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Evaluating tax reforms in a monetary economy
JEAN-FRAN(~OIS WEN Wilfrid Laurier University Waterloo, Ontario, Canada
DAVID R. F. LOVE Brock University St. Catharines, Ontario, Canada
Evaluating Tax Reforms in a Monetary Economy Hypothetical revenue-neutral tax reforms are conducted in a calibrated endogenous growth model in which money serves to economize on the time-costs of transacting. The modelincludes the cash-in-advance (CIA) and non-monetary frameworks as special cases of the parameterization. The results from our shopping-time model suggest that both the CIA and non-monetary models may somewhat underestimate the welfare benefits of lowering the income tax, while the growth effects of the tax reforms are almost the same across the models. We also examine the transitional dynamics resulting from the tax reforms.
1. Introduction Although the dynamic effects of alternative tax structures have been extensively studied, the conventional policy evaluations were arrived at using models that abstracted from money and inflation. (See Summers 1981; Chamley 1981; Judd 1987; Auerbach and Kotlikoff 1987; King and Rebelo 1990; Lucas 1990; and Devereux and Love 1994, 1995; for example. An exception is Cooley and Hansen 1992, who use a neo-classical monetary economy.) Including a role for money may be important for assessing tax structures because of the interaction between real and nominal variables. Smith (1996) augmented a one-sector version of an endogenous growth model with a cash-in-advance (CIA) motive for holding money, and showed analytically that there can arise important qualitative differences in the effects of taxation in monetary and non-monetary models. The direct growth effects of a change in tax policy can alter the inflation rate, for instance, which would in turn change the economic growth rate because of a portfolio substitution by households between transactions balances and physical capital. However, no systematic quantitative study has yet been undertaken on the welfare effects of alternative tax structures in a framework of money and growth. Journal of Macroeconomics, Summer 1998, Vol. 20, No. 3, pp. 487~508 Copyright © 1998 by Louisiana State University Press 0164-0704/98/$1.50
487
Jean-Franfois Wen and David R. F. Love In this paper, money is incorporated through a so-called shoppingtime technology into a two-sector endogenous growth model similar to King and Rebelo (1990) but with an elastic labor supply. We calculate the growth and welfare effects of hypothetical tax reforms based on exact numerical solutions of both the balanced growth paths and the transitional dynamics of the calibrated model. We then compare our results with a non-monetary version of the model, and also provide analytical derivations on the growth effects of taxation. The assumption that money serves to economize on the time-costs of transacting is sufficiently general as to include both the CIA constraint and a non-monetary economy as special cases of the model. Thus we can determine not only whether integrating money into a growth model is quantitatively significant for tax analysis, but also what differences arise between modeling the transactions demand for money as a CIA constraint or a timecost. One of the interesting features of the shopping-time approach (see McCallum 1989; Kimborough 1986; Guidotti and V6gh 1993; Lucas 1993; and Braun 1994, for example) is that there are always positive transactions costs associated with consumption, contrary to the CIA approach, in which the transactions costs associated ~vith cash goods are necessarily zero in equilibrium. Recent empirical support for the shopping-time approach is provided in Dowd (1990) and Love (1995). The trade-off at the margin between holding money for transacting or incurring shopping-time is closely related to the idea that resource-costly credit may be used as a means of payment, instead of cash. Transacting with credit may involve a loss of time, for instance, if the consumer's credit worthiness must be established prior to an exchange (see Lucas 1980 and Schreft 1992). Recent papers by Gfllman (1993), Marquis and Reffet (1994), Dotsey and Ireland (1996), and Lacker and Schreft (1996) find that endogenizing the cash-credit mix is important for assessing the welfare effects of the inflation tax. 1 The resource costs arising from fiscal tax distortions of the cashcredit margin have not previously been studied, but should yield results similar to our shopping-time model.2 The paper is organized as follows. The model is described in the next section. Then the competitive equilibrium is solved for and analytical results are provided on the growth effects of taxation. Numerical solutions for the 1Wang and Yip (I993) analyze the cost of inflation in a model where transactions time reduces the time available for human capital accumulation. Unlike the papers cited above, they include the welfare effects of the transition to a balanced growth path, as we do. 2In contrast, in the tax analysis of Cooley and Hansen (1992), credit goods are assumed to not entail a resource cost. However, excess burden can occur because the consumer's choice between cash and credit goods is distorted by the inflation tax. The authors find that taxing capital income is the least efficient tax policy.
488
Evaluating Tax Reforms
shopping-time, cash-in-advance, and non-monetary models are shown in the third section, followed by our conclusions.
2. The Model
Description The endogenous growth specification employed here is the same as in Gomme (1993) and Devereux and Love (1994, 1995), which are extensions of the model of King and Rebelo (1990) to the case of an elastic labor supply. A utility-maximizing representative household is assumed to have additive preferences that are isoelastic over consumption and leisure according to o~
U = ~
(i)
•tu(Ct, L t) ;
t=O
1 u(Ct , L t ) - - 1-~
, 1 - e)l - a , (CtLt
u(C t , L t) = ln(C;L~-~),
eE(0,1),
0 ~ 1 ;
a = 1.
