Martingale analysis of dynamic tax incidence in a nonstationary growth model

ELSEVIER Journal of Economic Dynamics and Control

21 (1997) 371-389

Martingale analysis of dynamic tax incidence in a nonstationary growth model

Sumit Joshi

Department of Economics, George Washington University, Washington, DC 20052, USA

(Received December 1994; final version received 1996)

Abstract

This paper develops a new martingale technique to study dynamic incidence of a capital income tax in nonstationary aggregate growth models. The stochastic Euler conditions are utilized to generate a submartingale process which is sensitive to a divergence between the no-tax and after-tax return to capital per unit. This submartingale process is instrumental in generalizing the classical result on the long-run incidence of a capital income tax in a stationary deterministic paradigm to nonstationary stochastic growth models. Since the Euler characterization is immediately available in aggregate growth models, the method developed here can also be utilized to study the dynamic incidence of other forms of taxation in general nonstationary aggregate growth models.

Key words: Capital income tax; Dynamic incidence; Nonstationarities; Martingales; Euler conditions JEL classi$cation: H22

1. Introduction

This paper contributes a new technique based on martingale methods to study the dynamic incidence of capital income taxation in nonstationary growth

This research was started in Summer 1993 and was made possible by a Junior Scholar Incentive Award granted by the George Washington University. I am extremely grateful to the referees of the journal for their comments and suggestions which have significantly improved the paper. I remain responsible for any errors.

01651889/97/$15.00 0 1997 Elsevier Science B.V. All rights reserved PII SO165-1889(96)00937-2

372 S. Joshi /Journal of Economic Dynamics and Control 21 (1997) 371-389

models.’ A classical result on the long-run incidence of a capital income tax, obtained in deterministic growth models with time-stationary preferences and technology, states that the tax is completely shifted to labour in the long run. From an economic policy viewpoint, this result indicates that a (small) change in the capital income tax will not affect after-tax return to capital in the long run. The definitive work in this regard is of Becker (1985). Given the economic significance of this classical result, it is worth investigating if the result also obtains in growth models which incorporate two essential elements of the real world - nonstationarities and uncertainty. With this in mind, it is illustrative to summarize Becker’s argument to highlight the potential problems in generalizing the classical result to the nonstationary stochastic case.

Let f be the neoclassical production function and 0 -C 6 < 1 the discount factor. Then, in the no-tax case, the steady state capital stock k* satisfies f’(k*) = l/6, i.e., the return to capital per unit in the long run is equal to the constant marginal rate of impatience. An elegant Equivalence Proposition due to Becker (1985) establishes that the perfect foresight equilibrium of a competitive economy in the presence of a tax 0 E (0,l) on capital income corresponds to the solution to a planner’s intertemporal maximization problem with the discount factor S(1 - 0). In the tax-perturbed case, therefore, the steady state capital stock k** is characterized by f’(k**) = l/6(1 - 0). Alternatively, (1 - O)f’(k**) = l/6, indicating that the net (or after-tax) return to capital per unit is independent of 19 and is equal to the same constant marginal rate of impatience as in the no-tax case. In other words, the tax on capital income is completely shifted to labour. Note, however, that since f is strictly concave, the steady state capital stock k** in the tax-perturbed case is smaller than the steady state capital stock k* of the no-tax case.

The problem in the stochastic case is that the steady state corresponds not to a capital stock but to a time-invariant probability distribution on the set of capital stocks. Such a steady state probability distribution does not have the same neat and manipulable characterization as the golden rule capital stock of the deterministic paradigm. Hence, in the stochastic case, it is a substantially harder task to demonstrate that the steady state probability distributions associated with the no-tax and after-tax return to capital per unit are identical.

The definitive analysis on dynamic incidence of a capital income tax in a stochastic environment is that of Danthine and Donaldson (1985). However, the primary emphasis there was on the effect of the capital income tax 8 on the

r Martingale techniques have been used extensively in the asset pricing literature (Duffie, 1988). In the optimal growth literature, it has been used to prove the late (or twisted) turnpike property that optimal programs from different initial stocks converge almost surely (Brock and Majumdar, 1978; Fiillmer and Majumdar, 1978). To the best of our knowledge, they have not been applied to the analysis of dynamic tax incidence.

S. Joshi /Journal of Economic Dynamics and Control 21 (1997) 371-389 373

mean and variance of the steady state probability distribution. The classical result on complete shifting of the tax was not treated. Further, the model assumed a stationary stochastic environment and stationary preferences and technology. The form of uncertainty was also rather specialized since it was assumed to affect production in a multiplicative fashion.

