Long-run invariance in economic dynamics: A note

Vol. 62 (1995), No. 2, pp. 215-226 Journal of Economics Zeitschrift for National6konomie

�9 Springer-Verlag 1995 - Printed in Austria

Long-run Invariance in Economic Dynamics: A Note

Sumit Joshi

Received March 16, 1995

A common thread in relatively disparate areas of economic dynamics is an invariance result demonstrating the convergence in a suitable metric of the time paths (or the invariance of the steady state) of a given variable to alternative specifications of some parameter. This note offers a unified approach to this phenomenon in the context of a general nonstationary growth model. The technique developed exploits the stochastic specification through a passage to martingale techniques and is general enough to incorporate nonclassical technologies. Since the paper nests the invariance result from diverse areas as special cases, in one stroke it extends the received analysis which has usu- ally been conducted in a deterministic stationary environment to a stochastic framework in which nonstationarities enter into the description of preferences, technology, and the stochastic environment.

1 Introduction

A common thread in relatively disparate areas of economic dynamics is an invariance result demonstrating the convergence in a suitable metric of the time paths (or the invariance of the steady state) of a given variable to alternative specifications of some parameter. Consider the following sample of dynamic results drawn from three relatively dis- tinct areas of econo/nics. In the theory of optimal growth, the variable of interest is the capital input and the parameter in question is the initial stock. The invariance result, called the late (or twisted) turnpike theorem, asserts that optimal programs of capital accumulation from different initial stocks converge in an appropriate metric (for instance, Majumdar and Zilcha, 1987). In the area of public finance, the variable of interest is the net or after-tax return to capital per unit and the parameter is the rate of capital income taxation. The invariance result takes the form of a classical proposition on dynamic incidence of a capital income tax couched in terms of steady states. It states that the capital

216 S. Joshi

income tax is completely shifted to the perfectly inelastic labor input in the long run. That is, the net return to capital per unit in the steady state is identical in the no-tax and tax-perturbed cases (for instance, Becker, 1985). In monetary theory, the variable of interest is once again the capital input and the parameter is the initial stock of money. The invariance result here claims that if real money balances are incorporated into the utility function, then the steady state of capital is invariant to a change in the initial stock of money (for instance, Brock, 1974).

The presence of the invariance phenomenon in various subdisci- plines of economics attests to its rather pervasive nature. However, a unified treatment of this principle is unavailable in the existing literature. This note provides such a unification by addressing the invariance issue in a very general framework which accommodates nonstationarities in the description of preferences, technology, and the stochastic environment and nests the three invariance results presented earlier (and others) as special cases. The merits of such a synthesis are the following:

(1) Many of the invariance results in the received literature are couched in terms of steady states. It has, therefore, been an open question if such invariance would obtain in a general nonstationary framework in which steady states do not exist. The issue is resolved here in one stroke for the invariance results in diverse areas by concentrat- ing attention instead on the asymptotic behavior of the dynamic time paths of the relevant variable. In particular, it is shown that the time paths of the variable of interest will converge in an appropriate metric if a sufficient uniformity condition is satisfied, thereby obtaining a nonstationary extension of the classical invariance results. Further, the generality in the specification of the model here allows it to be specialized to situations which have hitherto not been analyzed to verify whether invariance results are possible with respect to some parameter.

(2) The technique developed here is of independent interest for two reasons. First, it exploits the stochastic element to the maximum through a passage to martingale techniques. In fact, critical to the argument is a supermartingale value loss process derived through the stochastic Euler equations. This means that the stochastic environment is not a mere addendum but contributes crucially to the invariance result. Second, the methodology developed does not depend on the convexity of the technology. This allows an extension of the classical invariance results along the additional dimension of nonclassical technologies.

Long-run Invariance in Economic Dynamics 217

2 The Nonstationary Model

A compact metric space f2t with Borel ~r-field gt represents the possible states of the environment at date t > 1 and cot E f2t the state at date t > 1. The stochastic environment is represented by the measure space (g2, 5 t', v) where f2 = H ~ f2t, ~- = | is the o--field on f2 generated by the measurable cylindrical sets, and v is the measure on f2. 5ct is the sub ~r-field of 5 c induced by partial history till date t. ~0 denotes the coarse ~r-field {0, f2}.

