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Progressive taxation in sequential decisionmaking
Journal
of Public
Economics
16 (1981) 35552. North-Holland
Publishing
Company
PROGRESSIVE TAXATION IN SEQUENTIAL DECISIONMAKING Deterministic
and stochastic analysis
Steven A. LIPPMAN UCLA,
Received August
and John J. MCCALL
Los Angeles, CA 90024, USA
1980, revised version
received January
1981
This paper investigates the effect of progressive taxation on sequential decisionmaking. The decision is the archetypical problem of capital theory: determining the optimal cutting time for a tree that grows according to either a deterministic or stochastic process. For the deterministic model, the optimal cutting time is unaffected by proportional taxes, but decreases as taxes become more progressive. For the stochastic model, the major finding is that uncertainty can partially offset the inefftciency induced by progressive taxation; when this result holds the government benefits from increased uncertainty whereas the taxpayer is harmed. One noteworthy example of this is when the tree’s growth is governed by Brownian motion.
1. Introduction This paper studies the interplay between tax policy and sequential decisionmaking. The sequential decision is when to cut a tree (or sell a productive asset) whose growth obeys a law that may be deterministic or stochastic. This is, of course, the paradigm that has been used so successfully to motivate capital theory. The optimal cutting time depends on both the tax policy and the structure of the sequential process. Our intention is to assess the relation between efficiency and the degree of progressivity of the tax structure and between efficiency and the degree of uncertainty of the sequential decision process when decisionmakers have linear utility functions. The comparison of the optimal cutting times with and without taxes is utilized for discerning and measuring the efficiency of any particular tax structure. Using this measure of efficiency we find that, in general, the more progressive the tax structure the greater the inefficiency. However holding the tax policy fixed, increasing the uncertainty of the stochastic process can mitigate the inefficiency of progressive taxes. There are several distinct bodies of literature that relate to this problem; therefore, we begin the paper with a succinct survey in section 2. Following this literature survey, sections 3 and 4 present a deterministic model of efficiency and progressive taxation. The optimal cutting time is 000&0000/81/000&0000/$02.50
Q 1981 North-Holland
36
S.A. Lippman and J.J. McCall, Progressive taxation
investigated for proportional taxes and for progressive tax structures where progressivity is measured by the Arrow-Pratt index of absolute risk aversion. The major result in section 3 is that the cutting decision is unaffected by a regime with proportional taxes whereas progressive taxes hasten this decision, introducing an inefficiency. The impact of increased progressivity is considered in section 4 where it is shown that the optimal time to cut the tree decreases as the tax structure becomes more progressive. Section 5 contains a stochastic model of taxation and optimal cutting. The tree grows according to several different stochastic processes, the most important being Brownian motion. The primary finding is that uncertainty may act as a counterbalance to the inefficiency induced by progressive taxation. Under fairly reasonable conditions, it is shown that the government benefits from increased uncertainty whereas the taxpayer is injured. Other sources of uncertain income, including labor income, are added to the model, and we analyze the impact of increases in the certain and uncertain components of labor income. The concluding section lists several caveats and suggests additional research that appears promising.
2. Literature survey Earlier work on taxation and risk taking concentrated on the portfolio problem - the proper allocation of funds between risky and riskless assets - and ignored the timing problem - the appropriate time at which an asset should be bought or sold.’ In this paper we acknowledge the importance of the portfolio problem, but concentrate on the timing problem which surely must be included before any policy implications can be drawn from this work.2 Presumably it was the absence of this timing component in the stochastic portfolio theory that led Feldstein and Slemrod (1980) to conclude : Without an explicit theory of the optimal taxation of risky assets, it is not possible to say whether a move from equal tax rates on corporate and non-corporate investments (i.e., equal for any given taxpayer) to a system of unequal taxes causes a welfare gain or welfare loss.
‘See Arrow (1971), Domar and Musgrave (1944), Musgrave (1959), Tobin (1958), and especially Mossin (1958) and Stiglitz (1967). ‘In their recent book Atkinson and Stiglitz (1980) present a fairly general model of this portfolio problem. They distinguish between wealth and income taxation and also between private and social risk taking. In their analysis of income taxation they show that ‘there exists a critical value of the wealth elasticity of demand for the risky asset such that social risk-taking is increased for lower values and decreased for higher values’ (p. 106). They observe that this critical number depends on the length of the holding period. All of the analysis is conducted in a one-period setting, but this observation demonstrates the importance of the question: how long should the asset be held?
S.A. Lippman and J.J. McCall, Progressive taxation
37
One goal of this paper is to construct a stochastic theory in which one can begin to address welfare questions like these. It is well known that different types of tax policies have distinct effects on efficiency. For example, a tax on transactions induces a deadweight loss given by the familiar Harberger formula, whereas a true lump sum tax creates no deadweight loss. Furthermore, taxes that differ across industries For the most part the analysis generate costly resource reallocations. presented here ignores these important reallocations. That is, we do not measure the announcement effect created by taxing the timber industry differently from other industries nor do we measure the distorted capitallabor ratio induced by the particular tax rate on capital. Instead, inefficiency is simply measured by the deadweight loss created by not cutting the tree at the optimal time. The two types of tax studied are a progressive tax and a proportional tax. The effect of progressive income taxation on incentives is shown most dramatically in Baumol and Fischer (1979). They demonstrate that the quest for an equal income distribution through progressive taxes and transfers leads (in the limit) to a zero Gross National Product. We recognize the importance of these distributional effects but, because of their complexity, do not investigate them.3 In particular, we do not inquite as to the shape of the income distribution before and after the imposition of each progressive tax regime. In Brock, Rothschild and Stiglitz (1979), the theory of optimal stopping provides the foundation for a stochastic theory of capital. The main problem considered is the valuation of the asset (tree) in conjunction with its optimal selling (cutting) time. The value of the tree is continuously observable and is governed by a specific stochastic process. Their models’ are used to measure the effects of increased uncertainty on the optimal cutting time. They find it useful to distinguish between strictly increasing processes and more general processes such as Brownian motion. In our stochastic formulation of the growth process, which includes Brownian motion as a special case, the value of the tree is not monitored and, of course, our main focus is on the joint effect of uncertainty and progressive taxation on the optimal cutting time. Two recent applied papers are closely related to the models presented here. The first (Feldstein and Yitzhaki (1978)) is an econometric study of the relationship between capital gains taxations and the sale of common stock. They mention several proposed reforms and emphasize that both the choice among them and their practical importance hinges on the effect of taxes on investor’s decisions to sell assets. Their econometric analysis leads them to conclude that ‘investors are quite sensitive to tax considerations in their
3A comprehensive studies is presented
discussion of the theoretical and empirical issues that arise in redistribution in Lecture 9 of Atkinson and Stiglitz (1980).
38
S.A. Lippman and J.J. McCall, Progressive taxation
decisions to sell stock’. One of the proposals mentioned but not studied is to tax all realized capital gains at ordinary tax rates. This proposal is consistent with the Haig-Simons definition of income which, if strictly applied, would tax capital gains as they accumulate. 4 They anticipate (but do not prove) that sales should occur earlier if capital gains are taxed at ordinary rates. For many individuals the capital gains tax is roughly comparable to a proportional tax and hence this is an immediate consequence of our theorem 1. The Feldstein-Yitzhaki paper indicates other problems in financial analysis that are promising applications and extensions of the theory presented here. Finally, Schworm (1979) concludes that the optimal response to progressive as opposed to proportionate taxes is to replace equipment more frequently, thereby increasing maintenance expenditures and reducing taxes. Once again this behavior can be reexamined using the stochastic framework presented here. More importantly, an analysis like Schworm’s could enrich the neoclassical model used by Jorgenson (1963) in his pathbreaking research on investment behavior. Incorporating a model like this directly into the economic theory might provide valuable insights and justification for the sophisticated econometric methods used by Jorgenson and his coworkers.
3. Deterministic
models of efficiency
and progressivity
In much of the seminal work on capital theory [see Fisher (1930) and Wicksell (1901)] the tree is considered to be the basic asset. We continue this tradition and couch the discussion and analysis in the language of trees with cutting corresponding to selling a tree. Strictly speaking the results presented herein apply only to taxation and cutting policies for individual trees. The analysis requires some modification, presumably only cosmetic, if one is interested in the effect of taxation on the sale of a single financial asset and perhaps major changes if one wishes to investigate the relation between taxation and the management of a portfolio of financial assets. However, just as the simple Fisherian model clarified the underlying structure of capital theory, our hope is that the analysis presented here furnishes insight into these more complicated problems. In this section we assume that the growth of the tree is governed by a deterministic process. (Stochastic growth processes are studied in section 5.) Thus in the absence of taxes, establishing “Bailey (1969, p. 13) provides the rationale for these capital gains tax reforms: ‘Other things being equal, a growth of his wealth conveys the same benefit to a taxpayer whether it occurs through a diversion of ordinary income (for tax purposes) to the purchase of assets, through the sale of an asset at a profit and the purchase of a new one to replace it, or through the growth in value of an asset that is not sold at all. However, expected taxes differ according to the three cases, being zero for all accrued capital as long as they remain unrealized, and being lower for realized capital gains than for ordinary income.’
