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Social security as an economic stabilization program
D. ANDREW AUSTIN Bowdoin College Brunswick, Maine
Social Security as an Economic Stabilization Program* This paper analyzes the stabilizing properties of an unfunded social security system in an overlapping generations model with cyclical equilibria. This provides a welfare rationale for a social security system independent of the windfall gained by the first generation, or social insurance arguments. An unfunded social security system redirects intergenerational resource transfers from private savings, which are subject to endogenous fluctuations, towards a public system with an implicit rate of return of unity. In an economy with a cycle of order 2, a planner with a limit-of-means social welfare function will completely stabilize the economy,
1. Introduction
While the primary impetus for passage of the Social Security Act of 1935, which instituted a pay-as-you-go (PAYG) old age public pension system, was to alleviate the poverty and economic uncertainty facing the elderly, there may be another unanticipated benefit: a PAYG Social Security system may promote macroeconomic stability. This paper outlines how a PAYG social security system stabilizes an economy subject to endogenous cycles. This stabilization is not achieved through discretionary fiscal policy or income smoothing. Instead, the paper shows that a social security system may alter the excess demand functions for an economy in a way that reduces the magnitude of macroeconomic fluctuations. In an economy with identical individuals and perfect foresight, complete stabilization can always be achieved through introduction of an appropriate social security program. While many macroeconomists prefer explanations of aggregate fluctuations that rely on random fundamentals, others have presented plausible models with endogenous cycles, including Woodford (1990a) and Farmer and Guo (1994). Woodford (1990b) provides an extensive discussion of the applicability of several models which allow a role for endogenous fluctuations. These models obviously differ from the model presented below, which is based on Azariadis and Guesnerie (1986). However, as the Azariadis and Guesnerie model spawned these richer models, it is an appropriate place to *I wish to acknowledge useful comments from Michael Loewy, Costas Azariadis, Giles AuchmutT, Michael Ben-Gad, participants in the University of Houston/Rice Ulxiversity Macro seminar and two anonymous referees. Remaining errors are mine.
Journal of Macroeconomics, Spring 1999, Vol. 21, No. 2, pp. 309--333 Copyright © 1999 by Louisiana State University Press 0164-0704/99/$1.50
309
D. A n d r e w Austin
start the analysis of macroeconomic stabilization via social security. This paper aims to explain the effect of social security on macroeconomic fluctuations within the simplest possible model which allows such endogenous fluctuations. Section 2 describes a pure exchange overlapping generations (OLG) economy and a social security system. Section 3 shows how a social security system may enhance macroeconomic stability, and Section 4 describes the welfare effects of such a program. Section 5 concludes the paper.
2. The Model This OLG model follows Samuelson (1958) and Gale (1973). For simplicity the model only allows for pure exchange, although production could be included without changing any important features. Generation 0 is born old at t = 1 and is endowed with e z units of consumption and one unit of fiat money. A young generation is born in each of an infinite number of periods t = 1,2 . . . . . and is indexed by that period. All generations live two periods, except generation 0 which lives one period, and comprise N identical individuals. A member of generation t is endowed with e 1 units of consumption in t and e ~ units during period t + 1 (old age). The consumption good cannot be stored from one period to another, while fiat money is costlessly stored. Members of generation 0 have utility functions u°(Co), and all others have utility functions u(cr~, c°+a) where ctv is consumption in period t by a person born in period t, and cO+1 is consumption by that person when old. All utility functions are continuously differentiable, strictly quasiconcave and positively monotonic. Consumption when young is normal. Marginal utility tends to + ~ as consumption approaches 0. Let Pt be the price of money in period t. If pl > 0, then the first generation can trade its unit of money for an additional Pl units consumption, and i f p t > 1, a subsequent generation t faces the lifetime budget constraint
(e I - cYt)/pt + (e ~ - ct°+l)/pt+] >- O.
(1)
Each unit of saving by the young generation at t yields Rt -- Pt+ 1/Pt units of consumption at t + 1_ Per capita saving is thus a function of Rt; N
s(Rt) -----( I / N ) ' ~ i= i
argmax u(e 1 - s, e z + s ' R ) .
(9.)
0~---s"~el
Saving is assumed positive when R = 1, so that the economy is in the Sa310
Social Security as an Economic Stabilization Program
muelson case. This conforms to the stylized fact that young workers have higher incomes than older retirees, requiring private saving to ensure sufficient consumption in old age. All individuals perfectly foresee future prices. The excess demand of the young generation is the negative of the saving function, and the excess demand of the old is the dissaving of their unit of money. Therefore total excess demand at period t is D(p~, Pt+l)=-Pt - s(Rt) = Pt - s(pt+{pt).
(3)
DEFINITION 1. A perfect foresight monetary equilibrium is a sequence of prices {P}~=I such that Pt > 0 and D(pt, pt+l) = 0 Vt. The equilibrium condition D(pt,pt+l) = 0 implicitly defines a sequence of prices which constitute a perfect foresight monetary equilibrium. By fixing P~+ 1 and soMng for Pt a "backward-looking" equilibrium sequence of prices is derived. Let the mapping pt = f(pt+l) describe the backward-looking solution to (3), so that D(f(p~+l),pt+l) =- O. To be more specific _flPt+ 1)
~
S[Pt+1~(~t+l)]I
(4)
If consumption is a normal good in both periods there is a unique backwardlooking equilibrium for any given price level at time t + 1. Thus there is a continuum of backward-looking equilibria indexed by the price level at t + 1. Some of these equilibria may be periodic monetary equilibria, so that prices follow a deterministic cycle. DEFINITION 2. A monetary equilibrium is a nontrivial periodic equilibrium of order k / f P t = Pt+k V t >- 0 (k > 1) and if there is no positive integerj < k such that Pt = Pt+j. A trivially periodic equilibrium, in which the path of prices is constant, will be called a steady-state monetary equilibrium. DEFINITION 3. An equilibrium is a steady-state monetary equilibrium if p t =pt+l>-OVt>-O. This paper analyzes stationary equilibria, in which the values of real variables depend on the date only through the identity of the state at that date. Nonstationary equilibria, in which real values are nontrivially datedependent, are not analyzed. 311
D. Andrew Austin Next, locally stable equilibria are defined. Stability has played a large role in characterizing equilibria in this class of models, and will play an important role in the analysis below. Definitions of stability require a distinct_ion between the backward perfect foresight (b.p.f.) dynamics generated by the f(.) mapping and the forward perfect foresight dynamics, which can be approximated near a given orbit by taking the inverse of theft.) ([assuming f(-) is regular-valued). Thus local stability in the b.p,f, dynamics implies instability in the forward perfect foresight dynamics and vice versa. The dynamic analysis employs the b.p,f, dynamics because f(.) is single-valued whereas its inverse is not necessarily, and f(.) is closely related to the offer curve.
DEFINITION 4. A steady-state monetary equilibrium is locally stable in the forward perfect foresight (unstable in the b.p.f, dynamics) if df(p~ + l)/dp, + l <-1. Grandmont (1985, 1018) shows that local stability in forward perfect foresight dynamics of a steady, state equilibrium implies existence of a 2-cycle. An equivalent condition requires the elasticity of savings with respect to R be less than - 1/2 in the neighborhood ofR = 1.1 While many economists believe this elasticity is positive a priori, the empirical evidence on this question is not conclusive. 2 DEFINITION 5. A periodic equilibrium of order 2 (2-cycle) equilibrium is locally stable in the backward b,p.f, dynamics (unstable in the forward perfect foresight dynamics) if
dy[f(p +
• ~
<1.
Figure 1 illustrates an example of a periodic equilibrium of order 2, using the graph off(pt+ 1) and its reflectionfl(pt+ 1) about the 45 ° line. Suppose/5 is the price in period t. Thenf(~) is the price in t - 1, and if~ is a 1See Azariadis (1993, 368) for a discussion. Also see Grandmont's discussion of stability conditions for this class of models (p. 1007 et passim). aln par~cular, Boskin's 1978 study, which reported large positive elasticities, is regarded as discredited among many empirical researchers. Deaton (1992, 60.) concludes "the empirical results are as ambiguous as is the theory, or more positively, that the empirical results confirm the lack of invariance to time, place and other variables that the theory predicts."
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Social S e c u r i t y as an E c o n o m i c S t a b i l i z a t i o n P r o g r a m
D(p,+l,pO =0 p,
f(p) D(pt,P~+I) =0
f(p)
p.
~
p,+,
Figure 1.
coordinate of the intersection off(pt+l) and f f l(pt+l) then the price in t - 2 will be/~ = f-l[f(~5)]. Thus, the sequence {~5,f(/5)fi. . . . }, in which the solution of the excess demand function alternates between the two intersections off the 45 ° line, is a periodic equilibrium, and generates a sequence of interest factors {/t,1//~,/2. . . . } where/~ = f(fi)/~. A transfer of ~ units of consumption good from the young generation to the old generation represents a pay-as-you-go social security system. A planner, with preferences defined by a social welfare function, sets the level of this transfer. However the planner cannot bankrupt young individuals or impose negative taxes, so • E (O,el). After the social security program is in place, the budget constraints for each period imply consumption is (Co) = (e2 + z + pl),
for generation 0 ;
(ctY, cO+l) = (eI -- st -- z, e 2 + ~ + Rt'st),
for generations t > 0 .
(5a) (5b)
The lifetime budget constraint in a monetary equilibrium becomes
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D. A n d r e w
Austin
(el - x)/Pt + (e2 + x)/Pt+] -> crt/Pt + c°+i/pt+1,
(la)
and per capita saving is then a function R t and ~: N
s(Rt, x) -- ( 1 / N ) ' ~ i= 1
argmaxu(e 1 - ~ - s , e 2 + • + s ' R t ) .
