Growth effects of shifting from a graduated-rate tax system to a flat tax.

Cassou, Steven P. ; Lansing, Kevin J.

I. INTRODUCTION

Recently, the debate concerning fundamental U.S. tax reform has led to a number of proposals that involve a shift toward a consumption-based system. (1) One such proposal is the so-called flat tax of Hall and Rabushka (1995). The flat tax would apply a single tax rate to all labor income above a given threshold and to all capital income after fully expensing investment expenditures. Hall and Rabushka (1995) argue that the adoption of their proposal would provide an enormous boost to the U.S. economy by dramatically improving incentives to engage in productive activities and would save taxpayers hundreds of billions of dollars in compliance and administration costs. (2)

In this article, we develop a quantitative general equilibrium model to assess the growth effects of adopting a flat tax plan similar to the one proposed by Hall and Rabushka (1995). The model captures many of the features of the current U.S. tax code, such as graduated personal tax rates, a standard personal deduction, separate tax rates applied to personal and business income, double taxation of business income, and differential tax treatment of physical and human capital. Under appropriate parameter settings, the model can exhibit either endogenous or exogenous long-run growth. Our choice of functional forms facilitates a closed-form solution to the model. This allows us to characterize explicitly the economy's transition path following the reform.

A central issue in the debate over fundamental tax reform is the effect that such a reform would have on economic growth. The present analysis builds on the work of Stokey and Rebelo (1995), who use an endogenous growth framework to identify the key model features and parameters that are important for determining the quantitative impacts of distortionary taxes on long-run growth. (3) Our study differs from theirs and the bulk of the dynamic tax literature in one fundamental respect. Here we evaluate the growth effects of shifting from a graduatedrate tax system to a flat-rate system. Stokey and Rebelo (1995) consider only flat-rate systems in which the marginal tax rate is equal to the average tax rate.

We approximate the graduated-rate tax system in the U.S. economy by an empirical tax rate function that allows the personal tax rate to depend positively on household taxable income. In equilibrium, household decisions are influenced by both the level and slope of the tax schedule. To implement the flat tax reform, we shift the parameters of the tax rate function to flatten the marginal tax schedule while maintaining revenue neutrality. Our methodology treats the Hall-Rabushka proposal as one that in effect moves a representative household from one tax rate schedule to another. We view this setup as a reasonable approximation to gauge the growth effects of tax reform at the macro level. A more elaborate setup would of course allow for household heterogeneity at a given point in time. In this regard, we note that Caucutt et al. (2003) have recently examined the growth effects of graduated-rate taxes in a two-period overlapping generations model that includes both skilled and unskilled agents. In their model, a flatter tax schedule increases the fraction of skilled agents in the economy. The share of total output devoted to education rises accordingly and the economy's long-run growth rate is observed to increase. The quantitative implications of their model are difficult to compare to ours, however, because their two-period framework implies a very long time horizon between household decisions (about 30 years). Moreover, their model abstracts from many of the features of the U.S. tax code that we include here.

In addition to examining the consequences of flattening the marginal tax schedule, we investigate the growth effects attributable to other parts of the flat tax proposal, such as allowing full investment expensing, eliminating the double taxation of business income, and increasing the standard personal deduction. Although a flat tax may result in a lower marginal tax rate for some categories of income, it generally requires a higher average tax rate to pay for the more generous expensing and deduction features. This mitigates the resulting growth and level benefits. We also consider the tax treatment of human capital investment, an element that is not specifically addressed by Hall and Rabushka (1995). Judd (1998) argues that the Hall-Rabushka plan does not treat physical and human capital symmetrically and therefore is not equivalent to a pure consumption tax. We investigate whether this feature is quantitatively important.

In the endogenous growth version of the model, we find that adopting a flat tax can permanently increase per capita growth by 0.009 to 0.143 percentage points per year, depending on the elasticity of household labor supply. The small growth effects that we obtain are due to the form of the human capital technology where untaxed forgone earnings represent the largest input to production. Through a series of experiments, we decompose the long-run growth effect into the parts attributable to the flattening of the marginal tax schedule, the full expensing of physical capital investment, and the elimination of double taxation of business income. We find that the most important growth-enhancing element is the flattening of the marginal tax schedule. The full expensing of physical capital investment contributes only a small amount to the overall growth gain. Finally, we show that the elimination of double taxation of business income can actually reduce long-run growth when combined with full investment expensing. This is because full expensing largely eliminates the distortion associated with collecting revenue from double taxation. Doing away with double taxation forces the government to replace this revenue using a higher postreform tax rate. A higher postreform tax rate is harmful for growth because it discourages labor effort and time devoted to human capital accumulation.

We find that the asymmetric treatment of physical and human capital under the Hall-Rabushka type flat tax has only minor growth consequences. This is because nondeductible investment accounts for only a small fraction of the total costs of producing human capital. The largest input to the production of human capital is forgone earnings, which, as previously noted by Boskin (1977), is already implicitly expensed.

In the exogenous growth version of the model, adopting a flat tax yields a temporary growth gain but a permanent level shift. The transition dynamics can be quite lengthy, however. Following the adoption of a flat tax, the economy takes about 100 years to return to its prereform growth rate. In the years immediately following the reform, the economy's output trajectory is quite similar to that of the endogenous growth model. In both versions of the model, shifting to a flat tax initially produces a growth slowdown due to the immediate reaction of hours worked and time in school to the higher postreform tax rate that is needed to maintain revenue neutrality. The slowdown is only temporary, however. After about six years, the additional capital accumulation induced by changes in the tax code allows the postreform output trajectory to overtake the prereform trend.

The remainder of the article is organized as follows. Section II describes the model and the tax system. The incentive effects of the tax system are examined in section III. Section IV describes our calibration procedure. Section V presents our quantitative results. Section VI concludes. An appendix describes the solution of the model.

II. THE MODEL

The model economy consists of households, firms, and the government. We allow for variable leisure, investment adjustment costs, and a realistic version of the U.S. tax code. Our choice of functional forms, inspired by the work of Hercowitz and Sampson (1991), permits a closed-form solution of the model. The solution allows us to explicitly characterize the economy's transition path following a tax reform.

Households and Firms

We model the representative household as choosing [{[c.sub.t], [l.sub.t], [e.sub.t], [i.sub.ht], [i.sub.ht], [k.sub.t+1], [h.sub.t+1], [[tau].sub.pt]}.sup.[infinity].sub.[t=0] to maximize

(1) [[infinity].summation over t=0] [[beta].sup.t] log[[c.sub.t] - V([h.sub.t],[l.sub.t])], [beta] [member of] (0,1),

subject to

(2) [c.sub.t] + [i.sub.kt] + [i.sub.ht] = [r.sub.t][k.sub.t] + [w.sub.t][h.sub.t]([l.sub.t] - [e.sub.t]) - [T.sub.t],

(3) [T.sub.t] = [[tau].sub.pt][[w.sub.t][h.sub.t]([l.sub.t] - [e.sub.t]) - [D.sub.t] + [eta](1 - [[tau].sub.b])([r.sub.t][k.sub.t] - [[phi].sub.k][i.sub.kt] - [[phi].sub.h][i.sub.ht])] + [[tau].sub.b]([r.sub.t][k.sub.t] - [[phi].sub.k][i.sub.kt] - [[phi].sub.h][i.sub.ht]), [eta] [member of] [0,1], [[phi].sub.i] [member of] [0,1], i = k,h,

(4) [[tau].sub.pt] = F[[w.sub.t][h.sub.t]([l.sub.t] - [e.sub.t]), [r.sub.t][k.sub.t],[i.sub.kt], [i.sub.ht]],

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation (1) represents lifetime utility where [beta] is the discount factor; [c.sub.t] is private consumption; [l.sub.t] is time devoted to nonleisure activities, that is, work or education; and [h.sub.t] is the household's stock of human capital or knowledge. The disutility of nonleisure time is governed by the functional form

(7) V([h.sub.t],[l.sub.t]) = B[h.sub.t][l.sup.[gamma].sub.t], B > 0, [gamma] > 0,

which implies that forgone leisure is adjusted for "quality," as measured by [h.sub.t], reminiscent of the models of Becker (1965) and Heckman (1976). Alternatively, we may interpret V([h.sub.t],[l.sub.t]) as the reduced form of a more elaborate specification that incorporates home production (see Greenwood et al. 1995, 161). Our specification for V([h.sub.t],[l.sub.t]) facilitates a closed-form solution to the household's decision problem where tax policy can affect the time allocation decision. As [gamma] [right arrow] [infinity], the model reduces to one with fixed time allocations. The intertemporal elasticity of substitution in labor supply is given by 1/([gamma]- 1).

Equation (2) is the within-period budget constraint where [i.sub.kt] and [i.sub.ht] represent expenditures devoted to the accumulation of physical capital [k.sub.t] and human capital [h.sub.t]. We interpret [i.sub.ht] as private-sector expenditures on education, training, and research and development (R&D), which all contribute to a broadly defined stock of knowledge [h.sub.t]. Given a total time endowment normalized to one, households allocate their time across three activities: they supply labor effort to firms in the amount [l.sub.t] - [e.sub.t], devote time to human capital formation (learning) in the amount [e.sub.t], and spend the remainder of their time 1 - [l.sub.t] in leisure.

