The Macroeconomic Implications of Inequality Aversion

Journal of Macroeconomics, Spring 1997, Vol. 19, No. 2, pp. 363–380 363Copyright � 1997 by Louisiana State University Press0164-0704/97/$1.50

SIMON C. PARKERUniversity of Durham

Durham, England

The Macroeconomic Implications ofInequality Aversion*

This paper extends the social planner’s problem of optimal control of inflation and unemploy-ment to deal with the case where inequality also enters their objective function. The macro-economic implications of an aversion to inequality are traced both for the case where inequalityevolves deterministically, and for where it evolves as a joint deterministic and stochastic process.It is found that a positive amount of inequality will be optimal in equilibrium, and that aneconomy’s convergence to its steady state is faster, the greater the social planner’s aversion toinequality.

1. Introduction

An important but sometimes overlooked area of research in macro-economics relates to how macro variables drive income inequality. Followingthe seminal work of Blinder and Esaki (1978), numerous subsequent authorshave related empirically income inequality measures to macroeconomic de-terminants such as unemployment and inflation: see e.g. Weil (1984), Blankand Blinder (1986), Nolan (1987), Black, Hayes, and Slottje (1989), Slottje(1989) and Ashworth (1994). The aim of such research is essentially a de-scriptive one: given some hypothetical inflation and unemployment rates,say, then inequality should be predictable from an econometric model. Therole for active policy, however, has been less clearly articulated in this lit-erature, presumably because of a recognition that inflation and unemploy-ment, as well as inequality, enter the social planner’s welfare function. Thisis surprising in view of the existence of a well-established literature con-cerned with the optimal control of inflation and unemployment alone: e.g.,Nordhaus (1975), who has a social planner concerned with maximizing elec-toral appeal; and Taylor (1989), whose planner aims to minimize social loss.In both cases, the objective function is defined on inflation andunemployment.

The aim of this paper is to extend the optimal control approach to

*The author would like to thank John Ashworth and an anonymous referee for helpful com-ments on earlier drafts of the paper.

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include inequality as a potential source of social loss, and to bring togetherthe optimal control literature and the literature describing the macro deter-minants of inequality. The proposition that inequality can be an importantsource of social loss is well known in the welfare economics literature (Lam-bert 1993), but has not been applied to our knowledge in the area of optimalcontrol of inflation and unemployment. By including income inequality as apotential source of social loss, a more complete policy scenario can be ana-lyzed. In particular, instead of passively observing how inequality is deter-mined, it becomes possible to optimally control the evolution of inequalityover time, as well as of inflation and unemployment. It should be stressedthat the control of inequality studied in this paper is confined to the mac-roeconomic issues: microeconomic structure, such as the tax system, will bekept constant in order to concentrate on the macroeconomic issues. Thuscontrol of inequality emerges as part of the optimal control of the macroeconomy: issues of redistribution policy, for example, are not examined inthis paper.

Several constraints face the social planner in attempting to controlinequality, including the expectation-formation process, an expectations-augmented Phillips curve, and an inequality-efficiency trade-off. The latterconstraint recognizes that greater equality can damage incentives and hencereduce output. The set of constraints as a whole constitutes a simple mac-roeconomic model structured along mainstream monetarist/new classicallines. In common with the previous work in the inflation-unemploymentcontrol literature, particular functional forms are used in order to obtainquantitative results, but care is taken to use specifications which are well-established in the literature or are simple variants of them. One of the prin-cipal variants appears later in the paper, where it is recognized that a short-coming of the literature on the macro determinants of inequality is theprominent role played by stochastic factors in driving inequality. The modelis generalized to unify the deterministic and stochastic models within anencompassing specification, and the optimal policy is derived and then dis-cussed for this case.

The model generates some interesting, and sometimes perhaps sur-prising, results—for example, that inequality-averse governments may ac-tually bear down on inflation faster than more “right-wing” governments. Asummary of the results appears in the conclusion to the paper.