(2) (3)
0 < 13< 1 is the household's subjective rate of time preference; Ct is periodt consumption and L t is leisure. The time available for leisure is subject to the constraint (4)
L t = 1 - IKt -- 1Ht -- Zt,
where lze represents hours of work supplied to the final goods sector, lnt gives the hours devoted to human capital formation, and zt is the time spent transacting. The time spent transacting is described by the following technology: zt = ~b((1 + xOPtCJMt) ~=-d~((1 + ~Ovt) ~,
qb->0,
~>1,
(5)
where Pt is the money price of consumption goods, M t are nominal money balances at the start of the period, ~c is a tax on consumption goods, and vt is the velocity of money in terms of consumption. Thus the time spent transacting is increasing in consumption expenditures gross-of-tax, 3 and deereasaThis is the standard assumptionin the literature; see Guidotti and V6gh (1993),for example. 489
Jean-Franfois W e n and David R. F. Love
ing in real money balances. ~ > 1 implies that using money for transacting has diminishing returns, while qb is a scaling parameter. Note that setting ~b to zero collapses the model to a non-monetary version (velocity goes to infinity), while in the limit as ~ approaches infinity the transactions function is identical to a cash-in-advance constraint (velocity goes to one). Guidotti and V6gh (1993) show that assuming that transactions costs are homogeneous of degree zero in consumption and real money balances is consistent with well-accepted mieroeconomie foundations, as in, for example, the Baumol-Tobin paradigm. Since our model allows for continuous growth of consumption expenditures and real balances, homogeneity of degree zero is, in any ease, required if transactions time is to remain stationary in the long run. The period-t expenditures of the household must satisfy the following budget constraint (1 + ~c~)PtCt = (1 - "~l)WtHtl~ + (1 - "ck)Rt0tKt - Pt(Kt+l - (1 - 5~)K~) -
(Mt+l - Ms) + DEt + Tt,
(6)
where zz and zk and Tt are the labor income tax rate, the capital income tax rate, and a period-t lump-sum nominal transfer from the government, respectively. Wt is the nominal wage paid to an efficiency unit of labor as determined by H t, the stock of human capital; Rt is the nominal rental rate on physical capital. Kt is the stock of physical capital, 5~ is its depreciation rate, and 0t is the fraction of that stock that is rented to firms operating in the final-goods sector. Finally, DE t represents the nominal transactions balances costlessly transferred to the household through the deposit expansion mechanism, as further discussed below. The household produces and accumulates human capital according to Ht+ 1
-~
AH((1 -- Ot)Kt)~(1HtHt)1-~ + (1 - 8H)Ht,
(7)
where 0 < 7 < 1, A H > O, and 5 H is the depreciation rate on human capital. Output in the final-goods sector, Yt, is produced by firms under perfect competition using a technology of the form Y~ = A(Ot~)'~(l~Ht) 1-`~ '
(s)
where 0 < ot < -
1, A > 0. For profit maximization, W t = Pt(1 ot)A(OtKt)~(lr, tHt)-~ and Rt = PtmA(OtKt)C~-l(lmHt)l-% Market equilib-
rium in the goods market requires that, c,
490
+ K,+~ -
(1 -
~K)K~ = r ~ .
(9)
Evaluating Tax Reforms
The government in this economy sets the tax rates, ~c, ~l and ~, and determines the rate of expansion of the monetary base, while also setting the reserve requirements relating the monetary base to the supply of transactions balances according to the relationship Mt = (1/rr)MBt,
rr -< 1,
(10)
where MBt is the monetary base and rr is the deposit reserve ratio. This use of a deposit expansion mechanism in our model is intended only as an accounting device, allowing us to distinguish between the monetary base, which is relevant for measuring the seignorage revenue that enters the government's budget constraint, and the (potentially) broader monetary aggregate that serves to reduce the time-costs of transacting. 4 The monetary base is assumed to evolve according to: MBt+ 1 = ~tMBt,
(11)
where ~t is the exogenous rate of monetary expansion. Money market clearing requires that the demand for transactions balances be equal to the righthand side of (10). Both tax and seignorage revenues are assumed to be lump-sum redistributed to the household each period so that a balanced government budget is maintained: T t = ~:cPtCt + z~HtI~ + ?RtOtKt + (p - 1)MBt.
(12)
Finally, we suppose that the additions to transactions balances in excess of additions to the monetary base are also costlessly transferred to households in each period through a deposit expansion mechanism (the "money multiplier"). Thus, DEt = (Mt+l - Mr) - (MBt+I - MBt) = (1 - rr)(~t - 1)Mr.
(13)
Competitive Equilibrium A competitive equilibrium for this economy consists of the sequences {Ct, l~, lut, zt, Or, Kt, nt, MBt, Mr, P~, Wt, Rt, Tt, DEt}t%=0 satisfying: (i) household utility maximization subject to the conditions (2)-(7), the initial values for the state variables, and standard boundary conditions; (ii) profit
4Braun (1994) associatesonlythe monetarybase with transactions, whileCooleyand Hansen (1991) assume that only a fraction of M1 is relevant for transactions, and use this base also to c~dculateseignorage. 491
Jean-Franfois Wen and David R. F. Love
maximization by firms; (iii) government budget constraints; and (iv) goods and money market clearing. The first-order conditions for the household's optimization problem with respect to the variables C t, lse, lw, zt, Ot, Mr+ 1, K¢ + 1, and H t + 1, together with (5), (7), (9), and (10), implicitly define solutions for the variables Ct, Ira, lilt, zt, Or, Kt, mr, and ~tt, where m~ = Mt/P t and 1 + r~t = Pt+l/Pt are real transactions balances and gross inflation. Real wages, wt = W/Pt, and real returns, r t = Rt/Pt, are given residually. From the optimal consumption-leisure choice, one can calculate the effective tax rate ~ on an hour of labor income wtHt, which plays a critical role in the distinction between the shopping-time, cash-in-advance, and nonmonetary models. Thus, (1 - rt)/(1 + zc) (1
-
~) = 1 + ~(1 -
~)w,H~z/(1 + ~c)c,
(14)
The second term in the denominator of the right-hand side of (14) is the value of the household's time spent shopping for a marginal unit of consumption. If the parameter qb were set equal to zero, then zt = 0 and this term drops out, yielding the well-known expression for the leisureconsumption tax wedge in non-monetary models. In the extreme ease of a CIA constraint (lim~ --* ~), zt approaches zero "faster" than the rate of increase in {, so that the extra term vanishes asymptotically. Thus, the shopping-time model is distinct in that the consumption-leisure margin is endogenous. Another important feature of the model is the asset equilibrium condition, stating that transactions balances should be accumulated so as to equate the returns on money and capital. That is, (1 + (1 - "d)wt+lHt+l~z~+l/mt+l)/(1 + ~t+z) = (1 - ~K + (1 - ~k)rt+l) .
