Working Paper Series

Congressional Budget Office

Washington, DC

Median-voter Equilibria in the Neoclassical Growth Model under

Aggregation

Marina Azzimonti

University of Iowa

E-mail: marina-azzimonti@uiowa.edu

Eva de Francisco

Congressional Budget Office

E-mail: eva.defrancisco@cbo.gov

Per Krusell

Princeton University

E-mail: pkrusell@Princeton.edu

December 2005

2005-09

Working papers in this series are preliminary and are circulated to stimulate discussion and critical

comment. These papers are not subject to CBO’s formal review and editing processes. The analysis

and conclusions expressed in them are those of the authors and should not be interpreted as those

of the Congressional Budget Office. References in publications should be cleared with the authors.

Papers in this series can be obtained at www.cbo.gov (click on Publications and then on Working

and Technical Papers.)
Abstract

We study a dynamic version of Meltzer and Richard’s median-voter model where agents differ

in initial wealth. Taxes are proportional to total income, and they are redistributed as equal lumpsum transfers. Voting takes place every period and each consumer votes for the current tax

rate that maximizes his or her welfare. We characterize time-consistent (differentiable) Markovperfect equilibria in three ways. First, by restricting the class of utility functions, we show that

independently of the number of wealth types, the economy’s aggregate state can be summarized

by two statistics: mean and median wealth. Second, we derive the median voter’s first-order

condition and interpret it in terms of a tradeoff between distortions and net wealth transfers.

Third, we illustrate the key endogenous-taxation mechanisms using 1- and 2-period versions of

the model. Quantitatively, we find that the model in its baseline form cannot explain the large

wealth inequality that we observe in most economies.

Journal of Economic Literature Classification Numbers: E20, E60, E62, H21, H30.

Key Words: Political economy, dynamic macroeconomic model, endogenous voting, median voter, redistribution, inequality, and aggregation.
1 Introduction

Income taxes are arguably important determinants of aggregate economic performance, and

they are fundamentally endogenous. What determines taxes? A widely held belief is that the

desire to redistribute is a key explanatory factor, and furthermore that the amount of redistribution

is one of the central elements over which elections are decided. One way to evaluate this theory is

to construct a reasonably calibrated macroeconomic model and to compare its politico-economic

equilibrium to data. Dynamic models of political economy are complex objects of analysis,

however, and the development of the theory and the associated numerical analysis and empirical

methods are still in their infancy. The goal of this paper is to contribute to this methodological

development. We consider a theory based on endogenous redistribution between consumers of

different wealth types: agents vote on general income tax rates, with associated equal-per-capita

lump-sum transfers. The setting is a standard decentralized version of the one-sector neoclassical

growth model, and the political aggregation mechanism is majority voting. Taxes are proportional

to total income, and they are redistributed as equal lump-sum transfers. Voting takes place every

period and each consumer votes for the current tax rate that maximizes his or her welfare, and we

characterize time-consistent (differentiable) Markov-perfect equilibria.

The two most closely related papers in the literature are Meltzer and Richard (1981) and

Krusell and R ́ıos-Rull (1999). The former paper describes a static setup where distortionary

labor taxes are used to fund the transfers. Though conceptually constituting the core of modern

median-voter models, Meltzer and Richard’s setup is arguably not well-suited for quantitative

analysis in that it does not deal with the taxation of capital income and the associated distortions.

Krusell and R ́ıos-Rull (1999), on the other hand, is a fully dynamic model that considers the

1
taxation of capital income in a Meltzer-Richard kind of setting. Krusell and R ́ıos-Rull define

equilibrium in the dynamic model, provide a method for numerically finding equilibrium, and

report quantitative findings for a calibration of the model. Compared to that paper, in terms of

theory, the present work makes three contributions: (i) it develops an aggregation result that is

useful for simplifying the analysis, (ii) it derives a first-order condition for the median voter that

allows interpretation of the results, and (iii) by studying two “baby versions” of the fully dynamic

setup considered last—both a 1-period and a 2-period model—it offers a better understanding of

some of the intuitive mechanisms behind tax determination. In terms of implementation, we also

use an entirely different method for computing equilibria, one that is based directly on the median

voter’s first-order condition.

In addition, finally, this paper also studies a slightly different setting than the one used in

Krusell and R ́ıos-Rull (1999), who assume that there is an implementation lag for taxes, i.e., in

their setting, the tax voted on at t is implemented at t + k, with k > 0. This assumption means

that as far as its impact on capital accumulation, there is no immediate impact at all of raising the

tax rate at t—capital income is completely inelastic. The absence of an implementation lag in the

present work leads to significantly different quantitative results. In particular, the model predicts

that only a very narrow range of wealth distributions can be observed as a long-run outcome,

and the wealth inequality observed in most developed countries is outside this range. In short,

given the large wealth inequality observed, our model predicts that the median voter would tax

away most of these differences and that the economy would subsequently converge to a new

steady state with much lower inequality, i.e., we find that the model is unable to account for the

observed combination of taxes and inequality—the marginal benefit to the median voter of further

taxation by far exceeds the marginal cost. Therefore, we learn that a model that would have

2
greater quantitative success would need a larger cost of taxation or a smaller benefit. The setting

in Krusell and R ́ıos-Rull (1999) can better explain the data because their implementation lag is

one way of making taxation more costly, however, the nature of implementation lags in actual

tax constitutions is not apparent. One implication of the findings here is indeed that we need

quantitative measurement of any lags between political decision making and implementation:

these lags really matter quantitatively for what our theories predict.

Our aggregation characterization roughly is as follows: given consumer preferences of the

appropriate form, Markov-perfect equilibrium outcomes which arise as limits of finite-horizon

equilibria depend on the mean and the median asset holdings, and on nothing else. This finding is useful because it simplifies both the theoretical and numerical analysis. It is not surprising: with preferences that allow aggregation (for simplicity, we look at constant-elasticityof-intertemporal-substitution preferences here), so long as taxes do not depend on anything but

mean and median wealth, neither can prices nor aggregate quantities: marginal propensities to

save and work are equal across all consumers, so aggregates can be arrived at by summing up

across individuals; moreover, the propensities cannot depend on higher moments through taxes,

since taxes cannot depend on any other other moments by backwards induction.

The explicit first-order conditions for the median voter that we use in our analysis are significantly more complex, both conceptually and mathematically, than standard first-order conditions

from optimal control theory. They express how the median voter trades off the marginal benefits against the marginal costs of changes in income taxes. The tradeoff is expressed in terms

of distortions to labor-leisure and consumption-savings choices—“gaps”—on the one hand and

net transfer effects on the other. By measuring the size of each of the gaps in the calibrated

politico-economic equilibrium, one could provide an assessment of which tradeoffs matter the

3
most. Prior analysis focusing on first-order conditions in similar contexts include the work on

individual saving under time-inconsistent preferences, where reminiscent first-order conditions

have been derived (so-called “generalized Euler equations”; see Laibson (1997)), and some recent work on dynamic public finance (e.g., see Klein, Krusell, and R ́ıos-Rull (2003)). Azzimonti,

de Francisco, Krusell, and R ́ıos-Rull (2005) surveys these methods and their use in different applications, one of which is the present setup. That paper derives the first-order condition for the

present model from first principles; here, we show how to derive the condition only in the 1-

and 2-period models we consider; for the infinite-horizon version, we simply state the result and

focus on its contents.

Numerical solution of models of the kind studied in this paper is not straightforward either.

Steady states are impossible to find the way they are found in growth models with exogenous

policy, because the level of capital and the tax rate on income depend, via the first-order condition

of the median voter, on the derivatives of the equilibrium decision rules. This means that one

cannot specify a finite set of equations in levels only: levels depend on decision rule derivatives,

which in turn depend on higher-order derivatives of these same decision rules. One approach for

computation is to specify decision rules of some flexible parameterized functional form over some

domain and evaluate all the equilibrium conditions, including first-order condition of the median

voter, on this domain. Thus, one can iterate on the parameters of the functions in order to meet

some criterion. This approach seems feasible but cumbersome to implement, since we are dealing

with decision rules with more than one argument and since, in our recursive equilibrium definition

there are several such functions whose shape are unknown. In this paper, we instead continue the

development of an approach suggested in Krusell and Smith (2003) and later applied elsewhere

4
which is fast and which allows a systematic search over all possible steady states.1 This method,

which builds on approximating the decision rules with polynomials evaluated at the steady-state

point only, generates more equilibrium conditions—which are expressed as functional equations

(the unknown decision rules being the unknowns and their arguments being mean and median

wealth)—by successive differentiation and evaluation of these conditions at steady state.

Our focus on (differentiable) Markov-perfect equilibria, i.e., those where the state of the economy consists of payoff-relevant information only (see, e.g., Maskin and Tirole (2001)), by design rules out the study of the possible role of “reputation” in influencing political outcomes,

whereby voters would collectively make their current voting behavior depend on historical voting/policy outcomes (see, e.g., Bernheim and Nataraj (2002)). Thus, we study the “fundamental”

endogenous-tax equilibrium. Even if basic, such a framework, however, almost by definition

is very complex: it involves both economic and political dynamics. Most existing papers on the

topic are either based on computational work without much theoretical characterization (see, e.g.,

Krusell and R ́ıos-Rull (1999)) or on models which are not quantitatively satisfactory. In the latter

category, physical capital accumulation is typically ignored or studied in frameworks that do not

allow decreasing returns to capital (see, e.g., Persson and Tabellini (1994), Alesina and Rodrik

(1994), and Hassler, Rodriguez-Mora, Storesletten, and Zilibotti (2003)).

The paper is organized as follows. A 1-period model is analyzed in Section 2, a 2-period

model in Section 3, and an infinite-period model is finally discussed in Section 4. In each of

these sections, both equilibria with exogenous and endogenous taxes are discussed in turn, and

aggregation is explored. There is also emphasis, for each version of the model, on the first-order

condition from the median voter’s tax problem, since it changes nature depending on the time

1See, e.g., Krusell (2002) and Klein, Krusell, and R ́ıos-Rull (2003).

5
horizon. Moreover, each of the sections concludes with a functional-form example that admits

“almost” closed-form solutions. Section 5 concludes. Finally, an Appendix contains some formal

definitions and proofs.

2 The 1-period model

The 1-period model is a simple extension of Meltzer and Richard (1981) to include capital.

The 1-period model with capital makes taxation more beneficial, on net, for the median voter:

capital is inelastically supplied as the tax is decided upon. One can argue that by only focusing

on labor income, Meltzer and Richard ignored a key determinant behind large governments: the

existence of a large stock of unequally distributed wealth that can be taxed away and redistributed

at low cost. The taxation of capital in an intertemporal setting is more complex—taxation at t

distorts saving in earlier time periods, and because it redistributes it influences future taxation

as well—so it is important to build slowly toward the fully dynamic analysis. This goal is accomplished here by the use of first a 1-period, then a 2-period, and finally and infinite-horizon

model.

In the present section, we will (i) set up the basic environment, which will then be considered

in a dynamic context as well and (ii) to use a simple example to discuss aggregation, existence,

and how taxes are determined as a function of the primitives in this environment.

2.1 The environment: exogenous taxes

We describe the basic environment first, and then the decentralized equilibrium.

6
2.1.1 Environment

There is a set of agents who only differ in initial asset holdings. We will assume for simplicity

that the number of “types” is finite with measure μi for type i ∈ {1, 2, . . . , I}. Population size is

normalized to one: PI

i=1 μi = 1.

Utility of each agent is u(c, l), where c is consumption and l is leisure. Consumption and

leisure both have to be nonnegative.

Production takes place according to a production function which depends on capital and labor

and has constant returns to scale: Y = F(K, N) (we use capital letters to denote aggregates). In

the 1-period model, all of output is (privately) consumed: Y = C.

There is a constraint on the amount of time for each agent: each consumer has one unit, so that

li + ni = 1 for all i, where ni denotes the amount of hours worked. We will make assumptions

on primitives so that agents’ decision problems are strictly concave; hence, all agents of the same

type will make the same decisions and we can also write Li + Ni = 1, where Li and Ni reflect

common decisions regarding leisure and labor of all agents of type i. Therefore the aggregate

labor input is N =

PI

i=1 μiNi

.

2.1.2 Equilibrium

In a decentralized economy, consumers buy consumption goods and rent capital and sell their

labor services to firms under perfect competition. The rental rate for capital is denoted r and

the wage rate w, both in terms of consumption goods in the same period. In addition, we now

consider a government which taxes income at a proportional rate τ—capital and labor income

are taxed at the same rate—and makes equal lump-sum transfers T back to all consumers under

7
a balanced budget. Thus, a typical consumer i’s budget set B(Ai) reads

B(Ai) ≡

©

(c, l) ∈ R

2

+ : c = (1 − τ )[Air + w(1 − l)] + T

a

.