The paper is structured as follows. In Section 2, the basic model ispresented, which takes the case of a social planner who minimizes social losssubject to the given economic constraints. One of these is that inequality isdriven deterministically by a sub-set of the macro variables. The optimalsolution is derived, and policy implications are discussed. Section 3 gener-alizes the determination of inequality to become an Ito process: two pa-

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rameterizations are considered, and the results are interpreted. Section 4summarizes the results and concludes the paper.

2. Inequality as a Deterministic Process

This section starts by outlining the assumptions of the structure of theeconomy, when inequality is a deterministic process. A simple macro modelof the economy is then set out, the government’s objective is stated, andfinally its optimal plan is solved and discussed.

AssumptionsA1. An expectations-augmented Phillips curve exists;A2. Agents form adaptive expectations;A3. Inequality is determined by inflation surprises and unemployment; the

latter has a known growth rate (which may be positive or negative);A4. An efficiency-equality trade-off exists;A5. The government possesses a quadratic social loss function, defined on

unexpected inflation, unemployment, and inequality. Total social loss isattenuated by efficiency;

A6. The government’s intertemporal discount rate is positive and constant.

These six assumptions are either standard ones in the literature, ormodifications/extensions of them. A1 and A2 appear in the macro modelanalyzed by Taylor (1989). A3 is based on Blinder and Esaki (1978) andthose studies which posit inequality, as measured by some known index(see e.g., Morris and Preston 1986 and Cowell 1994) to be determined byinflation/unemployment-type macroeconomic variables. Inequality is astrictly increasing function of the inequality index. A4 is an important con-straint within any “monetarist/neoclassical” model where inequality plays a(potential) role in social preferences. A5 is a modification and extension ofTaylor (1989), who trades off inflation and unemployment with a quadraticloss function. The modification is that unexpected inflation, rather than in-flation per se, is felt to be undesirable. The extension, motivated by thesubject of this paper, is that social loss may also be caused by inequality.Offsetting these sources of social loss is the social gain of greater efficiency,measured by national income. For illustrative purposes and tractability, thisenters in a linear fashion. A6 follows directly Taylor and numerous otherstudies.

The Simple Macro ModelCorresponding to the foregoing assumptions, the following simple

model is proposed:

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p � �(p � p) � w , � � [0, 1] , (1)

p � cU � p , c � 0 , (2)

x � b (p � p) � b U , b � 0 , (3)1 2 2

U/U � k , (4)

Y � hx , h � 0 , (5)

2 2 2K[x, (p � p), U, Y] � � x � � (p � p) � � U � � Y ,1 2 3 4

∀ � � 0 , (6)j j

where

p, p � rates of actual and expected inflation;U, k � unemployment and its growth rate;

Y � income;x � inequality index;K � net social loss.

These equations provide a simple specification of the economy whichis intended to generate illustrative results. Other specifications could ofcourse be suggested; but no attempt is made here to investigate them. It isfelt that the above model is a reasonable, if stylized, representation of amonetarist/neoclassical model, which is capable of facilitating an analysis ofthe central issues.

Some of the parameters in the model bear straightforward interpre-tations. For example, c is the slope of the short-run Phillips curve; k is thegrowth rate of unemployment (which may be positive or negative); and hmeasures the steepness of the efficiency-equality trade-off (Equation (5) istaken directly from Lambert 1990, if x is taken as the Gini coefficient, andh is unity). The �s are the weights of the net social loss function: these couldbe normalized on one of the arguments of K, but that is not done here. Thebs map the macroeconomic variables on to inequality. In accordance withthe relevant literature, it is expected (and invariably found) that b2 � 0, i.e.,unemployment raises inequality. It might also be expected that unexpectedinflation would raise inequality, implying b1 � 0; but previous studies havefound mixed results using inflation variables (Ashworth 1994). All of thevariables explained in the list above are time-variable; explicit time notation,


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(t), is suppressed for clarity. A dot denotes a time derivative, and to simplifythe notation, is replaced by the symbol w in the subsequent analysis.p

The social planner’s objective, to minimize the net present value ofnet social loss given the above model, is stated as follows:

��rtmin E K[x, (p � p), U, Y]e dt , (7)�

0

subject to (1) through (6), with some given initial level of inequality x(0)� x0. Here, E is the expectations operator, and r is the social planner’sdiscount rate: r � 0. The problem can be analyzed in the optimal control(or calculus of variations) format if the derivative of (3) is taken: using (1),(2) and (4), this derivative is

x � [(b c � b )k/c�]w . (8)1 2

The Optimal PlanThe solution to the above problem is characterized by the following

pair of equations:

�1 �n t �11x*(t) � [x � �n ]e � �n , (9)0 2 2

w*(t) � � � n x*(t) , (10)2

where

1/22 21 4� (b c � b ) k1 1 22n � �r � r � � 0 , (11)1 � � 2 � �2 � c � �2 3

n � n c�/(b c � b )k , (12)2 1 1 2

h� [b c � b ]k� n h�4 1 2 2 4� � • � (13)� � � �1�2(n � r) � c � � c 2�1 2 3 1

are constants. See Appendix A for proof.

DiscussionEquation (9) describes the optimal policy path for inequality. It states

that the optimal plan is to reduce (increase) inequality over time towardsthe level � h�4/2�1, according as initial inequality is greater (less)�1�n2

than this level. That is, h�4/2�1, which is strictly positive, can be regarded

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as the social planner’s “desired” or “target” inequality level. As can be seen,it is directly proportional to both the steepness of the efficiency/equalitytrade-off, i.e. h, and the planner’s aversion to efficiency losses relative toequality gains, i.e. �4/�1—results which accord with intuition. The constancyof this long-run equilibrium level of inequality implies that the authoritiescan only influence long-run inequality, given their preference structure,through the steepness of the output/equality trade-off. For although h istreated as given in the analysis here, it clearly may be the outcome of eco-nomic factors such as the education system, optimal tax policy, etc. It isinteresting to speculate on the implications of these economic factors on hand thereby on long-run inequality. For example, an increase in the provisionof education may lead to a weakening of the trade-off (h falls) and hence areduction in long-run inequality.1 Other policy implications may also be sug-gested; clearly, however, a comprehensive and detailed analysis along theselines is beyond the scope of the present paper.

Equation (10) is a “feedback” rule expressing the optimal relationshipbetween the rate of change of expected inflation and inequality. Sincelimt��x*(t) � , it follows that limt��w*(t) � � � n2 limt��x*(t) ��1�n2

0. That is, the optimal policy leads to a steady-state equilibrium where therate of change of expected inflation converges to zero, i.e., expected inflationconverges on actual inflation.2 From Equation (2), this implies also thatlimt��U � 0, i.e. “unemployment will converge to zero.” Of course, thisimplication is a consequence of our definition of U, which in the model isimplicitly normalized around some “natural rate.” The model’s predictionshould therefore be read as “unemployment will converge on its naturalrate,” which may be non-zero for the usual reasons (frictional unemploy-ment, etc.). It may also be worth adding at this point that implicit normali-zation may also apply to the income and inequality variables. For example,the model predicts that income will settle at h2�4/2�1—which may be eitherhigher or lower than its initial level hx0—but any exogenous income levelnormalized to zero in the model must be added to this to find the effect ontotal real income. Likewise for inequality: any exogenous inequality levelwould need to be added to x* to obtain overall inequality. In all cases, thenormalization to zero of exogenous factors has been motivated by exposi-tional simplicity.

1However, the effects on income of this change are ambiguous a priori: q.v. below.2Since n2 cannot be signed unambiguously (because b1 could take either sign), it is unclear

a priori whether the convergence of the rate of change of expected inflation towards zero willtake place in an upwards or a downwards direction. The change in w at the time the policybegins is Dw � �n2x0, indicating either a positive or negative upwards initial shift; but what isclear from (9) and (10) is that any divergence between actual and expected inflation will beprogressively eliminated in succeeding time periods.