(15)
This arbitrage-based channel for the interaction of nominal and real magnitudes is the focus of the comparison between non-monetary and CIA models in the work of Smith (1996) and Gomme (1993). In the time-cost model, an additional unit of transactions balances acquired at time t provides a return at t + i of a unit of purchasing power plus the value of the marginal reduction in transactions costs, both adjusted for inflation. Note that (15) implicitly defines money demand in this economy as a function of the netof-tax wage, consumption, and the nominal interest rate. Balanced Growth Analysis It is useful to express the equilibrium in terms of a balanced-growth representation by defining the following variables: ht - - H t / K t , k t ~ Kt+ 1/Kt,
492
Evaluating Tax Reforms and (1 + gt) - Ct+JC~. Along a balanced-growth path, 1 + gt = kt = 1 + g gives the constant rate of growth of capital stocks, consumption, and real balanees; he, vt, lut, ln, and Ot are also constant. It is straightforward to shove5 that under balaneed-growth the stationary representation can be reduced to the following two equations in growth rates and total labor supply. (1 + g)(1-~(i-~))
=
~(~(1H + /K)n((1 _
,~/)(1-a)(1
_
.ck)c~)l-~l ..}_ 1 - - ~ K ) ,
(16)
where f~ = aA((1 - 7)AuA~4)n((1 - e~)7/a(1 - 7)) TM > 0, 1] = (1 - a)/ (1- a + 7)~(0,1),andle(1- ~)>0;
1, + IK = (1 - a)~(1 - (1 + (1 - ~){)qb(v(1 + ~))~) (1 -
a)e
+
(1 -
~)(1 -
~)(1
-
ag:(1 -
zk)/((1 -
3')0))(1 + f ) / ( 1
-
~z),
(17)
where O, v, and the composite term g:, are functions only of g, In + IK, and parameters, as given in the appendix. It is obvious from differentiating (16) with respect to each of the tax rates (holding the other tax rates constant) that dg/dx l, dg/dx k and dg/d~ ~ are all negative if the corresponding derivatives of the total labor supply with respect to the tax rates are negative. Devereux and Love (1995) show that these labor supply derivatives are indeed unambiguously negative in the case of the non-monetary model with 8n = fiK and a sufficiently low value for ~,6 but here the result is more complicated because of the velocity term v. In the appendix we show that in the monetary model the results do still hold unambiguously for the labor and capital income taxes; however, the growth effect of the consumption tax now becomes ambiguous, though it is negative for realistic parameter values, r Of course, the growth effects of revenueneutral tax reforms will depend on the relative magnitudes of these derivatives and on the sizes of the different tax bases.
3. Numerical Results
Calibration There are four categories of parameters requiring specification: (i) production function parameters, A, ~, ~K, An, 7, 8n; (ii) preference parameters, 5See our working paper for the fun derivation. 6The ambiguity of the theoretical result for low values of the intertemporal elasticity of substitution, which varies inverselywith ¢y,is similar to Smith's (1996) finding that the growth effect of a tax depends qualitativelyon the magnitude of ¢y. VTheambiguity of the growth effect of the consumption tax stems from the fact that, unlike in the income tax cases, a partial effect of the consumption tax is to reduce the velocity of money, which in turn raises the total supply of labor. 493
Jean-Franfois Wen and David R. F. Love
TABLE 1. e~ = 7 = [3 = qb =
0.360 0.310 0.997 183.0
Summary of Parameter Values
8K = 0.024 8H ---- 0.005 (~ = 2.000 ~ = 8.000
A AH e rr
= = = =
0.0705 0.0705 0.395 0.107
[3, e, c; (iii) transactions-cost function parameters, qb, ~; and (iv) government policy parameters, xk, xt, zc, g, and rr. Time periods are assumed to correspond to quarterly observations. The specification of the production function and preference parameters is standard, replicating long-run averages of post-World War II U.S. maeroeconomic data. s Table i summarizes the parameter values of the model. Regarding the calibration of the monetary aspects of the model, it is easy to show that Equation (15) generates the following money-demand relationship: mt = fI(NRt)-I/(I+~)((1 - "cl)wtHt)l/(l+~)((1 + "cc)Ct)~J
(18)
where NRt = (1 + gt)(1 + (1 - "~k)rt - 8K) - 1 is the net nominal interest rate, and f~ = (+{)v(l+g). ~ uniquely determines money demand elasticities in our model, providing an observable gauge for calibration of this transactions-cost parameter. Given {, the scale parameter qb can be chosen to pin down either time spent transacting, or a measure of velocity. Since velocity estimates are much more reliable, we calibrate qb according to this criterion and leave time spent transacting to be determined residually. We choose to calibrate using M2 as our measure of transactions balances for the following reasons. First, the money demand Equation (18) describes a long-run relationship, and recent empirical evidence (e.g., Haler and Jansen 1991; Mehra 1993) suggests that the existence of an equilibrium relationship for quarterly U.S. M1 is subject to doubt, but not so for M2. Second, it is well known that U.S. M1 velocities have displayed a significant trend in the past, largely attributable to new technologies, while M2 velocities are stationary. Hetzel and Mehra (1989) have suggested that as a result, M2 would be a better policy predictor. Finally, Mankiw and Summers (1986) show that 96% of M2 holdings can be related to consumption, supporting SHowever, unlike in the calibration of King and Rebelo (1990) and Gomme (1993), we do not assume identical technologies in the human-capital and final-goods sectors. In addition to being unlikely in reality, assuming identical technologies across sectors implies unrealistically fast adjustment transitions of just a single period, as well as implying that more time is allocated to human-capital production in the economy than to final goods production. Instead, our calibration assumes that human-capital production is relatively more labor-intensive.
