Capital gains taxation in an economy with an ‘Austrian sector’

Journal

of Public

Economics

CAPITAL

21 (1983) 215-256.

GAINS

North-Holland

TAXATION IN AN ECONOMY ‘AUSTRIAN SECTOR’ Daniel

WITH

AN

J. KOVENOCK

University of Wisconsin-Madison, Madison, WI 53706, USA

Michael

ROTHSCHILD*

University of Wisconsin-Madison, Madison, WI 53706, USA and National Bureau of Economic Rcscwrch

Received

August

1981, revised version

recrL$cd September

1982

This paper examines the effects of a proportional capital gains tax in an economy with an Austrian sector (with wine and trees) and an ordinary sector. We analyze the effect of capital gains taxation (on both an accrual and a realization basis) on the efficiency with which resources are used within the Austrian sector. Since time is the only input which can be varied in the Austrian sector, this amounts to looking at the effect of capital gains taxation on the harvesting time or selling time of assets. Accrual taxation decreases the selling time of Austrian assets. Realization taxation decreases the selling time of some Austrian assets and leaves it unchanged for others. Inflation further reduces the selling time of assets taxed on an accrual basis; often, but not always, inflation increases the selling time of Austrian assets taxed on a realization basis. We also examine the effect of the special tax treatment of capital gains at death and find that the current U.S. tax system, under which capital gains taxes are waived at death, encourages investors to hold assets longer. In contrast to these results - which suggest that the capital gains tax can reduce the holding period of an asset - we show that there is a sense in which such taxes (at least when levied on a realization basis) discourage transactions and increase holding periods. It is never profitable to change the ownership of an Austrian asset between the time of the original constraint and the ultimate harvesting of the asset for final use. Capital gains taxation diverts resources from the Austrian sector to the ordinary sector. The effects on the efficiency of the allocation of investment between sectors are complicated and not easy to summarize.

1. Introduction

an

This paper Austrian

examines the effects of a capital gains tax in an economy with sector and an ordinary sector. In the ordinary sector the

*We thank Alan Auerbach, David Bradford, Peter Diamond, Steve Lippman and Eytan Sheshinski for helpful conversations. Research support from the National Science Foundation and the University of Wisconsin Graduate School is gratefully acknowledged. Some of the work on this paper was done while Rothschild held the Oskar Morgenstem Distinguished Fellowship at Mathematics, Inc. The research reported here is part of the NBER’s research program in Taxation. Any opinions expressed are those of the authors and not those of the National Bureau of Economic Research.

0047-2727/83/.$3.00

0

1983, Elsevier Science Publishers

B.V. (North-Holland)

216

D.J. Kovenock and M. Rothschild, Capital xains taxation

investment process is straightforward; an investment of a dollar produces a stream of returns in the future. Assets in the Austrian sector do not yield a stream of services. They increase in value as they age; when they are harvested this growth stops. The standard examples of Austrian investments are wine and trees, and we shall use these terms to refer to two different kinds of Austrian assets. Trees are assets which require scarce resources i.e. ones which earn rents while they age. In considering when to cut down a tree it is important to take account of the fact that the land on which a tree now stands can be used to grow another tree once its present occupant is felled. Wine, in distinction, uses no scarce resources as it matures; the casks and cellars in which wine ages earn no rents.’ We focus on the effects of capital gains taxation on Austrian investments for two reasons. First, the effects are simple and straightforward. We are able to obtain exact expressions for the effect of capital gains taxes on the allocation of investment between sectors and for the efficiency of resource use within the Austrian sector. We extend some of this analysis to the case where the death of the investor (which has significant tax consequences in the U.S.) is uncertain. Secondly, a capital gains tax affects the timing of transactions. With Austrian assets these effects have real, easily-understood consequences. Many previous studies, empirical and theoretical, have, implicitly or explicitly, focused on the effect of capital gains taxation on purely financial assets. It seems clear from first principles that capital gains taxes should, at least in an inflationary world, operate as a kind of turnover tax, inhibiting the sales of stocks and bonds which show gains and encouraging the sale of assets on which losses have been incurred. Constantinides (1983) and Constantinides and Scholes (1980) have shown just how appealing the strategy of realizing losses and never selling gains can be when well-organized futures markets with low transactions costs exist. Feldstein, Slemrod and Yitzhaki (1980), Feldstein and Slemrod (1978), and Feldstein and Yitzhaki (1978) have presented evidence that this theory is correct - that the capital gains tax does indeed inhibit the sale of assets whose prices have increased since they were purchased. It is not clear, at least to us, what the real effect of such an inhibition is. In general, sales of stock are purely financial transactions and do not lead to investment or disinvestment of real assets. While it seems quite likely that the volume of transactions has some effect on the efficiency of the allocation of investment, to our knowledge no one has framed a theory which will allow us to determine the preferred volume of transactions. ‘Wine and trees are polar cases. What distinguishes them is whether the factor which earns a rent in production is destroyed in the process of production (wine) or is reusable (trees). A more general model would consider production with both factors; a still more general model would allow substitution between these factors. For an outline of such a model, see Samuelson (1976). We suspect that only the negative results of this paper will survive these generalizations.

D.J. Kovenock and M. Rothschild, Capital gains taxation

217

Without such a theory one cannot say whether discouraging transactions is good or bad. Our analysis assumes that capital gains taxes fall only on Austrian assets. Thus, we can only analyze the effect of the capital gains tax on Austrian assets. The ordinary sector is in our model for two reasons: first, we assume that it is sufficiently large that the after-tax rate of return is determined outside of the Austrian sector and is independent of the rate and form of the capital gains tax; secondly, having an ordinary sector allows us to examine the effect of capital gains taxation on the allocation of resources between Austrian and other investments. Section 2 sets out the basic model and analyzes the effects of capital gains taxation on Austrian investments. Taxation, in particular the capital gains tax, can affect economic efficiency in two quite different ways. First, taxes change the flow of investment between sectors. The intersectoral allocation of investment is efficient when the discounted value of all returns - public and private - from a dollar of investment is the same in all sectors. A social discount rate must be chosen to make such a comgarison. In this paper we use the gross or pretax rate of return in the ordinary sector. If there is a tax in the ordinary sector, and no taxes in the Austrian sector, then resources invested in the ordinary sector have a higher social rate of return than resources invested in either wine or trees. The other effect of capital gains taxation is on the allocation of resources within a sector. For Austrian investments, the resource whose use can be varied is time. Time is allocated optimally if the wine is sold, or the tree cut down, at the time which maximizes the discounted social value of resources in the Austrian sector. If income from the ordinary sector is taxed while there is no taxation in the Austrian sectors, Austrian assets are held too long. The absence of a capital gains tax induces a kind of lock-in effect. We analyze the effect of capital gains taxation on these two distortions. We consider two kinds of capital gains taxes: those levied on a realization basis are collected only when an asset is sold and those levied on an accrual basis are, at least in principle, collected whenever an asset’s market value changes. The results of our analysis may be summarized as follows.2 Imposing a capital gains tax (on either a realization or accrual basis) drives resources away from Austrian sectors and thus increases the rate of return on Austrian investments. If the tax rates in the Austrian sectors are the same as the rates in the ordinary sector, with one exception, the social rate of return on Austrian assets is less than the rate of return in the ordinary sector. The

‘In an earlier version of this paper we also considered the effect of uncertainty about the growth of Austrian assets; since the main conclusion of that exercise was that considering uncertainty did not change the character of our results, we have dropped it from our paper.

