The incidence of taxes

Journal

of Public

Economics

18 (1982) 161-194.

North-Holland

THE INCIDENCE

Company

OF TAXES

A dynamic macroeconomic

Stephen

Publishing

analysis

J. TURNOVSKY

Australian National University, Canberra A.C.T. 2600, Australia University of Illinois, Urbana, IL 61801, USA

Received

February

1981, revised version

received

September

1981

This paper analyzes the incidence of various tax changes using the intertemporal optimizing framework developed by Brock and Turnovsky. The model contains a reasonably complete specification of the corporate sector, in which firms have the choice of financial structure. It also allows for a range of government financial policies, although only one is considered in detail. Both the short-run and the long-run effects of changes in the personal income, capital gains, and corporate profit tax rates are analyzed. It is shown that the effects of tax changes (a) vary between the short and long run; (b) the effects of tax changes depend critically upon the financial structure adopted by forms; and (c) may also depend upon government financial policy.

1. Introduction The incidence of taxes has been a central issue in public finance for many years. Until recently, the standard framework for conducting incidence analysis has been the two-sector general equilibrium model pioneered by Harberger (1962). But despite the importance of this approach, it suffers from several limitations, as noted in the comprehensive survey by McLure (1975). In particular, being static, it is incapable of dealing with longer-run issues of tax incidence which arise in a dynamic context. Recently, several authors have considered the question of tax incidence using dynamic models which incorporate capital accumulation; see, for example, Feldstein (1974a, 1974b), Grierson (1975) Boadway (1979), Atkinson and Stiglitz (1980, lecture 8), and Homma (1981). One of the general conclusions to emerge from these studies is that the incidence of a particular tax in the long run may be very different from what it is in the short run. This is because of the ability of the taxed factor to shift at least part of the tax through induced changes in factor proportions in the economy. Consequently, the reassessment of tax incidence within a dynamic framework adds an important dimension to the discussion. 0047-2727/82/000&0000

$02.75 0

1982 North-Holland

162

S.J. Turnoosky, Thr incidence of’ta.ws

However, the existing dynamic analyses suffer from several limitations. In the first place, the models are all variants of the traditional neoclassical onesector growth model. They therefore focus exclusively on real phenomena and abstract from the existence of financial assets (either public or private) and inflation. Yet it is reasonable to argue that the effects of tax changes on real activity, and hence on real factor returns, will be closely related to their effects on the rate of inflation and the rates of return on different financial securities. Moreover, as we shall show below, the general macroeconomic effects of various tax changes, and their implications for tax incidence in particular, may be dependent upon the financial policy chosen by the government, an aspect which cannot be considered by these earlier studies. A second, related shortcoming of the existing literature is that it abstracts from the corporate sector. Yet it is well known from corporate finance theory that the relative tax rates on various classes of income are important determinants of the optimal financial structure chosen by firms. And this financial structure will be shown to be important in determining the incidence of certain tax changes. The third shortcoming is that savings behavior in the traditional onesector growth model underlying these analyses is specified arbitrarily, while the firms’ production decisions are derived from static optimization. The limitations of this approach became particularly severe when one wishes to augment the model to include a corporate sector embodying rational behavior. As we have just noted, the effects of alternative tax changes may depend critically upon the capital structure employed by firms, which in turn is a function of the tax structure and the differential tax treatments of different securities. To incorporate this aspect adequately requires all the behavioral relationships for both households and firms to be derived from their respective intertemporal optimizations. The purpose of the present paper is to analyze the incidence of various tax changes within a dynamic macroeconomic model which attempts to overcome these three criticisms. Specifically the model is an inflationary one, including both private and public financial securities and is based on an explicit intertemporal optimizing behavior. The framework we shall employ is that developed recently by Brock and Turnovsky (1981). This includes the following key features: (a) all demand and supply functions of households and firms are derived from maximizing behavior; (b) expectations are determined simultaneously with perfect foresight continually holding; and (c) all markets are continually cleared. Under these conditions all expectations will be ‘self-fulfilling’ and for this reason an equilibrium characterized by (a), (b), and (c) can be termed a ‘perfect foresight equilibrium’. In addition, the Brock-Turnovsky model has the following characteristics. The government can finance its deficit either by issuing bonds or by the creation of money. Likewise, firms may finance their investment plans either

S.J. Turnnvsky, The incidence of’ taxes

163

by issuing debt, by issuing equities, or out of retained earnings. Government bonds and both types of private securities are assumed to be perfect substitutes for one another. Three different types of taxes are considered. These include: (a) a tax on ‘ordinary’ personal income (income from wages, interest, and dividends); (b) a tax on income from capital gains on equities; and (c) a corporate profit tax. In doing so, the rates and structure are specified so as to approximate what might be considered to be ‘real world’ tax regimes; see, for example, Feldstein, Green and Sheshinski (1979) and Auerbach (1979). Finally, being a certainty model, except in special circumstances when the debt-to-equity ratio is indeterminate, the optimal corporate financial structure will consist of either all bond financing or all equity financing, depending upon relevant tax rates, in accordance with familiar propositions from finance theory; see Modigliani and Miller (1958) and Miller (1977). The earlier Brock-Turnovsky analysis was directed primarily at the development of an integrated framework within which alternative Because of space limitations macroeconomic policies could be considered. almost no attention was devoted to the analysis of specific policy issues. Yet this type of framework is a very natural one for examining issues like the incidence of alternative taxes, particularly their long-run effects. Indeed, a framework such as this makes it clear how the entire range of macroeconomic effects of tax changes are interrelated and must be considered in conjunction. By including government securities in the model, the range of tax incidence questions is expanded considerably. In the absence of such financial securities, a change in any one tax will require compensating adjustment in either some other tax or in government expenditure, in order for government budget balance to be maintained. In our expanded model, however, a budget deficit resulting from some single tax change can be accommodated by some appropriate change in the stock of government debt outstanding. Indeed, the incidence of any tax change may depend upon the financial policy chosen by the government to finance its deficit. The remainder of the paper is structured as follows. In section 2 we outline the Brock-Turnovsky model and summarize it in a form convenient for subsequent analysis. Sections 3 and 4 examine the short-run incidence of various tax changes. It is shown how in general these consist of direct effects, which are the usual comparative static effects, together with indirect effects, which operate through induced jumps in state variables and depend upon the dynamic properties of the system. Section 5 considers the long-run tax incidence under the assumption that the domestic monetary authorities follow a policy of steady monetary growth. The following section briefly considers some exercises of ‘differential tax incidence’ and also notes the case of an incrementally balanced budget. Section 7 briefly considers an

S.J. Turnmsky, The incidence of iuxes

164

alternative government the main conclusions.

financial

policy,

while

the final

section

summarizes

2. Review of framework Since the model has been developed in detail necessarily brief. We shall simply review its main to enable us to characterize the equilibrium contains three basic sectors - households, firms, of which are interrelated through their respective sectors shall be reviewed in turn, in all cases being expressed directly in real terms.

elsewhere, our treatment is features in sufficient depth relationships. The model and the government - all budget constraints. These the behavioral constraints

2.1. Household sector We assume that households can be aggregated into a single consolidated unit. The objective of this composite unit is to choose its consumption demand, its supply of labor, the rates at which it wishes to add to its real holdings of money balances, government bonds, corporate bonds, and equities, to maximize the intertemporal utility function’

5e-0’ subject

U(c,g, P.&‘)dt.

u,>o,

-I’@‘;+ b; + md)+ iqEd-

T,,

u,>o,

u,
to

B,(O)= B,o;

M(0) = M,;

lim exp -i 1+= (

0

B(s)ds h, 2 0; 1

(lb)

B,(O)= B,,;

f’“exp(

-qO)= JSJ,

-bO(s)d.s)

h,20;

(14

lim exp f + CL

‘We shall follow the convention of denoting letters and total derivatives by primes.