494
Evaluating Tax Reforms the intuition that a large portion of savings balances are accumulated for discretionary consumption expenditures on things like vacations and large durables. Hafer and Jansen (1991) and Mehra (1993) report long-rnn interest rate elasticities for U.S. quarterly M2 demand in the range of ( - 0 . 0 3 , - 0.19). Thus we choose the mid-point for the interest elasticity: - 1/(1 + ~) = -0.11; note that this implies a long-run consumption elasticity of ~/(1 + ~) = 0.88, which is comparable to the income elasticity of 0.89 reported by Haler and Jansen since consumption and income are proportional in our model in the long run. Quarterly M2 income velocity (where the price level includes consumption taxes) for the post World-War II U.S. has averaged 0.41. A choice of qb = 183 yields this value for our benchmark economy. Finally, the government policy parameters were chosen as follows. In accordance with the average quarterly rate of growth of the U.S. monetary base, we set g = 1.017, yielding an annual rate of inflation of 4.9%, given the benchmark real economic growth rate of 2% per year. The reserve ratio is set equal to the average ratio of the monetary base to M2; r r = 0.107. The benchmark tax rates ~w = 0.25 and ~c = 0.06 are based on Mendoza, Razin and Tesar (1994), and @ = 0.30 was chosen residually to yield the historical average ratio of total government spending to output of 0.31.
Balanced Growth Solutions. The benchmark case (~ = 0.30, rz = 0.25, xc = 0.06) is contrasted with three hypothetical revenue-neutral (unanticipated and permanent) tax reforms: (i) a lower labor income tax with a higher consumption tax; (ii) a lower capital income tax with a higher consumption tax; (iii) a lower general income tax with a higher consumption tax. In each case, the relevant income tax rate was reduced by ten percentage points and the consumption tax rate was raised to generate the same ratio of government revenues to GDP along the balanced growth path. Note that this means that the present value of total tax revenues may differ across tax regimes because of adjustments along the transition paths. 9 Thus to allow for a complete assessment of the alternative tax structures, changes in the present value of tax revenues are reported together with the changes in welfare. Although our focus is on quantifying the welfare and growth effects of 9Alternatively,conductingthe experimentsby holdingthe present value of tax revenuesconstant leads to differentbalanced growth ratios of tax revenue-to-GDPacross the tax regimes. Since we do not have a theory of the size of governmentin the model,we prefer to maintaina constant GDP-shareof tax revenue,rather than fixingits presentvalue. Our generalconclusions are independentof which definitionof revenue-neutralityis used. 495
J e a n - F r a n f o i s W e n and D a v i d R. F. L o v e
alternative tax regimes in a monetary endogenous growth model based on the "shopping-time" motive for holding money, we also present a parallel set of results for our approximations to the cash-in-advance model and the non-monetary model. This makes it easy to see to what extent and how monetary considerations can affect the results. Table 2 gives the balanced growth solutions for each tax regime. 1° Table 3 presents the welfare changes arising from each tax reform relative to the benchmark. The welfare change is measured as the constant stream of consumption required to eompensate a household for not carrying out the tax reform, n and is expressed as a percentage of the original consumption level. The welfare change is also reported as a percentage of the initial income level. The benchmark case for the shopping-time model predicts that roughly one-third of one percent of total available time 12 is spent transacting. This is one-tenth of the figure employed in D e n Haan (1990). W h e n evaluated at the effective wage rate, these transactions costs are equivalent to 1% of final-goods output, as compared to the 1% of a narrow measure of output used in Braun (1994). la Seignorage amounts to roughly 1.45% of total government revenues or 0.44% of output. 14 The main features of Table 2 are that the changes in economic growth associated with each tax reform are (almost) the same across the three types of models, but that the decline in leisure is comparatively small in the shopping-time model in response to a labor income tax cut. In the shopping1°Balanced growth paths are calculated using a Newton-Raphsonalgorithm on the balanced growth equation system. The solutions for the model's transitional dynamics are calculated by solving an appropriately specified two-point boundary problem derived from the optimality conditions of the model. This method closes the difference equation system that gives the transition paths of the economy by imposing initial values for the capital stocks, and by forcing the model to converge to its balanced growth path by some arbitrarily distant terminal date (usually 75 periods from t = 0). For a detailed and comprehensive explanation of this solution method and the algorithm we used, see Press, et al. (1994), Chapter 17. nLet {C~}~=0and {L/}t=0 be the post-tax-reform levels of consumption and leisure, and {Ct}~°=0 = {C0, (1 + g)Co, (1 + g)aCo.... } and {Lt}~'=0 = {L, L, L ...} the pre-tax-reform balanced growth paths of consumption and leisure, where g is the pre-tax-reform balanced growth rate. Then our measure of welfare change, C, is given implicitlyby E~=01~tu((1 + g)t(C0 + C), L) = E~=o~tu(C~,L~). In percentage terms, the welfare measure is C/C o. lZi.e., discretionarytime of about 16 hours per day. 13Ifwe interpret transactions time as reflecting lost leisure due to trips to the bank, as in Guidotti and V6gh (1993), then our model is generating about 3.5 minutes per day or 21 hours per year spent (per household) on banking activities, which seems reasonable. Evaluatingthis time at the U.S. average hourly wage times the civilian labor force yields one-half of a percent of GDP, using 1994 data from the Bureau of Labor Statistics. 14For a 5% inflation, Cooley and Hansen (1991) report seignorage revenues of 1.43-2.39% of government revenues. Braun (1994) reports seignoragerevenues equal to 0.47% of his narrow measure of output. 496