(2a)
O<--s<--e1
Similarly, total excess demand at period t is D(pt,
Pt+l, r) =- Pt - s(Rt, x) ,
(3a)
x] solves D~C(pt + l, "~), Pt + l, "¢] =- O. Changes in the social security tax rate x have two potential effects on each generation's well-being. First, in a 2-cycle equilibrium interest factors alternate between/] and 1//]. Thus, a larger social security transfer (which returns an implicit rate of return of unity) increases the well-being of the generation facing the low interest rate and decreases the well-being of the next because these agents are savers at all relevant interest rates. Second, changing the social security tax rate in a 2-cycle equilibrium will generally change the interest rate, as it will alter the supply and demand for savings. This effect will benefit whichever group of generations (even or odd) sees their real interest rate increase. The effect of changes in z on interest rates is discussed in the next section. In particular, under certain conditions raising the social security tax rate will stabilize interest rate fluctuations, thus increasing the welfare of generations that would have received lower rates of return on their private savings and decreasing the welfare of generations that would have received higher rates of return on private savings. The sharp opposition of interests between odd and even generations is due to the "short and simple" lifespans used in the model. A richer model with longer-lived individuals allows a less stark opposition of interests among generations. Moreover, a model which includes uncertainty and a richer demographic structure will almost certainly have richer dynamic behavior, which may match the behavior of actual business cycles more closely, at the cost of tractability. a n d f ( p t + l, z) =- s[pt + i / f ( p t + l),
3. Macroeconomic Stabilization and Endogenous Cycles Most of the economic analysis of social security has examined its effects on capital accumulation and intensity, the growth path of the economy, or the social insurance benefits accruing to individuals. There may be another 314
Social S e c u r i t y as a n E c o n o m i c S t a b i l i z a t i o n P r o g r a m
effect of the social security system not captured by that research. In particular, if expectations generate endogenous macroeconomic fluctuations, then the social security system may have an important stabilizing role. This section analyzes the stabilizing effects of an unfunded social security system. If a periodic equilibrium exists, as in Figure 1, the introduction of a social security system changes the shape of the aggregate excess demand function, and thereby reduces the magnitude of the economic fluctuations. Social security diverts intergenerational transfers from a private saving mechanism subject to endogenous cycles to a public system which pays an effective interest rate of unity. This effect is distinct from the usual theory of macroeconomic stabilization using discretionary fiscal policy intended to counteract the effects of unforeseen exogenous shocks. Theorem i shows the effect of a social security system on offer curves, which are used instead of excess demand functions for clarity. Theorem 1 shows that imposing a social security tax that does not force oversaving rotates the offer curve around (e 1 - s(1), e~ + s(1)), and thus rotates the excess demand function around the origin. Figure 2 shows the effect of a social security system. A higher social security tax rotates the excess demand curve around the origin, bringing the economy closer to the "classical" case. In an extreme situation, in which x = s(1), the social security system completely displaces private saving. The intuition is that movement of resources among generations through the social security system is unaffected by fluctuations in the interest rate, unlike private savings. Figure 2 shows an individual's intertemporal choice problem along with offer curves before and after introduction of a PAYG social security system. Figure 3 shows the corresponding excess demand curve. The excess demand curves are reflections of the offer curves, and the social security system shifts the point on the offer curve corresponding to the origin of the excess demand curve to the northeast. Let O(R) denote a point on the offer curve without social security, and let O(R;x) denote a point on the offer curve with a social security system with tax ~, so that: O(R) =- {crt, C°ll(Ctr, c°+1) = argmax u(crt, C°l)
s.t. ctr = e 1 - s and ct°+l = e 2 + R ' s } ;
(6a)
O(R; x) =- {crt, c°÷ll(ctr, c°+1) = argmax u(ctr, c°+1)
s.t. ctr = e 1 - s - x a n d c t ° + l = e 2 + R ' s + x } .
(6b)
The offer curve without [with] social security O[O(x)] consists of the col315
D. Andrew Austin CI+ I
R = 1
(eL-s(1), c2-s(l)) /
/
(eI-T,C2-I-T) (el,e2)
F 4
ct
Figure 2.
lection of 0(R) V R [0(R; ~) V R].3 Note that this maximization is dual to the maximization which defines the savings function in Equation (2). Let R be the interest rate that induces no net saving. Theorem i states the pre-social security offer curve (a) lies to the left of the offer curve after introduction of social security for interest factors between R__and 1, (b) these offer curves coincide at R = 1, and (c) the pre-social security offer curves he to the right of the offer curve after introduction of social security for interest factors R>I.
THEOREM 1. If O < r ~ s(1), so the social security does not force oversaving, then (a) V (c[', c°+1) ~ O(R; r) for R ~ (R, 1) 3 (cL c°+z) ~ O(R) such that c[ < c['; (b) 0(1) = 0(1; r); (c) V (c[', c°1) ~ O(R; r)for R ~ (1, oo) 3 (c[, cO+l) ~ O(R) such that c[ > c[. aNote that the graph of fiR) is identical to the set of points for which D(pt,p~ + 1) = O.
316
Social Security as an Economic Stabilization Program P~+l
D(pt,Pt+l)=O
/\l\
/;) /
/'
i/ t T
Figure 3.
PROOF. Because the social security tax offsets private saving exactly when R = 1 and 0 < * --- s(1), (b) follows immediately. This implies the offer curves intersect at R = 1. Suppose (a) is false_ Continuity of 0 and lira C°l = ~ a s R ~ ~ i m p l y 3 (ctY,ct+l) o ~ O(R) and some R < 1 such that C°+l = e 2 + z. Furthermore, c[ < e 1 - z, because otherwise z >- s(1). Continuity of the offer curves then implies offer curves intersect, that is, 0(R) = 0(B;z), for some R ~ (R,1). Let u(-) = a be the indifference curve through this intersection. By the definition of the offer curves and consumer maximization this indifference curve's slope must equal the slope of the chord between (el,e2) and the intersection, and the chord between (e 1 z,e 2 + z) and the intersection. However, because z > 0, the slopes of these chords are distinct, and the indifference curve cannot be tangent to both. This contradiction establishes (a). Part (c) follows from the same argument as (a). A government seeking to avoid the possibility of large fluctuations could reduce the amplitude of cycles by instituting a social security system. For the purposes of this section, stabilization is assumed desirable. Section 317
D. Andrew Austin
D(p,,p,+,)=0 P,+~ [D(p,,p,+fir)=0 \ 1
D(Pt+i,p,) =0
IV /
f
I
'
1 °
I
iI =
p
p,
Figure 4A.
4 considers the rationale for stabilization. Because social security taxes shift the lower portion of aggregate demand curve closer to the 45 ° line for Samuelson type economies, the scope for fluctuations will generally be smaller. In particular, if the average price level does not fall too much, the variation in interest rates will fall as social security taxes rise. This case is shown in Figure 4A. For some economies higher taxes may induce a larger variation in interest rates if the average price of money falls sharply (shown by f and f ' , which can be calculated by finding the intersection of the chord between equilibrium price vectors and the 45 ° line) relative to the movement closer to the 45 ° line, as shown in Figure 4B. Inspection of Figure 2 shows the offer curve O(Rt) for R -< Rt <-- 1 moves closer to the line with slope of - 1 through the endowment point (R is the autarkic interest factor). Similarly, the lower portion of the reflected offer curve moves closer to the 45 ° line, as shown in Figure 3. Furthermore, only the portion of the 45 ° line between the origin and the point pMAX, representing the steady-state monetary equilibrium with the highest price level, will he considered to ensure the following metric is well-defined.
318
Social Security as an Economic Stabilization Program R(x) Pt+ l
=
/
//
// / i
R(O) /
pt=pt+~
/
\
i
p
'~\
]
I
I
;,
p
I/R(O)
Figure 4B.
DEFINITION 6. (i) Define the lower portion of the reflected offer curve with social security tax r as
qDL(~) = {(Pt, Pt+l) 1D(pt, Pt÷l; x) = 0 andpt >-- Pt+l} • (ii) Define the lower portion of the offer curve with social security tax z as OL(~) = ~(R;~) such that R <~ R <- 1 where R is the autarkic interest rate. (iii) Let f L = [(p',p") I p' = p" <- pMAX} denote the 45 ° line. (iv) Let c/3 = {(crY,c°+1) I [crt - e 1] q- [ctO+l - - e2] = O, crt > O, ct°+l >~ O] denote the line with slope of - 1 through the endowment point.
The Hausdorff metric is a natural measure of distance between sets (see Munkres 1975, 279). DEFINITION 7. Let A,B be nonempty, closed, bounded subsets of R2+, and let d be the Euclidean metric. Define n(A, e) = U,~A Bd(a, e) where Bd(a, 319
D. Andrew Austin e) is an open ball of radius e centered on a. The Hausdorff metric is defined as:
H(A, B) = glb{elA C u(B, e) and B C u(A, ~) .
Let H1 = H[~(x),f L] be the Hausdorff distance between the lower portion of the reflected offer curve (of an economy in the Samuelson case) and the 45 ° line, in an economy with tax rate ~ ~ [0,1), and let H e = H[OL(z'),fL] be the Hausdorff distance between the lower portion of the reflected offer curve and the 45 ° line between the origin and pU~, in an economy with tax rate T' ~ [0,1) and T' > ~. Then H 1 > t12. THEOREM 2.
PrtoOF.
See Appendix.
DEFINITION 8. A function h:A ~ B is single-peaked if3 x* such that V x",x' ~A (a) x" > x' > x* implies h(x") < h(x*), and (b) x" < x' < x* implies h(x") < h(x') < h(x*). 1. Let 3q~o(pt+i) be the restriction of the function f(p~ +1,~)for some T° ~ [O,el]. Assume 3qo,(pt+1) is single-peaked Vr E [O,el]. ASSUMPTION
Single-peakedness off(') (for a given value of x) provides a major simplification, implying substitution effects are relatively stronger than income effects at low levels of R, and that income effects become increasingly large as R increases. Therefore, at some unique point income and substitution effects exactly offset each other. If f(-) were multi-peaked income and substitution effects would exactly offset each other at more than one interest factor. Grandmont (1985) shows that for a utility function separable in consumption when young and when old, if the degree of concavity for an old individual's subutility function is sufficiently greater than for a young individual, then the offer curve must be single-peaked. Furthermore, Grandmont finds additional conditions on the curvature off(') which along with the above convexity assumptions imply a unique stable cycle exists. 4 More generally, extending the results presented below to include multiple cycles is straightforward. aSee Grandmont (1985, 1024-26) on the negative Schwarzian derivative condition.