(lc)

partial

derivatives

by corresponding

lower case

S.J. Turnovsky, The incidence of taxes

165

where =real private consumption plans by households, = real government expenditure, taken to be fixed exogenously, = real supply of labor by households, = Md/P=demand for real money balances, m* M =nominal stock of money, P = nominal price of output, for real government bonds, b): =demand = nominal stock of government bonds, 4 =demand for real corporate bonds, % = nominal stock of corporate bonds, 4 E = number of shares outstanding, = relative price of equities in terms of current output, 4 = D/qE =dividend yield, taken to be parametrically given to the i household sector, D = real dividends, qEd =real stock demand for equities, W =real wage rate, =nominal interest rate on government bonds, r9 =nominal interest rate on private bonds, TP = instantaneous anticipated rate of inflation, P =actual rate of inflation in perfect foresight equilibrium, =personal income tax in real terms, specified more fully below, = consumers’ rate of time preference, 2 H(s) =instantaneous real rate of return to consumers at time s, to be determined endogenously below.

C

(1 I”

The utility function is assumed to be concave in its four arguments c,y, I, and m. The introduction of c and g as separate arguments reflects the assumption that consumers view private and public goods as imperfect substitutes. Labor yields disutility while the introduction of real money balances into the utility function is a convenient device for capturing the reasons for holding money in a certainty world. The household sector’s budget constraint is given by (1 b), which we have expressed in real flow terms. At each point of time the households are assumed to acquire income from a variety of sources. They supply labor to firms, at a real wage rate w; they earn interest income on their holdings of real private and government bonds; they make instantaneous capital gains or losses on their holdings of financial wealth denominated in nominal terms (money and bonds); they receive dividend payments at a rate i on their holding of equities. This rate is taken as parametrically given to households, but is one of the decision variables of the corporate sector. This income can be used in a variety of ways. They may use it to purchase real consumption

S.J. Turnovsky, The incidence of taxes

166

goods, to add to their real holdings of money, government bonds, corporate bonds, and equities (the relative price of which is q), and to pay taxes to the government. It is important to note that the decisions derived from this optimization procedure are planned demands (or supply in the case of labor). We have recorded this fact explicitly by the inclusion of the superscripts d(s). Finally, the restraints (Id) must be added if borrowing is allowed, in order to prevent the present value of debt from becoming as t--tm. 2.2. The corporate sector The firm is assumed

to be bound

y”= F(kd, Id)= ldf(kd/ld),

by the following f’>O,

constraints:

f”
(24

7c=yS-wld,

W

x=rpbp+D+RE+Tf,

(2c) (24

k(0) = k,;

E(0) = E,;

B,(O) = &J,

P)

where Id kd YS 71 b”, ES RE Tf

=real demand for labor by firms, =real demand for physical capital by firms, = real output, =real gross profit =supply of corporate bonds by firms, in real terms, =quantity of equities, issued by firms. =retained earnings, in real terms, =corporate profit taxes in real terms, specified more fully below.

All other symbols are as defined above. Eq. (2a) describes the production function, which is assumed to have the of positive but diminishing marginal usual neoclassical properties productivities and constant returns to scale. Eq. (2b) is the conventional definition of gross profits in real terms as being revenue less payments to labor. Eq. (2~) describes the allocation of gross profits. After paying corporate income taxes, this may be used to pay interest to bond holders, to pay dividends to stock holders, or retained with the firm. Eq. (2d) expresses the firm’s financial constraint. Any additions to capital stock must be financed out of retained earnings, by issuing additional equities or by issuing

167

S.J. Turnousky, The incidence of tuxe.7

the revenue on private additional bonds. The final term, pb,, is essentially bonds accruing to the firm by virtue of the fact that these bonds are presumed to be denominated in nominal terms. It is precisely analogous to the inflation tax generated on financial wealth issued by the government and which also appears in the household sector’s budget constraint. Finally, eqs. (2e) are initial conditions on the real stock of capital, the number of equities outstanding, and the nominal stock of corporate bonds. We define the market value of the firm’s securities outstanding at time t by

W = b,(t)+ 44 E(t)

(3)

(where we suppress superscripts) and shall assume that the firm’s objective is to maximize the initial real market value of its securities, V(O)= b,(O)+q(O)E. In Brock and Turnovsky it is shown how, given the constraints in (2a)(2e), the definition of Tr given below, and the optimality conditions for households, the firm’s objective function V(O) can be expressed as the following discounted flow:

(4)

where y(t) = real net cash flow, and x(t) = instantaneous cost of capital at time t. The firm’s optimization problem is to choose its production decisions k“ and Id, and financial decisions, the mix between debt and equity financing, II, and the dividend payout, i, to maximize the expression given in (4). The quantities y(t) and x(t) are defined by:

r(t)-(l-zp)Cf(k/l)-wll-k 2

x(t)Ee+(ty-Zp)(e+P)

l-z,

(54 (Q+P)~c+~(~,-~c) *

l-A+

1 -r,

(5b)

l-3,’

where ZP ZY

;

=rate of tax on =rate of tax on =rate of tax on = b,,/qE = firm’s

corporate profit, wages, interest income, and dividends, nominal capital gains on securities, debt-to-equity ratio.

All other symbols have been defined the superscripts have been dropped.

previously.

For notational

convenience,

S.J. Turnooskg, The incidence of’taxes

16X

The expression equivalent form:

x=

for

(w (

the

cost

of capital

(5b) can

be expressed

in

the

-~p)+(~y-~JhJ 2

+

1 -zy

> l+A

8+z,p+i(z,-T,)

1

(W

> 1 +I’

1 -z,

The quantity 0, which appears in (5b) and (5b’), is shown by Brock and Turnovsky to equal the consumer’s real net rate of return on a marginal dollar. For consumers to be in equilibrium this must be the same in all uses. The household’s optimality condition for consumption yield the familiar relationship:2

while the optimality equities are described

conditions by314

Hbi = [ri( 1 - ZY)- P]bi,

Substituting

(6b) (with i=p)

determining

for

bonds

and

(6b)

i = p, 9,

and (6~) into (5b’) yields

x = Cr,(1- zp)- plb,/V+

(i + d&W K

implying that x is simply a weighted average debt capital and equity capital to the firm. ‘This ‘The interest return

its demand

(5b”) of the real after-tax

costs

of

condition is essentially equivalent to the optimality rule obtained by Yaari (1964). net real rate of return to consumers on their holdings of bonds is the after-tax nominal less the rate of inflation. The net real rate of return on equities is the after-tax rate of on dividends, i(1 -TJ, plus the after-tax rate of nominal capitals gains (i/q+p) (I ~ TJ less the rate of inflation. Adding these three terms together yields the expression in (6~). 41t should be noted that the Hamiltonian for the consumer’s optimization problem is linear in the financial decision variables h,, h,, and E. In view of this, depending upon the precise tax structure assumed, some of these securities may or may not appear in strictly positive quantities in the equilibrium demands of the household sector. To allow for the possibility that some may turn out to be zero in equilibrium it is necessary to solve the optimization problem using Euler inequalities, which in effect are simply the analogues to the Kuhn-Tucker conditions in conventional non-linear programming. Thus, for example, the complete optimality condition for bonds is rAl -z,)-psfl and bJri(l -~~)-p-Q]=o. If the inequality is met strictly, then the corresponding decision variable bi=O, while conversely if b(>O, the corresponding constraint is satisfied with equality. For further discussion of this see the appendix to Brock and Turnovsky (1981).

