Dissertation proposal on financial systems and tax accounting- Havard Style

Tax Reform and Adjustment Costs: The Impact on Investment and Market Value

Author(s): Alan J. Auerbach

Source: International Economic Review , Nov., 1989, Vol. 30, No. 4 (Nov., 1989), pp. 939- 962

Published by: Wiley for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research, Osaka University

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INTERNATIONAL ECONOMIC REVIEW Vol. 30, No. 4, November 1989

TAX REFORM AND ADJUSTMENT COSTS: THE IMPACT ON

INVESTMENT AND MARKET VALUE*

BY ALAN J. AUERBACH 1

This paper provides an effective tax rate measure that is valid in the

presence of adjustment costs and anticipated tax changes and a measure of the

impact of tax changes on market value that may be decomposed into the

effects on discounted pure profits and normal returns to capital. Changes in the

value of capital may, in turn, be decomposed further into changes in the

marginal value of new capital and changes in the relative value of new and

existing capital. These new measures are used to evaluate tax changes similar

to those introduced by the recent U.S. tax reform.

1. INTRODUCTION

Changes in tax policy are thought to influence investment behavior, and

anticipated tax changes may do so as well. However, the impact of expected future

tax changes may push current investment in a different direction than would be

suggested by the long-run effects. For example, an anticipated increase in the

investment tax credit would be expected to decrease current investment, as firms

delay investment to take advantage of the credit. An important determinant of the

impact of current and future tax changes is the firm's technology. For example, if

it is very difficult for the firm to alter its investment plans, then anticipated tax

changes may have little impact on behavior. Thus, the structure of the taxation and

the structure of production interact in their effects on investment.

A second way in which tax changes and adjustment costs interact is through

changes in the value of the firm. Tax policies that encourage investment will also

increase the value of newly acquired capital for the firm facing adjustment costs.

The change in firm value that results depends not only on the magnitude of this

increase but also on the tax policy's relative treatment of old and new assets and

pure profits. Policies targeted at investment, such as investment tax credits, can

have quite different effects from tax rate reductions that apply to income from all

sources.

This paper derives analytical measures of the combined effects of tax changes

and adjustment costs on investment and market value. The measure of the impact

on market value allows one to distinguish among changes associated with pure

profits, after-tax returns to new capital, and the relative tax treatment of old and

new capital. The measure of the impact on investment is analogous to the "effective

* Manuscript received January 1988; revised July 1988.

l Financial support for this research was provided by the National Science Foundation. I am grateful

to Jim Hines, Jim Poterba, Larry Summers, two anonymous referees, and participants in seminars at

Harvard, Johns Hopkins, London School of Economics, Princeton and Toronto for useful comments on

previous drafts.

939

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940 ALAN J. AUERBACH

tax rate" found in various studies (e.g., Auerbach, 1983a; King and Fullerton,

1984). It is based on the same principle of estimating the impact of taxes on new

investment, but unlike earlier measures it is valid in the presence of adjustment

costs and anticipated tax changes. Using it, one can estimate how a complicated set

of current and future tax provisions affects current incentives. The ability to

measure current as well as long-run incentives is important. In the recent U.S. tax

reform debate, for example, attention was continually focused on the effective tax

rates that would eventually prevail under new law even though various phase-in

provisions were being considered that would have substantially affected investment

behavior in the short run.

Since the effects of tax changes depend on the nature of production, the measures

to be derived must be based on a specific model. The model chosen here, a

production function with convex adjustment costs, is commonly found in the "q"

theory literature and has been used to analyze the effects of tax reform in a number

of recent papers. This model has become popular in part because of its tractability,

but also because it embodies certain restrictions. These will be discussed below

where they are relevant.

2. THE MODEL

The behavior analyzed is that of a price-taking representative firm producing a

single output using one factor of production, capital, subject to a concave

production function F(-).2 The firm faces a corporate tax system represented by a

tax rate, an investment tax credit and a schedule of depreciation allowances. It

responds with perfect foresight to changes in these tax variables, subject to convex

costs of adjusting its capital stock. These costs impart an incentive for the

smoothing of investment.

For simplicity, we ignore personal taxes, corporate interest deductibility and

economic growth, and assume a uniform rate of inflation for output and capital

goods. Relaxing any of these restrictions would pose few analytical difficulties, but would add additional complexity without really altering the qualitative nature of the

paper's results.

There is no uncertainty in the model, and the firm's planning is done under

perfect foresight. Therefore, its objective is to maximize the present value of future

cash flows, discounted at the nominal, after-tax cost of capital, r, which is assumed constant:

(1) Vt = e-r(s-t)[psF(Ks) - psC(Is/Ks)Is - Ts] ds

2 The single-factor, decreasing-returns-to-scale production technology can, without any loss of generality, be viewed as a two-factor, constant-returns-to-scale technology with a fixed factor that earns

a competitive return equal to the firm's pure profits. Similar results would be obtained by assuming the

existence of a second variable factor, say labor, in fixed supply to the economy, since in equilibrium the

wage would have to adjust to provide the price-taking firm with zero profits. This is the approach taken

by Summers (1981), for example. The only difference is that in such a model there would be no pure

profits, for the "fixed factor" would be owned by workers rather than the firm.

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TAX REFORM, INVESTMENT AND VALUE 941

where

(2) Ts = rs[psF(Ks) - p,,C(I,,lK)I,,D(s, s - u) diu] - kpC(IsIKs)Is

is the tax bill at date S.3 Terms introduced in (1) are Ps, the price level, Is, gross investment, and KS. the capital stock, at date s. The investment unit cost function, CQ), has as its argument the rate of investment I/K. The tax variables at date s,

introduced in (2) are rS, the corporate tax rate, ks, the investment tax credit, and D(s, s - u), the depreciation allowance per dollar of date u capital expenditure.

Investment and capital are related by

(3) Is = Ks + ksg

where 8 is the (assumed to be geometric) rate of economic depreciation of capital.4 Because we will be linearizing the model around a long-run steady state, it may be

assumed without further loss of generality that the convex cost function CQ) is quadratic. A convenient normalization to adopt is that the derivative of the total

cost function C(I/K) * I with respect to I equals 1 in the steady state (k = 0). This derivative then has a direct interpretation as Tobin's q, greater than 1 when

investment exceeds its steady state value and less than 1 when it falls short. Given

(3), this normalization implies that:

(4) C(I/K) = 1 - ?8 + - OI/K.

