The Incidence of Deficit Finance with Imperfect Capital Markets.

Michael Ben-Gad [*]

The purpose of this paper is to examine the possible differential welfare implications of deficit finance using a portfolio allocation model. To analyze the incidence of changing the time path of taxation in an economy with heterogeneous agents, I develop a two-period, general equilibrium extension of work done previously to analyze the effects of taxation on risk-taking at the individual level. Constraints on short sales of assets are introduced, and fiscal policy, changing the timing of taxation, will indirectly determine which of these constraints bind as well as alter relative tax burdens. Changes in the timing of a flat-rate tax will also alter equilibrium asset returns, and because preferences are such that agents differ in their tolerance of risk, a Pareto frontier can be derived over a range of different levels of deficit finance.

1. Introduction

During the last 20 years there has been a considerable change in the way that economists view deficit finance. Whereas once researchers focused on the efficacy of deficit finance as an instrument of short-term countercyclical policy, today much of the attention, particularly of public finance economists, focuses on its longer-term redistributive effects across generations. Many scholars, including Diamond (1965), Yotsuzuka (1987), and Woodford (1990), have demonstrated that under certain circumstances, increases in deficit finance can be Pareto improving. Nonetheless Auerbach, Gokhale, and Kotlikoff (1991) have argued that recent fiscal policy in the United States has shifted the tax burden increasingly away from the elderly and toward members of younger cohorts in a way that is hard to justify on efficiency grounds. Similar work in other industrialized countries has found a similar trend. By contrast, within the large body of deficit finance literature, very little attention has been paid to the potential i ntragenerational redistribution of welfare that may result from changes in fiscal policy.

Similarly, there is an extensive literature comparing the intragenerational incidence of different tax regimes. This type of research (Fullerton and Rogers 1993; Kasten, Sammartino, and Toder 1994) measures the welfare gains and losses, for members of different income classes, of shifting the tax burden between different types of economic activities. What this literature does not address is the potential redistributive consequences of deficit finance--the intragenerational incidence of shifting the payment of a given tax from one period to the next.

This paper analyzes a two-period, portfolio choice model with imperfect capital markets to demonstrate that changes in the timing of a tax can redistribute welfare intragenerationally among agents with heterogeneous levels of initial wealth. These welfare changes are not driven by shifts in the relative tax burden. Instead, changes in fiscal policy induce changes in equilibrium asset returns that have differential effects on the agents in the economy. Finally, this paper will demonstrate that the different agents' policy preferences over the amount of deficit finance in the initial period are not necessarily monotonic in wealth.

The first work on the effects of taxation on portfolio choice was done by Domar and Musgrave (1944. The focus of this work was the effects of loss offsets on risk-taking at the level of the private individual. Mossin (1968) and Stiglitz (1969) extended this work to a model that includes agents with a general expected utility function within a two-state stochastic endowment economy. As in Domar and Musgrave, the focus of this work was the effects of government policy on an individuals' choices of assets in an economy in which the gross returns on these assets are fixed.

To analyze changes in the timing of taxation, this paper will use a two-period general equilibrium extension of the Mossin--Stiglitz model. To study the incidence of such changes, a government budget constraint is explicitly included. In this model, asset returns are endogenously determined and change with government policy. Market imperfections, constraints on short-selling of assets, are also introduced.

For reasons of moral hazard, adverse selection, or simply the higher likelihood that individuals will default, most people cannot borrow at the same rate of interest as the government or execute short sales of equities costlessly. [1] This limits their ability either to leverage their portfolios or to insure completely against all uncertainty. Fiscal policy, changing the timing of taxation, will indirectly determine which of the constraints on asset holdings bind, as well as alter relative tax burdens. However, the most important effect on welfare will come not from the small changes in the amounts of taxes paid, but rather from the effects of alternative fiscal policies on asset prices. In the presence of imperfect capital markets, if preferences are such that agents differ in their tolerance of risk, changes in the time path of a flat-rate tax will alter equilibrium asset returns, and produce a Pareto frontier over different levels of deficit finance.

During the last quarter century, United States fiscal policy has undergone considerable upheaval--the long post war decline in the debt burden was halted during the mid-1970s and the debt climbed considerably during the 1980s. Recently, the debt/GDP ratio has once again begun to decline, though its long-term future remains uncertain. More importantly, despite the recent declines in deficit spending, long-term projections of lifetime net tax payments for the members of future generations have remained almost unchanged since Auerbach, Gokhale, and Kotlikoff first introduced their method of generational accounting in 1991. Now as then, baseline fiscal policy appears unsustainable, and absent large cuts in spending, tax rates are likely to rise in the future. This work is a first step toward a sustained examination of the intragenerational incidence of these tax-postponing policies and also provides a possible insight into the type of political equilibria that may perpetuate them.

2. Model

Consider a closed economy that lasts two periods. Each consumer h [epsilon] H begins period 0 with an endowment of [[k.sup.h].sub.0] units of equities, a unit of which pays a dividend of one unit of consumption in the initial period. After receipt of the dividend, all agents simultaneously choose between the level of initial consumption [[c.sup.h].sub.0] and some portfolio of either government bonds [[b.sup.h].sub.1] or next-period equity holdings [[k.sup.h].sub.1], (net investment in equities is therefore [[k.sup.h].sub.1] - [[k.sup.h].sub.0]), which are exchanged at market prices p and q, respectively. In the second period, there are two different states, one in which the equity pays a dividend [d.sub.1]([s.sub.1]) in second-period consumption [[c.sup.h].sub.1]([s.sub.1]) and another in which the dividend is [d.sub.1]([s.sub.2]) in second-period consumption [[c.sup.h].sub.1]([s.sub.2]) (I will assume that [d.sub.1]([s.sub.2]) [greater than] 1 [greater than] [d.sub.1]([s.sub.1])). By contrast, the governmen t bond is risk free and pays one unit of consumption in either state during the second period. The total number of people in this economy is normalized to one, and so aggregate consumption is equal to one in the initial period and, depending on the state, to [d.sub.1]([s.sub.r]), [gamma] [epsilon] {1, 2} in period 1.