320
Social Security as an Economic Stabilization Program ASSUMPTION 2. For some social security transfer r ~ [O,s(1,0)] the economy has a 2-cycle which is locally stable in the backward perfect foresight dynamics and the 2-cycle is unique when it exists. 5 Theorem 3 considers the simplest possible case, in which the offer curve is single-peaked and a 2-cycle exists and provides sufficient conditions for the diminution of the cycle's amplitude. Success in stabilization will be measured by the amplitude of the cycle, defined as the sum of absolute values of interest rates over the cycle. DEFINITION 9. The amplitude of a 2-cycle for any j >- 1 is defined as
A(/~, R~+:)= IRi - 11 + I / ~ + : - 11. The amplitude metric can be easily extended to cycles of order k > 2 and the existence of multiple stationary k-cycles, which are not considered here. An interest factor R is the slope of the ray from the origin that passes through the equilibrium intertemporal price vector. Thus the amplitude corresponds to the angle between the rays from the origin to the two equilibrium intertemporal price vectors. As the examples in Figures 4A and 4B illustrate, the amplitude of a 2cycle may increase or decrease with the introduction of a social security system. The effects of raising the social security tax from a given level are qualitatively identical. Figure 4B shows that raising the social security tax from zero to x may increase the amplitude of a 2-cycle. This occurs if the reduction in savings (for a given Rt) by the young generation facing the lower interest rate exceeds the reduction in old-age consumption by the preceding generation (for the same Rt), then the interest rate must rise to clear the market. The following assumptions are sufficient to rule out such cases. Suppose {/~,f(~5,x)} are prices associated with a 2-cycle, where /5 < f(15,z), Assumption 3 requires that excess demand at/5 be associated with an elasticity of saving with respect to R is between zero and negative one. The set of prices which satisfy this condition comprise the backward-bending domain forf(pt+ i,x), and are denoted S(x). DEFINITION 10, Let S(z) = {Pt+i C_ R2+ I es(R;r) ~ [0,-I)}, where es(R;) --- n'sR/s(R,~), R = pt+Jf(Pt+i,T) and sR is the partial derivative of saving with respect to R. ~rhe stabilitycondition in Assumption4 should be understood to applyto the restriction of ]~pt÷l,Z)definedin Assumption 1 where a social securitysystemis in place.
321
D. Andrew Austin ASSUMPTION 3. Min[p~l,f(p~l,T)} ~ S(z) where pM and f(pM, z) are equi-
librium prices for a 2-cycle. Next, the introduction of a social security tax may increase economic fluctuations if generations facing the low interest rate either increase savings due to a strong income effect or decrease savings by a large amount relative to the reduction in savings by generations facing the high interest rate. Thus if the demand for fiat money in the low interest state increases enough or decreases relatively slightly in response to a social security tax, the interest factor may fall, thus increasing the amplitude of the economy. Assumption 4 rules out such cases. ASSUMPTION 4. Let [l = ptM+l/f(ptM+ l,'C) where pM < f(pM, z ) are equilibrium prices for a 2-cycle. Then
as (R,o,~"~) a
[ 1 + s,(/~)] [ 1 +
es(1//t)]
as(R, ~) at 1/4 A more transparent (albeit stronger) sufficient condition requires normality of consumption in both periods a n d / t E [1/2,1]. 6 6Assumption 4 and the Lemma (see Appendix) imply that [as(l~,~)/a~][s < 0 where t2 is the low interest factor, as shown at the conclusion of the proof of Theorem 2. Note this is not implied by normality. Normality in both periods and 1 ->/~ -> 1/2 do imply this inequality. Differentiating [e 1 - c~(Rt,e2)].Rt + [e2 - c°+l(Rt,eS)] ~ 0 , w.r.t, e2 and rearranging terms gives
ae~
/~
aez .
Normality ofctr and c°+1 implies 1
act
~t~ -> 'ae ~ >- 0 , and substituting in the identity
as(P~,~) acT(l%,~) a~e2 a "(1 - /~) -
322
1
Social Security as an Economic Stabilization Program While a planner can completely suppress fluctuations by setting = s(1), the marginal relationship between social security tax levels and volatility of interest rates provides a more robust basis for intervention. 7 Theorem 2 demonstrates that the above assumptions provide sufficient conditions for macroeconomic stabilization through increasing the size of a social security system. THEOREM 3. If Assumptions 1--4 hold, then the amplitude of the economy is decreasing in r, V r E [0, s(1,0)).
PROOF.
See Appendix.
4. Welfare
The possibility of stabilization does not imply its desirability. This section discusses the welfare implications of stabilization via social security. Partial stabilization of an economy with a periodic equilibrium is assumed to lead to a new periodic equilibrium of the same order, if one exists. The two-cycle case, as mentioned before, is an extreme case in which the economy swings from boom to bust, with odd generations benefiting from the cycle and even generations losing, or vice versa. Thus, any welfare justification for stabilization must rest upon a utilitarian social welfare argument. Suppose a planner can implement a social security system and choose the contribution level. Furthermore, suppose the planner must choose among date-invariant policies that support periodic or steady-state equilibria. This may be justified by the need for credible, and therefore inflexible, measures. Let the time-separable, strictly quasiconcave, strictly positively monotonic social welfare function (SWF) Wt(u t) represent the preferences of the planner for time t over utility of a representative member of generation t. Two possibilities are a limit-of-means function, can be shown to imply
as._0,~.(/~) E [0,-1] for/~ E [1/2,1]. O'c .~ 7If the planner does not completely stabilize the economy the analysis is complicated as a marginal change in taxes could set the economy on a new dynamic path, making comparative static analysis impossible. However, narrowing the lens between the reflected offer curves reduces the maximum possible amplitude.
323
D. Andrew Austin
W(u o, u 1,
1T
.) = am '
nf[- E re(u,)
T-~=
IT
t=o
] '
and a discounting function, eo
W(u °, u 1, ua,...) = W°(u °) + ~ 6t" Wt(ut), 6 E (0,1) , t=l
where Wt(u t) = W(u t) V t >-- 1. A limit-of-means function treats generations symmetrically, so that the effect of the generation 0 vanishes. Because preferences are convex the planner maximizes social welfare by equalizing utility across odd and even generations. This is accomplished by setting the social security tax sufficiently high to guarantee R = 1 for all t. Therefore a planner with a limit-of-means SWF will completely stabilize the economy. The discounting function treats generation 0 as potentially special, given their unique role in the model. The windfall to the first generation from the startup of the social security program has been prominent in policy debates because of the actuarial immaturity of the U.S. program, even though from a long-run standpoint the value of this windfall shrinks because only one initial generation has windfall gains but the infinite number of generations which follow do not. A social planner who cares about the initial old chooses a social security tax larger than the minimum tax needed to achieve stabilization, and induces an interest rate that exceeds the "golden rule" rate. Thus if stabilization redistributes wealth towards the first old generation this planner chooses a contribution rate which forces oversaving. If the planner does not care about the initial old and the economy starts out in the high-interest rate state (R1 > 1), complete stabilization is not optimal as the planner cares more about odd generations than even, and thus chooses an interest rate slightly above one. s This stabilization reduces the utility of generation zero if the fall in the price of money outweighs the social security benefit.
5. Discussion
In a richer model there are limits to stabilization. First, there may be costs to stabilization, including administraldve expenses, the possible effects on capital accumulation, and distortions induced by payroll taxes. Second, ~¢¢hen the planner restricts attention to stationary or cyclical paths, she may face a nonmarginal choice between R 1 = 1 and the interest rates associated with the smallest possible 2cycle. In other words, a discontinuity occurs where a bifurca~on in the path of the economy disappears.
324
Social Security as an Economic Stabilization Program
governments have incomplete information about the shape of the aggregate demand curve, and may wish to avoid overshooting in regard to the social security contribution level. In addition, governments may have other instruments for macroeconomic stabilization. In particular, monetary policy or a commodity standard policy could also achieve stabilization. However, the broad degree of participation in a social security system and its inherent inflexibility may make it a more credible instrument relative to policies which may be changed with the stroke of a pen. A social security system derives its fundamental credibility from its inflexibility, but allows little or no room for "fine-tuning." A full evaluation of the relative attractiveness of stabilization instruments would require weighing the costs of a social security system, such as those mentioned above, against the costs of monetary stabilization, including effects on production and distribution which will affect allocational efficiency (see Azariadis 1981). Of course, the net benefits ofmacroeconomic stabilization would be negative if social security taxes significantly distort work and saving incentives and sunspots are not much of a problem. 9 This evaluation is left for future research. Third, if generations are composed of heterogeneous individuals, a social security system may impose welfare costs on the individuals with larger old age endowments because it could force them to oversave. Analyzing this case would involve a straightforward extension of this model in which the stabilization and wealth redistribution effects are weighed against the welfare effects of oversaving. These results suggest a social security system may dampen the instability of a monetary economy subject to endogenous cycles. Given the size of the intergenerational transfers generated by social security, it may be an important stabilizing force, Received:August1994 Finalversion:June 1998
References Azariadis, Costas. "A Reexamination of Natural Rate Theory." American Economic Review 71 (December 1981): 946-60. • Intertemporal Macroeconomics. Cambridge, Mass.: Blackwell Publishers, 1993. Azariadis, Costas, and Roger Guesnerie. "Sunspots and Cycles." Review of Economic Studies 53 (October 1986): 725-37. Boskin, Michael. "Taxation, Saving and the Rate of Interest." Journal of Political Economy 86 (January 1978): 13-27. eThispointis due to CostasAzariadis. 325
D. Andrew Austin Deaton, Angus. Understanding Consumption. New York: Oxford University Press, 1992. Farmer, Roger, and Jang-Ting Guo. "Real Business Cycles and the Animal Spirits Hypothesis." Journal of Economic Theory 63 (January 1994): 4272. Grandmont, Jean-Michel. "On Endogenous Competitive Business Cycles." Econometrica 53 (September 1985): 995-1046. Gale, David. "Pure Exchange Equilibrium of Dynamic Economic Models." Journal of Economic Theory 6 (January 1973): 12-36. Munkres, James. Topology: A First Course. Englewood Cliffs, N.J.: PrenticeHall, 1975. Samuelson, Paul. "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money." Journal of Political Economy 66 (December 1958): 467-82. Woodford, Michael. "Learning to Believe in Sunspots." Econometrica 58 (March 1990a): 277-308. --. "Equilibrium Models of Endogenous Fluctuations: An Introduction." NBER Working Paper No. 3360, 1990b.
Appendix A
Table of Variable Definitions e1 e2
c?
= = = =
c~HY (~,e 2 ,u °) = ctHO (Rt,e2 ,U°) = =
D = e(Rt,e2,U°) = Pt
=
u°(co) =
R, = R =
326
Endowment received by young individuals. Endowment received by old individuals. Consumption in period t for person born in period t. Consumption in period t + i for person born in period t. Compensated consumption function in period t for person born in period t. Compensated consumption function in period t + 1 for person born in period t. Marginal utility of e2. Hessian for (Marshallian) consumption program. Expenditure function. Price of money in period t. Utility function for generation 0. Utility function for generation t -> 1. Interest factor. Autarkic interest factor.