Households receive a rental rate [r.sub.t] for each unit of physical capital used in production and earn a wage [w.sub.t] for each unit of effective labor employed by the firm. The goods-producing technology is given by

(8) [y.sub.t] = [A.sub.0][k.sup.[phi].sub.t][[h.sub.t]([l.sub.t] - [e.sub.t])].sup.1-[phi], [A.sup.0] > 0, [phi] [member of] (0,1),

where [y.sub.t] is per capita output and [h.sub.t]([l.sub.t] - [e.sub.t]) represents the effective labor input. (4) Profit maximization implies

(9) [r.sub.t] = [phi][y.sub.t]/[k.sub.t],

(10) [w.sub.t] = (1 - [phi])[y.sub.t]/[[h.sub.t]([l.sub.t] - [e.sub.t])].

Taxes paid to the government [T.sub.t] are given by equation (3) where [[tau].sub.pt] is the personal tax rate and [[tau].sub.b] is the business (or corporate) tax rate. Variables without time subscripts represent constants for all t. The personal tax rate can change over time as households make decisions that move them into a different tax bracket. This possibility is captured by the tax rate function F[*] in equation (4) which says that [[tau].sub.pt] depends on personal taxable income.

Personal taxable income is equal to labor income, less the standard deduction Dr, plus after-tax business income, which we assume is paid out each period in the form of dividends. We use the symbol [eta] [member of] [0, 1] to index the degree to which business income is taxed twice by the tax code. (5) Earnings forgone while in school [w.sub.t][h.sub.t][e.sub.t] are implicitly expensed under the current tax code (see Boskin 1977) and would receive the same treatment under all proposed reforms.

The symbols and [[phi].sub.k] and [[phi].sub.h] denote the fractions of investment in physical and human capital that can be expensed, or immediately deducted from business taxable income. For comparison with the U.S. tax system, [phi].sub.k] and [[phi].sub.h] can be interpreted as index numbers that summarize the various elements of the tax code that encourage saving or investment. Features that influence [[phi].sub.k] include the depreciation allowance for physical capital; the tax-deferred status of saving done through pensions, 401 (k)s, Keoughs, and retirement accounts; the favorable tax treatment of long-term capital gains; and the relatively tax-free status of service flows from owner-occupied housing. (6) Regarding [[phi].sub.h], firms may expense the costs of formal worker training, the wages of workers engaged in on-the-job training, job-related employee tuition, and expenditures for R&D. There is also a 20% tax credit for qualifying increases in R&D expenditures. (7)

Equations (5) and (6) describe the laws of motion for physical and human capital. When [[delta].sub.k] = [[delta].sub.k] = 1, the capital stocks depreciate completely each period, whereas 0 < [[delta].sub.i] < 1, for i = k, h implies that capital stocks are long-lasting. This nonlinear specification facilitates closed-form decision rules and can be viewed as reflecting adjustment costs as in Lucas and Prescott (1971). (8) Equation (5) implies that households can add to their stock of physical capital in only one way: through goods investment [i.sub.kt]. Equation (6) implies that human capital can be increased through goods investment [i.sub.ht] or by the allocation of household time [e.sub.t]. Introducing [i.sub.ht] as an input has an effect that is similar to introducing physical capital [k.sub.t] as an input because goods must be produced using physical capital. When [[delta].sub.h] = v = 0, the model collapses to one with exogenous labor-augmenting technological progress. In this case, the economy's long-run growth rate is given by log([h.sub.t+1]/[h.sub.t]) = log [A.sub.2].

The Tax System

The personal tax function, equation (4), allows for graduated rates and is assumed to take the following form

(11) [[[tau].sub.pt] = [[tau].sub.p][[w.sub.t][h.sub.t]([l.sub.t] - [e.sub.t]) - [D.sub.t] + [eta](1 - [tau].sub.b] x ([r.sub.t][k.sub.t] - [[phi].sub.k][i.sub.kt] - [[phi].sub.h][i.sub.ht)].sup.n] x [[w.sub.t][H.sub.t]([L.sub.t] - [E.sub.t]) - [D.sub.t] + [eta](1 - [[tau].sub.b]) x ([r.sub.t][K.sub.t] - [[phi].sub.k][I.sub.kt] - [[phi].sub.h][I.sub.ht].sup.-n],

where capital letters denote economy-wide averages and [[tau].sub.p] [member of] [0, 1] and n [greater than or equal to] 0 are parameters that govern the level and slope of the tax schedule. (9) Households view the economy-wide averages as outside their control. We impose the aggregate consistency conditions [H.sub.t]=[h.sub.t], [L.sub.t]=[l.sub.t], [E.sub.t]=[e.sub.t], [K.sub.t]=[k.sub.t], [I.sub.kt]=[i.sub.kt], and [I.sub.ht]=[i.sub.ht] after decisions are made. When n > 0, households with above-average taxable income face a higher personal tax rate than those with below-average taxable income. In contrast, when n = 0, all households face the same personal tax rate [[tau].sub.p] regardless of their taxable income. In our model, therefore, n > 0 represents a graduated-rate tax schedule and n=0 represents a horizontal or flat schedule.

The government sets the standard deduction [D.sub.t], which we model as a constant fraction of average pretax income:

(12) [D.sub.t] = [alpha][Y.sub.t], [alpha] [greater than or equal to] 0.

Because [D.sub.t] is a function of [Y.sub.t](as opposed to [y.sub.t]), households take the deduction as given.

For our analysis, it is useful to distinguish between the average and marginal tax rates on different categories of income. The average personal tax rate is defined as personal taxes paid divided by personal taxable income. The marginal personal tax rate is defined as the change in personal taxes paid divided by the change in personal taxable income, where we interpret "change" as an infinitesimally small amount. Intuitively, the marginal tax rate represents the rate applied to the last dollar earned. To derive the marginal personal tax rate, we multiply equation (11) by personal taxable income and then differentiate the resulting expression with respect to personal taxable income. After imposing the aggregate consistency conditions, we obtain [[tau].sub.p](n + 1) as the marginal personal tax rate. A similar procedure can be used to derive the marginal business tax rate. The expressions that govern the various equilibrium tax rates are

(13) Average personal tax rate = [[tau].sub.p],

(14) Marginal personal tax rate = [[tau].sub.p] (n + 1),

(15) Average business tax rate = [[tau].sub.b] + [eta](1 - [[tau].sub.p])[[tau].sub.p],

(16) Marginal business tax rate = [[tau].sub.b] + [eta](1 - [[tau].sub.p])[[tau].sub.p](n + 1]

When n >0, marginal tax rates on both categories of income are greater than the corresponding average tax rates. Notice that the business tax schedule exhibits progressivity only indirectly via the double taxation of business income.

Public Expenditures

The government sets the tax code parameters [[tau].sub.p], [[tau].sub.b], n, [eta], [[phi].sub.k], [[phi].sub.h] and [alpha] to finance a required level of per capita public expenditure [G.sub.t]. We assume that [G.sub.t] does not contribute to either production or household utility. The government budget constraint is given by

(17) [G.sub.t] = [[tau].sub.pt][[w.sub.t][H.sub.t]([L.sub.t] - [E.sub.t]) - [D.sub.t] + [eta](1 - [[tau].sub.b]) x ([r.sub.t][K.sub.t] - [[phi].sub.k][I.sub.kt] - [[phi].sub.h][I.sub.ht])] + [[tau].sub.b]([r.sub.t][K.sub.t] - [[phi].sub.k][I.sub.kt] - [[phi].sub.h][I.sub.ht]).

For simplicity, our specification imposes a period-by-period balanced budget. Per capita public expenditures are assumed to increase in fixed proportion to the average level of income in the economy, such that [G.sub.t] = [psi][Y.sub.t] where [psi][greater than or equal to]0. This formulation ensures that public expenditures remain a significant fraction of output as the economy grows. (10)

Consumption and Flat Taxes

The tax base under the current U.S. system is best described as a hybrid between income and consumption such that [[phi].sub.k],[[phi].sub.h] [member of] (0,1). If we impose [[phi].sub.k] = [[phi].sub.h] = 1 together with [[tau].sub.p] = [[tau].sub.b] = [tau] and n = [eta] = 0, then the tax system is equivalent to one that taxes consumption at the rate [[tau.sub.c] = [tau]/(1 - [tau]). (11) The Hall-Rabushka (1995) flat tax is sometimes referred to as a consumption tax. However, as noted by Judd (1998), the Hall-Rabushka plan calls for full expensing of new investment in physical capital ([[phi].sub.k] = 1) but contains no provisions to ensure equivalent treatment of human capital. Expenditures by individuals on education would not be deductible from taxable income under the Hall-Rabushka plan. In contrast, the Nunn-Dominici unlimited saving allowance tax proposal (for details see Weidenbaum 1996) includes a limited deduction for family expenditures on college tuition, vocational training, or remedial education. We model a HallRabushka-type flat tax by imposing [[phi].sub.k] = 1, [[tau].sub.p] = [[tau].sub.b] = [tau], and n = [eta] = 0, while holding [[phi].sub.h] unchanged at its prereform value.