In equilibrium, consumers’ holdings of assets have to add up to the total capital stock: PI

i=1 μiAi =

K. Consumer heterogeneity thus originates in ai 6= aj for all i 6= j. We define a competitive

equilibrium for a given government policy as follows:

Definition 1 Given a policy τ , a competitive equilibrium is a set of prices (w, r) together with

an allocation (K, N, T,(Ci, Ai, Li)i) satisfying the following conditions.

1. For all i, (Ci, Li) solves max(c,l)∈B(Ai) u(c, l).

2. w = FN (K, N) and r = FK(K, N), where N =

PI

i=1 μi(1 − Li) and K =

PI

i=1 μiAi

.

3. T = τ (Kr + Nw).

2.1.3 Aggregation

The equilibrium outcome for prices and output in this economy generally depends on the

entire distribution of assets and not just on the total capital stock. However, if the utility function

has a simple form—u(c, l) is homothetic—and consumers choices are interior, then only the total

capital stock matters. In other words, no matter how any given amount of capital is distributed in

the population, equilibrium prices and aggregate quantities are the same: there is “aggregation”.

The essential insight in the proof is simple, and the algebra that represents the following discussion will be left to the reader. Under homotheticity, independently of a consumer’s wealth

level, the marginal rate of substitution between the two goods—consumption and leisure—is a

8
function only of the ratio of the two goods (homogeneity) or of the ratio of two given affine

functions of each of the goods (homotheticity).2 When the consumer equalizes the marginal

rate of substitution (assuming interiority) to the net-of-tax relative price, consumption can be

expressed as an affine function of leisure, and the associated coefficients depend on prices but

not on wealth. Therefore, when budget balance is imposed, it follows that (i) leisure as well as

consumption must respond linearly to wealth and thus that (ii) the associated marginal propensities (coefficients in the linear expression) are the same for all agents, though of course they

depend on prices. Given that the resulting demand functions have the same marginal propensities to consume for all agents, any distribution of a given amount of capital leads to the same

total demand. This demand function can thus equivalently be derived from the assumption that

there is only one type of agent in the economy: an agent with utility function u and total wealth

K. In conclusion, given any equilibrium {(w, r),(K, N, T,(Ci, Ai, Li)i)} we can find another

equilibrium {(w, r),(K, N, T,(Cˆ

i

, Aˆ

i

,Lˆ

i)i)}, with P

i μiAˆ

i =

P

i μiAi = K (and, of course,

P

i μiCˆ

i =

P

i μiCi as well as P

i μiLˆ

i =

P

i μiLi). In all the examples we consider, we use

logarithmic utility—utility is a weighted average of log c and log l—which satisfies homotheticity.

Given the aggregation theorem, we can define competitive equilibrium outcome functions for

exogenous tax rates, namely the functions C ̃

i(K, τ ), L ̃i(K, τ ), N ̃i(K, τ ), N ̃(K, τ ), and T ̃(K, τ ).

These functions will be used below.

2More generally, aggregation applies if the marginal rate of substitution is a function of the ratio of two affine

functions of the consumption vectors.

9
2.2 Endogenous taxes and aggregation: politico-economic equilibrium

In order to focus as much as possible on the economics implied by the endogeneity of taxes, we

will simply assume that the “median consumer” is the politically pivotal one. We will, moreover,

take the median consumer to be the consumer with median wealth holdings. Later in this section,

we will briefly discuss whether these assumptions are warranted, i.e., whether there is an explicit

voting game with an equilibrium where these assumed properties are endogenous outcomes. We

can now proceed to the definition of a median-voter equilibrium. We use subscripts m to denote

the median agent.

Definition 2 Given (Ai)i, a median-voter equilibrium is a τ

∗ ≤ 1 and an associated competitive equilibrium {(w

∗

, r∗),(K, N∗, T∗,(C

∗

i

, Ai, L∗

i

)i)} such that there is no other τ ≤ 1 and

associated competitive equilibrium {(w, r),(K, N, T,(Ci, Ai, Li)i)} for which u(Cm, Lm) >

u(C

∗

m, L∗m).

Suppose that u is homothetic. Then given a vector (Ai)i, can a median-voter equilibrium, or

more precisely its prices and aggregate quantities, depend on anything but Am and K? If all

equilibria are characterized by interior solutions for all agents, the answer is no. Formally, one

needs to show that for any median-voter equilibrium associated to a vector (Ai)i, one could construct another median-voter equilibrium with identical prices and aggregate quantities associated

with any other vector (Aˆ

i)i as long as Aˆm = Am and P

i μiAˆ

i =

P

i μiAi = K. But it is now

straightforward to check that this must be the case, because (i) for any given K, the set of prices

and aggregate quantities in any equilibria have to coincide, and thus (ii) median utility has to be

identical across these equilibria, since prices are the same and the median agent’s wealth is the

same in the two cases. We summarize this finding as follows.

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Claim 3 Suppose that u is homothetic and that in any competitive equilibrium all agents’ solutions are interior. Then given values for Am and K, prices and aggregate quantities in a

median-voter equilibrium are independent of (Ai)i.

We can therefore define politico-economic equilibrium outcome functions as functions of

(K, Am): τ = Ψ(K, Am), N = N(K, Am), and T = T(K, Am).

3 They satisfy N(K, Am) =

N ̃(K, Ψ(K, Am)) and T(K, Am) = Ψ(K, Am)[r(Ψ(K, Am))K+w(Ψ(K, Am))N ̃(K, Ψ(K, Am))].

We now briefly discuss whether a median-voter equilibrium can be supported using majority

voting. First, note that in any equilibrium consumers differ only in one dimension: asset wealth.

Second, we restrict attention to the case that delivers aggregation, i.e., homothetic utility. Now

consider the indirect preferences over taxes of any agent:

u

3

C ̃

i(K, τ ), L ̃i(K, τ )

 ́

= u (bc(K, τ ) + dc(K, τ )Ai, bl(K, τ ) + dl(K, τ )Ai),

given homotheticity; it is crucial here to note that homotheticity implies that the coefficients

(bc, bl, dc, dl) do not depend on i. Hence, the derivative with respect to τ reads

u1i

μ

∂bc

∂τ + Ai

∂dc

∂τ ¶

+ u2i

μ

∂bl

∂τ + Ai

∂dl

∂τ ¶

.

This expression is positive if and only if

u1i

u2i

μ

∂bc

∂τ + Ai

∂dc

∂τ ¶

+

∂bl

∂τ + Ai

∂dl

∂τ > 0.

3Similarly, we can define Ci(K, Am) and Li(K, Am).

11
Noting that u1i/u2i

is equal to w(1 − τ ) for all i, because it has to be in any competitive

equilibrium, this expression is affine in Ai, with coefficients that are the same for all agents (and

that depend on prices and taxes). In particular, this means that if the agent with median asset

holdings has the first-order condition for taxes met with equality, so that he is indifferent in terms

of changing the tax rate at the margin; anyone poorer will prefer a higher tax and anyone richer

would prefer a lower tax. More generally, if an agent prefers a marginal tax increase over an

unchanged tax (in which case the inequality above is met), then so does anyone with a lower (or

higher, depending on the sign of w(1 − τ )

∂dc

∂τ +

∂dl

∂τ ) value for Ai

. Thus, half (or more) of the

population would support the median’s preferred choice over any alternative tax choice.

2.3 Analysis: the median voter’s first-order condition

In this model, there are several kinds of costs and benefits of taxing for the median voter.

On the cost side, income taxes generate well-known distortions to the decisions of the agents,

and these are taken into account by the median voter. The distortion to the median agent’s laborleisure decisions obviously matters directly to the median voter, but the distortions to the behavior

of other agents matter as well, though indirectly. In particular, the latter influence the provision

of inputs, which influence prices, and it is for this reason that the median agent cares about them.

This situation is parallel to that considered in Meltzer and Richard’s (1981) economy, though we

are casting the discussion more directly in terms of distortions here.

On the benefit side, the median agent seeks to use the gap in wealth between himself and the

mean agent to obtain transfers. In addition, the effect of taxation is to alter prices of capital and

labor, which also influences the median agent: he may be more or less dependent on one of the

12
sources of income than the mean agent, and thus would stand to benefit from some amount of

price distortion. We will discuss in turn each of the different effects of taxation that are relevant

to the median voter.

Given the median-voter theory, we can now derive the median voter’s first-order condition,

which is a necessary condition for a median-voter equilibrium (assuming that τ = 1 will not be

chosen; this will be guaranteed in the examples we look at). The median voter’s indirect utility is

given by

u

3

C ̃m(K, τ ), 1 − N ̃m(K, τ )

 ́

,

where

C ̃m(K, τ ) = 3r(K, τ )Am + w(K, τ )Nm(K, τ ) + τ

n

r(K, τ )[K − Am] + w(K, τ )[N ̃(K, τ ) − N ̃m(K, τ )]o ́

using r(K, τ ) ≡ Fk(K, N ̃(K, τ )) and w(K, τ ) ≡ Fn(K, N ̃(K, τ )). Taking derivatives with

respect to τ , we can write the resulting expression symbolically as

GAPn

dN ̃m

dτ + GAPred = 0.

The expression says that the optimal tax rate is chosen so that the weighted sum of distortions

(“gaps” for the labor-leisure choice and for optimal redistribution) created by such a policy is

equal to zero. The weights simply involve the induced changes in behavior, i.e., changes in labor

supply.

More precisely, GAPn is associated with the pure distortion on labor supply to the median

13
voter:

GAPn ≡ wu1m − u2m.

From the first-order condition of the consumer, this gap is zero if and only if taxes are zero;

moreover, in a Pareto optimum this gap must be zero, since w is the marginal product of labor.

The other gap in the median voter’s first-order condition is GAPred, which measures how an

increase in the marginal tax raises “redistribution”, and thus utility. It reads

GAPred = ucm1⁄2r (K − Am) + w

3

N ̃ − N ̃m

 ́

| {z }

direct

+

τ



w

d

h

N ̃ − N ̃m

i

dτ +

dr

dτ [K − Am] + dwdτ

h

N ̃ − N ̃m

i



 +

dr

dτ Am +

dw

dτ N ̃m







| {z }

indirect

.

There are two sorts of redistribution that take place. One is through the change in the direct net

redistribution that occurs. The direct net redistribution is

τ

n

r(τ ) [K − Am] + w(τ )

h

N ̃(τ ) − N ̃m(τ )

io .

It is influenced by taxes both through direct changes in demands from raising τ—the effects

referred to as “direct” in the expression—and via changes in prices—the remaining “indirect”

effects except for the last two terms.

Therefore, the median agent sees a net direct gain from taxation if he has lower asset holdings

than the mean agent has. We will, in line with all available data, indeed assume that median asset

holdings are lower than average asset holdings. Moreover, the mean agent is richer also in an

14
overall wealth sense, since he only differs from the median agent in his asset holdings (recall that

labor productivity, and thus the value of the sequence of time endowments, is equal among agents

in the benchmark model). Therefore, if leisure is a normal good, he would buy more leisure and

therefore work less than the median agent. This result means that the second direct effect of

taxation is detrimental to the median agent: he loses, on net, by redistribution of labor income.

The indirect effects include a standard, Meltzer-Richard channel: increased redistribution lowers the gap between the median and mean labor supply, because it moves the net-present-value

wealth of the two agents closer to each other. In this case, this is an effect in the median’s favor,

because the labor redistribution channel works against the median.

The indirect effects also include price effects. Here, a tax increase in the current period will

not affect the total capital stock, but it will reduce work effort, leading to a lower rental rate and a

higher wage. The median views both of these negatively: the lower rental rate is negative because

his asset holdings are lower than mean asset holdings, and the higher wage rate is also negative

because median labor supply exceeds mean labor supply.

Finally, the last two terms come from a second form of redistribution that occurs: through

changes in the composition of income due to price changes. Even in the absence of transfers,

a tax increase would lower N and thus increase w (and, given that we use a Cobb-Douglas

production function in our application, decrease r). Thus, consumers whose income has a larger

wage share in relative terms see an increased relative income share. This is the case for a median

consumer with less than average assets: such a consumer has below-average capital income and

above-average labor income, since poorer consumers buy less leisure and therefore work more.

An agent with mean wealth obtains no gain at all from the change in income composition.4

4The proof of this statement is as follows. The derivative of wN + rK = Fn(K, N)N + Fk(K, N)K with

15
It is apparent from the expression that if Am = K, i.e., if the median consumer has wealth

that exactly matches mean wealth, the first-order condition is met for a zero tax: the labor-leisure

distortion is minimized at this point, there is no change in the net transfer from changing the tax,

since the net transfer is always zero in this case, and finally, as argued in the previous paragraph,

there is no gain (or loss) to agents with mean asset holdings from changing the composition of

income through price changes.

