200310918443436246

73

0022-0531/01 $35.00 2001 Elsevier Science (USA)All rights reserved.

Journal of Economic Theory 105, 7398 (2002)doi:10.1006/jeth.2001.2840

On Non-existence of Markov Equilibria inCompetitive-Market Economies1

1 Paper presented at the meetings of the Society of Economic Dynamics (Sardinia, June1999), the Society of Economic Theory (Rhodes, July 1999), and the Latin American Meetingsof the Econometric Society (Cancun, August 1999). Comments and suggestions from severalconference participants have been extremely beneficial. The author is especially grateful toMordecai Kurz, Ellen McGrattan, Victor Rios-Rull, and Nancy Stokey for some insightfulremarks.

Manuel S. Santos

Department of Economics, Arizona State University, Tempe, Arizona [email protected]

Received July 18, 2000; final version received April 2, 2001;published online December 4, 2001

This paper presents some examples of regular dynamic economies with externali-ties and taxes that either lack existence of a Markov equilibrium or such equilib-rium is not continuous. These examples pose further challenges for the analysis andcomputation of these economies. Journal of Economic Literature ClassificationNumbers: C10, C62. 2001 Elsevier Science (USA)Key Words: competitive equilibrium; Markov equilibrium; externalities; taxes;

money.

1. INTRODUCTION

This paper presents some examples of dynamic competitive-marketeconomies with taxes and externalities in which either there is no Markovequilibrium or such equilibrium is not a continuous function of theunderlying state variables. The existence of a continuous Markov (orrecursive) equilibrium has been considered a minimal requirement for theanalysis and simulation of a dynamic model. Indeed, since the early workof Lucas and Prescott [22] and Prescott and Mehra [31], several papershave been devoted to study this existence problem. For instance, Lucas andStokey [23] and Giovannini and Labadie [15] show the existence ofMarkov equilibria for some monetary economies, and Bizer and Judd [5],

Coleman [10], and Greenwood and Huffman [16] address this sameexistence problem in economies with taxes and externalities. (Danthine andDonaldson [11] and Stokey et al. [38, Chaps. 1718] offer a detailedreview of analytical methods and specific contributions to this literature.)All these papers consider simple competitive economies with a represen-

tative consumer, and there are no known instances in which a Markovequilibrium fails to exist. In frictionless economies, continuous Markovequilibria can be shown to exist under relatively mild assumptions; here,the most common method of proof hinges upon the contraction propertyof the dynamic programming operator. As externalities, taxes, or money isintroduced into the analysis, one may still approach the problem of exis-tence of Markov equilibria using a recursive operator over finite horizons,but such an operator may fail to possess good convergence properties. Theaforementioned contributions are remarkable in that exploiting monotoni-city properties of the solution they substantiate the existence of continuousMarkov equilibria for some important families of unidimensional models.But it seems sensible to inquire to what extent these analyses may be gen-eralized. Our present results suggest that in certain directions there is notmuch scope for further generalizations.Examples of non-continuous Markov equilibria are often found in game-

theoretical settings (e.g., see Peleg and Yaari [30], Bernheim and Ray [4],and Leininger [21] for analyses of these equilibria in bequest models, andFudenberg and Tirole [14] for a general overview of this literature). Thebasic sources of these discontinuities are well understood and occur forreasons outside the scope of perfect competition. In several types of games,discontinuous choices arise from a loss of concavity of the objective func-tion originated by other players best responses. Similar results areobserved in the literature of dynamic contracts and incentives (e.g., Marcetand Marimon [24] and Rustichini [33]), where the loss of concavity stemsfrom the fulfillment of certain rationality conditions.There is by now a vast literature on the study of the dynamics of compe-

titive equilibrium solutions in economies with externalities, taxes, andmoney (see Benhabib and Farmer [2] for a recent update). In a thoughtprovoking paper, Kehoe et al. [18] illustrate that externalities and taxesmay affect substantially the qualitative dynamics of the system; in particu-lar, there could be a robust continuum of equilibria even in the presence ofa representative consumer. As these authors point out, these equilibria maybe characterized as solutions to a social planning problem with some addi-tional side constraints involving endogenous variables. Although this latterformulation of the problem may facilitate the analysis and simulation ofthe model, the possible effects of these side constraints on the equilibriumdynamics are not well understood. In some simple cases equilibriumsolutions in economies with distortions may be characterized by standard

74 MANUEL S. SANTOS

optimization problems (cf., [1, 11]), but in some other cases theseconstraints are non-redundant: No single strictly concave optimizationproblem can generate multiple solutions.Most of the literature on indeterminacy of equilibria has been concerned

with local properties around steady-state solutions. The local behavior of acertain orbit, however, does not necessarily preclude the existence of otherequilibrium paths comprising a Markov equilibrium. Hence, our papergoes a step further as a global argument is required to demonstrate that aMarkov equilibrium may fail to exist, or that such equilibrium may not bedescribed by a continuous function. Our main purpose here is to highlightsome theoretical possibilities that must be faced when analyzing andsimulating a dynamic model. It is our understanding from the work ofColeman [10] and Greenwood and Huffman [16] that the existence of acontinuous Markov equilibrium is guaranteed for a broad family of uni-dimensional growth models with taxes and externalities commonly used inapplied work. But as one of our examples illustrates, lack of existence ofthese equilibria seems to be a more pervasive phenomenon in multisectoreconomies.To provide some intuition for our results, in our economies the repre-

sentative agent solves a convex optimization problem, and hence theoptimal decision varies continuously with the vector of prices. In equilib-rium, however, prices are endogenously determined and may fail to beexpressed globally as continuous functions of the state variables. This lackof representation of prices as whole functions of state variables seems toapply to both stochastic and deterministic economies. Further subtledistinctions will be observed in the separate analyses of discrete- andcontinuous-time frameworks. Thus, in our family of continuous-timemodels with a single state variable, a Markov equilibrium is made up of afinite number of equilibrium trajectories or connected arcs. In discrete-timemodels, however, an equilibrium trajectory contains a countable number ofpoints, and hence a continuum of such trajectories is needed to conform aMarkov decision rule. Accordingly, additional types of discontinuities mayarise in a discrete-time setting.At this informal level of discussion, Table I is meant to elicit the signifi-

cance of the present findings. For the class of models considered in thispaper, it is relatively easy to establish that every finite-horizon economyalways contains a Markov equilibrium (i.e., see the first column of thetable). But as shown below (viz. Section 3.1), in the presence of an infinitehorizon, existence of this fundamental equilibrium concept is no longerguaranteed. There could be no fixed-point or stationary equilibrium solution.Regarding the second column of this table, note that every continuous-timefinite horizon economy has a unique continuous Markov equilibrium. It isillustrated in Section 2 below that non-continuous Markov equilibria may

NON-EXISTENCE OF MARKOV EQUILIBRIA 75

TABLE I

Existence of Markov Equilibriaa

Does a Markov equilibrium Is the Markov equilibriumClass of models always exist? always continuous?

