- Email: [email protected]
Stabilization
EUROPE~ ECONOMIC REVIEVV
i
European Economic
Review 41 (1997) 279-293
Stabilization Ailsa Riiell a,b,Oren Suss~an ’ ECARE,
CT *
Libre de Brwrelles, 39 Avenue F.D. Rooseoeit, B-1000 Brussels, Belgium b Tilburg Uniniuersity, Tilburg, The Netherlands University of the Negec and Monaster Centerfor Economic Research, P.O. Box 653, Beer-Sheua 84105, Israel
Urhersite
’ Ben-Gurion
Received
15 February
1994; revised 15 January 1996
Abstract We investigate the concepts of ‘activist macroeconomic policy’ and ‘stabilization’ within an optimal taxation framework when insurance markets are incomplete. JEL class~ficution: ES0; E60 Keywords:
Stabilization;
Feed-back
rules; Optimal taxation: Incomplete
insurance
markets
1. Intr~uction In this paper we conduct a welfare analysis of a macroeconomic stabilization within the conceptual framework of optimal taxation theory. We derive policy ’ an optimal tax scheme (on borrowing) for a non-monetary production economy with private info~ation and an aggregate demand shock. This scheme improves upon laissez faire risk-sharing. Moreover, the tax should be contingent on the magnitude of the realized macro shock, so that it can be interpreted as an ‘activist policy’. We show, however, that depending on the exact specification of preferences, the optimal tax may have a ‘stabilizing’ or ‘destabilizing’ effect. So, while
* Corresponding author. Fax: (+972) 7-647-2941. ’ In the broad sense of the word. The Palgrave defines: “the term ‘stabilization policy’ normally refers to deliberate changes in government policy instruments in response to changing macroeconomic conditions, in order to stabilize the economy” (Vines, 1987). Note that money is not mentioned. ~14-2921/97/$17.00 Copyright PI1 SOOl4-2921(96)00026-8
0 1997 Elsevier Science B.V. All rights reserved.
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Economic Review 41 (19971279-293
the analysis confirms the popular view that capital markets imperfections may call for some macroeconomic policy, it does not confirm another popular view, that such a policy will necessarily have a stabilizing effect. The basic structure of our model is taken from Diamond and Dybvig (1983). In a three-period economy, agents face a random intertemporal rate of substitution. In the interim period they find out whether they are patient or impatient; * their type is not publicly observable. The percentage of each type in the population is random as well - hence the aggregate demand shock. A high level of (interim) aggregate demand is a result of a high proportion of impatient agents in the population. There is a spot market for borrowing 3 (opened in period l), in which firms also participate in order to finance investment. Firms’ capital is observable and taxable, increasing the cost of capital above the households’ relevant interest rate. To understand the welfare implications of the tax, note that agents would insure themselves against the preference shock if type were publicly observable. In such an economy, competitive insurance contracts are available and income can be reallocated across types (when contracts are settled). This is impossible when the realization of agents’ types is private information. But then a tax on capital can reallocate income indirectly via interest rates. Consider (say) a lower householdrelevant interest rate. Wealth is reallocated away from patient agents who save much and have a large holding of interest-bearing assets, to impatient agents who save little and have a small (or even negative) holding of interest-bearing assets. An appropriate reallocation can improve ex ante risk sharing and is thus socially desirable. Note, however, that the tax is ex post distortionary as it drives a wedge between the marginal product of capital and the intertemporal rate of substitution. The crucial point is that the distortion of investment has only a second-order effect on welfare, while the redistribution has a first-order effect on welfare, and thus dominates for a small enough tax. 4 Hence, the optimal tax is non-zero. The above argument is consistent with both a negative and a positive tax. Whether wealth should be reallocated from impatient to patient agents or vice versa depends on the relative marginal utility of income across types and is determined by the exact specification of preferences. Since such reallocation affects the aggregate demand for consumption (impatient agents have a higher marginal propensity to consume), the macroeconomic effects of the optimal tax are ambiguous as well. To conclude, the mere existence of ‘imperfections’ provides no practical prescription as to the way an activist policy should be fine-tuned; stabilization per se is not inherently welfare-improving.
2 We generalize Diamond and Dybvig’s comer preferences; see Bhattacharya (1994). 3 Unlike in Diamond and Dybvig, banks are arbitraged away. See Jacklin (19891, Bhattacharya and Gale (1987) and recently Hellwig (1994). 4 The formal analysis of redistribution is a special case of Diamond and Mirrlees (19711, but the policy is motivated by the incompleteness of insurance markets (see Laffont, 1989).
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281
This result is compatible with some of the existing macroeconomic literature. Sargent and Wallace (1982) analyze an exchange economy with (deterministic) endowment fluctuations which affect the demand for credit. Money prices fluctuate in a laissez faire monetary equilibrium, fluctuations that can be stabilized by legal restrictions separating money and credit markets. It turns out, however, that this stabilized equilibrium is not Pareto-optimal. But in their model there are no credit markets imperfections, and there exists a laissez faire Pareto-optimal equilibrium. Stabilization is socially undesirable, but so is any sort of intervention. 5 (Note the difference: we show that even if there is a room for intervention, it is not necessarily stabilizing.) An alternative motivation to our work is that we link Sargent and Wallace with a branch of the literature which examines the equilibrium implications of some credit market imperfections (see Azariadis and Smith, 1993; Champ et al., 1991). In these models laissez faire may fail to achieve a constrained social optimum, But the question whether the optimal policy is stabilizing is not much emphasized. Also, these models often deal with exchange economies. Naturally, the potential role for capital taxation as a policy instrument is ignored. 6 It is noteworthy that our model contains none of the elements which often motivate intervention in the macro literature. We do not assume any form of price stickiness, and all of the markets (which open) clear at the equilibrium price. Also, given the allocation of information, no additional ad hoc constraint on contracts is imposed; agents may write and trade any (incentive compatible) contract they wish. It is important to note that the realization of the macro-shock is publicly known (as it is fully revealed by market prices), and agents are allowed to write contracts which are ‘indexed’ to the macro shock. The paper is organized as follows: the basic model is presented in Section 2. The benchmark first-best allocation (including transfers) is analyzed in Section 3, the laissez-faire equilibrium (without transfers) appears in Section 4. In Section 5 we show how government intervention can achieve a Pareto-improvement over laissez-fare. Section 6 concludes the paper.
2. The model Assumption A.l. Time. t = (0,1,2}. Consumption and production take place in periods 1 and 2. Uncertainty is resolved in period 1, whereupon decisions about lending, borrowing, consumption and saving take place. Agents are allowed to
5 Their main goal is to rehabilitate the free-banking real-bills doctrine. 6 Another, somewhat ignored, contribution is Weiss (1980). He analyzes a Lucas ‘islands’ environment in which the next period productivity shock distorts the present allocation of labor because some early informed agents are better informed than others. The monetary authority, by committing to respond ex post to the shock, makes nominal prices more informative and prevents the future disturbance from feeding in to the current allocation.