It is important to recognize that a flat tax (or a pure consumption tax) can still exhibit features that are considered progressive. The crucial element that governs the progressivity of a flat tax is the level of the standard deduction [D.sub.t]. To see this, consider an alternative tax rate defined as total taxes paid [T.sub.t] divided by gross pretax income [y.sub.t]. Using equations (3) and (11) with the flat tax parameter settings [[phi].sub.k] = 1, [[tau].sub.p] = [[tau].sub.b] = [tau], and n = [eta] = 0 yields

(18) [T.sub.t]/[y.sub.t] = [tau](1 - [i.sub.kt]/[y.sub.t] - [[phi].sub.h] [i.sub.ht]/[y.sub.i] - [D.sub.t]/[y.sub.t]).

If we assume that the standard deduction is held fixed at a specific dollar amount D (as is the case under the U.S. tax code), then the ratio D/[y.sub.t] declines as real income y, rises. If we further assume that the ratios [i.sub.kt]/[y.sub.t] and [i.sub.ht]/[y.sub.t] remain constant (as they do under balanced growth), then equation (18) says that the tax rate [T.sub.t]/[y.sub.t] will rise with income. In other words, households with higher incomes will face higher tax rates--a feature that is progressive.

In our model, the standard deduction is not held fixed but instead rises with the average level of income, as given by equation (12). This formulation ensures that the standard deduction continues to represent a significant fraction of income as the economy grows over time. Substituting equation (12) into equation (18) yields

(19) [T.sub.t]/[y.sub.t] = [tau](1 - [i.sub.kt]/[y.sub.t] - [[phi].sub.h] [i.sub.ht]/[y.sub.t] - [alpha][Y.sub.t]/[y.sub.t]),

which shows that households with aboveaverage incomes ([y.sub.t] > [Y.sub.t]) continue to face higher tax rates than those with below-average incomes ([y.sub.t] < [Y.sub.t]), even under a flat tax (again assuming that the ratios [i.sub.kt]/[y.sub.t] and [i.sub.ht]/[y.sub.t] remain constant).

III. INCENTIVE EFFECTS OF THE TAX SYSTEM

In this section, we show how the tax system affects various aspects of the equilibrium.

Equilibrium Decision Rules

In the appendix, we show that standard techniques yield the following closed-form expressions for the equilibrium decision rules:

(20) [i.sub.kt] = [a.sub.0](1 - [[tau].sub.k])[y.sub.t], [[tau].sub.k] [equivalent to] (1 - [phi].sub.k]) x [[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.p](n + 1)] /{1 - [[phi].sub.k][[[tau].sub.b] + [eta](1 - [[tau].sub.b]) [[tau].sub.p](n + 1)]},

(21) [i.sub.ht] = [b.sub.0](1 - [[tau].sub.h])[y.sub.t], [[tau].sub.h] [equivalent to] {(n + 1)[[tau].sub.p] - [[phi].sub.h][[[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.p] (n + 1)]}/{1 - [[phi].sub.h][[[tau].sub.b] + [eta](1 - [[tau].sub.b]) [[tau].sub.p](n + 1)]},

(22) [c.sub.t] = {1 - [[tau].sub.p](1 - [phi] - [alpha]) - [phi][[[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.b]] - [a.sub.0] (1 - [[tau].sub.k]) x [1 - [[phi].sub.k]([[tau].sub.b] + [eta] (1 - [[tau].sub.b])[[tau].sub.p])] - [b.sub.0](1 - [[tau].sub.h]) x [1 - [[phi].sub.h]([[tau].sub.b] + [eta](1 - [[tau].sub.b]) [[tau].sub.p])]}[y.sub.t],

(23) ([l.sub.t] - [e.sub.t]) = [A.sub.3][{[1 - [[tau].sub.p](n + 1)] x [([h.sub.t]/[k.sub.t]).sup.[phi]}.sup.1/([phi]+[gamma]-1)],

(24) [e.sub.t] = [A.sub.3][A.sub.4][{[1 - [[tau].sub.p](n + 1)] x [([h.sub.t]]/[k.sub.t]).sup.[phi]}.sup.1/([phi]+[gamma]-1),

where [a.sub.0], [b.sub.0], [A.sub.3], and [A.sub.4] represent combinations of deep parameters, and [y.sub.t] is equilibrium per capita output (or income). By substituting equation (23) into equation (8), we obtain the following expression for equilibrium per capita output:

(25) [y.sub.t] = [A.sub.0][A.sup.1-[phi].sub.3][k.sup.[phi][gamma] /([phi]+[gamma]-1).sub.t][h.sup.(1-[phi])([gamma]-1)/([phi]+ [gamma]-1).sub.t] x [[1 - [[tau].sub.p](n + 1)].sup.(1-[phi]) /([phi]+[gamma]-1)].

Private-sector investment depends on the effective marginal tax rates [[tau].sub.k] and [[tau].sub.h], which combine the statutory tax code parameters [[tau].sub.p], [[tau].sub.b], n, and [eta] with the expensing parameters [[phi].sub.k] and [[phi].sub.h]. A pure consumption tax implies [[tau].sub.k] = [[tau].sub.h] = 0. A labor supply distortion will continue to exist, however, as long as [gamma] < [infinity]. As [gamma][right arrow][infinity] this distortion will also be eliminated making a consumption tax equivalent to a lump-sum tax. All else equal, equation (25) implies that per capita output (or income) is a decreasing function of the ratio [h.sub.t]/[k.sub.t], a result that stems from the labor decision rule (23). The labor decision rule says that time devoted to market work declines as the household acquires more human capital relative to physical capital. The education decision rule (24) exhibits the same property. Intuitively, this occurs because higher levels of human capital raise the opportunity cost of time that is not devoted to leisure.

Transition Dynamics and Balanced Growth

The model's tractable nature allows us to characterize explicitly the economy's dynamic transition path for any set of initial conditions [k.sub.0] and [h.sub.0]. By substituting the decision rules and the expression for equilibrium output (25) into the laws of motion (5) and (6) we obtain

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Our specification of a goods-producing technology (8) that exhibits constant returns to scale in the two reproducible factors [k.sub.t] and [h.sub.t], together with the functional forms (1), (5), and (6), imply that the model possesses a unique balance growth path in which [k.sub.t], [h.sub.t], [y.sub.t], [c.sub.t], [i.sub.kt], and [i.sub.kt] all grow at the same constant rate. To derive an expression for the balanced growth ratio R = [h.sub.t]/[k.sub.t], we divide equation (27) by equation (26) and impose R = [h.sub.t+1]/[k.sub.t+1] = [h.sub.t]/[k.sub.t]. This procedure yields

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Per Capita Growth Rate

By taking log of equations (26) and (27), we obtain two equivalent expressions for the per capita balanced growth rate [mu]:

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where R is given by equation (28).

Equation (30) helps provide some insight into the robustness of results reported in the literature regarding the effects of distortionary taxes on long-run growth. Models that omit direct investment of goods in human capital or assume fixed time allocations will shutdown some channels through which tax policy can affect growth. For example, Lucas (1990) finds that distortionary taxes have very small growth effects in a model where the only inputs to the production of human capital are [h.sub.t] and household time. This case corresponds to our model with [[delta].sub.h] = 0. In the models of King and Rebelo (1990) and Kim (1998), the humancapital inputs are [h.sub.t] and [i.sub.ht]. This case corresponds to our model with v = 0. Our setup adheres to a commonly used specification where the human capital inputs are h.sub.t], [i.sub.ht] (or [k.sub.t]), and household time.

Revenue Neutrality

The flat-tax proposal of Hall and Rabushka is designed to be revenue neutral. The intent of the plan is to improve economic efficiency while leaving aside arguments about the appropriate size of government. (12) Given that a tax reform can affect the trend growth rate of all variables in our model, the concept of revenue neutrality used here is a relative one. Specifically, we hold tax revenues fixed relative to output. (13) Given that tax revenues must finance public expenditures, according to [G.sub.t] = [psi][Y.sub.t], revenue neutrality requires that [psi] remain unchanged as tax code parameters are varied. By substituting equations (9), (10), (12), (20), and (21) into equation (17) and then imposing the aggregate consistency conditions, we obtain the following relationship among the tax code parameters:

(31) [psi] = [[tau].sub.p](1 - [phi] - [alpha]) + [[[tau].sub.b] + [eta](1 -[[tau].sub.b]) [[tau].sub.p]][[phi] - [a.sub.0][[phi].sub.k](1 - [[tau].sub.k]) - [b.sub.0][[phi].sub.h](1 - [[tau].sub.h])],

where the effective marginal tax rates [[tau].sub.k] and [[tau].sub.h] are defined in equations (20) and (21). In our experiments, a particular reform may cause any number of tax code parameters to change simultaneously. Revenue neutrality is maintained by adjusting [[tau].sub.p] = [[tau].sub.b] = [tau] under the new system such that equation (31) is satisfied at the prereform value of [psi]. In general, adopting a flat tax requires a higher average tax rate to pay for the more generous expensing and deduction features and to compensate for the loss of revenue from doing away with double taxation.

IV. CALIBRATION

Parameter values are chosen so that the model's balanced growth path matches various characteristics identified from U.S. data. For parameters that are important for the results, a range of values is examined. A time period in the model is taken to be one year. We calibrate the prereform tax system to resemble the hybrid income-consumption tax system in the U.S. economy. As many authors have noted, the current system already allows a significant portion of U.S. saving to escape distortionary taxation.