Continuous-time,finite-horizon Yes Yes

Continuous-time,infinite-horizon No No

Discrete-time,finite-horizon Yes No

Discrete-time,infinite-horizon Unknown No

a These results apply for the class of regular economies considered in this paper (e.g.,economies that satisfy assumptions (A.1)(A.3)).

arise in unbounded horizons. Regarding discrete-time models, non-continuous Markov equilibria may be observed under both bounded andunbounded horizons. We will see, however, that certain types of disconti-nuities are characteristic of infinite-horizon economies.Finally, let us conclude this introduction with some related work on

dynamic social planning problems with non-convexities.2 Some early, key

2 There is a parallel line of research on dynamic models with optimal taxation; e.g., see [8].

contributions in this literature are Skiba [37], and Davidson and Harris[12]. These analyses are truly unidimensional in that optimal solutions arecharacterized from inspection of the phase diagram conformed by the Eulerequations.3 And, indeed, most results are obtained on a case by case basis.

3 See Ladron-de-Guevara et al. [20] for a recent attempt at generalization in a model withmultiple controls and state variables.

There is, however, a general result of particular interest to the presentwork: Steady states displaying complex eigenvalues are non-optimal andthe policy function may be discontinuous. A similar proposition on the non-continuity of the equilibrium function will emerge for some competitiveenvironments, even though the underlying arguments are quite different.The paper is structured as follows. Section 2 presents a simple economy

with a production externality in which from the equilibrium laws of motionone cannot construct a continuous Markov equilibrium. Section 3 relatesfurther developments in two economies with taxes. In the first examplethere are two production sectors that are taxed asymmetrically, and aMarkov equilibrium fails to exist. The second example describes a standard

76 MANUEL S. SANTOS

one-sector growth model with a decreasing tax rate on capital returns; inthis example every Markov equilibrium must be non-continuous. Furtherissues on the computation of these equilibria are examined in Section 4.This discussion is completed in our final section with some concludingremarks.

2. NON-EXISTENCE OF CONTINUOUS MARKOV EQUILIBRIAIN AN ECONOMYWITH EXTERNALITIES

This section considers a one-sector growth economy with an externalityin the production of the aggregate good in which there is no continuousMarkov equilibrium. All agents are identical in this economy, and sowithout loss of generality we shall focus on the equilibrium problem of arepresentative individual. For simplicity of exposition, a competitive equi-librium will be characterized as a solution to a dynamic optimizationproblem involving certain consistency conditions.The representative agent owns an initial stock of capital k0. The quantity

of output produced, y=f(k, ke), depends on the individuals stock ofcapital, k, and on her true perceptions about the average stock of capital inthe economy, ke. Output y is available for consumption, c, or investment, i.Capital is subject to a constant depreciation rate, d.Given a path of average stocks {ke(t)}t \ 0 the representative agent

chooses {c(t), i(t)}t \ 0 so as to solve the following optimization problem

max F.0

c(t)1s

1sert dt (P)

s.t. c(t)+i(t)=y(t)

y(t)=f(k(t), ke(t))

k(t)=i(t)dk(t)

k(0)=k0 given, s > 0, r > 0, d > 0,

c(t) \ 0, k(t) \ 0, t [0,.),

where k(t) denotes the time derivative of k at t.Production function f: R2+0 R+ is assumed to satisfy the following

conditions:

(A.1) Functionf is boundedand twicecontinuouslydifferentiableonR2++.


(A.2) For each given ke, the mapping f( , ke) is increasingly monotoneand concave.

(A.3) Let r(k)=f1(k, k) denote the derivative of f with respect to thefirst argument at (k, k). Then limkQ 0 r(k) > r+d.

A competitive equilibrium for this economy is defined as a triple of(absolutely continuous) functions {c(t), k(t), ke(t)}t \ 0 such that the pair{c(t), k(t)}t \ 0 solves problem (P) for ke(t)=k(t) for all t \ 0. It followsthat along an equilibrium path the following conditions are alwayssatisfied:

c(t)c(t)=1s[r(k(t))rd] (2.1)

k(t)=f(k(t), k(t))dk(t)c(t) (2.2)

limtQ.

ertc(t)s k(t)=0. (2.3)

Thus, a competitive equilibrium requires individual optimization, marketclearing, and consistency of beliefs. A steady-state equilibrium (cg, kg) is aconstant pair of positive solutions to Eqs. (2.1)(2.2).There are several studies on the equilibrium dynamics for growth models

with externalities (e.g., [2, 6], and references therein). Following Benhabiband Gali [3] we now consider an economy with three steady states. Thus,as illustrated in Fig. 1, the marginal productivity schedule, r(k), must cross

FIG. 1. Multiplicity of stationary solutions.