Based on (4), the relative prices of capital goods is:

pd[C(I/K)I]IdI (S) q = = I + (KIK).

p

With (2), (3) and (4) expression (1) may be rewritten as:

(6) Vt = f e-r(s-t)PS (1 -)F(Ks)

( 2 2 + Ks) (5Ks + Ks)(1-ks -FS) ds + At,

where

3 The expression for taxes in (2) treats all capital costs pC(IIK)I as part of capital expenditures for tax

purposes. This is consistent with the U.S. tax treatment calling for the addition of indirect costs (such as

installation) to basis. In reality, some of the indirect costs associated with adding capital such as retraining

of labor, would normally not be capitalized but simply deducted as an expense. This is a minor issue in

the current context.

4 The geometric decay assumption makes the model simpler to analyze, but does rule out certain

interesting phenomena, such as "echo" effects, whereby investment booms lead to subsequent booms in

investment simply to replace scrapped capital. A more general specification of depreciation would be

unlikely to alter the qualitative nature of the results below, which do not depend in any fundamental way

on the smoothness imparted by geometric decay.

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942 ALAN J. AUERBACH

(7) At= e-r (s)-5 f P11C(II,/Ku)I1D(s s - u) du ds

is predetermined at date t. At is the present value of tax savings due to depreciation of investments made before date t. The term Fs in expression (6) represents the present value of tax savings per dollar of date s investment:

rx

(8) vs = er(u-s) rUD(u, u - s) du.

The Euler condition for the optimal capital stock path based on expression (6)

may be written (letting p = r - plp be the real interest rate):

(9) F'(Kt) + Xt = ct = qt P + 6 )+ ('Ir ( - k - Ft)l(I - Tt)

where:

(10) Xt= 2 4(It/Kt)2(1- kt- Ft)/( -t)

Expression (9) differs from the standard Hall-Jorgenson cost of capital formula-

tion in two respects. Because tax changes are contemplated, it accounts for

changes in the relative capital goods price, q(l - k - F), caused not only by

changes in q, but also by changes in k and F. In addition, the full marginal return

to capital includes Xt, a reduction in current adjustment costs per unit of investment. This latter effect would be absent if adjustment costs depended only on

the level of investment, rather than the ratio of investment to the capital stock. The ratio specification is typically used in the empirical literature, and is more

convenient for the analysis of market value changes (e.g., Hayashi 1982). We discuss this further below.

Expressions (5), (9) and (10) yield a familiar system of first-order, nonlinear

differential equations in the capital stock, K, and the relative capital goods price, q,

which may be written (suppressing subscripts) as:

(lla) k (q- )

(I l) -F (K)I k F 2 0 5+ 0 ) +q(p +)+ q k+ F This two equation system in K and q is the one we wish to consider but its

nonlinearity makes a general analytic solution unavailable.

There have been three approaches taken in the literature to deal with this

problem. One may examine the system graphically using phase diagrams, as in Abel

(1982). While very helpful in understanding how the model works and how K and

q will respond to various tax changes, there are many cases in which the direction

of movement may be indeterminate, depending on the relative magnitude of

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TAX REFORM, INVESTMENT AND VALUE 943

parameters, which makes an analytic solution desirable. Moreover, the phase

diagram approach is ill-suited to studying effects on the value of the firm (i..e, "average" q). A second approach, used in Summers (1981) and Auerbach and

Hines (1987), utilizes numerical simulations. While in principle capable of charac-

terizing the sensitivity of the behavior of investment and market value to changes

in parameter values, this approach is unlikely to uncover the exact nature of

analytical relationships. The final approach is to obtain an analytic solution by

considering the behavior of the system near a steady-state equilibrium, where the

local behavior of K and q can be approximated by the version of (1la) and ( lib) linearized around the steady state.

This last approach is common in the literature on dynamic models. It has been

used in a related analysis by Judd (1985). The present specification differs from

Judd's in several respects, however, including a richer characterization of the tax

system and adjustment costs in production. Perhaps the most important contribu-

tions here are the derivation of explicit analytical expressions that summarize the

effects of very complicated tax changes on the incentive to invest, the extension of

the analysis to study impacts on market value, and the integration of the effects on market value and investment.

Linearizing (1la) and ( lib) around the steady state, one obtains (using the facts that q = 1 and k = r = k = 0 in the steady state):

K* (12a) K- (q- 1)

(12b) )F(K Ik* ( )* (K- K*) - 5(q - 1) + (p + 8)(q - 1)

F'(K*) F'(K*) k +

1 k* F* (I - k* - F*)2 1- -

where the "*" superscript denotes the steady-state value of a variable. In the steady state, (9) becomes:

(9') F'(K*) = + - )(1 - k*- F*)/(I - T*)

= (p + 3)(1 - k* - F*)/ (I -

where

(13) 3 = 5 I - ?

Hence expression (12b) may be rewritten

-F"(K*) -* (12b') FK= *) (p + 3)(K - K*) + p(q- 1) + (p + 3)( *

- (P + (k +F) -(k* +* 1F + (l k-t)

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944 ALAN J. AUERBACH

The term 3 defined in (13) is the rate of economic depreciation of the capital

stock, in the presence of adjustment costs.5 Expressions (12a) and (12b') form a

first-order linear system in K and q, which can be solved directly using the Laplace

transform technique, as in Judd (1985). To relate this model to the previous investment literature, however, an alternative approach is more useful. One can

represent (12a) and (12b') as a second-order linear equation in K by substituting q

from (12a) and q from the derivative of (12a) into (12b'). Doing so yields

a . (p +) a(p +) 1 (14) Kt -PK- Kt= - K*I1 --at

aJ where:

-F"(K*) K* (15) a F'(K*)

and

(k* + F*) - (kt + Ft) t 1 kt + Pt (16) at= - +p * -

I1-k* -F* I1-r* p + 1, -k* -F*'

The term a equals the elasticity of F' with respect to K, - d ln F'/d ln K, evaluated

at K*. (It will normally be different for different values of K.) This term is important in the translation of capital cost changes into capital stock changes and vice versa.

The term at represents the proportional deviation in the user cost of capital at time t from its long-run value defined in (9') due directly to taxation. (The user cost,

defined in (9), also depends on q and 4, which are themselves functions of the firm's

investment behavior unless 0 = 0). The characteristic roots of equation (14) are:

4a(p+ 4a(p +) p- p2+ P+ p2+

(17) 2 2 2

5 The total cost to the firm of new capital goods is (1 - j6 + 120IIK)I = (1 - 1/208)8K in the steady state. The steady-state value of the firm's capital stock is constant. Thus, depreciation, which is

the reduction in capital value plus expenditure on new capital goods, is (1 - 1/208)aK = AK. Another way of viewing the same result is that an increase in capital expenditure today, holding future expenditure

constant, yields an asset that depreciates at rate 8 plus additional capital at each date in the future because

of the reduced unit price of capital induced by the current expenditure. The increase (in the steady state)

is 1/2k82 per unit of capital, compounded at each date, yielding a net rate of depreciation of 8 - 1/2 82.