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282
meet at period 0 and draw up any (insurance) verifiability constraints.
contract
they wish, subject
to
Assumption A.2. Population. There is a measure 1 continuum of agents. Each is endowed with one unit of the good in period 1, which he may consume or lend. Each agent is also endowed with one unit of labor. Assumption A.3. Production. Firms F(k, n): capital, k, invested in period produce output. Denote f(k) = F(k, wages and interest rate are calculated aggregate) into w=f(k)
-kf’(k),
have a constant returns to scale technology 1, is combined (in period 2) with labor, n, to 1). G’tven the supply of labor, equilibrium by substituting capital stock (per capita and
r=f’(k).
(1)
Assumption A.4 Preferences. Type, p, is realized in period 1: either ‘impatient’, d, or ‘patient’, n; p = {d, n). A type d individual allocates more of his life-time income towards period 1 consumption than does a type n individual. The proportion of impatient individuals in the population is yrdxs, where s indicates ‘aggregate events’. We consider two specifications
of preferences
Es{Ep[ ypu( cf,“) + PU( c~‘~)]} EX{Ep[ U( cP~~) + ( P/-~P>u(
consistent
(specification
Cafe)]}
with the above story:
A),
(specification
(2.A) B),
(2.B)
and
(YdY) = (Y>l), 1 U(C) =-c
l-6
Y> 1,
(l-8) ’
6<
1.
(CRRA).
Once their type is realized, agents have the same ordinal time preferences under both specifications A and B. Thus, the ex post intertemporal rate of substitution (ITRS), and hence the (period 1) consumption function, is the same for both specifications A and B: cl =$(r;V).y,
d+
(forG<
I),
&>O,
(3)
for an agent with a preference factor yp, present value of lifetime income, y, and facing a gross interest rate r. The multiplicative structure of the consumption function follows from the homotheticity of preferences. On the other hand, the ex ante subjective rate of substitution across type-contingent, consumption depends on the specification of preferences (A or B). As a
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result, the indirect utility function, U( y, r; y “1, depends on realized type and the specification of preferences. The very reason why income reallocation is welfare improving (ex ante) is that the marginal utility of income differs across types: u,( r, y; r”)
= yPu’( cl) = T~u’(c~)
(specification
u,( r, y; yP) = u’( ci) = ( rP/yP)zk(
c2)
A),
(specification
(4-A) B).
(4-B)
Assumption AS. Information.
The individual’s type, whether d or n, is not observable. We also assume (realistically) that no information about individual agents’ consumption and saving patterns (which could reveal their type) is available. On the other hand, the realized aggregate proportion of impatient agents rrd*’ is revealed by equilib~um prices. Also, the amount of capital used by firms is observable, so that it can be taxed by the social planner.
3. A benchmark: The first best (complete markets) The social planner calculates the allocation that maximizes ex ante expected utility. Due to the expected utility assumption (separability) and the fact that resources cannot (technologically) be reallocated across aggregate events, the planner’s problem, say, for specification A, can be decomposed and solved separately for any given aggregate event s:
s.t
CTPJ(gJ
f
k” = 1)
(5)
f: 7r~‘sc2p’s=f( k”).
(6)
Wh$ rds-’ - 0 (or 1) all agents are identical, leaving a simple representative agent problem which is solved by equating the ITRS to the marginal product of capital. In general, when n- d~sE (0, 11,the first-order conditions are
U’(Cp) d( c;q = f’ ( k ” ) ( p/Jy( c;1J) = pzqcy> y
u’( c;‘~‘) = u’( c:,~)
u’( cf.“) = u’( CT;-‘)
(specification (specification
A), B)
.
(for both specifications),
(7) (84 (8.B)
Condition (7) states that the ITRS of both types is equated to the marginal product of capital. Conditions (8.A) and (8.B) state that the marginal utility of lifetime income is equated across types (see Eqs. (4.A) and (4.B)).
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Table 1 First-best allocation
Economic Review 41 (19971279-293
when pd.’ E (0, 1)
Specification
First-best income (after transfer)
Marginal utility of income (for given income)
A B
d.J > 1 > Y”.P Y d.s < 1 < y”~’ Y
o,(,;Yd)> u,(.;
yd)
u,(.;y”) <
c,(.;y”)
Obviously, this allocation can be implemented by a market economy with a spot market for loans if the social planner can (in period 1) observe types and reallocate income accordingly. 7 It follows from Eq. (8.A) and Eq. (6.A) that @” = c;.” and +.Y > +” (for specification A). That is, the present discounted value of lifetime consumption (and hence of income, transfers included) must be larger for type d agents than for type n agents. It is easy to see that without such redistribution, the marginal utility of lifetime income of an impatient agent is higher than that of a patient one. * These facts are summarized in Table 1. Thus, income is redistributed from types with low marginal utility of income to types with high marginal utility of income, up to the point where marginal utilities of income are equated across types. This implies transferring income from a patient to an impatient consumer under specification A, and vice versa under specification B. Under specification A, impatient agents would like their lifetime income to be relatively high. Under specification B, the opposite is true. The reason is that each agent’s preference parameter yP simultaneously determines both his intertemporal preferences and his marginal valuation of lifetime income. S~ci~cations A and B make opposite assumptions about the correlation between impatience and the marginal utility of lifetime income. Indeed, the direction of transfers is reversed when we move from specification A to B. A priori reasoning cannot determine whether a positive or a negative correlation is more plausible. We shall study the implications of both. The first-best allocation can also be attained without government intervention via a competitive insurance industry operating in period 0, provided that each agent’s type is observable and verifiable. In this case agents will have an incentive to draw up insurance contracts specifying net transfer payments contingent on both private and aggregate shocks. In equilibrium, insurance is purchased up to the point where type-contingent transfers equate the marginal utility of income.
7
After income is redistributed, the competitive loan market guarantees that the intertemporal rate of substitution for both types is equated with the marginal product of capital. * Take for example specification A: for a certain lifetime income 7, 4(y; rd) > QI(J; y”). so - for the same v. period 2 consumption is necessarily smaller for type d; it follows from Eq. f4.A) that the marginal utility of income is larger for type d. The result is reversed for specification B.
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4. Volatility
in the laissez-faire,
private-information
285
equilibrium
The competitive insurance industry described in the previous section requires direct observability of y p. The unobservability of type, by itself, does not preclude the existence of some insurance schemes, operating subject to incentive constraints, and utilizing whatever information is available. (Remember, however, that our crucial assumption that consumption allocation is also unobservable; otherwise agents may have signaled their type via the intertemporal allocation of consumption). Diamond and Dybvig (1983) analyze a demand-deposit scheme, where agents have an incentive to reveal their type truthfully. Such an institution cannot operate in our model. Note that there is no (period 1) market for borrowing in Diamond and Dybvig. In the absence of such a market, funds withdrawn can be used only for consumption. Once such a market is opened, transfers are evaluated at market prices. Obviously, all agents will identify themselves as belonging to the type that would maximizes the value of their transfer, and the scheme will be arbitraged away. 9 Note also that there is no trade in securities with payouts contingent upon the aggregate event: agents are all ex ante identical, and they would all want to hold equal amounts of such a security. If it is in zero net supply, individual holdings should be zero as well. We conclude that though no ad hoc restrictions are imposed on trade, financial trade is limited to the market for borrowing, where funds are traded as a result of realized differences in agents’ ITRS. The market clearing conditions (for each aggregate event s) are C7rpx”$( r”; yp) .yp,’ + k( r’) = 1,
(9)
P
where VP: Y
Pact= 1 + w.y/rs,
and factor prices are determined
by
f’( k”) = YS,
(10)
ws =f( k”) - k”f’( k”).