78 MANUEL S. SANTOS

FIG. 2. Dynamics of consumption and capital in an economy with externalities.

the constant function r+d at three given points, kgL, kgM, k

gH. Moreover,

Fig. 2 is intended to represent the phase diagram corresponding to Eqs.(2.1)(2.2) for this specific economy. Observe that steady states (cgL, k

gL)

and (cgH, kgH) are saddle-path stable, and steady state (c

gM, k

gM) is chosen to

be a spiral source. Benhabib and Gali [3] provide a discussion of certainassumptions generating this dynamic behavior. (These conditions canadditionally be envisaged from linearization of the system of Eqs. (2.1) and(2.2) at each particular steady state.)The thick lines in the diagram depict the stable arms WSL and W

SH of

steady states (cgL, kgL) and (c

gH, k

gH), which are joined by steady state

(cgM, kgM). These stable manifolds along with the three non-degenerate

steady states are equilibrium trajectories of our economy. Of course, as istypical in studies of the global equilibrium dynamics (cf. [37]) to ensurethat these are the only equilibrium solutions of our economy one wouldneed to rule out the possible existence of some other trajectories satisfying(2.1)(2.3). A global analysis of the equilibrium solutions, however, willnot be necessary for the present strategy of proof.If the model economy has a solution in feedback form, then we say that

there exists a Markov equilibrium. That is, a Markov equilibrium is a pairof decision rules c(k) and k=g(k) such that every trajectory {c(t), k(t)}t \ 0


generated by these feedback controls is a competitive equilibrium. Under(A.1)(A.3), we now claim that for the economy described in this sectionthese decision rules cannot be continuous functions.

Proposition 2.1. For the economy described in this section, there doesnot exist a continuous Markov equilibrium.

Proof. The proof of this proposition follows from the following steps:

(i) Point kgM cannot be a stationary solution under function g.Otherwise, given that kgM is a spiral source, continuity requires that thecontrol variable k=g(k) will circle around point kgM, generating an addi-tional countable number of steady state equilibria. This is not possible,since points kgL, k

gM, k

gH are assumed to be the only non-degenerate steady

state equilibria.(ii) The vector field points inward: g(k) > 0 for k < kgL and g(k) < 0

for k > kgH. Observe that (A.3) implies that g(k) > 0 for k sufficiently small;hence, by continuity, g(k) > 0 for every k < kgL. Moreover, the assertedboundedness of f in (A.1) and d > 0 imply that g(k) < 0 for some k largeenough; hence, by continuity, g(k) < 0 for every k > kgH.

(iii) For k < kgH every orbit starting at k under g cannot converge topoint kgH. Let k < k

gH. Then consider an equilibrium trajectory starting at a

point (c, k) converging to steady state (cgH, kgH). One can see from the

phase diagram (Fig. 2) that such a trajectory must belong to the stable armjoining points (cgM, k

gM) and (c

gH, k

gH). (Observe that for each plausible

equilibrium point (c, k), function g would be defined as k=g(k)=f(k, k)dk c.) But, analogously to (i), the continuity of our feedbackcontrols implies then that point k1 in Fig. 2 is an additional stationarypoint under g. This is therefore a contradiction, which validates this fact.In the same way, one can show

(iv) For k > kgL, every orbit starting at k under g cannot converge topoint kgL.

(v) Non-existence of a continuous decision rule, k=g(k). If g is acontinuous function, then (i) implies that kgL and k

gH are the only (non-

degenerate) stationary points under g. As the vector field points toward theinside (see (ii)) every orbit under g must converge to either kgL or k

gH. Then

(iii) entails that every orbit starting a point k < kgL must converge to kgL,

and (iv) entails that every orbit starting at a point k > kgH must converge topoint kgH. But (iii) and (iv) also imply that every orbit starting at point k inthe open interval (kgL, k

gH) will not converge to any stationary point, and

this stands in contradiction with the postulated continuity of function g. In

80 MANUEL S. SANTOS

other words, if g is a continuous function, then (ii)(iv) imply that thevector field points inward over the interval (kgL, k

gH). Then, the continuity

of g calls now for an additional stationary point in this interval, which is ofcourse in contradiction with (i).

The present proof of non-existence of a continuous Markov equilibriumshould be distinguished from more familiar results concerning spirallingtrajectories and non-continuous Markov equilibria. More specifically, tovalidate the non-existence of a continuous Markov equilibrium it is neces-sary to consider equilibrium functions over the whole domain; it is notenough to construct simply a non-continuous Markov equilibrium. Indeed,as an extension of the present analysis one could concoct instances ofeconomies that contain both continuous and non-continuous Markovequilibria. Therefore, the study of the local behavior of an equilibriumorbit would not be enough for present purposes.Also, it is relatively easy to prove that a finite-horizon version of

problem (P) would contain a continuous Markov equilibriumalbeit thecorresponding Markov equilibrium functions are non-stationary and mustbe parameterized by the time argument. Hence, within the family of modelswith a one dimensional state variable the non-existence of a continuousMarkov equilibrium is unique to the infinite-horizon model in that theremay be no single pair of continuous functions that can describe theequilibrium dynamics for consumption and investment.It is now readily seen from the phase diagram of Fig. 2 that the economy

contains a multiplicity of non-continuous Markov equilibria. These equi-libria can be derived from selecting appropriately portions of the upperpart of the manifoldWSL and of the lower part of the manifoldW

SH so as to

generate an equilibrium function k=g(k). For instance, let k be an arbi-trary point in the interval (k1, k2) of Fig. 2. Then consider the followingpaths: For k \ k, follow the lower part of WSH, and for k < k follow theupper part of WSL. This selection gives rise to an equilibrium function asthat depicted in Fig. 3. As a matter of fact, from these arguments one cansee that there is a continuum of these equilibrium functions with a point ofdiscontinuity k (k1, k2) in which the equilibrium function could be eithercontinuous from the right or continuous from the left.4

4 Of course, if the discontinuity is at point k1 the equilibrium function can only becontinuous from the left, and if the discontinuity is at point k2 the equilibrium function canonly be continuous from the right.

It should be stressed that in the present example every equilibriumtrajectory can only have one point of discontinuity, for equilibriumtrajectories can solely jump at time t=0 (i.e., at the initial point k) butnot at any other t > 0. In our economy the representative consumer is


FIG. 3. An equilibrium function with a discontinuity at point k.