39
S.A. Lippman and J.J. McCall, Progressive taxation
an optimal cutting time corresponds exactly to the Fisher problem (with no replanting) and the Faustmann problem (with replanting). In contrast to the earlier work on risk taking and taxation, all of our analysis assumes that the individual has a linear utility function.
3.1. Optimal
cutting
with no taxes
We begin with the familiar Faustmann problem.5 Let u(t) denote the revenue generated by cutting a tree of age t so that u is the pertinent production function and, by describing the tree’s growth, v implicitly reveals its value. The economic agent can own but one tree at any given point in time. However, he owns (nonmarketable) planting rights; accordingly, he can at no cost replant a new tree (i.e. a tree of age 0) whenever he cuts the existing tree. The agent seeks to maximize the present value of the after-tax revenues (retained profits) that can be generated from the repeated sale of trees over the infinite time horizon, where r>O is the continuous-time discount factor (e mar is the present value of one dollar received at time t).’ In the absence of taxes, the value V(t) of cutting and replanting every t units of time is simply e
at
t > 0,
W=pI!(t),
and the first-order
condition
(1)
is
V.(r)=&
-i+u(t)+L~‘(t) i
I=o.
(2)
We shall assume that V is unimodal so that t,, the solution of (2), is unique. In addition, we assume that u(O)50 and c’(O)< x if v(O)=0 lest it be possible to earn an inifinite amount of money in a finite amount of time via a nonstationary policy.’ It is easily shown [see Lippman and McCall (1980)] that “L”, the value of planting rights at time 0 satisfies V=sup { V(t): t>O]. Consequently, the existence of an optimal policy that is stationary can be inferred from the simple condition ‘l” < xi.
5See Hwshleifer (1971) and Silberberg (1978). ‘We assume that r is correctly chosen, that is, no allocative ‘For instance, if v(r)=~‘t, of x in 4 units of time.
then cutting
ineffkiencies
are induced.
10” trees at the age of 10e2”, n= 1.2,. .., yields a
return
40
3.2. Optimal
S.A. Lippman and J.J. McCall,
cutting
with progressive
Progressive
taxation
taxation
The focus of our interest is the impact of taxes on the optimal time to sell the asset (cutting time). Let F(x,A) be the marginal tax rate associated with the (taxable) income x when F( .,i) is the established tax structure, 1.>0. (If 2=0, set F(x,A)=O.) Of course, O
R(t)=-
-
l-e-”
Let t, be any solution
R’(t) =&
at
u(t)
t>O.
j Cl-F(x,i)]dx, 0 to the first-order
(3)
condition
-+“[)I~-F(x,A)]dx
=o.
+ [l -F(u(t),l)]u’(t) I
(4)
Our first result states the rather intuitive fact that the progressive nature of the tax structure induces the agent to cut his trees sooner than he would have in the absence of taxes. Thus, the introduction of a progrssiue tax is inefficient in that the sum of present value of revenues available to the agent and the taxing authority falls from V(t,) to V(t,). Were the tax merely proportional, i.e. F(x,A)=z for all x, then inspection of (3) reveals that no inefficiency is introduced by the tax structure (as t,= to). It has been observed repeatedly in the literature on taxation that progressive taxes are inefficient in that other taxes can raise the same revenue with less reduction in total output (work effort).’ In our model progressive taxes are inefficient with respect to both lump-sum taxes” and proportional sour definition of progressivity -- nondecreasing marginal tax rate differs from the standard definition in which progressivity corresponds to an increasing average tax rate. For a discussion of these differences see Atkinson and Stiglitz (1980, p. 29ff). (If we assume that F( .,i) 1, then F( .,I) is simply the distribution function of a is right-continuous and lim,, ~ F&i.)= nonnegative random variable.) ‘See Musgrave (1959), Baumol and Fisher (1979), and Atkinson and Stiglitz (1980). “‘In our model the lump-sum tax is associated with the planting rights and not with the sale of trees. Suppose, however, there is a fixed tax K>O paid either in advance or at the time of each sale. The present value of retained profits associated with the strategy of cutting at time t -e-‘I) or e-“‘[u(t)-K]/(l --em”), respectively, and in either case exhibits the is[e -“‘v(t)-K]/(l first-order condition -[m/(1 -e-a”)][v(t) -K] +u’(t)=O. Consequently, it is clear from (2) that the optimal cutting time, label it t,, satisfies t, > t, and t, increases (without bound) as K increases.
S.A. Lippmun and J.J.
McCall,
Progressive
41
taxation
taxes in that the latter do not affect the cutting time whereas the former do. (Of course, the lump-sum tax must not be so large as to render production unprofitable.) Theorem >F(O,i), Proof:
taxes 1. Progressive then t,O. From
(4) and the mean
v’(t)
=L_e-ari 1
c(
=-
1
inefficient.
value theorem,
In particular,
if F(v(t,),i)
we have (O=
Wi)
s [l -F(x,i)]dx/[l o
v(t;,)
_e-;xri
are
It then follows from (2) that
1 -F(L,l)
c!
1 -F(v(t,),i.)
V’(t,)>O,
-F(v(t,),iL)]
>-----u(t,). 1 -e-“’
whence
V(. ).
t, < to by the unimodality of Q.E.D.
We briefly digress to note that the optimal cutting time is a decreasing function of the discount factor (x. Consequently, the largest cutting time corresponds to the time-average case which is described in footnote 11. (This last fact is used in section 4.2 of Lippman and McCall (1980) in order to obtain various bounds.) Theorem 2. Fix i and denote the solution to (4) by T, so T, is the optimal cutting time as a function of the discount factor CLIf v’>O and ~“50, then Tj
From
(4) with (d/da)R’( T,) = 0, we obtain
’ -u;;~e:~tg” [l -F(v,i,)]v’ T;=
cl[l-F(v,~)]u’+~‘~F,(v,i)-[1-F(v,ib)lv”
Now k(zt)-l-e-“‘-ate-“’ ~0 for at>0 for z>O, whence TLO. Q.E.D. 4. The impact
of increased
as k(O)=0
and
k’(z)=ze-‘>O
progressivity
In seeking to determine the differential impact between two tax structures, we need the concept of one tax structure being more progressive than another. If all reasonable tax structures were such that the change in the
42
S.A. Lippman and J.J.
McCall, Progressive
taxation
marginal tax rate decreases, then F( .,3.) would be a nondecreasing concave function not unlike a utility function (at least in its mathematical properties). This suggests adapting the Arrow-Pratt measure of absolute risk aversion as a device for calibrating progressivity. In particular, we shall say that the tax structure F( ., jL2) is more progressive than F( ., AI) whenever
- F,,b, 4) > _LWl) F,(x,i,)
=
for all
x 2 0.
F,(x, j., 1 ’
(6)
We do not, however, require either side of (6) to be nonnegative, for in so doing we would exclude from consideration a great many interesting families of tax structures. The usual notion of ‘the degree of progressivity’ of a tax structure is inextricably tied to the measurement of after-tax inequality. As revealed in (6), our method of calibrating the degree of progressivity stands alone, though it is clear that this method is also consistent with the measurement of inequality. As can be inferred from the notation F( ‘,i), we shall be considering a number of different families of tax structures. For convenience each family is parameterized so that an increase in 2 represents an increase in progressivity. Once again we note that most of the previous literature on progressivity supports belief in a positive relation between progressivity and inefficiency. Indeed, Baumol and Fischer (1979) find that in the limit the inefficiency is such that zero G.N.P. ensues. Here we measure inefficiency by the deviation between t, and t,. Presently we shall verify that under certain regularity conditions the more progressive the tax structure the less efficient is the agent’s decision in that the cutting time decreases; that is, r>,< 0. Henceforth we assume that F,( ‘,j.) and F,(.Y.. I csiht. Then substituting (4) into (d/di)R’(t,)-0 yields t> =f’(i)/h(i),
(7)
where [U = u(t,); t = t,],
(8) and h(i)=[l-F(U
9;)]l-e-z’ ” --------“-Cl 3:
1 -emu* - ---F,(c, x
i)d2.