It is well known that the behavior of dynamic tax models is sensitive to assumptions about the labor supply elasticity. We therefore consider the values [gamma] = 6, 3, and 2.25, which imply intertemporal elasticities of substitution in labor supply of 0.2, 0.5, and 0.8, respectively. These elasticities are in the range of values obtained from empirical studies if one considers both males and females and adjustments along both intensive and extensive margins (see Eissa 1996; Mulligan 1999).

We choose [phi] = 0.36 to match the average share of capital income in U.S. gross domestic product (GDP), as estimated by Poterba (1997). The constants [A.sub.0], [A.sub.1], and [A.sub.2] are chosen to achieve the calibration targets of [mu] = 1.80%, [k.sub.t]/[y.sub.t] = 2.61, and [h.sub.t/[k.sub.t] = 13. Our measure of the U.S. physical capital stock includes structures, equipment, consumer durables, and residential components. Our target for [h.sub.t]/[k.sub.t] is based on the Jorgenson and Fraumeni (1989, table 5.33) capital stock estimates, which take into account the imputed value of human capital in nonmarket activities, such as school, leisure, or home production. (14)

The elasticity parameters [[delta].sub.k] and [[dalta].sub.h] are chosen so that the model matches the U.S. average ratios of [i.sub.kt]/[y.sub.t] = 0.22 and [i.sub.ht]/[y.sub.t] = 0.025. Consistent with our measure of physical capital, [i.sub.kt] includes structures, equipment, consumer durables, and residential components. Our measure of im includes private-sector expenditures on education, training, and R&D. (15)

We choose the discount factor [beta] to achieve an after-tax interest rate of 4% based on the estimates of Poterba (1997, table 1). (16) The elasticity parameter v in the human capital technology and the household preference parameter B are chosen to achieve the balanced-growth time allocations of [e.sub.t] = 0.12 and [l.sub.t]-[e.sub.t] = 0.17. These are the values estimated by Jones et al. (1993) for the U.S. economy.

The tax code parameters [[tau].sub.p] and n are estimated from the 1994 U.S. tax schedule for married taxpayers with no children who file IRS form 1040jointly. The tax schedule, taken from Mulligan (1997, table 5-2), displays 12 different tax brackets that derive from the combined effects of the federal individual income tax, the earned income tax credit, and employee and employer contributions to Social Security and Medicare. The Hall-Rabushka proposal does not call for any changes to Social Security or Medicare; the tax schedules for these programs are already flat for the average taxpayer. Nevertheless, these programs influence the slope of the prereform tax schedule. The marginal tax brackets from Mulligan (1997) are used to construct an average personal tax schedule, which is shown in Figure 1.

[FIGURE 1 OMITTED]

In Figure 2, we plot the U.S. average personal tax rate versus the income ratio, which we define as personal taxable income divided by its mean level. This ratio represents the empirical counterpart of the expression inside the square brackets in equation (11). In constructing Figure 2, we use a mean taxable income of $42,600, which is based on tax return data from 1994. (17) Using 350 data points (one for each $1,000 of taxable income), we perform a nonlinear least squares regression of the average personal tax rate on the corresponding income ratio. The regression yields

[FIGURE 2 OMITTED]

(32) Average Personal Tax Rate = 0.2528[(Income Ratio).sup.0.2144].

Comparing the expression to equation (11) Implies [[tau].sub.p] = 0.2528 and n 0.2144. (18) Figure 2 shows that the fitted relationship is close to the U.S. tax schedule for income ratios around 1.0 but flatter than the U.S. tax schedule for income ratios below 0.8 and above 1.2. This suggests that our calibration procedure yields a conservative estimate of the growth benefits of adopting a flat tax. Our experiments will examine the sensitivity of the computed growth effects to different values of the prereform slope parameter n.

The tax code parameter a is calibrated by dividing the total dollar amount of standard deductions taken in 1994 by U.S. GDP. This procedure yields [alpha] = 0.056. (19) Following Hall and Rabushka (1995), we set [[tau].sub.b] = 0.35 to match the statutory corporate tax rate. We set [eta] = 1 to reflect double taxation of business income. We adopt Auerbach's (1996, 51) estimate of [[tau].sub.k] = 0.16 to calibrate the baseline value of [[phi].sub.k] because his estimate takes into account the effective marginal tax rates for both residential and nonresidential capital. (20) A difficult parameter to pin down is [[phi].sub.h], which represents the fraction of private goods investment in human capital that is tax deductible. Recall that our measure of [i.sub.ht] includes private-sector expenditures on education, training, and R&D. Privately funded R&D investment (which is tax deductible) has averaged slightly more than 1% of GDP since 1954. Private expenditures for education and training (which are mostly not tax deductible) are roughly the same magnitude. We combine these observations to come up with an estimate of [[phi].sub.h] = 0.5. Later, we demonstrate that our quantitative results are not very sensitive to changes in [[phi].sub.h].

Table 1 summarizes the results of the calibration exercise.

Substituting the tax code parameters into the expression for [[tau].sub.h] in equation (21) yields [[tau].sub.h] = 0.044. With [[tau].sub.k] = 0.160 and [[tau].sub.h] = 0.044, the baseline tax structure is more favorable to human capital when it comes to private goods investment. Substituting the tax code parameters into equation (31) yields [psi] = 0.231, which says that public expenditures represent about 23% of output in the model. The figure is close to the U.S. average. The calibrated value of the marginal business tax rate in the model is [[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.p](n + 1) = 0.550. For comparison, McGrattan et al. (1997) report estimates of the average marginal tax rate on capital income for the period 1947 to 1992 Their end-of-sample estimate is 0.482.

>From equation (5), Tobin's q in the model is given by

(33) q = ([differential][k.sub.t+1]/[differential] [k.sub.t])/([differential][k.sub.t+1]/[differential][i.sub.kt] = (1 - [[delta].sub.k])[i.sub.kt]/([delta].sub.k][k.sub.t]) = 1.39.

Eberly (1997, table 1) estimates Tobin's q using U.S. firm-level data over the period 1981 to 1994. She obtains a mean estimate of 1.56 and a median estimate of 1.18. Comparing these figures to (33) suggests that our model provides a reasonable portrayal of U.S. investment fundamentals.

The baseline parameters imply that untaxed forgone earnings represent 95% of the total costs of producing human capital. (21) Only about 3% of the total costs of producing human capital are not tax deductible. (22) As a comparison, Clotfelter (1991, 72) estimates that forgone earnings represent 49%-79% of college education costs (tuition, room, board, and forgone earnings) for males and 41%-71% for females over various two-year periods from 1969 to 1988. Dupor et al. (1996) estimate an upper bound of 8% for the share of privately borne human capital costs which are not tax deductible.

V. QUANTITATIVE RESULTS

Figures 3 through 8 summarize the results of our tax reform experiments using the endogenous growth version of the model. To construct these figures, we use equation (29) to compute the balanced growth rate under the pre- and postreform tax systems. On the vertical axis of each figure, we plot the change in long-run growth [DELTA][mu] (in percentage points) that results from a sudden unannounced switch from the existing graduated-rate tax system to a revenue-neutral flat tax. We plot [DELTA][mu] for a range of values of the prereform slope parameter n. The estimated U.S. value of n = 0.2144 is highlighted in each figure. In addition, we plot [DELTA][mu] for three different values of [gamma], which correspond to the labor supply elasticities, [([gamma]-1).sub.-1] = {0.2, 0.5, 0.8}. For each value of n and [gamma] in the figures, we recalibrate the other parameters of the model so that the prereform economy continues to match the empirical facts described in section IV.

[FIGURES 3-8 OMITTED]

Table 2 summarizes the identifying characteristics of the tax systems that we consider.

Figure 3 depicts the results of our baseline reform experiment. This reform captures many of the features of the Hall-Rabushka proposal, such as equal tax rates applied to personal and business income ([[tau].sub.p] = [[tau].sub.b] = [tau]), a flat tax schedule (n = 0), no double taxation of business income ([eta] = 0), and full expensing of physical capital investment ([[phi].sub.k] = 1). The standard deduction parameter for this experiment is held fixed at its prereform value of [alpha] = 0.056.

Figure 3 shows that as the prereform tax schedule becomes steeper (n increases), the growth gains from adopting a flat tax become larger. When n = 0.2144, the prereform tax structure matches the estimated steepness of the U.S. tax schedule. At this point, the relevant prerefonn tax rates are:

Average personal tax rate = [[tau].sub.p] = 0.253,

Marginal personal tax rate = [[tau].sub.p](n+1)=0.307,

Average business tax rate = [[tau].sub.b] + [eta](1 - [[tau].sub.b]) x [[tau].sub.p] = 0.514,

Marginal business tax rate = [[tau].sub.b] + q(1 - [[tau].sub.b]) x [[tau].sub.p] (n + 1) = 0.550.

The postreform average and marginal tax rate is given by [[tau].sub.p] = [[tau].sub.b] = [tau] = 0.3437. Thus the baseline reform lowers the marginal business tax rate but raises the marginal personal tax rate. (23) When n=0.2144, the reform yields a growth gain of between 0.009 and 0.143 percentage points, depending on the elasticity of household labor supply. A less elastic labor supply implies a smaller growth gain because hours worked ([l.sub.t]-[e.sub.t]) and time in school ([e.sub.t]) are both less sensitive to changes in the after-tax wage. In shifting to a flat tax, hours worked increase by only 0.6% (from 0.170 to 0.171) when the elasticity is 0.2, but increase by 2.3% (from 0.170 to 0.174) when the elasticity is 0.8. In each case, time in school increases by the same percentage as hours worked. These results are consistent with the findings of other studies that show that the growth effects of distortionary taxation are sensitive to assumptions about the elasticity of household time allocations.