218

D.J. Kovenock

and M. Rothschild,

Capital gains taxation

exception is wine taxed on an accrual basis. When tax rates are the same, the social rate of return on wine taxed on an accrual basis is the same as the social rate of return in the ordinary sector. Since wine is a polar case, this suggests that from the point of view of intersectoral allocative efficiency, the capital gains tax rate ought to be higher than the effective tax rate on ordinary investments. The results on intrasectoral efficiency are somewhat more complicated. In all but one case the capital gains tax has a kind of reverse lock-in effect. The higher the tax rate the sooner the asset is sold. The exception is trees taxed on a realization basis. If investment opportunities are stationary, then selling time is independent of the tax rate, and longer than it should be.3 For wine intrasectoral efficiency is attained when wine is taxed on an accrual basis at the same rate as ordinary assets. However, when wine is taxed on a realization basis at this rate, it is still held too long before it is sold; in contrast, trees taxed on an accrual basis are cut down too quickly when tax rates are the same in the ordinary and Austrian sectors. Tables 1 and 2 below summarize these results, from which two conclusions emerge. First, economists have long maintained that many of the undesirable effects of the capital gains tax are due to its being levied on a realization rather than an accrual basis; our model suggests this is partially true; accrual taxation is fine for wine but not very good for trees. Our results make it clear that no rate of accrual taxation can simultaneously achieve intra and intersectoral efficiency for trees. Since wine is in a sense a polar case, we think the case for accrual taxation is somewhat weak.’ Secondly, it is clear that if capital gains are taxed on a realization basis the capital gains tax rate should be higher than the effective tax rate on ordinary investments. For accrual taxation our results give less guidance as to the desirable relative tax rates. For wine, the rates should be equal; for trees intersectoral efficiency suggests that the capital gains tax rate should be higher; intersectoral efficiency suggests it should be lower.5 In section 3, we show how these results are modified by inflation. The interaction of inflation with the tax system is complex and defies simple summary. Inflation sometimes alleviates and sometimes exacerbates the distortions and inefficiencies caused by the tax system. 3If investment opportunities are not stationary, then the effect of taxation on selling time depends on the nature of the deviations from stationarity in a complicated way. %ee footnote 1 above. ‘The comparisons we make between the rate of capital gains taxation and the effective rate of taxation in the ordinary sector (5, in Sections l-3) should be taken with a grain of salt. A major accomplishment of recent research on the taxation of capital has been the demonstration that rO is an extraordinarily complicated beast. As the endproduct of the complex interaction of many provisions of the tax law, r,, cannot be identified with anything so simple as the corporate income tax. Thus, it is hard to conceive of a realistic tax change which could set the capital gains tax rate in the Austrian sectors equal to the effective rate of taxation in the ordinary sector.

D.J. Kovenock and M. Rothschild, Capital gains taxation

219

In section 4 we show how the analysis is changed by taking account of the special ways in which capital gains are, or could be, taxed at death. We show how the present U.S. tax system, which allows automatic step up of basis at death, encourages people to hold on to assets longer than they otherwise would. Previous work on capital gains taxation has focused on the way in which capital gains taxes tend to inhibit transactions and to encourage investors to hold assets - at least assets which are growing in value - longer than they otherwise would. Our analysis suggests that the effects are diverse and complicated. The results summarized above showed that the capital gains tax and sometimes leave sometimes lengthen, could sometimes shorten, unchanged the length of time the owner of an Austrian asset would wait before he harvested it. However, we did obtain one result which shows that the capital gains tax discourages transactions in Austrian assets. In proposition 4 below we show that the owner of an Austrian asset who is subject to capital gains taxation on a realization basis will never sell it to He will always make higher profits by another intermediate producer. holding on to wine until it is sold to the consumer. Although proposition 4 is stated for wine, it also clearly holds for trees. Thus, the capital gains tax acts as a kind of turnover or transaction tax on Austrian assets and as such promotes vertical integration in industries which use Austrian processes. We have sacrificed a great deal of reality to keep our models simple and tractable. The capital gains tax is proportional and everyone pays the same rate. We ignore the complexities and arbitrage opportunities which are encouraged by progressive taxation at different rates.6 We also ignore the arbitrage opportunities which the options market permits. Constantinides (1983) and Constantinides and Scholes (1980) have argued that by using the options market, investors can avoid ever realizing capital gains. Sales of assets (which are taxable events) are dominated by the purchase and sale of options (based on the asset’s future value) which can be written in such a way that taxes are avoided or deferred. This argument loses some of its force in an Austrian model. Assets must be sold eventually or the tree rots or the wine goes bad. How the existence of an active options market would affect our results is an interesting question which we have not examined. We have also assumed that investors have a very simple goal. They maximize the discounted value of wealth. We assume a constant discount rate of r. When we treat uncertainty we assume that investors are risk neutral so that they maximize the expected discounted value of wealth. Thus we ignore both risk aversion and portfolio effects. We make this choice for two reasons. First, it is what we can do. Second, while it is not formally difficult to introduce risk aversion into some of our models, we are uncertain as to how to interpret the results. Suppose an investor’s portfolio contains 6Lippman

and McCall

(1981) analyze

a similar model in which taxes are progressive.

220

D.J. Kovenock

and M. Rothschild,

Capital gains taxation

assets which will be harvested at different dates; risk aversion is not adequately treated by assuming the same concave function values wealth received at each date. To handle risk aversion a more complex model presumably one with consumption - is needed. A further simplification is obtained by our treatment of death. We assume that investors place the same value on wealth which their heirs realize as on the wealth they receive if they harvest an asset. The subject of this paper is not new. An extensive literature discusses the effects of taxation on the allocation of resources to Austrian assets. Many of our results were first obtained by Gaffney (1967, 197G71). Because his papers were somewhat opaque and his model, particularly in its detailing of the relationships between Austrian and other sectors, not completely explicit, his results on what we call below inter and intrasectoral efficiency have often been misunderstood and rarely been accepted by later writers [see, for instance, the papers by Thomson and Goldstein (1971), Klemperer (1974), and Bentick (1980)]. A notable exception is Chisholm (1975) whose paper outlines a model similar to our own. Chisholm starts by defining tax neutrality and appealing to Samuelson’s Fundamental Theorem of Tax Rate Invariance [Samuelson (1964)] in specifying a neutral income tax.’ He then uses this tax as a benchmark with which to measure the effect of various forms of taxation in the Austrian sector. While Chisholm’s model is similar to our own, our analysis is more explicit. Our work supports and elaborates much of what Gaffney and Chisholm first stated. 2. The basic model 2.1.

The ordinary sector

A dollar invested in the ordinary sector yields a constant stream of gross returns y. We abstract from the entangling detail of the real world by assuming a simple proportional tax. Thus, after-tax returns are y( 1 - zO), where z, is the tax rate in the ordinary sector. The tax rate z, cannot be identified with any single parameter of the tax code such as the corporate profits tax. Auerbach (1979) and others have analyzed the complex way in which the provisions of the tax law combine to produce r’,. If the stream of returns is not constant, z, is even more complicated. 2.1.1. The after-tax rate of return We define the after-tax

rate of return

r as the solution

to

l=Ty(l-r,)e~‘“ds.

(1)

0

‘For a discussion of the relationship footnote 9 below and Kovenock (1982).

between

our

results

and

Samuelson’s

Theorem,

see

D.J. Kovenock

and M. Rothschild,

Capitul gains taxation

221

Thus, y = r/( 1 -r,). Since r is the rate of return available in the ordinary sector in a competitive economy, it is the rate which all investors will use to discount all future benefits. In a complete model r will be determined by the interaction between tastes (particularly time preferences), technology, and endowments. In this paper we analyze the effects of changes in parameters, particularly tax parameters, which do not affect the ordinary sector.

2.1.2. EfJiciency and discount rate. In order to assay the effects of taxes on economic efficiency, it is necessary to choose a discount rate. Economic theory suggests two candidates for this role. The first is the consumer’s private discount rate, which in equilibrium must equal the after-tax rate of return r. The argument for using r as the social discount rate is simply that it is the rate that consumers use to value changes in consumption (public or private) in different time periods. The second is the gross of tax rate of return, R=r/(l-7,); R is the discount rate which equates the total return from an investment of one dollar in the ordinary sector with its cost. The argument for using R as the social discount rate is that in the kind of second-best world in which taxes are levied the best welfare result which one can hope to obtain is productive efficiency. This requires that all investments earn the same rate of return. While not wishing to participate in this debate, we record here that we think that those who have advocated the use of R have much the better case. This is principally because of the difficulty which advocates of the use of r have had in formulating a general equilibrium model in which taxation has any real effect, in which r should be the social discount rate. [See Diamond’s (1968) criticisms of Marglin (1963).] In addition, the very general results of Diamond and Mirrlees on ‘Private constant returns and public shadow prices’ (1976) apply directly to our constant returns Austrian model.’

2.2. Austrian

sectors

In the economy there are also Austrian investment opportunities. Austrian investments are point input, point output processes. An investment at an initial time to produces an asset which has a value of X(t - to) if the asset is harvested and used for consumption at time t.

2.2.1. Wine and trees We consider

two kinds

of Austrian

production

processes.

In the first, and

‘Our conversion to this position is a recent one. Readers who feel we have made the wrong choice should consult an earlier version of this paper [Kovenock and Rothschild (1981)] where r rather than R is used to measure economic efficiency. We are grateful to David Bradford and Peter Diamond for patiently instructing us on these matters.

222

D.J. Kovenock and M. RotkrehiId,Capitalgains taxation

simpler, an initial investment produces an asset which increases in value as it ages.,We call the output of this process wine. The other Austrian output, a trek, is distinguished from wine by the fact that it uses a scarce resource, land, as it ages. When a tree is felled, another can be planted in its place. When wine is drunk, the bottle in which it has matured is discarded. We need terms to describe the time when an Austrian asset’s maturation is terminated and it is consumed. For trees ‘cutting time’ seems appropriate. For wine we shall call this time ‘selling time’. This usage is justified in proposition 4 below, where it is shown that the capital gains tax discourages transfers of wine before it is consumed. Since the argument of proposition 4 can be adapted to trees, we shall also occasionally use selling time to refer to trees and wine together.