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It should be stressed that the means by which the social planner steersthe macro economy (including inequality) in this model is via inflationaryexpectations, not via inequality through the tax-benefit system, for example.In the terminology of the Maximum Principle, the rate of change of infla-tionary expectations is the control variable; inequality is the state variable.Operational use of the model therefore presupposes that the social plannerdoes indeed possess the requisite tools to help determine private sectorinflationary expectations. Such tools would no doubt be primarily monetaryin nature; although since the monetary sector has been suppressed in themodel for simplicity, it is not possible to elaborate any further on this here.

In short, the solution predicts convergence of the economy to a steadystate where income settles at a new level, where expected inflation is equalto actual inflation, and where inequality converges to its “desired” value. Thespeed of convergence of all of these variables to their steady-state values isunambiguous, being given by n1. How is n1 determined? Taking first partialderivatives of n1 (Equation [11]) with respect to the structural parametersof the macro model, it is readily seen that the speed of convergence variesdirectly with the social planner’s degree of relative inequality aversion (i.e.,�1 relative to �2 and �3). However, it does not vary at all with respect to anaversion to efficiency losses (�4) or the size of the efficiency-equality trade-off (h). With the �s being exogenous taste parameters, taking different valuesdepending on the political color of the social planner, it is possible to inter-pret the above findings in terms of the sort of equilibrium favored by “leftwing” and “right wing” governments. Presumably, a left-wing governmentwill have a greater �1 value and a lower �4 value than a right-wing govern-ment. The implication of the foregoing is, perhaps surprisingly, that a “rightwing” government which cares a lot about efficiency losses relative toinequality-induced losses would take longer than a “left wing” governmentwith a stronger aversion to inequality to restore the economy to its steady-state equilibrium.

The reason the above result may seem somewhat surprising is theimplication that optimally behaving left-wing governments should aim toreduce inflation faster than optimally behaving right-wing ones. However,this result is, crucially, an outcome of the stylized “monetarist” structure ofthe model considered here. A “left-wing” government which did not acceptsuch a model, and believed that, say, monetary growth could permanentlyraise output (e.g., by augmenting Equation (5) with some function of thegrowth of the money supply), might easily end up with policies which gen-erate higher inflation than a “right-wing” government. But in the absence ofa relationship between output and monetary growth, the model here indi-cates that such an inflationary policy would certainly be sub-optimal.

Other comparative statics may also be performed. For example, a

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lower interest rate increases, ceteris paribus, the speed of adjustment to thesteady state, the intuition being that the lower the interest rate, the greaterthe social loss arising from disequilibria in future time periods, so providingan incentive for their rapid elimination. The speed of adjustment also variesdirectly with the strength of the deterministic inequality-macro variablesrelationship, i.e., the size of the bs (Equation [3]); but the effect on n1 ofthe slope of the short-run Phillips curve is ambiguous a priori.

A criticism which can be leveled at the model in its current form isthat it is purely deterministic. In practice, there is good reason to believethat inequality does not evolve in a smooth fashion as predicted by Equation(3), but is also subject to stochastic shocks. The impact of stochastic inno-vations on inequality has already been recognized by some researchers work-ing in the inequality literature. Work on this topic is briefly described in thefirst part of the next section, the remainder of which investigates the effectsof stochastic shocks to inequality on optimal macroeconomic policy.

3. Inequality as a Joint Deterministic and Stochastic Process

It has been assumed so far that inequality is a deterministic process,but there are good reasons to seek a relaxation of that assumption. Apartfrom the old income distribution literature, which explained inequality asthe outcome of various purely stochastic processes (Gibrat 1931; Champer-nowne 1937, 1953; Mandelbrot 1960), more recent research has argued thatinequality possesses a salient stochastic component. For example, Hayes etal. (1990) examined four well-known inequality measures using postwar U.S.data and were unable to reject the null hypothesis that they all evolve asrandom walks. Undoubtedly, part of the inspiration for recent work on sto-chastic determinants of inequality has been the increasing influence ofintegration/cointegration analysis on the time series inequality literature: inthis context, unit root tests by a number of authors have lent credence tothe notion that inequality measures are non-stationary series (see e.g. Ash-worth 1994; Parker 1996b). It therefore seems desirable to jointly model thedeterministic and stochastic elements of inequality. In the following, a spec-ification will be suggested which does this. Consistent with the analysis ofthe previous section, the analysis will be conducted within a continuous timeframework, drawing on the branch of mathematics known as the theory ofstochastic processes (see Dixit and Pindyck 1994, for an accessible andstraightforward introduction to this subject).