(11)
Substituting Eq. (3), Eq. (10) and Eq. (11) into Eq. (9), it is readily verified that r ’ is increasing in T*‘~, while k” is decreasing in T*,~. This can be explained intuitively with the aid of Fig. 1. Assume there are only two aggregate events (s = {h, 1}): a high aggregate demand for period 1 consumption state where ~*~“e(O,l), and a low (1) aggregate demand for period 1 consumption state where ~‘2~ = 0 and no-one is impatient. The demand for capital k is plotted against its cost p (for the time being r = p) with the origin at the right-hand side of the
9 See footnote 3 for references
to more detailed discussion
and proofs.
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,
r First-best Lsissez hfe both specifications First-best specification
B
k
The s = I state: xd,S = 0. The s = h state: 0 < ?ldms<: 1. Fig.
I. Aggregate consumption and investment: Two aggregate states.
picture. The aggregate demand for first-period consumption C = CPrr P*“4(r; yP)y3 is plotted against the interest rate r with the origin at the left-hand side of the picture. Both curves are downward sloping. At the h state, with a high proportion of impatient agents, there is higher demand for period I consumption and the Ch curve is higher than the C’ curve. The result is a higher price of period 1 consumption in terms of period 2 consumption (i.e. a higher interest rate t-1, and lower equilibrium investment, k. Note that since period 1 consumption (Eq. (3)) is the same for both specifications A and B, the spot market equilibrium does not depend on the specification of the utility function. We now compare the above spot market economy with a complete markets economy. We argue that under specification A and rrd,$ E (0, l), completeness of markets increases the interest rate while the opposite is true under specification B. This means that for the two-event (h, 1) case described in Fig. 1, lo under
10
Bear in mind that in the I case no-one is impatient.
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specification complete.
A (B) interest
rate volatility
is greater (lower)
287
when markets
are
Proposition 1. The complete markets and the incomplete markets economies have the same equilibrium when rrdxS= 0 (for both specifications A and B). Under specdfication A, the complete markets economy has a higher interest rate and less investment whenever rTTdzS E (0,l). The opposite is true under specification B. ProoJ: The two cases differ in that in the complete markets economy the settlement of insurance contracts signed in period 0 reallocates income across types in period 1. When 7~~2~= 0 no reallocation takes place, so that the complete and incomplete markets economies have the same equilibrium. To see the effect of the reallocation on the interest rate when rdXS E (0, l), we substitute the zero-profit condition for the insurance industry Cd~p3sxp’B = 0 into Eq. (9) and differentiate with respect ~~3’: dr” -=_ dxd,’
7Td3S[4( rS; 7”) - 4( rS; y”)]
Since for specification the interest rate. 0
> 0.
(12)
l), the reallocation
increases
. ) Ypus + CP~pSS+( .)[ -f(
~p+S~,(
A xd3’ > ~“2’ (see Table
kS)/rS]
The result can easily be understood with the aid of Fig. 1. Under specification A, income should be redistributed from patient type n to the impatient type d agents, i.e., from agents with a low marginal propensity to consume to agents with a high marginal propensity to consume. This will increase aggregate demand for period 1 consumption; indeed, the sign of Eq. (12) is determined by the difference in the marginal propensities to consume $(. ) of the two types (as well as by the negative sensitivity of aggregate demand to the interest factor). Hence, under specification A, the first-best aggregate demand for period 1 consumption will shift upwards (relative to the spot market economy), the interest rate will increase, and investment will fall. The opposite is true for specification B. Thus, completeness of markets increases the interest rate (for a given ~‘2~ E (0, I) under specification A, and decreases the interest rate under specification B.
5. Second-best:
Stabilization
We now consider a linear redistributive tax scheme. The social planner wishes to redistribute income towards the more impatient consumers. I’ Agents’ types are
” All discussion to.
in this section will treat specification
A unless specification
B is explicitly
referred
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unobservable, but the social planner is able to observe and tax (or subsidize) the capital stock, returning the tax revenue to all agents equally in the form of a lump-sum transfer. This enables the planner, indirectly, to redistribute income from type n to type d agents (or vice versa), as type d agents save less and therefore pay less tax. Note that the realization of the macro shock, its observation by the social planner, the adjustment of the tax and the determination of equilibrium - all take place simultaneously. Denote by 7 the proportional tax on interest payments so that the effective cost of capital is p = (1 + ~)r, while P is the interest rate which consumers face. Thus, a tax wedge is driven between the marginal product of capital and the intertempoml rate of substitution, distorting the allocation of resources. We shall therefore have to weigh the loss of welfare due to the intertemporal misallocation of resources against the welfare gain from the redistribution. The equilib~um conditions are: f’(F)
=ps,
pS=
(1 + 7)rS,
(13)
TS = 7. r”. k”, ys=
1+
Cgrp4+(
(14)
ws d- TS r”
(15)
’
r’; yJ’%“)ys + k” = 1.
(16)
7’” is the tax revenue (and the lump-sum transfer). It is assumed that the tax is collected and the lump-sum transfer is redistributed in period 2. ‘* By the constant returns assumption all output is paid out in the form of wages and pre-tax interest: f( k”) = ws + (1 + T)?k”.
(17)
Hence, Eq. (1.5) and Eq. (14) become: f(k”) ys = 1 - k( p”) + -.
(18)
rS
Appendix A shows that an increase in the tax rate 7 reduces investment lending rate and increases the cost of capital: dk”
z
CO,
dr” < 0 dr
and
and the
dpS -PO.
We can now state the main result of our paper Proposition 2. For spec~~catio~ A (B), a smuI1 enough (subsidy) improves welfare whenever n-d’S E (0,l).
” The
timing of the tax does not matter.
state-cu~ti~gent
tax
A. RBell, 0. Sussman/European
Pro06
Taking a total derivative
Economic Review 41 (1997) 279-293
289
of expected welfare in the h state
V” = [E,o( ~‘3 Y’; y”)], we get
Substituting
in Roy’s identity, u,
c, =y-r--,
Y we obtain
;=Epi”
y
( .? .y
P
)
.