solving a strictly concave maximization problem; hence, k must be a con-tinuous function of t for all t > 0 (cf. Fleming and Rishel [13, Chap. 1,Corollary 3.3]). Therefore, the convexity of the agents optimizationproblem precludes any other discontinuity of an equilibrium trajectoryafter time t=0. Moreover, as there is a single state variable, a functiondescribing a Markov equilibrium is conformed by a finite number of equi-librium trajectories or connected arcs. Thus, excepting initial jumps theequilibrium function is as smooth as the equilibrium trajectories (cf. Santosand Vila [35]). This is an important difference with respect to multi-dimensional models, or with respect to discrete-time models. For everyeconomy satisfying (A.1)(A.3), every Markov equilibrium would generallyhave at most a finite number of simple discontinuities (i.e., discontinuitiesof the first kind). In a discrete-time formulation of the model, however,current investment affects the stock of capital in discrete amounts, and soother types of discontinuities may in principle be observed.Finally, it may be helpful to call attention to the analogous analysis of

optimal solutions in non-concave social planning problems (cf. [12, 37]).In these optimization problems, a spiral source such as (cgM, k

gM) is not

optimalit yields less utility than other feasible trajectoriesand neitherare those trajectories starting at points arbitrarily close to (cgM, k

gM). As a

result, the policy function may be discontinuous. Since there is usually asolution that yields the highest utility, these optimal economies do notgenerally possess multiple equilibrium functions. In our decentralizedeconomy, however, all steady states and every trajectory converging toa steady state are competitive equilibria. These solutions can be appro-priately selected so as to generate a multiplicity of non-continuousequilibrium functions.

82 MANUEL S. SANTOS

3. ECONOMIES WITH TAXES

Under regular assumptions, Bizer and Judd [5], Coleman [10], andGreenwood and Huffman [16] have shown the existence of continuousMarkov equilibria in standard one-sector growth economies with taxes oncapital returns. This section contains two examples that illustrate thefragility of this well established result. The first example presents aneconomy with two sectors and asymmetric taxation; this economy fails topossess a Markov equilibrium. The second example reconsiders a one-sector economy with a decreasing tax rate on capital returns. Here, everyMarkov equilibrium must have one discontinuity point. One main conclu-sion from this section is that previous results on existence of continuousMarkov equilibria for one-sector economies with taxes appear to be ratherdifficult to extend to more complex settings.

3.1. An Economy with Physical and Human Capital and AsymmetricTaxation

Consider an endogenous growth economy with physical and humancapital and asymmetric taxation across sectors. The available stocks ofphysical capital, K, and human capital, H, can be allocated to the produc-tion of the physical good or to the production of education. Production ineach sector is determined by a simple CobbDouglas, constant returns toscale technology. Thus, let

X=AKaxH1ax , A > 0, 0 < a < 1 (3.1)

be the quantity produced of the aggregate good using the inputs vector(Kx, Hx), and let

Y=BKbyH1by B > 0, 0 < b < 1 (3.2)

be the quantity produced of the education good using the inputs vector(Ky, Hy).Let q refer to the price of human capital in terms of the aggregate good.

Then, for given physical capital returns, r, and wage rates, w, profit maxi-mization by firms in each sector implies that

aAKa1x H1ax [ rx, qbBKb1y H1by [ ry (3.3)

(1a) AKaxHax [ wx, q(1b) BKbyHby [ wy. (3.4)

(Here, of course, subscripts refer to the corresponding sector, and equalitymust hold for each condition whenever the corresponding vector (Kx, Hx)or (Ky, Hy) is positive.)


A representative consumer is also present in this economy. This agent isendowed with k0 units of physical capital and h0 units of human capital.At each moment in time the agent decides on the amounts to be allocatedfor consumption, c, physical capital investment, Ik, and human capitalinvestment, Ih, and on the fractions of physical capital, v, and of humancapital, u, to be devoted to the production of the physical good. Theremaining fractions of physical and human capital are devoted to theproduction of the education good. All physical capital returns are subjectto a single flat-rate tax, yk; in contrast, wages are taxed in the aggregategood sector at a rate, yh, and are subsidized in the education sector at arate, fh. The proceeds of taxation are rebated to the representativeindividual as lump-sum transfers, T.For given paths for good and factor prices and tax rates {q(t), rx(t),

ry(t), wx(t), wy(t), yk, yh, fh} the agent chooses {Ik(t), Ih(t), c(t), u(t), v(t)}so as to solve the following optimization problem

max F.0

c1s

1sert dt (P )

s.t. c+Ik+qIh [ (vrx+(1v) ry)(1 yk) k

+(uwx(1yh)+(1u) wy(1+fh)) h+T

k=Ikdk

h=Ih

c \ 0, k \ 0, h \ 0, 0 [ u [ 1, 0 [ v [ 1

k(0)=k0, h(0)=h0 given, r > 0, s > 0, d > 0, t [0,.).Here, d > 0 is the depreciation rate of physical capital; for convenience it

is assumed that human capital is not subject to depreciation. Now, thespecification of the government budget constraint and the market clearingconditions will complete our characterization of a competitive equilibrium.Revenues from factor taxation in both sectors are simply transferred asadditional income to the representative individual

T=ykrxKx+ykryKy+yhwxHxfhwyHy. (3.5)

Observe that these transfers depend on aggregate variables and not onthe decisions made by the representative agent. Finally, market clearingrequires at all times equilibrium in good and factor markets

c+Ik=X=AKaxH

1ax , Ih=Y=BK

byH

1by (3.6)

k=K=Kx+Ky, vk=Kx (3.7)h=H=Hx+Hy, uh=Hx.

84 MANUEL S. SANTOS

That is, aggregate quantities chosen by firms must be consistent with theagents choices.Formally, a competitive equilibrium for this economy is a path of prices

and tax rates {q(t), rx(t), ry(t), wx(t), wy(t), yk, yh, fh}t \ 0, a set of choicesfor aggregate production {Kx(t), Ky(t), Hx(t), Hy(t), X(t), Y(t)}t \ 0, and aset of choices for the representative agent {Ik(t), Ih(t), c(t), u(t), v(t)}t \ 0that solve problem (P ), and such that conditions (3.1)(3.7) are alwayssatisfied.5

5 Observe that this definition can encompass non-interior equilibria where at a certain timeeither good X or good Y is not produced. For instance, good Y will not be produced if rx \ ryand wx(1 yh) \ wy(1+fh) and one of these inequalities is strict.

A balanced growth path {k(t), h(t), c(t), u(t), v(t)}t \ 0 is a competitiveequilibrium such that {k(t), h(t), c(t)}t \ 0 grow at a constant rate n, and{u(t), v(t)}t \ 0 stay constant. If n > 0, then the balanced growth path is saidto be interior.In the present context, local stability properties of a balanced growth

path have been studied by Bond et al. [7]. The following simple refor-mulation of one of their results (see their Proposition 5) will be useful forour purposes.