-F(v,d)](l
-emat)u’
(9)
S.A. Lippman and J.J.
McCall,
Progressive
43
taxation
Because F 5 1 and F,zO, we need only assume v’(t)>0 and u”(t)~O for t 5 t, to ensure h(i)
3.
If v’>O, ~“~0, andf (i.)>O, then t;. < 0.
Thut is, the solution.
(10)
more
progressive
the
tax
structure
the
more
inefficient
the
Whereas the conclusion of theorem 3 -- a more progressive tax structure magnifies the extant inefficiencies - has important policy implications, its scope and applicability depends upon verifying the hypothesis f (>,)>O. In Lippman and McCall (1980) we have illustrated the efficacy of theorem 3 by demonstrating f (;l)>O whenever the family F( ‘,i) of tax structures is given by the Weibull or the Gamma (with integer shape parameter) cumulative distribution function. In addition, there is an intimate connection between the condition f(i)>0 and total positivity [see example 3 of Lippman and McCall (1980)]. For personal income taxes in the United States, the tax structure (as embodied in the tax tables) is a step function and is not concave. However, it is well-approximated by the concave structure”
F(x,i)= i with u=O.14
0,
x 5 3.4,
a+h[l_e-+3.4q,
x >
and b=0.56
(here income
is measured
“All of the analysis presented so far can be carried discounting. Thus, if the objective is to maximize
we obtain
the first-order
(11)
3.4, in thousands
out using
time averages
rather
than
condition
which lays bare the intuitive fact that the cutting time must be chosen retained revenue equals the time-average retained revenue. In addition, t;=f(j.)
of dollars).
such that
the marginal
lim h(i.). i a-o+
whence the analysis is unchanged. ‘*Of course this structure is only concave incomes under $3,400.
on the income
interval
[3.4, z);
we shall ignore
44
S.A. Lippman and J.J. McCall, Progressive taxation
This exponential approximation provides an exceedingly close fit with A =0.017. Indexing, which corresponds to decreasing the value of 2, is considered. The necessary and sufficient condition for indexing to be efficient (i.e. f(>.)>O) is obtained, and it is clear that this condition will often be satisfied. We also studied the effect of altering the maximum and minimum tax rates, and concluded that efficiency is enhanced by adopting a less progressive tax structure with fewer exclusions [see example 4 of Lippman and McCall (1980)].
5. Efficiency,
progressivity,
and production uncertainty
In analyzing the interaction between taxation and efficiency, it is essential that uncertainty be explicitly considered. In the deterministic model of sections 3 and 4 we saw how the tax structure affected the optimal cutting time. We shall soon see that the optimal cutting time is also influenced by the structure of the stochastic process governing the growth of value of the asset. To ignore uncertainty in the model’s specification is tantamount to attributing all of the fluctuations in cutting times to changes in the tax structure. Depending on the actual stochastic behavior, this could either overestimate or underestimate the efficiency of a particular tax structure. This section develops a sequential model of taxation in which the growth of the tree is governed by a fairly general stochastic process that includes Brownian motion as a special case. While we continue to characterize the asset of interest as purely physical (a tree), it should be emphasized that with minor modifications the analysis also applies to financial assets. There has been an extraordinary amount of research on the stochastic behavior of financial assets. It all began in 1900 with the seminal dissertation of Bachelier and, as is clear from scanning recent issues of finance journals, it has not yet reached its zenith. Most articles maintain that fluctuations in the market value of financial assets are best described by Browning motion. Renewed interest in Brownian motion stems from the ‘efficient market hypothesis,’ which, roughly speaking, prohibits clairvoyance and asserts that the stochastic process describing price movements has the Martingale property. Brownian motion possesses this property, is analytically tractable, and appears to have substantial empirical validity. The section commences with a detailed description of the stochastic production function that now replaces the deterministic function of sections 3 and 4. In order to concentrate on the influence of uncertainty, the progressivity of the tax structure is held constant as the level of uncertainty varies. Our first result (theorem 4) shows that the taxpayer’s welfare diminishes as uncertainty increases. The second result (theorem 4) states that the taxpayer and the tax collector (government) respond differently to increases in uncertainty. In particular, under certain mild assumptions (see
S.A. Lippman and J.J. McCall, Progressive taxation
45
theorems 5 and 6) and the optimal cutting time increases with uncertainty, thereby partially offsetting the inefficiency caused by progressivity. The later cutting time causes the government’s welfare to increase whie the taxpayer’s welfare declines. If this circumstance prevailed in the economy, it would pay the government to increase the uncertainty associated with production processes. Such an action would simultaneously counterbalance the inefficiency associated with progressivity and increase tax revenues. The stochastic process analyzed in theorem 6 is Brownian motion which enhances the importance of this conclusion. In particular, inflation and regulation can be regarded as a mechanism whereby industry’s production processes are rendered more variable. If this interpretation is valid and a Brownian economy obtains, then theorem 6 provides a novel rationale for inflation. Indeed, it is possible that a significant amount of risk enhancing interventions can be explained on these grounds. There are, however, nonBrownian stochastic processes and exponential tax structures for which the cutting time decreases with increased uncertainty, so there are efficiency losses and taxpayers and government alike are harmed by increased uncertainty. Two examples of this phenomenon are presented. This section concludes with a brief analysis of the impact of an uncertain labor income upon the optimal cutting time. 5.1. An analysis of production
uncertainty
We consider a sequential model of taxation in which the growth of the tree There are two distinct ways of is governed by a stochastic process. incorporating uncertainty into the decisionmaking. In the first the tree is cut in conjunction with continuous dynamic updating of knowledge about the state of the stochastic process, whereas in the second the cutting time is selected in advance. Thus in the first version the cutting time is a random variable while in the second it is simply a point in time. This section presents an analysis of the second version. (See Brock, Rothschild, and Stiglitz (1979) for a model with continuous monitoring.) We replace the deterministic production function u be the stochastic production function X, where X is given by X(Go)=u(t)+flZg(t),
0 2 0,
t > 0,
(12)
with EZ=O, EZ2 = 1, 020, and gZ0. We interpret g and IJ as the growth of the stochastic component of the tree’s price13 and the amount of uncertainty, 13The value of the tree at any time is composed of both quantity and price dimensions. The quantity of timber grows according to one stochastic process and the price of lumber is determined by a second stochastic process. We assume that the stochastic process discussed here describes the value of the tree at any time and hence reflects the influence of these two subsidiary stochastic processes.
S.A. Lippman and J.J. McCall, Progressive taxation
46
respectively. Our formulation enables us to consider both the additive well as the multiplicative case. Examples of the former include g(t)-1,
g(t)=t,
and
case as
g(t)=J;.
The last example, when coupled with the assumption that Z is normally distributed corresponds to Brownian motion. From the previous discussion it is clear that Brownian motion is the most compelling example to consider. Similarly, examples of the multiplicative case include g(t)=u(t),
g(t)=tv(t),
and
g(t)=fiv(t).
The first of these examples is the most interesting and frequently encountered because it corresponds to those situations in which uncertainty is proportional to the asset’s value; this example is explicitly considered in theorem 7. In order to isolate the impact of increased uncertainty, fix 1, and write F‘(x) in place of F(x,i). The agent now seeks to maximize - ZIr a(t,~)=~Ex’~B’[l-F(x)]dx.
(
0
Assume that q .,o) is unimodal unique solution to the first-order
Wt, 0) dt
e
-
so that condition
the optimal cutting [X =X(t, a)],
=w(t,0)+E[(v’(t) e-“’
1 -e-“’
-F(X))]
=o. 1
define the government’s e-“’
%P)=~E
time r0 is the
3f
+oZg’(t))(l
For convenience,
13)
(14)
take C!?and the total value V by
X(1.B)
s
0
F(x)dx
(15)
and ~~(t,,)=.~(t,.)+g(t,.)‘l_e~EX(t,”)=T/(t).
The following lemma is useful in carrying out the analysis. of Lippman and McCall (1981) for a proof of the lemma.)
(16) (See section
5
S.A.
Lemma
1.
and J.J.
Progressive
If k>O, k’
taxation
41
then
EZk(Z)
4.
(17)
The individual’s welfare decreases as CJincreases, that is,
$7 T,,cT)
(18)
if T,< t, and zA>O, then the government’s
take increases
with 0,
(19) ProoJ: Because k(Z) = 1 -F (v(t) + aZg(t)) of Z, we can conclude from lemma 1 that
is a positive
decreasing
function
g(tMZC1 -FW(t, a))11 is negative,
consequently,
r~r < CJ~yields
2(%,> ~1)~~(5,2’~1)>~(Za2’~.2). Similarly,
k(Z) =F(v(t)
I- lx G, say 0 = or. From
+ oZg(t))
is an increasing
the unimodality
function
of Z whence
of V and t,, < t, we can infer that
By hypothesis, zil > 0 so T,,ol Coupling both inequalities with rol
and
gz not
too
large.