The essential building block in stochastic modeling is the Wiener pro-cess, z, whose innovations dz are mean zero independently and normallydistributed random variables, with a standard deviation, r, which is poten-tially time-variable and dependent on economic variables. Consider the ex-


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treme case of Hayes et al. (1990), where inequality x is determined by purelystochastic factors. Then x is a stochastic process, written as dx � r(x, u, t)dz.Here r is a function potentially of x and t, and also of some other variable(s),u. With changes in x determined only by stochastic innovations, there isnothing to force x to revert to its mean: x will be a random walk.

At the other extreme is the purely deterministic process, such as (3)considered in the previous section. A general model which includes bothcomponents is known as an Ito process:

dx � [(b c � b )k/c�]w(t)dt � r(x, u, t)dz.1 2

In words, the term in square brackets is the expected rate of change ofinequality, but there is also a disturbance term. This equation will form thebasis of the analysis in the remainder of the paper.

For a variable to follow a pure random walk, it must, strictly speaking,be unbounded. In most economic applications, however, this issue is side-stepped by assuming that the variance of the stochastic term is too small topush the variable near one of the bounds.3 In such cases, the Brownianmotion process will continue to approximate almost perfectly the behaviorof the bounded variable: any “reflecting barrier” will not be hit. The samewill be assumed in the following about inequality measures (many of whichinvariably have a lower bounds of zero, although these can be easily trans-formed to become unbounded measures).

It remains to parameterize the standard deviation function r(•). Twoof the most popular cases investigated in stochastic models are (i) r is con-stant (a parameter); and (ii) r is a linear function of the state variable itself:r(•) � rx, where r is a parameter. Case (ii) is of interest in the presentcontext for those inequality measures (probably those in the class which donot possess an upper bound) where shocks tend to be larger the greaterinequality actually is. For example, in free-market economies, greater risk-taking entrepreneurial behavior can be assumed to increase inequality, in-comes, and the variability of incomes—and thereby the variability ofinequality.

To illustrate how recognition of stochastic factors impinges on optimalpolicy, both case (i) and case (ii) will be considered. It will be seen that theydiffer in ways which may carry important implications for optimal policy.Case (i) is the simplest to handle, and its properties can be summarized inthe following proposition:

3For example, Merton (1969) ignores the lower bound on individual wealth holdings; Parker(1996a) ignores that on self-employment returns. See also Malliaris and Brock (1982), whoenumerate many other studies which likewise posit strictly positive variables such as output andcapital possessing stochastic components.

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Proposition 1. (constant stochastic variance). The expected time paths ofinequality and the rate of change of expected inflation are as in the deter-ministic case. Fluctuations around the optimal deterministic paths are Gaus-sian: the variances of these stochastic disturbances for inequality and therate of change of expected inflation converge to r2/2n1 and ,2 2n r /2n2 1

respectively.

Proof. See Appendix B.

This proposition indicates that extending the model to allow for sto-chastic influences on inequality under case (i) leads to no more serious mod-ification of the previous results than introducing stochastic fluctuationsaround the deterministic solution paths. Interestingly, from the variance ex-pressions in Proposition 1, a faster speed of adjustment n1 will, ceteris par-ibus, reduce the dispersion of fluctuations of inequality around its steady-state equilibrium level; but it will also (by definition of n2) increase thedispersion of inflation surprises around zero.