¢D -q
= = = =
= = = =
zk xk zk zk
~k ~k ~k zk
= = = =
0.25, 0.25, 0.15, 0.15,
z~ r~ zc zc
= = = =
= = = =
0.25, 0.25, 0.15, 0.15,
zc zc ~ z~
0.30, 0.20, 0.30, 0.20,
~I xz ~z ~z
= = = =
0.25, 0.25, 0.15, 0.15,
xc ~c zc zc
Non-Monetary
0.30, zl 0.20, zl 0.30,~1 0.20,~l
= = = =
= = = =
Cash-in-Advance
rl zz zl ~z
Shopping-Time
0.30, 0.20, 0.30, 0.20,
0.06 0.123 0.167 0.238
0.06 0.123 0.167 0.238
0.06 0.123 0.167 0.238
0.0054 0.0056 0.0061 0.0063
0.0053 0.0055 0.0060 0.0062
0.0050 0.0052 0.0057 0.0059
g
6.524 5.677 6.230 5.445
6.558 5.706 6.260 5.471
6.662 5.795 6.354 5.552
h
0.5775 0.6092 0.5449 0.5773
0.5778 0.6095 0.5452 0.5776
0.5788 0.6104 0.5461 0.5784
0
Balanced Growth Paths of Alternative Tax Regimes
0.6290 0.5986 0.6085 0.5782
0.6289 0.5985 0.6085 0.5781
0.6287 0.5982 0.6082 0.5745
C/Y
0.0106 0.0109 0.0116 0.0119
0.0105 0.0107 0.0114 0.0117
0.0100 0.0102 0.0110 0.0112
r
0.4658 0.4640 0.4821 0.4803
0.4601 0.4583 0.4764 0.4746
0.4434 0.4416 0.4594 0.4576
In + lK
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
0.0036 0.0036 0.0035 0.0035
z
0.9275 0.8753 0.8424 0.7941
0.2431 0.2296 0.2204 0.2079
v
NOTES: g is the (quarterly) growth rate. The approximation to the CIA model uses ~ = 1000. The approximation to the non-monetary model uses = 10 -s. Income velocity of money equals v. (Y/C).Values for ~c differ between the models only at the fourth decimal point.
= = = =
zk zk zk zk
Tax R e g i m e
T A B L E 2.
Jean-Franfois Wen and David R. F. Love T A B L E 3.
PercentageWelfare and Revenue Changesfrom Tax Reforms
Tax Regime
Consumption Gain
Income Gain
PV Tax Revenue Change
3.43 6.46 10.03
2.16 4.06 6.31
2.94 -4.13 - 1.38
3.48 6.04 9.65
2.19 3.80 6.07
2.93 -4.14 -1.41
3.50 5.89 9.52
2.20 3.71 5.99
2.93 -4.15 - 1.41
Shopping-Time xk = 0.20, z ' = 0.25, ~ = 0.123 ~k = 0.30, rt = 0.15, xc = 0.167 zk = 0.20, zl = 0.15, ~c = 0.238
Cash-in-Advance xk = 0.20, xt = 0.25, z ~ = 0.123 ~k = 0.30, zl = 0.15, ~ = 0.167 = 0.20, rl = 0.15, z C = 0.238
Non-Monetary ~k = 0.20, "d = 0.25, xc = 0.123 zk = 0.30, z l = 0.15, ~c = 0.167 "ck = 0.20, ~l = 0.15, ~ = 0.238
time model, the reduction in zl increases the opportunity cost of time, inducing agents to hold more real money balances relative to consumption, which reduces time-costs by nearly 3% in the ease of the pure labor income tax cut, even though consumption is rising due to more labor supply. 15 Turning to Table 3, the welfare rankings of the tax reforms are consistent across the models and agree with the previous findings of Devereux and Love (1994) on the relatively high welfare gains to be achieved from reducing the labor income tax rate. The highlights in this table are that the CIA model generates welfare results that are quantitatively very similar to the non-monetary model, while the shopping-time model suggests that the gains from reducing the labor income tax rate (by 10 percentage points) are understated in the non-monetary model by about half a percent in terms of consumption. The source of this extra welfare gain in the shopping-time model is the permanent decrease in transactions time permitted by the reduction in the velocity of money. This adjustment channel is unavailable in a non-monetary framework, or in the CIA model where velocity is fixed. As a bottom hne, the simulations with the shopping-time model sug15Using the same data as in footnote (13), the reduction in transactions tame under a labor income tax cut amounts to a benefit of 0.0143 percent o£ GDP, or nearly $1 billion. 498
Evaluating Tax Reforms (Deviations from Base Case) 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
i
0
5
10
15
20
Figure 1. Interest Tax Cut: Growth
gest that reducing the general income tax rate by 10 percentage points, with revenue compensation achieved through a higher consumption tax rate, would provide a welfare gain of 10% in terms of consumption, or over 6% in terms of GDP. These figures slightly overstate the gains, since tax revenues would decrease during the transition to the new balanced growth path. The general income tax reform would also be associated with a 0.09% increase in the quarterly growth rate of GDP.
Transitional Dynamics Figures 1-8 depict the transition paths (expressed as deviations from the base case) for key variables following the tax reforms. The figures reveal important differences in the dynamics associated with reductions in the two factor taxes. They also show that the differences in the welfare effects across the three types of models are attributable to "level effects," rather than growth rate effects. Figures 1-2 depict the growth rate. Both labor and capital income tax cuts generate large short-run increases in growth, as physical capital is accumulated. However, the growth rate response is nearly identical in the monetary and non-monetary models. It is apparent from Figures 3-4, which plot the value of discounted utility, ~tu(Ct, L t ) , for each period, that the transitional dynamics resulting from the capital income tax cut generates welfare gains in every period, but 499
(Deviations ~ o m B a s e Case) ,
0.016
i
0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
5
Shopping Time Model
I
I
10
15
Cash in Advance . . . . . . Model
20 Non-Monetary Model
Figure 2. Wage Tax Cut: Growth
(Deviations from Base Case) , ~
0.06
,
0.05 0.04 0.03 0.02 0.01 0
0
5
10
Figure 3. Interest Tax Cut: Discounted Utility
500
15
2O
[
1
Deviations from Base Case) 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 0 __
5
Shopping Time Model
10 . . . . . . Cash in Advance Model
15 ~
20 Non-Monetary Model
I
I
Figure 4. Wage Tax Cut: Discounted Utility
(Deviations from Base Case) 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0
5
10
15
20
Figure 5. Interest Tax Cut: Leisure
501
(Deviations ~ o m B a s e Case) I
~
I
I 5
I 10
15
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 -0.018
--
0
Shopping Time Model .....