2.4 Numerical results for the example economy

We now consider an example: u(c, l) = α log c + (1 − α) log l and Cobb-Douglas production

with capital share θ.

2.4.1 Exogenous taxes

Consider competitive equilibria for different exogenous tax rates. In such case, we can find

total labor supply in closed form:

N ̃(τ ) = α(1 − τ )(1 − θ)

1 − α + α(1 − τ )(1 − θ)

.

Moreover, one can show that

C ̃m(τ ) = N ̃(τ )

1−θKθ

((1 − τ )(1 − αθ(1 − x)) + τ ),

respect to N equals both FnnN + Fn + FknK and, due to Euler’s theorem since total factor income equals total

production for a production function that is homogeneous of degree 1, Fn. This, in turn, means that FnnN + FknK,

i.e., the change in total income for the agent with mean asset and labor income only taking price effects into account,

must equal zero. Thus, since a tax change operates through the change in N, we have the desired result.

16
where x ≡ Am/K, whereas

N ̃m(τ ) = N ̃(τ )

μ

(1 − α)

1 − θx

1 − θ

+ α

¶

.

As for consumption and labor supply of agents with assets Ai, the same formulas apply, with

xi ≡ Ai/K in the place of x.

Notice that these expressions reflect interiority of all choices; corner solutions will never be

optimal here unless a consumer cannot obtain positive consumption even while working all the

time (setting leisure to zero). For an equilibrium to exist, therefore, the poorest consumer has

to have enough net-of tax wealth. As is clear from the expression above, for consumption (and

leisure) to be positive, we need (1 − τ )(1 − αθ(1 − x)) + τ > 0, where x ≡

mini Ai

K

. Thus,

no matter what x is, there is always a low enough tax rate for which this expression is violated.

Given any asset distribution, this puts a bound on the set of feasible tax rates:

τ > τ ≡ 1 −

1

αθ(1 − x)

.

Notice, thus, that although aggregation holds for all feasible values of the tax rate in this economy,

there is another kind of distribution dependence: whether an equilibrium exists or not depends

on the distribution of wealth, and not just on average wealth. More generally, if preferences were

such that corner solutions could apply, aggregation would break down at the point where one or

more types of (poor) agents chooses a corner for leisure: at that point, the marginal propensity to

work from decreasing taxes is 0, whereas the agent with mean wealth will have a strictly positive

propensity to work.

17
2.4.2 The median voter’s choice

First, note that since (i) consumption (of the median agent) is proportional to Kθ, (ii) labor

supply is a function of the ratio x of median to mean income, and (iii) utility is logarithmic, the

objective function for the median voter in terms of its choice variable, τ , does not depend on K:

the level of wealth factors out.5 This means that the tax choice will depend on x only. We will use

ψ(x) to denote this function ψ(Am/K) = Ψ(K, Am) whenever these assumptions are employed.

The first-order condition for the median can be expressed as a second-order polynomial equation in the unknown: τ , and since one of the resulting solutions violates non-negativity of consumption or leisure, it can be ignored throughout the analysis. Looking first at the case x < 1,

i.e., the empirically relevant case where median wealth is less than mean wealth, so that the median voter is poorer than average, we note that a positive tax rate is called for: it is strictly better

with a small positive tax than with a zero tax. A slight deviation from a zero tax rate to a positive

rate gives no first-order distortion loss, whereas it gives a first-order utility gain from the transfer

it induces and from the change in income composition it leads to. A hundred per cent tax rate, on

the other hand, is too high, because it leads to zero production and thus zero consumption for the

median consumer. Thus, an interior, positive tax rate will be chosen, and it will be the one that

solves the first-order condition stated.

For the case where the median consumer is richer than the mean, x > 1, we have the slightly

odd outcome that the lower bound on taxes may bind. To see this, notice that when x > 1, median

consumption as a fraction of mean output goes to infinity as the tax rate goes to minus infinity

5Had we included in total resources a stock of undepreciated capital that could be “eaten directly”, without

involving any labor input, this result would not have been possible to establish. In fact, in the multi-period versions

of the present model, the assumption that depreciation is 100% precisely fills the role of making the level effect on

taxes disappear.

18
(ignoring the lower bound on the tax rate). With large subsidies to working, the working hours of

the agent with mean asset holdings approach 1 in this case, with leisure going to zero. Because

x > 1, however, the formula for median working hours reveals that median leisure stays strictly

bounded below and away from 0, so median utility would increase without bound as taxes go to

minus infinity. Thus, were the median to be able to choose an infinite subsidy rate, he would.

This is the case where a corner solution may apply: taxes are not allowed to be below τ . Thus,

whether this lower bound is indeed the solution or an interior solution gives the highest utility

depends on parameter values; we found straightforward examples of both.

Figure 1 depicts the equilibrium tax as a function of mean to median asset holdings for two

alternative values of the capital share, θ; we set α at 0.3. The graph reveals that the tax rate is

indeed decreasing in x: the lower is median wealth relative to mean wealth, the higher is the

equilibrium tax rate. It also suggests that the equilibrium level of income taxes, for any wealth

distribution, increases with the capital share. This can be understood by analyzing the median’s

first-order condition. As θ increases, capital income is relatively more important in the agents’

budget constraints. Keeping in mind that the stock of capital is given and completely inelastic

in this static economy, taxes are perceived as being less distortionary when θ is high. The labor

gap is thus smaller for any given tax rate and net redistribution larger, inducing the median so

set taxes at a higher level. In other words, when the capital share is high, the median can extract

resources from richer agents at a low cost in terms of foregone labor income.

2.4.3 Concluding comments on existence

The nonexistence of a competitive equilibrium under exogenous taxes is not surprising: though

the exact conditions for existence are not entirely trivial to find, in essence nonexistence arises

19
0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

τ(Am/K)

Am/K

τ

θ=0.05

θ=0.4

Figure 1: Taxes as a function of the median to mean wealth

because the initial wealth distribution is too unfavorable to the poorest agents to allow positive

consumption. However, what about a median-voter (political) equilibrium: are there similar

existence problems? We will not discuss this issue here in detail, but in essence the answer is no.

The argument is that one can construct a set of tax rates that is nonempty and compact, and as

long as the median voter’s utility is continuous in the tax rate, a solution must exist to his choice

problem. Continuity is evident in the example economy due to the functional forms assumed.

Non-emptiness means that there are always tax rates that are feasible to choose for the median

agent—that are associated with competitive equilibria. In particular, we always assume that a tax

rate of 0 is feasible—a laissez-faire allocation is always feasible. Furthermore, the choice set is,

or can be made, closed and bounded.6

6Note that τ would lead to a maximum utility of minus infinity for the poorest agent. Therefore taxes that are

associated with the existence of competitive equilibrium need to be strictly greater than τ . Similarly, the choice

τ = 1 is not, strictly speaking, consistent with competitive equilibrium. Thus, the set of taxes that are associated

with competitive equilibria is an open set. The upper limit point will never be chosen by a median voter, and one can

therefore without loss of generality impose a slightly lower bound than 1. The lower limit point, in contrast, would

often be chosen, but in this case it does not seem restrictive to arbitrarily raise the lower bound ever so little in order

to render the set closed.

20
We will again note the possibility of existence problems in the 2-period model when taxes are

given exogenously, now for different reasons.

3 The 2-period model

The 2-period environment is a straightforward extension of what was described above. There

is production in both periods, with the same technology, and capital can be accumulated in the

usual neoclassical fashion. First-period output can be either consumed directly or installed as

second-period capital. Feasibility in this economy is summarized as follows.

I1 +

X

i

μiC1i = F(K1, N1),

K2 = I1,

X

i

μiC2i = F(K2, N2),

Nt = 1 −

X

i

μiLti, t = 1, 2

Cti, Lti ≥ 0, t = 1, 2; all i.

Notice that we are assuming full depreciation, i.e., that capital takes one period to install and

then cannot be used again.7 Utility is assumed to be time-additive and stationary for simplicity:

u(c1, l1) + βu(c2, l2).

The competitive environment will be discussed next.

7

It is conceptually straightforward to consider the more general case of less than full depreciation; the reason

we do not look at it here is that all the examples (numerical and analytical) rely on this assumption, since it admits

closed forms.

21
3.1 Exogenous taxes in the first period

In the model with fully endogenous taxes presented below, we will assume that taxes in period

t are voted on in period t, for t = 1, 2. That is, in period 1, the median voter cannot commit to

a period-2 tax rate. This assumption means that in period 1, the median voter has to consider

two effects of his choice of a τ1: (i) the effect on current utility and (ii) the effect on capital

accumulation and, thus, on period-2 utility. The second of these effects, moreover, involves how

τ2 will change in response to a change in the period-2 state variable induced by the choice of τ1

in period 1.

We will immediately restrict attention to a homothetic utility function u(c, l), as above, since

this allows us to write the second-period outcome for tax rates as a function of K2 and A2m

only: τ2 = Ψ2(K2, A2m). The function Ψ2 thus summarizes the endogenous tax-determination

mechanism described in Section 2.2. Similarly, we can summarize utility outcomes in the second

period with value functions where the aggregate state is (K2, A2m), and we can summarize aggregate equilibrium labor decisions with N2(K2, A2m), and aggregate equilibrium transfers with

T2(K2, A2m), as defined above.

We begin by discussing the 2-period economy with an exogenously given first-period tax rate

τ1, along with the given initial asset distribution, (A1i)i. Thus, we can define a competitive

equilibrium in the two-period model recursively: we define equilibrium for any given τ1 and an

endogenously determined τ2.

Definition 4 Given a policy τ1, a competitive equilibrium for period 1 of a 2-period economy is

a set of prices (w1, r1) together with an allocation (K1, N1, T1,(C1i, A1i, Li, A2i)i) satisfying the

following conditions.

22
1. For all i, (C1i

, L1i, A2i) solves

max

(c,l,a0)∈B1(A1i)

u(c, l) + βV (a

0

, K2, A2m)

where K2 =

PI

i=1 μiA2i

,

B1(A1i) ≡

©

(c, l, a0) ∈ R

2

+ × R : c + a

0 = [A1ir1 + w1(1 − l)](1 − τ1) + T1

a

,

and V (a

0

, K2, A2m) is the appropriate indirect utility function for period 2, i.e.,

V (a, K, Am) ≡

max

(c,l)∈R2

+

u([aFk(K, N2(K, Am))+(1−l)Fn(K, N2(K, Am))](1−Ψ2(K, Am))+T2(K, Am), l),

where N2 and T2 are equilibrium functions, as solved for in a 1-period economy.

2. w1 = Fn(K1, N1) and r1 = Fk(K1, N1), where N1 =

PI

i=1 μi(1 − L1i) and K1 =

PI

i=1 μiA1i

.

3. T1 = τ (K1r1 + N1w1).

3.1.1 Aggregation

Aggregation in this economy holds if present-value utility is homothetic jointly in all the

goods that the consumer derives utility from: (c1, l1, c2, l2). For this to be true, it is not sufficient that u be homothetic in (c, l). However, most of the utility functions employed by applied macroeconomists satisfy the required criterion. In particular, if period utility is a power

23
(including logarithmic, or exponential, or quadratic) function of a function of (c, l) that is homogeneous of degree 1, then the criterion is satisfied. In our example economy below, we use

u(c, l) = α log c + (1 − α) log l, which is in this class. This formulation leads to a recursive

formulation where consumers can be regarded as deriving utility in period 1 from three goods:

period-1 consumption and leisure and assets left for period 2, (c1, l1, a2), where the assets indirectly give utility through their provision of second-period consumption and leisure. Under the

stated assumptions, V will be such that present-value utility is homothetic in (c1, l1, a2).

8 Therefore, with the same kind of argument as was used for the 1-period model, we can deduce that

equilibria will not depend on the distribution of capital but merely on its mean K1 and on its

median A1m. The reason why A1m matters is that it influences A2m, and hence period-2 taxes.

As a result of these facts, we obtain period-1 equilibrium outcome functions that parallel

those we defined for the 1-period model: C ̃

1i(K1, A1m, τ1), L ̃1i(K1, A1m, τ1), N ̃1i(K1, A1m, τ1),

N ̃

1(K1, A1m, τ1), and T ̃1(K1, A1m, τ1). We also obtain, in this case, the equilibrium outcome

functions for savings: A2i = H ̃

i(K1, A1m, τ1).

3.2 Politico-economic equilibrium

We can now state the following definition.