Case (ii), where the stochastic variance is proportional to inequality, issomewhat more mathematically involved. The essential results are summa-rized in Proposition 2:

Proposition 2. (proportional stochastic variance). The optimal policy hasinequality converging to a lower desired equilibrium value, and convergingto it faster, than in the deterministic case. The expected path of inequality isgiven by the equation

�1 �f t �11E[x*(t)] � [x � Wf ]e � Wf , (14)0 2 2

where

1/22 21 4� (b c � b ) k1 1 22 2 2f � (r � r) � (r � r) � � n � 0 , (15)1 1� � 2 � �2 � c � �2 3

f � [f c�/(b c � b )k] � n , (16)2 1 1 2 2

n � r1W � � � � , (17)� �f � r1

are constants. A linear feedback rule still characterizes the optimal expectedinflation policy, but this rule is now w*(t) � W � f2x*(t). As in the deter-ministic case, the steady-state expected rate of change of expected inflation


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is zero, as is (normalized) unemployment, although actual values are nowsubject to stochastic shocks.

Proof. See Appendix C.

The intuition for the finding here that steady-state expected equilib-rium inequality will be lower than in the purely deterministic case, can beseen in terms of the social loss function. If the variance of shocks dependson inequality, lower inequality will reduce this variance, which is needed tokeep the loss component �1x

2 small. In other words, the extra stochasticcomponent provides additional penalization for high inequality. Just as be-fore, a higher degree of inequality aversion serves to increase the speed ofadjustment to equilibrium.

A new question arises in the context of proportional stochastic vari-ance, namely the stability of equilibrium in this case. As explained in thefinal part of Appendix C, the model is sufficiently general to produce bothstationary and non-stationary solution paths for inequality, and (normalized)unemployment and income; but the only economically sensible solution cor-responds to the stationary case. The interested reader is referred to thatappendix for the rationale.

To conclude, the effects of stochastic innovations on optimal policychoices depend centrally on whether the variance of the innovations does ordoes not depend on the level of inequality itself. If it does not, optimal policyis essentially unaltered from the purely deterministic case, with actual in-equality and undesired inflation fluctuating around the optimal policy pathsat all times. If it does, and if inequality follows a stationary process, then thegovernment should reduce inequality further and faster than it would oth-erwise do. Moreover, the greater the variance of the stochastic process inthis case, the further and the faster inequality should be reduced.

4. Conclusion

This paper has generated a relatively large number of results aboutthe macroeconomic implications of inequality aversion, and it is useful tosummarize the results together. They are:

1. In a purely deterministic setting, a social planner will ensure that (nor-malized) inequality converges to some positive desired value; (normal-ized) unemployment converges to zero; and expected inflation equalsactual inflation. Desired “equilibrium” inequality varies positively withboth the steepness of the efficiency/equality trade-off, and with the plan-ner’s aversion to efficiency losses relative to equality gains.

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2. In all cases, whether purely deterministic or whether inequality also em-bodies a stochastic component, the speed of adjustment to steady statevaries positively with the planner’s degree of inequality aversion, andnegatively with the discount rate.

3. Allowing for stochastic shocks to impinge on inequality, the theoreticalresults suggest that inequality will evolve as a stationary (mean-reverting)process.

4. When the variance of stochastic innovations varies proportionately withinequality itself, steady-state inequality will be lower, and the speed ofadjustment faster, than in the deterministic case.

All of these results pre-suppose that the social planner possesses the requisitetools to determine private sector inflationary expectations. While the policytransmission mechanism for this is not fully described in the present paper,an obvious extension would be to augment the model with a complete mon-etary sector, permitting analysis of means of control of inflationary expec-tations. This topic awaits future research.

Other extensions may also be suggested. One possibility is to allowmore variables in the economic system to evolve with stochastic components.Another would be to analyze the possibility of non-stationary inequality pro-cesses in more detail. Both of these extensions are weighty projects in theirown right and would be potentially mathematically complex: the former en-tailing the stochastic optimal control of dynamic systems, and the latter ne-cessitating numerical simulation of bounded control regions. Hopefully, thecurrent paper makes a start in the right direction.

Received: June 1995Final version: March 1997

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———. “A Model of Income Distribution.” Economic Journal 63 (June1953): 318–51.

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Appendix A.