Social Security as an Economic Stabilization Program s(R) = Per capita saving without a social security tax. Per capita saving with a social security tax. D(pt,pt + 1) = Excess d e m a n d function without a social security tax. D(pt,Pt + 1,z) = Excess d e m a n d funcldon with a social security tax. f(Pt+l) = Mapping of backward-looking solution for D(pt,Pt + 1) s(R,x) =
= 0. + 1, "l~) ---- Mapping
of backward-looking solution for D(pt,pt+l,x) = O. Social security tax rate. o(R) = Offer curve without a social security system at R. 3(R;~) Offer curve with a social security system and tax rate xatR. Offer curve without a social security system. o(~) = Offer curve with a social security system and tax rate
f(Pt
=
~L(~)
=
¢B= pMAX _-a(.)
=
~s(R) =
s(~) = w(.) =
The lower portion of the reflected offer curve with social security tax z. 45 ° line between the origin and pMAX. Line with slope of - 1 through the e n d o w m e n t point. Largest price level associated with steady-state monetary equilibrium. Amplitude metric for a stationary 2-cycle. Uncompensated elasticity of saving with respect to R. Set of prices such that ~s ~ [0, - 1). Social welfare function.
Appendix B PROOF FOR THEOREM 2. First, that Pt = s(';x) in equilibrium, s(';x) < el - t and (Pt,Pt+l) E ¢DL(z) imply Pt,Pt+l E [O,el - x]. Next let d = minp,_
0
@' - 0p' {(p~+l- p')~ + [f(Pt+l; ~) - p,]2} = -2(p~+1 - p') - 2[tip,+,; ~) - p'] = 0 ~Pt+l
-- P' -- p' + f ( p ~ + l ; x) = 0 ~ p '
Pt+l + .~Pt + 1; T)
2
Therefore
327
D. A n d r e w Austin 1 d{[pt+l,j~pt+l; x)], (p', p')} = ~ [ P t + l -- J ~ P t + l ; .~)]2 .
Because pt = f(pt+l,x) =- s[pt+.l/f(pt+l),~] = (e 1 - c r t
- ~) and
(e 1 - "c)/pt + (e 2 + x)/pt+l = crt/pt + C°+l/pt+1 P t + l " ( ex -- c Y -- "~)/Pt =
(e 1 - c[ - x)/pt --- c?+ 1 - e 2 - x
pt+l"s [Pt+l/~Pt+l), x]/pt = Pt+l = ct°+x - e 2 - -
"~,
then 1
d[ (p, + l, f[pt + l; x]), (p', p')] = ~ ((C°+1 - e ~ - x) --
(e I _
CY --
,~))2
=
21 ( c O
~ + c~ -
e~ -
d) ~
The last term is the distance between (c~(Rt; x),c°+ I(Rt;T)) and the closest point on % min d{[ctr(Rt; ~), C?+l(Rt; T)]; [ct, Ct+l] } s.t. [c; - e l] + [ct+ 1 - e 2] = O. 4.4 +1 (To see this, differentiate this expression w,r,t, x, find c[,c[+ 1 and substitute back into the Euclidean distance definition.) Next, define d*(~) as the furthest distance between a point on 0(~) and the closest point on % r d*(~) --- max 1 [C°+l(Rt; x) + ct(Rt; ~) __ e ~ -- el]2 Rt>l 2
By construction d*('c) = glb[slf L C u(q)L('c), s) and ¢DL(x) C u(fL S)] = glb[slf L C u(oL(x), S) and OL(x) C u(f L, s)];
d*(x) is characterized by the FOC:
328
Social Security as an Economic Stabilization Program
8.L
--
Oat
= [c[(at; x) + C ° l (at; z) - e 1 - e2] •
~!]
[8ct°+!(Rt; x) OcY(Rt; + - L Oat oa t
=0.
In the Samuelson case the first RHS term is strictly positive, so 8c,°÷l(at, z)
-sc[(at, ~)
aat
Oat
(al)
L e t [c[(x), c°+l(x)] solve (A1). Using results from utility maximization
-[st(•21-at
st(um - atu22) - ~.
D
--
Ull )
q-
at' ~-]
D
where D - -Ull + 2"u21'Rt - R~'u22 > 0 by convexity of u(') and L = OU*(c[, c°+i, e 2)
0e 2
> 0 by monotonicity. Simplifying this expression gives
Ull -- Ul~(1 -- at) -- atU22 --
(1
-
at)')~
st
Next, find sign for d[d*(.)]/d'c, where [c[(x), ctO+l('~)] is the point on 6~(x) furthest from cB. Differentiating and using the envelope theorem gives d[d*(.)]dz
[C°+l(X) + c[(x) - e 1 - e2] • 0
x) +
Ox
J
=[ct°+l(X) + c[(z) - e 1 - e2] •
I u12 --Oat iu2g" (1 - at) "q- -nil q-Oat'n21" ( 1 -
at)l "
Using condition (A1), the definition of savings and rearranging terms yields d[d*(')] - -
dx
-
=
[(1 + a t ) ' s t
- s,].
- k . R t . ( 1 - Rt) ~
D
(1 - Rt) 2 k D
st
< 0 for at ~ (0,1),
which establishes the r e s u l t l
329
D. Andrew Austin PROOF OF THEOREM 3. The proof is preceded by a lemma. LEMMA. Let p*+ 1(~) -- argmaxbC(pt+ 1,~)] and p* (z) -f(p*+ 1(z),z). Then Pc + 1 > p*+l(z) implies Pt+l E S(~). PROOF. Note the single-peakedness assumption implies p*+l(Z) is welldefined. The definition of p*(z) implies Of(Pt+l,z) • = sa = O. Op,+l P t + l f(Pt+t,~) + R t ' s R
There are three possibilities: either sR = 0, or the denominator approaches + ~ or - ~ . Suppose sa # O. First, f(pt+l,X) is bounded because the deftnition of the excess demand function and f(pt+l,X) imply that f(pt+l,z) = s[p~+/f(pt+l,X),~] <- e t. If I@ ~ ~ then Of(')/Opt+t approaches 1 for any finite R. Iff(pt+l,z) is maximized for R t = pt+z/f(pt+l,T) ---) ~ then s(Rt,~) 0, which contradicts the assumption that the economy is in the Samuelson case. Thus, sa = 0. Solving for sR gives
Of(p~ + t,~) .f(p~ + ~,~) Pt+l
SR =
1 - Rt" 0f(p,+ t,x) ' ~gt+l
This expression is negative by the single-peakedness assumption for Pt+t > p*+t(x). R, s(R, ~) >-- 0 implies es(R; x) < 0. Differentiating (5b) with respect to/It yields 0c°+ t( •) - s(-) + a~ -as(') ORt ORe
'
which reduces to
~,(.)
=
1 act°+:(.) s(.) aRt
:.
Define ct+t(Rt,e 2) -- argmax U(c[, c°+l) s.t. (e I - O ~ ) . a t + (8 2 - c°+i> --> 0 and define the Hicksian demand ctn°t(Rt,e2,U) ~ argmin [c°+t - (e i - c[)' Rt] s.t. U° - U(c[,ct°+ ~) >- 0. Then using the Slutsky decomposition gives
i [ocf°,< -) a'(') = s~:~')' L 330
Oe=
aRt ]
1,
Social Security as an Economic Stabilization Program
where e(Rt,e 2, U °) is the expenditure function. (For simplicity suppress the second argument of ~s.) The relevant Lagrangean is L = [Ct+l - (e 1 - crt)"Rt] + Z. [U ° - U(crt,ct°+l)] and the envelope theorem implies aL*
aRt
ae(-) -aRt
-
(e 1 - c~) = - s ( ' ) .
Substituting into the above equation yields 1 actH+Ol( • ) OctO+1(") s(') aRt + Oe2
G(')
1.
The first right-hand term is positive as it is a pure substitution effect and the second term is positive by normality of old-age consumption. Thus G(') >. -1.11 PROOF FOR MAIN RESULT OF THEOREM 3. Let (fi,f(fi,x)), and/5 < f(/5,x), be a stationary 2-cycle for the dynamic system generated by f(p,x), so /5 = f[f(/5,z),'c]. If no solution exists for/5 = f[f(/5,x + e),x + s] e > 0 and /5 ¢ f(/5,x + e) then the amplitude is defined by some strictly smaller cycle with amplitude, that is, the trivial cycle Pt = pt+l. If there is a solution, it may be parameterized as (A2)
#(~) -f~p(~),x),~]. Without loss of generality let/5(z) -< f(/5(x),'c) =/~, so that
R(x)=
P(~) < 1. f[/5(~),~]
(A3)
Note another associated solution exists for 1/R. Taking derivatives of (A2) and (A3), and collecting terms yields
[@,x) 1
dR(x) ~
~JL~-~
d~ 1
-
1
af[f(p),x]
I~J +
ax
1
1
(A4)
The denominator is positive by Assumption 2, as can be seen by using the chain rule and the definition of a 2-cycle locally stable in the forward perfect foresight dynamics. To evaluate the numerator first differentiate the definition off(pt+ 1,z) w.r.t, x, which after rearranging terms, gives 331
D. Andrew Austin
as(R,,x) /1. , :
Similarly, differentiate the definition off[f(pt+l,~),m] w.r.t, m, which, after rearranging terms, gives
af[f(pt + 1,m), m]
_af(pt+l,m) ] = sR(Rt,m) u[~"
pip-]
as(Rt,m) + am u~"
Using the previous equation and ~,(1//~) = (sRIII~)/[R • s(1/R)] = (snlu~)/R af[f(Pt+l'm)' "~]p - Os(Rt'z)
' Ll[/i"e~(11/1)] + as(Rt,'r)
Therefore af(p,x) .~(p5___2) afOC(p),x) as(i~,x)..[af(p,~) Ip a~ I~ + ~ ~ax ~ [~--~+11~
+ /1. e,(l/~)]. [ 1 + 1 (/1)] + as(Rt,x) .a~ u~ Using the following result af(pt+l,~)
sR
~t+l
f(pt+l,x) + Rt'sR
1
~(Rt)
the numerator is positive if as(R,x)
[/1. ~,(1//1)
4' L1 + ~,(l/J) + ~
E~(1//1)] [1
1 (/1)]> + -es
-
as(Rio) . a'c 1/ft
Note that normality of young consumption implies
[adI (~- 1)- 1]<
= LTe~l,, • 332
0 v/~ E (0,1],
Social Security as an Economic Stabilization Program
and Lemma 1 implies s~(/~) ~ (0,-1] and ss(1//~) ~ (0,- 1]. Rearranging terms giv6s as(R,z) [ 1 + £~(/~)] [ 1 + [ - R.a,(1//~) L2 +
Os(R,~) °q"C
£,(i//~)] ~J'
1/R
which holds by Assumption 4. Note the RHS is negative, so Os(R,x)/O~li~ must also be negative. For the fi < f(/5, x) case, note that high and low interest cases are exactly symmetric about the Pt = Pt+ 1 line. Thus the introduction of a social security system lowers the interest factor in this c a s e l
333
Social Security as an Economic Stabilization Program* This paper analyzes the stabilizing properties of an unfunded social security system in an overlapping generations model with cyclical equilibria. This provides a welfare rationale for a social security system independent of the windfall gained by the first generation, or social insurance arguments. An unfunded social security system redirects intergenerational resource transfers from private savings, which are subject to endogenous fluctuations, towards a public system with an implicit rate of return of unity. In an economy with a cycle of order 2, a planner with a limit-of-means social welfare function will completely stabilize the economy,
1. Introduction
While the primary impetus for passage of the Social Security Act of 1935, which instituted a pay-as-you-go (PAYG) old age public pension system, was to alleviate the poverty and economic uncertainty facing the elderly, there may be another unanticipated benefit: a PAYG Social Security system may promote macroeconomic stability. This paper outlines how a PAYG social security system stabilizes an economy subject to endogenous cycles. This stabilization is not achieved through discretionary fiscal policy or income smoothing. Instead, the paper shows that a social security system may alter the excess demand functions for an economy in a way that reduces the magnitude of macroeconomic fluctuations. In an economy with identical individuals and perfect foresight, complete stabilization can always be achieved through introduction of an appropriate social security program. While many macroeconomists prefer explanations of aggregate fluctuations that rely on random fundamentals, others have presented plausible models with endogenous cycles, including Woodford (1990a) and Farmer and Guo (1994). Woodford (1990b) provides an extensive discussion of the applicability of several models which allow a role for endogenous fluctuations. These models obviously differ from the model presented below, which is based on Azariadis and Guesnerie (1986). However, as the Azariadis and Guesnerie model spawned these richer models, it is an appropriate place to *I wish to acknowledge useful comments from Michael Loewy, Costas Azariadis, Giles AuchmutT, Michael Ben-Gad, participants in the University of Houston/Rice Ulxiversity Macro seminar and two anonymous referees. Remaining errors are mine.