S.J. Turnovsky, The incidence of taxes

169

The important thing to observe about (5a) and (5b) (or (5b’)) is that r(t) is a function of the real production decision variables, k and 1; the financial decision variables, A and i, are embodied in the cost of capital x. As a result of this, the two sets of decisions can be obtained sequentially. That is, 2 and i can first be chosen to minimize x; having determined the minimum cost of capital, the firm’s real production decisions can then be made. Taking the partial derivatives of (5b’) with respect to i and A yields

ax

sgnai =sgn (l-r, ax

7

Y

--z

>

(74

’

by - z,)(Q+ PI _ (0+ h, + i(z,- 4

(

sgna;l =sgn 1 -Z,

l-z,

) .

C’b)

From these two equations it is evident that the optimal dividend policy and the optimal capital structure will involve corner solutions.’ Assuming that the tax on capital gains is less than that on other kinds of personal income, (zC
side of (7b), is negative, side of (7b), is positive,

set i,= CC, i.c. set i =O. i.e.

Thus, the firm should employ all bond financing or all equity financing, depending upon the relevant tax rates. 6 This criterion is identical to that of Miller (1977) and yields the conclusions for a firm’s financial policy which turn out to be familiar from corporate linance theory.’ Thus, the firm’s minimum cost of capital, x*, may be expressed as

x*=e+min

(z,-5p)(e+P) (Q+Phf~~,--z,)

(

l-z,

’

1 -z,

1

(8)

5Note that in the absence of taxes x is independent of i. Dividend policy is therefore irrelevant, conlirming the well-known Miller and Modigliani (1961) proposition in this case. 6This raises problems of the compatibility of the equilibrium with the given initial stocks of bonds and equities. Two ways of circumventing this difficulty are indicated by Brock and Turnovsky. ‘The equivalence between the criterion given in (7b) and that of Miller (1977) is discussed by Brock and Turnovsky. Note that in the absence of taxes, x is independent of the debt-equity ratio, contirming the Modighani and Miller (1958) proposition.

S.J. Turnottsky. The incidence qf‘taxes

170

With all bond

financing

this reduces

x*=H(l --p)+(T,--JP 1 -r) while with all equity

x* =

financing

to8

=r,(l

it becomes

u+z,p+i&-TJ l-r,

-7,)-p,

i-+lj =- 4

’

where the second equalities in (8’) and (8”) are derived by using (6b) and (6c), respectively. Given the minimized cost of capital, the optimal production decisons are simply the marginal physical product conditions: (1 - ZP)F,(k, I) E (1 - t,)f’(k/l) F,(k, 1)= f(k//) - (k/l)f’(k/l)

=x*,

= w.

Pa)

V-4

This is to say, the after-tax marginal physical product of capital should be equated to the minimized cost of capital, while the marginal physical product of labor should be equated to the real wage. Also, the transversality conditions for the firm’s optimization can be used to establish the equality between the firm’s stock of capital on the one hand. and the value of its outstanding securities on the other:9 V=h,+qE=k.

(10)

2.3. The yowrnment The government is assumed to provide real goods and services, g, which it finances out of real tax receipts, or by issuing form of government debt. Its budget constraint is described in real terms by riz‘+ d; = g + rg h; - Th- Tf - (m” + h;)p,

(11)

where the superscript s denotes the planned supply by the government. This equation defines the real deficit net of the inflation tax by the right-hand side of (11) and asserts that this is financed either by increasing the real money “These expressions for the cost of capital are generally similar (1979), although there is a minor difference in (8”) stemming assumption regarding dividend policy. “For a derivation of this see Brock and Turnovsky.

to those given by Auerbach from a minor difference in

S.J. Turnnvsky, The incidence of taxes

171

supply or by increasing the real stock of government bonds. The choice between these two alternatives, or any other monetary policy for that matter, represents a policy decision which needs to be specified in order to close the model. Finally, we must specify the tax functions Th and TJ. These are hypothesized as follows:

Th= ~,[wl”+ rg b,d+ rp b; + iqEd] + r,(4 + qp)E, 7” = z,[y'- wld- rp b;],

(124

OSi;Ty<=1, (12b)

where for simplicity all tax structures are assumed to be linear. According to ‘ordinary’ personal income income from wages, interest, and dividends is taxed at the flat rate zY. Nominal capital gains on equities are assumed to be taxed at the rate z. which may or may not equal 5?, and indeed in many economies e,=O. Notice that (12a) implies that capital gains are realized at each point in time; i.e. the portfolio is continuously ‘rolled over’. Alternatively, one may view (12a) as representing taxes on unrealized capital gains. Turning to corporate income taxes, gross profit is assumed to to bond be taxed at the proportional rate zp, with the interest payments holders being fully deductible. In all cases full loss offset provisions are assumed.

(12a),

2.4. Perfect foresight

equilibrium

The perfect foresight equilibrium (PFE) we shall consider is defined as follows. First, consider the household sector’s maximization problem specified by (1aHld) with Th defined by (12a). Carrying out this maximization yields a set of demand functions for consumption and various securities together with a labor supply function, in terms of p, w, r,,, rg, etc. and other parameters which consumers take as given. Likewise, the corporate sector’s optimization problem, defined by (2aH2e) and (4), with Tf defined by (12b), yields a set of demand functions for capital and labor, and a set of supply functions for various securities together with output, which are also functions of w,rb, etc. which firms too treat as parametrically given. Thirdly, the government policy decision constrained by (11) generates supplies of money and government bonds and a demand for goods. The perfect foresight equilibrium is defined as a situation in which the planned demands for output, labor, and the various securities in the economy all equal their corresponding real supplies and, in addition, all anticipated variables are correctly forecast. In this case md=ms, etc. and

JPF

R

where no confusion can arise, we shall simply drop all superscripts. Thus, the quantity m say will denote the real money supply in a perfect foresight equilibrium and henceforth we shall focus our attention on these equilibrium quantities. 2.5. Determination

of short-run equilibrium

The household sector’s optimization problem is a standard calculus of variations and the details of the solution are discussed by Brock and Turnovsky. Carrying out the procedure and combining with the optimality conditions for firms, discussed briefly in subsection 2.2 above, the short-run perfect foresight equilibrium of the system can be described by the following sets of equations:

Uk 9, l,m) = - F,(k, I)( 1 - r,), UC@, g,tm)

(13b)

(13c)

x*=e+min

(

(~,-~.p)(~+P) l-T,

’

(H+P)‘C++d l-2,

1’

(134

c?=a(p-8),

(144

&=lf(k/l)-g-c,

(14b)

ri + hg = g +

t?b, - z,lf(k/lj - mp + [f3 -( 1 - z,)f’]k.

(14c)

Eq. (13a) is simply a short hand notation for the marginal utility of consumption. Eq. (13b) equates the marginal rate of substitution between consumption and leisure to the after-tax real wage, which in a PFE is simply the after-tax marginal physcial product of labor. The third equation requires that the marginal utility derived from holding a dollar in cash balances must equal the marginal utility that would be derived from spending the dollar on consumption. Given that 0 measures the after-tax real rate of return to consumers, this equation can also be interpreted as saying that the marginal rate of substitution between money and consumption equals the after-tax