As shown by Abel (1983), neutrality of the tax system in such a case would require netting these gains

against primary depreciation in computing depreciation allowances. That such a correction to the

measurement of economic depreciation is appropriate does not appear to be widely recognized in

discussions about measuring depreciation properly. Given typical estimated magnitudes of the propor-

tional adjustment cost parameter 0, the correction may be quite large. This would be especially true in the presence of growth, since the correction factor would then be 1I2>(n + 6)2, where n is the economy's

steady-state growth rate.

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TAX REFORM, INVESTMENT AND VALUE 945

Expression (9') implies that as long as the steady-state marginal product of capital

is positive, so is (p + 3). Then, since a and 0 are also positive, the model has one stable root (A 1 < 0) and one unstable root (A 2 > 0). This is the normal result in such

models. Given the initial value of K and the transversality condition ruling out the

explosion of q, there will be a unique saddlepath equilibrium for the system.

Incorporating the transversality condition into (14) yields a first-order equation in

K:

00A(-)a(p +) (18) Kt = AKt + jeA2(t) K* (I - - as) ds.

Expression (18) could, in turn, be solved for Kt using the initial condition with respect to the capital stock. However, it is more easily interpreted in its present

form.

Since A1A2 = [-a(p + 5)/0], (18) may be rewritten:

(19) Kt = (-A i)(K t-Kt),

where

(20) Kt = K

and

rx

(21) lt = A2 e-A2(5-t)as ds. t

Thus, the firm's investment behavior at time t may be described as a partial

adjustment process, at rate -A1, which closes the gap between the actual capital

stock, Kt, and the "desired" capital stock Kt. This desired capital stock differs from the long-run capital stock, K*, due to the existence of temporary tax

provisions during the period between date t and the steady state.

This partial adjustment approach is consistent with the traditional investment

literature (e.g., Hall and Jorgenson 1967), but goes further in characterizing the

determinants of the speed of adjustment, (-A 1), and being based on an optimal capital stock, K, that is consistent with the partial adjustment process. Since full

adjustment will not occur instantaneously, the target to which the firm adjusts

today depends on where it wishes to be in the future. The term f?t is a weighted average of the current and future tax effects, as, with weights summing to one and declining at rate A2. As 0 gets smaller, A2 increases (see (17)), making future tax effects less important in determining fl because of a reduced incentive to smooth

investment (as indicated by the coincident increase in -A1).

From expression (20), it follows that -(flt/a) is the proportional deviation of the desired stock capital stock from K* due to short-run tax factors. Given the

definition of a, it follows that fl represents the proportional increase in the short-run cost of capital per dollar due to tax changes. That is:

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946 ALAN J. AUERBACH

AF' I dF' K-K*

(22) F' = _ K _ (K-K*) =-a K*) fL

The current cost of capital effect, lt, combines future tax changes and adjust- ment costs in a particularly simple way. First one estimates the date s impact of tax

changes on the user cost of capital ignoring adjustment costs, as, for s > t, then weights these with the factors A 2e -A2(S-t) to account for the presence of adjustment

costs. The term lt differs from at in that the former includes cost of capital effects due to changes in the rate of investment that make q # 0. This difference would

vanish if 0 = 0, for then there would be no change in q due to investment. One may also express the effects of current and future tax policy on investment

in terms of an effective tax rate. Suppose we wished to know what tax rate, 0, if applied to true economic income without change over time, would yield the level of investment actually observed at the current date. This amounts to identifying the

steady-state tax rate on economic income that would make K the desired steady- state capital stock. Since expression (9') identifies the marginal product of the steady-state capital stock (and hence that capital stock itself) for any constant tax system, we may use it to solve implicitly for 0 by fixing the marginal product of

capital at F(K), and, consistent with the assumption of a true income tax, setting the investment tax credit to zero and the present value of depreciation allowances

to 3/p + 8 (see Auerbach 1983b). This yields a solution for 0 as a function of K, and an approximation of 0 near the steady state:

F'(K)-(p + 5) F' (23) F'(K)- = + * + Q(1 (F' -)

where 0* is the steady state effective tax rate.

With a constant tax system, Ql = 0 and 0 reduces to the standard effective tax rate

measure found in the literature (e.g., Auerbach, 1983a; King and Fullerton, 1984). The short run value will differ because of the anticipated tax changes accounted for by l.

These results apply in general for small changes in the tax system, and Ql and 0 are quite easily calculated. In addition, one may simplify the expressions for Ql in particular important cases, making possible the further analysis of the impact of anticipated tax changes in the next section.

3. THE IMPACT OF TAX REFORM

This section considers the impact, Ql, on the short-run cost of capital of anticipated temporary and permanent tax changes. It focuses on changes in the

corporate tax rate r and in the investment tax credit k, although other experiments, such as changes in the schedule of depreciation allowances, could also be examined.

A. Changes in the Investment Tax Credit. Suppose the investment tax credit

changes from k to k* at date T > t. Then, for s > T, as = 0 (see (16)). For s < T,

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TAX REFORM, INVESTMENT AND VALUE 947

k*-k Ak

(24) as=I-k* F* I k* F`

Through the effects of these terms on ft, a future increase in k leads to more investment today (i.e., reduces fl) to smooth the accumulation of the larger capital

stock desired after date T. However, when k increases it decreases the value of

existing capital, which does not qualify for the credit, relative to new capital, which

does. This effect, which discourages current investment if k is expected to increase,

is accounted for by the kt term appearing in (16). Because of the jump in k at T, kT is undefined. Its impact on flt is massed at T

in a T However, its effect can be calculated as the limit of the effects of the k terms

associated with a change from k to k* over an arbitrarily short interval around T.

This yields:

(25) A2 e-A2(T- t) Ak

which has the same sign as Ak, discouraging investment if Ak > 0. Combining (24)

and (25), using the definition of fl in (21) yields the total effect of the change in k:

(26) 1 A 2 )A -k* -r* ( + ^ -(T t))

If A2 > (p + 3), lt exceeds the value it would have for no change in k (i.e., for T -> oc). In this case, the expectation of an increase in the investment tax credit reduces current investment. The capital loss effect in (25) outweighs the smoothing

effect in (24). However, A2 may theoretically be less than (p + 8). From the

definition of A2 in (17), it follows that A2 > (p + 3) if and only if a < .5 and 4 is in the interval [(1 - \/1 - 2a)/8, (1 + \/1 - 2a)/8]. As shown below, this is the

same condition for the value of existing capital goods to increase with an increase

in the investment tax credit. The increase indicates that marginal q increases by

more than the gap between marginal and average q does. Since a capital gain occurs

in such an event, the anticipation of such a gain encourages current investment.6

To extend the analysis to the case of a temporary tax credit, imagine a second

shift from k* to k at some date T' < T before the shift back to k* at T. The shift at

T has the effect on lt just estimated, while the effect of the earlier shift is:

6 A similar ambiguity was found in a general equilibrium model without adjustment costs by Judd (1985). In his model, the interest rate is influenced by individual savings decisions. Such general

equilibrium effects, like adjustment costs, induce a smoothing of investment, although the cause, the

desire by households to smooth consumption, is quite different. Given the openness of the U.S. economy

and the size of the U.S. corporate sector, it is questionable whether consumption smoothing by U.S.

households can influence interest rates enough to induce significant smoothing of investment. In any

event, the predictions of the two models are generally consistent with each other.