The government’s gain associated with an increase in the level G of uncertainty stems from two disjoint sources. First, V(r,), the size of the pie
Lippman and J.J. McCall, Progressive taxation
48
that is to be divided between the government and the individual, grows with the level of uncertainty when the cutting time increases toward t,. Second, suppose the individual did not respond to the increase in uncertainty, leaving the cutting time unchanged. Then the mere fact that the tax structure is progressive means that the total tax bill is a convex increasing function so the expected taxes paid varies directly with the level CJof uncertainty. Whereas it was not surprising that the individual’s welfare decreases as the degree 0 of uncertainty increases, we were a bit surprised to discover that increased uncertainty can serve to ameliorate the inefficiency associated with a progressive tax structure. Whenever the conditions r, < t, and rb > 0 hold, the individual and the government will have opposite preferences in regard to the level of uncertainty. Using F(x)= 1 -e- Ax and either g(t)= 1 or g(t) =Jt, our experience with numerical examples suggests that the hypothesis ~,
5.2. Examples
wherein increases
in uncertainty
benefit the government
We now present two theorems in which the conditions that imply opposite preferences for uncertainty are satisfied.14 Henceforth, we assume that F” and g” exist, then substituting (14) into (d/do)%?(r,, D) = 0 yields
where [z-r,; g, g’, g”, v, v’ and evaluated at X(q,, o)], f(a)=E{clZg(l
-F)-
+ (1 -e-“‘)Zg(v’+
VI’ are evaluated
(1 -e-“‘)Zg’(l
at r,;
F and
F’ are
-F)
oZg’)F’}
(21)
and h(a)=E{-(l-e-“‘)a(v’+aZg’)(l-F)+(l-e-a’)
(22) Theorem
5.
If v’>O, ~“50, F”O.
14Theorems 5 and 6 also hold if the condition F”
S.A. Lippman and J.J.
McCall,
Progressive
49
taxation
Proof:
The progressivity of the tax structure ensures F’ZO so that h(.a)O, g’& 0 and u” 50, g”sO. Because F”
A heuristic explanation of theorem 5 is provided by the fact that, in conjunction with the progressivity of the tax structure, uncertainty introduces a ‘constant’ loss at each sale one which increases with 6. Therefore, increasing t, as B increases leads to fewer such losses per unit time. The heuristic does not apply to our next result (theorem 6), for there g(t) =J? so that uncertainty increases with the cutting time. Since J? corresponds to Brownian motion, however, we consider theorem 6 to be of considerable practical importance. Theorem increasing Proof
6. If v’>O, v” 50, F”
Clearly
h(0) ~0,
and EZ=O
und
g(t)=$,
implies f(O)=O,
then
q,
is
so TL=O. From
strictly
(21) we
infer
o=o
= E{
- ctF’(Zg)’
+ (1 - e-“‘)Z2gg’F’
+ (1 - e-ar)(Zg)2u’F” = {g2u’(l -e-“‘)F”+gF’[2(1
+ (1 -e-ar)Z2gg’F’} -e-“‘)g’-ccg]}EZ2
~0 (23)
because
g[2(l-e-“‘)g’-ccg]=l-e-“‘-ctr~O. Q.E.D.
5.3. Examples
wherein increases
in uncertainty
are deleterious
We begin by proving that increased uncertainty can cause the cutting time to decrease. This is done for the simple multiplicative case with a convex tax structure (i.e. F” > 0). Theorem
7.
If v’ > 0, VI’50,
a neighbourhood
F” > 0, and g = v, then 5, is strictly decreasing
in
of 0.
Proof. Inspection of (22) reveals that h(o)O, i t suffices to verify that 2(1- e-“)g’ - ug > 0. But from
50
S.A. Lippman and J.J.
21’> 0, the unimodality
McCall,
taxation
of K r,, < t, and (2) we obtain
I_e”g>ur-~u’~v a
2g’-
Progressive
e-are
yqJ>o. Q.E.D.
The following examples employ the concave (P’O - w h ence t: < 0 for 0 sufficiently small ~ is equivalent to demonstrating that k(t,) >O, where
k(t)-
-i.g(t)u’(t)(l
-e-“)+2(1
-e-a*)gf(t)-rg(t).
(24)
Using g(t)=t or g(t)=t’, with either u(t)=ln t or u(t)=r-K, it is not difficult [see Lippman and McCall (1980)] to provide ranges of numerical values of c(, A, and K for which k(t,) > 0.
5.4. Uncertain
labor income
Finally, we consider another aspect of uncertainty that impinges on the optimal cutting time. Here the taxpayer has labor income in addition to the income created by tree cutting, and uncertainty resides in the (labor) income stream generated by assets other than trees. Specifically we focus on the individual’s human capital and assume that each tax planning period the individual receives a labor income of y+ oZ, where a20 and EZ =O. Thus, labor income is the sum of random and nonrandom components. It can be shown [see theorem 9 of Lippman and McCall (1980)] that, increases in uncertainty decrease the under relatively mild conditions, optimal cutting time, thereby increasing the inefficiency of the tax structure. On the other hand, the impact of an increase in the certain component 4’ of the income stream is indeterminate [see theorem 10 of Lippman and McCall (1980)]: for convex tax structures the cutting time is reduced whereas the cutting time is increased for the concave approximation to the U.S. tax structure (see eq. (11)). 6. Conclusion The purpose of this paper was to introduce taxation into a sequential decisionmaking model. In this environment we analyzed the effects of and increased risk on the optimal progressivity, increased progressivity, cutting times. From this we then deduced the consequences for efficiency; in some cases we could distinguish between the welfare of the taxpayer and the
S.A. Lippman and J.J.
McCull, Progressive
taxation
51
welfare of the tax collector. When uncertainty is described by Brownian motion we showed that increased variability caused the cutting time to increase, thereby partially offsetting the inefficiency (earlier cutting) of progressive taxation. Furthermore, in this same Brownian environment, increased risk was detrimental to the taxpayer but beneficial for the tax collector. While these results are interesting, much remains to be done. The first task is to enlarge the scope of the one period portfolio problem to encompass the timing of sales and the effect of taxation on the sequential composition of the portfolio. Second, the manner in which uncertainty enters the models of section 5 should be extended to include continuous monitoring. That is, the cutting time should be a random variable that depends on current information about the state of the stochastic process. The third area for future research is to determine the income distribution associated with different progressive tax structures. Here, of course, the focus has been exclusively on efficiency issues. Perhaps this efficiency analysis can be meshed with the new literature on ‘optimum taxation’. Finally, it is clear that the valuation of assets is not exogenous to the tax structure as our analysis has assumed. Rather asset values should be endogenously determined within a model that specifies the tax structure. Thus a rather sophisticated equilibrium model must be designed before we can have confidence in our partial equilibrium conclusions.
Arrow, K.J., 1971. Essays in the theory of risk bearing (Markham,
Chicago). Atkinson, A.B. and J.E. Stiglitz, 1980, Lectures on public economics (McGraw-Hill, New York). Bailey, M., 1969, Capital gains and income taxation, in: A. Harberger, ed., The taxation of income from capital (Brookings, Washington). Baumol, W.J. and D. Fisher, 1979, The output distribution frontier: Alternatives of income taxes and transfer for strong equality goals, American Economic Review 69, 514525. Boskin, M.J., 1978, Taxation, saving, and the rate of interest, Journal of Political Economy 86, S3-S28. Brock, W., M. Rothschild and J.E. Stiglitz, 1979, Notes on stochastic capital theory, unpublished paper. Domar, E.D. and R.A. Musgrave, 1944, Proportional income taxation and risk-taking, Quarterly Journal of Economics 58, 388422. Feldstein, M., 1973, On the optimal progressivity of the income tax, Journal of Public Economics 2, 357-376. Feldstein, M., 1978, The welfare cost of capital income taxation, Journal of Political Economy 86, S299S52. Feldstein, M. and J. Slemrod, 1980, Personal ta::ation, portfolio choice, and the effect of the corporate income tax. Journal of Political Economy 88, 854866. Feldstein, M. and S. Yitzhaki, 1978. The effects of the capital gains tax on the selling and switching of common stock, Journal of Public Economics 9, 17-36. Fisher, I., 1930, The theory of interest (Macmillan, New York). Harberger, A.C., 1974, Taxation and welfare (Little, Brown, Boston). Hirshleifer, J., 1971, Investment, interest, and capital (Prentice-Hall, Englewood Cliffs). Jorgenson, D.W., 1963, Capital theory and investment behavior, American Economic Review 53, 247-259.