In each period and every state, the government consumes a fixed amount G, which it finances by a sequence of flat rate taxes [{[[tau].sub.0], [[tau].sub.1]([[S.sub.r])}.sub.r[epsilon][1,2]] on consumption (in this economy, there is aggregate risk and since the intertemporal government budget constraint must ultimately balance, the tax rate is also state dependent and inversely related to the performance of the equities).

Each consumer h maximizes two-period expected utility

[max.sub.[c.sup.h]] {[U.sup.h]([[c.sup.h].sub.0] + [pi][beta][U.sup.h]([[c.sup.h].sub.1]([s.sub.1])) + (1 - [pi])[beta][U.sup.h]([[c.sup.h].sub.1]([s.sub.2]))} [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] h [epsilon] H (1)

subject to the budget constraints

[[c.sup.h].sub.0] = q([[k.sup.h].sub.0] - [[k.sup.h].sub.1]) - [[pb.sup.h].sub.1] + [[k.sup.h].sub.0] / 1 + [[tau].sub.0] [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] h [epsilon] H (2)

[[c.sup.h].sub.1]([s.sub.r]) = [[b.sup.h].sub.1] + [[k.sup.h].sub.1]d([s.sub.r])/ 1 + [[tau].sub.1]([s.sub.r]) r [epsilon] {1,2} [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] h [epsilon] H. (3)

The government's budget constraints are

G = [[tau].sub.0] [[sigma].sub.h[epsilon]H] [[c.sup.h].sub.0] + p [[sigma].sub.h[epsilon]H] [[b.sup.h].sub.1] (4)

G = [[sigma].sub.h[epsilon]H] [[b.sup.h].sub.1] = [[tau].sub.1]([s.sub.r]) [[sigma].sub.h[epsilon]H] [[c.sup.h].sub.1]([s.sub.r]) r [epsilon] {1,2}. (5)

Finally, markets clear for each period and every state of the world:

[[sigma].sub.h[epsilon]H] [[c.sup.h].sub.0] + G = [[sigma].sub.h[epsilon]H] [[k.sup.h].sub.0] (6)

[[sigma].sub.h[epsilon]H] [[c.sup.h].sub.1]([s.sub.r]) + G = d([s.sub.r]) + G = d([s.sub.r]) [[sigma].sub.h[epsilon]H] [[k.sup.h].sub.1] r [epsilon] {1,2} (7)

[[sigma].sub.h[epsilon]H] [[k.sup.h].sub.1] = [[sigma].sub.h[epsilon]H] [[k.sup.h].sub.0] (8)

Because we prefer to remain in the realm of a two-state economy, only two linearly independent assets are necessary to ensure the completeness of markets. The terms bonds and equities are metaphors, and the variables [b.sub.1] and [k.sub.1] could easily be interpreted to include cash and savings accounts in the case of the former and a range of high-risk investments in the case of the latter.

PROPOSITION 1. If a fixed level of government expenditure is financed at least in part by debt, then in a two-period endowment economy with perfect capital markets, the prices of both assets decrease with the size of the deficit.

PROOF. See Appendix A.

Equations 1-8 are a model of an economy with flat-rate taxation and perfect capital markets. This economy has quasi-Ricardian neutrality: consumption remains invariant to fiscal policy but interest rates do not. As the government shifts the tax burden from the present to the future through increases in the debt, consumers will use their extra disposable income to buy extra assets to finance the new burden. In contrast to the behavior of models with lump-sum taxation, this increase in debt will lead to a decrease in the price of assets. Thus with flat-rate taxation, higher deficits in an intertemporal Walrasian economy will generate increases in interest rates reminiscent of the predictions of Keynsian models.

The aggregate quantity of equities is normalized to one, and if government spending is greater than tax revenues, the aggregate amount of debt at the end of the initial period will also be positive. This aggregate condition does not restrict the equilibrium values of individual asset holdings [[b.sup.h].sub.1] and [[k.sup.h].sub.1], and this implies that an individual with a high tolerance for risk can short-sell any amount of bonds to leverage his portfolio without any restriction.

For reasons of moral hazard and adverse selection, private individuals are not free to borrow without limit, even for investment purposes, and certainly not at the low rate of interest available to the government. Although buying securities with money borrowed from a broker is possible, the rate of interest charged is well above that of treasury bills, and margin requirements limit the quantity borrowed at even this relatively low interest rate. The implication is that the budget hyperplane described by Equations 2 and 3 should be replaced by a set of kinked line segments.

Few investors maintain margin accounts. Instead, I simplify the analysis by assuming that all individuals must maintain a minimum stock of each asset:

[[b.sup.h].sub.1] [greater than or equal to] [b.sup.-h], [[k.sup.h].sub.1] [greater than or equal to] [k.sup.-h] h [epsilon] H (9)

Setting [b.sup.-h] = 0 and [k.sup.-h] = 0, these become simple nonnegativity constraints that prevent agents from leveraging their portfolios by buying and holding negative quantities of bonds or hedging against risk by holding negative quantities of equities. The result that these constraints will bind for some agents and not for others, is an outcome of the heterogeneity of risk aversion generated by the type of preferences we will assume.