2.2.2. Values and rates of return without taxes Suppose that there are no taxes in the Austrian sector. In equilibrium assets invested in the Austrian sector must earn a rate of return equal to Y. If they earn a rate of return greater than r, the value of resources used to produce an Austrian asset will be bid up until their rate of return is just r. We use this principle to analyze the efficiency of the allocation of resources between the Austrian and the ordinary sectors and the efficiency of the choice of selling time within the Austrian sectors. (a) Wine. If B is the value or price of resources used to produce wine which has a value of X(t) when sold or harvested after t years, then B = max e r* X(t). We assume that X’(t)/X(t) is a decreasing function privately optimal selling time t* is the unique solution of

of t so that

the

X’( t)/X( t) = r. There are two efficiency questions we can ask about the allocation of resources to the wine sector. The first is: Are the resources in the wine sector social value? Under our used efficiently, or, is t* chosen to maximize assumptions the answer to this question is clearly, no. For, if R is the social discount rate, resources devoted to the production of wine will be used most efficiently if the selling time is f, where f is the solution to max,X(t)eeR’. Since f is the solution to X’( t)/X( t) = R,

D.J. Kovenock and M. Rothschild,Capitalgains tuwtion

223

f # t*. Since R >r, it must be that f< t*. In the next sections we will see how the introduction of a capital gains tax (which, as proposition 5 shows, decreases t*) can correct, and overcorrect, this tendency to keep wine too long. The second efficiency question is: Given the use of resources in the wine sector, is the allocation of resources between the wine sector and the ordinary sector socially optimal, or could productive efficiency be improved by shifting resources from one sector to another? If there are no taxes in the wine sector, then the intersectoral allocation is not optimal. For, in the ordinary sector an investment of resources with a market value of B produces an output with a social value of B. (Indeed, this equality is used to define the social discount rate R.) On the other hand, when resources with a market value of B are used in the wine sector, they produce wine that will be sold at time t* which has a social value of X(t*)epR’*. Since B=X(t*)e-r’*>X(t*)e-R’” it is clear that resources have a higher social value when used in the ordinary sector. As we shall see in section 2.3.1, imposing a capital gains tax on wine will shift resources from the wine sector to the ordinary sector. (b) Trees. For simplicity we restrict our attention for the most part to the stationary case. Suppose that a tree can be grown on a plot of land. It costs P to plant a tree on this plot. After u years the tree may be sold for V(u) dollars. If at that time another tree is planted and harvested in u years, and the cycle is repeated again and again, the net discounted social value of the trees grown on this land is:

-P+(V(u)-P)e-R”+(V(u)-P)e-R2”+

= -P+(V(u)-P)

f

...

eCRui = -P+(V(u)-P)/(eR”-

1).

i=l

In labelling this quantity the net social value of resources used in producing trees, we have assumed that land is the only factor earning rents in the production of trees. The resources used to plant trees earn no rents; they have alternative uses in the ordinary sector which in effect means they are in perfectly elastic supply. A more general model would allow both factors of production to earn rents; however, to avoid the joint cost problem, such a more general model would also have to allow for substitution between the two factors of production. To keep an already long and complicated paper manageable, we consider only the simpler case where land is the only factor of production earning rents in the production of trees. We now ask the two efficiency questions about the allocation of resources to and within the tree sector. Land will be used efficiently within the Austrian sector if the cutting time maximizes the social value of returns to

224

D.J. Kovcnock

land used maximizes

to

produce

and M. Ro/h.vchild,

trees.

Thus,

C‘aprtul piins

the

socially

taxation

optimal

cutting

time

-P+(V(u)-P)/(eR”-1).

ti

(2)

However, if there are no capital gains taxes the owner of land who plants a tree on it every u years will discount his profits at rate r, not R. He will receive _p+

Uu)-P e ru -1

and will set u* to maximize this expression. It is a consequence of the following lemma that if there are no capital gains taxes, trees are cut down too late. Lemma 1. Let F(u, s) =(V(u) -P)/(e”” - 1) and max, F(u, s). Then u’(s) CO. Proof: The implicit But

F,(u(s),s)=

function

theorem

V(u)(e” - 1) -(V(u)

implies

let u(s)

be the

that sign U’(S)=sign

- P)s es’

(eSd - 1)2

solution

to

F,,(u(s),s).

= 0.

Thus, VU es’ ~ ( I/- P) esU(1 + su) F,,(n(s), s) =

(es”-1)’

’

and sign F,,( U(S),s) = sign [ VU - ( V- P)( 1 + su)] = sign [ V’(u -(es’ - l)( 1 + su)/s es”)] = sign [us es’ - es’ - eSusu + 1 + su] =sign

[l+su-e”“]
We now examine the efficiency of the market allocation of land between the tree sector and the ordinary sector. To do this we consider land which has the same market value L when used in the ordinary sector and in the production of trees. We will show that the social value of the product of this

D.J. Kovenock

and M. Rothschild,

225

Capital gains taxation

land is higher in the ordinary sector. Thus, efficiency would be increased if scarce resources, in this case land, were diverted from the production of trees to the ordinary sector. Land in the ordinary sector with a market value of L produces a stream of returns with a social value of L. To see this, note that if the market value of land is L it must produce a stream of returns with an after-tax discounted value of L. Thus, if z is the flow rate of return, we must have L= z(1 - 7,) 1; e-*’ dt or z = RL. Thus, the social value of these returns is 2s; e mR’dt=L. We now show that land used to produce trees which has a market value of L will produce trees with a net discounted social value of less than L. The value of a plot of land used to grow trees is given by (3). Thus,

L= _,+w*)-p) e rut -1

(4)

.

However, the net social value of output is given by (2) evaluated at U* and is clearly less than L. Thus, if there is no taxation in the tree sector, resource allocation would improve if scarce resources (land) were shifted from the tree sector to the ordinary sector. We may summarize this discussion in the following: Proposition 1. If there are no taxes in the Austrian sector, the selling times of trees and wine are greater than they should be. Efficiency would be improved if resources were taken from the Austrian sector and transferred to the ordinary sector. In the realization

next sections and an accrual

we analyze how basis) and inflation

capital gains taxation affect these conclusions.

(on

a

2.3. Capital gains taxes In this section we analyze the effects of capital gains taxes which are levied on a realization and on an accrual basis. The results of these analyses are summarized in Tables 1 and 2. Table 1 shows how the two kinds of capital gains taxation affect the selling time of Austrian assets. Column 3 shows the first-order conditions which must be satisfied if there is to be a social optimum; column 4 shows the first-order conditions which will be satisfied when Austrian assets are owned and sold by profit-maximizing entrepreneurs. Column 5 compares the privately optimal selling time to the socially optimal selling time when there are no taxes in the Austrian sector. Column 6 shows the effect of an increase in taxation on selling time. Column 7 compares the private and social optima

Accrual

Realization

Mode of taxation (1)

v(u)= V(i)

,_

X(fJ

l’e-R’

1_e-Ra

l-r

X@,)

V(u,)

r l-r

-W,)

vu*)

v’(u*) -=

1_e-ru*

e’% - 1

Same as realization

ll*>li

Negative

None

u, _,$

t,=&

u*>li

t*2f

t*>f

Z=r(‘+ll;&)

X’(f)=R

Trees G=

Trees

Wine

Negative

(7) (6)

(5)

(2)

(4)

Asset

(3)

F.O.C. for social optimum

Optimality of selling time when 5=r,

efficiency Effect of increases in the tax rate r on selling time

1 on intrasectoral Optimality of selling time when 5=0

Table gains taxation

F.O.C. for selling time

Effect of capital

Accrual

Realization

(2)

(1)

Trees

exp[(z)lQ]

Wine

R,.

/

L(l-emR”*)+P

L(t_e-~q+p(l_re-~~*)

l--ze-” 0 I’\ \ l--z

~

e-’

Discounted social value of gross returns (3)

Trees

Wine

Asset

Mode of taxation

Effect of capital

Same as realization

Same as realization

emraR’*i 1

(4)

DSV when r=O

1
Table 2 gains taxation on intersectoral

l-r,

-em’“‘)+P

[

-r,e-‘“*)

i 1

L(1 -e-R”*)+P

I_(1 -emR”a)+P

ql

1

(l-r,)

-e-‘“*)+P(l

1--toe-“*

p-1

eCzOR”’ yl

e-roR,.

DSV when r=r, (9

efficiency

1 <1

Positive

Positive

Positive

Positive

Effect increases in z on intersectoral efficiency (6)

228

D.J. Kovenock

uncl M. Rothschild.