The choice of adjustment cost specification also matters. Under the level adjustment cost specification

used in an earlier version of this paper (C(I) instead of C(IIK)), A2 must exceed p + 8 and the ambiguity

disappears. Even in the present specification, however, this ambiguity turns out to be empirically

irrelevant.

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948 ALAN J. AUERBACH

(27) 1~ k*~ F*(1+e). -k* -r* p + ^

Combining (26) and (27) yields the full impact of the temporary change from T' to

T:

Ak A2-(P + ) A (28) fit k* F* (p + 3) (eA2(T-t)e-A2(T't))

Since T > T', this has the same sign as -Ak = k- k* if and only if A2 > (P + 8). In this case, an anticipated temporary increase in the investment tax credit from k* to k raises the current cost of capital. Once again, the desire to smooth higher

investment during the interval (T', T) is outweighed by the anticipated capital

losses at T' (net of the gains at T) caused by changes in the relative value of existing capital.

B. Anticipated Tax Rate Change. An important issue when considering the

impact of a change in - is the pattern of depreciation allowances that prevails during the tax reform. The extent to which such allowances are accelerated relative to economic depreciation is important in determining short-run investment incen-

tives and changes in market value. To allow for different degrees of acceleration,

assume that the depreciation allowance function D(@) is invariant with respect to time, and described by:

(29) D(v)= 8'e t v

where 8' is the rate of declining balance depreciation permitted for tax purposes. Thus, the present value of tax savings per dollar of date s investment is (from (8)):

00

(30) v s e Ur(l s)rI 'e-'(u-s) du. Js

Note that these tax savings are discounted at the nominal interest rate, r = p + XT

(where XT = plp), since depreciation allowances are expressed in nominal terms and not indexed for inflation. Under a constant tax system (and hence in the steady

state), F = rz, where z = +8'+V is the present value of the depreciation allowances themselves.

Suppose the tax rate is currently (at date t) equal to T, and will remain so until switching permanently to r* and date T > t. This will affect all three terms on the

right-hand side of (16), the expression for at. Solving for Ft from (30) yields:

(31) Ft = {Jijl - e (P+T+8')(T-t)I + *e (p+ +8')(T-t)}z

= - Arz[l - e (p+T+8')(T-t)I

where Ar = r* - -. Differentiating (31) with respect to t yields:

(32) Pt = ATz(p + iT + ')e-(P + X + 8')(T-t)

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TAX REFORM, INVESTMENT AND VALUE 949

which has the same sign as Ar, since depreciation allowances increase in value as

t -- T, for r* > -.

The last impact on at of the impending change in r is the direct effect on after-tax cash flows. Combining these three effects yields:

1 z

= - T* + - k* - p *

AT -\Z e(P + 8)(T -t)d -e(8 + 8-)(T -t) p + 3I1 - k* - F* dt

Integrating at to obtain ft, according to (21), yields:

(34) Qlt = -AT( * - e--r) 1 k-

K 2 5 K IT e (p + iT + 8 )(T -t) eA AT t) A2 -(P + IT + 1 P +

If 8 = 8' + , the second term in (34) vanishes, leaving

(35) fl t +i tLOc i:l -)(+ e )- i+T6)Tt A(t) l1 - T* I -k*-rF*)

I1 - k - F

= - -r) (1 - e-A2(T-t) (1 -k* -F*

I - T*

In this special case (and assuming also that A(1 - k - F)/ [(1 - r)] has the same sign

as Ar), this expression calls for a higher value of ft, and a lower value of current investment, the sooner a tax increase occurs (and a higher value the sooner a tax

cut occurs). The intuition is that, after T, the desired capital stock will be lower.

Given the incentive to smooth investment, the firm will reduce investment immediately. The sooner the tax change, the stronger the incentive to smooth

investment. The strength of the smoothing incentive depends on A2. If there were no adjustment costs, A2 would be infinite (see (17)); the firm would wait until date

T to reduce its capital stock.

If 8 < 8' + T, as is more common in actual tax systems, depreciation allowances decline more rapidly than actual depreciation. This acceleration may be a legislated one (i.e., 8' < 8), but may also be attributable in part to the fact that depreciation allowances on existing assets decline over time due to inflation. In this case, this second term in (34) reinforces the reduction in current investment associated with

a future tax increase. This occurs because, with accelerated depreciation, old

capital's quasirents will be more exposed to the tax increase than new capital's quasirents. As with a prospective change in the investment tax credit, this

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950 ALAN J. AUERBACH

anticipated increase in the relative tax burden of existing capital is associated with

future capital losses and discourages investment today. What is different here is

that this tax on old capital is associated with a policy that discourages new

investment (an increase in r) rather than one that encourages it (an increase in k).

With accelerated depreciation, it is also possible for an anticipated tax increase

to discourage investment more than an immediate one. Put another way, a delayed

cut in - may increase investment more than an immediate one. This possibility was demonstrated by Abel (1982) for the case of instantaneous tax depreciation (8' =

0),7 but there is a much weaker condition sufficient for the result to hold. Consider the effect on lt of an increase in T:

dflt -A (T - ) f _ __ _ 8'+ IT - (36) dT= ArA2e A [l*+lk*-* ( ( p+ )

z I p + IT + at \t+ IT - t\ 1- k*~ i'* A2 2~7I~. '))(p+8 ~)(e(A2-(P+ X+ 6'))(T-t) - 1)}. 1- k*-rF* WA 2 -(P + 7T + ) P +

For a delay in a tax increase to reduce current investment, this derivative must have

the same sign as Ar. At T = t, the second part of (36) equals zero, so the condition

that sgn (df1t/dT) = sgn (Ar) is (given the definitions of F and z):

(p + 8)(1 - k* - F*)

(37) 1- < The intuition for this result is quite simple. At time t, the new investment's tax base

is negative if its gross marginal product of capital is less than its depreciation

allowance. The left-hand side of (37) is the long-run cost of capital per dollar under

the new tax system, to which the marginal product will eventually converge, while the right-hand side of (37) is the instantaneous depreciation allowance per dollar.