Cash in Advance " . . . . . Model
20
Non-Monetary ~Model
Figure 6. Wage Tax Cut: Leisure
(Deviations from Base Case)
0
I
I
-0.0002 -0.0004 -0.0006 -0.0008 -0.001 [
-0.0012 0
5
.....
I
I
10
15
Figure 7. Interest Tax Cut: Consumption 502
20
[
Evaluating Tax Reforms (Deviations from Base Case)
0
I
I
I
5
10
15
-0.0002 -0.0004 -0.0006 -0.0008 -0.001 -0.0012 -0.0014 0 _ _ S h o p p i n g Time Model
. . . . . . Cash in Advance Model
20
~Non-Monetary Model
[
Figure 8. Wage Tax Cut: Consumption
that there are substantial losses initially from a cut in the labor income tax. 16 However, the comparatively higher long-run economic growth rate associated with the labor income tax cut more than offsets the transitory losses, so that the labor income tax reform generates the greatest total improvement in welfare. Also note that there are differences in the adjustment paths for discounted utihty across the three types of models. The dynamics of leisure and consumption, the ultimate sources of welfare gains, are given in Figures 5-8. The paths of leisure and consumption in the shopping-time model lay above the corresponding paths in the non-monetary and CIA models, especially in the case of the labor income tax reform.
4, Conclusions Using a calibrated endogenous growth monetary model, where money serves to reduce the time-costs of transacting, we find that a ten percentage point reduction in the income tax rate, with revenues replaced by consump16In fact, the path of discounted utility lies below the base case balanced growth path in each of the periods depicted in Figure 4. Of course, eventuallythe welfare change turns positive.
503
Jean-Francois Wen and David R. F. Love tion taxes, yields a welfare gain of about 10% of consumption, and a 0.36 percentage point increase in the annual growth rate. The parametric form of the transactions technology used in this paper allows for a comparison of effects of tax reforms across broad classes of macroeconomic models: time-cost, cash-in-advance, and non-monetary. We find that the growth effects of taxation are almost unaffected by monetary considerations relative to the traditional non-monetary analysis. However, the benefit from reducing the labor income tax is (slightly) understated in both the non-monetary and cash-in-advance models, compared to the shopping-time specification. This extra welfare gain in the shopping-time model arises because of a decrease in transactions costs associated with a fall in the velocity of money. On the whole, we conclude that the policy evaluations stemming from non-monetary tax models are robust to the inclusion of a role for money at moderate levels of inflation. However, the shopping-time specification could be modified to assume that investment in capital also involves a time-cost. Stockman (1981) shows that the superneutrality of money depends on whether a cash-in-advance constraint applies to consumption or investment. Thus the differences between the evaluation of tax policies conducted in monetary and non-monetary settings could be more significant in that case. Received: October 1996 Final version: June 1997
References Auerbach, Alan J., and Laurence J. Kotlikoff. Dynamic Fiscal Policy. Cambridge: Cambridge University Press, 1987. Braun, Anton R. "Another Attempt to Quantify the Benefits of Reducing Inflation." Federal Reserve Bank of Minneapolis Quarterly Review Fall (1994): 17-25. Chamley, Christophe. "The Welfare Cost of Capital Income Taxation in a Growing Economy." Journal of Political Economy 89 (1981): 468-95. Cooley, Thomas, and Gary Hansen. "The Welfare Costs of Moderate Inflations." Journal of Money, Credit, and Banking 23 (1991): 483-503. • "Tax Distortions in a Neoclassical Monetary Economy." Journal of Economic Theory 58 (1992): 290-316. Den Haan, Wouter J. "The Optimal Inflation Path in a Sidranski-Type Model with Uncertainty." Journal of Monetary Economics 25 (1990): 389-409. Devereux, Michael B., and David R. F. Love. "The Effects of Factor Tax-
504
Evaluating Tax Reforms ation in a Two-Sector Model of Endogenous Growth." CanadianJournal of Economics 28 (1994): 509-335. --. "The Dynamic Effects of Government Spending Policies in a TwoSector Endogenous Growth Model."Journalof Money, Credit, and Banking 27 (1995): 167-82. Dotsey, Michael, and Peter Ireland. "The Welfare Costs of Inflation in General Equilibrium." Journal of Monetary Economics 37 (1996): 29-47. Dowd, Kevin. "The Value of Time and the Transactions Demand for Money." Journal of Money, Credit, and Banking 22 (1990): 51-64. Gillman, Max. "The Welfare Cost of Inflation in a Cash-in-Advance Economy with Costly Credit." Journal of Monetary Economics 31 (1993): 97115. Gomme, Paul. "Money and