S.J. Turnousky. The incidence qf taxes

173

nominal rate of return. Eqs. (13d) and (13e) restate the marginal productivity condition for capital and the minimum cost of capital, respectively. These five equations may be used to solve for the short-run solutions of the five variables 1,c,p, 8, and x* in terms of the dynamically evolving variables a, k, and m. The first of the dynamic equations is simply a restatement of (6a), derived from the consumer’s optimality problem. Eq. (14b) describes the rate of capital accumulation required to maintain product market equilibrium, while (14~) is the government budget constraint. To obtain this form of the constraint, use has been made of the optimality conditions for both firms and households, as well as the linear homogeneity of the production function; see the specification of the dynamics is Brock and Turnovsky. lo Finally completed by the introduction of &me government financial policy (i.e. rules for m and/or b,).” The system of equations in (13) and (14) provide the basis for the shortrun effects of tax changes undertaken in section 3. At this point several points should be noted. First, the equilibrium real stocks of corporate bonds and equities are determined from the balance sheet constraint (lo), together with the optimality conditions (7b). Apart from knife-edge cases, these imply either h,= k or qE = k, depending upon whether the optimal financial structure calls for all bond or all equity financing. There are also technical issues regarding initial jumps in real stocks through jumps in the price level, but these are not discussed here. Secondly, the nominal rates of return on the financial securities can be summarized from (6b) and (6~) as r&l -z,)-p=#=max[r,(l

-z,)-p,$l-TJ (130

From this equation it is seen that the nominal rate of interest paid on government bonds must be such as to equate their real after-tax rate of return to investors to the real after-tax rate of return on the existing private security. If bond financing is optimal for firms, then rg = r,,. Otherwise, in the case of equity financing, r9 is determined by the rate of return on equities. Thirdly, it is clear how the introduction of government financial assets allows a much greater degree of flexibility with respect to the specification of “‘The derivation and interpretation of this equation is rather complicated and is discussed further by Brock and Turnovsky. “The household and corporate sectors’ budget constraints are embodied in the optimality conditions, as well as the government budget constraint, contained in the equilibrium system described by (13) and (14). They require explicit consideration only if one wishes to focus on the flow of inside assets between these two parts of the private sector. Also, equilibrium conditions in the various markets are embodied in the notion of perfect foresight equilibrium described by this set of equations.

174

S.J. Turnomky,

The incidence

of

taxes

tax policy than is the case in a purely ‘real’ model. In effect, there are six policy parameters, ry,rprz,,y, hj, and B,, any five of which can be specified independently. 2.6. Determination

of steady-state

equilibrium

The steady state of the system is reached when dc= ri =ti=hg=O, implying that (I= fi and F(k, 1)=c+y. Accordingly, the long-run equilibrium of the

(15a)

(t5b)

(15c)

(15d) which involves the five variables k, l,p,m, and parameters y, rY, rD, and r,. In addition, analogous of return on the financial assets satisfy r&l -rY)-p=fi=max[r,(l

-r&p,$l

b,, as well as the policy to (14f), the nominal rates

-rJ

+4(1 -~Jlq-~,Pl. The set of equations (15) forms incidence of section 5-l below.

the basis for our long-run

(154 discussion

of tax

3. Short-run effects of tax changes: Direct effects To consider the short-run (impact) effects of various tax changes, we focus on the short-run equilibrium conditions described by eqs. (13) and (14). We shall consider government financial policies which allow at least one of the financial variables m and/or b, to evolve gradually as required to finance the government deficit. This enables us to analyze the effects of the various changes taken individually. Within this class of financial policy, which is really quite broad, the direct impact effects of these individual tax changes are independent of the financial mix chosen by the government to finance the

S.J. Turnovsky, The incidence

oftaxes

175

deficit. Also, comparative changes or questions of ‘differential tax incidence’ can be derived as a combination of the effects we obtain. By contrast, if for example the government chooses to peg both m and b,, so that tijz=hg=O, the real deficit as defined by the right-hand side of (14~) must always be in balance. In this case, any change in one tax rate must be accompanied by a compensating change in some other tax rate or in government expenditure so as to ensure government budget balance is maintained, just as is required in the real models; see, for example, Grierson (1975). Since we have argued that the macroeconomic effects of any tax change are inevitably interrelated, we shall consider the effects of the tax changes on the following variables: wu, the after-tax real wage; x*, the after-tax real cost of capital; 1 the level of employment, which in the short run is equivalent to output; c, the rate of consumption; It, the rate of capital accumulation, rg, the nominal rate of interest on government bonds; p, the rate of inflation; and 0 the net after-tax rate of return to investors (consumers). As noted previously, the short-run solutions for these variables can be expressed in terms of the dynamically evolving variables, together with the relevant tax parameters. Thus, for example, we may solve eqs. (13a) and (13b) to yield

I= Kz,,g;m,a,4,

(16)

c = c(z,,9; m,ff,k),

and likewise for the other variables. Provided that the dynamics described by the system (14aH14c) is stable (in the conventional sense of all eigenvalues having negative real parts), at any point of time the state variables m,a, and k can be taken as predetermined, in which case the full impact effects of the various tax changes on the system are described by the partial derivatives ailaT,, etc. However, it is familiar that systems embodying perfect foresight (rational expectations) are typically associated with some unstable roots, leading to saddle-point behavior. l2 Unfortunately, a complete stability analysis of the system (14) under general assumptions concerning government financial policy, turns out to be intractable. Nevertheless, the existence of unstable roots can be established for simplified, but plausible, cases. For example, Brock and Turnovsky (1981) show that in the special case where (i) labor is fully employed; (ii) the monetary authorities maintain a steady rate of monetary growth; and (iii) the real stock of government bonds is maintained fixed, with the deficit being financed by an endogenous lump sum tax, then the dynamics of the system is governed by a third-order system having two unstable and one stable root. A similar root structure can be obtained if the “The literature Brock (1974).

documenting

this is vast. One reference

pertinent

to the present

analysis

is

monetary authorities adopt a policy of maintaining the real stock of money. These examples, together with the existing literature, suggest that the existence of unstable roots should be taken as being the more representative behavior.i3 The essential point about the existence of unstable roots is that in order for the transversality conditions underlying the optimizations in the model to be met, certain state variables must undergo endoyenously determined jumps whenever exogenous changes, such as tax rate changes, are imposed on the system. The effect of these jumps is to ensure that the system is always on some stable 10cus.‘~ The number of variables required to undergo such jumps depends upon the number of unstable roots in the system, while their identity should be apparent from the economic context. But whenever some exogenous change leading to such jumps occurs, the corresponding state variable cannot be treated as predetermined. In the present analysis, where two unstable roots seems likely, it is reasonable to assume that the jumps are undertaken by m and a; k, being the stock of physical capital, is constrained to evolve continuously. Thus, allowing for these jumps, the complete impact effect of a change in zY at time 0 say on employment is given by

dI(0)

p=aZ+-p d%

az

ai am(O)

y am aZ,

al au(o) +aCr ___ aT, .

(17)

This consists of the direct effect, al/&,, together with the indirect effects, measured by the remaining terms of (17) which operate through the endogenous jumps in m(0) and ~(0) resulting from the tax change. Analogous expressions apply with respect to the other variables and the other tax changes. The direct effects are obtained by conventional comparative static analysis and are discussed in the remainder of this section. The indirect effects involve an explicit consideration of the dynamics of the system. These turn out to be much more complex and indeterminate and some analysis of them is given in section 4. The qualitative responses of the direct impact effects of the various tax changes are summarized in table 1. Some of these effects are invariant with respect to the firm’s financial structure, while others depend upon whether the corporate financial structure involves all bond or equity financing. We structure our comments accordingly. 13At the same time we should note that perfect foresight can be consistent with stability under appropriate assumptions regarding the financing of the government deficit; see Turnovsky and Nguyen (1980) for an example. “‘See, for example, Brock (1974).

177

S.J. Turnovsky, The incidence of taxes Table Direct short-run

A. Effects independent of 1. Employment (r) 1 2. 3. 4. 5. 6.

1

effects of alternative

corporatefinancial structure
Output (v) Consumption (c) Rate of capital accumulation (L) After-tax real wages (w,) After-tax cost of capital (x*) Interest rate on government bonds (r+,)

B. Effects under all bond financing (7) After-tax real rate of return to investors (0) (8) Rate of inflation (p) C. Effects under all equity financing (7) After-tax real rate of return to investors (0) (8) Rate of inflation (p) Notes (i) $=B+p-if-s,). (ii) (?) denotes probable

3.1.

tax changes.