I can more easily analyze the model by rearranging the first-order conditions of the individuals' maximization problem in Equations 1 to 3 as an intertemporal Euler condition

E[[beta][U.sup.h]'([[c.sup.h].sub.1](s))/(1 + [[tau].sub.1](s))[U.sup.h]'([[c.sup.h].sub.0])] = p/1 + [[tau].sub.0] (10)

and an intratemporal Euler condition

(1 - [pi])[U.sup.[h.sup.t]]([[c.sup.h].sub.1]([s.sub.2]))/[pi][U.sup.[h.su p.t]]([[c.sup.h].sub.1]([s.sub.1)) = (1 + [[tau].sub.1]([s.sub.2]))(pd([s.sub.1]) - q)/(1 + [[tau].sub.1]([s.sub.2]))(pd([s.sub.2]) - q) (11)

The left-hand side of the intratemporal condition of Equation 11 equals [[delta][[c.sup.h].sub.1]([s.sub.1])]/[[delta][[C.sup.h].sub.1]([s.su b.2])]/U=U, the slope of the indifference curve in the period 1 state space (see Figure 1). The right-hand side represents the intratemporal budget constraint. For all individuals whose nonnegativity constraints are slack, both of these Euler equations remain valid. On the other hand, the condition in Equation 11 will no longer be an equality for a given individual if one of this person's nonnegativity constraints is binding. Those people for whom the left-hand side of Equation 11 is greater (in absolute value) than the right-hand side (in absolute value) buy no bonds, consume at the lower right comer of their budget constraints, and would choose to borrow at the prevailing rate of interest and leverage their stock portfolios if that were possible. Conversely, violation of the equality in the other direction implies that a person chooses to allocate period 1 consumption on the upper left-hand comer of the budget constraint. These people would move to a higher level of utility if they were allowed to move further toward the 45[degrees] certain consumption ray. This means hedging against future risk (taxes are still uncertain) by holding a negative amount of the equity.

The existence of quantity constraints on assets has profound implications for the effects of government policy. Suppose the constraint [[b.sup.j].sub.1] [equal to or greater than] 0 is binding for person j. In the absence of the constraint, this person will choose to borrow at the same interest rate as the government to buy more equities. Imposition of the constraint makes this impossible. If the government postpones taxes to the next period through deficit finance of present expenditure, the present value of the tax burden for j may not necessarily change very much. However, postponement of taxation means that more equities can now be purchased in the initial period, slackening the borrowing constraint. In the next period, everyone faces higher taxes to repay bond holders both their principal and interest. Nonetheless, the interest on these bonds is less than the expected return on the equities, implying an ex ante welfare gain for j. Deficit finance becomes a low-interest loan for those interested in leve raging their portfolio with riskier, high-return assets (Figure 2). In the absence of changes in equilibrium asset prices, j will prefer higher deficits, at least until the constraint is no longer binding. Of course in general, equilibrium policy changes do affect asset returns, and potential benefits derived from slackening of the nonnegativity constraints may be overwhelmed by these changes.

In this model, I assume that individuals differ in their choices of assets because they differ in their taste for risk. To model heterogeneous risk aversion, I could assume that different individuals have different utility functions, each implying a different aversion to risk. As an alternative, I will assume that all share the same utility function--one that produces different rates of relative risk aversion at different levels of consumption. From the large set of possible utility functions that share this property, the extended power utility function was chosen because it allows the model to be solved numerically without the need to resort to approximations and because of its similarity to more familiar functional forms.

The extended power utility function is defined as U(c) = [([delta] + [rho]c)].sup.1-(1-1/[rho])]/([rho] - 1), [rho] [greater than] 0, [delta] [not equal to] 0, c [greater than] max [-[delta]/[rho], 0] and can be thought of as a generalization of the better known Stone-Geary utility function. The rate of relative risk aversion is [R.sub.R](c) c/([delta] + [rho]c). If [delta] [greater than] 0, the rate of relative risk aversion increases with consumption, if [delta] [less than] 0, the rate decreases with consumption, and if [delta] = 0 we have the familiar constant relative risk aversion utility function with the Arrow-Pratt rate of relative risk aversion equal to lip. [2] Increasing the size of the absolute values of either parameter will lower the level of relative risk aversion at all levels of consumption. The term -[delta]/[rho] can be interpreted as a consumption floor--people will forego high-return investments if bad outcome means they may be pushed close to this low-level of subsistence.

In this paper, I will assume that the rate of relative risk aversion is decreasing in consumption, a case to which Stiglitz (1969a) paid special attention. Data from the Survey of Consumer Finances, Federal Reserve (1983), clearly demonstrate that people who invest in stocks have both higher levels of wealth and income than people who either have no financial assets or invest only in low-risk bonds and savings accounts (Tables 1, 2)) [3]

The income expansion path of the extended power utility function is generally nonlinear in the state space. The demand functions it generates are nonaggregable, there is generally no representative consumer in this model even if all preferences are identical, and therefore, there is no possibility of achieving a closed-form solution for equilibrium prices, consumption, and portfolio decisions for all the agents. Instead, the model is simulated numerically, and the decision variables and prices are calculated for different levels of the initial rate of taxation. [4]

Government expenditure G is fixed at 0.2 or 20% of the initial resources in the economy. I set the subjective discount rate [beta] = 0.95, and the economy is assumed to have a total initial capital of [k.sub.0] = 1, paying a dividend of [d.sub.0] = 1 in the initial period. I will assume that the same capital will produce a dividend of d([s.sub.1]) = 0.99063 or d([s.sub.2]) = 1.05345, each with the probability [pi] = 0.5. The simulation thus corresponds to an economy whose aggregate consumption is growing at an annual rate of 2.204% with a standard deviation of 3.141%, as was the case for the United States between 1953 and 1985 (Mehra and Prescott 1985).