Definition 5 Given (A1i)i, a median-voter equilibrium in the 2-period economy is

• a τ

∗

1 ≤ 1 and an associated competitive equilibrium {(w

∗

1

, r∗

1

),(K1, N∗

1

, T∗

1

,(C

∗

1i

, A1i, L∗

1i

,

A∗

2i

)i)} such that there is no other τ ≤ 1 and associated competitive equilibrium {(w1, r1),

(K1, N1, T1,(C1i, A1i, L1i, A2i)i)} for which u(C1m, L1m)+βV (A1m, K2, A2m) > u(C

∗

1m, L∗

1m)+

8The algebraic proof of this claim is straightforward.

24
βV (A∗

1m, K∗2

, A∗

2m); and

• a τ

∗

2 which is a median-voter equilibrium, as defined above, for (A∗2i

)i.

Again, this definition is a general one that applies for any period utility function u, and thus in

general politico-economic outcomes could depend nontrivially on the whole initial distribution of

asset holdings. Fortunately, if u has certain properties, we can extend our aggregation result from

the 1-period model—that outcomes depend only on mean and median wealth—to the 2-period

model by use of recursive methods. In particular, given that the result applies in the last period,

and given intertemporal preferences in a certain class, the proof techniques for the last period can

be used for the period immediately prior to it, and so on. So define

Assumption 6 Suppose that u(c, l) = f1(f2(c, l)), where f1 is a power function, logarithmic,

exponential, or quadratic and f2 is homogeneous of degree 1.

We thus have the following.

Claim 7 For the 2-period model, suppose that u(c, l) satisfies Assumption 6 and that in any

median-voter equilibrium all agents’ solutions are interior. Then with A1m and K1 fixed, prices

and aggregate quantities in a median-voter equilibrium are independent of (A1i)i.

The arguments underlying the proof can be used for models with any time horizon, including

an infinite one. For infinite-horizon models, one can also imagine that other kinds of equilibria

exist, namely those where other moments of the asset distribution matter due to “self-fulfilling

expectations.” We are not aware of models where such equilibria in fact do exist, however.

Aggregate and median outcomes in the 2-period median-voter equilibrium are summarized by

the functions Ψt(K, Am), Nt(K, Am), Ntm(K, Am), H(K, Am), Hm(K, Am), and Tt(K, Am),

25
for t = 1, 2. These are defined similarly to those in the 1-period model, e.g., Hm(K, Am) =

H ̃ (K, Am, Ψ1(K, Am)).

The definition of a median-voter equilibrium in the 2-period model is stated in terms of sequences. Alternatively, an equilibrium can be defined directly as a set of functions. The infinitehorizon equilibrium definition below will be stated in terms of functions only, and it can be

thought of as the limit of outcome functions dated t obtained from a sequence of finite-horizon

models as those described here, using the same recursive technique, i.e., solving backwards from

the last period.

Finally, though not in focus here, one needs to verify that the median-voter theorem also

applies in the 2-period model. We will not develop this argument in detail, but it follows the idea

behind the argument in the 1-period model: in consumption unit terms, if someone prefers to

increase the tax rate slightly, so does anyone with a lower asset level, due to the linearity in Ai

that follows from the functional forms we employ.

3.3 The median voter’s first-order conditions

The problem faced by the second period’s median voter is analogous to the one presented for

the one-period economy. Since the economy only lasts two periods, and given the levels of K2

and Am2, the trade-offs faced are exactly the same than those faced in the static case. We can

solve for the implied tax function in period two by using the GEE presented in section 2.3, which

as discussed above will take the form Ψ2(K2, Am2).

The median voter in the first period chooses taxes taking into account how the winner of

the next election—his future self in this case—will choose taxes tomorrow (the Ψ2 function).

26
Therefore, when finding the optimal level for τ1, he must consider how this will affect current

period savings (summarized by the functions H ̃m and H ̃ ), which by modifying the level of assets

that the next incumbent inherits will influence next period’s economic as well as political (tax)

outcomes.

Like in the one-period economy, the median voter will trade off distortions away from the

first-best—gaps—that are introduced by redistributive policies. There are a larger number of

distortions in a 2-period economy than in a 1-period economy. In particular, a wedge is introduced

in the first period’s savings decision. The final result is a first-order condition for the median voter

in the first period—a “generalized Euler equation”, or GEE—that can be written as a weighted

sum of gaps that involve wedges in both periods:

GAP1n

dN ̃

1m

dτ1

+ GAPred + GAPa

0

1m

dH ̃m

dτ1

| {z }

t=1

+ β

"

GAP2n

dN ̃

2m

dτ1

+ GAP2red#

| {z }

t=2

= 0

The derivation is straightforward. An increase in the tax rate (τ1) will, first, decrease labor

supply in the first period which has a per-unit cost of GAP1n: the static labor-leisure distortion

(recall that the gaps are distortions from the perspective of the median agent). Second, it has a

static redistributive gain like that discussed in the context of the 1-period model: GAP1red. Third,

by changing savings—since less time will be spent working and savings under our assumptions

will be increasing in income/wealth: dH ̃m

dτ1

< 0—there is also an intertemporal distortion, since

the presence of 2nd-period taxes on total income will distort savings in the direction of being too

low: GAPa

0

1m

, defined as ut1 − βut+1,1rt+1, all evaluated for the median agent and for t = 1.

Fourth, because the increase in τ1 will induce changes in assets and thus in second-period tax

rates, we also have static costs and benefits for the median voter in period 2. The GAP2red is

27
GAP2red = ucm2

1⁄2

dΨ2

dτ1

h

r2

3

H ̃ − H ̃m

 ́

+ w2 (N2 − Nm2)

i

| {z }

direct

+

Ψ2

"

w2

d [N2 − Nm2]

dτ1

+

dr2

dτ1

h

H ̃ − H ̃m

i

+

dw2

dτ1

[N2 − Nm2] + r2

Ã

dH ̃

dτ1

−

dH ̃m

dτ1

!# +

dr2

dτ1

H ̃m +

dw2

dτ1

Nm2

)

| {z }

indirect

.

Changes in τ1, by affecting savings in the first period, trigger changes in second period taxes

(since the optimal choices of the government at that point depend on the states inherited). The

median voter in period one realizes that this will cause a direct effect on redistribution next period.

The direct effect is then the change in net redistribution (keeping asset holdings constant) due to

induced changes in future taxes.

The first three terms in the indirect effect are analogous to those in the static one period economy: there is a decrease in the labor gap between the mean and the median, plus an effect through

changes in prices. The fourth term appears because next period’s asset holding is elastic, which

in this case is a negative effect of raising current taxes (savings of the median and the mean move

closer to each other, thus lowering the net transfer to the median). The last two terms, as in the

one period case, reflect changes in the median’s current income due to changes in prices.

3.3.1 Tax manipulation

The lack of commitment and the fact that only current taxes can be chosen implies that the

government today perceives tomorrow’s capital to be more elastic than what future governments

will. Hence, it has an incentive to strategically influence next period’s taxation decision. This

effect appears in the GEE but is only visible indirectly: by changing the current tax rate, the

current median influences savings, and thus future taxes, in order to change current expectations

28
of consumers. In this class of models, time inconsistency of fiscal policy is limited to the effect

on private agents’ expectations: if these expectations (which determine current behavior) could

be controlled separately, there would be no disagreement between the current and the future

governments/median voters, i.e., conditional on arriving at a state (A0

m, K0

) tomorrow, there is

no remaining disagreement. As an example, taxes on capital income next period lowers savings

this period. If the government could affect expectations today so as to make consumers believe

that taxes next period will be very low, that would be desirable: it would increase savings and

reduce the distortion. However, the assumption that the consumers’ expectations are rational

is a constraint on the government, and in this model it can thus only partially influence these

expectations by altering the current tax, and hence future variables, in the direction of making the

distortions less costly.

3.4 The 2-period example economy

We consider the same parametric example as in the 1-period model, with the only new parameter being β, the discount factor.

3.4.1 Exogenous taxes in the first period: preliminaries

For the second-period outcomes, we inherit all the results from the earlier analysis. In the first

period, we obtain savings that satisfy

K2 = h2Kθ

1N

1−θ

1

A2m = h2mKθ

1N

1−θ

1

,

29
where thus h2 and h2m are the fractions of total first-period output saved by the average agent

and the median agent, respectively, and where

N1 =

α(1 − θ)(1 − τ1)

(1 − α)(1 − h2) + α(1 − θ)(1 − τ1)

.

It is straightforward to solve for h2 given τ2:

h2 =

βθ(1 − τ2)

1 + βθ(1 − τ2)

.

However, τ2 here depends on h2m/h2: it is endogenous. In particular, we know that τ2 =

ψ(h2m/h2). Furthermore,

h2m

h2

= 1 −

1 + βθ(1 − τ2)

1 + β

(1 − τ1)(1 − x1)

(1 − τ2)

,

where x1 stands for A1m/K1. That is, we have

A2m

K2

= 1 −

1 + βθ(1 − τ2)

1 + β

(1 − τ1)(1 −

A1m

K1

)

(1 − τ2)

= 1 − S + S

A1m

K1

,

where S ≡

1+βθ(1−τ2)

1+β

1−τ1

1−τ2

. Note now that if tax rates are (positive and) decreasing over time,

i.e., τ2 ≤ τ1, we find 0 < S < 1, so that inequality—as measured by the ratio of median to mean

asset holdings—must decrease over time.

Because ψ does not have a convenient closed form, there is no closed-form solution for h2 or

30
h2m/h2 here. One finds, in addition, that

N1m =

(1 − α)[h2m − h2] + (1 − τ1) [1 − θ (α + (1 − α)x1)]

(1 − θ)(1 − τ1)

N1.

What is actually possible to derive is the difference in savings:

h2m − h2 = −(1 − τ1)

βθ(1 − x1)

1 + β

.

Here, we see that the difference between median and mean savings does not depend on the period2 tax rate, so this is indeed a closed form. This expression can then be used in turn to solve, in

closed form, for

N1m

N1

=

(1 + β)(1 − θ(α + (1 − α)x1)) − βθ(1 − x1)(1 − α)

(1 + β)(1 − θ)

.

3.4.2 Exogenous taxes in the first period: existence and uniqueness

Reexpressing the solution for h2 above as a fixed-point problem, we see that we need to find a

solution to

LHS(h2) ≡ h2 =

1

A

1−ψ 1+ B

h2

+ 1

≡ RHS(h2)

in the interval [0, 1], where we recall that A = 1/(βθ) and B = (1 − τ1)

βθ(x1−1)

1+β

. Three cases

appear. We presume throughout what we found for the 1-period model, i.e., that ψ2 is decreasing.

The case with x1 = 1. Here, we have B = 0 and since ψ(1) = 0 (no taxation when the

median equals the mean), we obtain RHS(h2) = 1/(1 + A) = βθ/(1 + βθ) so that is what

savings must be.

31
The case with x1 > 1. In this case, first we need to deal with the issue of non-existence in

period 2. So suppose that we “resolve” that problem by, whenever the median voter would

want to choose a corner solution τ2 = τ , we define this as the outcome (even though strictly

speaking the poorest agent does not have a well-defined utility maximization problem in this

case). Notice here that τ actually depends on the period-2 wealth ratio between the poorest agent

and the mean agent, x1, which in turn is influenced by mean savings: the absolute (level) gap

is constant, so the higher the savings of the mean, the lower is the poor-to-mean asset ratio in

period 2. Specifically, we have that h2 − h2 = B ≡ (1 − τ1)

βθ(1−x1)

1+β

, where h2is the savings of

the poorest and x1is the period-1 ratio of the asset holdings of the poorest to those of the mean.

Since 1 − x1 = (h2 − h2)/h2 = (1 − τ1)

βθ(1−x1)

h2(1+β)

, this means that τ is decreasing in h2, because

it satisfies

τ = 1 −

1

αθ(1 − x1

)

= 1 −

1

αθ(1 − τ1)

βθ(1−x1

)

h2(1+β)

= 1 − h2 ·

1 + β

αβθ2(1 − τ1)(1 − x1

)

.

In fact, for h2 = 0, we must have x1 = −∞, which implies τ = 1: the only feasible tax rate in

order for the poorest to have non-negative consumption is a tax rate of 1. The implied expression

for RHS(h2), which is the value for RHS(h2) in the case the corner constraint for taxes binds

in the second period, is

RHS(h2) = 1

A

h2·

1+β

αβθ2(1−τ1)(1−x1)

+ 1

=

1

Aαβθ2(1−τ1)(1−x1

)

h2(1+β) + 1

.