Proposition 3.1. Assume that the above economy has an interiorbalanced growth path. Let ((1b) a/b(1a))((1+fh)/(1yh)) > 1 >(1b) a/b(1a). Then, the balanced growth path is locally unstable.

Raurich [32] and Ortigueira [27] provide extensions of these results toenvironments with active government policies.A Markov equilibrium is a list of functions {Ik, Ih, c, u, v} that depend

on (k, h) such that every trajectory generated by these decision rules is acompetitive equilibrium. Since the utility function follows a power law andeach production sector features constant returns to scale, there is norestriction of generality to impose that functions Ik, Ih, and c be homoge-neous of degree one in (k, h), and that functions u and v be homogeneousof degree zero. Then our state variable is in fact the ratio m=kh , and ourmain objective is to study if there is a Markov equilibrium so that the lawof motion of variable m can be expressed by a function m=g(m). We havethe following:

Theorem 3.2. Under the conditions of Proposition 3.1, for the economydescribed in this section there does not exist a Markov equilibrium.

It should be stressed that the theorem asserts the non-existence of a(continuous or non-continuous) Markov equilibrium. As discussed in thepreceding section, equilibrium trajectories must be continuous functions of


t for all t > 0. Then the strategy of proof is to show that this continuityproperty of equilibrium trajectories, together with the existence of aunique, locally unstable steady-state solution, is not compatible with theexistence of a Markov equilibrium.

Proof of Theorem 3.2. It is shown in the Appendix that if there is aninterior balanced growth path, then this is unique and there is not anyother boundary balanced growth path. Now, assume the existence of anequilibrium function m=g(m). Under the present assumptions one canreadily establish that the vector field must point inward: g(m) > 0 for mclose to zero and g(m) < 0 for m sufficiently large.Then pick a point m. As illustrated in the preceding section every equi-

librium trajectory starting at m must vary continuously with t for all t > 0.As the vector field points towards the inside, every initial condition munder g must converge to a singular point mg. Moreover, one can showthat mg is a steady-state solution for our economy.6 We have therefore

6 This statement follows from inspection of Eqs. (13a)(14) in [7]. This equations systemhas a recursive structure, and the relative price q is constant over the transitional dynamics.

reached a contradiction, since there is at most one stationary solution mg,and such a steady state is locally unstable. The proof is complete. L

Within the space of feasible parameter values, one can readily see thatthe conditions of Proposition 3.1 will hold for an open set of economies.Indeed, it should be clear that the results in this paper concerning eithernon-existence of a Markov equilibrium or non-existence of a continuousMarkov equilibrium are usually robust to small perturbations of theprimitives. In general, these examples are not isolated, and small changesin parameter values will not restore existence of a continuous Markovequilibrium.The following example is intended to shed light on the preceding analysis.

Example 3.3. Let

X=AKaxH1ax , Y=BK

byH

1by

a=0.3, b=0.4, yh=0.4, fh=0.2.

Then one can check that the parametric condition stated in Proposition 3.1is satisfied:

(1b) ab(1a)

1+fh1yh

> 1 >(1b) ab(1a)

.

86 MANUEL S. SANTOS

Now, following our derivation in the Appendix the existence of an interiorbalanced growth path is guaranteed by a selection of appropriate values forparameters A, B, s, r, and d. These latter parameters, however, do not havea key role in the qualitative dynamics about a balanced growth path. Thelocal stability around a balanced growth path is driven by the rankings ofphysical factor intensities and factor income shares across sectors. Asexplained in Bond et al. [7], taxation may distort the ranking of factorintensities and income shares. Thus, the present tax rates make sector Xphysical capital intensive, even though this capital commands a higherrelative income share in sector Y. As a result of this reversal in the rankingsof factor intensities and income shares, the Rybczynski and StolperSamuelson effects work in opposite directions; and the saddle-path stabilityof the stationary solution no longer holds. Then under the aforementionedforward-continuity property of equilibrium trajectories we have establishedthat there cannot exist a Markov equilibrium.

3.2. A One-Sector Economy with a Decreasing Tax Rate on Capital Returns

It is known from the work of Coleman [10] and Greenwood andHuffman [16] that for regular one-sector economies with flat or increasingtax rates on capital returns there always exists a continuous Markov equi-librium. Our purpose now is to show that continuity may be lost if theaverage tax rate on capital returns is a decreasing function of the capitalstock.Time is discrete, t=0, 1, 2, ..., in this economy. There exist but a single

consumer, who is endowed with k0 units of capital, and a unique firmwhich operates a one-sector technology described by a function, y=f(k).The firm is owned by the consumer, who gets all its revenues as eitherprofits, p, or returns from capital, r. Capital returns are taxed at a ratey(kt). The tax rate is allowed to depend on the capital stock of theeconomy.The consumer chooses {ct, xt}t \ 0 in order to solve the following

optimization problem

max C.

t=0c tu(ct)

s.t. ct+xt [ pt+(1yt) rtkt+Ttkt+1=xt+(1d) kt

k0 given, 0 < c < 1, 0 [ d [ 1

ct \ 0, xt \ 0, t=0, 1, 2, ...


As in our two previous examples, the agent has perfect foresight andconsiders that yt, rt, and Tt are exogenously given. These variables areviewed as functions of aggregate quantities. The firm takes rt as given andchooses Kt in order to maximize one-period profits, pt=f(Kt)rtKt. Thegovernment sets Tt=ytrtKt. In a competitive equilibrium the good andfactor markets must clear, so that

xt+ct=f(Kt) and kt=Kt, t=0, 1, 2, ... (3.8)

At an interior solution, these equilibrium identities and utility maximiza-tion conform to the Euler equation

u(ct)+cu(ct+1)[f(kt+1)(1y(kt+1))+(1d)]=0. (3.9)

Under regular assumptions, if the tax rate y is a non-decreasing functionof k, then (3.9) has at most a unique stationary solution, kg. But if y is adecreasing function of k, then there could be multiple steady states. Wenow consider a simple parameterization in which there are three stationarysolutions. Let

u(c)=log c, f(k)=k1/3

c=0.95, d=1

with the continuous, piecewise linear tax schedule

y(k)=0.10 if k [ 0.1600020.0510(k0.165002) if 0.160002 [ k [ 0.1700020 if k \ 0.170002.

