Are tax rates too volatile?

• Author: Hess, Gregory D.

1. Introduction

   The way in which a government chooses to raise tax revenue over time has attracted considerable attention both in the tax literature, and the political business cycle literature. The tax literature has emphasized planning problems where the government chooses taxes and a path for government debt to minimize the excess burden of taxes over time (see Barro [21 and Lucas and Stokey [15]). Alternatively, political business cycle models focus on the fiscal policies pursued by incumbent government to either improve their reelection prospects or to constrain the policies that future government in power will pursue (see Alesina and Tabellini [1], Hess [11], Persson and Svensson [21], Rogoff and Sibert [23]).

   This paper addresses the question of whether tax rates fluctuate "in excess" of movements in economic fundamentals. I consider, below, a simple theory of optimal taxation which implies that tax rates should follow a random walk. Under this null hypothesis, variance bounds on the intertemporal government budget constraint are derived. The methodology of evaluating these variance bounds corresponds to the stock market volatility literature pioneered by Leroy and Porter [13] and Shiller [28].

   Using the methodology of Mankiw, Romer and Shapiro (hereafter M-R-S) [16; 17], these bounds are calculated for United States data from 1870-1989. There are two main benefits to using the M-R-S methodology. First, it uses non-central rather than central variances which eliminates the bias due to estimating the sample means. Second, the M-R-S methodology allows the forcing variables (government expenditures and seignorage) to be potentially non-stationary series. The inability of earlier volatility tests literature to allow for non-stationarity in the forcing variables was a considerable drawback.

   It is found that broad movements in tax rates that correspond to relatively large permanent changes in government expenditures are adequately "smoothed". However, it appears that tax rates have been excessively volatile in the United States both in the time period before World War I and after World War II. This suggests that either governments in power have manipulated taxes to fulfill political objective or that the random theory of taxation is an over-simplification of optimal taxation.

   The outline of the paper is as follows. Section H presents the simple linear-quadratic model of taxation and derives the inequality bounds for the volatility tests. Section HI presents the calculation of the inequality bounds and discusses the results. In section IV, results from Monte Carlo Isimulations of die model, under a range of specifications, show the robustness of the empirical results. Section V explores both business cycle and political business cycle explanations for the excess volatility of taxes. I conclude in section VI.

2. The Linear-Quadratic Model

   The linear-quadratic model of intertemporal taxation assumes there exists an infinitely lived Benevolent Social Planner (BSP) who chooses taxes to smooth over time the excess burden of taxation. The BSP chooses tax rates and a path for government debt to minimize die expected discounted value of tax distortions, subject to a sequence of budget constraints, an initial condition, and a transversality condition:

   [Mathematical Expression Omitted]

   where [a.sub.1] and [a.sub.2 ] and [b.sub.t] are tax revenue, government expenditures, seignorage, and government debt outstanding at the end of time t, respectively, as ratios of national output. The time discount factor and the real interest rate factor are [beta] and R, respectively. [I.sub.t] is the information set available to the Benevolent social planner at time t. Among other things, [I.sub.t] contains {[g.sub.j]}.sup.t.sub.j] = - [infinity], {[s.sub.j]}.sup.t.sub.j] = - [infinity] and [b.sub.t-1]. Assuming E[[beta]R] [is nearly equal to] 1, the sequence of optimality conditions is:

   [E.sub.t]{[tau].sub.t + 1]\[I.sub.t]} = [[tau].sub.t .] (5)

   Expression (5) is the null hypothesis that tax rates should follow a random walk.(1) Bizer and Durlauf [4], and Sahasakul [24] have explored this relationship and have found it to be violated. In order to analyze the fundamental sources of this rejection, I consider the question of whether tax rates are too volatile as compared to the expected present value of the BSP's net obligations.