Technology is given by a sequence of possibly time-varying production functions { 3~" N+ x ff2t+ 1 x R+ --+ R + } where ft, tO allow maximum generality, is allowed to depend on the parameter 0. It is assumed for each t > 0 and 0 e IR+ that f t is continuous on R+ x f2t+l. Further Vcoz+l, ft(O, cot+l, O) = O, f t (k , cot+l, O) is twice continuously differentiable at k > 0, and f ' t > 0 at k > 0 where f / = Oft~Ok. Given an initial stock s > 0, the real-valued nonnegative {grt}-adapted process (k, c) =-- {(kt, ct)} is a feasible progi'am from s given parameter value 0 if k0 + co < s and kt+l + ct+l < f t (k t , cot+l, 0) v-a.s. ' i t > 0. Let cb(s, {f t}, O) denote the set of all feasible programs from s given 0

R+. From continuity of f t and compactness of f2t, there exists a pure accumulation process from s, {k]'~ which is independent of co and is

such that for any {(kt, Ct)} in ~(s , {3~}, 0), 0 _< kt, ct < kt "~ v-a.s. Preferences are represented by the sequence of felicity functions

{ ut: R 2 --+ R+ } where once again, in the interests of generality, fe- licities are allowed to depend on the parameter 0. It is assumed that for each t > 0 and 0 c R+, ut(c, O) is continuous and strictly con-

'(c, 0) exists and is strictly positive for c > 0, cave for c > O, u t and u't(c, O) --+ +(x~ as c $ 0 . A program {(kt ~ ct~ in q~(s, {f t}, O) is said to be optimal if for any other program {(kt, ct)} ~ Cb(s, {ft}, 0), limsuPN~oo y~U E[ut(ct , O) - ut(c~ 0)] _< 0. The existence of an optimal program follows standard arguments (see Majumdar and Zilcha,

0 0 1987, theorem 1). From the Inada condition on ut, c t , k t > 0 for a.e. co "it > 0. The Euler equations characterizing {(kt ~ cO)} for t _> 0 are

, o , o O ) f t ( k ~ cot+l , O) II ~'t] v-a.s . ut (c t , O) = E[Ut+l (Ct+l , , ( 1 )

The random variable whose asymptotic behavior is of interest is specified as follows

o = ht(kOt, 0), t 0, 1,2, (2) Zt C O t + l , = . . - ,

where ht is assumed to be continuous on R+ x f~t+J for each t _> 0

218 S. Joshi

given 0. This general form can be specialized to the particular case under consideration. For instance, if the asymptotic behavior of the optimal program {(k], c])} with respect to a perturbation in the initial stock s is the objective of analysis, then ht (k], cOt+l, s) - k~. If the behavior of the after-tax return to capital per unit with respect to a change in the rate of capital income taxation 0 is the issue, then ht(k~ cot+l, 0) ----- (1 - 0~ r ' t k ~ , ~ t , t , cot+l). Specification (2), therefore, allows the analysis to be conducted in very general terms. Now, define the shadow price process {pt ~ } associated with the optimal program {(kt ~ cO)} as pO , o = ut(c t ) Yt > 0. Note that shadow prices are strictly t

positive and bounded for a.e. co because of the interiority of the optimal program. Assuming that the function in (2) is invertible in k for any co E f2 and 0 c R+, let kt ~ = gt(z~ cot+l, 0), t >_ O. Further let

' , 0), cot+l, 0), t 0, 1, 2, Ft(z~ cot+l, 0) ~ ft (gt(z~ cot+l, = . . . .

The Euler equations then become

pO o o 0) II ~t] v-a.s, t > 0 (3) = E[Pt+lFt ( z t , cOt+l, _ .

3 A Value Loss Result

Consider two alternative parameter values 0, g 6 R+, 0 ~ y and any e > 0. This section constructs a supermartingale value loss process {l(t} which is sensitive to an e-divergence between zt~ and ztY(co) measured in terms of the metric I[ II for any t > 0 and any co ~ f2. Consider the Borel set S~ = {co: IIz~ -z[(co)l l > e} and partition it into the two following 5Or-measurable sets:

A~ = {co: IIz~ z[(co)ll > e,

F,(z ~ o9,§ # F,(z[, co,+l, • } ,

~7 = {co: Ilz~ - z[(co)ll > e,

Ft(zOt , cot+l, O) = Ft(z[, c o t + l , Y) } �9

This partition incorporates the two possible reactions of Ft to an e-divergence between z ~ (co) and z[ (co) evaluated in terms of the metric [I I1.

Lemma 1: Under the stated assumptions on technology and prefer-


ences, there exists a supermartingale process {Wt} such that E[Wt -

wt+l II :5] > 0 o n A ~ .