This is an increasing concave function in h2 whose value is strictly less than 1. Moreover, it is

easy to see that its derivative at h2 = 0 equals 1+β

Aαβθ2(1−τ1)(1−x1) =

1+β

αθ(1−τ1)(1−x1) > 1, unless τ1 is

32
a large enough subsidy. This is an important feature: RHS(h2) starts out at zero with a derivative

above the 45-degree line, i.e., with RHS(h2) > LHS(h2) for small h2. Because RHS(h2) is

concave and less than 1 it then intersects LHS(h2) at a unique value of h2 less than 1 (this value

can be solved for analytically).

Next, consider the issue of when the corner constraint will actually bind in period 2. For small

enough h2s, it has to bind, because then inequality in the second period is extreme: we saw above

that the only feasible period-2 tax rate at h2 = 0 is 1. So then either it ceases to bind for some

h2 < 1, or it binds for all h2 ≤ 1. In the former case, we can see that, since a higher value for h2

decreases h2m/h2 = 1 + B

h2

(since B > 0), ψ(1 + B

h2

) must be increasing in h2 if the tax in the

second period is interior, so RHS(h2) is a decreasing function, and again there must be a unique

solution for h2. Thus, graphically, either of the following cases must apply.

1 1 1

0 0 0 1 1 1

RHS

RHS

RHS

RHS

RHS

LHS LHS LHS Figure 2: Three cases

Finally, here, one of course needs to check not only that period-2 consumption is positive for

the poorest consumer but that period-1 consumption is positive as well. However, because of

the form of preferences considered here, period-1 consumption is positive if and only if period-2

33
consumption is positive, and if and only if leisure (in either period) is positive.

The case with x1 < 1. Now, since B < 0, an increase in h2 must raise h2m/h2, and ψ(1 + B

h2

)

is consequently decreasing, leading RHS(h2) to be an increasing function globally. Here as well,

RHS(0) = 0, but for a different reason than in the x1 > 1 case: here, the median consumer is

infinitely poor relative to the mean, and chooses a tax equal to 100%; in the x1 > 1 case with

x = 0, the median is infinitely rich compared to the poorest person and then becomes restricted

to using a 100% tax rate, because no other tax rate leaves the poorest person with non-negative

consumption.9 Now the issue of existence of solutions with h2 ∈ (0, 1) is harder to explore. For

example, whether RHS(h2) is above or below LHS(h2) in the neighborhood of h2 = 0 depends

on what ψ2(x1) looks like there, and it may be that whether one curve is above the other depends

on parameters.

We can say something about how RHS(h2) depends on primitives, however. Suppose that τ1

falls. Then 1 + B

h2

falls for any given h2 (the median-mean gap increases), and ψ(1 + B

h2

) will

then increase. This leads RHS(h2) to fall. We can thus imagine a picture where RHS(h2) is

concave and has a unique intersection with LHS(h2), but there will be a point where τ1 is low

enough that RHS(h2) falls sufficiently that it is everywhere below the 45-degree line for h2 > 0.

As τ1 falls continuously in this case, the intersection point h2 moves toward zero and eventually

zero savings is the only “solution”. Figure 3 illustrates this possibility.

The intuitive reasoning behind non-existence here is as follows. As we have seen above, the

median will save less than the mean, and the lower is τ1, the larger is the mean-median difference.

This implies that next period, there is still significant inequality, which (as we know) will lead to

9The two cases are, in some sense the same: in both cases, the tax rate has to be 100% because of the poorest

person, and the only difference is that when x1 < 1, the median person is the poorest person, whereas when x1 > 1,

someone else is.

34
0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LHS and RHS

h

p

RHS,LHS

45o line

τ

1

=0.9

τ

1

=0.5

Parameters:

θ=0.9,

β=0.9,

α=0.9,

x

1

=0.1

Figure 3: Existence and τ1

a high tax rate τ2. This high tax will make the return on saving lower, the lower is τ1, and hence

saving will fall as τ1 is lowered. Ordinarily there would be a limit to this mechanism, because as

savings fall, the return to capital rises in period 2, thus inducing more savings, so that savings are

bounded away from zero. The return does increase here as well, but a rise in the returns to capital

can also have another effect: it can make the median agents poorer in period 2. The reason is

that when mean savings are close enough to zero, median savings have to be negative—see the

formula for the difference above—and then a high interest rate means that the debt service is

more costly for the median agent, thus lowering his before-tax income. The lower is τ1, the more

savings fall, and for a low enough τ1, there is no period-2 tax rate that can maintain positive

consumption for the median agent. As τ1 falls, thus, we see mean savings fall toward zero, along

with a rise in τ2 toward one, consumption of both consumers going to zero in period 2, along with

work effort.

All of this is a political-economy effect: wealth inequality in the second period rises as the

35
before-tax return to capital rises, translating to a higher tax rate on savings via popular vote. In

fact, as mean savings go to zero, period-2 wealth inequality, as measured by the ratio of mean

to median asset income, becomes infinite, since the interest rate the median agents have to pay

goes to infinity, and their borrowing is strictly positive and bounded away from zero. The rise in

taxes overtakes the rise in the before-tax return to capital leading to a net return to savings equal

to zero. The reason for this effect is that the tax going to one also makes the labor input go to

zero, independently of what the level of capital is.

Fundamentally, the mechanism that leads to non-existence of a competitive equilibrium has to

do with the period-2 tax rate being endogenous and not committed to in advance; in particular,

we think of a situation without commitment where the endogeneity of the tax rate is given by

what the government will choose in period 2.

A final interesting issue is whether the solution is always unique here, which it is in the case

with x1 > 1. What ensures, for example, that RHS(h2) is concave? The simple parametric

model we are exploring here turns out to be well-behaved for all the parameter configurations

we have explored. But more generally it seems hard to rule out cases with two solutions for

h2 in (0,1). The idea is easy to express intuitively: if there is an expectation of higher taxes

tomorrow, all savings fall. But because savings of all consumers fall, and by a similar amount

(because the difference between savings of different groups do not depend on the tax tomorrow),

wealth inequality measured as the ratio of median to mean wealth will increase, justifying the tax

tomorrow being higher. Or, if we think taxes will be very low, savings are high, leading to lower

wealth inequality in the future, consistently with expecting a lower tax rate, i.e., we would have

“self-fulfilling expectations”.

36
3.4.3 The median voter’s choice in the first period: decreasing tax rates

Finally, we report what the median voter would choose among all feasible values for τ1, i.e.,

for values for τ1 for which there is an equilibrium in the 2-period economy, taking into account

the endogeneity of taxes in the second period. Of course, the choice set here is smaller than in

the case with commitment, not just because τ2 is not subject to choice, but because a narrower

range of values for τ1 is available: there are values for τ1 that, together with specific choices of τ2,

would lead to equilibria, but would not be consistent with an equilibrium when τ2 has to be taken

as given through ψ2. We find an equilibrium using both first-order conditions and with global

search, and the following graphs describe our findings for the parameter configuration α = 0.3,

β = 0.9, and θ either equal to 0.4 or to 0.05.

0 0.5 1 1.5 2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

τ1(x1) for θ=0.4 and θ=0.05

x

1

τ1

θ=0.05

θ=0.4

Figure 4: Taxes as a function of the median to mean wealth

We see from Figure 4 that, as in the 1-period model, the tax rate is globally decreasing in the

median-mean wealth ratio. We also see that the (absolute) rate of taxation is higher the more

important capital is in production: θ = 0.4 leads to much larger tax rates than θ = 0.05. Of

course, as θ is increased, the share of labor, and the costs of distortions, also decrease, making it

37
less costly to tax.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

τ1(x1) and τ2(x1) for θ=0.4

x

1

τ1 and τ2

τ2

τ1

Figure 5: Tax rates in the two periods

Figure 5 shows the tax rates in the two periods as a function of the initial median-to-mean

asset ratio. It shows that taxation (assuming that median income is below mean income) is lower

in period 2 than in period 1. The source of this finding is not just the finite time horizon, but also

the fact that capital income is inelastic. Under commitment, clearly, the rate of taxation should

optimally—from the perspective of “optimal redistribution to the median voter”—decrease over

time, since it is costlier to tax in the future than in the present. Here, the median voter cannot

commit to the future tax rate, but on the other hand an increase in the current tax rate would

automatically, by reducing inequality, lead to a lower future tax rate. Thus, the median voter

can still enact a decreasing tax path. Thus the equilibrium features heavy initial taxation and

subsequent lower inequality and less taxation.

Figure 6 illustrates the resulting path of inequality. The solid line in the figure depicts the

mean-median ratio in the second period as a function of the initial ratio, while the dotted one is

just the 45 degree line. Due to the decrease in inequality, there will indeed be less taxation in

the future: more current taxation leads to less taxation in the future. This result is an important

38
0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

2

(x1) for θ=0.4

x

1

x

2

x

2

(x1)

45o line

Figure 6: Median-mean wealth ratio tomorrow as a function of that today

insight that will have implications for the infinite-horizon setup: though in principle a range of

different wealth distributions are possible as steady states in that model—each one associated

with a different tax rate—this range is limited: with significant initial inequality, there will be

high initial taxes and a decreasing path of taxes and inequality, leading to a limited set of longrun values for the median-mean wealth ratio. In Figure 6, for example, we see that even if the

median agent is infinitely asset poor relative to the average agent, A1m/K1 = 0, the same period2 ratio will be over 0.45.

4 The ∞-period model

The infinite-horizon version of the model is an important extension because it allows us to ask

questions about the long-run level of taxes and inequality, i.e., just in the way the neoclassical

growth model and its balanced growth path, or steady state, is used for describing, say, the postwar path of U.S. output and capital accumulation, so can we use the extension of the same model

to discuss the forces behind taxation and, in turn, inequality. In particular, one can make quanti39
tative assessments. In this paper, the focus is more on methods than on quantitative evaluation,

though we do discuss some important implications of the present model. In particular, a more

satisfactory quantitative model would involve inequality in labor income/productivity as well.10

The infinite-horizon model, as we shall show, inherits all the essential features of the 1- and 2-

period models. Some new mechanisms appear—the dynamic components of the median voter’s

tradeoffs become richer—but the core of the political redistribution mechanism remains. Computationally, it raises some new issues, and we discuss these in some detail.

We first define recursive competitive equilibria for a given tax function Ψ. In the main text, we

will presume that equilibria depend on the state vector (K, Am) only, and not on other aspects of

the asset distribution; in the Appendix, we make the formal statement to this effect: we show that

the set of all equilibria contains the set of equilibria where aggregate outcomes depend only on

(K, Am).

11 Equilibria involve decision rules for savings and leisure of each of the types of agents

as a function of the state vector. Moreover, for compactness, in the text we will not describe the

competitive agent’s dynamic programming problem, which would necessitate also including the

individual state variable. Instead, we will only define aggregate decision rules.

In order to express aggregation as compactly as possible, we also define competitive equilibrium using another concepts of wealth, namely the present-value human and transfer wealth (net

of taxes), E, and relative total net present-value wealth of the mean- and median-asset agents,

λ. The latter variable is very convenient: it will dictate, and equal, the relative consumption and

leisure levels of the mean- and median-asset agents.

Definition 8 A recursive competitive equilibrium with aggregation for a given tax function

10This extension will be considered in future work.

11As pointed out before, this still leaves open the possibility that there are self-fulfilling equilibria where other

moments of the asset distribution matter for aggregate outcomes.

40
Ψ(Am, K) is a set of functions hm(Am, K), nm(Am, K), h(Am, K), n(Am, K), R(K, N), W(K, N),

λ(Am, K), E(Am, K), and T(Am, K) with the following properties.

1. hm and nm solve, for all (Am, K),

uc(cm(Am, K), 1−nm(Am, K)) = βR(K0, n(A

0

m, K0

))(1−Ψ(A

0

m, K0

))uc(cm(A

0

m, K0

), 1−nm(A

0

m, K0

)

and

uc(cm(Am, K), 1−nm(Am, K))W(K, n(Am, K))(1−Ψ(Am, K)) = ul(cm(Am, K), 1−nm(Am, K)),

where

cm(Am, K) ≡ Am+[AmR(K, n(Am, K))+nm(Am, K)W(K, n(Am, K))](1−Ψ(Am, K)))+T(Am, K)

A

0

m ≡ hm(Am, K),

and

K0 ≡ h(Am, K).

2. h(Am, K) and n(Am, K) satisfy, for all (Am, K),

h(Am, K) ≡ R(K, n(Am, K))(1 − Ψ(Am, K)))(K − λ(Am, K)Am)+

+[W(K, n(Am, K))(1−Ψ(Am, K))+T(Am, K)](1−λ(Am, K))+λ(Am, K)hm(Am, K)

41
and

1 − n(Am, K) = λ(Am, K)(1 − nm(Am, K)).

3. W and R satisfy W(K, N) = F1(K, N) and R = F2(K, N) for all (K, N).

4. T satisfies T(Am, K) = Ψ(Am, K)(R(K, n(Am, K))K + W(K, n(Am, K))n(Am, K))

for all (K, Am).