- sgn(U,,) o

0

0

0 0 0
0 0 0 0 0

by firms O(?)

sgn(p) - sgn(p)

O(?)

o

by firms -sgn* sgn *

sign.

Effects invariant with respect to corporate financial structure

Consider first the marginal utility conditions (13a) and (13b). Taking m, c(, and k to be predetermined, these two equations jointly determine c and 1 as functions of the personal income tax rate ~~ together with 9; these variables are therefore independent of all other tax rates. Moreover, since they are determined independently of the cost of capital, the determination of c and 1 remains the same irrespective of the financial structure chosen by firms. (i) Change in z,,: Differentiating (13a) and (13b) with respect to z,,, we may readily show from the concavity of the utility function that al/&, ~0. From this result a number of other effects immediately follow. First, with k fixed in the short run, output falls. Also, the capital-labor ratio rises, thereby causing the marginal physical product of capital to fall. As a consequence of this, the before-tax real wage, defined by

w = F,(k 1)= f (k/l)-(k/W ‘WL rises, while the after-tax real wage w, =(1 - z,)w falls. The after-tax cost of capital, X* = (1 - z,) f ‘(k/l), also falls. Thus, we deduce that an increase in the personal income tax rate leads to falls in both the after-tax real wage rate and in the after-tax cost of capital.

S.J. Turnovsky, The incidence

17x

Using the index suggested by Feldstein the burden of the tax, we can show

of tuxe.7

(1974a) to consider

the incidence

1dw, UC, U,, - Cl 1dw, + kdx* = U,, U,, - Ug + (1 - rp)f”Ucc Ctk2/13

of

(18)

which, using the fact that

al

4,

aw=u,, u,,- u,:>o, can be written

as

dwa

1

dw, + kdx* = I - (1 - qjyk2p)

(19)

a yaw.

This result can be shown to be equivalent to Feldstein’s corresponding result (his eq. (16)) in the case where capital is assumed to be fixed in the short run (as we assume here). r5 Thus it is seen that in general the direct short-run burden of an increase in the personal income tax rate is shared between labor and capital, with the proportion of the tax borne by labor varying inversely with labor’s responsiveness to the real wage. In the optimizing framework we have here, this depends upon the concavity of the utility function in consumption and labor. From (13a) the effect of an increase consumption is given by

in the personal

income

tax rate

on

ac -u,, a1 u,, c72,’

-_=

az,

If one assumes, reasonably, that the marginal utility of consumption increases with leisure time and therefore varies inversely with labor (U,,
of Feldstein’s

eq. (16), noting

the definitions

S.J. Turnovsky,

The incidence of taxes

overall short-run effects of the tax change. Given o”l/&, < 0 and dc/&,> follows from the product market equilibrium condition that

179

0, it

That is, an increase in the personal income tax rate will cause the instantaneous rate of capital accumulation to fall. If, in addition to U,, ~0, we argue that U,,O, which also seem intuitively reasonable, it follows from (13~) that the changes in c and 1 the resulting from the tax increase will cause U, to rise, thereby increasing after-tax nominal rate of return (6)+p). We may then deduce from (13f) that the rate of interest on government bonds rg will rise by even more. (ii) Change in z,,: Since c and 1 are independent of rP, it follows that an increase in rP will leave both employment and output unchanged in the short run. The capital-labor ratio therefore remains unchanged so that the marginal physical product of capital remains fixed. As a further consequence, both the before-tax and after-tax real wage also remain fixed, while the aftertax real cost of capital falls directly by an amount f”dr,. Because capital is fixed in the short run, while labor is unresponsive to the particular tax, the situation is in effect the same as in the traditional static fixed factor model, with the tax being borne fully by the taxed factor. Note further that because c and I remain fixed instantaneously, the rate of capital accumulation also remains fixed, as does the rate of interest on government bonds. (iii) Change in r,: In the case of all-equity financing, when a capital gains tax becomes relevant, essentially the same general conclusions apply. Shortrun consumption, employment, output, capital accumulation, and nominal interest rate on government bonds remain fixed. With the capital-labor ratio fixed, the marginal physical product of labor remains fixed. The before-tax and after-tax real wage rates remain unchanged, as does now the after-tax real cost of capital as well. The short-run effect on the remaining variables, 19,the real rate of return to investors (consumers), and p, the rate of inflation, depend upon the financial structure of firms and the two cases of bond-financing and equity-financing need to be considered separately. 3.2. All bond financing

by firms

In this case the relevant

cost of capital

is given by (8’), which we may write

as

x*,u -q(H+P)_p l-z,

(204

S.J. Turnovsky. The incidence of taxes

180

and combine

with eq. (13d) to yield

.o > f”

;

=f!?&”

(13d’)

l-r,

(i) Change in T+,:Differentiating

ap_p(l-r&

F-

(0+&l

A

ae

-(l

l2 Urn- Urd U,,) 7

Y -zP)

(8+p)A

A +

wclUJ”k

( (1+

-&L

-= dt,

(13a), (13b), (13~) and (13d’) we can show

(214

U,,j”“k

+wa

(1 - zyy

l2

w(z,- z,.)cud urn, - urn, UC,1 (1- ~,)U - zy)

!

(21’4

’

where A E U,, U,, - U,"l+ Ucc(x(1 - r,,)f”’ k2/13 > 0. If we impose the sign restrictions introduced alone on the partial derivatives of U, then ap/&,>O. And if in addition we assume rP
u,(m)la = 0 + P, and with both m and a predetermined in the short run, it follows that aH/&, + ap/azy =O, i.e. the responses of p and tl are exactly offsetting. The fall in 1 resulting from the increase in rY leads to a fall in x*, as noted above. However, looking at (20a) we see that with 8(0+p)/&,=O, the increase in r,, causes the first component to rise. Hence, in order for the real cost of capital “Note requiring

that

in signing

the nominal

these effects we are

interest rate on government

assuming

(O+p)20,

which

bonds to be non-negative.

is equivalent

IO

181

S.J. Turnovsky, The incidence cf taxes

to undergo the necessary fall, the rate of inflation must increase, causing the net rate of return to bond holders to fall. (ii) Change in rP: The effects of an increase in the corporate profit tax rate zP on p and 8 are much simpler, being given by the expressions

ap

-P

(224

K=-’

l-r,

ad

P

VW

q-1-r,'

Assuming that the rate of inflation is positive, an increase in zP will be deflationary in the short-run leading to a higher real rate of return for bond holders. These qualitative effects are therefore exactly opposite to those just obtained for an increase in the personal income tax rate. The reason is that since c and I are independent of zP, it follows from (13~) that d(d+p)/az,=O. And since the marginal physical product of capital has been shown to be independent of zP, it follows from (13d’) that any increase in zP must be offset by a reduction in p in order for the appropriate beforetax cost of capital to remain constant. 3.3. All equity financing byfirms The relevant

cost of capital

is now described

x*=~+P++Tc) l-r, and which combined

(1

by (8”) which we rewrite as

(2Ob)

-p2

with (13d) gives the marginal

product

condition

0 =e+p;J4p.