Proof" Let rt = min[Ft(zt ~ cot+l, 0), Ft(zY, cot+l, Y)], I ~ 0, and note that rt is strictly positive for a.e. co and 5vt+l-measurable for all t. From the Euler equations we obtain:

p0 = E[pt0+l Ft(zOt, cot+l, 0) I I -~ t ] ~ E[P~ II Ft] v-a.s. , (4)

PY = E[P• cot+l, V)II-~,] __ E[pY+lrt II St] v-a.s. . (5)

Suppose IIz~ - zY(co)ll > e for some co. If co r A~, then Fttz~ co,+l, O) # Ft(zY, cot+l, ~V). I f Ft(zf, cot+l, 0) > Ft(zY, cot+l, V), then (4) holds as a strict inequality and (5) as an equality. The opposite is true if Ft(z~ , cot+l, O) < F~(zYt , cot+l, V). Let

t

pro = 1, zrt+l = I'-I ri, t > 0 . 0

Sincezrt is .7"t-measurable, multiplying (4) and (5) by prt yields

p Opr, >_ E[pO+~pr,+~ II f , ] v - a . s . , (6) ?,

Pt prt > E[P~+lprt+l ]].Yt] v-a.s.. (7)

Once again, (6) holds as a strict inequality and (7) as an equality for those co 6 A~ for which Ft(z ~ cot+~, O) > Ft(zY, cot+l, V) while the opposite is true for co 6 Af for which E,(zt ~ cot+l, 0) < Ft(z~, cot+l, Y). Letting Wt = p~ +PYPrt, t > O, it follows from (6) and (7) that {Wt} is a supermartingale process satisfying the supermartingale inequality strictly for all co c A~. []

Lemma 2: Under the stated assumptions on technology and preferences, there exists a supermartingale process {It} such that E[Yt - Y,+I II 3 ] > 0 on o.,~.

Proof" The proof will exploit the fact that for any co ~ ~Pt e, zt ~ Ft (zt ~ cot+l, O) ~ z~ Ft(zy, wt+l, g). Recalling the coordinatewise bound on optimal programs by the pure accumulation process, there exists a se-

220 S. Joshi

i < m] v-a.s., i = 0, ?/. Lett ing quence {M~} such that for each t > O, zt _

Mt = m a x { M ~ MtY}, z~/Mt < 1 v-a.s., i = O,g. Now let:

qt = min z , cot+l, 0) , Ft(zY, cot+l, ~/ , t = O, 1, 2 . . . . . t

Then, f rom the Euler equations it fol lows

I/z I 1 pO >_ E ptO+l --~ttFt(z~ .f't

> E[P~ II-~'t] v-a.s. ,

_ ( Z g , py > E PY+I F t , t cot+l, Y) f ' t

> E[PYt+lqt II f t ] v-a . s . .

(8)

(9)

The second inequality in (8) is strict while the second inequality in (9) is an equality if zt ~ (co) > zY (co) and co c ~ . The opposite is true for

t those co 6 ~7 for which zY(co) > zt~ Let #0 = 1 and/Zt+l = I-Ioqi, t > 0. Then, s ince /z t is strictly positive for a.e. co and 5Or-measurable

o , ( l o ) P~ #t > E[Pt+ltZt+l I] 5vt] v-a.s.

~' . ( 1 1 ) Pt t-tt > E[PY+llZt+I II ~ t ] v-a.s.

Once again, the inequality in (10) is strict while (11) holds as an equal- s Y 0 ity if zt ~ (co) > zt y (co) and co 6 qJt. The opposite is true if zt (co) > zt (co)

and co c ~ . Lett ing Yt 0 • = P t l z t + P t # t , t > O, the process {It} is a su- permart ingale satisfying the supermartingale inequality strictly on ~ .

[]

The two lemmas are now combined by letting Vt = Wt + I t , t > O. For any co c S~ we have

E[Wt - Wt+l II ~t ] (co) > 0, E[Yt - Yt+l II f t ] (co) > 0 if co ~ A t ,

E[Wt - Wt+l II ~-t](co) = 0, E[Yt - Yt+l II ~-t](co) > 0 i f co 6 ~ t �9

Therefore, E[Vt - Vt+I II ~'t](co) > 0 for all co 6 S[ = A~ U qJT. Hence, for any co 6 S~, a value loss 6t(co, s) > 0 can be determined such that E[Vt - Vt+l II ~-t](co) -> 6t(co, s). Note that the construction


of this value loss process did not entail any concavity restriction on the production functions. A general value loss result characterizing the asymptotic behavior of {zt ~ } and {z[} can now be proved which extends FOllmer and Majumdar (1978, theorem 3.1) to the general nonstationary case considered here. Consider the X-value loss set, Ax = { (t, co): 3t(co, e) > k}, and define the total time spent by {Vt} in Ax for any given co as Tx(co) = y]~o x(Ax)( t , co). Here, x (A) is the indicator function of the measurable set A. It is now shown that Tx has finite expectation, i.e., {Vt} will almost surely leave Ax after at most a finite number of visits. Hence, the value loss St(co, e) can exceed k v-a.s, for at most a finite number of time periods. Since the magnitude of the value loss depends on the divergence between the {zt ~ } and {z[}, the next result implicitly places restrictions on the asymptotic behavior of these variables.