Because differences in risk aversion are driven by differences in consumption, I match the model to some distribution of resources in the economy. Data on the distribution of net worth of households by monthly income quintiles is obtained from the Survey of Income and Program Participation (SIPP) for 1991 (Eller 1994, p. viii, table B). The population is divided into income quintiles whose initial endowments of wealth are {[k.sub.0](1), [k.sub.0](2), [k.sub.0](3), [k.sub.0](4), [k.sub.0](5)} {0.6, 2, 3, 5, 12.4}. These numbers are obtained by rounding up the median net worth of each quintile as reported in the SIPP to the nearest $1000 and then dividing by 10,000 (Eller 1994, p. xii, table F).

Setting [delta] = -0.3 and p = 0.7 means that the consumption floor will be just under three-quarters of the initial endowment of the poorest quintile and a third of the official poverty threshold for a family of four ($12,812 in 1991). As consumption is increasing, the rate of relative risk aversion is decreasing, reaching a minimum of 1.42857, close to the lower bound of empirical estimates.

In Figure 3a, I plot the equilibrium bond holdings as a fraction of total savings for the case when the constraints (Eqn. 9) are not imposed (equity holdings/total savings is simply one minus this number). When the budget is balanced, the government supplies no bonds to the market, and therefore in this closed economy, the aggregate amount of bonds is zero. Nonetheless, in this Walrasian economy, the members of the two richest quintiles are free to issue private bonds that will be bought by the members of the other three quintiles. With the increase in the government's deficit, the aggregate supply of bonds increases, and the members of all the quintiles will increase their net holdings of these assets to offset the increase in the future tax burden. The poorest quintile will choose to hold negative amounts of the equity for every level of deficit.

Because the aggregate supply of equities is invariant to policy changes, equilibrium holdings of equities are less sensitive to the size of the deficit. Nonetheless, the members of the two richest quintiles will increase their equity holdings with the size of the deficit, while the others will lower theirs.

In Figure 3b, I plot the equilibrium asset holdings for the case when the constraints (Eqn. 9) are imposed. The baseline case of zero deficits implies that taxes in the initial period and for the two states in the following period will be {[[tau].sub.0],[[tau].sub.1]([S.sub.1),[[tau].sub.1][S.sub.2]} = {0.25, 0.25296, 0.23434}. As initial taxes are lowered, the members of the first quintile will lower their holdings of equities and replace them with some of the extra bonds the government issues to compensate for the revenue shortfall. When the deficit is higher than 2.357% of GDP, the first quintile will be constrained from holding negative quantities of equities. Although they will continue to increase their bond holdings with the increase in the debt, this will occur at a slower pace as the substitution of equities for bonds becomes impossible. All other quintiles will choose to always hold some positive level of equities for all fiscal policies. On the other hand, they will all have binding constraints on their bond holdings for some level of the deficit. Naturally, the wealthiest, who are also the least risk averse, will want to leverage their portfolios the most. Only if the deficit is higher than 8.2% of GDP will they choose to hold bonds in positive quantities. For the second, third, and fourth quintiles, the relevant deficits will be 2.356, 3.4, and 5.22% of GDP. Additionally, between 2.356 and 2.357%, there is a region of multiple equilibria, and this generates the discontinuities in Figure 4 (see Appendix B and Figure 7 for details).

As in the unconstrained case, expected after tax returns for both assets will increase. For relatively low levels of debt, where [b.sup.1] [greater than] 0, [b.sup.i] = 0, i [epsilon] {2, 3, 4, 5}, [K.sup.j] [greater than] 0, j [epsilon] {1, 2, 3, 4, 5}, the return on bonds rises steeply. In this region, where nearly everyone's bond holdings are zero, the risk premium will be relatively high (compared to the risk premium generated by most macroeconomic models), even though only a small number of people, all of whom are poor, are very risk averse.

3. Welfare Calculations

A simple way to look at the welfare implications of different levels of deficits would be to measure how changes in fiscal policy affect tax burdens. Although the tax itself is imposed at a flat rate, consumption allocations will shift with changes in fiscal policy, which implies a redistribution in the overall intertemporal tax burden. We measure this by substituting Equation 5 into Equation 4 and dividing the expected present value of taxes paid by the initial endowment:

[T.sup.h] = [[tau].sub.0][[C.sup.h].sub.0] + pE[[[tau].sub.1](s)[[C.sup.h].sub.1](s)]/(1 + p)[[k.sup.h].sub.0] h [epsilon] H (12)

Even though the tax is imposed at a flat rate, the burden does shift with policy. Because the more risk averse are also more interested in smoothing consumption over time and because the economy is growing, higher taxes in the initial period will increase the tax burden for the first three quintiles (Figure 5). Therefore in terms of taxes paid, deficit spending can be characterized as mostly progressive in nature. Calculating tax burdens is a common way of measuring the welfare effects of policy. However, in an economy in which prices are affected and constraints slackened by different policies, we may not be able to rely on tax burdens to capture the true incidence of policy. In fact, I will demonstrate that this simple measure of the tax burden captures only a small part of the real welfare implications of changing the timing of taxation and can easily be overwhelmed by the price effects.