Growth Revisited: Measuring the Costs of Inflation in an Endogenous Growth Model." Journal of Monetary Economics 32 (1993): 51-78. Guidotti, Pablo E., and Carlos A. V6gh. "The Optimal Inflation Tax When Money Reduces Transactions Costs: A Reconsideration."Journal of Monetary Economics 31 (1993): 189-205. Hafer, R. W., and Dennis W. Jansen. "The Demand for Money in the U.S.: Evidence from Cointegration Tests."Journal of Money, Credit, and Banking 23 (1991): 155-68. Hetzel, Robert L., and Yash P. Mehra. "The Behavior of Money Demand in the 1980's." Journal of Money, Credit, and Banking 21 (1989): 45563. Judd, Kenneth L. "The Welfare Cost of Factor Taxation in a Perfect Foresight Model." Journal of Political Economy 95 (1987): 675-709. Kimborough, Kent P. "Inflation, Employment, and Welfare in the Presence of Transactions Costs." Journal of Money, Credit, and Banking 18 (1986): 129-39. King, Robert G., and Sergio T. Rebelo. "Public Policy and Economic Growth: Developing Neoclassical Implications."Journalof PoliticalEconomy 98 (1990): $126-50. Lacker, Jeffrey M., and Stacey L. Schreft. "Money and Credit as a Means of Payment." Journal of Monetary Economics 38 (1996): 3-23. Love, David R. F. "The Transactions Demand for Money and the Value of Time: Some Empirical Evidence." Brock University discussion paper no. 1995-02. Love, David R. F., and Jean-Frangois Wen. "Inflation, Welfare, and the Time-Costs of Transacting." Brock University discussion paper no. 199505. Lucas, Robert E. "Equilibrium in a Pure Currency Economy." In Models of
505
Jean-Franfois Wen and David R. F. Love Monetary Economies, edited by J. H. Karaken and N. Wallace. Minneapolis Minn: Federal Reserve Bank of Minneapolis, 1980. . "Supply-Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-315. --. "On the Welfare Cost of Inflation." University of Chicago, Department of Economics, 1993. Mimeo. Mankiw, Gregory N., and Lawrence H. Summers. "Money Demand and the Effects of Fiscal Policies." Journal of Money, Credit, and Banking 18 (1986): 415-29. Marquis, Milton H., and Kevin L. Reffett. "New Technology Spillovers into the Payment System." The Economic Journal (September 1994): 112338. McCallum, Bennett T. Monetary Policy: Theory and Practice. New York: Macmillan Publishing Company, 1989. Mehra, Yash P. "The Stability of the M2 Demand Function: Evidence from an Error-Correction Model." Journal of Money, Credit, and Banking 25 (1993): 455-60. Mendoza, Enrique G., Assif Razin, and Linda L. Tesar. "Effective Tax Rates in Macroeconomics, Cross-Country Estimates of Tax Rates on Factor Incomes and Consumption." Journal of Monetary Economics 34 (1994): 297-323. Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing (C + + version). Cambridge: Cambridge University Press, 1992. Schreft, Stacey L. "Welfare-Improving Credit Controls." Journal of Monetary Economics 30 (1992): 57-72. Smith, Todd R. "Money, Taxes, and Endogenous Growth."Journal of Macroeconomics 18 (1996): 449-62. Stockman, Alan C. "Anticipated Inflation and the Capital Stock in a CashIn-Advance Economy." Journal of Monetary Economics 8 (1981): 38793. Summers, Lawrence. "Capital Taxation and Accumulation in a Life-Cycle Growth Model." American Economic Review 71 (1981): 533-44. Wang, P., and C. K. Yip. "Real Effects of Money and Welfare Costs of Inflation in an Endogenously Growing Economy with Transactions Costs." Research paper 9311, Research Department, Federal Reserve Bank of Dallas.
Appendix Derivatives with Respect to Taxes Proof That d(l~ + dlK)/d~k < 0 and d(l~ + l~)/d~l) < 0 (and therefore that dg/dxk < 0 and dg/dxz < 0) when 6~ = 5K = ~ < 1: 506
Evaluating Tax Reforms W r i t e (17) as l H + lK = W A w h e r e , ~7 A = FI(~, 0, .~k, .el, .cc)
(1 - c~)e --=
[-
(1
a)e + (1
-
e)(1
-
-
ar{(1 - xk)] (1 + ~ ) ;
r{)[1
(1
-
T = F2(v, "if) -- 1 - (1 + (1 - e)~)qb(v(1 + xc))~ ;
13(1 - 7)(g + 5) r{ = Fz(g) =- (1 + g)1-~(1-~) _ it(1 _ 8 ) , 0 = F4(e, ~ , ~)---
(19)
7) 0 /j (1-- zz)
-
~(1 - ~)(1 - ~?)(1 - ~)
( 1 - a)7-~(1- ~) + a(1 - 7)(1- ~ ) ( 1 - ~k),
(20)
(21)
(22)
v = F~((A" W), ~, g, 0, x ~, ~1, .f)
I( ---
or(l--_ "gk)~.l{(1--~)(ZH-l-iK)(~t(1-b 1
0(1 - 3') ] \
g)~(a--1) __ ~))]1](~--1)
(1 - :Z)l~b(1 + "f)~
(23)
A > 0 since the t e r m 1 - otr{(1 - xk)/(1 - 7)0 = Ct/Yt is positive, and t h e r e f o r e qJ > 0 for positive l a b o r supply. It is easy to show that r{ = 1H/(lH + IK) a n d so 1 - r{ > 0. N o w w e n e e d to s h o w that d(~P • A)/dz k < 0 and d ( q ~. A ) d z z < 0. Totally differentiating lH + 1K = T A yields:
d(W.a) = ~P ((OF1/Or{)dr{ + (OF1/O0) dO + (OFJO~) dx k (+)
(+)
(?)
(-)
(?)
(-)
(+)
+ (OFa/O~) dx 1 + (OF1/O~c) dz ~ )
(-)
(+)
(-)
(+)
+ A ((OFJOv)dv + (OFJz c) dx ~ ) ~-~ ~ ~.~ ~,. ~.~ ;
(+)
(-)
(?)