(
.f’(k”) 1
But by Eq. (181, we may write dy” -= d7
--
[
rs
1 .---.dk” d7
f(k”) ( rs)2
dr” dr ’
(19)
Using Eq. (19) and Eq. (18) again, dV” dr
We can use the market clearing condition dV”
~dXS(~;3S - 1 + k”)
dr
r’
-=-
(16) to get
+7(.;yd)-uy(.:yn)]
.dr”
d7
(20) The second term on the right-hand side is negligible for a small enough 7, because then the marginal product of capital is close to the lending rate. Under specification A the first term is positive: as stated in Table 1, u,(. ; rd> > u,(. ; yn), < 0. Thus a small enough tax improves social while l3 c:” > 1 - k” and dr”/dr welfare. Under specification B u,(.; rd) < u,(.; y”) so that a small enough subsidy raises welfare. 0
13
Because
cf.” > c;,”
andCD~TTPh(~f,h ~ 1+ kh) = 0.
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The two terms on the right-hand side of Eq. (20) capture the reallocation and distribution effects of the tax, respectively. The social planner can reallocate income from patient to impatient agents by taxing investment. The effect of such redistribution on welfare is non-zero when different types have different marginal utilities of income. But such a manipulation of prices is distortionary, as captured by the second term on the right-hand side of Eq. (20). The tax (or subsidy) will always have a negative effect on welfare via the distortionary effect; but the appropriate tax will always improve welfare via the reallocation effect. As shown above the first-order distortionary effect of the tax is zero, so that the reallocation effect always dominates around r= 0. Thus, a welfare improving tax can always be found. At some point, however, the second-order effect of the distortion will be large enough to nullify the welfare gain from the reallocation. Therefore the tax should be levied only up to the point at which the gain from the reallocation outweighs the welfare loss generated by the distortion. Note that according to Proposition 2, for specification A, the optimally derived tax is positive for any fld,$ E (0,l). It does not say much about how the tax varies across ‘aggregate events’. Deriving analytical results here is quite difficult.
0
0.2
0.4
0.8
0.6
Proportion of “Impatient Type”
Fig. 2. The optimal tax scheme, specification
A.
1.0
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291
Instead, we simulate the (specification A) model for the following parameters: f(k) = k”.25, y = 2, 6 = 0.6, and /3 = l/1.2. The results are presented in Fig. 2. The aggregate event, i.e. the percentage of impatient agents rrdxS, is plotted on the horizontal axis, while the optimally derived tax is plotted on the vertical axis. The tax increases up to a maximum of over 8% at rrdxs = 0.5, and then falls.
6. Discussion We have studied a model with macro-economic demand shocks. Unlike the case of a representative agent framework, the shock affects different agents in different ways. We find that the government can improve risk-sharing among agents by taxing (or subsidizing) investment. The optimal tax scheme is contingent on the aggregate shock, so that macro-economic conditions feed back into policy responses. We interpret such a feed-back rule as an activist macro-economic policy and argue that an appropriately designed activist policy can be welfare improving. Note, again, that unlike much of the macro literature on intervention, there is no price stickiness (nor any ad hoc constraints on contracts), expectations are ‘rational’, and the government has no advantage in information over the private sector. The crucial point is that this reallocation does not require any specific information about the identity (type) of any individual. It is exactly because such information is not available that the transfers cannot be achieved by private insurance. But the social planner, with his power to tax can achieve the appropriate reallocation via taxation-induced price changes. Is the activist policy stabilizing in any sense? There is no unambiguous answer. Even within our very specific example, the answer depends on the structure of the preference shock. Consider again the two event (h,l) case discussed above with specification A preferences. In the 1 event (rd3s = 0) the optimal tax is zero. In the h event (~‘3~ E (0, l)), aggregate period 1 consumption is high and hence investment is low. The tax is described intuitively by the wedge r* inserted between the k and Ch curves (see Fig. l), and decreases the household-relevant interest rate below the cost of capital. It follows from Proposition 2 that investment should be taxed in the high aggregate consumption event (no intervention is required in the low aggregate consumption event). This tax policy can be described as an ‘accommodating’ policy: in the event of high aggregate consumption, the policy shifts some additional resources from investment to consumption. The policy is ‘destabilizing’ in the sense that it magnifies the effect of the aggregate shock on period 1 consumption and investment. In contrast, under specification B the planner should ‘lean against the wind’ and decrease the effect of the shock on consumption, sheltering investment from the shock. Thus, the optimal second-best policy may or may not stabilize aggregate consumption and investment. If there exists some general ‘rule’ about the management of an activist policy, it seems to be that aggregate consumption and investment should be ‘pushed’ in the direction
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292
of their first-best magnitudes. Remember, however, that first-best consumption or investment may fluctuate more (or less) than the spot market equilibrium consumption. What are the implications of the second-best policy for interest rate ‘volatility’? again they are ambiguous. In a laissez-faire economy volatility may be higher or lower than in the first-best, depending on the specification of preferences (see Fig. 1). However, the second-best policy pushes the lending rate away from its first best level: under specification A, the first-best lending rate is higher than the spot market lending rate, but the second-best lending rate is lower than the spot market rate. The ‘rule’ stated above for consumption fails to hold for the lending rate: second-best policy pushes the lending rate in the opposite direction to its first-best level. Note, however, that the lending rate and the cost of capital move in opposite directions due to the tax wedge. Stating those ‘rules’ in terms of the former rather than the latter is wholly arbitrary.
Acknowledgements A substantial part of this paper was written at the Gerzensee 1992 Summer Symposium. We would like to thank two anonymous referees for very helpful comments. Sussman would like to thank the Scheinburn Foundation for financial support.
Appendix
A. The effect of a tax increase on r, p and k
Throughout this appendix we drop the s superscript for notational convenience. Using Eq. (1 S), the equilibrium condition (16) may be rewritten as CD(r) = ?P(Y,7) where G(r)
=C7Tp+(r;yp),
is the aggregate propensity l-k I_k+f(k),r
S(r,r)=
Qr
to consume
with
where k is given by f’(k) = (1 + r)r. From the equilibrium condition dr d7- --.
%
in the first period and Tr>O,
and
qT>O,
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It is readily seen that dr/dr
293
< 0. We also calculate
dr -= dr
(note: the numerator is clearly negative while the denominator T -1 since @< 1 and T> -1). Obviously, it follows that dp/dT > 0.
is positive even for
References Azariadis, Costas and Bruce D. Smith, 1993, Adverse selection in the overlapping generations model: The case of pure exchange, Journal of Economic Theory 60, 277-305. Bhattacharya, Sudipto, 1994, The economics of bank regulation, Mimeo. Bhattacharya, Sudipto and Douglas Gale, 1987, Preference shocks, liquidity and central bank policy, In: W.A. Barnet and K.J. Singleton, eds., Preference shocks, liquidity and central bank policy (Cambridge University Press, Cambridge). Champ, B. Bruce Smith and Steven Williamson, 1991, Currency elasticity and banking panics: Theory and evidence, Working paper (Rochester Center for Economic Research, Rochester, NY). Diamond, Douglas and Philip Dybvig, 1983, Bank runs, deposit insurance, and liquidity, Journal of Political Economy 91, 401-419. Diamond Peter A. and James A. Mirrlees, 1971, Optimal taxation and public production, I: Production efficiency and II: Tax rules, American Economic Review 61, 8-27. Hellwig, Martin, 1994, Liquidity provision, banking and the allocation of interest rate risk, European Economic Review 38, 1363-1389. Jacklin, Charles J., 1989, Banks and risk sharing: Instability and coordination, In: Bhattacharya and Constantinides, eds., Vol. 2 (Rowman and Littlefield, Totowa, NJ). Laffont, Jean-Jacques, 1989, The economics of uncertainty and information (MIT Press, Cambridge, MA). Sargent, Thomas J. and Neil Wallace, 1982, The real-bills doctrine versus the quantity theory: A reconsideration, Journal of Political Economy 90, 1212-1236. Vines, David, 1987, Stabilization policy, In: The new Palgrave dictionary (MacMillan, London). Weiss, Laurence, 1980, The role for active monetary policy in a rational expectations model, Journal of Political Economy 88, 221-233.
