Economics Letters 2 (1979) 121-124 0 North-Holland Publishing Company

PARETO VS. WICKSELL ON ADJUSTMENT A Reformulation

COSTS IN CONSUMPTION

Jess BENHABIB University of Southern Received

March

California, Los Angeles, CA 90007,

USA

1979

Pareto’s adjustment cost model generating cyclical consumption patterns and Wicksell’s observation that it contains mathematical errors are analyzed. An alternative model capturing the essence of Pareto’s model and meeting Wicksell’s criticisms is proposed where it is shown that consumption and leisure patterns may be cyclical over time.

Pareto often expressed concern about the relation between utility levels and the dynamic path of consumption [see for example, Pareto (1892-93, 1906)]. ’ In his Cours d’Economie Politique (1896, vol. II, ch. IV, sect. 928) he discussed the dynamic paths of savings and consumption. He constructed a mathematical theory of cyclical consumption patterns that arises from single period utility maximization. He then explained business cycles as the ‘resonance’ that results when individual cyclical behavior is aggregated. Wicksell (1907, 1913) took a critical interest in Pareto’s theory and discovered important mathematical errors in his analysis. After correcting the errors, he dismissed the possibility of cyclical consumption patterns under the assumptions made by Pareto. We will propose a model similar to Pareto’s where we reverse one of his assumptions. We will assume that utility depends positively on the levels of consumption and leisure and positively (negatively) on increases (decreases) in consumption and leisure. We will show that Pareto’s results on cyclical consumption can be rectified in our modified model. Wicksell (19 13) describes Pareto’s model as follows: ‘It is assumed that a person who daily receives a certain sum of money, r,, saves part of it, and devotes the rest to the consumption of - for the sake of simplicity - a single type of good (A). The satisfaction pertaining to the

’ Unfortunately, Pareto tried to relate the ‘integrability’ question that arises with static tastes to the dynamic problem of changing tastes where changes depend on the path of consumption. On this point see Samuelson (1950). 121

122

J. Benhabib /Pareto

vs. Wicksell on adjustment costs

savings on the one hand, and to the consumption on the other, depends therefore solely on the amount of money saved daily, r,, and the amount of goods consumed, r,, or may be considered as functions (the marginal utilities), C& and #e, of each of these quantities. The former satisfaction tends to increase savings, the latter consumption. But, as Pareto correctly points out, even if these two forces are not in equilibrium, the consequent changes in consumption, etc., will not take place at once, but only after some delay. In other words, forces of inertia (‘the force of habit’) come into operation, and analogously to the so-called resistive forces of mechanics, oppose every change in consumption or saving, and are only gradually overcome.’ The forces of habit, or the marginal disutilities of changing consumption and savings are given by f, and f,. If pa is the price of consumption good, the first-order conditions for maximizing utility, which depend on savings and consumption, require that the net marginal utility (per dollar) of increasing consumption be equal to the marginal utility of increasing savings,

The budget constraint

is

raPa +re = r,,

where r, is consumption, r, is savings and r, is a constant level of income. Note that r, is assumed to be independent of,past values of r,, which is another weak point of the analysis. Pareto puts r, = (Y+ x where CYis constant and x is a variable, small in absolute value. He then expands the difference f,- paf,in powers of x up to the second. He makes his first error by assuming that the coefficient of the first power of x, [(l/p,) G’(o) + r,+‘(e)],where E = r, - crp,, can be made equal to zero by an appropriate choice of CY.Note that (l/p,) @‘(LX) t @I(E)is the sum of derivatives of marginal utilities, and therefore negative. Pareto wrote it as a difference and Wicksell caught the error. Pareto also argued that the resistive forces, f,and f,, are proportional to a power of the changing ‘speeds’ of consumption and savings. Since resistive forces should operate both upwards and downwards, Pareto argued that f,and f,should involve even powered terms of the time derivatives of the changes in consumption and savings. Substituting these in eq. (1) and making use of the Taylor expansion up to second order, Pareto obtained a differential equation in x, where the solution x (t) converges to a cyclical path of constant amplitude. Unfortunately there is another error in the above analysis. If utility depends on even powered terms of the time derivatives of the changes, marginal utilities (or disutilities in this case) will involve odd powered terms. Wicksell (19 13) observed that ‘. . . even the greatest of mathematicians are not always immune from error . . .‘. Pareto assumed that there are positive costs of adjustment to increasing or decreasing consumption. It seems plausible however that the increases in consump-