52
S.A. Lippman and J.J. McCall, Progressive
taxation
Lippman, S.A. and J.J. McCall, 1980, Progressive taxation in sequential decisionmaking: Deterministic and stochastic analysis, Western Management Science Institute Working Paper No. 304, UCLA, p. 58. Lippman, S.A. and J.J. McCall, 1981, The economics of uncertainty: Selected topics and probabilistic methods, in: K.J. Arrow and M. Intriligator, eds., Handbook of mathematical economics (North-Holland, Amsterdam). Mossin, J., 1968, Taxation and risk-taking: An expected utility approach, Economica 35, 74-82. Musgrave, R.A., 1959, The theory of public finance (McGraw-Hill, New York). Musgrave, R.A. and P.B. Musgrave, 1973, Public finance in theory and practice (McGraw-Hill, New York). Naslund, B., 1968, Some effects of taxes on risk-taking, Review of Economic Studies 35, 2899306. Schworm, W.E., 1979, Tax policy, capital use, and investment incentives, Journal of Public Economics 12, 191-204. Silberberg, E., 1978, The structure of economics (McGraw-Hill, New York). Stiglitz, J.E., 1967, The effects of income, wealth and capital gains taxation on risk-taking, Quarterly Journal of Economics 83, 263-283. Tobin, J., 1958, Liquidity preference as behavior toward risk, Review of Economic Studies 25, 65586. Wicksell, K., 1901, Lectures on political economy, volume 1. English edition, 1934 (George Routledge and Sons, London).
of Public
Economics
16 (1981) 35552. North-Holland
Publishing
Company
PROGRESSIVE TAXATION IN SEQUENTIAL DECISIONMAKING Deterministic
and stochastic analysis
Steven A. LIPPMAN UCLA,
Received August
and John J. MCCALL
Los Angeles, CA 90024, USA
1980, revised version
received January
1981
This paper investigates the effect of progressive taxation on sequential decisionmaking. The decision is the archetypical problem of capital theory: determining the optimal cutting time for a tree that grows according to either a deterministic or stochastic process. For the deterministic model, the optimal cutting time is unaffected by proportional taxes, but decreases as taxes become more progressive. For the stochastic model, the major finding is that uncertainty can partially offset the inefftciency induced by progressive taxation; when this result holds the government benefits from increased uncertainty whereas the taxpayer is harmed. One noteworthy example of this is when the tree’s growth is governed by Brownian motion.
1. Introduction This paper studies the interplay between tax policy and sequential decisionmaking. The sequential decision is when to cut a tree (or sell a productive asset) whose growth obeys a law that may be deterministic or stochastic. This is, of course, the paradigm that has been used so successfully to motivate capital theory. The optimal cutting time depends on both the tax policy and the structure of the sequential process. Our intention is to assess the relation between efficiency and the degree of progressivity of the tax structure and between efficiency and the degree of uncertainty of the sequential decision process when decisionmakers have linear utility functions. The comparison of the optimal cutting times with and without taxes is utilized for discerning and measuring the efficiency of any particular tax structure. Using this measure of efficiency we find that, in general, the more progressive the tax structure the greater the inefficiency. However holding the tax policy fixed, increasing the uncertainty of the stochastic process can mitigate the inefficiency of progressive taxes. There are several distinct bodies of literature that relate to this problem; therefore, we begin the paper with a succinct survey in section 2. Following this literature survey, sections 3 and 4 present a deterministic model of efficiency and progressive taxation. The optimal cutting time is 000&0000/81/000&0000/$02.50
Q 1981 North-Holland
36
S.A. Lippman and J.J. McCall, Progressive taxation
investigated for proportional taxes and for progressive tax structures where progressivity is measured by the Arrow-Pratt index of absolute risk aversion. The major result in section 3 is that the cutting decision is unaffected by a regime with proportional taxes whereas progressive taxes hasten this decision, introducing an inefficiency. The impact of increased progressivity is considered in section 4 where it is shown that the optimal time to cut the tree decreases as the tax structure becomes more progressive. Section 5 contains a stochastic model of taxation and optimal cutting. The tree grows according to several different stochastic processes, the most important being Brownian motion. The primary finding is that uncertainty may act as a counterbalance to the inefficiency induced by progressive taxation. Under fairly reasonable conditions, it is shown that the government benefits from increased uncertainty whereas the taxpayer is injured. Other sources of uncertain income, including labor income, are added to the model, and we analyze the impact of increases in the certain and uncertain components of labor income. The concluding section lists several caveats and suggests additional research that appears promising.
2. Literature survey Earlier work on taxation and risk taking concentrated on the portfolio problem - the proper allocation of funds between risky and riskless assets - and ignored the timing problem - the appropriate time at which an asset should be bought or sold.’ In this paper we acknowledge the importance of the portfolio problem, but concentrate on the timing problem which surely must be included before any policy implications can be drawn from this work.2 Presumably it was the absence of this timing component in the stochastic portfolio theory that led Feldstein and Slemrod (1980) to conclude : Without an explicit theory of the optimal taxation of risky assets, it is not possible to say whether a move from equal tax rates on corporate and non-corporate investments (i.e., equal for any given taxpayer) to a system of unequal taxes causes a welfare gain or welfare loss.
‘See Arrow (1971), Domar and Musgrave (1944), Musgrave (1959), Tobin (1958), and especially Mossin (1958) and Stiglitz (1967). ‘In their recent book Atkinson and Stiglitz (1980) present a fairly general model of this portfolio problem. They distinguish between wealth and income taxation and also between private and social risk taking. In their analysis of income taxation they show that ‘there exists a critical value of the wealth elasticity of demand for the risky asset such that social risk-taking is increased for lower values and decreased for higher values’ (p. 106). They observe that this critical number depends on the length of the holding period. All of the analysis is conducted in a one-period setting, but this observation demonstrates the importance of the question: how long should the asset be held?
S.A. Lippman and J.J. McCall, Progressive taxation
37
One goal of this paper is to construct a stochastic theory in which one can begin to address welfare questions like these. It is well known that different types of tax policies have distinct effects on efficiency. For example, a tax on transactions induces a deadweight loss given by the familiar Harberger formula, whereas a true lump sum tax creates no deadweight loss. Furthermore, taxes that differ across industries For the most part the analysis generate costly resource reallocations. presented here ignores these important reallocations. That is, we do not measure the announcement effect created by taxing the timber industry differently from other industries nor do we measure the distorted capitallabor ratio induced by the particular tax rate on capital. Instead, inefficiency is simply measured by the deadweight loss created by not cutting the tree at the optimal time. The two types of tax studied are a progressive tax and a proportional tax. The effect of progressive income taxation on incentives is shown most dramatically in Baumol and Fischer (1979). They demonstrate that the quest for an equal income distribution through progressive taxes and transfers leads (in the limit) to a zero Gross National Product. We recognize the importance of these distributional effects but, because of their complexity, do not investigate them.3 In particular, we do not inquite as to the shape of the income distribution before and after the imposition of each progressive tax regime. In Brock, Rothschild and Stiglitz (1979), the theory of optimal stopping provides the foundation for a stochastic theory of capital. The main problem considered is the valuation of the asset (tree) in conjunction with its optimal selling (cutting) time. The value of the tree is continuously observable and is governed by a specific stochastic process. Their models’ are used to measure the effects of increased uncertainty on the optimal cutting time. They find it useful to distinguish between strictly increasing processes and more general processes such as Brownian motion. In our stochastic formulation of the growth process, which includes Brownian motion as a special case, the value of the tree is not monitored and, of course, our main focus is on the joint effect of uncertainty and progressive taxation on the optimal cutting time. Two recent applied papers are closely related to the models presented here. The first (Feldstein and Yitzhaki (1978)) is an econometric study of the relationship between capital gains taxations and the sale of common stock. They mention several proposed reforms and emphasize that both the choice among them and their practical importance hinges on the effect of taxes on investor’s decisions to sell assets. Their econometric analysis leads them to conclude that ‘investors are quite sensitive to tax considerations in their
3A comprehensive studies is presented
discussion of the theoretical and empirical issues that arise in redistribution in Lecture 9 of Atkinson and Stiglitz (1980).