I calculate numerically the utility changes as equivalent variations from the baseline point of a balanced budget. This means calculating utility for each type when the budget is balanced and then calculating the size of the initial endowment necessary to keep individuals at this level while allowing them to choose new portfolio allocations at the new tax rates (yet still imposing the nonnegativity constraints). I then solve for different values of [[k.sup.h].sub.0], [[b.sup.h].sub.1], and [[k.sup.h].sub.1] using the first-order conditions of Equations 13 and 14 and the utility (Eqn. 15) for a range of different tax policies and their corresponding asset prices:

pU'([[C.sup.h].sub.0]/1 + [[tau].sub.0] + [beta]E[U'([[C.sup.h].sub.1](s))/1 + [[tau].sub.1(s)] = 0 (13)

qU'([[C.sup.h].sub.0]/1 + [[tau].sub.0] + [beta]E[d(s)U'([[C.sup.h].sub.1](s))/1 + [[tau].sub.1](s)] = 0 (14)

U([[C.sup.h].sub.0]) + [beta]E[U([[C.sup.h].sub.1](s))] - [U.sup.h] = 0 (15)

where [[C.sup.h].sub.0] = [q([k.sub.0] - [k.sub.1] - [pb.sup.1] + [d.sub.0][k.sub.0]/(1 + [[tau].sub.0]) and [[C.sup.h].sub.1](s) = [[b.sub.1] + [k.sub.1]d(s)]/[1 + [[tau].sub.1](s)].

I plot the equivalent variation in percentage terms as ([k.sub.0] - [k.sub.0])/[k.sub.0] 100 for each individual in Figure 5. This represents the amount that each individual would be willing to pay, as a percentage of initial endowment, to be moved from a policy of balanced budgets to different levels of budget deficit. The larger the term, the greater the increase in utility the policy implies when compared to the level that obtains when the budget is balanced. Because in general equilibrium the government's budget constraint must be satisfied, it makes no sense to vary the three tax rates independently.

The welfare implications of different fiscal policies are several orders of magnitude greater, and often of opposite sign when measured as equivalent variations rather than as tax burdens. Aside from the first quintile, the equivalent variations follow a similar pattern--welfare increases with the size of the deficit until it reaches an optimum between 1.3 and 1.4% of GDP, after which it declines back to zero. For the second quintile, there is a second region in which welfare is increasing. For quintiles two to five, the predicted size of the welfare changes are small, never more than 0.004% of initial wealth. The welfare calculations for the members of the first quintile follow a very different pattern because their asset allocations are usually very different. Welfare increases monotonically and quickly as the deficit rises until its constraint on short sales of equities is binding, after which it rises and then falls gently. The size of the welfare increase is also much larger than for the other quintiles .

The size of the welfare gains in Figure 5 are small, at least in part because of the conservative assumptions that were made about the population's aversion to risk. Decreasing the size of the parameters [rho] and [delta] proportionally will yield a model with higher relative risk aversion in the limit, and although the pattern of portfolio choices will remain very similar, potential welfare gains will be higher. Also, if we impose a larger lower bound on [b.sub.1], the magnitude of the welfare effects will also increase (along with changes in equilibrium allocations). [5] More importantly, the tax changes are only for a year, sustained deficits over many years like those in the United States during the 1980s will again increase the size of the equivalent variations.

The postponement of taxation to the future slackens the nonnegativity constraints on both assets--balanced budgets in this economy are not Pareto optimal. At the same time, relative returns of the assets are also changing. Thus, increases in the the deficit both lengthen the budget constraint in period 1 state space and also make these constraints steeper. This change in the slope of the constraint is what makes the welfare predictions hard to predict in advance.

According to these results, in general equilibrium, those whose constraints are slackened by higher deficits, the less risk averse, will not always advocate higher debt. At some point, the additional benefit they receive from being able to buy more equities when taxes are postponed is offset by the decrease in the equity premium. For members of the first quintile, the constraint on equities becomes more binding with increases in the deficit. Nonetheless, their welfare will generally improve with the size of the deficit as the equity premium drops. The fact that the equivalent variation is always positive implies that a balanced budget is Pareto dominated by deficit spending. For the top four-fifths of the population, the optimal deficit will be between 1.3% of aggregate income for members of the second quintile and 1.37% for members of the richest, fifth quintile.

4. Changes in the Distribution of Income

How sensitive are these results to the distribution of wealth that I have assumed for this economy? Using the very aggregated data from the SIPP, the economy has the top 20% of the population owning 53.9% of total capital while the bottom quintile owns 2.6%. This corresponds to a Gini coefficient of 0.463. Suppose wealth were more evenly distributed. The model is altered so that only 8% have initial wealth of 0.6 or 12.4, 16% have initial wealth of 2 or 5, and 52% have wealth of 3. This more egalitarian economy will have a Gini coefficient of 0.257 and the equivalent variations are plotted on the left side of Figure 6. The poorest group, now only 8% of the population, will prefer a deficit of 5.76% of total income. Similarly, the other groups will also prefer smaller deficits--around 0.64% of GDP.

To simulate an economy with greater inequality, I reconfigure the size of the wealth groups so that 37% of the population has endowments of 0.6 and another 37% endowments of 12.4, with the remaining 26% evenly divided between 2, 3, and 5. The Gini coefficient will be 0.63, and the level of debt preferred by each income group will increase, as in the right side of Figure 6. The poorest group will prefer a debt of 5.53%, while the others will favor deficits that range from 1.97% of aggregate income to 2.06%. As before, the political equilibrium will most likely be determined by the members of the fourth quintile. [6] These individuals will prefer a policy that sets the deficit at 2.03 8% of GDP. If we ignore high deficit levels where the equivalent variations are not single peaked, we find that this model predicts a positive correlation between the equilibrium level size of deficits and the amount of economic inequality.