_Tp)f’

;

(13d”)

E

(i) Change in r;

ap

This yields the following 1

-=z %

TA

-_w(u,, ( l-z,

effects on p and 0:

Lr~~-LI,,U,,)+WS(l-T~Uccf”k l-r,

w5,(Uc,

> u,,_

UmLU,,)_wU(l

(23a)

-r$U.cf”kJ. ,

G-1

S.J. Turnoosky, The incidence of taxes

182

These expressions are generally similar to those obtained under bond financing. Under the conditions noted above, an increase in 5,. will be inflationary, reducing the real rate of return to investors, who in this case are now stock holders. Furthermore, the intuition for these results is as in the previous case and need not be repeated. (ii) Change in zP: These are simply

g= f’>O, P

a0

-= aTP

(244

-f’
GW

That is, an increase in the corporate profit tax rate will be unambiguously inflationary, reducing the real rate of return to investors. These effects, which are now qualitatively the same as those for a personal income tax, are opposite to those obtained for an increase in rp under bond financing. The reason for the present effects is straightforward. As shown above, an increase in rp leads to a fall in the cost of capital x *. With c and 1 being independent of rp, (13~) implies that d(8+p)/az, =O, so that from (20b) the first component of the real cost of capital remains unchanged. Thus, in order for the real cost of capital to undergo the required fall, the rate of inflation must rise. (iii) Change in z,: Finally, the effects of an increase in the capital gains tax are given by

ap

e+p-ifi -zy)

ig=

(l-r,)2

ae

-(O+p-Q--J)

z=

(1-Q

’

.

(2W

The quantities c,l, and x* have all been shown to be independent of r,. Thus, any change in the first component of x* in (20b) arising from the increase in r, must be offset by an appropriate change in the inflation rate. This response is seen to be ambiguous in sign. Using the optimality conditions for consumers derived by Brock and Turnovsky (1981), the quantity 0 + p - T(1 -rJ appearing in (25a) and (25b) can be shown to be of the same sign as the nominal capital gains (4/q +p) being earned on holdings of equities. This enables us to interpret these effects as saying that an increase in the rate of capital gains tax will be inflationary or deflationary in the short run, depending upon whether the rate of nominal capital gains being earned on equities is positive or negative.

S.J. i%rnovsky, The incidence cf taxes

183

Comparing the results we have been discussing in subsections 3.2 and 3.3, we see the importance of allowing for the corporate financial structure in assessing the direct impact effects of tax changes on the short-run rate of inflation and rate of return. In the case of the personal income tax rate, the differences involve only the quantitative magnitudes, whereas in the case of the corporate profit tax the differences are actually qualitative. Moreover, the effects we have been discussing relate to infinitesimally small tax changes, so that the existing corporate financial structure remains in existence both before and after the tax change. But sufficiently large changes in tax rates will induce changes in the optimal corporate financial structure, making some of the changes more complex. This can be illustrated most clearly with z,,. Suppose initially zP < z,, so that (7b) is negative. In this case firms will employ all bond financing, so that an increase in z, will be deflationary. As z, continues to increase, eventually (7b) will become positive and firms will be induced to switch to all equity financing. At this point further increases in zP will begin to be inflationary.

4. Short-run effects of tax changes: Indirect effects It should be stressed that the effects discussed in section 3, being measured by partial derivatives, are only the direct effects of the various tax changes. These will also be the complete short-run effects if the system is inherently stable, although we have argued that this is unlikely. Otherwise, to obtain the complete short-run effects we must consider the remaining terms in expressions such as (17), which operate through the induced initial jumps in certain state variables, required to eliminate unstable roots. To do this requires more detailed information on the dynamic properties of the system and this in turn depends upon the government financial policy, as well as the corporate financial structure. Space limitations preclude a complete analysis of this aspect, which in any event turns out to be very cumbersome. In this section, therefore, we simply outline a procedure for determining these induced short-run responses for the most plausible root structure. We assume that the monetary authorities adopt the familiar and simple policy of pegging the rate of growth of the nominal money supply in accordance with the rule &l/M = ,u.

The dynamics of m,a, and k, which determine the short-run system, are given by (14a) and (14b) together with ti=(p-p)m

(26) state

of the

(26’)

184

S.J.

Turnocsky,

and the set of short-run differential equations about the form17 -

The

equilibrium the steady

-au,

au, mp,

- mp,

F,l,-c,

F,l,-c,

incidencr

of/axes

conditions (13). Linearizing these state, we may write the dynamics in

-au, - mpk

(27)

F,l,-c,+F,

where 0,~ CM/&, etc. can be derived by differentiating (13), and 6, rii, and 6 denote steady state values and are determined as in subsection 2.6. It is evident that the corporate financial structure will influence the dynamics of the system through the partial derivatives Bi and pi (i = a, m, k). Let us assume, on the basis of the special case analyzed by Brock and Turnovsky, that the dynamic system (27) has two unstable and one stable root. Once the unstable roots are eliminated by invoking the transversality conditions, the solutions for k,m, and c( must be of the form k(t) = (k, - E)eA1’ +

15,

(284

m(t) = (m(0) - ri?)e”’ + ri?,

(28b)

cc(t)= (cc(O) - fl)e”l’ + ii,

(28c)

where i, ~0 is the only stable eigenvalue; k, is the exogenously given initial stock of capital; and m(0) and cc(O)are the endogenous initial values of m and c( which need to be determined. Differentiating (28) with respect to t and substituting into (27) yields -d&-A,

! -

mp,

FJ, - c,

-

cru,

-mp,-2, FJ, - cm

-&I, - mpk FJ, - ck + F, - i,

(29)

confirming that A1 is an eigenvalue of (27), with (a-& m-fi, k-l?) being the corresponding eigenvector. Eq. (29) which holds for all t, involves three linearly dependent relationships between the three elements of the “Note that since 6, does not interact with either the short-run equilibrium or the dynamic equations (14a), (14b), and (26’) we ignore the dynamics of b,, allowing it to be determined residually by the government budget constraint. As noted by Brock and Turnovsky, this does raise a technical point in that in order for the solution for h,(t) to be consistent with the consumer’s transversality condition for bonds it is generally necessary to assume that the monetary authorities pick on appropriate initial stock of bonds by undertaking an initial open market exchange of money for bonds.

S.J. Turnovsky, The incidence of taxes

eigenvector. Thus, from the first two equations of (29) at time t=O, express the solutions for a(O) and m(0) in the following form:

185

we can

a(O)=k+$,,

(304

m(0)= Kl + ($2,

(3Ob)

where

These equations express the initial values of CIand m in terms of their steadystate values, plus rather complex multiples of (k, - k), which depend upon ;1, and various short-run partial derivatives. l8 We may now combine (30) with the results of the previous section to obtain the complete effects of a particular tax change. Thus, for example, the effect of an increase in rY on employment, (17) is now given by

(31)

The effect of 7,, on the steady-state money stock can be derived from the analysis of section 5. However, under the assumptions of that section, namely additivity of the consumer utility function, I, =O, in which case (31) reduces to

(31’) Under these same conditions it can be readily established that l,>O and dC/dz,>O so that one component of the indirect effect offsets the direct effect. The increase in z,,, by reducing steady-state output and consumption, increases the marginal utility of consumption. This in turn causes an increase in labor supply and employment. In addition there is the term l,&#~,/dt,, which in principle can be calculated from (30a), although in practice this “The fact that the inital values m(0) and a(O) depend upon their respective steady-state values and the initial conditions of the sluggishly evolving variable, k, is a familiar feature of rational expectations models. See, for example, Gray and Turnovsky (1979) for a similar type of solution in another context (exchange rates).

turns out to be intractable given by

to do. The analogous

response

in consumption

is

(32) where c,
5. Steady-state

effects of tax changes under the monetary growth rule

The steady-state effects of the alternative tax changes can be obtained by analyzing eqs. (15). As we have noted previously, these involve the five variables k, l,p, m, and b,, in addition to the tax parameters rY, r,,, z,, and government expenditure y. It is important to observe that because m and/or b, are endogenously determined in the long run, the steady state of the system will depend upon the financial policy adopted by the government. Thus, before the long-run effects of tax changes can be analyzed it is necessary to specify some debt management policy for the government, and in this section we shall retain the monetary growth rule (26). As we will illustrate in section 7 below, some of the impacts we shall consider are sensitive to this specification of monetary policy and need not obtain under alternative policy specifications. From eq. (26’) the steady-state rate of inflation p= p. Also, in the steady state the after-tax real rate of return to investors, 8, is driven to the social rate of discount, /?. Thus, the two variables p and t3 whose responses to the various tax changes are the most complex in the short run, turn out to be very simply determined in the steady state. They are both in effect pegged exogenously and are therefore independent of all tax parametersi The ‘“The result that the steady-state after-tax real rate of return 0 equals the social rate of return /I; essentially the convergence of the optimal growth path to the moditied golden rule familiar from the optimal growth literature; see, for example, Burmeister and Dobeil (1970). In the absence of taxes and with zero population growth this requires the marginal product of capital to converge to the rate of social discount. Note that because of the existence of taxes in our model, the after-tax rate of return to investors deviates from the after-tax cost of capital.