Cqitul

guins lu.wrion

when the capital gains tax rate is the same as the tax rate in the ordinary sector. Table 2 shows how capital gains taxation affects the intersectoral allocation of scarce resources. Column 3 gives a formula for the discounted social value (henceforth DSV) of gross returns to one dollar invested in trees or wine. This formula is a function of the tax rate in the ordinary sector, the capital gains tax rate, and the mode of taxation. Resources are allocated efficiently when the formula in column 3 has a value of unity. Column 4 gives the value of the DSV when capital gains are not taxed. Column 5 records the value of the DSV when tax rates in the Austrian sectors are the same as tax rates in the ordinary sector. Column 6 documents the fact that taxing Austrian assets drives resources from the Austrian sector which (at least when z < 7,) increases efficiency. These tables tell a relatively simple story. If untaxed, Austrian assets are held too long; the social rate of return to investments in Austrian assets is less than the social rate of return to assets invested in other sectors. Taxation can alleviate these inefficiencies. However, if taxation is levied on a realization basis, it is not enough to tax capital gains at the same rate as ordinary investment income; the inefficiencies remain. For accrual taxation the situation is different, and in one case more hopeful. For wine, there are no distortions. if taxes are levied on an accrual basis at the same rate as prevails in the ordinary sector. However, accrual taxation does not work so well for trees; if capital gains are taxed at the same rate as ordinary investments, trees are cut down too soon (indicating that taxes are too high) while resources invested in trees earn a lower rate of return than they would in the ordinary sector (indicating that taxes are too low). The details of the analysis are spelled out below. The skeptical reader who wants to see how inflation disturbs this relatively neat pattern should refer to table 3. 2.3.1. Wine taxed on a realization

basis

Consider an investment in wine which if harvested and sold at time t will yield X(t). If the wine initially costs B, capital gains are [X(t)-B]. If capital gains are taxed when realized at rate z, net proceeds are (X(t)-B)(l--)+B=X(t)(l-z)+zB.

(5)

The cutting time is chosen to maximize the discounted value of these net proceeds. In equilibrium investments earn the competitive rate of return so that B satisfies

B=maxe-”

[X(t)(l-7)+7B]=e-“*[X(t*)(l-7)+7B],

(6)

229

D.J. Kovenock and M. Rothschild. Capital gains taxation

where the selling time t* maximizes the discounted value of the investor’s after-tax profits. In interpreting (6) it should be kept in mind that r, r, and X( .) are parameters while B and t* are chosen to satisfy (6). If r and X( .) are held constant, B and t* are functions of z. B is the value of the investment opportunity represented by X( .). If the real cost of the resources used to produce X( .) is less than B, then these resources will be used to produce wine; competition will cause the value of these resources to rise to B. If they cost more than B, the wine which has the value stream X( .) will not be produced. The discounted social value of the total returns to an investment of B dollars in wine is X(t*) e PRf*. We now show that if z szO, resources invested in the wine sector produce less than resources invested in the ordinary sector. Proposition 2. The invested in wine is

discounted

social

value

of total

returns from

a dollar

- roRf* -Ze=Rt* J(t*,T,T,)=e

1-z

If z 5 zO, then the discounted Proof

Rearrange X(t*)

(7)

. social value of total returns is less than one.

(6) to obtain

e-Rf*=B(e~‘oR’*-~e-Rt*)/(l

-t).

Divide both sides of this equation by B to obtain (7). Clearly, J(O,r,rJ = 1 so it will suffice to prove that JJt*, z, z,) ~0 for r 5 z, and t* > 0. However, it is easy to see that Jt*(t*, z, z,) has the same sign as (z emRcl mrO)f* -7,) which is negative whenever r 5 7,. If z >zO, then returns to investments in wine may or may not be greater than returns to ordinary investments. Since different kinds of wines may have different selling times (different t*) it is quite possible that some wine investments may have greater and some lesser total social returns than ordinary investments. In interpreting the condition z 62, it is well to keep in mind what a complicated parameter r, is. The value of B determines the allocation of resources to the wine sector. As B increases, more resources are devoted to the production of wine. Whether or not this increases or decreases economic well-being depends on (7). In contrast, the parameter the efficiency with which t * determines resources in the wine sector are used. The value of all resources devoted to wine grows according to the function X( .). Since the social discount rate is R, the wine should be allowed to mature until f, where f maximizes eCRfX(t). Assuming, as we do, that X’(t)/X(t) is a decreasing function, f is the unique

230

solution

D.J. Kovenock and M. Rothschild, Capital gains taxation

to x’(t)/X(t) = R.

However, the selling return, not social return;

(8) time for wine, t*, is chosen to maximize private t* satisfies not (8) but (6), thus t* must satisfy

X’(t*) -=r x0*1

(9)

Clearly, the right-hand side of (9) is an increasing function wine is sold sooner. We can rewrite (9) in terms of R as

of r; as taxes rise

X’(t*)/X(t*)=R[(l-z,)/(l-r)][l-z(l-(B/X(t*)))].

(10)

Wine will be sold too soon or too late depending on whether side of (10) is greater than or less than R. Clearly, if z =r,, then

the left-hand

X’(t*)/X(t*)=R(l-z(l-(B/X(t*))))
one, X’(t*)/X(t*) may be greater B, part A, show. We summarize

Proposition 3. If z SzO, then t* S-C for z su.ciently than or less than f.

high, t* may be greater

In interpreting this proposition it should be remembered that while each wine investment may have a different socially optimal selling time f and a different privately optimal selling time t *, for all such investments if ~57, the privately optimal time is too long. Were it to decrease, the resources allocated to the wine sector would be used more efficiently and the discounted social value of the output of the wine sector would increase. A high basis increases the value of an asset subject to capital gains taxation. It is conceivable that this effect could be strong enough to encourage turnover of assets like wine. Working against this is the fact that capital gains taxes are also turnover taxes and thus they inhibit transactions. We show in proposition 4 that this latter effect dominates. Proposition

4. No wine is sold before it is consumed.

Proof Suppose first that the basis is 0. In this case the wine will be consumed at the same time as it would be were there no tax. The value of

D.J. Kovenock and M. Rothschild. Capital gains taxation

231

holding the wine to maturity is IVr(O)=maxe-r’X(t)(l-r)=(1-s)maxe-r’X(t)=(1-r)Wo. t

f

If the person sells the asset immediately for a price S he will net, after taxes, only (1 -r)S. Thus, he will be better off selling only if S( 1 -r) 2 (1 -r)W” or if Sz W”. However, it is easy to see that no person facing the capital gains tax will pay more than W” to buy the asset. A prospective buyer is willing to pay only S, where S is a solution to S=max[X(t)(l-r)+rS]e-“=e-“*X(t*)(l-r)+rSe-”*. Thus, S[l --Ze-“*I =eerf*X(t*)(l -7) or 1

S=e-‘t*X(t*)

‘,Irte
If the basis is B, then W’(B)> W’(O), so a sale would have to realize even more than W” for it to be preferable to holding the asset until maturity. This argument considers a sale only at time t=O, but it obviously generalizes to sales at other times. This proof depends heavily on our assumption that all investors face the same capital gains tax. The capital gains tax affects both the allocation of resources between sectors, through B, and the effkiency with which resources in the wine sector are used, through t*. The signs of these effects are determinate. Proposition 5. Both B and t* are decreasing functions of z. Proof:

(1) Let G(B,z,t)=e-“(X(t)(l-z)+zB), ~(B,z)=argmaxG(B,r,t), f

and

H(B,~)=G(B,T,&B,T))-B.

In equilibrium H(B, z) =O, so dB/dr = -Hz/H,.

But

H,=C,+G,$-l=G,-l=rexp(-r4(B,r))-l
step follows from the envelope

theorem,

G,(B, z, c#J(B,t)) =O.

232

D.J. Kovenock

and M. Rothschild,

Capital gains taxation

Similarly, H, = G, + G,d4/dz = G, = (exp( -r&B, Thus, letting dB dr=

t))(B - X(&B, 7)))).

t* = $(I?, r), e-*‘*(II-X(t*)) f-_ze-lf*


’

(2) To calculate dt*/dz we note that B is a function of r and maximizes F(t, r) = e -“(X(t)( 1 -z) + rII(r)) so that t* must satisfy O=Fl(t*,z)= -rF+e-“*X’(t*)(l

-z).

t*

(12)

Thus, dt*/dz = - F,,IF,, and, since F, 1 ~0 by the second-order for maximizing F with respect to t, it follows that sign dt*/dz=sign using (6), (1 l), and (12) we have that F,,=

that

conditions F,,. But

-rF2-e-“*X’(t*) = -re-“‘(B-X+zdB/ds)-e-“*X’

X-B+

=e-*‘*r

[

7(X-B) *f* e --z

1 -~

rB

1-z

=‘[&-$&+&]] -rB

=-
--z

Propositions 2, 3, and 5 imply that capital gains in the wine sector should be taxed more heavily than investment income in the ordinary sector. As long as rzz, both intra and intersectoral effkiency increase as r increases.