5. Political Equilibria

Figure 5 displays the welfare improvements that result from various levels of deficit finance when the distribution of wealth matches data from the SIPP. The members of the first, fourth, and fifth quintiles have preferences that are single-peaked over the entire range of policies that are feasible. The remaining two quintiles, the second and third, do not have single-peaked preferences and so application of the median voter theorem is not possible unless we restrict our attention to relatively modest levels of deficit finance. Nonetheless, if we rule out very high deficits the theorem is potentially useful--within the region of 0-3% of GDP, all preferences are single-peaked and a political equilibrium can be predicted.

The utility of the first quintile is monotonically increasing in the size of the deficit throughout this region. For the remaining people, the size of the welfare-optimizing deficit is increasing in wealth (Table 3); for example, the members of the fifth quintile would prefer slightly higher deficits than would be chosen by members of the second, third, and fourth quintiles. Therefore, if I rule out very high levels of deficit finance, the baseline model predicts that the members of the second and third quintile will find common cause in favor of keeping deficits no higher than 1.305% of GDP against both the richest and poorest members of society who will prefer deficits at least 0.06% higher. The median voters in this economy will be not the members of the third, median quintile, but rather the relatively well-off people in the fourth quintile.

The relatively large welfare gain that accrues to the poor from a big postponement in taxation certainly corresponds nicely with a common perception that parties that represent their interests are more likely to tolerate higher deficits and to be less committed to the virtues of balanced budgets. On the other hand, the common prejudice that the rich are likely to be the most committed to the most conservative fiscal policies is not demonstrated here.

Another result in Table 3 is that as inequality rises, the welfare-optimizing deficit is decreasing for the poorest portion of the population but increasing for everyone else. Therefore, as inequality rises, the fiscal policies that maximize welfare for both the richest and poorest converge. If the distribution of wealth is made more equal (left-hand side of Figure 6), the members of the second, third, fourth, and fifth wealth groups will support deficits between 0.0648 and 0.068% of GDP. More importantly, members of the third group now constitute a majority on their own, and it is their choice of 0.066% of GDP that is likely to prevail.

Suppose the distribution of wealth is made less equal by increasing the size of the population at either extreme of the distribution (right-hand side of Figure 6). On one side are the members of the middle second, third, and fourth groups, each favoring deficits between 1.968 and 2.04% of GDP and together constituting 26% of the population. By contrast, the members of the poorest group favor a deficit of 5.526% of GDP and constitute 37% of the population.

Mediating between these two factions will be the members of the richest group, also 37% of the population. In this case it will be the richest, whose members favor setting the deficit to 2.056% of GDP whose welfare-optimizing choice will prevail.

6. Conclusion

Much of the literature in the field of public finance has focused on comparing the incidence properties of different types of taxes. Certainly, shifting the burden of financing government between different tax regimes--between taxation of corporate profits and taxation of labor income, for example--will typically have a far more profound welfare effect, both direct and indirect, than those that were demonstrated here. Nonetheless, this paper has demonstrated that shifts in the timing of a given tax, even the simple type of taxes modeled here, will imply changes to general equilibrium prices--changes that generate differential welfare effects when capital markets are imperfect and the population is heterogeneous.

Postponing taxes permits people with the greatest wealth to leverage their asset portfolios. However, beyond a relatively small amount of deficit finance, this benefit is quickly overwhelmed by the general equilibrium effects that lower the relative rate of return of the risky assets that these people are purchasing. By contrast, until the deficit is relatively high, the risk-averse bond-holders benefit when taxes are postponed. Although we can reject balanced budgets or very small deficits as Pareto inefficient, increases in deficit finance beyond a certain point will imply a redistribution of welfare along a Pareto frontier. Furthermore, the greater the inequality of initial wealth, the higher the government's deficit must be for the policy to still be efficient.

This paper also demonstrates that the optimal fiscal policy for different agents in the economy may not be a monotonic function of initial wealth. In the baseline model the members of the poorest and richest groups are likely to be paired together in a coalition that favors higher deficits. The members of the median wealth group will not be the median voters. Instead, it is the policy preference of the members of the next to wealthiest group that is likely to be implemented. Furthermore, the initial distribution of wealth will affect which policy is ultimately chosen. The model predicts that in an economy with less equal distribution of wealth, the deficit is likely to be larger.

Unlike big shifts in the tax code, shifts in the timing of taxation must be addressed by policy makers whenever a new budget is considered. In a standard representative agents model without market imperfections, there will typically exist an optimal policy that minimizes excess burden. If agents' utility function is time-additive separable, minimizing the excess burden can typically be accomplished by equalizing tax rates over time. This model demonstrates that if there is no representative agent because preferences are nonhomothetic and agents are heterogeneous in the size of their initial endowments, optimal fiscal policies will be replaced by a Pareto frontier. Perhaps it is not surprising that observed fiscal policies are hard to justify as socially optimal in the context of a standard identical agents framework. Political parties, once in power, may have good reason to increase or decrease the amount of deficit finance they choose in accordance with the interests of their supporters.

(*.) Department of Economics, University of Houston, Houston, TX 77204-5882, USA; E-mail [e-mail].

I thank Andrew Austin, Gary Becker, Steve Craig, Michael Loewy, Michael Palumbo, David Papell, Raaj Sah, Michael Woodford, and two anonymous referees for their comments and suggestions. I have benefited greatly from comments made by participants at various workshops at the University of Chicago as well as the participants of seminars at the Federal Reserve Bank of New York, The University of Colorado--Boulder, The Hebrew University of Jerusalem, The University of Houston, The University of Maryland--Baltimore County, Tel Aviv University, Texas A&M University, and The Virginia Polytechnic Institute and State University. I am grateful for financial support from the Henry J. Morgenthau Jr. Memorial Fund, the Bradley Foundation, and the University of Chicago. All errors are my own.