S.J. Turnovsk~,The incidence of taxes

steady-state responses of the remaining variables do corporate finance structure, so the two cases need separately. 5.1.

All bond financing

187

depend to be

upon the considered

by firms

In this case (7b) is negative and the steady state reduces to the following four equations determining the four variables k, 1, m, and b,:

(33b)

(*

_zp)f’

5 J(l -4+kw_x*

0 1

g + Bb, - r,lf(k/l)

(33c)

2

l-r,

- pm + [B - (1 - z,)f’(k/l)]

k = 0.

(334

The long-run equilibrium in the case where firms engage in all bond financing can be obtained in the following recursive manner. First, given the parameters /?, p, r,,, and z,,, (33~) yields the marginal physical product of capital. With the linear homogeneity of the production function, this establishes the capital-labor ratio, which in turn determines the real wage. Having determined k/l, the two marginal rate of substitution conditions, (33a) and (33b), together determine the employment of labor and the real stock of money balances. With kfl and 1 now fixed, the real stock of capital is known, while the level of output y follows from the production function. The government budget then determines the real stock of government bonds required to balance the budget. The qualitative effects of the changes in the tax rates are summarized in table 2, part A. Those effects which stem directly via the capital-labor ratio from the marginal product condition (33d) (the effects on w,, x* and k/l) hold quite generally. The responses of k, 1, and y, however, are based on the simplifying assumption that the consumer’s utility function is additive in the three arguments c, 1, and rn.” (i) Change in z,: From (33~) it is seen that an increase in the personal income tax rate will raise the steady-state net cost of capital x*, thus reversing the initial direct effect and causing the capital-labor ratio to fall. As “The same general propositions can be established additively separable in m and U,, < 0.

under

the weaker

restriction

that

U is

S..J. Turnorsky, The rrzcidence of tuxes

188

Table 2 Steady-state

A. Effects under I, After-tax real 2. After-tax cost 3. Capital-labor 4. Employment 5. output (y) 6. Capital stock B. I. 2. 3. 4. 5. 6.

Effects under After-tax real After-tax cost Capital&labor Employment output (y) Capital stock

effects of alternative

all bond financing by firms wage (w,) of capital (x*) ratio (k/l) (r) (k) all equityfinancing wage (M.,) of capital (x*) ratio (k/L) (r) (k)

tax changes:

Fixed monetary

o
sgn(p)
o

+

growth.

&)I

by jirms

Notes (i) Sufficient condition for dl/dT, < 0 is rd + c/y > 0. (ii) Effects on l,y, and k are all obtained for the case where U is additive

-sgnti sgn li/ -sgn$ ~ sgn[$(ur -sgni -w*

+ (,iy)]

in (.I, and m.

a consequence, the before-tax real wage rate falls, with the after-tax real wage rate falling even further. Thus, whereas in the short run an increase in rY is shared between capital and wages, in the long run it is more than fully borne by wages. This result, like the corresponding short-run effect, is identical to that obtained by Feldstein, and the following simple explanation can be ratio given. The impact effect of an increase in rp is to raise the capital-labor by reducing the employment of labor. At the same time, the higher rate of taxation causes the rate of capital accumulation to fall. Over time this leads to a reduction in the capitalHabor ratio, thereby pushing up the marginal physical product of capital and reducing the marginal physical product of labor. To consider the effects of the increase in 7,. on y and 1, it is convenient to take differentials of (33a), together with the production function. Invoking the simplifying assumption that U is additive, we have

U,, dl+ w, U,, dy = - U, dw,,

dy-fdl=ldj; where dw, ~0 and dj’
S.J. Turnovsky, The incidence of taxes

189

induced by the tax increase. The fact that output must fall can be established by contradiction. If, instead, dy>O, it follows from the marginal utility condition (34a) that 1 must necessarily fall. However, it then follows from the production function (34b) that a fall in both f and 1 is inconsistent with the assumed rise in y. Hence output must fall. The effect on employment is, however, indeterminate. Because of the lower capital-labor ratio, the reduced output is consistent with either increased or decreased employment. Indeed, the response will depend upon, among other things, the nature of the production function. By calculating dl/dr, formally from (33), it seems likely that employment will fall, a sufficient but not necessary condition for this to be so being cTr+c/y>O,

(35)

where c = elasticity of substitution, r = Urcc/Uc = elasticity of the marginal utility of consumption, and c/y=average propensity to consume. The lefthand side of (35) would appear to be highly indeterminate in sign. The likelihood of (35) being met increases with the average propensity to consume, but decreases with the magnitude of I or the elasticity of substitution. However, irrespective of the response of 1, the reduction in the capital-labor ratio can be shown to be sufficient to ensure that the total stock of capital must fall. (ii) Change in zp: Turning now to an increase in the corporate profit tax rate, it is seen from (3%) that an increase in zP will lower the after-tax real cost of capital, The effect on the capital-labor ratio depends upon the sign of the monetary growth rate, ~1.If this is positive, the marginal physical product of capital will rise, increasing both the before-tax and after-tax real wage rates. In this case the increase in rP will be more than fully reflected in a lower after-tax cost of capital, with wage earners actually benefiting from the increase. Only if the monetary stock is contracting (p ~0) do wage earners share in the burden. The effect of a change in zg on y can be derived from (34a) (34b). In the present case, since the responses of both w, and f depend upon 11, the same applies to y. Assuming p> 0, so that output increases, it can be shown by direct analysis of the equilibrium condition (33) that the response of employment depends upon (~r+c/y). Also, the increase in the capital-labor ratio which occurs when pu>O is sufficient to ensure that the total stock of capital always increases. Comparing table 2, part A with table 1, part A, it is seen that the long-run effects of an increase in the profit tax rate may be very different from what they are in the short run. Indeed, provided that it is accompanied by a policy of steady monetary expansion, it may turn out to be quite stimulating, through the reduction in the cost of capital it generates.

S.J. Turnovsky, The incidence of taxes

190

5.2.