Although the matter is complicated by the fact that the actual value of F' won't

equal the left-hand side of (37) immediately (because of adjustment costs), the intuition is that if the underlying tax base is negative just after investment, then a

delay in a tax cut lowers taxes and raises investment.

A quite similar result to this is that firms facing a system of limited loss offset will prefer to be in a taxable position when depreciation allowances are sufficiently

accelerated (Auerbach 1983a, 1986). In that case, the tax rates change over time

due to changes in the firm's taxable income rather than the tax law itself.

To summarize the results of this section, an anticipated tax change influences

current investment in two ways. The first relates to the desire to smooth the investment path to a new desired long-run capital stock. As the enactment date, T,

becomes more distant, the current impact on investment of this incentive declines.

The effect disappears entirely when there are no adjustment costs. The second

effect, which is present even when there are no adjustment costs, comes from

7 Abel actually considered the case where only a fraction of new investment could be written off immediately, so that z = 8/(p + 8) despite the accelerated write-off. The crucial issue, however, is the

timing of the allowances.

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TAX REFORM, INVESTMENT AND VALUE 951

anticipated capital gains or losses associated with changes in the relative treatment

of new and old capital goods. When there is an impending tax cut (Ar < 0) and

accelerated depreciation (8' + iT > 8), a delay of the tax cut may increase current

investment if initial depreciation allowances are sufficiently large. Eventually,

however, the first effect must dominate, since foregoing a tax cut forever clearly

cannot encourage investment.

4. NUMERICAL SIMULATIONS OF EFFECTIVE TAX RATES

This section uses the expressions derived above to illustrate the effects, fl, on the

cost of capital associated with tax reforms such as those recently enacted in the

U.S., and translates these estimates into the effective tax rates on current

investment with expression (23).

Performing these experiments requires values for several economic and techno-

logical parameters. The real discount rate is set at 4 percent,8 as is the inflation rate.

For the constant elasticity specification F(K) = AKT, the parameter a is constant and equal to 1 - y (see (15)). Since this production function corresponds to the

Cobb-Douglas function with other factors (such as labor) held constant, one may

view a as the complement of the capital share of gross output. Since depreciation

in the U.S. is typically about 10 percent of GNP, and capital's share of net income

is about one quarter, it is reasonable to set a = .65. This value of a guarantees that

A2 > (p + 3), so that an anticipated cut in the investment tax credit stimulates

investment. The remaining technological parameters, 8 and 4, are varied to estimate the impact of tax changes for different assets under different adjustment

cost conditions. Values of 8 = .03 and .10 are used to represent structures and equipment, respectively, and values of 4 = .5 and 20 are considered. We use such distinct values for 4 because of the remaining uncertainty about its "true" value.

The higher value of 4 is much more consistent with findings in the empirical literature (indeed, most are even higher), though it has been frequently argued that

such estimates may be biased upward because of the noise with which the proper

value of q is estimated using market value data. This is a plausible argument, given

the extremely slow speed of adjustment implied by setting 4 = 20. As shown in

Table 1, the adjustment coefficient (-A 1), which can be interpreted as the fraction of the gap between actual and desired capital stocks closed within a year, equals .02

for equipment and .03 for structures. This sluggish behavior is consistent with a

very steeply rising marginal cost of capital goods; when 4 = 20, a one percentage point increase in the growth rate of the capital stock increases the marginal cost of

capital goods to the firm by 20 percent. However, it is at odds with speeds of

adjustment found in the literature when partial adjustment models are estimated

directly (e.g. Clark 1979). Though such results are not easily translatable into the

8 The use of a single real discount rate for different cash flows is consistent with the assumption of no uncertainty. In a more general model with flows having different risk characteristics, this would no longer

be the case. There is also some question concerning whether firms do in fact use discount rates that are

in line with theory (e.g. Summers 1987). Such issues are certainly relevant but lie beyond the scope of this

paper.

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952 ALAN J. AUERBACH

TABLE 1

THE EFFECTS OF TAX REFORM ON INVESTMENT INCENTIVES

Effective Tax Rates

New Law

Case 8 -Al A2 Old law 1985 1986 1987 After 1987

Equipment (8 = 1,8'= .2,k= .1) k*= 0, r* = .34

= 20 0 .02 .06 .055 -.185 .071 .113 .128 = .5 .0975 .40 .44 .167 -8.793 .292 .321 .336

k*= .1, r* = .34 =20 0 .02 .06 .055 -.144 -.074 -.023 -.004

+ = .5 .0975 .40 .44 .167 -.235 -.145 -.056 -.015 k* = 0, r* = .46

= 20 0 .02 .06 .055 -.028 .196 .196 .196 = .5 .0975 .40 .44 .167 -.385 .456 .456 .456

Structures (8 = .03,8' = .05, k= k* = 0, i- .34)

= 20 .021 .03 .07 .444 .356 .339 .329 .326 = .5 .0296 .28 .32 .478 .496 .422 .373 .356

All simulations assume - = .46, p = .04, T = .04, and a = .65. Where k* = 0, the change in k occurs in 1986. Where r* = .34, the change occurs in equal stages in 1987 and 1988.

current specification, it is useful to consider how sensitive the results are to the

magnitude of adjustment costs.9

The Tax Reform Act of 1986 was enacted after several years of discussion, during

which it became progressively more likely that the reform would occur. Earlier in

the 1980s, important changes in investment incentives were introduced in 1981,

1982 and 1984. In 1981, as well as in 1986, the reform eventually introduced was

discussed and debated for at least two years. Thus, it is quite important to consider

the potential impact of anticipated tax changes on current investment.

The 1986 reform lowered the statutory corporate tax rate from .46 to .34 and

repealed the investment tax credit, which had been .10 for equipment only. Thus,

r* = .34 and k* = 0. The provisions of the Act called for the investment credit to

be removed retroactively, dating back to the beginning of 1986, and for the tax rate

to be lowered to .40 in 1987 and .34 in 1988. Thus, certainly in 1986 and 1987, and

perhaps before, there have been anticipated tax changes causing effective tax rates

to differ from those which would apply under either the new or the old tax law.

To assess the impact of these expectations, Table 1 presents effective tax rates

(based on expression (23)) for equipment and structures investments undertaken in

1985, 1986, 1987 and the new steady state (post-1987) assuming the provisions of the

tax reform were known with certainty in each year. Also presented for comparison

9 There is also some uncertainty about the appropriate form of the adjustment cost function. As mentioned above, one can conceive of a specification in which the level rather than the rate of change of

investment determines adjustment costs. Indeed, given the smoothness of K over time, it is empirically

difficult to distinguish between estimates based on the ratio of I to K and those based on the level of I

alone. Calculations analogous to those present in Table 1 for this alternative adjustment cost specification

were qualitatively similar to the results reported.