All of the reforms we consider exhibit relatively small growth effects due to the form of the human capital technology where untaxed forgone earnings represent the largest input to production. The computed growth effects would have been even smaller if we had adopted a utility function with more curvature than logarithmic, implying higher risk aversion. Small growth effects from tax reform are consistent with the findings of many authors, including Lucas (1990), Devereux and Love (1994), and Stokey and Rebelo (1995). Nevertheless, it is important to recognize that quantitative theoretical models cannot decide the question of whether distortionary taxes have large or small growth effects. The answers will always depend on the chosen form of the human capital technology, the degree of risk aversion, and the values of other key parameters, such as the labor supply elasticity. (24)

The far left in Figure 3 corresponds to n = 0 in the prereform tax system. This point isolates the growth gains attributable to the other (nonslope) features of the flat tax reform. The vertical intercept shows that the combined effects of the other elements produces a negative growth gain. We will elaborate further on this point in our discussion of Figures 5 and 6.

Figures 4 through 6 illustrate variants of the baseline reform in which a key element is not included. Figure 4 corresponds to a reform that does not include full expensing of physical-capital investment. Figure 5 corresponds to a reform that does not eliminate double taxation of business income. Figure 6 corresponds to a reform that does not flatten the personal tax schedule.

Comparing Figure 4 to Figure 3 shows that leaving out full expensing of physical capital investment reduces the growth gain by only a small amount. This is because the prereform value of [[phi].sub.k] = 0.844 (which derives from our calibration target of [[tau].sub.k] - 0.16) captures the fact that a large fraction of U.S. saving is done through vehicles like pensions, retirement accounts, and 401(k) plans, which already receive consumption tax treatment. The growth effect of increasing 0k from 0.844 to 1.0 is small.

Figure 5 shows that a reform that does not eliminate double taxation is actually beneficial for growth--provided the reform continues to include full expensing of physical capital investment. The intuition for this result is straightforward. From equation (20), we see that setting [[phi].sub.k] = 1 yields [[tau].sub.k] = 0. This sharply reduces the tax distortion to physical-capital investment although a tax distortion still exists via the expression for [y.sub.t] in equation (25). When [[phi].sub.k] = 1, double taxation of business income is a low-distortion source of revenue that allows the government to implement the reform with a lower postreform tax rate. (25) From Table 2, we see that [[tau].sub.p] = [[tau].sub.b] = [tau] = 0.3169 for this tax reform, which is the lowest postreform tax rate of all the experiments we consider. The lower postreform tax rate leads to faster growth by encouraging labor effort and time devoted to human capital accumulation. Figure 6 shows that a reform that does not flatten the slope of the personal tax schedule is harmful for growth. This is consistent with the findings discussed earlier; the other elements of the reform shrink the tax base and force the government to impose a higher postreform tax rate to maintain revenue neutrality.

[FIGURE 6 OMITTED]

Figure 7 illustrates the growth effects of switching to a pure consumption tax. The reform is accomplished by allowing full expensing for both physical and human capital investment such that [[phi].sub.k] = [[phi].sub.h] = 1. This experiment is aimed at investigating the quantitative implications of the point made by Judd (1998) that the Hall-Rabushka proposal is actually biased in favor of physical capital. When n = 0.2144, Figure 7 shows that switching to a pure consumption tax yields a long-run growth gain of between 0.031 and 0.156 percentage points. Figure 3 shows that switching to a Hall-Rabushka type flat tax yields a long-run growth gain of between 0.009 and 0.143 percentage points. The long-run growth gains implied by the two reforms differ by only about 0.02 percentage points. These similar outcomes are not surprising given that nondeductible goods investment in human capital (1 - [[phi].sub.h])[i.sub.ht] accounts for only a small fraction of the total costs of producing human capital in our model.

[FIGURE 7 OMITTED]

In Figure 8, we allow the standard deduction parameter to change from its prereform value of [alpha] = 0.056. The figure plots [DELTA][mu] versus the ratio [[alpha].sub.new]/[[alpha].sub.old] where [[alpha].sub.new] and [[alpha].sub.old] represent the standard deduction parameters under the flat tax and the existing U.S. tax system, respectively. We hold [[alpha].sub.old] fixed at 0.056 and allow [[alpha].sub.new] to vary from 0 to 0.168. As the ratio [[alpha].sub.new]/[[alpha].sub.old] increases, the growth gains become smaller as revenue neutrality necessitates a higher postreform tax rate. At the far right when [[alpha].sub.new]/[[alpha].sub.old] = 3, the postreform tax rate is [[tau].sub.p] = [[tau].sub.b] = [tau] = 0.4119. At this point, adopting a flat tax reduces long-run growth. In general, the desired features of progressivity that are achieved through a more-generous standard deduction come at the cost of sacrificed growth gains.

As a supplement to the information provided in the figures, Table 3 summarizes the balanced growth properties of the various tax systems for the case of the intermediate labor supply elasticity, [([gamma] - 1).sub. -1] = 0.5.

Finally, Figure 9-12 compare the transition dynamics of the endogenous and exogenous growth versions of the model. To compute the transition paths for the exogenous growth model, we set [[delta].sub.h] = v = 0 and [A.sub.2] = exp(0.018) = 1.018. All other parameters are unchanged from before. Each simulation starts from the economy's balanced growth path with [h.sub.0]/[k.sub.0] = 13. We choose h0 such that per capita output [y.sub.t] is equal to 1.0 in the period preceding the reform. Figures 9 and 10 depict the output and growth rate trajectories for the baseline flat tax reform (shown earlier in Figure 3) with the intermediate labor supply elasticity, [([gamma] - 1).sub. -1] = 0.5. Figures 11 and 12 depict the corresponding trajectories for the reform with no change in the double taxation parameter [eta] (shown earlier in Figure 5).

In Figures 9 and 10, we see that shifting to a flat tax initially produces a growth slowdown in both the endogenous and exogenous growth versions of the model. This is due to the immediate reaction of hours worked and time in school to the higher postreform tax rate that is needed to maintain revenue neutrality. The slowdown is only temporary, however, because the reform's investment incentives eventually cause the output trajectory to overtake the prereform trend after about six years. Nevertheless, in the real-world, such an outcome could pose a challenge to the political feasibility of the reform. If elected policy makers are myopic relative to the average household, they may be unwilling to endure a near-term growth slowdown to achieve the permanent benefits of a flat tax. (26)

[FIGURE 9-10 OMITTED]

The simulations show that it takes a long time (about 100 years) for the output trajectory of the endogenous growth model to overtake the output trajectory of the exogenous growth model. The exogenous growth model exhibits a temporary output advantage because households do not need to divert resources to accumulate human capital. This allows physical capital to be accumulated more quickly. Figure 10 shows that adopting a flat tax produces a permanent growth gain in the endogenous growth model but only a temporary growth gain (with a permanent level shift) in the exogenous growth model. The transition dynamics of the exogenous growth model turn out to be quite lengthy, however. Following the adoption of a flat tax, it takes about 100 years for the economy to return to its prereform growth rate. In the years immediately following the reform, the economy's output trajectory is quite similar to that of the endogenous growth model.

In Figures 11 and 12, we see that a reform that does not eliminate double taxation of business income produces a milder slowdown in the periods immediately following the adoption of the reform. This result is consistent with our earlier discussion; double taxation is beneficial for growth when combined with full investment expensing. The simulations show that it now takes less time (about 20 years) for the output trajectory of the endogenous growth model to overtake the output trajectory of the exogenous growth model.

[FIGURE 11-12 OMITTED]

A quantitative summary of the transition paths is presented in Table 4.

VI. CONCLUSION

This article developed a simple theoretical framework to assess the growth effects of shifting from a graduated-rate tax system to a flat tax. The model captures the incentive effects of rising marginal tax rates on household decisions to consume, work, learn, and invest. These decisions, in turn, influence the rate of economic growth. Our baseline reform experiment predicts that a revenue-neutral flat tax can permanently increase per capita growth by 0.009 to 0.143 percentage points per year relative to a graduated-rate tax system calibrated to match features of the U.S. tax code. The flattening of the marginal tax schedule is the most important element of the reform for enhancing growth. Allowing full expensing of physical capital investment contributes only a small amount to the overall growth gain. The elimination of double taxation of business income is actually harmful for growth when combined with the full expensing provision.

Although our model is admittedly an abstract and simplified representation of the vastly complex U.S. tax code, we believe it provides some useful insight into the potential growth benefits of currently proposed tax reforms. By imposing the discipline of general equilibrium, we have attempted to take into account the macroeconomic repercussions that may be induced by a major overhaul of the U.S. tax system. Some important caveats regarding the interpretation of our results are in order, however. First, although our model provides a theoretical framework for evaluating claims regarding the growth benefits of adopting a flat tax, the empirical evidence regarding the links between tax policy and long-run growth is somewhat inconclusive. (27) Second, our results, like those of Stokey and Rebelo (1995), show that the growth effects of distortionary taxes are sensitive to the choice of parameter values. In particular, the labor supply elasticity has a substantial impact on the growth implications of the reform. Third, because our model abstracts from stochastic shocks, we allow no role for the tax code in providing implicit insurance against income uncertainty. (28) Fourth, our model assumes that all households are identical and therefore cannot be used to address distributional issues. (29) That said, the model and tax reform characteristics that we have identified as important for determining the magnitude of the growth effects will also be present in more complicated frameworks.