Journal of Macroeconomics, Spring 1999, Vol. 21, No. 2, pp. 309--333 Copyright © 1999 by Louisiana State University Press 0164-0704/99/$1.50
309
D. A n d r e w Austin
start the analysis of macroeconomic stabilization via social security. This paper aims to explain the effect of social security on macroeconomic fluctuations within the simplest possible model which allows such endogenous fluctuations. Section 2 describes a pure exchange overlapping generations (OLG) economy and a social security system. Section 3 shows how a social security system may enhance macroeconomic stability, and Section 4 describes the welfare effects of such a program. Section 5 concludes the paper.
2. The Model This OLG model follows Samuelson (1958) and Gale (1973). For simplicity the model only allows for pure exchange, although production could be included without changing any important features. Generation 0 is born old at t = 1 and is endowed with e z units of consumption and one unit of fiat money. A young generation is born in each of an infinite number of periods t = 1,2 . . . . . and is indexed by that period. All generations live two periods, except generation 0 which lives one period, and comprise N identical individuals. A member of generation t is endowed with e 1 units of consumption in t and e ~ units during period t + 1 (old age). The consumption good cannot be stored from one period to another, while fiat money is costlessly stored. Members of generation 0 have utility functions u°(Co), and all others have utility functions u(cr~, c°+a) where ctv is consumption in period t by a person born in period t, and cO+1 is consumption by that person when old. All utility functions are continuously differentiable, strictly quasiconcave and positively monotonic. Consumption when young is normal. Marginal utility tends to + ~ as consumption approaches 0. Let Pt be the price of money in period t. If pl > 0, then the first generation can trade its unit of money for an additional Pl units consumption, and i f p t > 1, a subsequent generation t faces the lifetime budget constraint
(e I - cYt)/pt + (e ~ - ct°+l)/pt+] >- O.
(1)
Each unit of saving by the young generation at t yields Rt -- Pt+ 1/Pt units of consumption at t + 1_ Per capita saving is thus a function of Rt; N
s(Rt) -----( I / N ) ' ~ i= i
argmax u(e 1 - s, e z + s ' R ) .
(9.)
0~---s"~el
Saving is assumed positive when R = 1, so that the economy is in the Sa310
Social Security as an Economic Stabilization Program
muelson case. This conforms to the stylized fact that young workers have higher incomes than older retirees, requiring private saving to ensure sufficient consumption in old age. All individuals perfectly foresee future prices. The excess demand of the young generation is the negative of the saving function, and the excess demand of the old is the dissaving of their unit of money. Therefore total excess demand at period t is D(p~, Pt+l)=-Pt - s(Rt) = Pt - s(pt+{pt).
(3)
DEFINITION 1. A perfect foresight monetary equilibrium is a sequence of prices {P}~=I such that Pt > 0 and D(pt, pt+l) = 0 Vt. The equilibrium condition D(pt,pt+l) = 0 implicitly defines a sequence of prices which constitute a perfect foresight monetary equilibrium. By fixing P~+ 1 and soMng for Pt a "backward-looking" equilibrium sequence of prices is derived. Let the mapping pt = f(pt+l) describe the backward-looking solution to (3), so that D(f(p~+l),pt+l) =- O. To be more specific _flPt+ 1)
~
S[Pt+1~(~t+l)]I
(4)
If consumption is a normal good in both periods there is a unique backwardlooking equilibrium for any given price level at time t + 1. Thus there is a continuum of backward-looking equilibria indexed by the price level at t + 1. Some of these equilibria may be periodic monetary equilibria, so that prices follow a deterministic cycle. DEFINITION 2. A monetary equilibrium is a nontrivial periodic equilibrium of order k / f P t = Pt+k V t >- 0 (k > 1) and if there is no positive integerj < k such that Pt = Pt+j. A trivially periodic equilibrium, in which the path of prices is constant, will be called a steady-state monetary equilibrium. DEFINITION 3. An equilibrium is a steady-state monetary equilibrium if p t =pt+l>-OVt>-O. This paper analyzes stationary equilibria, in which the values of real variables depend on the date only through the identity of the state at that date. Nonstationary equilibria, in which real values are nontrivially datedependent, are not analyzed. 311
D. Andrew Austin Next, locally stable equilibria are defined. Stability has played a large role in characterizing equilibria in this class of models, and will play an important role in the analysis below. Definitions of stability require a distinct_ion between the backward perfect foresight (b.p.f.) dynamics generated by the f(.) mapping and the forward perfect foresight dynamics, which can be approximated near a given orbit by taking the inverse of theft.) ([assuming f(-) is regular-valued). Thus local stability in the b.p,f, dynamics implies instability in the forward perfect foresight dynamics and vice versa. The dynamic analysis employs the b.p,f, dynamics because f(.) is single-valued whereas its inverse is not necessarily, and f(.) is closely related to the offer curve.
DEFINITION 4. A steady-state monetary equilibrium is locally stable in the forward perfect foresight (unstable in the b.p.f, dynamics) if df(p~ + l)/dp, + l <-1. Grandmont (1985, 1018) shows that local stability in forward perfect foresight dynamics of a steady, state equilibrium implies existence of a 2-cycle. An equivalent condition requires the elasticity of savings with respect to R be less than - 1/2 in the neighborhood ofR = 1.1 While many economists believe this elasticity is positive a priori, the empirical evidence on this question is not conclusive. 2 DEFINITION 5. A periodic equilibrium of order 2 (2-cycle) equilibrium is locally stable in the backward b,p.f, dynamics (unstable in the forward perfect foresight dynamics) if
dy[f(p +
• ~
<1.
Figure 1 illustrates an example of a periodic equilibrium of order 2, using the graph off(pt+ 1) and its reflectionfl(pt+ 1) about the 45 ° line. Suppose/5 is the price in period t. Thenf(~) is the price in t - 1, and if~ is a 1See Azariadis (1993, 368) for a discussion. Also see Grandmont's discussion of stability conditions for this class of models (p. 1007 et passim). aln par~cular, Boskin's 1978 study, which reported large positive elasticities, is regarded as discredited among many empirical researchers. Deaton (1992, 60.) concludes "the empirical results are as ambiguous as is the theory, or more positively, that the empirical results confirm the lack of invariance to time, place and other variables that the theory predicts."
312
Social S e c u r i t y as an E c o n o m i c S t a b i l i z a t i o n P r o g r a m
D(p,+l,pO =0 p,
f(p) D(pt,P~+I) =0
f(p)
p.
~
p,+,
Figure 1.
coordinate of the intersection off(pt+l) and f f l(pt+l) then the price in t - 2 will be/~ = f-l[f(~5)]. Thus, the sequence {~5,f(/5)fi. . . . }, in which the solution of the excess demand function alternates between the two intersections off the 45 ° line, is a periodic equilibrium, and generates a sequence of interest factors {/t,1//~,/2. . . . } where/~ = f(fi)/~. A transfer of ~ units of consumption good from the young generation to the old generation represents a pay-as-you-go social security system. A planner, with preferences defined by a social welfare function, sets the level of this transfer. However the planner cannot bankrupt young individuals or impose negative taxes, so • E (O,el). After the social security program is in place, the budget constraints for each period imply consumption is (Co) = (e2 + z + pl),
for generation 0 ;
(ctY, cO+l) = (eI -- st -- z, e 2 + ~ + Rt'st),
for generations t > 0 .
(5a) (5b)
The lifetime budget constraint in a monetary equilibrium becomes
313
D. A n d r e w
Austin
(el - x)/Pt + (e2 + x)/Pt+] -> crt/Pt + c°+i/pt+1,
(la)
and per capita saving is then a function R t and ~: N
s(Rt, x) -- ( 1 / N ) ' ~ i= 1
argmaxu(e 1 - ~ - s , e 2 + • + s ' R t ) .