The major motivation behind this paper is the issue of how general is the classical result on complete shifting of a capital income tax. In particular, does the classical result obtain in a general growth model permitting nonstationarities and uncertainty? The presence of nonstationarities vitiates classical steady state analysis because stochastic steady states may not exist. The classical result on shifting, therefore, has to be couched in terms of the asymptotic behaviour of the dynamic paths of the no-tax and after-tax return to capital per unit. This paper establishes that the classical result does generalize to the nonstationary framework by demonstrating that the no-tax and after-tax return to capital per unit converge almost surely. Hence, the tax is completely shifted to labour in the long run. The analysis has the desirable characteristic of concentrating attention on the entire dynamic time path of the return to capital per unit rather than restricting the study to a comparison of stochastic steady states. In fact, it is shown that classical steady state comparisons follow as a corollary by specializing the general framework of this paper to the stationary case. This extends the stochastic analysis of Danthine and Donaldson (1985) to include the classical result with a general (not necessarily multiplicative) specification of uncertainty.

The proofs of all lemmas and theorems are relegated to an Appendix.

2. The recursive competitive equilibrium

This paper follows the rational expectations recursive competitive equilibrium model of Danthine and Donaldson (1985) and Prescott and Mehra (1980). The underlying stochastic aggregate growth model follows Majumdar and Zilcha (1987) in allowing nonstationarities in preferences and in the evolution of the stochastic environment. We note at the outset an important point regarding notation. Following Prescott and Mehra (1980), we distinguish between aggregate (or economy-wide) capital, _kr, and the representative consumer’s holding of capital,2 k,. The model comprises of the following elements:

‘As noted in Danthine and Donaldson (1985, p. 258, Fn. 1) and Prescott and Mehra (1980, Section 4), a correct specification of the consumer’s problem requires that k, should be allowed to vary independently of k,. In equilibrium, however, given the subsequent assumption that consumers are identical and distributed on the interval [0, 11, it will follow that k, = k,.


2. I. The stochastic environment

Let the closed interval [a, fi] in Iw + , 0 < a < /? < co, endowed with the Bore1 a-field 6, represent the possible states of the environment at date t 2 1 and o1 E [a, p] the state at date t 2 1. The stochastic environment is given by the probability space (Q, 9, v) where 52 = flp [a, /?I, 5 = @FS is the o-field on Sz generated by the measurable cylindrical sets, and v is the probability measure on 0. No i.i.d restrictions are imposed thereby allowing possible nonstationarities in the evolution of the environment. (St”l> denotes the filtration on D where S$ is the sub-u-field on 52 induced by partial coarsest a-field (8, Q}.

history till date t. F0 denotes the

2.2. The production sector

The production sector is represented by a unique competitive firm with production function f: R, x [a,/?] + Iw,. Given the realization of any state o, in period t and the gross (or before-tax) return to capital per unit,3 R,, the firm’s problem in period t 2 1 is to choose a nonnegative capital stock k, in order to solve:4

sup_%,,4 - R&t, k,ER+.

The assumptions on the production function are:

(P-1)

P.2)

(P-3)

(P.4)

(P-5)

f is continuous on R + x [a, /3].

f(0, 0) = 0 for all 0 E [a, /?I.

For each o E [a, /I], f' E i3f/i3k exists and is strictly increasing in k. Further, f’ is continuous on R+ x [a,/?].

For each o E [a,fi], f” E @f/&* exists. Further, f” < 0 at k > 0.

f’(k, 0) + + co as kJ0 for all w E [a, j3 J.

3Note that the gross return to capital per unit, R,, is not exogenous. Rather, in the recursive competitive equilibrium, R, is such that the demand for capital in period t by the firm is equal to the aggregate supply of capital in period t by the consumers. On this point, also see the discussion in Prescott and Mehra (1980, pp. 1370-1371).

4The fact that the firm maximizes profits in each period rather than a discounted sum of profits does not imply any kind of myopic behavior. Profit maximization in each period by a price-taking firm is a standard way to model a recursive competitive equilibrium (for instance, Danthine and Donaldson, 1985, p. 256; Prescott and Mehra, 1980, p. 1367).

S. Joshi 1 Journal of Economic Dynamics and Control 21 (1997) 371-389 375

(P.6) There exists a constant k” E (0,~) such that f(k, w) I ( >)k if k 2 ( <)k” for all 0 E [a,/?].

(P.7) There exists a constant CJ > 0 such that infoEta,B1 f’(k”, o) 2 0.

Under these assumptions, the optimal interior solution to the firm’s maximization problem is characterized by R, =f’(&,w,).