Proof of the Optimal Plan (9) and (10)First substitute the equations of the macro model into (6), and re-

arrange to obtain K(x, w) � [�h�4x � �1x2 � gw2], where g � (�2 �

�3c�2)/�2 � 0. Likewise, the transition Equation (8) can be rewritten as x

� hw, where h � (b1c � b2)k/c�. The Bellman equation for this autono-mous infinite-horizon problem is

r •V(x) � min{K(x, w) � V�(x)hw} (18)w

2 2� �h� x � � x � [hV�(x)] /4g , (19)4 1

using the fact that the minimizing w for (18) is

w � �hV�(x)/2g . (20)

Equation (19) is a non-linear first-order differential equation, whose solutionis of the form V(x) � Ax2 � Bx � C: putting this and its squared firstderivative into (19) definitizes the constants of V(x) as A � [�r �

]g/2h2 � �, B � �h�4g/(h2A � gr) and C � �h2B2/4rg.2 2r � 4� h /g� 1

The positive root of A is the only feasible solution for this problem; and A� 0 ⇒ B � 0 since h�4 � 0. Putting V�(x) � 2Ax � B into (20) yields thesolution

�1w* � �(B � 2Ax)h/2g � (h� h/2g(n � r)) � n h x , (21)4 1 1


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where n1 is defined by (11). Use of the definitions of g and h above thenyields the solution (10) in the text, with the definitions of n2 and � given by(12) and (13). (The final expression on the right-hand side of [13] is obtainedby observing that gn1(n1 � r) � h2�1). Finally, (9) is obtained as the solutionof the differential equation x/h � � � n2x. �

Appendix B.

Proof of Proposition 1Using the notation of the proof in Appendix A above, write the tran-

sition equation as dx � hwdt � rdz; the Hamilton-Jacobi-Bellman equationis (Malliaris and Brock 1982, 114)

2r •V(x) � min{K(x, w) � V�(x)hw � V�(x)r /2} (22)w

2 2 2� �h� x � � x � [hV�(x)] /(4g) � V�(x)r /2 , (23)4 1

since the minimizing w is given by (20) as before. Equation (23) is a non-linear second-order differential equation, whose solution is the same quad-ratic as before: V(x) � Ax2 � Bx � C. The only difference is that thedefinitized constant C is now augmented by the amount Ar2. But becauseC does not enter the solution as given by (20), the solution (10) (the feedbackrule) remains unchanged from before.

Now x*(t) is the solution of the stochastic differential equation dx �(�n1x � h�)dt � rdz, which is

t�1 �n t �1 n s1 1x*(t) � [x � �n ]e � �n � e rdz(s) ,0 2 2 �

0

where the stochastic integral captures the cumulative stochastic deviationsfrom the deterministic path. Its expected value is zero, so the expected valueof x*(t) is given by (9) as before. Since the feedback rule is also unchanged,the expected value of w also remains unchanged.

Turning to the second part of the Proposition, the variance of x* is

2r �2n t1Var[x*(t)] � (1 � e ) ,2n1

and since the stochastic integral in the x*(t) expression is Gaussian,

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378

asy�1 2x*(t) � �(Tn , r /2n )2 1

(taking limits of the process: see e.g. Arnold 1974, chap. 8). Thus the varianceof x* around its steady state is r2/2n1; the variance of w is then computedusing the feedback rule. �

Appendix C.

Proof of Proposition 2The Hamilton-Jacobi-Bellman equation for this problem follows the

format of that given in the proof in Appendix B above, except that (rx)2

replaces r2. The solution V(x) and Equation (20) have unchanged generalforms; but whereas B and C are unchanged, A takes the new value A � [(r2

� r) � ]g/2h2. As before, w* � �(B � 2Ax)h/2g,2 2 2(r � r) � 4� h /g� 1

which becomes

h� h4 �1w* � � h f x , (24)12g(f � r)1

using (20) and the definition of A above; where f1 � h2A/g; and where B� h�4/(f1 � r). Using the definitions (17) and (16) of W and f2 respectively,gives the feedback rule of w*(t) in the body of Proposition 2; and x*(t) isfound as the solution to the stochastic differential equation dx � (hW� f1x)dt � rxdz. The solution to this equation is