(2a)
O<--s<--e1
Similarly, total excess demand at period t is D(pt,
Pt+l, r) =- Pt - s(Rt, x) ,
(3a)
x] solves D~C(pt + l, "~), Pt + l, "¢] =- O. Changes in the social security tax rate x have two potential effects on each generation's well-being. First, in a 2-cycle equilibrium interest factors alternate between/] and 1//]. Thus, a larger social security transfer (which returns an implicit rate of return of unity) increases the well-being of the generation facing the low interest rate and decreases the well-being of the next because these agents are savers at all relevant interest rates. Second, changing the social security tax rate in a 2-cycle equilibrium will generally change the interest rate, as it will alter the supply and demand for savings. This effect will benefit whichever group of generations (even or odd) sees their real interest rate increase. The effect of changes in z on interest rates is discussed in the next section. In particular, under certain conditions raising the social security tax rate will stabilize interest rate fluctuations, thus increasing the welfare of generations that would have received lower rates of return on their private savings and decreasing the welfare of generations that would have received higher rates of return on private savings. The sharp opposition of interests between odd and even generations is due to the "short and simple" lifespans used in the model. A richer model with longer-lived individuals allows a less stark opposition of interests among generations. Moreover, a model which includes uncertainty and a richer demographic structure will almost certainly have richer dynamic behavior, which may match the behavior of actual business cycles more closely, at the cost of tractability. a n d f ( p t + l, z) =- s[pt + i / f ( p t + l),
3. Macroeconomic Stabilization and Endogenous Cycles Most of the economic analysis of social security has examined its effects on capital accumulation and intensity, the growth path of the economy, or the social insurance benefits accruing to individuals. There may be another 314
Social S e c u r i t y as a n E c o n o m i c S t a b i l i z a t i o n P r o g r a m
effect of the social security system not captured by that research. In particular, if expectations generate endogenous macroeconomic fluctuations, then the social security system may have an important stabilizing role. This section analyzes the stabilizing effects of an unfunded social security system. If a periodic equilibrium exists, as in Figure 1, the introduction of a social security system changes the shape of the aggregate excess demand function, and thereby reduces the magnitude of the economic fluctuations. Social security diverts intergenerational transfers from a private saving mechanism subject to endogenous cycles to a public system which pays an effective interest rate of unity. This effect is distinct from the usual theory of macroeconomic stabilization using discretionary fiscal policy intended to counteract the effects of unforeseen exogenous shocks. Theorem i shows the effect of a social security system on offer curves, which are used instead of excess demand functions for clarity. Theorem 1 shows that imposing a social security tax that does not force oversaving rotates the offer curve around (e 1 - s(1), e~ + s(1)), and thus rotates the excess demand function around the origin. Figure 2 shows the effect of a social security system. A higher social security tax rotates the excess demand curve around the origin, bringing the economy closer to the "classical" case. In an extreme situation, in which x = s(1), the social security system completely displaces private saving. The intuition is that movement of resources among generations through the social security system is unaffected by fluctuations in the interest rate, unlike private savings. Figure 2 shows an individual's intertemporal choice problem along with offer curves before and after introduction of a PAYG social security system. Figure 3 shows the corresponding excess demand curve. The excess demand curves are reflections of the offer curves, and the social security system shifts the point on the offer curve corresponding to the origin of the excess demand curve to the northeast. Let O(R) denote a point on the offer curve without social security, and let O(R;x) denote a point on the offer curve with a social security system with tax ~, so that: O(R) =- {crt, C°ll(Ctr, c°+1) = argmax u(crt, C°l)
s.t. ctr = e 1 - s and ct°+l = e 2 + R ' s } ;
(6a)
O(R; x) =- {crt, c°÷ll(ctr, c°+1) = argmax u(ctr, c°+1)
s.t. ctr = e 1 - s - x a n d c t ° + l = e 2 + R ' s + x } .
(6b)
The offer curve without [with] social security O[O(x)] consists of the col315
D. Andrew Austin CI+ I
R = 1
(eL-s(1), c2-s(l)) /
/
(eI-T,C2-I-T) (el,e2)
F 4
ct
Figure 2.
lection of 0(R) V R [0(R; ~) V R].3 Note that this maximization is dual to the maximization which defines the savings function in Equation (2). Let R be the interest rate that induces no net saving. Theorem i states the pre-social security offer curve (a) lies to the left of the offer curve after introduction of social security for interest factors between R__and 1, (b) these offer curves coincide at R = 1, and (c) the pre-social security offer curves he to the right of the offer curve after introduction of social security for interest factors R>I.
THEOREM 1. If O < r ~ s(1), so the social security does not force oversaving, then (a) V (c[', c°+1) ~ O(R; r) for R ~ (R, 1) 3 (cL c°+z) ~ O(R) such that c[ < c['; (b) 0(1) = 0(1; r); (c) V (c[', c°1) ~ O(R; r)for R ~ (1, oo) 3 (c[, cO+l) ~ O(R) such that c[ > c[. aNote that the graph of fiR) is identical to the set of points for which D(pt,p~ + 1) = O.
316
Social Security as an Economic Stabilization Program P~+l
D(pt,Pt+l)=O
/\l\
/;) /
/'
i/ t T
Figure 3.
PROOF. Because the social security tax offsets private saving exactly when R = 1 and 0 < * --- s(1), (b) follows immediately. This implies the offer curves intersect at R = 1. Suppose (a) is false_ Continuity of 0 and lira C°l = ~ a s R ~ ~ i m p l y 3 (ctY,ct+l) o ~ O(R) and some R < 1 such that C°+l = e 2 + z. Furthermore, c[ < e 1 - z, because otherwise z >- s(1). Continuity of the offer curves then implies offer curves intersect, that is, 0(R) = 0(B;z), for some R ~ (R,1). Let u(-) = a be the indifference curve through this intersection. By the definition of the offer curves and consumer maximization this indifference curve's slope must equal the slope of the chord between (el,e2) and the intersection, and the chord between (e 1 z,e 2 + z) and the intersection. However, because z > 0, the slopes of these chords are distinct, and the indifference curve cannot be tangent to both. This contradiction establishes (a). Part (c) follows from the same argument as (a). A government seeking to avoid the possibility of large fluctuations could reduce the amplitude of cycles by instituting a social security system. For the purposes of this section, stabilization is assumed desirable. Section 317
D. Andrew Austin
D(p,,p,+,)=0 P,+~ [D(p,,p,+fir)=0 \ 1
D(Pt+i,p,) =0
IV /
f
I
'
1 °
I
iI =
p
p,
Figure 4A.
4 considers the rationale for stabilization. Because social security taxes shift the lower portion of aggregate demand curve closer to the 45 ° line for Samuelson type economies, the scope for fluctuations will generally be smaller. In particular, if the average price level does not fall too much, the variation in interest rates will fall as social security taxes rise. This case is shown in Figure 4A. For some economies higher taxes may induce a larger variation in interest rates if the average price of money falls sharply (shown by f and f ' , which can be calculated by finding the intersection of the chord between equilibrium price vectors and the 45 ° line) relative to the movement closer to the 45 ° line, as shown in Figure 4B. Inspection of Figure 2 shows the offer curve O(Rt) for R -< Rt <-- 1 moves closer to the line with slope of - 1 through the endowment point (R is the autarkic interest factor). Similarly, the lower portion of the reflected offer curve moves closer to the 45 ° line, as shown in Figure 3. Furthermore, only the portion of the 45 ° line between the origin and the point pMAX, representing the steady-state monetary equilibrium with the highest price level, will he considered to ensure the following metric is well-defined.
318
Social Security as an Economic Stabilization Program R(x) Pt+ l
=
/
//
// / i
R(O) /
pt=pt+~
/
\
i
p
'~\
]
I
I
;,
p
I/R(O)
Figure 4B.
DEFINITION 6. (i) Define the lower portion of the reflected offer curve with social security tax r as
qDL(~) = {(Pt, Pt+l) 1D(pt, Pt÷l; x) = 0 andpt >-- Pt+l} • (ii) Define the lower portion of the offer curve with social security tax z as OL(~) = ~(R;~) such that R <~ R <- 1 where R is the autarkic interest rate. (iii) Let f L = [(p',p") I p' = p" <- pMAX} denote the 45 ° line. (iv) Let c/3 = {(crY,c°+1) I [crt - e 1] q- [ctO+l - - e2] = O, crt > O, ct°+l >~ O] denote the line with slope of - 1 through the endowment point.
The Hausdorff metric is a natural measure of distance between sets (see Munkres 1975, 279). DEFINITION 7. Let A,B be nonempty, closed, bounded subsets of R2+, and let d be the Euclidean metric. Define n(A, e) = U,~A Bd(a, e) where Bd(a, 319
D. Andrew Austin e) is an open ball of radius e centered on a. The Hausdorff metric is defined as:
H(A, B) = glb{elA C u(B, e) and B C u(A, ~) .
Let H1 = H[~(x),f L] be the Hausdorff distance between the lower portion of the reflected offer curve (of an economy in the Samuelson case) and the 45 ° line, in an economy with tax rate ~ ~ [0,1), and let H e = H[OL(z'),fL] be the Hausdorff distance between the lower portion of the reflected offer curve and the 45 ° line between the origin and pU~, in an economy with tax rate T' ~ [0,1) and T' > ~. Then H 1 > t12. THEOREM 2.
PrtoOF.
See Appendix.
DEFINITION 8. A function h:A ~ B is single-peaked if3 x* such that V x",x' ~A (a) x" > x' > x* implies h(x") < h(x*), and (b) x" < x' < x* implies h(x") < h(x') < h(x*). 1. Let 3q~o(pt+i) be the restriction of the function f(p~ +1,~)for some T° ~ [O,el]. Assume 3qo,(pt+1) is single-peaked Vr E [O,el]. ASSUMPTION
Single-peakedness off(') (for a given value of x) provides a major simplification, implying substitution effects are relatively stronger than income effects at low levels of R, and that income effects become increasingly large as R increases. Therefore, at some unique point income and substitution effects exactly offset each other. If f(-) were multi-peaked income and substitution effects would exactly offset each other at more than one interest factor. Grandmont (1985) shows that for a utility function separable in consumption when young and when old, if the degree of concavity for an old individual's subutility function is sufficiently greater than for a young individual, then the offer curve must be single-peaked. Furthermore, Grandmont finds additional conditions on the curvature off(') which along with the above convexity assumptions imply a unique stable cycle exists. 4 More generally, extending the results presented below to include multiple cycles is straightforward. aSee Grandmont (1985, 1024-26) on the negative Schwarzian derivative condition.