The above assumptions are fairly standard in the optimal growth literature. Note that (P.6) assumes the existence of a maximum sustainable capital stock. Such an assumption is commonly imposed in growth models to ensure that the capital accumulation process is uniformly bounded from above.s (P.7) requires that the marginal product of capital at the maximum sustainable level be bounded away from the origin for all realizations of the stochastic environment. Consider the following examples of production functions where uncertainty enters the production function either multiplicatively or additively:

f(k, 4 = dkh 0.l E C%Bl, (1)

f(k,4 - g(k) + 0, OJ E c%Bl. (4

Here g: R, + R, is a standard neoclassical production function.6 Danthine and Donaldson (1985) have used specification (1) in their analysis. It is easily verified that (1) and (2) satisfy (P.l)-(P.7).

2.3. The consumption sector

The consumption sector comprises of a continuum of identical consumers distributed on the interval [0, 11. The representative consumer’s preferences are described by the sequence of felicity functions (~1,: R, + R,}, t 2 0. It is assumed that for each t 2 0:

(U.l) u,(c) is continuous on R,.

(U.2) u,(c) is strictly concave for c 2 0.

(U.3) u:(c) exists and is strictly positive for c > 0.

(U.4) u;(c) + + co as ~10.

‘In stochastic growth models, it has been used in both convex models (Brock and Mirman, 1972) and nonconvex models (Majumdar, Mitra, and Nyarko, 1989).

6That is, g is twice differentiable, strictly increasing, strictly concave, with liml,,, g’(b) = 0 and admitting a maximum sustainable capital stock defined implicitly by g(k”) = km.


The consumer inelastically supplies labour; hence labour-leisure choices are not considered. Letting s E (0, co) be the given initial stock, a nonnegative {R}-adapted stochastic process {(c,, k,, 1)} is a feasible program if:

co + kl = s, (3)

c,+kt+l = (1 - e)R,k, + W, + gt, v - a.s., t 2 1. (4)

Here, c, and k, are respectively the period t consumption level and capital holdings of the individual; W, and gt are respectively the period t wage rate and the transfer payment from the government. The government’s role in this model is to tax capital at rate 8 and then distribute the tax revenue in a lump sum fashion. That is,

gt = e&k, = w-‘(k,,Nh. (5)

Wages are given residually by

wt =f(k,,4 -f’(k,,w)k,, (6)

Substituting in the budget equation given by (4) yields that with probability one for all t:

ct+k,+l = (1 - @_f-‘(kt, w&t +f(kt, 4

-f’(kt,Q-4k, + W’(k,,w)kt. (7)

In any period t 2 1, the consumer’s capital holding, k,, the aggregate capital, kr, and the realized state of the environment, CD,, determine the consumer’s income (noting that income in period 0 is s). The consumer then has to allocate this income between current consumption, ct, and capital holdings for the next period, k,.,. Let Q(s) denote the set of all feasible programs, that is, those programs which satisfy (3) and (7). From Assumption (P.6), for any {(co k,, 1)} in G(s),

0 5 c,,k,+, I km, v - a.s., t = 0,1,2, ... . P-3)

Let 0 < S < 1 be the discount factor. The consumer’s problem is given by

SUP E : W(ct), k, k+ 1)) E W. 0

S. Joshi 1 Journal of Economic Dynamics and Control 21 (1997) 371-389 311

Note that in the special case where u, = u, it follows from (U.3) that for any

{(CA,+ I)> E @P(s):

E f 6’u(c,) I 5 6’u(k”) = u(k”)/(l - 6) < co. 0 0

Therefore, optimality can be defined in terms of maximizing the infinite discounted sum of felicities over the set of feasible programs. However, with time-varying felicity functions, the discounted sum of one-period utilities may not converge. Since this paper is not primarily concerned with the issue of existence of optimal programs, we simplify the consumer’s problem by imposing a bounded value assumption:7

(B) f 6’u,(k”) < co. 0

Note that the results of this paper continue to be true without the above assumption.* The existence of an optimal program, { (cf, kf, ,)}, solving the consumer’s problem follows standard arguments. From the Inada condition on ut, the optimal program is interior, i.e., c: > 0 and kf, 1 > 0 for a.e. w and for all t 2 0. The stochastic Euler equations and the transversality condition character- izing the optimal program for t 2 0 are

uXcf> = W4+ I(cf+ d1 - WC+ 1 II PA v - a.s., (9)

lim G’u:(cf)kf = 0. t-m (10)

‘Such a bounded value assumption has been used by Bhattacharya and Majumdar (1981, Assump- tion A.3.4) and Mitra and Nyarko (1991, Condition E) to show existence of optimal programs.