5. λ(Am, K) satisfies, for all (Am, K),

λ(Am, K) = K[R(K, n(Am, K))(1 − Ψ(Am, K))] + E(Am, K)

Am[R(K, n(Am, K))(1 − Ψ(Am, K))] + E(Am, K)

.

6. E(Am, K) satisfies, for all (Am, K),

E(A

0

m, K0

) =

[E(Am, K)−W(K, n(Am, K))(1−Ψ(Am, K))−T(Am, K)]R(K0, n(A

0

m, K0

))(1−Ψ(A

0

m, K0

)),

where

A

0

m ≡ hm(Am, K)

and

K0 ≡ h(Am, K).

Thus, E adds up the present value of the stream of total labor endowments, net of taxes, and

the stream of transfers; these are the same for all agents, so E is a part of wealth that is common to

42
all. The differences between agents are then captured with λ, which expresses net-present-value

wealth ratios when current assets, net of taxes, are included.

Notice that this equilibrium definition is stated entirely in terms of functions: each condition

in the equilibrium definition is required to hold for all values of the arguments of the functions,

thus defining a set of functional equations.

Also, note that in this setup the ranking of asset between types stays the same over time. This

follows rather straightforwardly from assuming normal goods—which our typical assumptions

on additive time separability and the concavity of u imply—so that a higher asset holding in a

given period translates into higher asset holdings (and thus more consumption) for the future as

well. If one considers heterogeneity in preferences or in labor productivity, this result will no

longer necessarily hold, and the asset ranking can change over time.

4.1 Endogenous policy: recursive majority-voting equilibrium

We now define a Markov-perfect equilibrium by requiring that the government policy function

Ψ be the preferred choice by the agent whose asset holdings are median in the distribution. The

function Ψ is stationary—due to future being infinite at any point in time, it does not have a time

subscript—and we think of it as a limit of the sequence of first-period outcome functions Ψ0T

where T represents the time horizon. As in the 1- and 2-period models, and as in Krusell and

R ́ıos-Rull (1999), we first define equilibrium by characterizing one-period deviations from the

policy rule Ψ(Am, K): in the current period, the tax rate is τ , whereas all future tax rates are

given by the rule Ψ, evaluated at the asset distributions that will result from current taxation at

τ . We do this in order to be able to state the median voter’s problem: the median voter needs

43
to consider all possible current τ values, and their associated competitive equilibria, in order

to see which one is best. The political equilibrium will then require that Ψ(Am, K) is the best

among all policies τ , for all (Am, K). To define behavior in the one-period deviations we add the

argument τ and tildes to all associated functions. The resulting equilibrium is defined as follows.

The one-period deviation will require functions that depend on τ and these will also satisfy the

aggregation property; the proof, moreover, is parallel. A key element of this definition will

be the value function of the median voter for a one-period deviation, V ̃ (Am, K, τ ), along with

h ̃m(Am, A, τ  ̄ ), h ̃(Am, A, τ  ̄ ), n ̃m(Am, A, τ  ̄ ) and n ̃(Am, A, τ  ̄ ). A more formal definition follows.

Definition 9 The recursive competitive equilibrium with aggregation and a one-period tax deviation is a set of functions V (Am, K), V ̃ (Am, K, τ ), hm(Am, K), h ̃m(Am, K, τ ), nm(Am, K),

n ̃m(Am, K, τ ), h(Am, K), h ̃(Am, K, τ ), n(Am, K), n ̃(Am, K, τ ), R(K, N), W(K, N), λ(Am, K),

λ ̃(Am, K, τ ), E(Am, K), E ̃(Am, K, τ ), T(Am, K) and T ̃(Am, K, τ ) with the following properties:

1. V ̃ (Am, K, τ ) solves:

V ̃ (Am, K, τ ) = max

nm,A0m

uc(cm, 1 − nm) + βV (A

0

m, K0

)

subject to

cm ≡ Am + [AmR ̃(K, n ̃(Am, K, τ )) + nmW ̃ (K, n ̃(Am, K, τ ))](1 − τ )+

T ̃(Am, K, τ ) − A

0

m.

44
2. h ̃m and n ̃m attain the argmax above.

3. h ̃(Am, K, τ ) and n ̃(Am, K, τ ) satisfy, for all (Am, K, τ ),

h ̃(Am, K, τ ) ≡ (1 + R ̃(K, n ̃(Am, K, τ )))(1 − τ ))(K − λ ̃(Am, K, τ )Am)+

[W ̃ (K, n ̃(Am, K, τ ))(1−τ )+T ̃(Am, K, τ )](1−λ ̃(Am, K, τ ))+λ ̃(Am, K, τ )hm(Am, K, τ )

and

(1 − n ̃(Am, K, τ )) = λ ̃(Am, K, τ )(1 − n ̃m(Am, K, τ )).

4. T ̃ satisfies T ̃(Am, K, τ ) = τ (R(K, n ̃(Am, K, τ ))K+W(K, n ̃(Am, K, τ )) ̃n(Am, K, τ )) for

all (K, Am, τ ).

5. λ ̃(Am, K, τ ) satisfies, for all (Am, K, τ ),

λ ̃(Am, K, τ ) = K[R(K, n ̃(Am, K, τ ))(1 − τ )] + E ̃(Am, K, τ )

Am[1 + R(K, n(Am, K, τ ))(1 − τ )] + E ̃(Am, K, τ )

.

6. E ̃(Am, K, τ ) satisfies, for all (Am, K, τ ),

E(A

0

m, K0

) = [E ̃(Am, K, τ ) − W(K, n(Am, K, τ ))(1 − τ )−

T ̃(Am, K, τ )]R(K0, n(A

0

m, K0

))(1 − ψ(A

0

m/K0

)),

45
where

A

0

m ≡ hm(Am, K)

and

K0 ≡ h(Am, K).

7. V (Am, K), hm(Am, K), nm(Am, K), h(Am, K), n(Am, K), R(K, N), W(K, N), λ(Am, K),

E(Am, K) and T(Am, K) constitute a recursive competitive equilibrium with aggregation.

Finally, given a recursive competitive equilibrium with aggregation with a one-period tax deviation, it is straightforward to define a Markov-perfect median voter equilibrium with aggregation. Such an equilibrium thus has as its key requirement that

ψ(Am, K) = arg max

τ

V ̃ (Am, K, τ )

for all (Am, K).

4.2 The problem of the median voter: the Generalized Euler Equation

We saw in the 1- and 2-period model that the first-order necessary condition of the median

voter amounted to an equation setting a weighted sum of “gaps” to zero. In the 1-period model,

these gaps were the labor-leisure gap and a redistribution gap; in the 2-period model, these gaps

reappeared, one for each time period, and in addition an intertemporal, or savings, gap appeared.

In a 3-period model, the first-order condition for the median voter in the first of the three periods

would involve three static gaps: one for each of the three periods. Models with 4 and more periods, however, also involve only gaps in three periods: the current one and two consecutive ones.

46
Thus, as we shall see here, the infinite-horizon model delivers a first-order condition of the median voter that consists of three consecutive periods of static gaps, together with two consecutive

periods of savings gaps, but no more. The intuition for this finding will be discussed below. The

detailed derivation can be found in Azzimonti, de Francisco, Krusell, and R ́ıos-Rull (2005).

The GEE thus reads

GAPa

0m

dh ̃m

dτ + GAPlm

dn ̃m

dτ + GAPred

| {z }

t=1

+ β

"

GAPa

00m

dh ̃0

m

dτ + GAPl

0m

dn ̃

0

m

dτ + GAPred0

#

| {z }

t=2

+

β

2

·

GAPl

00m

dn ̃

00

m

dτ + GAPred00 ̧

| {z }

t=3

= 0. (1)

There are no conceptual news in this condition—the definitions of gaps are inherited from

the 2-period model—so the only remaining issue is the one of why three and only three periods

appear. This can be understood by thinking of the GEE as resulting from a variational experiment.

The key insight in this regard is that there are two state variables and only one control in the

median voter’s maximization problem. Suppose the median agent kept (Am, K) and (A00

m, K00)

fixed and optimally varied the controls in between, as in a parallel of what occurs in a standard

dynamic optimization problem. The controls would be τ and τ

0

, or, alternatively, the vector

(A0

m, K0

). The problem with this experiments is that there are not enough degrees of freedom

for a variational experiment: the two controls are completely pinned down by the end conditions,

(A00

m, K00), and cannot be varied beyond that! This is why a variational experiment here has to

involve keeping (Am, K) and (A000

m, K000) fixed and optimally varying τ , τ

0

, and τ

00, where there

47
now is one degree of freedom and utility can be maximized. As a consequence, the GEE must

contain terms also dated two periods from the current period.

4.3 The infinite-horizon example economy

Using the same functional forms as in the 2-period case, we will again be able to use the result

that levels do not matter: the endogenous variables that are nontrivially determined in equilibrium

depend only on the ratio of median to mean assets. This is very convenient since we only need to

keep track of the evolution of one state variable.

To proceed, we will first solve for the recursive competitive equilibrium. Then we will characterize the competitive equilibrium under a one-period deviation. Finally, we will use the GEE

derived before to solve for the tax rule using numerical methods.

4.3.1 Equilibrium for a given tax rule

Given a tax rule, ψ(Am/K), it is possible to find a closed-form solution to the decision rules in

the recursive competitive equilibrium. The results are summarized in the following proposition.

Proposition 10 The recursive competitive equilibrium for the parameterized economy is characterized as follows.

1. The value function satisfies:

V (Am, K) = S(Am/K) + B log(K)

48
where

S(Am/K) = ©α ln £n

1−θ

[(1 − ψ(Am/K))(αAm/K + (1 − α))θ(1 − β) + 1 − θ) + ψ(Am/K)]¤

+(1 − α) ln[1 − nm(Am/K)] + βθα

1 − θβ ln £

βθ(1 − ψ(Am/K)n(Am/K)

1−θ

¤

3⁄4

1

1 − β

and

B =

θα

1 − θβ .

2. The median agent’s savings and working decisions are

hm(Am, K) = β(1 − ψ(Am/K))R(Am, K)Am

and

nm(Am, K) = (1 − θ) + (1 − α)θ(1 − Am/K)(1 − β)

1 − θ

n(Am, K).

3. The savings and working decisions of the mean-asset agent are

h(Am, K) = βθ(1 − ψ(Am/K))n(Am, K)

1−θKθ

and

n(Am, K) = α(1 − ψ(Am/K))(1 − θ)

(1 − ψ(Am/K))(1 − αθ) + (1 − α) [ψ(Am/K) − βθ(1 − ψ(Am/K))].

4. The functions W and R satisfy W(Am, K) = (1 − θ)n(Am, K)

−θKθ and R(Am, K) =

49
θn(Am, K)

1−θKθ−1

.

5. The function T satisfies T(Am, K) = ψ(Am/K)[R(Am, K)K + W(Am, K)n(Am, K)].

6. The function λ satisfies

λ(Am, K) = 1 − βθ(1 − ψ(Am/K))

1 − βθ(1 − ψ(Am/K)) − (1 − β)αθ(1 − ψ(Am/K)) ¡1 −

Am

K

¢.

7. The function E(Am, K) satisfies, for all (Am, K),

E(Am, K) = R(Am, K)K(1 − ψ(Am/K))1 − λ(Am, K)Am/K

λ(Am, K) − 1

.

This equilibrium has several interesting properties. First, notice that the ratio between the

mean and the median capital holdings is constant over time, i.e., A0

m/K0 = hm(Am, K)/h(Am, K) =

Am/K. In this economy, all agents’ assets will grow at the same rate until reaching the steady

state. Hence, the distribution of income remains unchanged along the transition path. More importantly, a constant ratio in equilibrium implies that the tax rate does not vary over time. In the

2-period model, in contrast, we saw that equal tax rates over time implied that asset positions

moved closer together. However, in the 2-period model, a measure of wealth that includes all

sources of income would have remained constant across the two periods. The reason for assets

moving closer together, thus, was that asset wealth was a larger fraction of total wealth in the final

period than in the first period, when human wealth derives from two periods of income. That is,

50
a given total equality requires larger asset inequality in the first period, since human wealth is

larger then and equal across agents.

Second, mean savings are a constant proportion of total output: the rate given by βθ(1 −

ψ(Am/K)). Given the aggregate paths and the constant tax rule, the median agent chooses to

save a constant fraction of after-tax capital income, the function being linear in his capital income

RAm

12. The fact that the level of transfers and the wage income do not affect savings is a result of

assuming logarithmic utility and Cobb-Douglas production with full depreciation, where income

and substitution effects offset each other.