Theorem 1: Under the stated assumptions on technology and preferences, ETx < oo.

Proof." For any N > 1, let the truncation of Tz be TU(co) ---- y]U_ 1 X (Ax)(t, co). Using Lemmas 1 and 2

~.x(Az)(t , co) = X < St(co, ~) < E[Vt - Vt+l tl Ft](co)

if (t, co) e Ax ,

;~X (Ax)(t, co) = 0 < St(co, e) < E[Vt - Vt+111 5t](co)

if (t, co) ~ Ax .

It now follows that

dl) z

N

f Z kX (A~)(t, co)dv t = l

N

t = l

N

= Z f [ v t - Vt+l] dv from Billingsley (1979, p. 395) t = l

f r 0 / y < V0 dv = 2[u0(c0) + Uo(C o)] ,

222 S. Joshi

since Vr+l > 0 for a.e. co. Since the above holds for all N > 1, it follows from the Monotone Convergence Theorem that

X T)~ dv = )v dv < 2[Uo(Co)

4 The Convergence Result

For any co ~ S~, the supermartingale process {Vt} gives a strictly positive value loss which, in addition to s, may depend on the tuple (t, co). The almost sure convergence of {zt ~ } and {zt y } in II II requires a strength- ening of this sensitivity property of { Vt } via the strong restriction of a

Uniformity Assumption: For any s > 0 there exists a a(s) > 0 such that for any t _> 0 and any co 6 f2

if IIz~~ > s, then E{V~-Vt+I II ~-~](co) ~ ~t(co, s) ~ ~(~).

Example: In the context of optimal growth theory, similar uniformity restrictions have been used for instance in F611mer and Majumdar (1978). Since the use of uniformity assumptions in areas other than optimal growth theory is relatively unknown, this example uses the area of capital income taxation to show how the broad methodology outlined in Lemmas 1 and 2 can be used to construct a process satisfying this assumption. The application to a particular instance, and the choice of a particular metric, requires a slight variation in the technique developed in the previous section. The divergence between the net returns to capital is measured using a variation of the metric used in Majumdar and Zilcha (1987). It is defined for any co 6 f2 and net returns r(co), r'(co) as:

IIr (co) - r'(co)II = Ir(co) - r'(co) I

max{r (co), r'(co)}

Let 0 < 0, g < 1 be two alternative levels of capital income taxation i = ( l _ i ) r,~k i and r t a t , t, cot+~), i = O, y, be the after-tax return to capital

per unit. Partition the set of all co for which Ilr~ - ry(co)[I > s into the Yt-measurable sets


A ; = {o~: r, ~ - r l (~o) > ~r, ~

r7 = {~o: r l (~o) - r, 0( ,o) > ~ r f (o~) }.

If 0 is the rate of capital income taxation, then from the Equivalence Principle of Becker (1985), the competitive equilibrium with tax rate 0 corresponds to the planner's problem:

OG

sup E ~ [ ~ ( 1 - o)]tut(ct), {(k~, ct)} ~ ~ ( s , {it}, 0 ) , t=0

where 0 < ~ < 1 is the discount factor. Letting pO = yut(cOt), the stochastic Euler equations for t >_ 0 are

pO o 0 = E[Pt+lrt+l I[ Set] v-a.s. (12)

Let q, = min[rt ~ r [ ] , t > 1 and let ~o = 1 and #t = I-It1 qi, t >_ 1. Note that #t is strictly positive almost surely and 5t-t-measurable. Using (12) we obtain

0 ~ 0 0 ~ 0 0 Pt rCt-lrt = E[Pt+lrh-lrt rt+l ]l Ut] v-a.s. (13)

Note that letting ~ot = ~?t-1 r~ t _> 1, it follows that

0~0 0 0 P t t > E[Pt+l~t+l ]l St'l] v-a.s. , (14)

indicating that the process {pOlO} is a supermartingale. Returning to (13), for any A c Ut follows

fa 0~0 P t t dv

= fanA~

> fAoa~

fA UA = P t + l t + l dv + e naf

I " 0 ~ O 0 j 0 ~ 0 0

P t + l T r t _ l r t r t+ 1 dv + P t + l T r t _ l r t r t+ 1 dv JA ]a pO ~0 dv p ~ l T r t _ l ( r t y q- 6r~)rOt+l d v q- t+, ,+1

0 ~ 0 0 P t + l T r t _ l r t r t+ 1 d v .