38
S.A. Lippman and J.J. McCall, Progressive taxation
decisions to sell stock’. One of the proposals mentioned but not studied is to tax all realized capital gains at ordinary tax rates. This proposal is consistent with the Haig-Simons definition of income which, if strictly applied, would tax capital gains as they accumulate. 4 They anticipate (but do not prove) that sales should occur earlier if capital gains are taxed at ordinary rates. For many individuals the capital gains tax is roughly comparable to a proportional tax and hence this is an immediate consequence of our theorem 1. The Feldstein-Yitzhaki paper indicates other problems in financial analysis that are promising applications and extensions of the theory presented here. Finally, Schworm (1979) concludes that the optimal response to progressive as opposed to proportionate taxes is to replace equipment more frequently, thereby increasing maintenance expenditures and reducing taxes. Once again this behavior can be reexamined using the stochastic framework presented here. More importantly, an analysis like Schworm’s could enrich the neoclassical model used by Jorgenson (1963) in his pathbreaking research on investment behavior. Incorporating a model like this directly into the economic theory might provide valuable insights and justification for the sophisticated econometric methods used by Jorgenson and his coworkers.
3. Deterministic
models of efficiency
and progressivity
In much of the seminal work on capital theory [see Fisher (1930) and Wicksell (1901)] the tree is considered to be the basic asset. We continue this tradition and couch the discussion and analysis in the language of trees with cutting corresponding to selling a tree. Strictly speaking the results presented herein apply only to taxation and cutting policies for individual trees. The analysis requires some modification, presumably only cosmetic, if one is interested in the effect of taxation on the sale of a single financial asset and perhaps major changes if one wishes to investigate the relation between taxation and the management of a portfolio of financial assets. However, just as the simple Fisherian model clarified the underlying structure of capital theory, our hope is that the analysis presented here furnishes insight into these more complicated problems. In this section we assume that the growth of the tree is governed by a deterministic process. (Stochastic growth processes are studied in section 5.) Thus in the absence of taxes, establishing “Bailey (1969, p. 13) provides the rationale for these capital gains tax reforms: ‘Other things being equal, a growth of his wealth conveys the same benefit to a taxpayer whether it occurs through a diversion of ordinary income (for tax purposes) to the purchase of assets, through the sale of an asset at a profit and the purchase of a new one to replace it, or through the growth in value of an asset that is not sold at all. However, expected taxes differ according to the three cases, being zero for all accrued capital as long as they remain unrealized, and being lower for realized capital gains than for ordinary income.’
39
S.A. Lippman and J.J. McCall, Progressive taxation
an optimal cutting time corresponds exactly to the Fisher problem (with no replanting) and the Faustmann problem (with replanting). In contrast to the earlier work on risk taking and taxation, all of our analysis assumes that the individual has a linear utility function.
3.1. Optimal
cutting
with no taxes
We begin with the familiar Faustmann problem.5 Let u(t) denote the revenue generated by cutting a tree of age t so that u is the pertinent production function and, by describing the tree’s growth, v implicitly reveals its value. The economic agent can own but one tree at any given point in time. However, he owns (nonmarketable) planting rights; accordingly, he can at no cost replant a new tree (i.e. a tree of age 0) whenever he cuts the existing tree. The agent seeks to maximize the present value of the after-tax revenues (retained profits) that can be generated from the repeated sale of trees over the infinite time horizon, where r>O is the continuous-time discount factor (e mar is the present value of one dollar received at time t).’ In the absence of taxes, the value V(t) of cutting and replanting every t units of time is simply e
at
t > 0,
W=pI!(t),
and the first-order
condition
(1)
is
V.(r)=&
-i+u(t)+L~‘(t) i
I=o.
(2)
We shall assume that V is unimodal so that t,, the solution of (2), is unique. In addition, we assume that u(O)50 and c’(O)< x if v(O)=0 lest it be possible to earn an inifinite amount of money in a finite amount of time via a nonstationary policy.’ It is easily shown [see Lippman and McCall (1980)] that “L”, the value of planting rights at time 0 satisfies V=sup { V(t): t>O]. Consequently, the existence of an optimal policy that is stationary can be inferred from the simple condition ‘l” < xi.
5See Hwshleifer (1971) and Silberberg (1978). ‘We assume that r is correctly chosen, that is, no allocative ‘For instance, if v(r)=~‘t, of x in 4 units of time.
then cutting
ineffkiencies
are induced.
10” trees at the age of 10e2”, n= 1.2,. .., yields a
return
40
3.2. Optimal
S.A. Lippman and J.J. McCall,
cutting
with progressive
Progressive
taxation
taxation
The focus of our interest is the impact of taxes on the optimal time to sell the asset (cutting time). Let F(x,A) be the marginal tax rate associated with the (taxable) income x when F( .,i) is the established tax structure, 1.>0. (If 2=0, set F(x,A)=O.) Of course, O
R(t)=-
-
l-e-”
Let t, be any solution
R’(t) =&
at
u(t)
t>O.
j Cl-F(x,i)]dx, 0 to the first-order
(3)
condition
-+“[)I~-F(x,A)]dx
=o.
+ [l -F(u(t),l)]u’(t) I
(4)
Our first result states the rather intuitive fact that the progressive nature of the tax structure induces the agent to cut his trees sooner than he would have in the absence of taxes. Thus, the introduction of a progrssiue tax is inefficient in that the sum of present value of revenues available to the agent and the taxing authority falls from V(t,) to V(t,). Were the tax merely proportional, i.e. F(x,A)=z for all x, then inspection of (3) reveals that no inefficiency is introduced by the tax structure (as t,= to). It has been observed repeatedly in the literature on taxation that progressive taxes are inefficient in that other taxes can raise the same revenue with less reduction in total output (work effort).’ In our model progressive taxes are inefficient with respect to both lump-sum taxes” and proportional sour definition of progressivity -- nondecreasing marginal tax rate differs from the standard definition in which progressivity corresponds to an increasing average tax rate. For a discussion of these differences see Atkinson and Stiglitz (1980, p. 29ff). (If we assume that F( .,i) 1, then F( .,I) is simply the distribution function of a is right-continuous and lim,, ~ F&i.)= nonnegative random variable.) ‘See Musgrave (1959), Baumol and Fisher (1979), and Atkinson and Stiglitz (1980). “‘In our model the lump-sum tax is associated with the planting rights and not with the sale of trees. Suppose, however, there is a fixed tax K>O paid either in advance or at the time of each sale. The present value of retained profits associated with the strategy of cutting at time t -e-‘I) or e-“‘[u(t)-K]/(l --em”), respectively, and in either case exhibits the is[e -“‘v(t)-K]/(l first-order condition -[m/(1 -e-a”)][v(t) -K] +u’(t)=O. Consequently, it is clear from (2) that the optimal cutting time, label it t,, satisfies t, > t, and t, increases (without bound) as K increases.
S.A. Lippmun and J.J.
McCall,
Progressive
41
taxation
taxes in that the latter do not affect the cutting time whereas the former do. (Of course, the lump-sum tax must not be so large as to render production unprofitable.) Theorem >F(O,i), Proof:
taxes 1. Progressive then t,
(4) and the mean
v’(t)
=L_e-ari 1
c(
=-
1
inefficient.
value theorem,
In particular,
if F(v(t,),i)
we have (O=
Wi)
s [l -F(x,i)]dx/[l o
v(t;,)
_e-;xri
are
It then follows from (2) that
1 -F(L,l)
c!
1 -F(v(t,),i.)
V’(t,)>O,
-F(v(t,),iL)]
>-----u(t,). 1 -e-“’
whence
V(. ).
t, < to by the unimodality of Q.E.D.
We briefly digress to note that the optimal cutting time is a decreasing function of the discount factor (x. Consequently, the largest cutting time corresponds to the time-average case which is described in footnote 11. (This last fact is used in section 4.2 of Lippman and McCall (1980) in order to obtain various bounds.) Theorem 2. Fix i and denote the solution to (4) by T, so T, is the optimal cutting time as a function of the discount factor CLIf v’>O and ~“50, then Tj
From
(4) with (d/da)R’( T,) = 0, we obtain
’ -u;;~e:~tg” [l -F(v,i,)]v’ T;=
cl[l-F(v,~)]u’+~‘~F,(v,i)-[1-F(v,ib)lv”
Now k(zt)-l-e-“‘-ate-“’ ~0 for at>0 for z>O, whence TL
of increased
as k(O)=0
and
k’(z)=ze-‘>O
progressivity
In seeking to determine the differential impact between two tax structures, we need the concept of one tax structure being more progressive than another. If all reasonable tax structures were such that the change in the
42
S.A. Lippman and J.J.