Third, aggregate labor is constant and only depends on the ratio of mean to median capital via

the tax rate. This is also a result of the above functional-form assumptions.

Finally, if the median agent is currently poorer than the mean (i.e., Am < K), the relative

present value wealth is greater than one: λ(Am, K) > 1. Hence, he will remain poorer in the

future despite of the existence of lump-sum transfers. We can also see that the median voter will

consume less and work more than the average agent in equilibrium.

4.3.2 One-period deviations

In order to find the equilibrium tax rate, we need to characterize the decision rules in a competitive equilibrium after a one-period deviation. It is important to find out how agents will react

to a deviation of τ from the rule ψ(Am/K) by the current government. Since we are studying

a one-period deviation only, future governments are assumed to follow ψ(Am/K). Therefore,

agents’ decision rules depend on both τ and ψ(Am, K).

Proposition 11 The recursive competitive equilibrium with aggregation and a one-period tax

12Krusell, Kurus ̧c ̧u, and Smith (1999) find a similar result in the context of hyperbolic discounting.

51
deviation for the parameterized economy is characterized as follows.

1. The median and mean agents’ savings and working decisions are given implicitly by

h ̃m(Am, K, τ ) = β(1−τ )R ̃(Am, K)Am+βθ hτ − ψ[h ̃m(Am, K, τ )/h ̃(Am, K, τ )]iKθn ̃(Am, K, τ )

1−θ

,

n ̃m(Am, K, τ ) = (1 − θ) + (1 − α)θ(1 − Am/K)(1 − β)

1 − θ

n ̃(Am, K, τ ),

h ̃(Am, K, τ ) = βθ(1 − ψ[h ̃m(Am, K, τ )/h ̃(Am, K, τ )]Kθn ̃(Am, K, τ )

1−θ

and

n ̃(Am, K, τ ) = (1 − θ)(1 − τ )α

(1 − τ )(1 − αθ) + (1 − α)[τ − βθ(1 − ψ[h ̃m(Am, K, τ )/h ̃(Am, K, τ )]]

.

2. The functions W ̃ and R ̃ satisfy W ̃ (Am, K, τ ) = (1−θ) ̃n(Am, K, τ )

−θKθ and R ̃(Am, K, τ ) =

θn ̃(Am, K, τ )

1−θKθ−1

.

3. The function T ̃ satisfies T ̃(Am, K, τ ) = τ [R ̃(Am, K, τ )K + W ̃ (Am, K, τ ) ̃n(Am, K, τ )].

4. The function λ ̃ satisfies

λ ̃(Am, K, τ ) = 1 − βθ(1 − ψ[h ̃m(Am, K, τ )/h ̃(Am, K, τ )]

1 − βθ(1 − ψ[h ̃m(Am, K, τ )/h ̃(Am, K, τ )] − (1 − τ )αθ(1 − Am/K)(1 − β)

.

52
5. The function E ̃ satisfies

E ̃(Am, K, τ ) = R ̃(Am, K, τ )K(1 − τ )

1 − λ ̃(Am, K, τ )Am/K

λ ̃(Am, K, τ ) − 1

.

It is straightforward to show that if τ = ψ(Am/K), the equations above collapse to those in

Proposition 8: if there were no deviation, we would be back in a recursive competitive equilibrium

for the given tax function.

When the government deviates, agents react by modifying their savings and working decisions

in the current period. For example, the median savings’ rule is adjusted by the difference between

current and future taxes. The first term of hm(Am, K, τ ) takes the same form as in the equilibrium

with no deviation: savings are linear in capital income net of taxes. The second term increases or

decreases savings depending on the spread between taxes today and tomorrow. While the ratio of

mean to median hours worked remains unchanged (not only over time but also compared to the

previous case), the levels are modified when there is a deviation. Since aggregate labor supply

changes, so do wages and interest rates. Moreover, individual savings decisions are affected as

well, so aggregate capital in the future will change via that indirect effect.

In contrast to the case of labor decisions, the ratio of median to mean capital is no longer

constant over time. In particular,

A0

m

K0

=

(1 − τ )Am/K + τ − ψ(A0

m/K0

)

1 − ψ(A0

m/K0

)

6=

Am

K

. (2)

The deviation in the tax rule results in a deviation in relative capital holdings, which in turn

affects future taxes. So even if future governments decide not to deviate from the equilibrium

53
path, the change in their inherited state of the world causes them to adjust the tax rate. We see

from the formula that, as in the 2-period model, if the current tax rate is higher than the future

tax rate, inequality decreases. Note that in all periods after tomorrow, the capital ratio remains

unchanged, that is, A0m

K0 =

A00m

K00 =

A000 m

K000 and so on. Thus, if a deviation from ψ is considered and

the current tax is raised, it will lead to a higher tax today than tomorrow: not only is the current

tax higher, but since inequality will fall, the tax in the next period, and all periods hence, will be

lower as well.

4.3.3 The median voter’s problem

Now we will characterize the tax rule chosen by the median voter for this economy. The

median voter’s decision problem is thus to choose τ to maximize

α log([AmR(K, n ̃(Am, K, τ ))+ ̃nm(Am, K, τ )W(K, n ̃(Am, K, τ ))](1−τ ))+T ̃(Am, K, τ )−h ̃m(Am, K, τ ))

+(1 − α) log(1 − n ̃m(K, n ̃(Am, K, τ )) + βV (h ̃m(Am, K, τ ), h ̃(Am, K, τ )),

with V (Am, K) defined above. If, for every (Am, K), ψ(Am, K) solves this problem, we have a

Markov-perfect equilibrium. Of course, since decision rules, as well as V , depend on ψ here, we

are looking at a nontrivial fixed-point problem in the function ψ. Unfortunately, no closed-form

solution is available for the function ψ(Am/K). It is a function of one variable only, so it is in

that sense not a higher-dimensional problem than that of solving the neoclassical growth model

for the optimal savings function (in a case where no analytical solution is available). On the other

hand, there is not, as far as we are aware, any general guarantee of existence or uniqueness unlike

in the case of the standard growth model.

54
4.3.4 Numerical solution method

Our main aim is to find the set of steady states. With the tax rate set at an arbitrary constant τ ̄,

it is well known that the present model has a unique steady state in terms of the level of average

capital, and in addition that there is a continuum of associated steady-state asset distributions; see

Krusell and R ́ıos-Rull (1999) for a discussion. Here, in principle, there is also the possibility of a

continuum of steady states, though each one would be associated with its own tax rate (along with

an average capital stock). The conjecture is thus that there is a ψ(x), at least in the neighborhood

of x = 1, such that the median voter chooses τ = ψ(x) at x and, thus, to remain at a constant tax

rate and constant level of inequality x.

We proceed to find a candidate ψ function by analyzing the GEE. A problem here is that

it depends on the derivatives of the rules that determine savings and hours worked, and these

functions in turn depend on the equilibrium tax rule itself. So the GEE constitutes a functional

equation in ψ that also contains the derivative of ψ, through the derivatives of the equilibrium

decision rules. One implication is that standard linearization methods cannot be applied. Of

course, the equilibrium functions can be arbitrarily well approximated by linear functions around

the steady state. However, the steady state cannot be found without knowing (some of) the

derivatives of these functions, as in standard dynamic problems. In particular, because the median

voter’s first-order condition involves derivatives of the ψ, one cannot solve for steady-state levels

independently of solving for higher-order features of these functions.

We use an algorithm that can be viewed as an extension of linearization: a version of that

is outlined in Krusell and Smith (2001), where it has proven useful both in terms of speed and

accuracy.13 This algorithm is feasible to implement for this problem, fast, and does not require

13The method was applied there to a consumption-savings problem under time-inconsistent preferences and has

55
significant work beyond deriving the equilibrium first-order conditions as functional equations,

a task which was accomplished in the previous section. Judd (2005) discusses this method in

some detail and points to a potential pitfall—multiplicity of candidate solutions—but also points

to ways to discriminate between these. Here, we use the method to find a candidate rule ψ

and then verify with global methods, in the neighborhood of x, that it indeed is (close to) an

optimal choice for the median voter. The essential idea behind the method is to approximate

the equilibrium function with a polynomial evaluated at a single point: the steady-state point.

Thus, a 0th-order approximation would let all the derivatives be zero and the steady state could

be found from the system of first-order conditions. A 1st-order (linear) approximation would

involve more unknown parameters—first derivatives in addition to levels—of the five functions.

The additional equations needed to pin down these unknown parameters are obtained by partial

differentiation, with respect to each argument Am and K, of each of the equilibrium functional

equations. With this procedure, successively higher-order polynomial approximations are thus

rather straightforward to derive, and convergence is obtained when the addition of higher orders

does not alter the steady state more than by a very small amount. Differentiation of the functional

equations is extremely tedious to implement with pencil and paper, but it can be automated using

a symbolic math program (one of which is available as part of MATLAB).

In this paper, we will stop at a linear approximation of the key functions, which we later verify

is a good approximation by global search. We will thus in particular use the approximation

ψ (Am/K) = x0 + x1

Am

K

. (3)

also been employed in an optimal-public-expenditure problem without commitment (Klein, Krusell, and R ́ıos-Rull

(2003)).

56
As an intermediate step, we need to find the rule that determines the evolution of relative asset

holdings under a one period deviation of τ from equilibrium taxes. We will approximate equation

(2) using a linear function as well.14 Thus, we let

h ̃m (Am, K, τ )

h ̃ (Am, K, τ )

= x2 + x3

Am

K

+ x4τ. (4)

We can replace our guess on ψ into the first-order conditions with respect to capital of the

mean and the median agent and obtain a system (of two equations) determining savings per unit

of output for the two types of voters:

h ̃m0 = βθ h(1 − τ )Am/K + τ − ψ(h ̃mo/h ̃

o)

i

,

h ̃

o = βθ h

1 − ψ(h ̃mo/h ̃

o)

i

,

Since h ̃mo/h ̃

o = h ̃m/h ̃, we can solve the system above to obtain one equation governing the

evolution of relative asset holdings as a function of the current ratio of median to mean assets, and

the tax deviation τ . These equations together with the GEE will be used to solve for the unknown

coefficients. The perturbation method is used to solve for the parameters of these polynomials.

We have five unknowns, x0, x1, x2, x3 and x4. The first derivative of the GEE determines the

change in the tax rule when the capital ratio changes (using the implicit function theorem). We

can obtain the remaining two equations by taking the derivative of the ratio of asset holdings

with respect to the current ratio and its derivative with respect to taxes. The system can then

14Given ψ, the linearization of the rule governing the evolution of relative asset holdings is straightforward and

does not involve derivatives.

57
be solved using a standard non-linear equation solver. Thus, these equations deliver values for

the parameters around the chosen Am/K ratio. We will now look at the numerical solution to a

specific example to further characterize the tax rule.

4.3.5 Numerical results

We again use α = 0.3 and β = 0.9 to compute the solution in the infinite-horizon economy.

As a benchmark, we set θ = 0.05, and then consider the case where θ = 0.4.

<(Am/K) for T=0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Am/K

<

Figure 7: Equilibrium tax function (θ = 0.05)

Figure 7 shows that the tax function, as in the finite-horizon case, is decreasing in the medianmean ratio. A relatively poor pivotal voter would push forward high taxes so as to redistribute

resources in his favor. We can also observe that even when wealth inequality is large, taxes are

not too distortionary; for example, when the ratio is 0.7 the optimal tax rate does not exceed

5%. This is clearly due to the low productivity of capital assumed for this economy (recall that

58
θ = 0.05). It is useful then to analyze the behavior of taxes for the case with θ = 0.4. The

resulting tax rule is depicted in Figure 8.

-0.2

0

0.2

0.4

0.6

0.8

1

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

Am/K

<

<(Am/K) for T=0.4

Figure 8: Equilibrium tax function (θ = 0.4)

As in the one and two-period examples, as the economy becomes more productive, the benefits

of taxing the current stock of capital increase. It is interesting to notice as soon as we move away

from the representative-agent case to one where the median agent is poorer than the mean, the tax

rate increases very steeply. When the ratio is only 0.95, taxes are almost 100%. Intuitively, steady

states, which we are studying here, are long-run outcomes, and we saw that even in a two-period

model taxes fall over time, and inequality declines. Thus, with many time periods, only a small

set of values for median-mean wealth are possible: too much initial inequality would be taxed

away over time. And the small set of values for Am/K that are feasible steady states are then

59
associated to high rates of taxation: inequality is “almost” taxed away. Thus, with a reasonable

value for θ—similar to that used in most macroeconomic calibrations, the model predicts taxes

that are too high, and a range of feasible values for asset inequality that is nowhere near the level

of asset inequality observed in data.