i
European Economic
Review 41 (1997) 279-293
Stabilization Ailsa Riiell a,b,Oren Suss~an ’ ECARE,
CT *
Libre de Brwrelles, 39 Avenue F.D. Rooseoeit, B-1000 Brussels, Belgium b Tilburg Uniniuersity, Tilburg, The Netherlands University of the Negec and Monaster Centerfor Economic Research, P.O. Box 653, Beer-Sheua 84105, Israel
Urhersite
’ Ben-Gurion
Received
15 February
1994; revised 15 January 1996
Abstract We investigate the concepts of ‘activist macroeconomic policy’ and ‘stabilization’ within an optimal taxation framework when insurance markets are incomplete. JEL class~ficution: ES0; E60 Keywords:
Stabilization;
Feed-back
rules; Optimal taxation: Incomplete
insurance
markets
1. Intr~uction In this paper we conduct a welfare analysis of a macroeconomic stabilization within the conceptual framework of optimal taxation theory. We derive policy ’ an optimal tax scheme (on borrowing) for a non-monetary production economy with private info~ation and an aggregate demand shock. This scheme improves upon laissez faire risk-sharing. Moreover, the tax should be contingent on the magnitude of the realized macro shock, so that it can be interpreted as an ‘activist policy’. We show, however, that depending on the exact specification of preferences, the optimal tax may have a ‘stabilizing’ or ‘destabilizing’ effect. So, while
* Corresponding author. Fax: (+972) 7-647-2941. ’ In the broad sense of the word. The Palgrave defines: “the term ‘stabilization policy’ normally refers to deliberate changes in government policy instruments in response to changing macroeconomic conditions, in order to stabilize the economy” (Vines, 1987). Note that money is not mentioned. ~14-2921/97/$17.00 Copyright PI1 SOOl4-2921(96)00026-8
0 1997 Elsevier Science B.V. All rights reserved.
280
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Economic Review 41 (19971279-293
the analysis confirms the popular view that capital markets imperfections may call for some macroeconomic policy, it does not confirm another popular view, that such a policy will necessarily have a stabilizing effect. The basic structure of our model is taken from Diamond and Dybvig (1983). In a three-period economy, agents face a random intertemporal rate of substitution. In the interim period they find out whether they are patient or impatient; * their type is not publicly observable. The percentage of each type in the population is random as well - hence the aggregate demand shock. A high level of (interim) aggregate demand is a result of a high proportion of impatient agents in the population. There is a spot market for borrowing 3 (opened in period l), in which firms also participate in order to finance investment. Firms’ capital is observable and taxable, increasing the cost of capital above the households’ relevant interest rate. To understand the welfare implications of the tax, note that agents would insure themselves against the preference shock if type were publicly observable. In such an economy, competitive insurance contracts are available and income can be reallocated across types (when contracts are settled). This is impossible when the realization of agents’ types is private information. But then a tax on capital can reallocate income indirectly via interest rates. Consider (say) a lower householdrelevant interest rate. Wealth is reallocated away from patient agents who save much and have a large holding of interest-bearing assets, to impatient agents who save little and have a small (or even negative) holding of interest-bearing assets. An appropriate reallocation can improve ex ante risk sharing and is thus socially desirable. Note, however, that the tax is ex post distortionary as it drives a wedge between the marginal product of capital and the intertemporal rate of substitution. The crucial point is that the distortion of investment has only a second-order effect on welfare, while the redistribution has a first-order effect on welfare, and thus dominates for a small enough tax. 4 Hence, the optimal tax is non-zero. The above argument is consistent with both a negative and a positive tax. Whether wealth should be reallocated from impatient to patient agents or vice versa depends on the relative marginal utility of income across types and is determined by the exact specification of preferences. Since such reallocation affects the aggregate demand for consumption (impatient agents have a higher marginal propensity to consume), the macroeconomic effects of the optimal tax are ambiguous as well. To conclude, the mere existence of ‘imperfections’ provides no practical prescription as to the way an activist policy should be fine-tuned; stabilization per se is not inherently welfare-improving.
2 We generalize Diamond and Dybvig’s comer preferences; see Bhattacharya (1994). 3 Unlike in Diamond and Dybvig, banks are arbitraged away. See Jacklin (19891, Bhattacharya and Gale (1987) and recently Hellwig (1994). 4 The formal analysis of redistribution is a special case of Diamond and Mirrlees (19711, but the policy is motivated by the incompleteness of insurance markets (see Laffont, 1989).
A. Riiell, 0. Sussman / European Economic Review 41 (1997) 279-293
281
This result is compatible with some of the existing macroeconomic literature. Sargent and Wallace (1982) analyze an exchange economy with (deterministic) endowment fluctuations which affect the demand for credit. Money prices fluctuate in a laissez faire monetary equilibrium, fluctuations that can be stabilized by legal restrictions separating money and credit markets. It turns out, however, that this stabilized equilibrium is not Pareto-optimal. But in their model there are no credit markets imperfections, and there exists a laissez faire Pareto-optimal equilibrium. Stabilization is socially undesirable, but so is any sort of intervention. 5 (Note the difference: we show that even if there is a room for intervention, it is not necessarily stabilizing.) An alternative motivation to our work is that we link Sargent and Wallace with a branch of the literature which examines the equilibrium implications of some credit market imperfections (see Azariadis and Smith, 1993; Champ et al., 1991). In these models laissez faire may fail to achieve a constrained social optimum, But the question whether the optimal policy is stabilizing is not much emphasized. Also, these models often deal with exchange economies. Naturally, the potential role for capital taxation as a policy instrument is ignored. 6 It is noteworthy that our model contains none of the elements which often motivate intervention in the macro literature. We do not assume any form of price stickiness, and all of the markets (which open) clear at the equilibrium price. Also, given the allocation of information, no additional ad hoc constraint on contracts is imposed; agents may write and trade any (incentive compatible) contract they wish. It is important to note that the realization of the macro-shock is publicly known (as it is fully revealed by market prices), and agents are allowed to write contracts which are ‘indexed’ to the macro shock. The paper is organized as follows: the basic model is presented in Section 2. The benchmark first-best allocation (including transfers) is analyzed in Section 3, the laissez-faire equilibrium (without transfers) appears in Section 4. In Section 5 we show how government intervention can achieve a Pareto-improvement over laissez-fare. Section 6 concludes the paper.