All equity ,$nancing

by firms

For all equity financing to be optimal, (7b) must be positive, in which case the steady-state values of k, l,m, and b, are determined by eqs. (33a), (33b), and (33d), with (33~) being replaced by (1

_T

)f’

P

k J+w+~~,-%)

0 1

(33c’)

l-z,

The steady state is attained in much the same way as in subsection 5.1, although the relevant cost of capital determining the capital-labor ratio is now given by (33~‘). The qualitative effects of the various tax changes are summarized in table 2, part B, with the responses of k, 1, and y again being based on an additive utility function. (i) Chunge in zY: The effects of an increase in zY are virtually identical to those obtained with all bond financing. Indeed, the economic reasoning explaining them is parallel and need not be repeated. (ii) Change in z,: An increase in rP leaves the after-tax cost of capital unchanged. As a consequence, it follows from (33~‘) that the marginal product of capital must rise, leading to a fall in the capital-labor ratio. This in turn means that the after-tax real wage must also fall. This fall in w,, together with that in the output-labor ratio induced by the fall in the capital-labor ratio, can be shown, using (34), to imply a fall in output. The effect on employment now varies inversely with (ar +c/y), while the total stock of capital declines unambiguously. Thus, as in the case of all bond financing, the long-run effects of an increase in zP are very different from their short-run effects. However, in contrast to the personal income tax, the long-run effects of a corporate profit tax depend critically upon the financial structure employed by the firms. Whereas an increase in rP is generally stimulating if all bond financing is employed, it is generally contractionary with all equity financing. (iii) Change in z,: Finally, the effects of an increase in TV on the after-tax capital costs depends upon $, which we have identified before as reflecting the rate of nominal capital gains being earned on equities. If these are positive, then net capital costs increase, reducing the capital-labor ratio, and hence the after-tax real wage. As before, the fall in w, andf which occur lead to a fall in output, while the effect on employment depends upon (ar+c/y). If (35) holds then 1 will fall, while whatever the response in 1, the total stock of capital will decline. The effects on warx*, (k/l), y, and k are all reversed if $
analyses

of sections

3-5

have

dealt

with

changes

in individual

tax

S.J. Turnovsky, The incidence of taxes

191

rates, As a result of these tax changes tax revenue is altered, and with government expenditure held constant the resulting (change in) government deficit is financed by an appropriate adjustment in the stock of government debt. The framework we have developed is most convenient for considering alternative tax policies and in this section two such policies are briefly summarized. We shall retain our assumption that the monetary authorities maintain a policy of steady monetary growth and restrict our remarks to the long run. 6.1. Equal tax yield The tax changes we have been discussing will in general contribute differently to tax revenue. We consider a situation in which one tax is substituted for another to maintain a given tax yield. Taking the case of all equity financing, steady-state tax receipts are given by

= q-’

k + zY[y -f’

k + Zk]+ zC[B+ p - ql - Q]k/( 1 - rJ.

(36)

This consists of tax on corporate profit (f’ k) taxed at the rate rP; tax on wage income (y-f’k) and dividend income (2) both taxed at tv; and tax on capital gains ([/?+p-
+ [y-f

‘k + ik] dr, + (p + P- g kdr,.

(37)

One interesting result that can be established is that starting from zero initial tax rates the effects of all three possible combinations of pairwise tax substitutions (2, for r,,,r,, for r,,r, for rJ is borne fully by capital costs; the after-tax wage rate remains unchanged. This is also true in the case where firms employ all bond financing and an income tax is substituted for a profit tax. However, it is important to note that these results have been obtained only in a very special case and are unlikely to carry over to the more general case where tax rates are initially non-zero. 6.2. Incrementally balanced budget

Another variant one can consider is where the tax receipts generated by the tax change are spent so that dg=dT It is interesting to observe that the effects of such an incrementally balanced budget on the cost of capital and

S.J. Turnwsky.

192

The incidence

of taxes

after-tax real wage rate will be identical to those discussed in section 4. The reason for this is simply that the critical equation determining these effects is the marginal physical product relationship, (33~) and (33c’), which is independent of g. The effects on output, employment, and total capital stock will also be identical to those obtained previously if and only if private and public goods are perceived by the private sector as being perfect substitutes. In this case c and g enter the utility function additively as c +g, which in steady-state equilibrium is simply If(k/l). Thus, in the steady state the marginal utilities are independent of changes in g. The only effect of a change dg is to influence the stock of real bonds required to balance the government budget, but this has no effects on the variables we have been considering. However, these latter effects cease to be identical if c and g are imperfect substitutes, since g will then enter the utility function directly, implying that the change in g resulting from the tax revenue will generate changes in I, y and k through its effects on the relevant marginal utilities.

7. An alternative government monetary policy We have remarked previously how the incidence of taxes will be dependent upon government financial policy. To illustrate this we shall consider an alternative policy in which the monetary authorities choose to peg the real stock of money m= riizl Focusing on the long run, the steady-state equilibrium conditions (15) now determine the four variables k, 1, p, and b,. In particular, the rate of inflation p becomes an endogenous variable, influencing the cost of capital in different ways depending upon the tax structure. The capital-labor ratio and the rate of inflation are therefore jointly determined. We shall not discuss this policy in any detail except to illustrate one example where a difference does arise from the previous policy of a fixed monetary growth rule. Consider an increase in zP on the assumption that firms employ all equity financing. With the tixed monetary growth policy of section 5, this was shown to leave the cost of capital unaffected; the capitallabor ratio falls, with real wages falling as well. With the present policy, the effect on the cost of capital is described by dx* q==-

z,

dp

(38)

1 - rC dz,’

which depends upon the effects of the tax change on the steady-state rate of inflation. Assuming that the utility function is additive it can be established that an increase in zP will reduce both the capital-labor ratio and the rate of inflation, so that both the cost of capital and after-tax real wages now fall. “Even though the direct instrument of monetary control adjustment of this variable the real stock can be pegged.

is the nominal

stock M, by judicious

S.J. Turnovsky, The incidence OJ taxes

193

8. Conclusions This paper has analyzed the short-run and long-run incidence of alternative taxes in a dynamic macro model. The framework we have employed possesses three characteristics: (a) all behavioral relationships are derived from optimizing behavior; (b) the model contains a reasonably complete corporate sector in which firms have the choice of financial structure; and (c) in contrast to existing models of tax incidence, the model contains a more fully developed government sector, allowing for a range of government financial policies, although only one has been considered in detail. The following general conclusions can be drawn. (i) The incidence of taxes may differ substantially in the long run from what they are in the short run, thus confirming the need to allow for the accumulation of capital and other dynamic processes. This is true for all the tax changes we have considered and is irrespective of whether firms finance their investments through issuing bonds or issuing equities. (ii) The short-run effects of tax changes can be broken down into direct effects and indirect effects, The former are just the usual short-run comparative static effects, given the predetermined state variables. The latter occur through the induced jumps in the state variables which occur whenever the system is subject to an exogenous disturbance and are required to eliminate the effects of any unstable roots in the dynamic system. (iii) The direct short-run effects of an increase in the personal income tax rate are qualitatively similar under either bond financing or equity financing by corporations. The same tends to be true of the corporate income tax, although there are some differences. Provided that there is an ongoing inflation, the initial effect of an increase in rp tends to be deflationary under bond financing whereas with equity financing it is unambiguously inflationary. In the former case the rate of return to investors rises, while in the latter case it falls. (iv) In the long run, the effects of an increase in rY are again generally similar for both forms of corporate financial structure. However, in this case the effects of an increase in zp tend to diverge markedly. If firms finance through bonds and if the monetary authorities adopt a policy of steady monetary growth, an increase in rg tends to be expansionary. If firms finance through equities, the effects are generally contractionary, irrespective of the monetary growth rate. (v) Whereas in the short run the direct incidence of taxes are invariant with respect to government financial policy over a wide range of policies, the indirect effects are always dependent upon the policy specification. In the long run the chosen government policy can also be important in determining the effects of tax increases. These results have important policy implications. They imply that in order tax change one needs to know the to assess the effects of a particular

194

S.J. Turnousky,

The incidence q( taxes

financial structure adopted by both corporations and the government. This is particularly true for the corporate profit tax rate. Grierson (1975) has drawn attention to the opposition by labor to the reduction in corporate taxes. In our model this opposition may well be justified if firms employ all bond financing and the monetary policy is one of steady growth. In that case a reduction in the corporate profit rate will in the long run lower the capitallabor ratio, resulting in a reduced after-tax real wage.

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