Under this parameterization, Euler equation (3.9) has three interior sta-tionary solutions, kgL=0.152148, k

gM=0.165002, k

gH=0.178198. Observe

that the tax rate is constant around steady states kgL and kgH, and hence

these steady states are saddle-path stable. The slope of the tax schedule atkgM has been chosen so that this point is a spiral source. In the presentmodel, this is always possible.7 The dotted lines in Fig. 4 depict the local

7 Steady states displaying complex eigenvalues may likewise be generated under alternativetax schedules. For instance, a variable subsidy on capital returns may give rise to a spiral sink.

equilibrium dynamics for this economy. Observe from this figure that thecorresponding stable manifoldsWSL andW

SH are joined by point (k

gM, k

gM).

As in our preceding examples, it is now contended that the equilibriumlaw of motion of capital cannot be described by a continuous function,kt+1=g(kt). Observe that the qualitative dynamics of this example are like

88 MANUEL S. SANTOS

FIG. 4. PEA and accurate equilibrium solutions.

those of our illustration in Section 2. In a discrete-time model, however, thedynamics could be more complex (i.e., one-dimensional discrete-timemodels may generate periodic orbits or more complicated recurrent paths),and hence the corresponding arguments for proving non-existence of acontinuous decision rule become more delicate.

Proposition 3.4. For the economy described in this section, there doesnot exist a continuous Markov equilibrium.

Proof. (i) The vector field points inward and so kg=g(kg) for somekg > 0. If g is a continuous function, then one can show that g(k) > k for knear k=0. Also, under our production function f(k)=k1/3 it follows thatg(k) < k for k large enough. Therefore, there must be some stationarysolution kg=g(kg) for kg > 0.

(ii) Point kgM is not a stationary solution under g. At points near thesteady-state solution kgM, the dynamics are well approximated by thelinearization of (3.9). The eigenvalues of this linear system are the complexconjugate pair l and l, with l=0.189604+1.03541i. Let r=(0.1896042+1.035412)1/2=1.05263 and h=arc cos0.189604r =1.75191. Then, making anappropriate coordinate change around the point (kgM, k

gM), every nearby

equilibrium point (k, g(k)) undergoes under this linear system a coun-terclockwise rotation through h=1.75191 radians followed by a stretchingout of the distance to the steady state (kgM, k

gM) by a factor of r (cf. [17,

p. 56]). Thus, for k0 sufficiently close to kgM, every equilibrium orbit {kt}

.t=0

generated by function g will cross a number of times the 45-degree line (i.e.,


this orbit will contain several pairs of points kt > kgM and kt > k

gM such that

g(kt) < kt and kt > g(kt)). If kgM=g(k

gM), and g is a continuous function,

then there must be a countable number of steady states arbitrarily close tokgM. Consequently, k

gM ] g(kgM).

(iii) If kgH=g(kgH), then there exists a neighborhood U of (k

gH, k

gH)

such that [graph(g) 5 U] WSH. If kgH=g(kgH), then the continuity of gand point (i) above imply that every trajectory under g starting at k > kgHmust converge to kgH. Therefore, the graph of g belongs to W

SH for every

point k \ kgH. Suppose now that k < kgH. Let lmaxH > 1 > lminH > 0 be theeigenvalues of the linearization of (3.9) at kgH. By the l-Lemma (cf. [29,p. 84]), there is no loss of generality in assuming that the left-side slope ofg at kgH is well defined and it is either l

maxH or l

minH . If the left-side slope is

lmaxH , then k > g(k) for k close to kgH. In view of (i) and (ii) above, we must

then have convergence to the lower steady state so that kgL=g(kgL). More-

over, the continuity of g entails that the right-side slope of g at kgL islmaxL > 1, where l

maxL is the corresponding eigenvalue of the linearization of

(3.9) at kgL. But this is impossible, since g is a continuous function, andk > g(k) for k in the open interval (kgL, k

gH). This contradiction then

establishes that the left-side slope of g at kgH is lminH , and hence the graph of

g belongs toWSH for k < kgH close to k

gH. In the same way one can prove

(iv) If kgL=g(kgL), then there exists a neighborhood U of (k

gL, k

gL) such

that [graph(g) 5 U] WSL.(v) Non-existence of a continuous decision rule kt+1=g(kt). It is now

easy to see that (i)(iv) imply that function g must have at least one pointof discontinuity. Thus, (i)(ii) entail that kgL and k

gH are the only plausible

stationary solutions under g. Moreover, by (iii) and (iv) both kgH and kgL

should be stationary solutions, since neither WSH nor WSL alone can

encompass graph(g). Then, as in Section 2 we may now conclude from (ii)that g must have a point of discontinuity in the interval [k1, k2] of Fig. 4.The result is thus established.

In contrast to the continuous-time framework, it should be observed thata discrete-time finite-horizon economy may contain a non-continuousMarkov equilibrium. Indeed, even in the case of a finite-horizon for eachinitial condition k0 there could be a continuum of solutions to the systemof Euler equations (3.9). Therefore, standard assumptions can onlyguarantee the upper-semicontinuity of the equilibrium correspondence,even if attention is restricted to Markov equilibria. Consequently, severaltypes of discontinuities may arise for equilibrium decision rules of discrete-time models.Finally, let us mention that we have been unable to establish a discrete-

time counterpart of the example of the preceding section. That is, an

90 MANUEL S. SANTOS

example of an economy which always contains a competitive equilibrium,but which nevertheless lacks existence of a Markov equilibrium. Again, oneshould appreciate here a major difference with respect to the one-dimen-sional continuous-time framework. In continuous-time economies, the non-existence of a Markov equilibrium relied on the forward continuity of theequilibrium trajectories. This argument, however, does not carry throughto a discrete-time framework, since in a discrete-time model every equilib-rium orbit is just made of a countable number of points. Thus, an equilib-rium function may well display an infinite number of discontinuities.