2. The model Assumption A.l. Time. t = (0,1,2}. Consumption and production take place in periods 1 and 2. Uncertainty is resolved in period 1, whereupon decisions about lending, borrowing, consumption and saving take place. Agents are allowed to
5 Their main goal is to rehabilitate the free-banking real-bills doctrine. 6 Another, somewhat ignored, contribution is Weiss (1980). He analyzes a Lucas ‘islands’ environment in which the next period productivity shock distorts the present allocation of labor because some early informed agents are better informed than others. The monetary authority, by committing to respond ex post to the shock, makes nominal prices more informative and prevents the future disturbance from feeding in to the current allocation.
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282
meet at period 0 and draw up any (insurance) verifiability constraints.
contract
they wish, subject
to
Assumption A.2. Population. There is a measure 1 continuum of agents. Each is endowed with one unit of the good in period 1, which he may consume or lend. Each agent is also endowed with one unit of labor. Assumption A.3. Production. Firms F(k, n): capital, k, invested in period produce output. Denote f(k) = F(k, wages and interest rate are calculated aggregate) into w=f(k)
-kf’(k),
have a constant returns to scale technology 1, is combined (in period 2) with labor, n, to 1). G’tven the supply of labor, equilibrium by substituting capital stock (per capita and
r=f’(k).
(1)
Assumption A.4 Preferences. Type, p, is realized in period 1: either ‘impatient’, d, or ‘patient’, n; p = {d, n). A type d individual allocates more of his life-time income towards period 1 consumption than does a type n individual. The proportion of impatient individuals in the population is yrdxs, where s indicates ‘aggregate events’. We consider two specifications
of preferences
Es{Ep[ ypu( cf,“) + PU( c~‘~)]} EX{Ep[ U( cP~~) + ( P/-~P>u(
consistent
(specification
Cafe)]}
with the above story:
A),
(specification
(2.A) B),
(2.B)
and
(YdY) = (Y>l), 1 U(C) =-c
l-6
Y> 1,
(l-8) ’
6<
1.
(CRRA).
Once their type is realized, agents have the same ordinal time preferences under both specifications A and B. Thus, the ex post intertemporal rate of substitution (ITRS), and hence the (period 1) consumption function, is the same for both specifications A and B: cl =$(r;V).y,
d+
(forG<
I),
&>O,
(3)
for an agent with a preference factor yp, present value of lifetime income, y, and facing a gross interest rate r. The multiplicative structure of the consumption function follows from the homotheticity of preferences. On the other hand, the ex ante subjective rate of substitution across type-contingent, consumption depends on the specification of preferences (A or B). As a
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283
result, the indirect utility function, U( y, r; y “1, depends on realized type and the specification of preferences. The very reason why income reallocation is welfare improving (ex ante) is that the marginal utility of income differs across types: u,( r, y; r”)
= yPu’( cl) = T~u’(c~)
(specification
u,( r, y; yP) = u’( ci) = ( rP/yP)zk(
c2)
A),
(specification
(4-A) B).
(4-B)
Assumption AS. Information.
The individual’s type, whether d or n, is not observable. We also assume (realistically) that no information about individual agents’ consumption and saving patterns (which could reveal their type) is available. On the other hand, the realized aggregate proportion of impatient agents rrd*’ is revealed by equilib~um prices. Also, the amount of capital used by firms is observable, so that it can be taxed by the social planner.
3. A benchmark: The first best (complete markets) The social planner calculates the allocation that maximizes ex ante expected utility. Due to the expected utility assumption (separability) and the fact that resources cannot (technologically) be reallocated across aggregate events, the planner’s problem, say, for specification A, can be decomposed and solved separately for any given aggregate event s:
s.t
CTPJ(gJ
f
k” = 1)
(5)
f: 7r~‘sc2p’s=f( k”).
(6)
Wh$ rds-’ - 0 (or 1) all agents are identical, leaving a simple representative agent problem which is solved by equating the ITRS to the marginal product of capital. In general, when n- d~sE (0, 11,the first-order conditions are
U’(Cp) d( c;q = f’ ( k ” ) ( p/Jy( c;1J) = pzqcy> y
u’( c;‘~‘) = u’( c:,~)
u’( cf.“) = u’( CT;-‘)
(specification (specification
A), B)
.
(for both specifications),
(7) (84 (8.B)
Condition (7) states that the ITRS of both types is equated to the marginal product of capital. Conditions (8.A) and (8.B) state that the marginal utility of lifetime income is equated across types (see Eqs. (4.A) and (4.B)).
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Table 1 First-best allocation
Economic Review 41 (19971279-293
when pd.’ E (0, 1)
Specification
First-best income (after transfer)
Marginal utility of income (for given income)
A B
d.J > 1 > Y”.P Y d.s < 1 < y”~’ Y
o,(,;Yd)> u,(.;
yd)
u,(.;y”) <
c,(.;y”)
Obviously, this allocation can be implemented by a market economy with a spot market for loans if the social planner can (in period 1) observe types and reallocate income accordingly. 7 It follows from Eq. (8.A) and Eq. (6.A) that @” = c;.” and +.Y > +” (for specification A). That is, the present discounted value of lifetime consumption (and hence of income, transfers included) must be larger for type d agents than for type n agents. It is easy to see that without such redistribution, the marginal utility of lifetime income of an impatient agent is higher than that of a patient one. * These facts are summarized in Table 1. Thus, income is redistributed from types with low marginal utility of income to types with high marginal utility of income, up to the point where marginal utilities of income are equated across types. This implies transferring income from a patient to an impatient consumer under specification A, and vice versa under specification B. Under specification A, impatient agents would like their lifetime income to be relatively high. Under specification B, the opposite is true. The reason is that each agent’s preference parameter yP simultaneously determines both his intertemporal preferences and his marginal valuation of lifetime income. S~ci~cations A and B make opposite assumptions about the correlation between impatience and the marginal utility of lifetime income. Indeed, the direction of transfers is reversed when we move from specification A to B. A priori reasoning cannot determine whether a positive or a negative correlation is more plausible. We shall study the implications of both. The first-best allocation can also be attained without government intervention via a competitive insurance industry operating in period 0, provided that each agent’s type is observable and verifiable. In this case agents will have an incentive to draw up insurance contracts specifying net transfer payments contingent on both private and aggregate shocks. In equilibrium, insurance is purchased up to the point where type-contingent transfers equate the marginal utility of income.
7
After income is redistributed, the competitive loan market guarantees that the intertemporal rate of substitution for both types is equated with the marginal product of capital. * Take for example specification A: for a certain lifetime income 7, 4(y; rd) > QI(J; y”). so - for the same v. period 2 consumption is necessarily smaller for type d; it follows from Eq. f4.A) that the marginal utility of income is larger for type d. The result is reversed for specification B.