*In the general case where the discounted sum of felicities may diverge, optimality can be defined in terms of catching up of partial discounted utility sums. In particular, a program {(cf, k:, 1)} is optimal in @J(S) if for any other feasible program {(c,, k,+ ,)} in G(s):

lim :‘, i a’[u,(cl) - u,(c,)] 2 0. - 0

Existence, using the catching up criterion, can be proved by using the limiting argument of Majumdar and Zilcha (1987) or Mitra and Nyarko (1991). The first step is to prove the existence of optimal programs in the finite-horizon model. It is then shown that the optimal program is monotonically increasing in the length of the planning horizon. Letting the horizon go to infinity and using the bound on feasible programs given by (8) then yields a limit program. Finally it is shown that the limit program is optimal in the sense of catching up of partial utility sums.

378 S. Joshi /Journal of Economic Dynamics and Control .?I (1997) 371-389

Using the fact that R, =f’(k,,o,) from the firm’s problem, the Euler equations can be rewritten as

4(d) = ~EC4+I(cf+~)(1 - fW’(kf+l,~,+l) II FJ, v - as. (11)

Therefore, Eqs. (10) and (11) incorporate the optimal decisions of the representative consumer as well as the competitive firm.

2.4. The O-recursive competitive equilibrium and the equivalence principle

Given the solution ((cf, kf,,)} to the representative consumer’s problem, aggregate capital holdings and consumption are given by k, = JA k, dm and ct = ji c,dm, where m represents the distribution of consumers on [0, 11. In equilibrium, we must have aggregate demand equal to supply of the good:

ceo + k; = s, (12)

cf + kf+ r =f(kf,o,), v - a.s., t = 1,2, . . . . (13)

Since all consumers are identical, the optimal consumption levels and capital holdings do not vary across consumers. Therefore, as in Danthine and Donaldson (1985, Fn. 1) and Prescott and Mehra (1980, Section 4), at the optimal solution , kf = kf j: dm = kf and c: = cfj: dm = cf. Substituting this in (3) and (7) then shows that (12) and (13) hold and we have equality between aggregate demand and supply with probability one in each period. We now have the following:

Definition: Given the initial stock s and the tax rate 8 on income from capital, a sequence {(c:, k:+ ,)} of aggregate consumption levels and capital stocks is a rational expectations O-recursive competitive equilibrium if it satisfies the optimality conditions for the consumer and the representative firm (that is, (10) and (11) hold) and is such that the planned demand for capital and consumption is equal to the supply of output with probability one in each period (that is, (12) and (13) hold).

Of fundamental importance in the literature is the following result of Becker (1985):

The Equivalence Principle: The aggregate sequence {(c:, kf+ r)} of a &recursive competitive equilibrium is identical to the solution to a planner’s maximization

S. Joshi / Journal of Economic Dynamics and Control 21 (1997) 371-389 379

problem given by

SUP E f CW - @l’u,(ct) such that c0 + kI I s and 0

c, + k 1+1 If(k,,o,), v -as., t r 1

The proof of the above proposition follows the arguments of Prescott and Mehra (1980, Sections 5,6,7). For completeness, we record the following:

Theorem 1. Under Assumptions (P.l)-(P.4), (P.d), (U.l)-(U.3), and (B), a tI- recursive competitive equilibrium exists.

3. The uniform value loss process and the invariance result

This section generalizes the classical result on dynamic incidence of a tax on capital income to nonstationary stochastic environments. It makes fundamental use of the stochastic Euler equations given by (11). Define the competitive price process, (pf}, associated with the B-competitive equilibrium, {(~f,kf)+ ,)I, as pf = u;(c~) for all t 2 0. Note that prices are strictly positive and bounded for a.e. w because of interiority of optimal programs. The net return to capital per unit is given by rf = 6(1 - @f’(kf,o,), t 2 0. Note that the discount factor, 6, is incorporated into the net return to capital.’ The Euler equations can be rewritten for t 2 0 as

P: = E CP~+ I rf+ I II Ft19

Now define the process {xf}

I

v - a.s.

as

(14)

Note that n: is strictly positive and Ft-measurable. Therefore, multiplying (14) by rrj) and using a well-known property of conditional expectations,” the Euler

‘This does not involve any loss of generality in proving the almost sure convergence of the no-tax

and after-tax net return to capital. This is because for any E > 0, IT, - r;j 2 ( I)& is equivalent to I&, - &;I 2 ( 1)6s = E’.

“If a random variable X is F,-measurable and the random variables Y and XY are integrable, then

E [XY 1) St] = XE[ Y 11 F,]. For instance, Billingsley (1979, Thm. 34.3).


equations for t 2 0 become

From (15), the process {pfn$ is a martingale” with jpfrrfdv = {ptntdv = u&).