ABBREVIATIONS

GDP: Gross Domestic Product

R&D: Research and Development

APPENDIX

This appendix outlines the derivation of the equilibrium household decision rules. The Lagrangian for the household's problem can be written as

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [[lambda].sub.t] is the Lagrange multiplier associated with the household budget constraint (2). To conserve space, we have not substituted out [i.sub.kt] and [i.sub.ih] using equations (5) and (6). We make use of these substitutions to derive the following first-order conditions with respect to the indicated variables:

(A2a) [k.sub.t+1]: [[lambda].sub.t] [1 - [[phi].sub.k] [[tau].sub.h] - [eta](1 - [[tau].sub.b])[[phi].sub.k][[tau].sub.pt] (n + 1)][i.sub.kt]/([[delta].sub.k][k.sub.t+1] = [beta][[lambda].sub.t+1] {[1 - [[tau].sub.b] - [eta](1 - [[tau].sub.b][[tau].sub.pt] x 9n + 1) [r.sub.t+1] + [1 - [[phi].sub.k][[tau].sub.b] - [eta](1 + [[tau].sub.b]) [[phi].sub.k] x [[tau].sub.pt+1](n + 1)](1 + [[delta].sub.k]) [i.sub.kt+1]/([delta].sub.k][k.sub.t+1])},

(A2b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(A2c) (c.sub.t) : [[lambda].sub.t] = 1/([c.sub.t] - B[h.sub.t] [l.sup.[gamma].sub.t]),

(A2d) [l.sub.t] : [[lambda].sub.t][1 - [[tau].sub.pt](n + 1)][w.sub.t] [h.sub.t] = B[gamma][h.sub.t][l.sup.[gamma]-1.sub.t]/[c.sub.t] - B[h.sub.t] [l.sup.[gamma].sub.t],

(A2e) [e.sub.t] : [1 - [[tau].sub.pt](n + 1)][w.sub.t][h.sub.t] = [1 - [[phi].sub.h][[tau].sub.b] -[eta](1 - [[tau].sub.b])[[phi].sub.h] [[tau].sub.pt](n + 1)]v[i.sub.ht]/([[delta].sub.h][e.sub.t]),

where [[tau].sub.pt], is given by equation (11). The transversality conditions are [lim.sub.t[right arrow][infinity]] [[beta].sup.t][[lambda].sub.t] [k.sub.t+1] = 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The first term on the right side of equation (A2b) shows that households take into account the influence of human capital on the amount of quality-adjusted leisure. In particular, higher levels of human capital will raise the opportunity cost of time that is not devoted to leisure.

The equilibrium decision rules are obtained using the method of undetermined coefficients. To solve for the decision rules, we first impose the aggregate consistency conditions [H.sub.t] = [h.sub.t], [L.sub.t] = [l.sub.t], [E.sub.t] = [e.sub.t], [K.sub.t] = [k.sub.t], [I.sub.kt] = [i.sub.kt], and [I.sub.ht] = [i.sub.ht]. These conditions imply [[tau].sub.pt] = [[tau].sub.p] for all t. We make the conjecture that the decision rules take the form:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(A5) [[lambda].sup.-1.sub.t] = [d.sub.0][y.sub.t],

(A6) ([l.sub.t] - [e.sub.t) = [f.sub.0][l.sub.t],

where [a.sub.0], [b.sub.0], [d.sub.0], and [f.sub.0] are constants to be determined, and [y.sub.t] is equilibrium per capita output. By substituting the conjectured decision rules and the profit maximization conditions (9) and (10) into the first-order conditions for [k.sub.t+1] and [h.sub.t+1] and also making use of equations (A2c) and (A2d), we obtain

(A7) [a.sub.0] = [theta][[delta].sub.k]/([rho] + [[delta].sub.k]),

(A8) [b.sub.0] = (1 - [theta)[[delta].sub.h][1 + ([gamma][f.sub.0]).sup.-1]]/([rho] + [[delta].sub.,

where [rho] [equivalent to] 1/[beta] the household's rate of time preference.

Substituting the expression for [w.sub.t]= from equation (10) into the first-order condition (A2e) and then making use of equations (A4), (A6), and (A8) yields

(A9) [f.sub.o] = ([rho] + [[delta].sub.h] + v/[gamma])/([rho] + [[delta].sub.h] + v).

Substituting equation (A9) back into equation (A8) and solving for [b.sub.0] yields

(A10) [b.sub.0] = (1 - [theta])[[delta].sub.h](1 - [[gamma].sup.-1] /(p + [[delta].sub.h]+v/[gamma]).

A convenient property of the utility function (1) is that It can be solved for independently of the marginal utility of income [lambda]. Substituting the expression for [w.sub.t] from equation (10) into the first-order condition for [l.sub.t] and then making use of equations (A2c), (A6), and (A9) yields

(A11) [l.sub.t] = [{[A.sub.0](1 - [theta])(B[gamma]).sup.-1][1 + v(1 - [[gamma].sup.-1])/([rho] + [[delta].sub.h] + v/[gamma])][theta] [1 - [[tau].sub.p](n + t)] x [([h.sub.i]/[k.sub.t]).sup.-[theta]] s}.sup.1/([theta]+[gamma]-1]

Equation (A11) can now be used to solve for ([l.sub.t] - [e.sub.t] and [e.sub.r using the conjectured relationships ([l.sub.t] - [e.sub.t]) = [f.sub.o][l.sub.t] and [e.sub.t] = ([f.sup.-1.sub.0] - 1)([l.sub.t] - [e.sub.t]). The results are

(A12) ([l.sub.t] - [e.sub.t]) = [A.sub.3][{[1 - [[tau].sub.p](n + 1)] ([h.sub.t]/[k.sub.t]).sup.-[theta]}.sup.1/([theta]+[gamma]-1),

A3 [equivalent to] {[A.sub.0](1 - [theta][(B[gamma]).sup.-1] [[1 + v(1 - [[gamma].sup.-1])/([rho] + [[delta].sub.h] + v/[gamma])].sup.1-[gamma]]}.sup.1/([theta]+[gamma]-1)],

(A13) [e.sub.t] = [A.sub.3][A.sub.4][{[1 - [[tau].sub.p](n + 1)] [([h.sub.t]/[k.sub.t]).sup.-[theta]]}.sup.1/([theta]+[gamma]-1)], [A.sub.4] [equivalent to] v(1 - [[gamma].sup.-1])/([rho] + [[delta].sub.h] + v/[gamma]).

To obtain equilibrium consumption, we substitute the investment decision rules (A3) and (A4) and the profit maximization conditions (9) and (10) into the household budget constraint (2). After collecting terms, this procedure yields

(A14) ct = {1 - [[tau].sub.p](1 - [theta] - [alpha]) - [theta] [[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.p] - [a.sub.0] (1 - [[tau].sub.k][1 - [[phi].sub.k]([[tau].sub.b] + [eta] (1 - [[tau].sub.b])[[tau].sub.p])] - [b.sub.0](1 - [[tau].sub.h] [1 - [[phi].sub.h]([[tau].sub.b] + [eta](1 - [[tau].sub.b]) [[tau].sub.p])]}[y.sub.t].

The next step is to verify that the conjectured forms of equations (A3)-(A6) are correct by showing that [d.sub.0] is in fact constant. We use the first-order condition for [c.sub.t] to obtain

(A15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where the second equality replaces B[h.sub.t][l.sup.[gamma].sub.t] by an equivalent expression that is obtained by combining equations (A2c) and (A2d). Substituting the consumption decision rule (A14) into the expression and solving for [d.sub.0] yields

(A16) [d.sub.0] = 1 - [[tau].sub.p](1 + [theta] - [alpha]) - [theta][[tau].sub.n] + [eta](1 + [[tau].sub.b][[tau].sub.p]} - [a.sub.0](1 - [[tau].sub.k]{1 - [[phi].sub.k][[tau].sub.b] + [eta] (1 - [[tau].sub.b])[[tau].sub.p]]} - [b.sub.0](1 - [[tau].sub.h] {1 - [[phi].sub.k][[tau].sub.b] + [eta](1 - [[tau].sub.b])[[tau].sub.p]]} - [(0 - [theta])/([gamma][f.sub.0])][1 - [[tau].sub.p](n + 1)],

which is a constant and thus verifies our conjecture.