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

−6.744

−6.742

−6.74

−6.738

−6.736

−6.734

−6.732

−6.73

−6.728

−6.726

Am/K

V and Vm

Welfare

(θ=0.05)

Vm

V

Vm

*=V *

Figure 9: Welfare (θ = 0.05)

In Figure 9, finally, we have graphed the welfare function for the average agent and the median

voter (assuming that the asset holdings of the average agent are always one (A = 1)), as a function

of the asset holdings of the median voter. The welfare of the median voter is monotonically

increasing in Am. However, the welfare of the average agent with A = 1 is increasing in Am

for Am < A, but decreasing in Am for Am > A, so that a maximum is reached when there is

no inequality (Am = A). Production is always increasing in Am since taxes decrease and they

become subsidies when Am > A so there is overaccumulation of capital in this case. However,

subsidies are also distortionary and they decrease the welfare of the average agent.

60
5 Concluding comments

In this paper, we have developed finite-horizon models of endogenous redistribution using

the median-voter construct, and we have explored the infinite-horizon version of the setup as

well. The analysis demonstrates first that, under assumptions about the utility function that are

common in the applied macroeconomic literature, an aggregation result applies: the aggregate

politico-economic equilibrium outcomes, i.e., taxes, output, prices, etc., depend on the mean

level of assets and on the median asset holding, and on no other aspect of the asset distribution.

This result facilitates tractability considerably; dynamic models with forward-looking, rational

agents rapidly become more complex as the number of state variables grows. Thus, it would for

example be feasible to study the economy considered here with aggregate productivity shocks

and thereby analyze any “political business cycles” arising from median-voter tax determination

in a quantitative context.15

The aggregation result requires complete markets, and in the present context—which does not

have uncertainty—this just means that all agents can borrow and lend at the same rate. Under

uncertainty, aggregation would require complete insurance markets. We know, however, from

Krusell and Smith (1998), that a setting with idiosyncratic shocks and no insurance markets but

precautionary savings using one asset leads to “approximate aggregation”. Thus we conjecture

that politico-economic equilibria in such a model would approximately depend only on an aggregate state vector (K, Am) and not (almost at all) on any other moments of the asset distribution.

We hope to explore this kind of setup in future work.

Second, we used first-order conditions of the median voter to interpret how taxes are chosen.

15The addition of an exogenous state variable—aggregate productivity—does make the analysis more difficult but

would be entirely feasible.

61
We thus showed that the tax choice can be viewed as a tradeoff between direct redistribution effects and distortions to the labor-leisure and the consumption-savings choices in three consecutive

time periods. One noteworthy point is that the consumption-savings distortion is a consideration

for the median voter, even though capital income is inelastic ex post; recall that there is no commitment in advance to the tax choice. The reason is that the current tax influences savings from

the present to the future, since it influences total resources available.

Third and finally, we used numerical methods to find a set of steady states for the infinitehorizon model. We compared the same parametric setup in a 1-period model, a 2-period model,

and an infinite-horizon model. We found for the 2-period model that inequality, as measured by

the median-to-mean wealth ratio, falls substantively between periods 1 and 2, as do tax rates.

As an implication, the set of steady states is quite narrow, at least for parameter values that are

close to those used in the macroeconomic literature. More specifically, we found that only very

modest levels of inequality could be supported as long-run outcomes of the model. This indicates

that models that have a chance of generating inequality/tax combinations that resemble those we

observe in most developed countries would need different ingredients. One possibility is that

explored in Krusell and R ́ıos-Rull (1999), namely, that there is an implementation lag for taxes,

so that taxes are perceived as more distortionary when they are chosen. Another possibility is

that inequality in labor productivity/wages, which is abstracted from here, would improve the

quantitative performance of the model. In general, features that make it more costly to tax, or

less beneficial to redistribute, would be required to improve the quantitative performance of the

model.

62
References

[1] Alesina, A. and Rodrik, D., 1994, Distributive Politics and Economic Growth, Quarterly

Journal of Economics 109, 2, 465-90.

[2] Azzimonti, M., de Francisco, E., Krusell, P., and J.-V. R ́ıos-Rull, 2005, Public Policy Making without Commitment, Working Paper.

[3] Bernheim, B.D. and Nataraj, S., 2002, A Solution Concept for Majority Rule in Dynamic

Settings, Working Paper.

[4] Hassler, J., Rodriguez Mora, J.V., Storesletten, K., and Zilibotti, F., 2003, The Survival of

the Welfare State, American Economic Review, 93, 1, 87-112.

[5] Judd, K.L., 2005, Existence, Uniqueness, and Computational Theory for Time-Consistent

Equilibria: A Hyperbolic Discounting Example, Working Paper.

[6] Klein, P., Krusell, P. and J.-V. R ́ıos-Rull, J.V., 2003, Time-Consistent Public Expenditures,

Working Paper.

[7] Krusell, P., 2002, Time-Consistent Redistribution, European Economic Review, 46, 4-5,

755-69.

[8] Krusell, P., Kurus ̧c ̧u B., and Smith, A., 1999, Tax Policy With Quasi-Geometric Discounting, Working Paper.

[9] Krusell, P., Martin, F., and J.-V. R ́ıos-Rull, 2003, Time-Consistent Debt, Working Paper.

[10] Krusell, P. and J.-V. R ́ıos-Rull, 1999, On the Size of Government: Political Economy in the

Neoclassical Growth Model, American Economic Review, 89, 5.

63
[11] Krusell, P. and Smith, A., 1998, Income and Wealth Heterogeneity in the Macroeconomy,

Journal of Political Economy, 106, 5, 867-896.

[12] Krusell, P. and Smith, A., 2003, Consumption-Savings Decisions with Quasi-Geometric

Discounting, Econometrica, 71, 365-375.

[13] Laibson, D., 1997, Golden Eggs and Hyperbolic Discounting, Quarterly Journal of Economics 112, 2, 443-77.

[14] Maskin, E. and Tirole, J., 2001, Markov Perfect Equilibrium, Journal of Economic Theory,

100, 2, 191-219.

[15] Meltzer, A.H. and Richard, S.F., 1981, A Rational Theory of the Size of Government, Journal of Political Economy, October, 89, 5, 914-27.

[16] Persson, T. and Tabellini, G., 1994, Is Inequality Harmful for Growth?, American Economic

Review, 84, 3, 600-621.

Appendix

The state variable of the economy is the distribution of asset holdings, which we denote

A ≡ (A1, . . . , AI ). Thus let H(a, A) be a function specifying the law of motion of the asset

holding of an individual agent with beginning-of-period holdings a: a

0 = H(a, A) (primes denote next-period values). Similarly, we let the leisure choice of a given agent be L(a, A), with the

associated aggregate labor supply function N(A). Let Ψ(A) be the tax rate on income imposed

by the government. The function for transfers, T(A), is specified residually to obey government

budget balance.

64
Definition 12 A recursive competitive equilibrium is a set of functions V (a, A), H(a, A), N(A),

L(a, A), R(K, N), W(K, N), Ψ(A), and T(A) with the following properties.

1. V solves

V (a, A) = max

l,a0

u(c, l) + βV (a

0

, A0)

subject to

c + a

0 = W(K, N(A))(1 − l)(1 − Ψ(A)) + R(K, N(A))(1 − Ψ(A)))a + T(A)

and A0 = (H(A1, A), . . . , H(AI , A)) for all (a, A), where K ≡

PI

i=1 μiAi

.

2. H and L attain the argmax above.

3. N satisfies N(A) = PI

i=1 μi(1 − L(Ai

, A)).

4. W and R satisfy W(K, N) = FN (K, N) and R = FK(K, N) for all (K, N).

5. T satisfies T(A) = Ψ(A)(R(K, N(A))K + W(K, N(A))N(A)) for all A, where K ≡

PI

i=1 μiAi

.

Next, we look at one-period deviations.

Definition 13 A recursive competitive equilibrium with a one-period tax deviation is a set

of functions V (a, A), V ̃ (a, A, τ ), H(a, A), H ̃ (a, A, τ ), N(A), N ̃(A, τ ), L(a, A), L ̃(a, A, τ ),

R(K, N), W(K, N), Ψ(A), T(A), and T ̃(A, τ ) with the following properties.

1. V (a, A), H(a, A), N(A), L(a, A), R(K, N), W(K, N), Ψ(A), and T(A) is a recursive

competitive equilibrium.

65
2. For all τ , V ̃ satisfies

V ̃ (a, A, τ ) = max

l,a0

u(c, l) + βV (a

0

, A0)

subject to

c + a

0 = W(K, N ̃(A, τ ))(1 − l)(1 − τ ) + R(K, N ̃(A, τ ))(1 − τ ))a + T ̃(A, τ )

and A0 = (H ̃ (A1, A, τ ), . . . , H ̃ (AI , A), τ ) for all (a, A), where K ≡

PI

i=1 μiAi

.

3. H ̃ and L ̃ attain the argmax above.

4. N ̃ satisfies N ̃(A, τ ) = PI

i=1 μi(1 − L ̃(Ai

, A, τ )).

5. T ̃ satisfies T ̃(A, τ ) = τ (R(K, N ̃(A, τ ))K + W(K, N ̃(A, τ ))N ̃(A, τ )) for all A, where

K ≡

PI

i=1 μiAi

.

Here, note that the deviation equilibrium will satisfy H(a, A) = H ̃ (a, A, Ψ(A)), N(A) =

N ̃(A, Ψ(A)), and L(a, A) = L ̃(a, A, Ψ(A)).

We can now state a definition of a Markov-perfect median-voter equilibrium. Let m denote

the median type; Am is thus the median asset holding.

Definition 14 A Markov-perfect median-voter equilibrium is a set of functions V (a, A), V ̃ (a, A, τ ),

H(a, A), H ̃ (a, A, τ ), N(A), N ̃(A, τ ), L(a, A), L ̃(a, A, τ ), R(K, N), W(K, N), Ψ(A), T(A),

and T ̃(A, τ ) which is a recursive competitive equilibrium with a one-period tax deviation and

66
which satisfies the following property:

Ψ(A) = arg max

τ

V ̃ (Am, A, τ )

for all A.

Suppose now that we have found a recursive equilibrium with aggregation as defined in the

main text, i.e., suppose that we have functions that satisfy all the stated conditions. We can then

use these functions in order to construct additional functions, specifying behavior and utility of

agents with arbitrary asset holdings, that together with the given functions meet all the conditions

of the earlier, general definition of a recursive competitive equilibrium. This constitutes our

aggregation theorem, which thus reads as follows:

Proposition 15 Suppose that u(c, l) satisfies Assumption 6 and that in any median-voter equilibrium all agents’ solutions are interior. Given a recursive competitive equilibrium with aggregation, thus satisfying Definition 8,

1. define

λ(a, Am, K) ≡

a[R(K, n(Am, K))(1 − Ψ(Am, K))] + E(Am, K)

Am[R(K, n(Am, K))(1 − Ψ(Am, K))] + E(Am, K)

;

2. define

H(a, A) ≡ R(K, n(Am, K))(1 − Ψ(Am, K)))(a − λ(a, Am, K)Am)+

+[W(K, n(Am, K))(1−Ψ(Am, K))+T(Am, K)](1−λ(a, Am, K))+λ(a, Am, K)hm(Am, K);

67
3. define

L(a, A) ≡ λ(a, Am, K)nm(Am, K)

and

N(A) ≡ n(Am, K);

4. Let K =

PμiAi where Ai ∈ A

5. define Ψ(A) ≡ Ψ(Am, K) and T(A) ≡ T(Am K); and

6. solve, for all (a, A), for

V (a, A) = u(c(a, A), 1 − L(a, A)) + βV (H(a, A), A0),

where A0

i = H(Ai

, A) for all i and

c(a, A) ≡ a+[aR(K, n(Am, K))+W(K, n(Am, K))λ(a, Am, K)nm(Am, K)](1−Ψ(Am, K)))+

T(Am, K) − H(a, A).

Then V (a, A), H(a, A), N(A), L(a, A), R(K, N), W(K, N), Ψ(A), and T(A) constitute a

recursive competitive equilibrium, i.e., they satisfy Definition 12.

Proof. Note first that the solution for V is well-defined: it is the fixed point of a contraction

mapping. Given concavity of the consumer’s problem in Definition 12, the first-order conditions,

which appear in Definition 8, are sufficient for maximization. The remainder of the proof uses

the functional-form version of the first-order conditions, which imply that all consumption goods

(consumption and leisure at all points in time) are a constant fraction of net-present value wealth

(asset holdings plus the present value of non-asset wealth). This feature allows us to show that if

68
the first-order conditions hold for the median agent—and they do by assumption—they also hold

for agents with all other asset levels. The details of the manipulations required to demonstrate

this are straightforward and only involve algebraic manipulations.

69