McCall, Progressive
taxation
marginal tax rate decreases, then F( .,3.) would be a nondecreasing concave function not unlike a utility function (at least in its mathematical properties). This suggests adapting the Arrow-Pratt measure of absolute risk aversion as a device for calibrating progressivity. In particular, we shall say that the tax structure F( ., jL2) is more progressive than F( ., AI) whenever
- F,,b, 4) > _LWl) F,(x,i,)
=
for all
x 2 0.
F,(x, j., 1 ’
(6)
We do not, however, require either side of (6) to be nonnegative, for in so doing we would exclude from consideration a great many interesting families of tax structures. The usual notion of ‘the degree of progressivity’ of a tax structure is inextricably tied to the measurement of after-tax inequality. As revealed in (6), our method of calibrating the degree of progressivity stands alone, though it is clear that this method is also consistent with the measurement of inequality. As can be inferred from the notation F( ‘,i), we shall be considering a number of different families of tax structures. For convenience each family is parameterized so that an increase in 2 represents an increase in progressivity. Once again we note that most of the previous literature on progressivity supports belief in a positive relation between progressivity and inefficiency. Indeed, Baumol and Fischer (1979) find that in the limit the inefficiency is such that zero G.N.P. ensues. Here we measure inefficiency by the deviation between t, and t,. Presently we shall verify that under certain regularity conditions the more progressive the tax structure the less efficient is the agent’s decision in that the cutting time decreases; that is, r>,< 0. Henceforth we assume that F,( ‘,j.) and F,(.Y.. I csiht. Then substituting (4) into (d/di)R’(t,)-0 yields t> =f’(i)/h(i),
(7)
where [U = u(t,); t = t,],
(8) and h(i)=[l-F(U
9;)]l-e-z’ ” --------“-Cl 3:
1 -emu* - ---F,(c, x
i)d2.
-F(v,d)](l
-emat)u’
(9)
S.A. Lippman and J.J.
McCall,
Progressive
43
taxation
Because F 5 1 and F,zO, we need only assume v’(t)>0 and u”(t)~O for t 5 t, to ensure h(i)
3.
If v’>O, ~“~0, andf (i.)>O, then t;. < 0.
Thut is, the solution.
(10)
more
progressive
the
tax
structure
the
more
inefficient
the
Whereas the conclusion of theorem 3 -- a more progressive tax structure magnifies the extant inefficiencies - has important policy implications, its scope and applicability depends upon verifying the hypothesis f (>,)>O. In Lippman and McCall (1980) we have illustrated the efficacy of theorem 3 by demonstrating f (;l)>O whenever the family F( ‘,i) of tax structures is given by the Weibull or the Gamma (with integer shape parameter) cumulative distribution function. In addition, there is an intimate connection between the condition f(i)>0 and total positivity [see example 3 of Lippman and McCall (1980)]. For personal income taxes in the United States, the tax structure (as embodied in the tax tables) is a step function and is not concave. However, it is well-approximated by the concave structure”
F(x,i)= i with u=O.14
0,
x 5 3.4,
a+h[l_e-+3.4q,
x >
and b=0.56
(here income
is measured
“All of the analysis presented so far can be carried discounting. Thus, if the objective is to maximize
we obtain
the first-order
(11)
3.4, in thousands
out using
time averages
rather
than
condition
which lays bare the intuitive fact that the cutting time must be chosen retained revenue equals the time-average retained revenue. In addition, t;=f(j.)
of dollars).
such that
the marginal
lim h(i.). i a-o+
whence the analysis is unchanged. ‘*Of course this structure is only concave incomes under $3,400.
on the income
interval
[3.4, z);
we shall ignore
44
S.A. Lippman and J.J. McCall, Progressive taxation
This exponential approximation provides an exceedingly close fit with A =0.017. Indexing, which corresponds to decreasing the value of 2, is considered. The necessary and sufficient condition for indexing to be efficient (i.e. f(>.)>O) is obtained, and it is clear that this condition will often be satisfied. We also studied the effect of altering the maximum and minimum tax rates, and concluded that efficiency is enhanced by adopting a less progressive tax structure with fewer exclusions [see example 4 of Lippman and McCall (1980)].
5. Efficiency,
progressivity,
and production uncertainty
In analyzing the interaction between taxation and efficiency, it is essential that uncertainty be explicitly considered. In the deterministic model of sections 3 and 4 we saw how the tax structure affected the optimal cutting time. We shall soon see that the optimal cutting time is also influenced by the structure of the stochastic process governing the growth of value of the asset. To ignore uncertainty in the model’s specification is tantamount to attributing all of the fluctuations in cutting times to changes in the tax structure. Depending on the actual stochastic behavior, this could either overestimate or underestimate the efficiency of a particular tax structure. This section develops a sequential model of taxation in which the growth of the tree is governed by a fairly general stochastic process that includes Brownian motion as a special case. While we continue to characterize the asset of interest as purely physical (a tree), it should be emphasized that with minor modifications the analysis also applies to financial assets. There has been an extraordinary amount of research on the stochastic behavior of financial assets. It all began in 1900 with the seminal dissertation of Bachelier and, as is clear from scanning recent issues of finance journals, it has not yet reached its zenith. Most articles maintain that fluctuations in the market value of financial assets are best described by Browning motion. Renewed interest in Brownian motion stems from the ‘efficient market hypothesis,’ which, roughly speaking, prohibits clairvoyance and asserts that the stochastic process describing price movements has the Martingale property. Brownian motion possesses this property, is analytically tractable, and appears to have substantial empirical validity. The section commences with a detailed description of the stochastic production function that now replaces the deterministic function of sections 3 and 4. In order to concentrate on the influence of uncertainty, the progressivity of the tax structure is held constant as the level of uncertainty varies. Our first result (theorem 4) shows that the taxpayer’s welfare diminishes as uncertainty increases. The second result (theorem 4) states that the taxpayer and the tax collector (government) respond differently to increases in uncertainty. In particular, under certain mild assumptions (see
S.A. Lippman and J.J. McCall, Progressive taxation
45
theorems 5 and 6) and the optimal cutting time increases with uncertainty, thereby partially offsetting the inefficiency caused by progressivity. The later cutting time causes the government’s welfare to increase whie the taxpayer’s welfare declines. If this circumstance prevailed in the economy, it would pay the government to increase the uncertainty associated with production processes. Such an action would simultaneously counterbalance the inefficiency associated with progressivity and increase tax revenues. The stochastic process analyzed in theorem 6 is Brownian motion which enhances the importance of this conclusion. In particular, inflation and regulation can be regarded as a mechanism whereby industry’s production processes are rendered more variable. If this interpretation is valid and a Brownian economy obtains, then theorem 6 provides a novel rationale for inflation. Indeed, it is possible that a significant amount of risk enhancing interventions can be explained on these grounds. There are, however, nonBrownian stochastic processes and exponential tax structures for which the cutting time decreases with increased uncertainty, so there are efficiency losses and taxpayers and government alike are harmed by increased uncertainty. Two examples of this phenomenon are presented. This section concludes with a brief analysis of the impact of an uncertain labor income upon the optimal cutting time. 5.1. An analysis of production
uncertainty
We consider a sequential model of taxation in which the growth of the tree There are two distinct ways of is governed by a stochastic process. incorporating uncertainty into the decisionmaking. In the first the tree is cut in conjunction with continuous dynamic updating of knowledge about the state of the stochastic process, whereas in the second the cutting time is selected in advance. Thus in the first version the cutting time is a random variable while in the second it is simply a point in time. This section presents an analysis of the second version. (See Brock, Rothschild, and Stiglitz (1979) for a model with continuous monitoring.) We replace the deterministic production function u be the stochastic production function X, where X is given by X(Go)=u(t)+flZg(t),
0 2 0,
t > 0,
(12)
with EZ=O, EZ2 = 1, 020, and gZ0. We interpret g and IJ as the growth of the stochastic component of the tree’s price13 and the amount of uncertainty, 13The value of the tree at any time is composed of both quantity and price dimensions. The quantity of timber grows according to one stochastic process and the price of lumber is determined by a second stochastic process. We assume that the stochastic process discussed here describes the value of the tree at any time and hence reflects the influence of these two subsidiary stochastic processes.
S.A. Lippman and J.J. McCall, Progressive taxation
46
respectively. Our formulation enables us to consider both the additive well as the multiplicative case. Examples of the former include g(t)-1,
g(t)=t,
and
case as
g(t)=J;.
The last example, when coupled with the assumption that Z is normally distributed corresponds to Brownian motion. From the previous discussion it is clear that Brownian motion is the most compelling example to consider. Similarly, examples of the multiplicative case include g(t)=u(t),
g(t)=tv(t),
and
g(t)=fiv(t).