TABLE 1

Calibrated Parameter Values

 Labor Supply
 Elasticity

Parameter 0.2 0.5 0.8 Empirical Fact to Match

[gamma] 6.000 3.000 2.250 Labor supply elasticity
[theta] 0.360 0.360 0.360 Average share of physical
 capital in output = 0.36
[A.sub.0] 0.231 0.231 0.231 Average per capita output
 growth [mu] = 1.80%
[A.sub.1] 1.173 1.173 1.173 Average [k.sub.t]/[y.sub.t]
 = 2.61
[A.sub.2] 1.078 1.123 1.233 Average [h.sub.t]/[k.sub.k]
 = 13
[[delta].sub.k] 0.057 0.057 0.057 Average [i.sub.kt]/[y.sub.t]
 = 0.22
[[delta].sub.h] 0.001 0.002 0.004 Average [i.sub.ht]/[y.sub.t]
 = 0.025
v 0.022 0.039 0.075 Fraction of time in school
 or training [e.sub.t]
 = 0.12
B 6.248 0.305 0.161 Fraction of time in market
 work ([l.sub.t] -
 [e.sub.t]) = 0.17
[beta] 0.979 0.979 0.979 After-tax interest rate = 4%
[[tau].sub.p] 0.253 0.253 0.253 Fitted value from 1994 U.S.
 tax schedule
n 0.214 0.214 0.214 Fitted value from 1994 U.S.
 tax schedule
[alpha] 0.056 0.056 0.056 Ratio of 1994 U.S. standard
 deductions to GDP
[eta] 1.000 1.000 1.000 Double taxation of business
 income
[[tau].sub.b] 0.350 0.350 0.350 Statutory corporate tax rate
[[phi].sub.k] 0.844 0.844 0.844 Effective marginal capital
 tax rate [[tau].sub.k] = 0.16
[[phi].sub.h] 0.500 0.500 0.500 Fraction of tax-deductible
 goods investment in human
 capital

TABLE 2

Characteristics of the Tax Systems

 Tax Code
 Parameter
Tax System Values

Graduated-rate system (U.S.) [[tau].sub.p] 0.253,
 [[tau].sub.p] = 0.350 n=0.214
Baseline fat tax reform [[tau].sub.p] = [[tau].
 sub.b] = [tau] = 0.3427 n=0
Reform with no change in [[tau].sub.p] = [[tau].
 [[phi].sub.k] sub.b] = [tau] = 0.3174 n=0
Reform with no change in [[tau].sub.p] = [[tau].
 [eta] sub.b] = [tau] = 0.3169 n=0
Reform with no change in n [[tau].sub.p] = [[tau].
 sub.b] = [tau] = 0.3431 n=0.214
Pure consumption tax reform [[tau].sub.p] = [[tau].
 sub.b] = [tau] = 0.3520 n=0
Reform with change in [[tau].sub.p] = [[tau].
 [alpha] sub.b] = [tau] [??]
 0.3437 n=0

Tax System Tax Code Parameter Values

Graduated-rate system (U.S.) [eta] = 1 [[phi].sub.k] = 0.844
Baseline fat tax reform [eta] = 0 [[phi].sub.k] = 1
Reform with no change in
 [[phi].sub.k] [eta] = 0 [[phi].sub.k] = 0.844
Reform with no change in
 [eta] [eta] = 1 [[phi].sub.k] = 1
Reform with no change in n [eta] = 0 [[phi].sub.k] = 1
Pure consumption tax reform [eta] = 0 [[phi].sub.k] = 1
Reform with change in
 [alpha] [eta] = 0 [[phi].sub.k] = 1

Tax System Tax Code Parameter Values

Graduated-rate system (U.S.) [[phi].sub.h] = 0.5 [alpha] = 0.056
Baseline fat tax reform [[phi].sub.h] = 0.5 [alpha] = 0.056
Reform with no change in
 [[phi].sub.k] [[phi].sub.h] = 0.5 [alpha] = 0.056
Reform with no change in
 [eta] [[phi].sub.h] = 0.5 [alpha] = 0.056
Reform with no change in n [[phi].sub.h] = 0.5 [alpha] = 0.056
Pure consumption tax reform [[phi].sub.h] = 1 [alpha] = 0.056
Reform with change in
 [alpha] [[phi].sub.h] = 0.5 [alpha] [??] 0.056

TABLE 3

Balanced-Growth Properties of Tax Systems with Intermediate Labor
Supply Elasticity

Tax System Tax Rate [mu] %

Graduated-rate system (U.S.) [[tau].sub.t]=0.253 1.800
 [[tau].sub.b]=0.35
Baseline flat tax reform [tau] = 0.3437 1.857
Reform with no change in
 [[phi].sub.k] [tau] = 0.3174 1.854
Reform with no change in
 [eta] [tau] = 0.3169 1.953
Reform with no change in n [tau] = 0.3431 1.638
Pure consumption tax reform [tau] = 0.3520 1.878
Reform with [[alpha].sub.new]
 /[[alpha].sub.old]=3 [tau] = 0.4119 1.660

 [c.sub.t]/ [i.sub.kt]/ [i.sub.ht]/
Tax System [y.sub.t] [y.sub.t] [y.sub.t]

Graduated-rate system (U.S.) 0.524 0.220 0.025
Baseline flat tax reform 0.487 0.262 0.021
Reform with no change in
 [[phi].sub.k] 0.504 0.244 0.021
Reform with no change in
 [eta] 0.483 0.262 0.024
Reform with no change in n 0.489 0.262 0.018
Pure consumption tax reform 0.481 0.262 0.026
Reform with [[alpha].sub.new]
 /[[alpha].sub.old]=3 0.488 0.262 0.019

 [k.sub.t]/ [h.sub.t]/ [l.sub.t]/
Tax System [y.sub.t] [k.sub.t] [e.sub.t]

Graduated-rate system (U.S.) 2.61 13.0 0.170
Baseline flat tax reform 3.08 9.86 0.173
Reform with no change in
 [[phi].sub.k] 2.87 11.0 0.173
Reform with no change in
 [eta] 3.02 9.98 0.176
Reform with no change in n 3.20 9.75 0.165
Pure consumption tax reform 3.06 10.0 0.172
Reform with [[alpha].sub.new]
 /[[alpha].sub.old]=3 3.18 9.78 0.166

Tax System [e.sub.t]

Graduated-rate system (U.S.) 0.120

Baseline flat tax reform 0.122
Reform with no change in
 [[phi].sub.k] 0.122
Reform with no change in
 [eta] 0.124
Reform with no change in n 0.117
Pure consumption tax reform 0.121
Reform with [[alpha].sub.new]
 /[[alpha].sub.old]=3 0.117

Notes: The computations assume [gamma]=3 such that
[([gamma]-1).sup.-1]=0.5. The remaining nontax parameters are shown in
the middle column of Table 1. [mu]=per capita output growth rate,
[[tau].sub.k]=effective tax rate for physical capital investment,
[[tau].sub.h]=effective tax rate for human capital investment,
[c.sub.t]/[y.sub.t]=ratio of consumption to output, [i.sub.kt]/
[[y.sub.t]=ratio of physical capital investment to output, [i.sub.kt]/
[y.sub.t]=ratio of human capital investment to output, [k.sub.t]/
[y.sub.t]=ratio of physical capital stock to output, [h.sub.t]/
[k.sub.t]=ratio of human to physical capital stocks, [l.sub.t]-
[e.sub.t]=fraction of time spent in working, [e.sub.t]=fraction of
time spent in school/training.

TABLE 4

Transition Paths for Selected Reforms with Intermediate Labor Supply
Elasticity

 Year log [k.sub.t]/ [h.sub.t]/
 ([y.sub.t]/ [y.sub.t] [k.sub.t]
 [y.sub.t-1])
 (%)

Graduated-
 rate system
 (U.S.) 0 1.800 2.61 13.0
Baseline flat
 tax reform
 (endogenous
 growth
 model) 1 0.324 2.65 13.0
 5 2.114 2.71 12.5
 10 2.068 2.77 12.0
 50 1.902 3.01 10.3
 [infinity] 1.857 3.08 9.86
Baseline flat
 tax reform
 (exogenous
 growth
 model) 1 0.324 2.65 13.0
 5 2.180 2.70 12.6
 10 2.125 2.76 12.1
 50 1.892 3.00 10.3
 [infinity] 1.800 3.11 9.69
Reform with
 no change
 in [eta]
 (endogenous
 growth
 model) 1 1.410 2.62 13.0
 5 2.200 2.67 12.5
 10 2.157 2.73 12.0
 50 1.997 2.96 10.4
 [infinity] 1.954 3.02 9.98
Reform with
 no change
 in [eta]
 (exogenous
 growth
 model) 1 1.410 2.62 13.0
 5 2.206 2.67 12.5
 10 2.147 2.73 12.0
 50 1.898 3.00 10.2
 [infinity] 1.800 3.11 9.49

 [l.sub.t]/ [e.sub.t]
 [e.sub.t]

Graduated-
 rate system
 (U.S.) 0.170 0.120
Baseline flat
 tax reform
 (endogenous
 growth
 model) 0.166 0.117
 0.167 0.118
 0.168 0.119
 0.172 0.122
 0.173 0.122
Baseline flat
 tax reform
 (exogenous
 growth
 model) 0.166 0.117
 0.167 0.118
 0.168 0.119
 0.172 0.121
 0.174 0.123
Reform with
 no change
 in [eta]
 (endogenous
 growth
 model) 0.169 0.119
 0.170 0.120
 0.171 0.121
 0.175 0.123
 0.176 0.124
Reform with
 no change
 in [eta]
 (exogenous
 growth
 model) 0.169 0.119
 0.170 0.120
 0.171 0.121
 0.175 0.124
 0.177 0.125

Notes: The computations assume [gamma]=3 such that
[([gamma] - 1).sup.-1]=0.5. The remaining nontax parameters are shown
in the middle column of Table 1. log([y.sub.t]/[y.sub.t-1])=per capita
output growth rate in year t, [k.sub.t]/[y.sub.t]=ratio of physical
capital stock to output, [h.sub.t]/[k.sub.t]=ratio of human to physical
capital stocks, [l.sub.t] - [e.sub.t]=fraction of time spent working,
[e.sub.t]=fraction of time spent in school/training.