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TAX REFORM, INVESTMENT AND VALUE 953

are the effective tax rates under the old system with no changes anticipated.'0 The

term 8', the depreciation rate allowed for tax purposes, based on historical cost, is

set equal to .20 for equipment and .05 for structures. This parameter choice reflects

the fact that each asset has accelerated depreciation under both old and new tax

systems, relative to the direct measures of economic depreciation, 5. II

For purposes of comparison, simulations that consider the impact on equipment

of the changes in r and k alone are also presented. Results for each simulation

include the values of 3, the true rate of economic depreciation, -A 1, the speed of

adjustment, and A 2, the discount factor applied to the terms a, (s > t) in computing Qt (see (21)).

As many analyses of the recent tax changes have concluded, the long-run

effective tax rates rise for equipment (when the investment tax credit is removed)

and fall for structures, with both new rates very close to the statutory tax rate. The

fact that 0* r* does not, however, deny the presence of accelerated depreciation. It simply indicates that the acceleration via 8' is roughly offset in present value by the lack of inflation indexing. Indeed, the measure of acceleration that matters in

the current context is 8' + iT, since this is the rate at which real depreciation

allowances decline. Thus, a delay in the tax cut provision need not by itself increase

the current effective tax rate.

Since the tax change is delayed, the short-run results are different. For

structures, the present value of depreciation allowances is small, so the positive

effect of investment due to increased value of depreciation deductions does not

outweigh the negative effect of a reduction in the long-run cost of capital. For

equipment, the acceleration of depreciation itself, described in the second set of

simulations, is enough to make a delay in the tax cut increase 1985 and 1986

investment and lower the concurrent effective tax rates. This is because the present

value of depreciation allowances, z, is much higher for this more rapidly depreci-

ating asset. The removal of the investment tax credit described in the third set of

simulations increases the incentive for 1985 investment as firms wish to invest more

to take advantage of the investment tax credit. These two effects combined give the

announced policy shifts a powerful effect on 1985 investment, although by 1986,

after the credit removal, the effective tax rate exceeds its value under old law.

It is not necessary that reductions in the effective tax rate associated with a delay in tax changes also reduce tax revenues. Indeed, the delayed reduction in r

encourages equipment investment while raising revenue. This is because, while taxes collected on new investment may be reduced (if 8' exceeds the marginal product of capital), taxes collected on existing assets will be increased by keeping

the higher tax rate. This ability to increase current investment and revenue at the

same time is another way of presenting the fact that windfalls are being given

existing assets when the tax rate is cut. Such windfalls have been seen as an

10 The tax rate for the old law would be slightly higher if based on the linear approximation around the new steady state. The rates .055, .167, .444 and .478 would be replaced by .064, .170, .500, .547. This

explains why the structures' 1985 tax rate for 0 = .5 appears higher than the rate under old law. 11 The 1986 Act did include changes in depreciation provisions. These were less important than the

changes in i and k, and are not considered here.

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954 ALAN J. AUERBACH

important shortcoming of the tax reform, since if revenue is being held constant the

effective tax rate on new investment rises. However, one must stop short of

characterizing as superior or more efficient policies that increase current invest-

ment without decreasing current tax revenue, since taxing existing capital may well

have an impact on expectations about the shape of future "reforms." Nevertheless,

the short-run impact of tax reforms, as well as their long-run consequences, should

be considered in light of the frequency with which new provisions have been

introduced.

5. TAX REFORM AND MARKET VALUE

Tax changes affect the value of the firm as well as the incentive to invest. The

close relation of these two effects has already been brought out in showing how

anticipated capital gains and losses are incorporated into current incentives. It is

also possible to calculate how tax reforms affect the value of the firm as a whole,

not just investments undertaken at a specific date.

At any given time, the value of the firm will depend not only on the capital stock

but also on the previous path of capital accumulation, since depreciation allow-

ances do not follow economic depreciation. This section's analysis is limited to

cases in which previous accumulation has been in a steady state. Thus, the tax

reform experiment is one not only near a steady state, but where the steady state

has been disturbed.

With a single factor of production, capital, and decreasing returns to scale, there

are two sources of firm value in the absence of taxes: normal returns to capital and

pure economic profits. As always, it is possible to reinterpret a decreasing returns

technology with one factor as a constant returns technology with two factors, the

second being a fixed factor owned by the firm that "earns" the economic profits as

a factor reward. This is especially helpful in the current context, for then it is

possible to apply the result of Hayashi (1982), adjusted for taxes by Summers

(1981), that the value of the firm's capital stock per unit equals the marginal cost of

new capital, adjusted for differences in tax attributes. Thus, the firm's value has two

components: this tax-adjusted value of marginal q, multiplied by the capital stock,

plus the discounted value of pure profits. This can be expressed per unit of capital,

yielding a value of average q that includes not only the tax adjusted value of capital

but also the discounted profits per unit of capital. One can then consider the effects

of tax reform on the total as well as the components, a particularly useful exercise

if one wishes to consider the effects of tax reform on the value of the firm under

different assumptions about whether the firm has any pure profits. To begin, consider the value of the firm's capital stock. The marginal price of new

capital goods is, from (5), q = 1 + 4kIK. This capital receives investment credits kq per unit and depreciation allowances worth Fq, and yields a stream of after-tax

quasirents in the future. Hence, the existing capital stock, K, which has the same

future productivity per unit, must be worth qK = q(l - k - F)K + A, where A is the present value (in terms of taxes saved) of this capital stock's depreciation

deductions. Using this relation, one may obtain, after a few steps (see the

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TAX REFORM, INVESTMENT AND VALUE 955

Appendix) an expression for the deviation of qK from its steady state value at time zero due to an unannounced tax policy change:

(38a) AqK = qK _q = Ako - a, - 5)

rx Qo~~~~~~~~~~~''

x fArte-(P + 8'+T)t dt + A 10(l - k* - F*)- a

where Ako = ko - k* and Art = rt - r*. This value will reflect both changes in marginal q (through fl0) and changes in the relative valuation of new and old capital.

Next, consider the impact of tax reform on the discounted value of pure profits relative to the capital stock, qP. If, for purposes of analysis, one assumes that a is constant (i.e., the production function is of the Cobb-Douglas form), then (see the Appendix) the deviation of qp from its steady-state value is:

(p +)(1- k*-*) 1a r (38b) AqP= t(1kt - d* a Jt

+ A1 f e-Pt(1 - it) e eA (t-s)fs ds dtl.