233

D.J. Kovenock and M. Rothschild, Capital gains taxation

2.3.2. Trees taxed on a realization

basis

The analysis for trees is in some ways simpler. As Lippman and McCall (1980) have observed, the privately optimal cutting time u* is independent of the rate at which a proportional capital gains tax is levied. We model the effect of the capital gains tax by assuming the owner of a tree pays a tax on the increase in the tree’s value from P, its initial cost, to V(u). Thus, cash flow at time u is assuming he plants another tree, his after-tax (V(u) - P)( l-t) + P - P = (V(u) - P)( 1 - 2). Thus (6) is changed to

+e -‘2”(V(u)-P)(l-r)+ = _p+

v(“)-p e I” -1

...

(13)

(l-+

It is worth noting that this does not hold for all tree investments; it is a consequence of the assumption that investment opportunities in the tree sector are stationary. If a plot of land will support a sequence of trees each associated with a planting cost Pi and a potential value stream l$, the sequence of cutting times {u*} = (uT, ~4,. . .) will be chosen to maximize L(u)=-PP,+e-‘“‘[(l/,(u,)-PJ(l-r)+(P,-P,)]

=

[(~(Ui)-pi)(l-z)+(pi-pi+I)l~ -pd$l(exp-r(kil ‘k))

The optimal sequence {u*} will be independent of T if and - for all practical purposes - only if Pi = P for all i. Unfortunately, even though cutting time is independent of r, it will in general be too long. Proposition 6. If planting costs are constant, Furthermore, u* > li.

u* is a constant

independent

of z.

The introduction of a capital gains tax on a realization basis changes only slightly the analysis of social returns in the tree sector given in section 2.2.2 above. There we proved that if z =O, the social return to resources used in the tree sector is at the margin less than the social return to the use of resources in the ordinary sector. As z increases, private returns in the tree sector fall; resources shift to the ordinary sector, improving efficiency. We now show that efficiency is not attained unless T > 7,. J.P.E.

E

234

D.J. Kovenock and M. Rothschild, Capital gains laxalion

Proposition 7. The social return to resources used in the tree sector is less than, equal to, or greater than the social return to resources in the ordinary sector as T is less than, equal to, or greater than ?, the solution to (1 _ T)(er”*i( l prd -

Although Z is a function

1) = (eru* -

1).

(14)

of V( .) and P, S>z, for all values of these parameters.

Proof: Again social returns from the use of land to grow a tree for u* years is V(u*)eeR”*. The social cost of the resources used to produce this tree is L(u*)( 1 - eeRu*) + P, while in equilibrium

Some manipulation

shows that the ratio of social benefits

to social costs is

vu*) v(u*)( 1 - z)a + P( 1 - (1 - z)cX)’ where CL = (eRu*- l)/(e”* - 1). Since V(u*) > P, it is clear that

or as (1

_~)(ery*l(l-ro)

_ 1) s(eru* - 1).

(15)

Clearly, the left-hand side of (15) is a decreasing function of z so there is one and only one value of z which makes (15) an equality. (This analysis is legitimate since u* is independent of 7.) Let 7 be this taxation rate. We now show that z,ey-l. will suffice to show that F’(z,) > 0, for 0
j7(~,)

=

_(eYI(l-k4

_

1) +(l

_~o)eYi(l-r~)----_

(1 Y7J2 e-Y/'l-k4_l

=eYl'l-'o' [

since emx > 1 --x for all x>O.

+

(lT?.)

1

‘O,

D.J. Kovenock and M. Rothschild, Capital gains taxation

235

2.3.3. Wine taxed on an accrual basis We begin our analysis by noting that the effect of accrual taxation on wine is to force investors to behave as if their discount rate were r/(l-z) rather than r. Thus, if tax rates in the two sectors are the same, private investors will make decisions using R as their discount rate. Proposition 8. If taxed on an accrual basis at rate z, wine with a growth path .) will be harvested at time t,, where t, satisfies

of X(

x’(ta) R(l - 4

z(TJ=(1-z) The discounted value to the investor of such wine is X(t,)exp [ - R(l -z&J (1 -r)]. Furthermore, dt,ldz <0 and t,< t*, where t* is the harvesting time when wine is taxed at realization at rate z. Proof: (1) The key to the proof is our assumption that taxes are levied on increases in the market value of the wine. That is, if W(t) is the value of the wine at time t, increases in W(t) must satisfy (16) This formula assumes that the investor bought the wine for W(t) t and sold it for W(t + At) in period t + At realizing an after-tax ( 1 -z) W(t + At) + z W(t); if At is sufficiently small, there is no difference taxation on an accrual basis and taxation on a realization basis. Taylor series expansion to evaluate the right-hand side of discarding all terms of order (At)’ or greater we see that W(t)(l -z)-

in period return of between Using a (16) and

W(t)r=O

01

W’

_w-1-r.

r

This first-order differential equation describes the evolution of W(t), the value of the wine. A boundary condition is that W(t,) = X(t,) if the wine is sold at time t,. Thus, the discounted value of wine taxed on an accrual basis at rate 7 which will be sold at t, is just X(t,) ePcricl -r))ta. Clearly t, should be chosen to maximize this value.

D.J. Kovenock and M. Rothschild, Capital gains taxation

236

(2) That dt,/dT < 0 follows immediately from the fact that d(r/( 1 -z))/dz ~0. (3) Let t* be the selling time when taxed on a realization basis with basis B. Then

zX(t*) -=r l+-X(t*) I[ 1 x’(t*)

z

[

=y

B

1 -Z X(t*)


XYLJ

( > 1+-

z

=-_=-

1-Z

1+---1 -7 X(t*)

(1 LT)

X(t,)

Since X’(t)/X(t) is a decreasing function of t, t, < t*. Social and private discount rates are the same and accrual taxation leads to the correct allocation of resour&s between the wine and the ordinary sector only when t = 5,. Proposition 9. The discounted social value of a dollar of resources invested in wine is

exp[

(E)Ri,],

Proof: Consider one dollar invested in at t, and worth X(t,) at that time. returns is clearly X(&J eCRfa. However, after-tax returns is equal to one dollar

Substituting,

exp

wine which goes to produce wine sold The discounted social value of gross in equilibrium the discounted value of so

we find gross social returns

[

=RtRt, 1-z

from one dollar

1 .

Whether the social returns from investments in wine than returns in the ordinary sector depends on whether SRt,

exp 1

This in turn depends

to be

1

21.

on whether

are greater

or less

(17) z 2 7,.

231

D.J. Kovenock and M. Rothschild, Capital gains taxation

2.3.4. Trees taxed on an accrual basis The effects of accrual taxation on trees are also straightforward; however, the results are not as favourable for trees as the results for wine. We showed in proposition 1 that if there is no taxation, then trees are held too long; the privately optimal cutting time is greater than the socially optimal cutting time. We show in proposition 10 that accrual taxation can correct this departure from efficiency as the privately optimal cutting time U, is a decreasing function of the accrual tax rate. However, in contrast to the result for wine, if the capital gains tax rate is greater than or equal to the tax rate in the ordinary sector, things are not set right. In this case the cutting time is less than it should be. We also showed in proposition 1 that if there were no capital gains tax, resources used in the tree sector had a lower social product than in the ordinary sector. Imposition of a capital gains tax corrects this distortion by driving marginal resources out of the tree sector into the ordinary sector. However, equating the tax rates in the two sectors does not bring about a sufficient correction. If both sectors were taxed at the same rate, marginal resources devoted to trees would have a higher social product in the ordinary sector. Together, these two results imply that, in contrast to wine, no rate of accrual taxation can simultaneously bring about the efficient allocation of resources within the tree sector and between the tree sector and the ordinary sector.9 We begin by noting that the analysis of growth yields an after-tax return with a discounted value of V(u)e -“‘i(1 -‘I. If planting a tree costs P, a plot of land on which trees are planted yields returns which are equivalent to the payment of V(U) e-U’i(’ -I) - P at times 0, U, 2u,. . . . Thus, the value of land is L(u) = V(u) eeuri(l -I) -P l-_e-rU

(18)

’

The privately optimal cutting time, u,, is chosen to maximize the right-hand side of (18). It is immediately clear from (18) that U, is not, as in the realization case, independent of the rate of capital gains taxation. Proposition an accrual z=s,>O,

10. The privately optimal cutting time for trees which are taxed on basis

is a decreasing

function

of the tax

rate.

Furthermore,

if

then u,
91n the case of wine, the combination of income taxation in the ordinary sector and accrual taxation at the same rate in the Austrian sector represents a neutral tax scheme in the sense of Samuelson. For trees, neutrality is attained when accrual taxation in the Austrian sector takes as its base increments in land value and not just increments in the value of the tree that the land supports. For an analysis of this type of accrual taxation and its relation to property taxation, see Kovenock (1982).

D.J.