320
Social Security as an Economic Stabilization Program ASSUMPTION 2. For some social security transfer r ~ [O,s(1,0)] the economy has a 2-cycle which is locally stable in the backward perfect foresight dynamics and the 2-cycle is unique when it exists. 5 Theorem 3 considers the simplest possible case, in which the offer curve is single-peaked and a 2-cycle exists and provides sufficient conditions for the diminution of the cycle's amplitude. Success in stabilization will be measured by the amplitude of the cycle, defined as the sum of absolute values of interest rates over the cycle. DEFINITION 9. The amplitude of a 2-cycle for any j >- 1 is defined as
A(/~, R~+:)= IRi - 11 + I / ~ + : - 11. The amplitude metric can be easily extended to cycles of order k > 2 and the existence of multiple stationary k-cycles, which are not considered here. An interest factor R is the slope of the ray from the origin that passes through the equilibrium intertemporal price vector. Thus the amplitude corresponds to the angle between the rays from the origin to the two equilibrium intertemporal price vectors. As the examples in Figures 4A and 4B illustrate, the amplitude of a 2cycle may increase or decrease with the introduction of a social security system. The effects of raising the social security tax from a given level are qualitatively identical. Figure 4B shows that raising the social security tax from zero to x may increase the amplitude of a 2-cycle. This occurs if the reduction in savings (for a given Rt) by the young generation facing the lower interest rate exceeds the reduction in old-age consumption by the preceding generation (for the same Rt), then the interest rate must rise to clear the market. The following assumptions are sufficient to rule out such cases. Suppose {/~,f(~5,x)} are prices associated with a 2-cycle, where /5 < f(15,z), Assumption 3 requires that excess demand at/5 be associated with an elasticity of saving with respect to R is between zero and negative one. The set of prices which satisfy this condition comprise the backward-bending domain forf(pt+ i,x), and are denoted S(x). DEFINITION 10, Let S(z) = {Pt+i C_ R2+ I es(R;r) ~ [0,-I)}, where es(R;) --- n'sR/s(R,~), R = pt+Jf(Pt+i,T) and sR is the partial derivative of saving with respect to R. ~rhe stabilitycondition in Assumption4 should be understood to applyto the restriction of ]~pt÷l,Z)definedin Assumption 1 where a social securitysystemis in place.
321
D. Andrew Austin ASSUMPTION 3. Min[p~l,f(p~l,T)} ~ S(z) where pM and f(pM, z) are equi-
librium prices for a 2-cycle. Next, the introduction of a social security tax may increase economic fluctuations if generations facing the low interest rate either increase savings due to a strong income effect or decrease savings by a large amount relative to the reduction in savings by generations facing the high interest rate. Thus if the demand for fiat money in the low interest state increases enough or decreases relatively slightly in response to a social security tax, the interest factor may fall, thus increasing the amplitude of the economy. Assumption 4 rules out such cases. ASSUMPTION 4. Let [l = ptM+l/f(ptM+ l,'C) where pM < f(pM, z ) are equilibrium prices for a 2-cycle. Then
as (R,o,~"~) a
[ 1 + s,(/~)] [ 1 +
es(1//t)]
as(R, ~) at 1/4 A more transparent (albeit stronger) sufficient condition requires normality of consumption in both periods a n d / t E [1/2,1]. 6 6Assumption 4 and the Lemma (see Appendix) imply that [as(l~,~)/a~][s < 0 where t2 is the low interest factor, as shown at the conclusion of the proof of Theorem 2. Note this is not implied by normality. Normality in both periods and 1 ->/~ -> 1/2 do imply this inequality. Differentiating [e 1 - c~(Rt,e2)].Rt + [e2 - c°+l(Rt,eS)] ~ 0 , w.r.t, e2 and rearranging terms gives
ae~
/~
aez .
Normality ofctr and c°+1 implies 1
act
~t~ -> 'ae ~ >- 0 , and substituting in the identity
as(P~,~) acT(l%,~) a~e2 a "(1 - /~) -
322
1
Social Security as an Economic Stabilization Program While a planner can completely suppress fluctuations by setting = s(1), the marginal relationship between social security tax levels and volatility of interest rates provides a more robust basis for intervention. 7 Theorem 2 demonstrates that the above assumptions provide sufficient conditions for macroeconomic stabilization through increasing the size of a social security system. THEOREM 3. If Assumptions 1--4 hold, then the amplitude of the economy is decreasing in r, V r E [0, s(1,0)).
PROOF.
See Appendix.
4. Welfare
The possibility of stabilization does not imply its desirability. This section discusses the welfare implications of stabilization via social security. Partial stabilization of an economy with a periodic equilibrium is assumed to lead to a new periodic equilibrium of the same order, if one exists. The two-cycle case, as mentioned before, is an extreme case in which the economy swings from boom to bust, with odd generations benefiting from the cycle and even generations losing, or vice versa. Thus, any welfare justification for stabilization must rest upon a utilitarian social welfare argument. Suppose a planner can implement a social security system and choose the contribution level. Furthermore, suppose the planner must choose among date-invariant policies that support periodic or steady-state equilibria. This may be justified by the need for credible, and therefore inflexible, measures. Let the time-separable, strictly quasiconcave, strictly positively monotonic social welfare function (SWF) Wt(u t) represent the preferences of the planner for time t over utility of a representative member of generation t. Two possibilities are a limit-of-means function, can be shown to imply
as._0,~.(/~) E [0,-1] for/~ E [1/2,1]. O'c .~ 7If the planner does not completely stabilize the economy the analysis is complicated as a marginal change in taxes could set the economy on a new dynamic path, making comparative static analysis impossible. However, narrowing the lens between the reflected offer curves reduces the maximum possible amplitude.
323
D. Andrew Austin
W(u o, u 1,
1T
.) = am '
nf[- E re(u,)
T-~=
IT
t=o
] '
and a discounting function, eo
W(u °, u 1, ua,...) = W°(u °) + ~ 6t" Wt(ut), 6 E (0,1) , t=l
where Wt(u t) = W(u t) V t >-- 1. A limit-of-means function treats generations symmetrically, so that the effect of the generation 0 vanishes. Because preferences are convex the planner maximizes social welfare by equalizing utility across odd and even generations. This is accomplished by setting the social security tax sufficiently high to guarantee R = 1 for all t. Therefore a planner with a limit-of-means SWF will completely stabilize the economy. The discounting function treats generation 0 as potentially special, given their unique role in the model. The windfall to the first generation from the startup of the social security program has been prominent in policy debates because of the actuarial immaturity of the U.S. program, even though from a long-run standpoint the value of this windfall shrinks because only one initial generation has windfall gains but the infinite number of generations which follow do not. A social planner who cares about the initial old chooses a social security tax larger than the minimum tax needed to achieve stabilization, and induces an interest rate that exceeds the "golden rule" rate. Thus if stabilization redistributes wealth towards the first old generation this planner chooses a contribution rate which forces oversaving. If the planner does not care about the initial old and the economy starts out in the high-interest rate state (R1 > 1), complete stabilization is not optimal as the planner cares more about odd generations than even, and thus chooses an interest rate slightly above one. s This stabilization reduces the utility of generation zero if the fall in the price of money outweighs the social security benefit.
5. Discussion
In a richer model there are limits to stabilization. First, there may be costs to stabilization, including administraldve expenses, the possible effects on capital accumulation, and distortions induced by payroll taxes. Second, ~¢¢hen the planner restricts attention to stationary or cyclical paths, she may face a nonmarginal choice between R 1 = 1 and the interest rates associated with the smallest possible 2cycle. In other words, a discontinuity occurs where a bifurca~on in the path of the economy disappears.
324
Social Security as an Economic Stabilization Program
governments have incomplete information about the shape of the aggregate demand curve, and may wish to avoid overshooting in regard to the social security contribution level. In addition, governments may have other instruments for macroeconomic stabilization. In particular, monetary policy or a commodity standard policy could also achieve stabilization. However, the broad degree of participation in a social security system and its inherent inflexibility may make it a more credible instrument relative to policies which may be changed with the stroke of a pen. A social security system derives its fundamental credibility from its inflexibility, but allows little or no room for "fine-tuning." A full evaluation of the relative attractiveness of stabilization instruments would require weighing the costs of a social security system, such as those mentioned above, against the costs of monetary stabilization, including effects on production and distribution which will affect allocational efficiency (see Azariadis 1981). Of course, the net benefits ofmacroeconomic stabilization would be negative if social security taxes significantly distort work and saving incentives and sunspots are not much of a problem. 9 This evaluation is left for future research. Third, if generations are composed of heterogeneous individuals, a social security system may impose welfare costs on the individuals with larger old age endowments because it could force them to oversave. Analyzing this case would involve a straightforward extension of this model in which the stabilization and wealth redistribution effects are weighed against the welfare effects of oversaving. These results suggest a social security system may dampen the instability of a monetary economy subject to endogenous cycles. Given the size of the intergenerational transfers generated by social security, it may be an important stabilizing force, Received:August1994 Finalversion:June 1998
References Azariadis, Costas. "A Reexamination of Natural Rate Theory." American Economic Review 71 (December 1981): 946-60. • Intertemporal Macroeconomics. Cambridge, Mass.: Blackwell Publishers, 1993. Azariadis, Costas, and Roger Guesnerie. "Sunspots and Cycles." Review of Economic Studies 53 (October 1986): 725-37. Boskin, Michael. "Taxation, Saving and the Rate of Interest." Journal of Political Economy 86 (January 1978): 13-27. eThispointis due to CostasAzariadis. 325
D. Andrew Austin Deaton, Angus. Understanding Consumption. New York: Oxford University Press, 1992. Farmer, Roger, and Jang-Ting Guo. "Real Business Cycles and the Animal Spirits Hypothesis." Journal of Economic Theory 63 (January 1994): 4272. Grandmont, Jean-Michel. "On Endogenous Competitive Business Cycles." Econometrica 53 (September 1985): 995-1046. Gale, David. "Pure Exchange Equilibrium of Dynamic Economic Models." Journal of Economic Theory 6 (January 1973): 12-36. Munkres, James. Topology: A First Course. Englewood Cliffs, N.J.: PrenticeHall, 1975. Samuelson, Paul. "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money." Journal of Political Economy 66 (December 1958): 467-82. Woodford, Michael. "Learning to Believe in Sunspots." Econometrica 58 (March 1990a): 277-308. --. "Equilibrium Models of Endogenous Fluctuations: An Introduction." NBER Working Paper No. 3360, 1990b.
Appendix A
Table of Variable Definitions e1 e2
c?
= = = =
c~HY (~,e 2 ,u °) = ctHO (Rt,e2 ,U°) = =
D = e(Rt,e2,U°) = Pt
=
u°(co) =
R, = R =
326
Endowment received by young individuals. Endowment received by old individuals. Consumption in period t for person born in period t. Consumption in period t + i for person born in period t. Compensated consumption function in period t for person born in period t. Compensated consumption function in period t + 1 for person born in period t. Marginal utility of e2. Hessian for (Marshallian) consumption program. Expenditure function. Price of money in period t. Utility function for generation 0. Utility function for generation t -> 1. Interest factor. Autarkic interest factor.
Social Security as an Economic Stabilization Program s(R) = Per capita saving without a social security tax. Per capita saving with a social security tax. D(pt,pt + 1) = Excess d e m a n d function without a social security tax. D(pt,Pt + 1,z) = Excess d e m a n d funcldon with a social security tax. f(Pt+l) = Mapping of backward-looking solution for D(pt,Pt + 1) s(R,x) =
= 0. + 1, "l~) ---- Mapping
of backward-looking solution for D(pt,pt+l,x) = O. Social security tax rate. o(R) = Offer curve without a social security system at R. 3(R;~) Offer curve with a social security system and tax rate xatR. Offer curve without a social security system. o(~) = Offer curve with a social security system and tax rate
f(Pt
=
~L(~)
=
¢B= pMAX _-a(.)