In the no-tax case (i.e., 8 = 0), it is well-known that the solution to the planner’s problem and the recursive competitive equilibrium coincide (Prescott and Mehra, 1980, Sections 5,6,7). The stochastic Euler equations corresponding to the recursive competitive equilibrium, {(cp, kf+ ,)}, in the no-tax case are given for t 2 0 by

uXcP) = sEC~;+~(cP+~)f’(k,o+,,0~+~) II FJ, v - a.s., (16)

while the transversality condition is similar to (10). The return to capital per unit (gross and net) for this case is given by rp+ 1 = Gf’(kp+ l,w,+l). Going through the same machinations as in the tax-perturbed case yields for t 2 0:

pi’4 = W6’+ ld’+l II FJ, v - a.s., (17)

where the process {rep} is defined as rt8 = 1, 7rp = fli rp, t 2 1. From (17), the process {pt)$) is a martingale with EpFaF = u6(& Using the two martingale processes, we can prove the following:

Theorem 2. Assume (PJ)-(P.7) and (U.Z)-(U.4) hold. For any T 2 1, there exists a uniform value loss submartingale process {ZT; 0 I t 5 T} which is bounded from above in expectation by t&,(&J + ub(&) and which displays the following sensitivity property. For any E > 0 there corresponds a y(e) > 0 such that for any 0 I t s T - 1 and any o E 52:

Theorem 2 is a technical result which asserts that in any period and for any realization of the environment, if the no-tax and after-tax net returns to capital diverge by more than E, the process, (22 0 I t I T} records a strictly positive value loss of Y(E) which is independent of the tuple (t, T,o). With the aid of this process, we can now prove:

“A {P,}-adapted process {X,} is a martingale if EIXt+I 11 SF,] = X,, v - a.s. For martingale processes, j X,+ 1 dv=jX,dv=jX,,dvforallt=1,2,... (for instance, Billingsley, 1979, p. 408).

S. Joshi /Journal ofEconomic Dynamics and Control 21 (1997) 371-389 381

Theorem 3 (Long-run dynamic incidence). Let {I-:} and {I:} be the processes describing the no-tax and after-tax return to capital per unit respectively. Under Assumptions (PA)-(P.7), (U.I)-(U.4), and (B), jr: - $1 + 0, v - a.s., as t + 03.

The economic significance of Theorem 3 lies in demonstrating that there is a complete shifting of the tax on capital income to the labour input in the long run even in the presence of uncertainty and nonstationarities in preferences and the stochastic environment.

4. Classical steady state analysis

The classical result on dynamic incidence of a capital income tax was couched in terms of the equivalence between the steady state no-tax and after-tax return to capital in a deterministic stationary environment. No corresponding result is available in the existing literature for a stochastic economy. This section bridges the gap in the literature by showing the equivalence between the stochastic steady states of the no-tax and after-tax return to capital as a corollary of Theorem 3.

The steady state equivalence result will draw upon an ergodic theorem of Brock and Mirman (1972) describing the limiting behaviour of the sequence of probability measures generated by a recursive competitive equilibrium. An application of this theorem will require the stationarity assumption:”

(U.0) u, = u for all t 2 1.

Assumption (U.O), along with (P.3) and (P.6), implies that the discounted sum of utilities is finite for any program in Q(s). Therefore, the value boundedness condition, (B), can now be dispensed with. Assuming (P.l)-(P.4), (P.6), and (U.O)-(U.3), it follows from Theorem 1 that a e-recursive competitive equilibrium, {(c:, ICY+ J}, exists. The sequence of probability measures on [w + generated by { kf} is defined for any Bore1 set A in R+ as

p;(A) 3 v( {ox kf E A}), t2 1. (19)

r * Brock and Mirman (1972) imposed the additional assumption that i?f(k, w)/&s > 0 for all k > 0.

It was subsequently shown in Mirman and Zilcha (1975, p. 334) that this monotonicity condition was not required to prove the ergodic theorem. It was also noted by Brock and Mirman (1972, p. 486) that the ergodic theorem generalized to the case where the evolution of the stochastic environment did not follow an i.i.d. process.

382 S. Joshi / Journal of Economic Dynamics and Control 21 (1997) 371-389

We note for completeness the following:

Lemma 1. ,uf is a well-defined probability measure on the Bore1 subsets of R,.

When summarizing the argument of Becker (1985) in the Introduction, we used the notion of a steady state (or golden rule) capital stock to which the optimal program of capital accumulation converged in the long run. In the uncertainty case, we need the existence of a stochastic steady state which is ensured by the following ergodic result:

Lemma 2. Under Assumptions (P.I)-(P.6) and (U.O)-(U.4), $ converges weak- ly13 to a stationary measure pe on an ergodic set [E,iZ], 0 < k < iz -C 03.