Received October 1998; accepted June 1999.

(1.) Yotsuzuka (1987) developed a two-period, two-agent open economy to demonstrate that if the market imperfection results from adverse selection and private lenders are not allowed to pool information, the equilibrium level of lending will be suboptimal in the absence of deficit finance.

(2.) Arrow (1965) argues that absolute risk aversion should decrease with wealth, whereas relative risk aversion should increase with wealth, with both hovering around the value of one. This, however, need not be true if limited to finite levels of wealth (Stiglitz 1969b).

(3.) Stiglitz cites as evidence Lampman's (1962) study of estate tax returns. Lampman's work gave a detailed, statistically significant portrait of the wealthiest Americans. The data set was collected from federal estate tax returns, which in 1953 were only required for estates larger than $60,000. The decendents reporting were therefore within the upper 2% of all decendents for that year. His findings demonstrate a direct relationship between the size of an estate and the portion that was held in stock and an inverse relationship between estate size and the portion of the estate left in cash (Table 2).

(4.) The simulations use the Gauss-Newton solution method to find, for a given [[tau].sub.0], solutions to the system of nonlinear equations that solve for the equilibrium values: {[{[[b.sup.h].sub.1]}.sup.[5.sub.h=1]], [{[[k.sup.h].sub.1]}.sup.[5.sub.h=1], [P.sub.0], [Q.sub.0], [[tau].sub.1]([S.sub.1]), [[tau].sub.1]([S.sub.2])}.

(5.) This would be analogous to a cash-in-advance constraint, and I could consider part of the bonds printed by the government to be seignorage rather than debt.

(6.) Because the equivalent variations are only single-peaked for a limited range of policies, I can only be certain of the existence of an equilibrium if I rule out higher level of deficit.

References

Arrow, Kenneth J. 1965. Aspects of the theory of risk-bearing. Helsinki: Academic Book Store.

Auerbach, Alan J., Jagdeesh Gokhale, and Laurence J. Kotlikoff. 1991. Generational accounts: A meaningful alternative to deficit accounting. In Tax policy and the economy 5, edited by David Bradford. Cambridge, MA: MIT Press, pp. 55-110.

Barro, Robert J. 1974. Are government bonds net wealth? Journal of Political Economy 82:1095-117.

Diamond, Peter A. 1965. National debt and neo-classical economic growth. American Economic Review 55:1125-50.

Domar, Evsey D., and Richard A. Musgrave. 1944. Proportional income taxation and risk-taking. Quarterly Journal of Economics 56:388-422.

Eller, T. J. 1994. Household wealth and asset ownership: 1991. Current population reports P70-34. Washington, DC: U.S. Department of Commerce, Bureau of the Census.

The Federal Reserve, Board of Governors. 1983. Survey of Consumer Finances. Washington, DC: Federal Reserve.

Fullerton, Don, and Diane Lim Rogers. 1993. Who bears the lifetime tax burden? Washington, DC: Brookings Institution.

Kasten, Richard, Frank Sammartino, and Eric Toder. 1994. Trends in federal tax progressivity, 1980-1993. In Tax progressivity and income inequality, edited by Joel Slemrod. New York: Cambridge University Press.

Lampman, Robert J. 1962. The share of top wealth-holders in national wealth. Princeton, NJ: Princeton University Press.

Mehra, Rajnish, and Edward C. Prescott. 1985. The equity premium: A puzzle. Journal of Monetary Economics 15:145-61.

Mossin, Jan. 1968. Taxation and risk-taking: An expected utility approach. Economica 35:74-82.

Stiglisz, Joseph E. 1969a. The effects of income, wealth, and capital gains taxation on risk-taking. Quarterly Journal of Economics 83:262-83.

Stiglitz, Joseph E. 1969b. Review of: Aspects of the Theory of Risk Bearing--Yrjo Jahnsson Lectures (Kenneth J. Arrow). Econometrica 37:742-3.

Woodford, Michael. 1990. Public debt as private liquidity. American Economic Review 80:382-8.

Yotsuzuka, Toshiki. 1987. Ricardian equivalence in the presence of capital market imperfections. Journal of Monetary Economics 20:411-36.

               The 1983 Survey of 4437 Households Represents
                      a Weighted Sample of 81,902,288
                          Households in the U.S.
                         Portfolio Behavior of
                           U.S. Households
                             No Financial      Low-Risk Asset
                                Assets          Holdings Only  Stockholders
Net Worth                       $17,357            $81,507     $340,802
Income                          $11,852            $26,593      $47,528
Percentage of Population             23.8               57.9         18.3
Observations                        688               2039         1710
              Percentage of Gross Estates in Selected Assets
                  and Liabilities for Three Gross Estate
               Sizes in Three Age Groups of Male Top Wealth-
                               Holders, 1953
                                    Gross Estate Size (X $1000)
Age Group                                      60-70            150-200
Percentage in U.S. Government Bonds
 30-40                                          1.1               5.6
 55-60                                          7.8               5.6
 75-80                                          8.3               7.8
Percentage in Stock
 30-40                                          8.4              17.3
 55-60                                         13.9              27.7
 75-80                                         18.2              33.0
Age Group                           500-1000
Percentage in U.S. Government Bonds
 30-40                                1.4
 55-60                                6.0
 75-80                                6.0
Percentage in Stock
 30-40                               73.6
 55-60                               46.9
 75-80                               53.2
Source: Lampman (1962, pp. 179-81, table 82).
            The Distribution of Wealth and Political Equilibria
                               Wealth Groups
                                     1           2         3
Gini Coefficient = 0.257
  Optimizing Deficit/GDP          0.10444     0.00648    0.0066 [a]
  Welfare Gain (%)                0.28805     0.0018676  0.001949
  Percentage of the Population    8          16         52
Gini Coefficient = 0.463
  Optimizing Deficit/GDP            0.076     0.013048   0.01332
  Welfare Gain (%)                0.29425     0.003803   0.00398
  Percentage of the Population   20          20         20
Gini Coefficient = 0.63
  Optimizing Deficit/GDP          0.05526     0.01968    0.02008
  Welfare Gain (%)                0.29933     0.005785   0.0060564
  Percentage of the Population   37           8.67       8.67
                                  4             5
Gini Coefficient = 0.257
  Optimizing Deficit/GDP        0.00672       0.0068
  Welfare Gain (%)              0.001998      0.0020225
  Percentage of the Population 16            8
Gini Coefficient = 0.463
  Optimizing Deficit/GDP        0.013504 [a]  0.013648
  Welfare Gain (%)              0.004089      0.00416
  Percentage of the Population 20            20
Gini Coefficient = 0.63
  Optimizing Deficit/GDP        0.0204        0.02056 [a]
  Welfare Gain (%)              0.006235      0.006468
  Percentage of the Population  8.67         37
(a.)The political equilibrium choice of Deficit/GDP.