4. SOME THOUGHTS ON THE SIMULATION OF THESEECONOMIES

A model may be simulated even though it may not possess a continuousMarkov equilibrium. Indeed, in our previous examples all equilibria can becharacterized as solutions to systems of Euler equations satisfying certaintransversality conditions, and often these paths can be readily ascertained.But matters may be more complex in models with several state variables orin the presence of uncertainty.Several numerical procedures have been proposed for the computation of

continuous Markov equilibria (cf. [11]). For most algorithms, however, asolution is not known to exist. Existence of a continuous Markov equilib-rium is guaranteed in the iterative schemes considered by Bizer and Judd[5], Coleman [10], and Greenwood and Huffman [16], provided that acertain monotonicity condition is satisfied. These methods have not beenextended to models with several endogenous state variables.Some numerical methods may preserve the continuity of the value and

policy functions at each iteration. But it should be stressed that a key stepin the computation of solutions for infinite-horizon models is to show thatthe limiting functions are also continuous and constitute true equilibriumdecision rules. Therefore, typical procedures for stopping the algorithm(say, whenever two consecutive policy functions gn and gn+1 fall within agiven tolerance bound, ||gngn+1 || [ E) may lead to the wrong belief thatthe true policy function is continuous. The numerical experiment presentedbelow should warn us against the use of computational methods withoutbasic knowledge of assumed qualitative properties of solutions includingexistence of a Markov equilibrium.As is well understood, these pitfalls can also be encountered in quadratic

economies. Thus, not every linear Euler equation system has a correspond-ing continuous equilibrium function. A solution always exists for thelinearization of Eulers equation in regular concave optimization problems(cf. Scheinkman [36]), but it would not exist for the linearizations of our


previous unidimensional examples at those steady states displayingcomplex eigenvalues. For a linear system of Euler equations to be solvablecertain stability conditions need to be satisfied (cf. [25]), and in theabsence of those conditions a finite-iteration approximation scheme mayproduce deceptive outcomes.Our examples in Sections 2 and 3 do not possess a continuous Markov

equilibrium. Hence, it may be inappropriate to compute equilibrium func-tions using algorithms with continuous solutions. Moreover, application ofthese algorithms may not unveil existing discontinuities in the equilibriumlaw of motion.As a simple numerical experiment, a version of the PEA algorithm with

collocation, as described in Christiano and Fisher [9], has been applied toour parameterized economy in Section 3.2. The algorithm produced stableoutcomes in the sense that successive higher-order interpolants led toessentially the same solutionno major variations were discerned in thesolution coefficients from changes in the polynomial degree of the inter-polant. Again, from this parsimonious behavior one may be led to believethat this economy contains a continuous Markov equilibrium. Figure 4displays a representative computed equilibrium function under the PEAalgorithm with collocation. But as shown in the same figure the computedpolicy function deviates sensibly from the accurate solution. For the com-puted PEA solution, we checked the size of the Euler equation residuals (cf.Santos [34]), and the residuals were fairly large (i.e., of the order of 103).Moreover, the residuals remained of roughly the same size under higher-order interpolations. Thus, evaluation of the residuals may provide a goodindication that the algorithm is not performing well, or that it is inadequatefor the simulation of the model.8

8 Although an evaluation of the residuals may alert us that the algorithm is not performingwell, it is not clear that this criterion may always be effective. Since the model is not concave,small residuals may be associated with large deviations from the true policy function. Whatseems to be true is that the residuals may yield a better indication of accuracy than the pre-viously mentioned check on the stability of the solution coefficients over different interpola-tion schemes.

Further difficulties arise in attempting to compute equilibrium solutionsfor the economy of Section 3.1. This is a deterministic model with a uniquestationary solution, but such a steady state is unstable. Numerical solutionof the model seems a more laborious undertaking, since no equilibriumtrajectory converges to a balanced growth path. Furthermore, as shown inOrtigueira and Santos [28], apart from the stationary solution there is noequilibrium trajectory that lies in the interior at all times; that is, along anynon-stationary equilibrium orbit the economy is constantly switching fromperiods in which both production sectors are active to periods in which

92 MANUEL S. SANTOS

only one of the sectors is active.9 The most obvious numerical procedure

9 Computation of equilibria in the other two economies will amount to tracing back thosetrajectories converging to a steady state, and this task can be accomplished by relatively fastnumerical methods.

for the computation of equilibria in the present economy that comes tomind is simply to single out those solutions to the system of Euler equa-tions that satisfy feasibility and the transversality condition at infinity. Butthis seems to lead to a rather lengthy searching procedure, where sortingout the desired trajectories involves some kind of trial-and-error method orwhat is known as the shooting method.From a computational standpointand for other purposes as wellit

may be of importance to establish links between the existence of Markovequilibria and qualitative properties of the model such as existence anduniqueness of stationary solutions, local stability, and indeterminacy ofsolutions. Let us mention that monotonicity together with determinacy ofequilibrium solutions is a key property for validating the existence of acontinuous Markov equilibrium. But most local properties play a muchweaker role, since a multiplicity of continuous and non-continuous Markovequilibria may coexist in a dynamic model. For instance, in our two pre-vious examples with three steady states there could be other configurationsin which none of the stable arms sprinkles from the middle steady state.Then there could be two continuous Markov equilibria even if the middlesteady state is a spiral. Therefore, local properties of solutions cannotusually provide definite clues, since as already pointed out globalarguments are needed to rule out the existence of a continuous Markovequilibrium.Our discussion so far has been confined to simple deterministic models.

Here, lack of existence of a continuous Markov equilibrium may nothamper the analysis and simulation of the model. Sometimes there areways to characterize an equilibrium solution which are easily amenable tocomputation. Further difficulties, however, may originate in multidimen-sional models or in stochastic environments, where it may be much harderto ascertain the limiting behavior of the orbits, the existence of cycles,limit sets, or invariant distributions, and as to whether or not there is acontinuous Markov equilibrium.

5. CONCLUDING REMARKS

This paper contains three examples of regular dynamic economies whichlack existence of continuous Markov equilibria. In the first example, thesource of non-existence stems from a pronounced production externality.


