + (MRSMN/$ MRSJN/$)d$

  • Therefore both the transfer in apples dA and the transfer in money d$ contribute to the change in the size of the nut pie Nuts—as long as the pie is measured in some third commodity not involved in the transfers. There is no illusory foothold to recommend either dA or d$ by itself on “efficiency” grounds.

  • But when we change the numeraire to one of the goods involved in the transfers, then that term drops out courtesy of numeraire illusion. For instance, now take $ as the numeraire.

  • $ = (MRSJ$/A MRSM$/A)dA + (MRSM$/$ MRSJ$/$)d$

  • = (MRSJ$/A MRSM$/A)dA + (1 1)d$

  • = (MRSJ$/A MRSM$/A)dA.

Reversed numeraires + 3rd numeraire

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