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4. Volatility
in the laissez-faire,
private-information
285
equilibrium
The competitive insurance industry described in the previous section requires direct observability of y p. The unobservability of type, by itself, does not preclude the existence of some insurance schemes, operating subject to incentive constraints, and utilizing whatever information is available. (Remember, however, that our crucial assumption that consumption allocation is also unobservable; otherwise agents may have signaled their type via the intertemporal allocation of consumption). Diamond and Dybvig (1983) analyze a demand-deposit scheme, where agents have an incentive to reveal their type truthfully. Such an institution cannot operate in our model. Note that there is no (period 1) market for borrowing in Diamond and Dybvig. In the absence of such a market, funds withdrawn can be used only for consumption. Once such a market is opened, transfers are evaluated at market prices. Obviously, all agents will identify themselves as belonging to the type that would maximizes the value of their transfer, and the scheme will be arbitraged away. 9 Note also that there is no trade in securities with payouts contingent upon the aggregate event: agents are all ex ante identical, and they would all want to hold equal amounts of such a security. If it is in zero net supply, individual holdings should be zero as well. We conclude that though no ad hoc restrictions are imposed on trade, financial trade is limited to the market for borrowing, where funds are traded as a result of realized differences in agents’ ITRS. The market clearing conditions (for each aggregate event s) are C7rpx”$( r”; yp) .yp,’ + k( r’) = 1,
(9)
P
where VP: Y
Pact= 1 + w.y/rs,
and factor prices are determined
by
f’( k”) = YS,
(10)
ws =f( k”) - k”f’( k”).
(11)
Substituting Eq. (3), Eq. (10) and Eq. (11) into Eq. (9), it is readily verified that r ’ is increasing in T*‘~, while k” is decreasing in T*,~. This can be explained intuitively with the aid of Fig. 1. Assume there are only two aggregate events (s = {h, 1}): a high aggregate demand for period 1 consumption state where ~*~“e(O,l), and a low (1) aggregate demand for period 1 consumption state where ~‘2~ = 0 and no-one is impatient. The demand for capital k is plotted against its cost p (for the time being r = p) with the origin at the right-hand side of the
9 See footnote 3 for references
to more detailed discussion
and proofs.
A. R&11, 0. Sussman/European
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Economic Reuiew 41 (19971279-293
,
r First-best Lsissez hfe both specifications First-best specification
B
k
The s = I state: xd,S = 0. The s = h state: 0 < ?ldms<: 1. Fig.
I. Aggregate consumption and investment: Two aggregate states.
picture. The aggregate demand for first-period consumption C = CPrr P*“4(r; yP)y3 is plotted against the interest rate r with the origin at the left-hand side of the picture. Both curves are downward sloping. At the h state, with a high proportion of impatient agents, there is higher demand for period I consumption and the Ch curve is higher than the C’ curve. The result is a higher price of period 1 consumption in terms of period 2 consumption (i.e. a higher interest rate t-1, and lower equilibrium investment, k. Note that since period 1 consumption (Eq. (3)) is the same for both specifications A and B, the spot market equilibrium does not depend on the specification of the utility function. We now compare the above spot market economy with a complete markets economy. We argue that under specification A and rrd,$ E (0, l), completeness of markets increases the interest rate while the opposite is true under specification B. This means that for the two-event (h, 1) case described in Fig. 1, lo under
10
Bear in mind that in the I case no-one is impatient.
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specification complete.
A (B) interest
rate volatility
is greater (lower)
287
when markets
are
Proposition 1. The complete markets and the incomplete markets economies have the same equilibrium when rrdxS= 0 (for both specifications A and B). Under specdfication A, the complete markets economy has a higher interest rate and less investment whenever rTTdzS E (0,l). The opposite is true under specification B. ProoJ: The two cases differ in that in the complete markets economy the settlement of insurance contracts signed in period 0 reallocates income across types in period 1. When 7~~2~= 0 no reallocation takes place, so that the complete and incomplete markets economies have the same equilibrium. To see the effect of the reallocation on the interest rate when rdXS E (0, l), we substitute the zero-profit condition for the insurance industry Cd~p3sxp’B = 0 into Eq. (9) and differentiate with respect ~~3’: dr” -=_ dxd,’
7Td3S[4( rS; 7”) - 4( rS; y”)]
Since for specification the interest rate. 0
> 0.
(12)
l), the reallocation
increases
. ) Ypus + CP~pSS+( .)[ -f(
~p+S~,(
A xd3’ > ~“2’ (see Table
kS)/rS]
The result can easily be understood with the aid of Fig. 1. Under specification A, income should be redistributed from patient type n to the impatient type d agents, i.e., from agents with a low marginal propensity to consume to agents with a high marginal propensity to consume. This will increase aggregate demand for period 1 consumption; indeed, the sign of Eq. (12) is determined by the difference in the marginal propensities to consume $(. ) of the two types (as well as by the negative sensitivity of aggregate demand to the interest factor). Hence, under specification A, the first-best aggregate demand for period 1 consumption will shift upwards (relative to the spot market economy), the interest rate will increase, and investment will fall. The opposite is true for specification B. Thus, completeness of markets increases the interest rate (for a given ~‘2~ E (0, I) under specification A, and decreases the interest rate under specification B.
5. Second-best:
Stabilization
We now consider a linear redistributive tax scheme. The social planner wishes to redistribute income towards the more impatient consumers. I’ Agents’ types are
” All discussion to.
in this section will treat specification
A unless specification
B is explicitly
referred
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Economic Review 41 (I9971 279-293
unobservable, but the social planner is able to observe and tax (or subsidize) the capital stock, returning the tax revenue to all agents equally in the form of a lump-sum transfer. This enables the planner, indirectly, to redistribute income from type n to type d agents (or vice versa), as type d agents save less and therefore pay less tax. Note that the realization of the macro shock, its observation by the social planner, the adjustment of the tax and the determination of equilibrium - all take place simultaneously. Denote by 7 the proportional tax on interest payments so that the effective cost of capital is p = (1 + ~)r, while P is the interest rate which consumers face. Thus, a tax wedge is driven between the marginal product of capital and the intertempoml rate of substitution, distorting the allocation of resources. We shall therefore have to weigh the loss of welfare due to the intertemporal misallocation of resources against the welfare gain from the redistribution. The equilib~um conditions are: f’(F)
=ps,
pS=
(1 + 7)rS,
(13)
TS = 7. r”. k”, ys=
1+
Cgrp4+(
(14)
ws d- TS r”
(15)
’
r’; yJ’%“)ys + k” = 1.
(16)
7’” is the tax revenue (and the lump-sum transfer). It is assumed that the tax is collected and the lump-sum transfer is redistributed in period 2. ‘* By the constant returns assumption all output is paid out in the form of wages and pre-tax interest: f( k”) = ws + (1 + T)?k”.
(17)
Hence, Eq. (1.5) and Eq. (14) become: f(k”) ys = 1 - k( p”) + -.
(18)
rS
Appendix A shows that an increase in the tax rate 7 reduces investment lending rate and increases the cost of capital: dk”
z
CO,
dr” < 0 dr
and
and the
dpS -PO.