2r(z(t)�z(0)) 2 �(f �(r /2))t1x*(t) � [x e �W(f /f (f � (r /2)))]e0 1 2 1

2� W(f /f (f � (r /2))) ,1 2 1

which is a Gaussian variate by virtue of the Wiener z terms. Its expectedvalue is given by (14) in the body of Proposition 2 (as are the definitions ofthe constants in the above); its variance is

2�1 2 2 (r �2f )t1Var[x*(t)] � (Wf ) [(2f /(f � r )) � 1] � c.e2 1 1

�1 �2f t �1 �11� (x � Wf )e � 2Wf (x � Wf )0 2 2 0 2

2 �f t1• [(f /(f � r )) � 1]e , (25)1 1

where


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�1 22Wf (x � Wf ) 2f W1 0 2 12c � x � � ,0 � 2 2 2 �f (f � r ) f (2f � r )2 1 2 1

is a constant.Comparing (11) through (13) with (15) through (17), it is clear that

the presence of the stochastic term (with r � 0) implies: f1 � n1, f2 � n2,and W � �. Now limtr�E[x*(t)] � is the new expected steady state�1Wf2

equilibrium inequality level. By the above, this must be smaller than thedeterministic equilibrium value of . Also f1 � n1 implies that the speed�1�n2

of adjustment to equilibrium is faster in the present case than in the purelydeterministic case. Finally, from the feedback rule, it is obvious thatlimtr�E[w*(t)] � W � f2limtr�E[x*(t)] � 0; and expected (normalized)unemployment therefore also goes to zero by the Phillips relation. �

The Stationarity IssueConcerning the stationarity of the solution paths, consider Equation

(25) above. There are four terms on its right-hand side, the first being con-stant, and the third and fourth going to zero as t r �. What happens to thesecond term as t r � depends on the sign of (r2 � 2f1), which is a prioriambiguous. A positive sign implies an exploding variance, i.e. inequality isnon-stationary; a negative sign or equality implies that the variance of in-equality will converge to a constant value. In the latter case, inequality willbecome “stationary in the wide sense” in the limit, since also limtr�E[x*(t)]� (constant). The intuition behind the ambiguity of the variance�1Wf2

is seen more clearly if one regards the evolution of the variance being subjectto two offsetting forces: a “widening” stochastic force, and a “narrowing”deterministic (or “economic”) force. Left to itself, the stochastic elementwould impart non-stationarity to inequality; but �f1 plays a “mean-reverting” role.4 The question is whether the deterministic or the stochasticforce dominates.

From the foregoing, and using (15), the condition for stationarity is

2 24� (b c � b ) k1 1 22 �2r � r � 2r .� 2 �� c � �2 3

This condition will obviously hold in two cases: r very small (i.e., close to0�), and r very large. On the basis of the discussion at the start of this section,

4For example, compare �f1 in the stochastic differential equation in Appendix C, and �n1

in the stochastic differential equation in Appendix B, with �� in the Ornstein-Uhlenbeckprocess: the fact of � � 0 causes mean reversion in that process (Uhlenbeck and Ornstein1930).

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the former is an attractive proposition but the latter is not, since the needfor reflecting barriers (which are not built into our model) is absent in thecase of the former but becomes essential for the latter. This conclusion issupported by the fact that limrr� � 0: i.e., r very large implies, un-�1Wf2

realistically, that desired inequality must tend to zero. The upshot of thisdiscussion is therefore that we would expect inequality—and hence also therate of change of expected inflation, and normalized unemployment andincome—to be stationary processes in practice. In fact, this is not contra-dicted by the econometric results which infer that inequality follows a ran-dom walk. It is well known that many genuinely stationary time series canlook like random walks in short samples (see e.g., Dixit and Pindyck 1994chap. 3, for example): unit root tests have notoriously low power (Banerjeeet al. 1986).

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The Macroeconomic Implications of Inequality Aversion