(1.) For detailed descriptions and analyses of the various proposals, see the two conference volumes Frontiers of Tax Reform (Boskin 1996) and Economic Effects of Fundamental Tax Reform (Aaron and Gale 1996), and two U.S. government publications: Joint Committee on Taxation (1997) and U.S. Congressional Budget Office (1997).

(2.) A similar reform proposal was originally put forth by Friedman (1962, 175).

(3.) Other research that examines the quantitative effects of tax reform on long-run growth includes King and Rebelo (1990), Lucas (1990), Rebelo (1991), Jones et al. (1993) Pecorino (1993; 1994), Devereux and Love (1994), Laitner (1995), Milesi-Ferreni and Roubini (1998), Glomm and Ravikumar (1998), Ortigueira (1998), Kim (1998), Cassou and Lansing (2003), and Caucutt et al. (2003), among others.

(4.) Alternatively, we could model households as competitive entrepreneurs that operate their own production technology. Under this setup, we would replace [r.sub.t][k.sub.t] in equations (2)-(4) with the term [y.sub.t] - [w.sub.t][h.sub.t] ([l.sub.t]-[e.sub.t]). The resulting equilibrium decision rules would be identical to those derived here.

(5.) If business income was not paid out as dividends but instead reinvested by the firm, then the household would accrue capital gains from the resulting increase in the firm's stock price. Any realized capital gains would also be subject to a second round of taxation at the personal level.

(6.) The investment tax credit for equipment was abolished by the Tax Reform Act of 1986.

(7.) For further details on the tax treatment of human capital, see Quigley and Smolensky (1990) and Steurle (1996).

(8.) Kim (2003) shows that equation (5) can be viewed as a special case of a more general specification where [k.sub.t+1] = [A.sub.1][[(1 - [delta].sub.k])[k.sup.1-[sigma].sub.t] + [[delta].sub.k][([i.sub.kt]/[[delta].sub.k]).sup.1-[sigma]]].sup.1 /(1-[sigma]). Our setup implies [sigma] = 1, whereas a linear law of motion with no adjustment costs would imply [sigma] = 0. Aside from reflecting adjustment costs, our setup can be viewed as capturing the behavior of an aggregate stock that is measured by adding up different types of capital (structures, equipment, consumer durables, residential) that each display different depreciation characteristics.

(9.) Guo and Lansing (1998) employ a simplified version of equation (11) to study government stabilization policy.

(10.) Cassou and Lansing (2003) show that the common practice of modeling public expenditures as entirely wasteful (as opposed to providing utility) can lead to a substantial downward bias in the computed welfare gain from a growth-enhancing tax reform if these wasteful expenditures are held fixed relative to the size of the economy (as we do here). Our formulation for [G.sub.t] does not pose any problems for the present analysis because we are only interested in computing the growth and level effects of the tax reform, not welfare effects.

(11.) In this case, the equilibrium version of the household budget constraint (2) can be written as: [c.sub.t] + [[tau].sub.c]([c.sub.t] - [D.sub.t]) + [i.sub.kt] + [i.sub.ht] = [Y.sub.t], where ([c.sub.t] - [D.sub.t]) represents taxable consumption expenditures.

(12.) Cassou and Lansing (2003) show that the appropriate size of government can be an important factor in determining the welfare gains from fundamental tax reform.

(13.) Altig et al. (2001) employ a similar definition of revenue neutrality by holding tax revenues fixed when measured in effective units of labor. In our model, effective labor is given by hi ([l.sub.t] - [e.sub.t]), which grows at the same rate as output along the economy's balanced growth path.

(14.) Studies that restrict their attention to market activities obtain estimates of [h.sub.t]/[k.sub.t] [approximately equal to] 3. See Davies and Whalley (1991, appendix) for a review of various studies that estimate the aggregate value of human capital.

(15.) Private expenditures on education are from the Citibase series GAESE. R&D expenditure data are from National Patterns of R&D Resources: 1994, National Science Foundation (1995), table B-15 and p. 10. Physical capital and investment data are from Fixed Reproducible Tangible Wealth in the United States, U.S. Bureau of Economic Analysis (1998).

(16.) The after-tax interest rate r is defined by introducing privately issued real bonds (which exist in zero net supply) into the household budget constaint. The balanced-growth version of the first-order condition for bonds implies r = exp([mu] - log[beta]) - 1.

(17.) There were 42,228,108 joint returns filed in 1994, which accounted for a total taxable income of $1,800,054,965,000. Dividing the second number by the first yields a mean taxable income across all joint returns of $42,627. See U.S. Internal Revenue Service (1997, table 1, 25).

(18.) Gouveia and Strauss (1994) estimate U.S. average personal tax rates from 1979 to 1989 using a functional form that employs three parameters: Average Personal Tax Rate = b{1 - [[s[(Economic Income).sup.p] + 1].sup.-1/p}, where Economic Income is expressed in thousands of dollars. In our model, this specification would cause the asymptotic average and marginal tax rates to both equal b, because household income grows over time.

(19.) In 1994, total standard deductions were $397.1 billion. See U.S. Internal Revenue (1997, table 7, 197).

(20.) An alternative calibration strategy would be to choose [[phi].sub.k] such that the magnitude of expensed investment [[phi].sub.k][i.sub.kt] coincides with a measure of capital depreciation [delta][k.sub.t] where a standard linear law of motion for physical capital would imply [delta] = 1 - exp([mu]) + [i.sub.k]/k. Given the U.S. value of [i.sub.k]/k = 0.084, the alternative calibration strategy would yield [[phi].sub.k] = 0.785 and [[tau].sub.k] = 0.208. Use of these values would increase the growth gain of adopting a flat tax by about 0.01 to 0.04 percentage points.

(21.) Forgone earnings are given by [w.sub.t][h.sub.t][e.sub.t]. The total costs of producing human capital are given by [w.sub.t][h.sub.t][e.sub.t] + [i.sub.ht]. For our calibration, [w.sub.t][h.sub.t] [e.sub.t]/([w.sub.t][h.sub.t][e.sub.t] + [i.sub.ht]) = 0.948.

(22.) The total costs of producing human capital are given by ([w.sub.t][h.sub.t][e.sub.t] + [i.sub.ht]). The nontax-deductible portion of these costs are given by (1 - [[phi].sub.h])[i.sub.ht]. For our calibration (1 - [[phi].sub.h])[i.sub.ht]/([w.sub.t][h.sub.t][e.sub.t] + [i.sub.ht]) = 0.026.

(23.) In 1994 about two-thirds of U.S. taxpayers faced a federal income tax rate of 15%. If there was no change in the standard deduction (as we assume in the baseline reform experiment) then these taxpayers would have experienced an increase in their marginal tax rate under the Hall-Rabushka proposal, which specifies a flat tax rate of 19 percent. See Hall and Rabushka (1996, 44).

(24.) In an earlier version of this article (Cassou and Lansing 1996), we showed that it is possible to obtain much larger long-run growth effects using a model with log utility and an Arrow-Romer-type spillover mechanism for human capital accumulation. This type of human capital technology involves no forgone earnings.

(25.) A related result is obtained by Guo and Lansing (1997). They show that an accelerated depreciation allowance combined with a positive tax rate applied to all capital-type income is a low-distortion source of revenue because it can partially confiscate economic profits.

(26.) Arrow and Kurz (1970) consider the possibility of futurity divergence between the policy maker and private agents and discuss the implications for fiscal policy.

(27.) Padovano and Galli (2001) find that measures of effective marginal tax rates are negatively correlated with growth in a panel of 23 Organisation for Economic Co-operation and Development (OECD) countries from 1950 to 1990. They argue that most previous studies employ measures of effective average tax rates. This poses problems of collinearity with government expenditures that may either enhance or retard growth. A similar point is made by Kneller et al. (1999), who find that distortionary taxation reduces growth in a study of 22 OECD countries from 1970 to 1995.

(28.) See Eaton and Rosen (1980) and Hamilton (1987) for models that address this issue.

(29.) Altig et al. (2001) and Ventura (1999) consider the distributional consequences of tax reform in life-cycle models.

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KEVIN P. CASSOU and KEVIN J. LANSING *

* For helpful comments and suggestions, we thank William Gale, Greg Huffman, Kenneth Judd, Rodi Manuelli, Alvin Rabushka, B. Ravikumar, Ben Russo, and participants in the CEPR/Tilburg University Conference on Dynamic Aspects of Taxation, Tilburg, 8-10 September 2000. Part of this research was conducted while Lansing was a national fellow at the Hoover Institution, whose hospitality is gratefully acknowledged.

Cassou: Associate Professor, Department of Economics, Kansas State University, 327 Waters Hall, Manhattan, KS 66506. Phone 1-785-532-6342. Fax 1-785-532-6919, E-mail scassou@ksu.edu

Lansing: Senior Economist, Research Department, Federal Reserve Bank of San Francisco, P.O. Box 7702, San Francisco, CA 94120-7702. Phone 1-415-974-2393, Fax 1-415-977-4031, E-mail kevin.j.lansing@sf.frb.org