=
~s(R) =
s(~) = w(.) =
The lower portion of the reflected offer curve with social security tax z. 45 ° line between the origin and pMAX. Line with slope of - 1 through the e n d o w m e n t point. Largest price level associated with steady-state monetary equilibrium. Amplitude metric for a stationary 2-cycle. Uncompensated elasticity of saving with respect to R. Set of prices such that ~s ~ [0, - 1). Social welfare function.
Appendix B PROOF FOR THEOREM 2. First, that Pt = s(';x) in equilibrium, s(';x) < el - t and (Pt,Pt+l) E ¢DL(z) imply Pt,Pt+l E [O,el - x]. Next let d = minp,_
0
@' - 0p' {(p~+l- p')~ + [f(Pt+l; ~) - p,]2} = -2(p~+1 - p') - 2[tip,+,; ~) - p'] = 0 ~Pt+l
-- P' -- p' + f ( p ~ + l ; x) = 0 ~ p '
Pt+l + .~Pt + 1; T)
2
Therefore
327
D. A n d r e w Austin 1 d{[pt+l,j~pt+l; x)], (p', p')} = ~ [ P t + l -- J ~ P t + l ; .~)]2 .
Because pt = f(pt+l,x) =- s[pt+.l/f(pt+l),~] = (e 1 - c r t
- ~) and
(e 1 - "c)/pt + (e 2 + x)/pt+l = crt/pt + C°+l/pt+1 P t + l " ( ex -- c Y -- "~)/Pt =
(e 1 - c[ - x)/pt --- c?+ 1 - e 2 - x
pt+l"s [Pt+l/~Pt+l), x]/pt = Pt+l = ct°+x - e 2 - -
"~,
then 1
d[ (p, + l, f[pt + l; x]), (p', p')] = ~ ((C°+1 - e ~ - x) --
(e I _
CY --
,~))2
=
21 ( c O
~ + c~ -
e~ -
d) ~
The last term is the distance between (c~(Rt; x),c°+ I(Rt;T)) and the closest point on % min d{[ctr(Rt; ~), C?+l(Rt; T)]; [ct, Ct+l] } s.t. [c; - e l] + [ct+ 1 - e 2] = O. 4.4 +1 (To see this, differentiate this expression w,r,t, x, find c[,c[+ 1 and substitute back into the Euclidean distance definition.) Next, define d*(~) as the furthest distance between a point on 0(~) and the closest point on % r d*(~) --- max 1 [C°+l(Rt; x) + ct(Rt; ~) __ e ~ -- el]2 Rt>l 2
By construction d*('c) = glb[slf L C u(q)L('c), s) and ¢DL(x) C u(fL S)] = glb[slf L C u(oL(x), S) and OL(x) C u(f L, s)];
d*(x) is characterized by the FOC:
328
Social Security as an Economic Stabilization Program
8.L
--
Oat
= [c[(at; x) + C ° l (at; z) - e 1 - e2] •
~!]
[8ct°+!(Rt; x) OcY(Rt; + - L Oat oa t
=0.
In the Samuelson case the first RHS term is strictly positive, so 8c,°÷l(at, z)
-sc[(at, ~)
aat
Oat
(al)
L e t [c[(x), c°+l(x)] solve (A1). Using results from utility maximization
-[st(•21-at
st(um - atu22) - ~.
D
--
Ull )
q-
at' ~-]
D
where D - -Ull + 2"u21'Rt - R~'u22 > 0 by convexity of u(') and L = OU*(c[, c°+i, e 2)
0e 2
> 0 by monotonicity. Simplifying this expression gives
Ull -- Ul~(1 -- at) -- atU22 --
(1
-
at)')~
st
Next, find sign for d[d*(.)]/d'c, where [c[(x), ctO+l('~)] is the point on 6~(x) furthest from cB. Differentiating and using the envelope theorem gives d[d*(.)]dz
[C°+l(X) + c[(x) - e 1 - e2] • 0
x) +
Ox
J
=[ct°+l(X) + c[(z) - e 1 - e2] •
I u12 --Oat iu2g" (1 - at) "q- -nil q-Oat'n21" ( 1 -
at)l "
Using condition (A1), the definition of savings and rearranging terms yields d[d*(')] - -
dx
-
=
[(1 + a t ) ' s t
- s,].
- k . R t . ( 1 - Rt) ~
D
(1 - Rt) 2 k D
st
< 0 for at ~ (0,1),
which establishes the r e s u l t l
329
D. Andrew Austin PROOF OF THEOREM 3. The proof is preceded by a lemma. LEMMA. Let p*+ 1(~) -- argmaxbC(pt+ 1,~)] and p* (z) -f(p*+ 1(z),z). Then Pc + 1 > p*+l(z) implies Pt+l E S(~). PROOF. Note the single-peakedness assumption implies p*+l(Z) is welldefined. The definition of p*(z) implies Of(Pt+l,z) • = sa = O. Op,+l P t + l f(Pt+t,~) + R t ' s R
There are three possibilities: either sR = 0, or the denominator approaches + ~ or - ~ . Suppose sa # O. First, f(pt+l,X) is bounded because the deftnition of the excess demand function and f(pt+l,X) imply that f(pt+l,z) = s[p~+/f(pt+l,X),~] <- e t. If I@ ~ ~ then Of(')/Opt+t approaches 1 for any finite R. Iff(pt+l,z) is maximized for R t = pt+z/f(pt+l,T) ---) ~ then s(Rt,~) 0, which contradicts the assumption that the economy is in the Samuelson case. Thus, sa = 0. Solving for sR gives
Of(p~ + t,~) .f(p~ + ~,~) Pt+l
SR =
1 - Rt" 0f(p,+ t,x) ' ~gt+l
This expression is negative by the single-peakedness assumption for Pt+t > p*+t(x). R, s(R, ~) >-- 0 implies es(R; x) < 0. Differentiating (5b) with respect to/It yields 0c°+ t( •) - s(-) + a~ -as(') ORt ORe
'
which reduces to
~,(.)
=
1 act°+:(.) s(.) aRt
:.
Define ct+t(Rt,e 2) -- argmax U(c[, c°+l) s.t. (e I - O ~ ) . a t + (8 2 - c°+i> --> 0 and define the Hicksian demand ctn°t(Rt,e2,U) ~ argmin [c°+t - (e i - c[)' Rt] s.t. U° - U(c[,ct°+ ~) >- 0. Then using the Slutsky decomposition gives
i [ocf°,< -) a'(') = s~:~')' L 330
Oe=
aRt ]
1,
Social Security as an Economic Stabilization Program
where e(Rt,e 2, U °) is the expenditure function. (For simplicity suppress the second argument of ~s.) The relevant Lagrangean is L = [Ct+l - (e 1 - crt)"Rt] + Z. [U ° - U(crt,ct°+l)] and the envelope theorem implies aL*
aRt
ae(-) -aRt
-
(e 1 - c~) = - s ( ' ) .
Substituting into the above equation yields 1 actH+Ol( • ) OctO+1(") s(') aRt + Oe2
G(')
1.
The first right-hand term is positive as it is a pure substitution effect and the second term is positive by normality of old-age consumption. Thus G(') >. -1.11 PROOF FOR MAIN RESULT OF THEOREM 3. Let (fi,f(fi,x)), and/5 < f(/5,x), be a stationary 2-cycle for the dynamic system generated by f(p,x), so /5 = f[f(/5,z),'c]. If no solution exists for/5 = f[f(/5,x + e),x + s] e > 0 and /5 ¢ f(/5,x + e) then the amplitude is defined by some strictly smaller cycle with amplitude, that is, the trivial cycle Pt = pt+l. If there is a solution, it may be parameterized as (A2)
#(~) -f~p(~),x),~]. Without loss of generality let/5(z) -< f(/5(x),'c) =/~, so that
R(x)=
P(~) < 1. f[/5(~),~]
(A3)
Note another associated solution exists for 1/R. Taking derivatives of (A2) and (A3), and collecting terms yields
[@,x) 1
dR(x) ~
~JL~-~
d~ 1
-
1
af[f(p),x]
I~J +
ax
1
1
(A4)
The denominator is positive by Assumption 2, as can be seen by using the chain rule and the definition of a 2-cycle locally stable in the forward perfect foresight dynamics. To evaluate the numerator first differentiate the definition off(pt+ 1,z) w.r.t, x, which after rearranging terms, gives 331
D. Andrew Austin
as(R,,x) /1. , :
Similarly, differentiate the definition off[f(pt+l,~),m] w.r.t, m, which, after rearranging terms, gives
af[f(pt + 1,m), m]
_af(pt+l,m) ] = sR(Rt,m) u[~"
pip-]
as(Rt,m) + am u~"
Using the previous equation and ~,(1//~) = (sRIII~)/[R • s(1/R)] = (snlu~)/R af[f(Pt+l'm)' "~]p - Os(Rt'z)
' Ll[/i"e~(11/1)] + as(Rt,'r)
Therefore af(p,x) .~(p5___2) afOC(p),x) as(i~,x)..[af(p,~) Ip a~ I~ + ~ ~ax ~ [~--~+11~
+ /1. e,(l/~)]. [ 1 + 1 (/1)] + as(Rt,x) .a~ u~ Using the following result af(pt+l,~)
sR
~t+l
f(pt+l,x) + Rt'sR
1
~(Rt)
the numerator is positive if as(R,x)
[/1. ~,(1//1)
4' L1 + ~,(l/J) + ~
E~(1//1)] [1
1 (/1)]> + -es
-
as(Rio) . a'c 1/ft
Note that normality of young consumption implies
[adI (~- 1)- 1]<
= LTe~l,, • 332
0 v/~ E (0,1],
Social Security as an Economic Stabilization Program
and Lemma 1 implies s~(/~) ~ (0,-1] and ss(1//~) ~ (0,- 1]. Rearranging terms giv6s as(R,z) [ 1 + £~(/~)] [ 1 + [ - R.a,(1//~) L2 +
Os(R,~) °q"C
£,(i//~)] ~J'
1/R
which holds by Assumption 4. Note the RHS is negative, so Os(R,x)/O~li~ must also be negative. For the fi < f(/5, x) case, note that high and low interest cases are exactly symmetric about the Pt = Pt+ 1 line. Thus the introduction of a social security system lowers the interest factor in this c a s e l
333