The first of these examples is the most interesting and frequently encountered because it corresponds to those situations in which uncertainty is proportional to the asset’s value; this example is explicitly considered in theorem 7. In order to isolate the impact of increased uncertainty, fix 1, and write F‘(x) in place of F(x,i). The agent now seeks to maximize - ZIr a(t,~)=~Ex’~B’[l-F(x)]dx.
(
0
Assume that q .,o) is unimodal unique solution to the first-order
Wt, 0) dt
e
-
so that condition
the optimal cutting [X =X(t, a)],
=w(t,0)+E[(v’(t) e-“’
1 -e-“’
-F(X))]
=o. 1
define the government’s e-“’
%P)=~E
time r0 is the
3f
+oZg’(t))(l
For convenience,
13)
(14)
take C!?and the total value V by
X(1.B)
s
0
F(x)dx
(15)
and ~~(t,,)=.~(t,.)+g(t,.)‘l_e~EX(t,”)=T/(t).
The following lemma is useful in carrying out the analysis. of Lippman and McCall (1981) for a proof of the lemma.)
(16) (See section
5
S.A.
Lemma
1.
and J.J.
Progressive
If k>O, k’
taxation
41
then
EZk(Z)
4.
(17)
The individual’s welfare decreases as CJincreases, that is,
$7 T,,cT)
(18)
if T,< t, and zA>O, then the government’s
take increases
with 0,
(19) ProoJ: Because k(Z) = 1 -F (v(t) + aZg(t)) of Z, we can conclude from lemma 1 that
is a positive
decreasing
function
g(tMZC1 -FW(t, a))11 is negative,
consequently,
r~r < CJ~yields
2(%,> ~1)~~(5,2’~1)>~(Za2’~.2). Similarly,
k(Z) =F(v(t)
I- lx G, say 0 = or. From
+ oZg(t))
is an increasing
the unimodality
function
of Z whence
of V and t,, < t, we can infer that
By hypothesis, zil > 0 so T,,
and
gz not
too
large.
The government’s gain associated with an increase in the level G of uncertainty stems from two disjoint sources. First, V(r,), the size of the pie
Lippman and J.J. McCall, Progressive taxation
48
that is to be divided between the government and the individual, grows with the level of uncertainty when the cutting time increases toward t,. Second, suppose the individual did not respond to the increase in uncertainty, leaving the cutting time unchanged. Then the mere fact that the tax structure is progressive means that the total tax bill is a convex increasing function so the expected taxes paid varies directly with the level CJof uncertainty. Whereas it was not surprising that the individual’s welfare decreases as the degree 0 of uncertainty increases, we were a bit surprised to discover that increased uncertainty can serve to ameliorate the inefficiency associated with a progressive tax structure. Whenever the conditions r, < t, and rb > 0 hold, the individual and the government will have opposite preferences in regard to the level of uncertainty. Using F(x)= 1 -e- Ax and either g(t)= 1 or g(t) =Jt, our experience with numerical examples suggests that the hypothesis ~,
5.2. Examples
wherein increases
in uncertainty
benefit the government
We now present two theorems in which the conditions that imply opposite preferences for uncertainty are satisfied.14 Henceforth, we assume that F” and g” exist, then substituting (14) into (d/do)%?(r,, D) = 0 yields
where [z-r,; g, g’, g”, v, v’ and evaluated at X(q,, o)], f(a)=E{clZg(l
-F)-
+ (1 -e-“‘)Zg(v’+
VI’ are evaluated
(1 -e-“‘)Zg’(l
at r,;
F and
F’ are
-F)
oZg’)F’}
(21)
and h(a)=E{-(l-e-“‘)a(v’+aZg’)(l-F)+(l-e-a’)
(22) Theorem
5.
If v’>O, ~“50, F”
14Theorems 5 and 6 also hold if the condition F”
S.A. Lippman and J.J.
McCall,
Progressive
49
taxation
Proof:
The progressivity of the tax structure ensures F’ZO so that h(.a)
A heuristic explanation of theorem 5 is provided by the fact that, in conjunction with the progressivity of the tax structure, uncertainty introduces a ‘constant’ loss at each sale one which increases with 6. Therefore, increasing t, as B increases leads to fewer such losses per unit time. The heuristic does not apply to our next result (theorem 6), for there g(t) =J? so that uncertainty increases with the cutting time. Since J? corresponds to Brownian motion, however, we consider theorem 6 to be of considerable practical importance. Theorem increasing Proof
6. If v’>O, v” 50, F”
Clearly
h(0) ~0,
and EZ=O
und
g(t)=$,
implies f(O)=O,
then
q,
is
so TL=O. From
strictly
(21) we
infer
o=o
= E{
- ctF’(Zg)’
+ (1 - e-“‘)Z2gg’F’
+ (1 - e-ar)(Zg)2u’F” = {g2u’(l -e-“‘)F”+gF’[2(1
+ (1 -e-ar)Z2gg’F’} -e-“‘)g’-ccg]}EZ2
~0 (23)
because
g[2(l-e-“‘)g’-ccg]=l-e-“‘-ctr~O. Q.E.D.
5.3. Examples
wherein increases
in uncertainty
are deleterious
We begin by proving that increased uncertainty can cause the cutting time to decrease. This is done for the simple multiplicative case with a convex tax structure (i.e. F” > 0). Theorem
7.
If v’ > 0, VI’50,
a neighbourhood
F” > 0, and g = v, then 5, is strictly decreasing
in
of 0.
Proof. Inspection of (22) reveals that h(o)
50
S.A. Lippman and J.J.
21’> 0, the unimodality
McCall,
taxation
of K r,, < t, and (2) we obtain
I_e”g>ur-~u’~v a
2g’-
Progressive
e-are
yqJ>o. Q.E.D.
The following examples employ the concave (P’
k(t)-
-i.g(t)u’(t)(l
-e-“)+2(1
-e-a*)gf(t)-rg(t).
(24)
Using g(t)=t or g(t)=t’, with either u(t)=ln t or u(t)=r-K, it is not difficult [see Lippman and McCall (1980)] to provide ranges of numerical values of c(, A, and K for which k(t,) > 0.
5.4. Uncertain
labor income
Finally, we consider another aspect of uncertainty that impinges on the optimal cutting time. Here the taxpayer has labor income in addition to the income created by tree cutting, and uncertainty resides in the (labor) income stream generated by assets other than trees. Specifically we focus on the individual’s human capital and assume that each tax planning period the individual receives a labor income of y+ oZ, where a20 and EZ =O. Thus, labor income is the sum of random and nonrandom components. It can be shown [see theorem 9 of Lippman and McCall (1980)] that, increases in uncertainty decrease the under relatively mild conditions, optimal cutting time, thereby increasing the inefficiency of the tax structure. On the other hand, the impact of an increase in the certain component 4’ of the income stream is indeterminate [see theorem 10 of Lippman and McCall (1980)]: for convex tax structures the cutting time is reduced whereas the cutting time is increased for the concave approximation to the U.S. tax structure (see eq. (11)). 6. Conclusion The purpose of this paper was to introduce taxation into a sequential decisionmaking model. In this environment we analyzed the effects of and increased risk on the optimal progressivity, increased progressivity, cutting times. From this we then deduced the consequences for efficiency; in some cases we could distinguish between the welfare of the taxpayer and the
S.A. Lippman and J.J.
McCull, Progressive
taxation
51
welfare of the tax collector. When uncertainty is described by Brownian motion we showed that increased variability caused the cutting time to increase, thereby partially offsetting the inefficiency (earlier cutting) of progressive taxation. Furthermore, in this same Brownian environment, increased risk was detrimental to the taxpayer but beneficial for the tax collector. While these results are interesting, much remains to be done. The first task is to enlarge the scope of the one period portfolio problem to encompass the timing of sales and the effect of taxation on the sequential composition of the portfolio. Second, the manner in which uncertainty enters the models of section 5 should be extended to include continuous monitoring. That is, the cutting time should be a random variable that depends on current information about the state of the stochastic process. The third area for future research is to determine the income distribution associated with different progressive tax structures. Here, of course, the focus has been exclusively on efficiency issues. Perhaps this efficiency analysis can be meshed with the new literature on ‘optimum taxation’. Finally, it is clear that the valuation of assets is not exogenous to the tax structure as our analysis has assumed. Rather asset values should be endogenously determined within a model that specifies the tax structure. Thus a rather sophisticated equilibrium model must be designed before we can have confidence in our partial equilibrium conclusions.
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taxation
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