Jo Jo~~~~~~~~~~~~~~

Expressions (38a) and (38b) provide the component changes in market value resulting from any change in tax policy initiated at date zero. For an immediate, permanent tax change,12 they simplify considerably. In that case, it is easy to show that:

1 /1~~~ - k* - F* (39) - 1 k* -Fr* k -

which is simply the proportional change in the long-run cost of capital, (p + 8)(1 - k - F)/ (1 - r). Substituting (39) into (38a) and (38b) (and using the facts that A + A2 = p and A 1A2 = -a(p + 8)/1) yields:

(40a) AqK 4 = A k - (z(* )z [ A ) - Ak + ATz(z) - ]

(40b) AqP = [Ak - Ar -F* z + P +2)

aT (1 - k* - F*P +

12 Since perterbations around the steady state are being assumed, these "permanent" changes are, strictly speaking, temporary changes of a very long duration.

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956 ALAN J. AUERBACH

(40c) AqT =AqK+ AqP= Ak p

A4 [(4(*Fr*)( 1 )(+)a ( a + 'IT ]

From these expressions, a number of points about the effects of changes in r and

k may be made. Each tax change affects the value of the capital stock, qK, in two

ways, represented by the two bracketed terms in (40a). The first is the change due

to the change in marginal q, the second the change in the relative value of new and

existing assets. Any policy that increases marginal q (a cut in r or an increase in k)

increases the first term, while with accelerated depreciation, the credit increase and

the tax cut affect the second term in opposite directions. The credit increase causes

a capital loss by increasing the distinction between old and new capital, while the

tax cut narrows the difference associated with differences in prospective depreci-

ation allowances.

A second difference between the two policies appears in expression (40b), the

impact on the present value of pure profits. This is due to the extra windfall given

to the firm by a tax cut as the result of the reduced taxation of existing profits. Since

A2 ? p (for F' and hence p + 3 > 0), policies of either type that encourage

investment also increase profits through an expansion of output. This profit

increase depends on the assumption that the firm faces fixed output prices. In a

more general model, with other factors of production or profits bid down by

declining output prices, one might expect all or part of this increase in profits to be

absent. For example, if the firm's production function exhibited constant returns to

scale in capital and labor, with labor in fixed supply, the profits would be replaced

by tax-deductible returns to labor. Wages would be affected by the tax reform but

profits would always be zero. (See footnote 2.)

The division of changes in the total value of the firm between changes in the value

of capital and changes in the value of profits depends on the technology of

adjustment. For an investment tax credit, the total change in the value of the firm

is simply Ak(31p), the discounted value of additional investment credits. With high

adjustment costs, A2 -* p, so this appears entirely as an increase in qK; there is little change in output or profits and the firm simply receives the additional credits

as a windfall to capital. At the other extreme, with no adjustment costs, AqK = -Ak, as the value of marginal q doesn't change at all. At the critical intermediate

value of A2 = p + 8, the effect on qK is zero, as the two effects in (40a) cancel. As already indicated, A2 > p + 8 is the "normal" case, empirically.

For a tax cut, the situation is more complicated, depending on the extent to

which depreciation allowances are accelerated. Total value increases by more per

unit increase in marginal q than in the case of the investment tax credit for three

reasons: reduced taxation of normal returns to existing capital, reduced taxation of

the component of the tax base associated with prior accelerated depreciation, and

reduced taxation of preexisting profits. The first two effects are present in (40a), the

last in (40b). Even with no adjustment costs (A2 = o?), the value of the capital stock

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TAX REFORM, INVESTMENT AND VALUE 957

increases because of the reduced tax on the base associated with prior accelerated

depreciation, by Ai-z{1 - [1/(8' + [r)]}.

Since much of the criticism of the recent tax reform has focused on shifts in the

tax burden between new and old capital, it is interesting to consider the effects on

the value of the existing capital stock, qK, of such a change.

Table 2 presents calculations of AqK based on expression (38a) for the same tax

reform considered in Table 1, the removal of the 10 percent investment tax credit

for equipment in 1986 and the cut in the corporate tax rate from .46 to .34 in equal

steps in 1987 and 1988. The calculations use the same economic parameters as

before (8 = .1, 8' = .2 for equipment, 8 = .03, 8' = .05 for structures, and p =

X = .04). For each case, the change in qK is broken down into four components:

the changes in marginal q and the gap between marginal and average q caused by

the changes in k and r.

Three sets of estimates are presented for each asset and value of 4. The first is

for 1985, and assumes that the tax change was first anticipated in that year; the

second is for 1986 under the assumption that the tax change was foreseen only then;

the last is also for 1986 under the same expectations assumptions but for a different

tax change, an immediate rather than delayed reduction in r.

Consider first the estimate for 1986 for the law as passed, including the tax cut

phase-in. As expected, the combined effects coming through marginal q are

negative for equipment and positive for structures, with the effects larger when

adjustment costs are large. This impact of adjustment costs is especially strong for

structures because of the lower rate at which structures depreciate; existing capital

gets the benefit of increased after-tax returns over a longer period. The "windfall"

effects represented by increases in the value of old relative to new capital are

positive for both parts of the policy, the removal of the tax credit and the reduction

in the tax rate. For equipment, each part of the policy raises the market value of

capital, with the total impact relatively insensitive to the size of adjustment costs.

For structures, the total increase value is substantially higher with higher adjust-

ment costs.

The effects of the delay in r differ between structures and equipment. Eliminating

the phase-in increases the windfall for both assets. It increases q further through the

marginal q effect on structures, but has the opposite effect on equipment. This is

consistent with the results in Table 1 which showed that equipment investment is

encouraged by the phase-in while structures investment is discouraged. As pointed

out above, the effect of the phase-in is to reduce the increase in the value of

equipment while at the same time encouraging equipment investment.

Anticipation of the policy change in 1985 would have led to smaller increases in

value than for the case where information was received in 1986. For equipment,

both the anticipated removal of the investment tax credit and the prospective tax

rate cut would have driven marginal q up even more, but this increase is more than

offset by the reduction in windfalls to existing capital. For structures, the longer

delay in the tax reduction makes both windfalls and marginal q smaller than for the

1986 simulations.

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958 ALAN J. AUERBACH

TABLE 2

THE EFFECTS OF TAX REFORM ON THE VALUE OF EXISTING CAPITAL

Proportional Change in Capital Value Resulting From:

Removal of Investment Credit Cut in Corporate Tax Rate

New-Old New-Old Marginal Capital Overall Marginal Capital Overall

Asset Year Effect Effect Effect Effect Effect Effect Total

Equipment = 20

1985 .033 .033 .057 .043 .100 .132 1986 -.065 .100 .035 .048 .057 .105 .140 1986 -.065 .100 .035 .027 .086 .113 .147

(no phase in) =.5

1985 .044 .044 .016 .026 .042 .086 1986 - .031 .100 .069 .017 .034 .051 .120 1986 -.031 .100 .069 .013 .051 .064 .133

(no phase in) Structures

= 20

1985 .032 .102 .134 .134 1986 .033 .109 .143 .143 1986 .035 .121 .157 .157

(no phase in)

J= .5 1985 .026 .013 .040 .040 1986 .028 .019 .047 .047 1986 .031 .030 .061 .061

(no phase in)

Calculations are based on expression (38a) with the parameters used for simulations presented in Table 1.