The importance of the ergodic theorem is that it helps us to uniformly bound the after-tax return to capital as shown in the following:

Lemma 3. Under Assumptions (P.l)-(P.6) and (U.O)-(U.4), there exists constants 0 < a6 < be < co such that rf E [ae, be], v - a.s., for each t 2 1.

Lemma 3 shows that the after-tax return to capital cannot become arbitrarily small or arbitrarily large with time. Now define the sequence of probability measures (A:} associated with the process (r:} for any Bore1 subset A of 58, as

1:(A) s v({ox rf E A}). (20)

That A:, t L 1, is a well-defined probability measure on the Bore1 a-algebra on R, follows as in Lemma 1. The final result we need to show steady state equivalence is:

Lemma 4. Under Assumptions (P.I)-(P.6) and (U.O)-(U.4), the sequence (A:> converges weakly.

Lemma 4 shows that there exists a stochastic steady state corresponding to the after-tax return to capital. Counterparts of Lemmas 1,2,3, and 4 follow in an analogous way for the no-tax return to capital. Now, in order to show the equivalence between the stochastic steady states, we will need a distance function on the set of probability measures defined on the Bore1 subsets of R + . Given

1 3 That is, ja. g dp: + JR. g d# as t -P co for all g E C(W +), where C(R +) denotes the set of bounded continuous functions on R+. Alternative characterizations of weak convergence of measures are available in Billingsley (1968, Thm. 2.1, p. 11).


any two probability measures I and I’ on the Bore1 a-algebra on R + , define the ‘distance’ between these measures by the metric p as follows:14

where 2 is the set of all real-valued bounded functions on R, satisfying the Lipschitz continuity condition Is(x) - g(y)1 < Ix - yl. The next result demon- strates that the stochastic steady states associated with the processes {I:} and (r!] are identical. It therefore offers a counterpart to the steady state equivalence result of Becker (1985) obtained in a stationary deterministic paradigm and an extension of the stationary stochastic analysis of Danthine and Donaldson (1985).

Theorem 4 (Stochastic steady state equivalence). Let (rp) and (I:) be the processes describing the no-tax and after-tax return to capital per unit respectively. Let (1:) and {A:} be the sequences of probability measures associated with these processes. Under Assumptions (P. I)-(P. 7) and (U 0)-(U.4), p($, 1:) + 0 as t + al.

It may be noted that comparative dynamics can be conducted in the framework of this paper in a manner similar to Danthine and Donaldson (1985) by adapting the elegant lattice programming techniques of Hopenhayn and Prescott (1992).

5. Conclusion

This paper had two main objectives. The first objective was to demonstrate that the classical result of a tax on capital income being completely shifted to labour in the long run also obtains in a very general stochastic growth model with nonstationarities in preferences, and the stochastic environment. The second objective was to develop a new approach based on the tools of probability theory thereby exploiting to the fullest extent the stochastic framework. The martingale methods which have been developed are based on the Euler conditions which characterize the interior optimal programs of any growth model. Therefore, the technique developed here may also be useful in analyzing the incidence of other taxes in a stochastic growth model.

“‘This metric is also used by Brock and Majumdar (1978) to examine the convergence of probability measures associated with the optimal programs of capital accumulation in a stochastic growth model.


Appendix

Proof of Theorem I: From the Equivalence Principle, it suffices to show the existence of an optimal solution to the planner’s problem (since it is identical to the aggregate sequence of consumption levels and capital stocks of a &recursive competitive equilibrium). The existence of an optimal solution follows the arguments developed in Bhattacharya and Majumdar (1981) or Majumdar and Zilcha (1987) or Mitra and Nyarko (1991). n

Proof of Theorem 2: Consider any T 2 1 and E > 0. Partition the set of all o for which [r:(w) - rp(w)l > E into the following sets:

A: = {w: r:(w) - r:(o) > c}, r: = (0: r,“(o) - r:(o) > 6).

The two sets are S,-measurable since from (P.3) the net returns to capital are Ft-measurable. Define the process {qr} as follows:

q1 = min [rl, $1, t = 1,2, . . . ,T.

Further, let

QO = 1, Qt = h qi, t = 1,2, . . . , T. i=l

Note that QI is strictly positive with probability one and pt-measurable. From (14):

~fQ~-~rf = Wd’+~Q,-~rfrf+~ II Ftl, v - a.s.