We can now state the main result of our paper Proposition 2. For spec~~catio~ A (B), a smuI1 enough (subsidy) improves welfare whenever n-d’S E (0,l).
” The
timing of the tax does not matter.
state-cu~ti~gent
tax
A. RBell, 0. Sussman/European
Pro06
Taking a total derivative
Economic Review 41 (1997) 279-293
289
of expected welfare in the h state
V” = [E,o( ~‘3 Y’; y”)], we get
Substituting
in Roy’s identity, u,
c, =y-r--,
Y we obtain
;=Epi”
y
( .? .y
P
)
.
(
.f’(k”) 1
But by Eq. (181, we may write dy” -= d7
--
[
rs
1 .---.dk” d7
f(k”) ( rs)2
dr” dr ’
(19)
Using Eq. (19) and Eq. (18) again, dV” dr
We can use the market clearing condition dV”
~dXS(~;3S - 1 + k”)
dr
r’
-=-
(16) to get
+7(.;yd)-uy(.:yn)]
.dr”
d7
(20) The second term on the right-hand side is negligible for a small enough 7, because then the marginal product of capital is close to the lending rate. Under specification A the first term is positive: as stated in Table 1, u,(. ; rd> > u,(. ; yn), < 0. Thus a small enough tax improves social while l3 c:” > 1 - k” and dr”/dr welfare. Under specification B u,(.; rd) < u,(.; y”) so that a small enough subsidy raises welfare. 0
13
Because
cf.” > c;,”
andCD~TTPh(~f,h ~ 1+ kh) = 0.
290
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The two terms on the right-hand side of Eq. (20) capture the reallocation and distribution effects of the tax, respectively. The social planner can reallocate income from patient to impatient agents by taxing investment. The effect of such redistribution on welfare is non-zero when different types have different marginal utilities of income. But such a manipulation of prices is distortionary, as captured by the second term on the right-hand side of Eq. (20). The tax (or subsidy) will always have a negative effect on welfare via the distortionary effect; but the appropriate tax will always improve welfare via the reallocation effect. As shown above the first-order distortionary effect of the tax is zero, so that the reallocation effect always dominates around r= 0. Thus, a welfare improving tax can always be found. At some point, however, the second-order effect of the distortion will be large enough to nullify the welfare gain from the reallocation. Therefore the tax should be levied only up to the point at which the gain from the reallocation outweighs the welfare loss generated by the distortion. Note that according to Proposition 2, for specification A, the optimally derived tax is positive for any fld,$ E (0,l). It does not say much about how the tax varies across ‘aggregate events’. Deriving analytical results here is quite difficult.
0
0.2
0.4
0.8
0.6
Proportion of “Impatient Type”
Fig. 2. The optimal tax scheme, specification
A.
1.0
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291
Instead, we simulate the (specification A) model for the following parameters: f(k) = k”.25, y = 2, 6 = 0.6, and /3 = l/1.2. The results are presented in Fig. 2. The aggregate event, i.e. the percentage of impatient agents rrdxS, is plotted on the horizontal axis, while the optimally derived tax is plotted on the vertical axis. The tax increases up to a maximum of over 8% at rrdxs = 0.5, and then falls.
6. Discussion We have studied a model with macro-economic demand shocks. Unlike the case of a representative agent framework, the shock affects different agents in different ways. We find that the government can improve risk-sharing among agents by taxing (or subsidizing) investment. The optimal tax scheme is contingent on the aggregate shock, so that macro-economic conditions feed back into policy responses. We interpret such a feed-back rule as an activist macro-economic policy and argue that an appropriately designed activist policy can be welfare improving. Note, again, that unlike much of the macro literature on intervention, there is no price stickiness (nor any ad hoc constraints on contracts), expectations are ‘rational’, and the government has no advantage in information over the private sector. The crucial point is that this reallocation does not require any specific information about the identity (type) of any individual. It is exactly because such information is not available that the transfers cannot be achieved by private insurance. But the social planner, with his power to tax can achieve the appropriate reallocation via taxation-induced price changes. Is the activist policy stabilizing in any sense? There is no unambiguous answer. Even within our very specific example, the answer depends on the structure of the preference shock. Consider again the two event (h,l) case discussed above with specification A preferences. In the 1 event (rd3s = 0) the optimal tax is zero. In the h event (~‘3~ E (0, l)), aggregate period 1 consumption is high and hence investment is low. The tax is described intuitively by the wedge r* inserted between the k and Ch curves (see Fig. l), and decreases the household-relevant interest rate below the cost of capital. It follows from Proposition 2 that investment should be taxed in the high aggregate consumption event (no intervention is required in the low aggregate consumption event). This tax policy can be described as an ‘accommodating’ policy: in the event of high aggregate consumption, the policy shifts some additional resources from investment to consumption. The policy is ‘destabilizing’ in the sense that it magnifies the effect of the aggregate shock on period 1 consumption and investment. In contrast, under specification B the planner should ‘lean against the wind’ and decrease the effect of the shock on consumption, sheltering investment from the shock. Thus, the optimal second-best policy may or may not stabilize aggregate consumption and investment. If there exists some general ‘rule’ about the management of an activist policy, it seems to be that aggregate consumption and investment should be ‘pushed’ in the direction
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292
of their first-best magnitudes. Remember, however, that first-best consumption or investment may fluctuate more (or less) than the spot market equilibrium consumption. What are the implications of the second-best policy for interest rate ‘volatility’? again they are ambiguous. In a laissez-faire economy volatility may be higher or lower than in the first-best, depending on the specification of preferences (see Fig. 1). However, the second-best policy pushes the lending rate away from its first best level: under specification A, the first-best lending rate is higher than the spot market lending rate, but the second-best lending rate is lower than the spot market rate. The ‘rule’ stated above for consumption fails to hold for the lending rate: second-best policy pushes the lending rate in the opposite direction to its first-best level. Note, however, that the lending rate and the cost of capital move in opposite directions due to the tax wedge. Stating those ‘rules’ in terms of the former rather than the latter is wholly arbitrary.
Acknowledgements A substantial part of this paper was written at the Gerzensee 1992 Summer Symposium. We would like to thank two anonymous referees for very helpful comments. Sussman would like to thank the Scheinburn Foundation for financial support.
Appendix
A. The effect of a tax increase on r, p and k
Throughout this appendix we drop the s superscript for notational convenience. Using Eq. (1 S), the equilibrium condition (16) may be rewritten as CD(r) = ?P(Y,7) where G(r)
=C7Tp+(r;yp),
is the aggregate propensity l-k I_k+f(k),r
S(r,r)=
Qr
to consume
with
where k is given by f’(k) = (1 + r)r. From the equilibrium condition dr d7- --.
%
in the first period and Tr>O,
and
qT>O,
A. Rb’ell, 0. Sussman / European Economic Review 41 (1997) 279-293
It is readily seen that dr/dr
293
< 0. We also calculate
dr -= dr
(note: the numerator is clearly negative while the denominator T
is positive even for
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