J. Benhabib /Pareto

vs. Wicksell on adjustment costs

123

tion or leisure may generate increases in the level of well-being if the agent is sensitive to improvements over the past. We will construct a specific example of cyclical consumption and leisure (rather than argue with approximations that use Taylor expansions). We will assume that utility is positively related to the levels of consumption and leisure and positively (negatively) related to the increases (decreases) in consumption and leisure. Note that in this model the maximizing agent is myopic because he does not or cannot take into account the effect of his present choice on the achievable utility levels tomorrow. ’ Thus this model is in the tradition of the models used by Weizsacker (197 l), Pollak (1976), and Hammond (1976). Let the utility function be U(t) = V(E,) + X(i,> + Y(?, - fr-_h) + Z(i, - it_/& where C, is consumption given by

(2)

and it is leisure at time t. Let the budget constraint

be

&=2-i*, where the price of 2 and the wages are taken to be unity and maximum work time is taken as 2. It is assumed that all income is consumed. Let ?, = F + ct and It = r+ I, where c and fare constants that we set equal to unity. The budget constraint then becomes (since C= r= 1) Ct = -1t . Since past values of consumption and leisure are given, maximizing respect to ct, lt yields (since dc, = -dZt), v’-X’=Z’-

y’,

utility with

(3)

where the primes above the variables denote derivatives with respect to ct. We now specify the functions V, X, Y, Z as follows: v(t) = ct - +c:, X(t) = 31; = -fc:

Y(t) = gc, - ct_fJ3, )

Z(t) = $(Zt - lt-h)3 3

where V(c? = X(r> = 0 without loss of generality. We restrict the above functions to hold for -1 < cy < 1 so that Et, & > 0 and the marginal utilities are non-negative as well. The first order condition (3) now reduces to 1 - c: - z: = 2(1, - Et-h)2 - (Ct - Ct_#.

(4)

Since ct = --It, this further reduces to (1 - 2$)=

(cy - Cr_#

.

(4’)

2 Note that this can be optimal behavior if the discounting of tomorrow’s utility is large enough. In such a case the agent does the best he can today and always ignores tomorrow.

J. Benhabib /Pareto

124

vs. Wicksell on adjustment costs

The above is the first order difference equation that Pareto obtains in his model with consumption and savings. (As we pointed out before however, his analysis is incorrect.) The solution of the above difference equation will converge to the solution of the differential equation below if we let h + 0, 3 (dc/dt)2

= (1 - 2~:) .

The solution of this differential c(t) = $j

equation

is

sin@ + 2t),

(6)

where

and k is an arbitrary constant. Thus consumption and leisure oscillate around C and L with deviations of at most I/&!. It seems that periods of high consumption and hard work alternate with those of modest consumption and laziness. It should be possible to generalize this analysis to allow the individual to carry an inventory of wealth to smooth his consumption path, with the oscillations then reflected in his wealth stock and labor supply.

References Hammond, P.J., 1976, Endogenous tastes and stable long-run choice, Journal of Economic Theory 13,329-340. Pareto, V., Cours’ d’iconomie politique (F. Rouge, ed., Lausanne, 1896-97). Oeuvres completes, Vol. I, Travaux de droit, d’economie, de sociologic et de sciences politiques, No. 26 (Librairie Droz, Geneva, 1964). Pareto, V., Ophelimity in nonclosed cycles, in: J.S. Chipman, L. Hurwicz, M.K. Richter and H.F. Sonnenschein, eds., Preferences, utility, and demand (Harcourt Brace Jovanovich, New York, 1971) 370-386. Pollak, R.A., 1976, Habit formation and long-run utility functions, Journal of Economic Theory 13,272-297. Samuelson, P.A., 1950, The problem of integrability in utility theory, Economica N.S. 17, Nov., 355-385. Weizsacker, C.C. von, 1971, Notes on endogenous change of tastes, Journal of Economic Theory 3,345-372. Wicksell, K., The enigma of business cycles, in: International Economic Papers, No. 3 (MacMillan, London, 1953) 58-75. Wicksell, K., Vilfredo Pareto’s manuel d’dconomie politique, in: Knut Wicksell, ed., Selected papers on economic theory (George Allen & Unwin, London, 1958) 159-175.

3 A rigorous proof would be contained in the proof of the Cauchy-Peano existence of solutions to a first order differential equation.

theorem

showing

the