224 S. Joshi

Since the above is true for all A 6 St, it follows that v-a.s.:

pOlO 0 0 e 0 ~ 0 0 t t - - E [ P t + l ~ t + l 11 fit] > 8 X ( A t ) E [ P t + l Y r t - l r t r t + 1 [1 ~ t ]

e o o = e x ( A t )p t~ t �9

(15)

From the Mean Value Theorem, there exists a random variable ht such that Pt-~176 - > ht >_ E[Pt+l~t+ 1 0 0 U .Tt] for a.e. co and:

log(p~ ~ -- log(E[P~176 [l .Ut]) = p O l O 0 0

t t - - E [ P t + l ~ t + l 11 .fit] ht

(16)

Using Jensen's inequality for conditional expectations on (14) (Billings- ley, 1979, p. 399) yields

log(p~ ~ > log(E[P~176 [1 .Tt])

>_ E[log(p~176 .Tt] p-a.s. (17)

This indicates that the process {log(p~176 is a supermartingale. Com- bining (15), (16), and (17) yields

log(pt0r 0 0 - E[log(Pt+l~t+l) [[ Ut]

>_ log(p~ -- log(E[Pf+,~t~ II 5ct])

o o E[_O ~.o P t ~t - - [ J t + l q t + l II Y,]

h t

pOlO _ E[pO+l~tO+l ][ .fit] t t

An identical argument yields v-a.s.

l o g ( p y ~ y - E [ l o g ( p t Y + l ~ y + l ) I I > x(FT), f f = 7 c t _ l r [ .

lo V Y Letting Mt = log(pt~ ~ + g(p, ~t ), it now follows that

E[Mt - - M t + l [[-~"t] > ~X(At U F t ) .

Therefore, for any period t and realization co for which []r~ - r[(co)ll > e, the process {Mr} records a strictly positive value loss of e > 0 which is independent of (t, co).


Theorem 2: Under the Uniformity Assumption, Ilzt ~ - zt y II ~ 0 v-a.s.

Proof." From the Uniformity Assumption follows

<_ f v,+, ii Ftldv .

Further, since {Vt} is a supermartingale, the random variable E[Vt - gt+l II : r t ] is almost surely nonnegative. For any A ~ Ut, since f E[Vt- gt+l II 2:tl dv = 0 when v(A) = 0, it follows that

f x(f2 - S[)E[Vt - II f t ] d v 0 V,+l >_ I

Combining and using Billingsley (1979, p. 395) we have

~(e)v(s;) <_ f e[v,- v,+, [1 . . T ' t ] d v = f [v, Vt+l]dv .

Summing up, and noting that Vt is strictly positive for a.e. co:

T T

~(~) F , ~(sb <_ F , ely, - v t + , ] = E [ V o - v ~ - + , ] t = 0 t = 0

t V < E [ V o l = 2 [ u ~ ( c ~ T = l , 2 . . . . .

Hence, }--~ v(S[) < ec. Now, from the first Borel-Cantelli lemma (Billingsley, 1979, theorem 4.3) it follows that v(lim s u P t ~ $7) = 0.

[]

5 Conclusion

This paper offered a synthesis of long-run invariance results which have been pervasive in different areas of economic dynamics but have lacked a unified coherent approach. This synthesis was achieved in the context of an aggregate growth model in which the Euler characterization of optimal programs could be used to generate a supermartingale value loss process. An open issue is to extend the framework to multisector growth models where presumably, in the context of convex preferences and technology, the competitive conditions derived through separation arguments would have to be used.

226 Joshi: Long-run Invariance

References

Becker, R. A. (1985): "Capital Income Taxation and Perfect Foresight." Jour- nal of Public Economics 26: 147-167.

Billingsley, P. (1979): Probability and Measure. NewYork: Wiley. Brock, W.A. (1974): "Money and Growth: the Case of Long Run Perfect

Foresight." International Economic Review 15: 750-777. F611mer, H., and Majumdar, M. (1978): "On the Asymptotic Behaviour of

Stochastic Economic Processes." Journal of Mathematical Economics 5: 275-287.

Majumdar, M., and Zilcha, I. (1987): "Optimal Growth in a Stochastic En- vironment: Some Sensitivity and Turnpike Results." Journal of Economic Theory 43: 116-133.

Address of author: Sumit Joshi, Department of Economics, The George Washington University, 2201 G Street N.W., Washington, D.C. 20052, USA.

Long-run invariance in economic dynamics: A note

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