For any A E .Ft,

pj)Qt_lrfdv = I pi’+ I Qt- 1 r%‘+ 1 dv AnA:

+ P:+ I Qt - I r%+ 1 dv

> d+lQr-ltry + 4ri’+ldv

+ s P:+ I Q, r,“+ I dv AM?-A:)

= s I$‘+ 1 Qd+ 1 dv + 8 I d+lQt--lrf+ldv, A AhIf

S. Joshi / Journal of Economic Dynamics and Control 21 (1997) 371-389 385

where the inequality is a consequence of the definition of A: and the fact that Q,_ 1 r, 2 Qt, v - a.s. Since the above is true for all A E pt, it follows that v - a.s. for t = 1,2, . . . ,T,

= WC)Qt- IP~‘> 64.1)

where X(A) is the indicator function of A. Let qr = minwsR min, I is T Ui(k”)Qi_ 1 and note that it is well-defined as a consequence of (P.3) and the compactness of Sz. Further, u, is a strictly positive constant over Sz from (P.7), and qt G ql+ 1 by construction. Finally, letting rf = QI- l~f/qt, it follows from (A.l) that:

p!r: - ECp;+ I l!+ 1 II St1 2 cd4L v - a.s., t=1,2 ,..., T. 64.2)

Recalling the martingale process given by (15), construct the 0 I t I T} as follows:

Xi = p; = I&(&), XT = &c; - p;r;, t = 1,2, . . . , T.

From (15) and (A.2), it then follows that:

E[XT+ 1 II FJ - XT 2 .sx(Af), v - a.s., t = 1,2, . . . , T.

Therefore, {XT; 0 I t I T} is a submartingale with

jX;dv 5 jp:n:dv = u&“o).

An identical argument yields the existence of a submartingale 0 I t I T} such that

E[Y?+, II Fc] - YT 2 EX(~:), v - a.s., t = 1,2, . . , T,

jY:dv I [p:n:dv = ub(&).

process {XT;

(A.3)

process { Y T;

(A.4)

Letting 2: s XT + YT, 0 I t I T, it follows from (A.3) and (A.4) that

E[ZT - ZT+ 1 II 9’J 2 .~x(n:uT$ v - a.s., t = 1,2, . . . , T.


Therefore, for any period 1 I t I T and realization o E Q for which [r:(o) - r:(o)1 > E, the process (2:; 0 I t 5 T} records a strictly positive value loss of y(s) = E which is independent of (t, T,w). Further, 2: is bounded from above in expectation by t&(&J + u&i). n

Proof of Theorem 3: Consider the setI S: = {w E 0: [r:(o) - r$‘(o)l > E} and note that it is Yt-measurable. l6 Using Theorem 2, for any T 2 1 and any O<t<T-1,

Therefore, taking expectations and noting that J &4)dv = JA dv = v(A),

Further, since (Z?} is a submartingale,

I ECZT+ I -Z:II&]dv>O. 0-g

It now follows that

= s ECZ,T,, - ZtT II St1 dv R

= [Z;+ 1 - Z;] dv,

“Since the paper is concerned with the asymptotic behaviour of the net return to capital, without loss of generality we can let St = 62.

*sThis follows from (P.3), the {9,}-adaptability of the aggregate programs {(c:,b:+ 1)} and {(r;:, Sf+ 1)}, and the fact that the difference of two measurable functions is also measurable.

S. Joshi J Journal of Economic Dynamics and Control 21 (1997) 371-389 387

where the last equality is a consequence of Billingsley (1979, p. 395). Summing up, and noting that Z,’ is positive for a.e. o:

T-l

Y(E) 1 v(S:) I E[Z: - Z,‘] I E[Z;] < r&(&J + ub(c;), T = 1,2, . . . . r=o

(A.5)

Since the right-hand side of (A.5) is independent of T, letting T -+ co yields

Y(E) jJ v(s:) 5 ub(ceo) + ub(c:) < 00. t=o

It now follows from the first Borel-Cantelli lemma in Billingsley (1979, Thm. 4.3) that v(lim supl _,ao S:) = 0. Therefore, there is no o which belongs to infinite- ly many sets Sf except possibly those belonging to a set of probability measure zero. This proves the result. n

Proof of Lemma 1: For each t 2 1, PLI) is well-defined on the Bore1 a-algebra on R, from the P,-measurability of kf. Since v is a probability measure on 52, for any Bore1 set A, p!(A) = v( {w: kf E A}) E [0, 11. Since kf > 0 for a.e. o, p:(O) = 0 and ,~f(lR+) = 1. Now, let {A,} b e any disjoint sequence of Bore1 subsets of lR +. Then, the sequence {B,}, B, = {w: kf E A,}, is a disjoint sequence of Ft-measurable subsets of Q. Note that