238

Kovrnock

and M. Rothschild,

Capital gains taxafion

Proof (1) By definition

24,= argmax

”

Assuming

V(u) exp ( - i-u/( 1 - z)) - P l-e_‘” .

an interior

solution,

the first-order

(19)

conditions

imply

P 5 V(u,) exp ( - ru, /( l-z))

$(u,, 7) =

{[ V’(4 ew x [l

(20)

(s)-w@+0)]

-e-‘“a]-

--P+

V(u,)exp

(s)]veLrUa)

[

(21) Thus,

and, since i?$~/&, < 0 by second-order

conditions,

du, 84 sign dz = sign x

Rearranging

the first-order

PU =L~eerUaexp V(k)

condition

& (

(21), we find that

+e-‘“a-1 >

Thus aexp(&)+e~rU~-

1).

D.J. Kovenock and M. Rothschild, Capital gains taxation

From

(20) we know that P 5 V(u,) exp ( - ru,/( 1 -z)),

239

which implies

(22) The right-hand side of (22) attains a maximum over u,rzO when u,r=O. At this point (u,r + 1) eprua - 1 =O. Thus, du,/dz 50. (2) To prove the second part of the proposition, rearrange eq. (21) and set z, equal to Z. Define 0 to be the common value of r and z,. Thus,

(23) If H=O then obviously u,=ti. (23) as a function of 8. Then

H’(B) =

Let H(H) be the value of the right-hand

side of

~[l-~e~u~l.[l_~R"-I eR(l-o)u. (e a

R(l-0)~.

a-1)2

Since V(u,) e mRua2 P from (20) sign H’( 0) = sign [ 1- eR( ’ =sign[l where d[l +(xz=z,>o,

x=(1 -8)Ru,. 1) e”]/dx=xe” ld,
+(x-

~0

1)exl,

When I x=0, 1+(x-l)e”=O. Furthermore, for x >O. Thus, H’(B) >O which implies that for

We now analyze the social returns taxation and show that tax neutrality allocation of resources.

to investment in trees under accrual (r =zO) does not lead to an efficient

Proposition Il. Ifcapital gains are taxed on an accrual basis at rate z then the value of land used raising trees is a decreasing function of z. If T=T,, then the social value of the return to marginal resources used in the tree sector is less than the return in the ordinary sector. Proof (1) In equilibrium

the value of land is (24)

240

D.J. Kovenock

and M. Rothschild,

Cupirul

gains taxation

which is clearly a decreasing function of z. [Since u, is chosen to maximize L(u,) we may ignore the effect of T on U, in signing dL(u,(z),z)/dr]. (2) The social value of resources used to produce a tree is L(1 -ePRua) +P. The social return from an investment in a single tree is V’(u,)emR”a. Market equilibrium [(24) above] implies that

V(u,)e

*“~“‘-“=L(l-e-*“.)+p,

Thus V(u,) e R”a= exp{[R(z-z,)/(l

Dividing

When

-z)]~,)[L(l-e~‘“a)+P].

each side by L( 1 - e - Rua)+ P gives us

z=~~, the right-hand L(1 -epr”a)+P L(l-e~R”a)+P

side equals

< 1.

3. Inflation 3.1. The ordinary sector We model inflation in the ordinary invested yields a stream of gross returns v, is now the solution to

sector by assuming y(s)e v . The after-tax

that a dollar rate of return,

(25)

l=~y(s)e~S(l-r,)e-““ds. 0

If r is the after-tax rate of return in the ordinary sector without straightforward to show that v= r+y; Y+Y] is the rate which use to discount all future benefits. 3.2.

inflation, investors

it is will

The Austrian sectors

Our analysis of the effects of inflation on the Austrian sectors focuses on the two distortions examined earlier. Table 3 summarizes the effects of inflation on selling time and value of gross returns for wine and trees under both accrual taxation and taxation on a realization basis. The details of the

D.J. Kovenock and M. Rothschild. Capital gains taxation

Effects of inflation

Mode of taxation

Asset

Effect of increased inflation on selling time

Wine

Ambiguous for small q. positive as q-+m

Trees

Positive

Wine

Negative

Trees

Negative

Table 3 on efficiency of resource

241

allocation

Discounted social value of gross returns when z = q,

e-zORI*_Se-CR+q)r* 21

1-T

Realization

1<

~(l-e-‘“*)+P(l_se-l’+“‘“‘)

Accrual

>1

L(l-e-R”‘)+P

exp

(v > Gt”

>l

L(1 -e-‘“‘)+P L(l-e-R”.)+p

I

3 l

derivation of some of the entries in table 3 are tedious and are given in appendix B. Here we summarize and explain the results. Under accrual taxation, the analysis of the effects of inflation is straightforward, but the effects themselves defy simple summary. All that can be said is that inflation complicates sometimes exacerbating and sometimes alleviating - the distortions induced by taxation. If the inflation rate is yl and the tax rate is r, wine taxed on an accrual basis is exactly the same as wine which is untaxed but discounted at a rate (r + tp)/( l-z). From (25) inflation can be modeled by assuming that the discount rate rises to r + q while the nominal value of wine sold at time t equals X(t)e”‘. Thus, wine harvested at time t has a discounted after-tax value of

X(r)e”‘exp[

-Er]=X(r)exp[

The selling time of wine, t,, maximizes the first-order condition

X’(L) x(t,)=Tz

r+rp

-=I].

the expression

above.

This gives us

242

D.J. Kovcw~ck und M. Rothschild,

Capital gains taution

The right-hand side of this equation is increasing in q. Thus, t, is decreasing in I?. The calculation of the DSV of gross returns under inflation is similar to that of proposition 9, only now the discount rate R + y is substituted for R. Consider one dollar invested in wine which goes to produce wine harvested at t, and worth X(t,) e qfa at that time. The DSV of gross returns is clearly the discounted value of after-tax returns X(t,)ePR’a. However, in equilibrium equals one dollar, so X(t,) =exp {[(r+ u]r)/( 1 -r)]t,}. Substituting, we find that the DSV of gross returns from a one dollar investment equals a= exp{[(R(T - z,) + v)/( 1 - r)]t&.

exp{[(r+r]z)/(l-z)]t,}~e-R’

When there is no capital gains tax (r=O), this equals emRr2d< 1. When rates in the two sectors are equal (r =zJ, this equals exp {[(r]z)/(l -z)]t,} > 1. Thus, under inflation, setting r equal to z, does not correct intersectoral distortion as in proposition 9 above. In appendix B, part B, we show that e”l”” P’))l., can be increasing or decreasing in q, the rate of inflation. Thus, an increase in the rate of inflation can alleviate or exacerbate tax-induced distortions. Trees are again more complicated than wine but the effects of inflation are qualitatively the same. Under inflation, an investor in a tree taxed on an accrual basis chooses u, to solve

”

V(u)exp

max

This is similar

I/(u)exp substituted

-

1

R(l-r,)+?ru 1-r

_R

[ l-exp[-R(l-z,)u]

to the case without

c -

RU -cJ+v 1-Z

inflation u

’ but with

1

for

In appendix B, part C, u, is shown to be decreasing in n. We now solve for the DSV of gross returns to one dollar Market equilibrium under inflation gives us

invested

in trees.

D.J. Kovenock and M. Rothschild, Capital gains tu.rution

Multiplying

243

each side by

we get R(T

[L(l-eP’“a)+P]exp

-

7,) + y-c

1-r

ua

= V(u,) eeRua,

which implies

When z = 0, this equals L(l-e-‘“a)+P [ L(1 -ePR”a)+P

<1, 1eeRroua If r = r,, this equals

as when there is no inflation.

e”w’

1 -r)lua

L(l -e-*“*)+P L(l-e-R”a)+P<

, I ’

Since u,-+t, when L(u,)-+O, the examples in appendix B, part B, can also be used to show that this expression can either increase or decrease as y increases. The analysis of inflation with taxation at realization is more complicated. Consider first wine. The discounted value of the after-tax return from wine with basis B. sold at date t is

=e -“X(t)(l The owner solves

-z) +e-tV+‘)‘7B.

of wine with basis B will choose

max W( t, 7, r~). t

(26) a harvesting

time t*(z,n)

which

244

D.J. Kovenock and M. Rothschild, Capital gains taxation

The first-order

which t* must satisfy is

condition

@(t*,z, q) =e-(q+r)t* { -rX(t*)

evt*(l -7)

-(y + r)zB + X’(t*) eqf*(1 - z)} = 0. Let W*(z, y) = W(t*(T, y), z, q). In equilibrium rate of return so that

investments

(27) earn

a competitive

B = B(z, rf) = W*(z, q) = max e -(‘+q)z{X(t)e~t(l-z)+zB(z,~)}.

f

Rearranging

B(z

Applying

(28)

we find that

> yl)

=

X(t*(5 Ml -4 Ce”’ _

the implicit

ze

-“‘*I

function

(29)

.

theorem

dt* sign-=sign[(q+r)t*-1 dq

to (27) we find that

+~e-(~+‘)‘*].