6. CONCLUSION

This paper has presented an analytical discussion of the impact of tax reforms on

current investment and market value, taking account not only of the nature of the

tax law but also the production and adjustment cost technology.

Its main contribution has been the derivation of analytical expressions for the

impact of future tax provisions on the value of the firm and the user cost of capital.

These expressions are helpful in understanding the impact of particular tax changes

and the importance of investment smoothing and announcement effects.

The analysis has incorporated several simplifications regarding taxation. For

example, in recent years, tax losses and other constraints have been an important

phenomenon. The impact of such constraints varies across assets and can either

encourage or discourage investment (Auerbach, 1983a, 1986; Auerbach and Pot-

erba, 1987a; Altshuler and Auerbach, forthcoming). In an environment without

perfect loss offset, the effects of immediate or delayed tax reforms may be more

complicated than those portrayed here. In addition, there are several components

of the corporate tax beyond those considered here. As shown by Auerbach and

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TAX REFORM, INVESTMENT AND VALUE 959

Poterba (1987b), the increases in corporate tax collections resulting from the 1986

law are attributable largely to these other factors, which include changes in

accounting rules and the treatment of financial intermediaries. Ignoring some of

these provisions may be justified to the extent that one wishes to focus on fixed

investment, rather than other corporate activities, but there are undoubtedly

further effects than those captured here, effects which may be very indirect and

hence more difficult to describe and analyze.

In ignoring corporate financial policy and personal taxes, we have implicitly

assumed a separation of real and financial decisions. Moreover, even if this

separation is valid, personal tax changes may influence the incentive to invest by

influencing the rate of return investors require of firms. A full analysis incorporating

personal taxes would require one to deal with important unresolved issues, such as

the determinants of financial equilibrium and dividend policy. Without undertaking

such a task, one would add little to the analysis already provided in this paper

simply by assuming a fixed debt-equity ratio and dividend payout rate, for example.

The entire analysis would still hold, except for a simple adjustment to the corporate

discount rate (see Auerbach 1983b).

One should also consider the restrictions of the assumed pattern of adjustment

costs. Aside from the issue already discussed concerning whether such costs are

based on the level or the rate of investment, there is also the question of whether

capital becomes immediately productive, as assumed here, or is subject also to

delivery lags. Intuition suggests that the presence of such additional lags should not

change the basic nature of the results (except those concerning temporary tax cuts

and tax credits that apply during the initial lag period), since the key characteristics

of the model, partial adjustment of investment to a moving target and hence

fluctuations in the shadow value of new capital, remain. But, clearly, extensions in

this direction would be quite useful. The "q" theory of investment does not seem

to be fully adequate for explaining investment behavior.

In considering assets individually, one ignores the potential heterogeneity of

adjustment costs and the spillover effects that changes in one type of investment

may have on another through complementarity in the production function and

shared adjustment costs. A multiple capital stock model is too complicated for the

derivation of interpretable analytical expressions, although Auerbach and Hines

(1987) have obtained results similar to those reported here in a numerical simulation

model with two capital stocks. The similarity of such simulation results also

suggests that the approximation error involved in model linearization is not too severe.

Finally, for this paper's analysis to be useful in evaluating alternative tax

incentive policies, a better positive model of the dynamic process of tax reform is

needed. Prospective tax changes are never certain, and probably depend on future

economic conditions. Incorporating uncertainty about future tax variables and

studying the basis of policy decisions in an important topic for future research. A

preliminary attempt at extending this paper's model in such a direction may be

found in Auerbach and Hines (1988).

University of Pennsylvania and NBER, U.S.A.

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960 ALAN J. AUERBACH

APPENDIX

This Appendix derives expressions (38a) and (38b) for the changes in the market

value of capital and pure profits.

qK Since it has been assumed that this capital was accumulated in a steady state, a constant amount of capital, 8K, was purchased at each prior date, at an

average price of (1 - 2 4 ). This average cost is relevant for calculating the total

value of depreciation allowances, since allowances are based on total capital

expenditures. Thus, the value of depreciation allowances on existing capital at the

current date, zero, is:

(A1) A =f I K(- -2 F)(- dt = 8K f r-) dt

where rf-t) is, as before, the present value of depreciation allowances remaining for an asset of age -t. If one assumes, as above, that depreciation allowances are

at a constant proportional rate 8' but not indexed, then (Al) may be rewritten:

(A2) A = AK F (8 +T) tdt= =5 + FK.

It follows that the average value of the capital stock per unit of replacement cost is:

(A3) qK= q(l-k F) + A/K= q(l-k F) + F. 8' + 'IT

To simplify expression (A3), note that, in the steady state, K = K*, the optimal

capital stock under the steady state's tax system. Thus, (19) and (20) may be

combined to yield, at t = 0,

(A4) K= (-A )(K-K*) = (-Al(a)K*.

Using (A4) and the definition of q, one may rewrite (A3) as:

(A5) qK = (I k -(F 8' - T)) A+ 1(l -k-F)flIa.

In the steady state, k = k*, F = r*z and fl = 0. Combined with (A5), and using (30), this yields expression (38a).

qP. By construction, profits equal the after-tax quasirents in each year in

excess of the capital stock's marginal product, or, normalized by the current capital

stock,

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TAX REFORM, INVESTMENT AND VALUE 961

I rx (A6) qP = f e -Pt (1- rt)[F(Kt) - KtF'(Kt)] dt.

For a small change in tax policy around the steady state, the change in qP at time zero is:

(A7) /q= e Pt{-Art[F(K*) - K*F'(K*)] K*

- (1 - *)K*F"(K*)(Kt -K*)} dt

where Art = rt - r*. This has two components, due to the change in the taxation of existing profits, and the change in profits. The first component depends on the production function itself, not just its local properties.

If a is assumed constant, then given (9') and the definition of a in (15), (A7) may also be written:

(p + )(1 - k*- F*) [ a r0

(A8) \qp e -(1- r*) -1 JePt Art dt

+ ae f e-Pt(1 - (t) Kt K) dtl

An expression for Kt is obtained by solving the first-order differential equation (19), using the initial condition that Ko = K*:

rt

(A9) Kt= eAIt K*-A1 f e-A1s Ks ds

which, given the definition of Ks in (20), yields:

Kt_-K* A1 ft (AIO) =-J e l e ds.

Substitution of (AIO) into (A8) yields a solution for AqP in terms of exogenous parameters alone, (38b).

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962 ALAN J. AUERBACH

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