0: IL; E (-j A, = (j {Ox lc: E A,} = fj B,. n=l n=l n=l

Since v is countably additive, it follows that

Therefore, PLI) is a probability measure on the Bore1 subsets of R,. n

Proof of Lemma 2: See Brock and Mirman (1972, Thm. 4.1) or Mirman and Zilcha (1975, Thm. 2).” W

“Both these papers present the ergodic theorem in terms of convergence of the distribution functions associated with the sequence {k:}. Noting, however, that the distribution function, F:(k), and the probability measure, p:, are related as

F,(k) = ,({a: k: < k}) = A’(CO,k)),

the ergodic result could equally well be couched in terms of the sequence {PI}.


Proof of Lemma 3: From (P.3) and the compactness of [LX,~], ae =

inf,,[,,fi, S(1 - O)f’(k”,w) exists from the Weierstrass theorem. From (8) and the concavity off,

rf = 6(1 - O)f’(k,B,q) 2 6(1 - B)f’(k”,o,) 2 ae, v - a.s., t = 1,2, . . . .

From Lemma 2, there exists an integer t(O) < co such that ky E [“k,%], v - a.s., for all t 2 t(13). Now let r = max{rl): 1 I t I t(O)} and note once again from the concavity off that

r: I max{f,6(1 - O)f’(R,o)}, v - a.s., t = 1,2, . . . . 64.6)

From (P.3), S(1 - e)f’(& ) co is continuous on [a,/I]. Also from (P.3), rf is continuous on [a, /3]. Therefore, f, since it is the (pointwise) maximum of a finite set of continuous functions, is continuous” on l’jy’ [a, /.I]. It follows from the Weierstrass theorem that the right-hand side of (A.6) is bounded from above by some be < 00. n

Proof of Lemma 4: Note that 6(1 - O)f’(k, w is continuous in k for any given ) o from (P.3). Further, from Lemma 2, the sequence of probability measures associated with { kf} converges weakly. The result now follows from Billingsley (1968, Corol. 1, p. 31, or Thm. 5.5, p. 34). n

Proof of Theorem 4: Using a result on change of variables from Billingsley (1968, p. 223), it follows that

P(xx) = SUP WY

= sup gs9

“This also follows from the Bergs Maximum theorem by noting that any finite set of integers is compact in the discrete topology on the set of positive integers. An alternative approach to Lemma 3 is to note that the long-run convergence to the steady state is not affected by what transpires over a finite number of time periods. Note also that, except possibly for a finite number of periods, r: lies in the interval [a(1 - O)f’(&o),~S(l - O)f’(E,o)] for all o (by suitably redefining net return to capital over sets of zero measure). The bounds now follow from (P.3) and the compactness of [a, fl.


Since the after-tax and no-tax return to capital belong to compact intervals (Lemma 3), their absolute difference also belongs to a compact interval. It now follows from Theorem 3 and the Lebesgue Dominated Convergence theorem that{,Ir:-$ldv+Oas t-co. n

References

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Billingsley, P., 1979, Probability and measure (Wiley, New York, NY). Billingsley, P., 1968, Convergence of probability measures (Wiley, New York, NY). Brock, W.A. and M. Majumdar, 1978, Global asymptotic stability results for multisector models of

optimal growth under uncertainty when future utilities are discounted, Journal of Economic Theory 18225-243.

Brock, W.A. and L.J. Mirman, 1972, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory 4, 479-513.

Danthine, J.P and J.B. Donaldson, 1985, A note on the effects of capital income taxation on the dynamics of a competitive economy, Journal of Public Economics 28,255-265.

Duffie, D., 1988, Security markets: Stochastic models (Academic Press, Boston, MA). Follmer, H. and M. Majumdar, 1978, On the asymptotic behaviour of stochastic economic

processes, Journal of Mathematical Economics $275-287. Hopehayn, H.H. and E.C. Prescott, 1992, Stochastic monotonicity and stationary distributions for

dynamic economies, Econometrica 60, 1387- 1406. Majumdar, M. and I. Zilcha, 1987, Optimal growth in a stochastic environment: Some sensitivity

and turnpike results, Journal of Economic Theory 43, 116- 133. Majumdar, M., T. Mitra, and Y. Nyarko, 1989, Dynamic optimization under uncertainty:

Non-convex feasible set, in: G.R. Feiwel, ed., Joan Robinson and modem economic theory (Macmillan, London).

Mirman, L.J. and I. Zilcha, 1975, On optimal growth under uncertainty, Journal of Economic Theory 11, 329-339.

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Martingale analysis of dynamic tax incidence in a nonstationary growth model

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