(30)

At low rates of inflation increases in q can increase or decrease selling time, depending on the evolution of X( .). In appendix B, part D, we provide examples where dt*/d?$,,o is greater than, less than, and equal to zero. As q approaches cc the problem the owner of wine faces converges to the problem he would face if his basis were zero and there were no inflation. That is, W(t,q 03) =e-l’X(t)(l

-7).

We now examine the DSV of gross returns on a one dollar investment in wine. An investment of one dollar in wine yields an after-tax return with a discounted value to the investor of (31) where r= R( 1 -zJ and wine is sold at t*. The DSV of the gross benefits generated by this dollar is X(t*)emR’*. Setting (31) equal to 1 and rearranging, we find

eeR’*X(t*)

Note

ce

r”Rf* _

ze

-

CR+ VP*

1-Z . that if z =O, inflation has no effect and the DSV is less than

one. If

245

D.J. Kovenock and M. Rothschild, Capital gains taxalion

r=r,

the DSV can be greater Rt*

--I

J(T,T,,q)=e o ;l’i

or less than one. To see this, let -(R+q)f*

Remember that at low rates of inflation, t* can increase or decrease in q. At high rates of inflation, t* increases in y. Assume z = z,. When y =O, J(z, z, q) < 1 as was shown above. As ‘1 approaches cc, J(r, rO, q) approaches e PToR7/(1- 7), w h ere t” is the harvesting time of wine taxed at realization with a zero basis. t” satisfies X’(o/X(I?) =R( 1 -zJ. The following examples show that lim,, m J(z, r, q) can be greater or less than one: (i) Let

X(t) =

0

t 5 d,

At -d)

t > d,

% satisfies X’(Q/X(I) = R( 1 - r,) l/R( 1- r,). Thus,

where d > 0.

which

implies,

in

this

case,

f= d +

that

exp[ -rRd-&]

exp[ -rR(d+&)] =

lim J(z, r, q) = q-30

1-z

1-z

.

For a fixed R and z#O, if d is sufficiently large this is less than one. (ii) Now let X(t) = pt + k, where k > 0. In this case

I=

[assume

1 R(l-7,)

that k

k P -z,)

in order to obtain

an interior

Thus,

cxp[ -%+?I

exp[ -rR(&-2)1 lim J(z, r, y) = q+m

solution].

= l-r

l-2

As k approaches p/R( l-z) this expression approaches l/( 1 -z), which is greater than one when z > 0. Thus, for k sufficiently large, lim,, n J(r, r, y) > 1. The effect of inflation on trees taxed on a realization basis is also complicated. In appendix B, part E, we show that as inflation increases, the cutting time of trees increases; inflation induces inefficiency. The proof of this fact is tedious and is given in appendix B. However, there is a simple intuitive explanation. Inflation and taxation interact to make it appear to the

246

D.J. Kovenock and M. Rothschild, Capital gains taxation

investor that maximize

planting

costs

have

increased.

The

investor

chooses

u to

While in any period he must pay planting costs of P, when he pays taxes he costs have can only exclude Pe -V from his tax base. Effective planting increased. The natural response is to use less planting by increasing the time between plantings. While it is easy to derive an expression for the DSV of a one dollar investment in trees when there is inflation, we have been unable to determine the effect of increases in the inflation rate on the value of this expression. In equilibrium L(1 -e-‘“*) Rearranging

+ P=epcr+s)u*[equ*

and multiplying

V(u*)( 1 -r)

+zP].

each side by e” pR)u*:

Thus, e

RI,* qu*)

L(1 -epRU*)+P

e - r”Ru*

L(1

-ee-‘“*)+P(l

-~e-(‘+‘IJ”*)

=p

(l-z)

L(l-epR”*)+P

1’

Since u* is increasing in y it seems difficult to determine the effect of an increase in y on the DSV of gross returns to a one dollar social investment. When z = 0, the DSV is less than one.

4. Death and taxes Death complicates. Until 1976, in the United States, when the holder of an asset died, the estate paid no capital gains tax; if the investor’s heirs sold the asset in the future the basis of the asset used to compute capital gains tax liability was its value on the date of the investor’s death. From the point of view of the estate, death was equivalent to a sale on the date of death which escaped capital gains taxation. Although death may not have had much else to recommend it, it was a dandy way to avoid taxes. This system was

D.J. Kovenock and M. Rothschild, Capital gains taxation

247

changed in 1976, and the U.S. began to move toward a system like that analyzed in section 1 in which the heir’s basis would be the basis of the original purchaser of the asset. In 1980, this movement stopped; the U.S. has now reverted to the pre-1976 system. In this section we analyze the effect of the capital gains tax on the selling time and value of Austrian assets when death alters the way in which capital gains are taxed. Special treatment of capital gains at death alters our conclusions about the effect of the capital gains taxes on selling time. Let F(t) be the probability that the owner of wine lives until at least period t. Then F(0) = 1, F(a) =O, and F(t) is decreasing. For simplicity we assume F(t) is differentiable and let f(t)=F’(t). If the probability of death is constant, then F(t) = epyt and f(t) = -y emy’. Let zd be the rate of capital gains taxation at death. We consider two values of zd. z,=O corresponds to the present U.S. system - which is known as automatic step up in basis, or step up for short. Under step up, capital gains taxes are avoided at death. Another possible treatment of capital gains at death is what is known as constructive realization. Under this system the estate is presumed to sell the asset at death and to pay tax at the ordinary capital gains rate on the gains which the sale realizes. Under constructive realization zd=z. We compare these two systems to the system analyzed in the previous section which is referred to as a carryover system. If an investor plans to sell an asset at t, the expected discounted value of the after-tax proceeds is

J(t)=e~“r’(t)(X(t)(l-r)+rL1)-~e-’”(X(s)(l-~,)+~d~)~(s)ds.

(32)

0

The first term in (32) represents the discounted after-tax value he will receive if he lives until t. The second term is the discounted after-tax value which his estate will receive. This formula is not quite correct for two reasons. First, the pretax value the estate realizes will in general be more than X(t) since the value of the asset will exceed X(t). Determining exactly what this value is seems very difficult. Secondly, in contrast to the analysis of section 2 above, B here is an arbitrary parameter. Although in principle B should satisfy a zero profit condition, it is hard to write down this condition for this model because investors’ mortality functions will influence their profits. It is hard to figure out what should be the ‘market’ mortality function. We think neither of these simplifications will affect our analysis. The planned selling time t* is chosen to maximize J(t); t* satisfies z--zd 1 +I-zF(t*)

f(t*)

1_

[

-1 B

X(t*)

.

(33)

248

D.J. Kovenock

and M. Rothschild,

Capital gains taxation

Proposition 12. For capital gains taxation at the same rate, planned selling time is the same under constructive realization as under carryover. Planned selling time is greater under step up. Proof: Sincef(t)

< 0, this follows immediately

from (33).

Note that planned selling time is not the same as actual selling time. Under either step up or constructive realization, an investor’s heirs may plan to sell an asset at a different time than the original investor planned. This is because the heirs will have a different, higher basis which will cause them to plan to sell sooner than the original investor. They also will have a different mortality function which will, under step up, also affect their planned selling time. It is easy to see from (32) that increases in either r or zd will decrease the value of an asset. The effects of increases in r on planned selling time are more complicated. Since planned selling time is the same under constructive realization as under carryover, dt*/dr
z

P

l+I--cI/o+v(u*)(l-r)

L

~-?lf(U*) 1_ +-1 --z F(u*) I [

v(u*) 1. P

Appendix A: Frequently used notation y(s) =Rate of flow of returns from investments in the ordinary sector s years after the initial investment. X(t) = Value of wine if consumed t years after it is laid down. B =Base or initial value of investment in wine. V(u)=Value of a tree if cut down u years after planting. P = Planting cost of a tree. L =Value of land used to grow a sequence of trees. =Selling time which maximizes discounted social value of total returns F to investments in wine. =Cutting time which maximizes discounted social value of total returns li to investment in trees. time which maximizes discounted private after-tax returns to t* =Selling investments in wine taxed on a realization basis.

U*

r, % r R r0 Z ‘d

D.J. Kovenock and M. Rothschild, Capital gains taxation

249

=Cutting time which maximizes discounted value of after-tax returns investments in trees taxed on a realization basis. =Privately optimal selling time for wine taxed on an accrual basis. =Privately optimal selling time for trees taxed on an accrual basis. = After-tax rate of return. =Gross rate of return in the ordinary sector. =Rate of taxation in ordinary sector. =Rate of taxation in Austrian sectors. =Rate of taxation of capital gains at death.

to

Appendix B A. We wish to show that as T approaches E. From eq. (10):

Rearranging

one, t* may be greater

or less than

(6) we find that

B ----= X(t*)

e -I**( 1 - r)

’

1 -rem”*

which upon insertion

into (10) gives us

X’(t*) Y R(l X(t*)=l_-Ze-r”=l_-ze-rt*. The following example approaches one. Let

X(t) =

shows

-d that

0,

t 5 d,

A-4

t>d.