   Substituting expression (2) forward and using expressions (3) and (4), the intertemporal government budget constraint is:

   [Mathematical Expression Omitted]

   The left hand side of (6) is the present value of expected current and future government expenditures and the right hand side is the present value of expected current and future tax revenue.(2) Under the null hypothesis that tax rates are optimal, expression (5) is substituted into (6) to obtain the optimal tax rate at time t as a function of the ratios of the amount of debt outstanding and the expected present value of current and fumm govemnient spending less seignorage to national output:

   [Mathematical Expression Omitted]

   Define [tau.sup.p.sub.t] to be the "perfect foresight" or "ex-port rational" tax rate. This is the tax rate that should have prevailed if taxes were optimal and the BSP had perfect over government expenditures and seignorage. Since expectations are a linear operator, the perfect foresight over government rate is:

   [Mathematical Expression Omitted]

   In contrast to the perfect foresight tax rate, let the "naive" tax rate be a tax rate that is based upon an ad hoc forecast of the model's forcing variables. This "rule of thumb" uses an information set at time t, [H.sub.t], that is a strict subset of the information set at time t, [I.sub.t], that rational agents actually possess. For purposes of this exercise, the "naive" processes for {[g.sub.t ]} and {[s.sub.t]} are:

   [E.sub.t]{[g.sub.t+k]\[H.sub.t]} = [g.sub.t] k = 1,2,... (9a)

   [E.sub.t]{[s.sub.t+k]\[H.sub.t]} = [s.sub.t] k = 1,2,... (9b)

   Substituting these ad hoc processes into expression (7) yields the "naive" tax rate:

   [tau.sup.N.sub.g] = [g.sub.t] - [s.sub.t] + (1 - [beta])[b.sub.t-1]. (10)

   Following M-R-S, expressions (7), (8), and (10) are related as foflows. First the identity:

   ([tau.sup.p.sub.t] - [tau.sup.N.sub.t]) = ([tau.sup.p.sub.t] - [tau.sub.t]

   + ([tau.sub.t] - [tau.sup.N.sub.t]). (11)

   Squaring both sides and taking expectations with respect to the appropriate conditioning information sets yields:

   [Mathematical Expression Omitted]

   where use is made of ([tau.sup.p.sub.t] - [tau.sub.]) being uncorrelated with [I.sub.t], and hence is uncorrelated with ([tau.sub.t] - [tau.sup.N.sub.t]). [W.sub.t] is any scale variable that is known at time t, and is included primarily to deal with trending in the series which can lead to heteroskedasticity problems in estimation. Since both right hand side terms of (12) are positive, the left hand side term must be greater than each right hand side component. Using the law of iterated expectations, the model's theoretical implications are summarized as follows:

   [Mathematical Expression Omitted]

   From expression (13a), the actual tax rate better reflects the economic fundamentals of optimal taxation than a tax rate based upon a naive forecast of the future forcing variables. From expression (13b), the naive forecast better predicts the actual tax rate than the perfect foresight tax rate. This is because the naive tax rate is based on [H.sub.1] and the realized tax rate is based upon [I.sub.t], where [H.sub.t] [subset] [I.sub.t]. However, the perfect foresight tax rate is based upon the true path of realized forcing variables which supercedes the information contained in [I.sub.t]. Expression (13c) is a restatement of expression (12). A rejection of equality (13c) would be due to the non-orthogonality of ([tau.sup.p.sub.t] - [tau.sub.t]) and ([tau.sup.N.sub.t]), namely that the forecast effors using the perfect foresight tax rate to predict the actual tax rate and the forecast errors using the naive tax rate to predict the actual tax rate are correlated. If, for example, this correlation were positive, then [S.sub.3] would be negative.

   To make the perfect foresight tax rate operational, expression (8) must be augmented to reflect that the infinite path of government expenditures and seignorage is unobservable. To circumvent this problem, the future forcing variables that are as yet unrealized are substituted out. This is achieved by using (8) to solve for [tau.sup.P.sub.T], which under the null hypothesis of optimal taxation allows one to solve out [Mathematical Expression Omitted]. After algebraic manipulations, the perfect foresight tax rate, is re-wtitten as:

   [Mathematical Expression Omitted]

3. Results

   Figure 1 presents plots of the tax rate, government spending ratio, the end of period government debt outstanding ratio and the seignorage ratio. The plot of...