Nevertheless, this economy displays a multiplicity of non-continuousMarkov equilibria. The second example describes a two-sector model withphysical and human capital and asymmetric taxation in the labor markets.It is shown that for a certain range of parameter values there is no Markovequilibrium. Finally, the third example reconsiders a one-sector model witha decreasing tax rate on capital returns. As in the first example, everyMarkov equilibrium must be non-continuous.Of course, the most intriguing example is that of Section 3.1 of non-

existence of a Markov equilibrium. It remains to investigate the possiblenon-existence of Markov equilibria in discrete-time economies. A majoranalytical difficulty in the discrete-time framework is that an equilibriumorbit is made up of a countable number of points, and so every Markovequilibrium comprises a continuum of equilibrium orbits. In contrast, inthe continuous-time economies recasted in the first two examples, anequilibrium orbit varies continuously with time, and so every Markovequilibrium is conformed by a finite number of equilibrium orbits.In all these economies, each individual agent is facing a convex optimi-

zation problem, and hence the sources of non-existence of a continuousMarkov equilibrium are to be found on the side effects that these distortionsinflict upon the dynamics of the equilibrium system. These results highlightimportant differences between optimal and non-optimal economies, andattest against what appears to be a widely held belief that competitive-market economies may be suitably respecified as optimization problems. Inthe presence of simple tax schemes or externalities an optimization problemcharacterizing equilibrium solutions must satisfy certain endogenous con-sistencyconditions,whichmayprecludetheexistenceofaMarkovequilibrium.These findings should stimulate further advances in the analysis and

simulation of economic models. Thus, there has been extensive work on theexistence of continuous Markov equilibria in competitive economies (cf.[11, 37]), and the present analysis should help elucidate the conditionsunder which these equilibria may exist. Second, our work poses furtherchallenges to the computation of equilibria in competitive economies. Ourdiscussion in the preceding section suggests that there is no dominantnumerical method for the analysis and simulation of dynamic economies,and hence the search for a suitable numerical procedure should begin withan analysis of the characterization of equilibrium solutions, along with allother additional theory embedded in the model. Of course, there are situa-tions in which the model is not easily amenable to theoretical analysis, andwhere the application of general-purpose, standard numerical methods maybe most valuable. The present paper, however, adds a word of caution tothis basic view, and illustrates that the most powerful numerical techniquesmay be inadequate in those situations where paradoxically their use wouldbe most badly needed.

94 MANUEL S. SANTOS

APPENDIX

In this appendix, we show that for the economy described in Section 3.1there is at most one balanced growth path. Thus, if there is one interiorbalanced growth path, there cannot be any further boundary stationarysolutions. There are several related papers (e.g., [7, 19, 26, 32]) in whichone can find a weaker version of this result, namely, that the economy canpossess at most one interior balanced growth path. These papers restrictthe analysis to interior solutions, leaving out the existence of boundarystationary solutions. For the purposes of our global analysis, however, it isnecessary to consider all possible stationary solutions.Let kx=

ukuh and ky=

(1v) k(1u) h . Then, following Ladron-de-Guevara et al.

[19] and Stokey and Rebelo [39], an interior balanced growth path{kgx , k

gy , (

ck)

g, ug, ng} must satisfy the conditions

r+sng+d=(1yk) aAkga1x (A.1)

r+sng=(1+fh)(1b) Bkgby (A.2)

(1yh)(1a) kgx

a=(1+fh)(1b) k

gy

b(A.3)

(1ug) Bkgby =ng (A.4)

ug 1hk2g Akgax =d+ng+1 ck2g. (A.5)

Equations (A.1)(A.3) uniquely determine the growth rate, ng > 0, andthe capital ratios, kgx , k

gy . Then, (A.4) yields a unique value for u

g, andconsidering that kh=ukx+(1u) ky the consumption capital ratio (

ck)

g isobtained from (A.5). Observe that the level variables c, k, h grow at theconstant rate ng, and the ratios ck and

kh remain constant along the balanced

growth path. Hence, in what follows we fix h=1.Good and factor prices can now be read off from these quantities. Thus,

hg1=cgs is the shadow price of the aggregate consumption good, and

hg2=hg1q

g is the shadow price of human capital, where

qg=aAkga1xbBkgb1y

=(1yh)(1a) Ak

gax

(1+fh)(1b) Bkgby

. (A.6)

These two equalities follow from the required equality of factor returnsacross sectors, rx=ry and (1 yh) wx=(1+fh) wy. Using these identities,we can rewrite (A.2) as

qg=hg2hg1=wgx(1 yh)r+sng

. (A.7)


As (A.1)(A.5) imply that there is at most one interior balanced growthpath, it now remains to prove that in such a case there cannot be a non-interior stationary solution.Observe that it cannot be optimal to specialize in human capital produc-

tion. Hence, at a non-interior balanced growth path all factors must bedevoted to the production of the physical good, and the human capitalsector will remain inactive. Thus, if {kx, c} is a non-interior stationarysolution, the following restrictions on good and factor prices must besatisfied

r+d=(1 yk) aAka1x (A.8)

rh2=(1 yh) h1wx (A.9)

rx \ ry (A.10)

(1 yh) wx \ (1+fh) wy. (A.11)

Since b > a and rgx > rx it follows from the StolperSamuelson effect (cf.(A.6)) that (A.10) and (A.11) are only possible if q < qg. On the other hand,(A.1) and (A.8) imply that kx > k

gx , and so wx > w

gx . Hence, (1yh) w

gx/

(r+sng) < (1yh) wx/r. Then, (A.7) and (A.9) entail that qg=hg2/h

g1

< h2/h1=q. But, as already argued, qg < q cannot hold true under rgx > rx

and inequalities (A.10) and (A.11).This contradiction shows that if there is an interior balanced growth

path, there cannot be a boundary stationary point. As a matter of fact, it isnow easily established that each economy in this class can only have onestationary solution. Equations (A.1)(A.5) guarantee the uniqueness of aninterior balanced growth path, and Eqs. (A.8)(A.9) guarantee theuniqueness of a stationary boundary point. Moreover, the precedingarguments rule out the coexistence of both stationary solutions.

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98 MANUEL S. SANTOS

1. INTRODUCTION 2. NON-EXISTENCE OF CONTINUOUS MARKOV EQUILIBRIA IN AN ECONOMY WITH EXTERNALITIESFIG. 1FIG. 2FIG. 3

3. ECONOMIES WITH TAXESFIG. 4

4. SOME THOUGHTS ON THE SIMULATION OF THESE ECONOMIES5. CONCLUDING REMARKSAPPENDIXREFERENCES

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