(A.11 t* may

be greater

or less than

f as z

Then from (A.l) -=

P

p(t*-d)

Y

l-rep”*’

which, after some rearrangement, rt+ze

-“=rd+

1

implies

that t* solves

64.2)

D.J. Kovenock

250

and M. Rothschild,

Capital

gains tavafion

The right-hand side of (A.2) is independent of t. The left-hand side of (A.2) is increasing in t for t>O. A simple calculation shows that with X( .) given as above, F=d + l/R. But then

~r?=rd+~+~e~‘d-‘/R=rd+l+(~e’-‘d~lf’”)-_Zo).

rf+ze

R

For d sufficiently large and z, #O, the expression in parentheses is negative. Thus, in this case, rt^+ ze mrf< rd + 1 and, since rt + se -*’ is increasing in t, (A.2) implies that t*>f. On the other hand, if d =O, (A.2) becomes rt* +z eert*= 1 and E equals l/R. Thus,

For z close to one and r, sufficiently small, one. Since rt + Te-” is increasing in t, &> t*. B. (1) Let C(v) =et”‘/(‘-‘)“~

where t, satisfies

X’(L) -=R+E. X(&J Let X(t) = pt. Then

x’(ta)_&J+$ X(L) which implies t, =

ta that l-r

R(1 -T)+qz’

Thus

With C(0) = 1 and rR(1 -r) “(‘)

= [R( 1 -r)

+ yz12

C(V) ’ 0

this expression

is greater

than

D.J. Kovenock and M. Rothschild, Capital gains taxation

251

we find that inflation increases the DSV of a one dollar investment above the optimal level of one dollar. As y approaches a3, C(y) approaches e. (2) For a more dramatic example of this effect, let t 5 d,

0, A-4,

X(t) =

d>O.

t > d,

Again t, solves

X’(L) X(L)

-=R+E,

which implies

that 1-T

ta=d+R(l-z)+qz’

Substituting

into C(q), we find that

” R(l -z)+qz

1

From the calculation in paragraph 1, it is apparent that C’(y) >O. Furthermore, lim,,, C(q) = co. (3) For an example where C’(y) can be negative for q > 0 and 0 < z < 1, let X(t) = pt + k, where

cl-+

/LR(~ -2)’

R(l --)+?,/z

‘k’[R(l-~)+qz]2.

In this case, ta =

1-z R(l--)+qz

k -->o p

and

v C(r) =exp R(l--)+qz

--- v (l-z)

1

k

1-1 ’

D.J. Kovenock and M. Rothschild, Capital gains taxation

252

It is easy to show that

1_-z p1

z -__-.

From

the restrictions

k

on k, C’(q) < 0.

C. Proposition A.I. The privately optimal cutting time of trees accrual basis is decreasing in q, the rate of inflation. Proof: The privately

Assuming

an interior

optimal

solution,

cutting

taxed

on an

time u, solves

the first-order

conditions

imply

(A.3)

64.4)

Thus,

253

D.J. Kovenock and M. Rothschild, Capital gains taxation

and, since i&b/au, < 0 from the second-order

conditions,

sign$=sign$=signV(u,){(e-‘“a-l)[s+l]

.

+&[r(l-re~‘“a)+r~(l-e-‘“a)] I Rearranging

the first-order

=$

condition

(A.4), we find that

Pre-’ a

Thus, .

du,

.

“_Pre-‘“aexp sign dy = sign i Vu,)

[(~)u~]+c-?-l

From (A.3) we know that

which implies u, ~Pre-“aexp[(~)u,]+e’“,-Ic(u,r+i)e-’”a-~. Vu,) The right-hand side of (A.5) attains a maximum over u,rzO this point (u,r + 1) eerua - 1= 0. Thus, duJdq s 0.

(AS)

when u,r=O.

At

basis, at D. In this section we show that with wine taxed on a realization low rates of inflation an increase in q can increase or decrease selling time. is greater than, less than, and equal to We give examples where dt*/dyl,,, zero. (1) Let

X(t) =

0,

tsd,

At-d),

t>d,

where d > 0.

D.J. Kovenock

254

and M. Rolhschild,

Capital gains taxation

Then from (27), with y=O,

and from (28) and (29) W*(r,O)=F(l-z)exp

From

-l-dr+(l_r)P

rzB

(30) this implies that

sign[%iqZJ=sign[rd-&F+rcrp(

-l-k+&)].

Inserting the condition B = W*(z, 0) from (28), we see that the right-hand of the above equation equals sign [rd]. Since rd > 0, dt*/dy/,= 0 > 0. (2) Let X(t) = pt + c, where c >=0. Setting q = 0, (27) implies that

side

and from (28) and (29),

Al -4

w*(T, 0) =p

Inserting

r

exp

this value of t* into (30) and setting

sign[g/VZJ=sign[

ye= 0, we find

-F-&+ieXp(

-1 +:+s)]-

Setting B= W*(Z, 0), the right-hand side of this equation becomes sign [ - cr/p]. Thus, if c equals zero, dt*/dy 111 = 0 = 0, while if c is greater than zero, dt*/dy I,,=,-,
A.2. For

time, II*, is increusing Proof.

An investor

trees

taxed

at realizution,

the privately

optimal

cutting

in rf.

with a plot

of land

which

can support

one tree forever

D.J. Kovenock and M. Rothschild, Capital gains taxalion

chooses

to cut each tree planted

max --P+ ”

2

255

after u* years, where u* solves

e-‘“i[V(u)(l-r)+rPe-‘r”-P]

i=l

= max V(u)(l -z) + P[z e-“” -e”] e ‘11-1 U Assuming

an interior

solution,

a first-order

condition

for this problem

is

Thus du* - a407 -==--a4/au* drl and, since &$/au*~0 from the second-order

Rearranging,

conditions,

we find that

[(r + q)u* - l] e’“’ + (1 -vu*)

= qu*(e’“* -l)+(ru*-l)e’“*+l 2 rp*(e’“*

- 1) 2 0.

Thus, du*fdr] 2 0.

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256

D.J. Kovenock and M. Rothschild, Capital gains faxalion

Diamond, P., 1968, The opportunity costs of public investment: Comment, Quarterly Journal of Economics 82, 682-688. Diamond, P. and J. Mirrlees, 1976, Private constant returns and public shadow prices, Review of Economic Studies 43, 4147. Feldstein, M. and J. Slemrod, 1978, Inflation and the excess taxation of capital gains on corporate stock, National Tax Journal 31, 1077118. 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. Feldstein, M., J. Slemrod and S. Yitzhaki, 1980, The effects of taxation on the selling of corporate stock and the realization of capital gains, Quarterly Journal of Economics 94, 777791. Gaffney, M., 1967, Tax-induced slow turnover of capital, Western Economic Journal 5(4), 30% 323. Gaffney, M. 197@71, Tax-induced slow turnover of capital. Part I. American Journal of Economics and Sociology 29(l), 25-32. Part II, AJES 29(2), 179-197. Part III, AJES 29(3), 277-287. Part IV, AJES 29(4). 4099424. Part V. AJES 30(l)./. 105-l 11. Klemperer, W.D., 1974, Forests and the property tax. A reexamination, National Tax Journal 27(4), 645-651. Kovenock, D.J., 1982, Property and income taxation in an economy with an Austrian sector, University of Wisconsin-Madison, Social Systems Research Institute Workshop Series No. 8227. Kovenock, D.J. and M. Rothschild, 1981, Capital gains taxation in an economy with an Austrian sector, National Bureau of Economic Research Working Paper No. 758. Lippman, S.A. and J.J. McCall, 1981, Progressive taxation in sequential decisionmaking: Deterministic and stochastic analysis, Journal of Public Economics 16, 35-52. Marglin, S.A., 1963, The opportunity costs of public investment, Quarterly Journal of Economics 77, 274289. Samuelson, P.A., 1964, Tax deductibility of economic depreciation to insure invariant valuation, Journal of Political Economy 72, 6044606. Reprinted in The Collected Scientific Papers of Paul A. Samuelson, vol. 3 (MIT Press, Cambridge, MA, 1972) 571-573. Samuelson, P.A., 1976, Economics of forestry in an evolving society, Economic Inquiry 14, 466492. Reprinted in The Collected Scientific Papers of Paul A. Samuelson, vol. 4 (MIT Press, Cambridge, MA, 1977) 146172. Thomson, P. and H.N. Goldstein, 1971, Time and taxes, Western Economic Journal 9(l), 27745.