Appendix A: Proof of Propsoition 1

Combining Equation 2 with Equation 3 and using Equations 4 and 5 to solve for the state r tax rate [[tau].sub.1]([S.sub.r]), I can restate the consumers' problem as maximizing Equation I with respect to a single intertemporal budget constraint:

[v.sub.0][[C.sup.h].sub.0] + [[[sigma].sup.2].sub.r=1] [v.sub.1]([S.sub.r])[[C.sup.h].sub.1]([S.sub.r]) = [[K.sup.h].sub.0] (16)

Equilibrium will be the consumption allocation and Arrow-Debreu prices of contingent claims and [v.sub.0], [v.sub.1]([S.sub.1]), and [v.sub.1]([S.sub.2]) that solve this maximization and satisfy the market-clearing conditions of Equations 6, 7, and 8. Similar to the debt neutrality results in Barro (1974), for a given level of government spending, these equilibria can be defined without reference to fiscal policy (i.e., the choice of [[tau].sub.0]). Furthermore, for the generic set of initial endowments that correspond to regular economies, the number of possible equilibria is finite. Nonetheless, each local equilibrium will correspond to a possibly infinite set of fiscal policies that uniquely determine asset prices p and q and asset allocations.

Because the equilibria of this economy are at least locally unique, I can fix contingent claims prices [v.sub.0], [v.sub.l]([s.sub.r]) and consumption allocations and test how changes in fiscal policy will affect p and q

[v.sub.0] = 1 + [[tau].sub.0]/q + [d.sub.0] [right arrow] q = 1 + [[tau].sub.0]/[v.sub.0] - [d.sub.0] [right arrow] [delta]q/[delta][[tau].sub.0] [greater than] 0

[v.sub.1]([s.sub.r]) = (pd([s.sub.r]) + (G - [[tau].sub.0](1 - G)))(p[v.sub.0]d([s.sub.t]) + ([v.sub.0] - (1 + [[tau].sub.0])))/(1 + [[tau].sub.0])(d([s.sub.r]) - G)(d([s.sub.t]) - d([s.sub.r]))p

where t [not equal to] r represents two different states in the second period.

Employing the implicit function theorem:

[delta]p/[delta][[tau].sub.0] = p(1 + q)[v.sub.0]((pd([s.sub.r]) + (G - [[tau].sub.0](1 - G)))(pd([s.sub.t]) + 1) + (1 + [[tau].sub.0])(1 - G)(pd([s.sub.t]) - q))/p(1 + [[[tau].sub.0].sup.2](q(G - [[tau].sub.0](1 - G)) + [p.sup.2]d([s.sub.t])d([s.sub.r]))

If the budget is in deficit, G - [[tau].sub.0](1 - G) [greater than] 0.

If assets are held in positive amounts by anyone, pd([s.sub.t]) - q [greater than] 0 and the dividends are different in the two states d([s.sub.t]) [greater than] d([s.sub.r]), t [not equal to] r [right arrow] [delta]p/[delta][[tau].sub.0] [greater than] 0. Therefore, the prices of bonds and equities are increasing with the rate of initial taxation and inversely related to deficit finance of a given level of spending. QED.

Appendix B: The Region of Multiple Equilibria

In the small region between 2.356 and 2.357%, the constraint on short sales of bonds is binding for the three richest quintiles (Figure 7). Also, at least one additional constraint on either the equity holdings of the poorest or the bond holdings of the second quintile will be binding. Thus three equilibria will exist for each level of deficit: one for the case of [b.sup.1] [greater than] 0, i [epsilon] {1, 2}, [b.sup.i] = 0, i [epsilon] {3, 4, 5}, [k.sup.1] = 0, [k.sup.j] [greater than] 0, j [epsilon] {2, 3, 4, 5}; one for the case of [b.sup.1] [greater than] 0, [b.sup.i] = 0, i [epsilon] {2, 3, 4, 5}, [k.sup.j] [greater than] 0, j [epsilon] {1, 2, 3, 4, 5}; and one for the unstable case of [b.sup.1] [greater than] 0, [b.sup.i] = 0, i [epsilon] {2, 3, 4, 5}, [k.sup.1] = 0, [k.sup.j] [greater than] 0, j [epsilon] {2, 3, 4, 5}. In this region, asset prices will be highly volatile with respect to small changes in policy corresponding to the discontinuities in Figure 4.