(-)
(+)
(24)
w h e r e the sign o f the t e r m is p r o v i d e d u n d e r n e a t h each expression. E x c e p t
17The full derivation of expressions (16), (17), and the solutions for g:, 0, and v are provided in Love and Wen (1995). 507
Jean-Francois Wen and David R. F. Love in the cases of the three terms with (?) below them, the signs are self-evident from partially differentiating the functions F1 • • • F5 Suppose for now that dg < O, d¢: < 0 and dz c = 0. Then it is easy to see that dO > 0 (by totally differentiating F4(¢~, zl, zk)) and dv > 0 (by totally differentiating Fs((A" qJ), g:, g, 0, ,ok, rl, xc)). Filling these signs in for (?) in expression (24), one sees that the sign for d(1H + lK) would be negative, which would in turn be consistent with dg < O. Given dg < O, it is clear from (21) that dR < 0 if a~3g > 0. It is straightforward from differentiating ~ = F3(g) to show that the latter inequality holds for r~ -< i as well as for larger values for ¢r within some upper bound. Finally, the growth derivative with respect to the consumption tax is theoretically ambiguous because of one term in the total derivative of v; that is, 3v/az ~ < 0 (see Equation [23]). In fact, substituting v = F~(.) into the expression z = qb(v(1 + zc))~ reveals that transactions time couldfaU as z~ rises, if i < ~ < 2. In that case, labor supply would tend to increase, raising the marginal product of capital, investment, and growth. This positive growth effect resulting from an increase in z~ does not arise in our simulations, as we set ~ = 8.
508
DAVID R. F. LOVE Brock University St. Catharines, Ontario, Canada
Evaluating Tax Reforms in a Monetary Economy Hypothetical revenue-neutral tax reforms are conducted in a calibrated endogenous growth model in which money serves to economize on the time-costs of transacting. The modelincludes the cash-in-advance (CIA) and non-monetary frameworks as special cases of the parameterization. The results from our shopping-time model suggest that both the CIA and non-monetary models may somewhat underestimate the welfare benefits of lowering the income tax, while the growth effects of the tax reforms are almost the same across the models. We also examine the transitional dynamics resulting from the tax reforms.
1. Introduction Although the dynamic effects of alternative tax structures have been extensively studied, the conventional policy evaluations were arrived at using models that abstracted from money and inflation. (See Summers 1981; Chamley 1981; Judd 1987; Auerbach and Kotlikoff 1987; King and Rebelo 1990; Lucas 1990; and Devereux and Love 1994, 1995; for example. An exception is Cooley and Hansen 1992, who use a neo-classical monetary economy.) Including a role for money may be important for assessing tax structures because of the interaction between real and nominal variables. Smith (1996) augmented a one-sector version of an endogenous growth model with a cash-in-advance (CIA) motive for holding money, and showed analytically that there can arise important qualitative differences in the effects of taxation in monetary and non-monetary models. The direct growth effects of a change in tax policy can alter the inflation rate, for instance, which would in turn change the economic growth rate because of a portfolio substitution by households between transactions balances and physical capital. However, no systematic quantitative study has yet been undertaken on the welfare effects of alternative tax structures in a framework of money and growth. Journal of Macroeconomics, Summer 1998, Vol. 20, No. 3, pp. 487~508 Copyright © 1998 by Louisiana State University Press 0164-0704/98/$1.50
487
Jean-Franfois Wen and David R. F. Love In this paper, money is incorporated through a so-called shoppingtime technology into a two-sector endogenous growth model similar to King and Rebelo (1990) but with an elastic labor supply. We calculate the growth and welfare effects of hypothetical tax reforms based on exact numerical solutions of both the balanced growth paths and the transitional dynamics of the calibrated model. We then compare our results with a non-monetary version of the model, and also provide analytical derivations on the growth effects of taxation. The assumption that money serves to economize on the time-costs of transacting is sufficiently general as to include both the CIA constraint and a non-monetary economy as special cases of the model. Thus we can determine not only whether integrating money into a growth model is quantitatively significant for tax analysis, but also what differences arise between modeling the transactions demand for money as a CIA constraint or a timecost. One of the interesting features of the shopping-time approach (see McCallum 1989; Kimborough 1986; Guidotti and V6gh 1993; Lucas 1993; and Braun 1994, for example) is that there are always positive transactions costs associated with consumption, contrary to the CIA approach, in which the transactions costs associated ~vith cash goods are necessarily zero in equilibrium. Recent empirical support for the shopping-time approach is provided in Dowd (1990) and Love (1995). The trade-off at the margin between holding money for transacting or incurring shopping-time is closely related to the idea that resource-costly credit may be used as a means of payment, instead of cash. Transacting with credit may involve a loss of time, for instance, if the consumer's credit worthiness must be established prior to an exchange (see Lucas 1980 and Schreft 1992). Recent papers by Gfllman (1993), Marquis and Reffet (1994), Dotsey and Ireland (1996), and Lacker and Schreft (1996) find that endogenizing the cash-credit mix is important for assessing the welfare effects of the inflation tax. 1 The resource costs arising from fiscal tax distortions of the cashcredit margin have not previously been studied, but should yield results similar to our shopping-time model.2 The paper is organized as follows. The model is described in the next section. Then the competitive equilibrium is solved for and analytical results are provided on the growth effects of taxation. Numerical solutions for the 1Wang and Yip (I993) analyze the cost of inflation in a model where transactions time reduces the time available for human capital accumulation. Unlike the papers cited above, they include the welfare effects of the transition to a balanced growth path, as we do. 2In contrast, in the tax analysis of Cooley and Hansen (1992), credit goods are assumed to not entail a resource cost. However, excess burden can occur because the consumer's choice between cash and credit goods is distorted by the inflation tax. The authors find that taxing capital income is the least efficient tax policy.
488
Evaluating Tax Reforms
shopping-time, cash-in-advance, and non-monetary models are shown in the third section, followed by our conclusions.
2. The Model
Description The endogenous growth specification employed here is the same as in Gomme (1993) and Devereux and Love (1994, 1995), which are extensions of the model of King and Rebelo (1990) to the case of an elastic labor supply. A utility-maximizing representative household is assumed to have additive preferences that are isoelastic over consumption and leisure according to o~
U = ~
(i)
•tu(Ct, L t) ;
t=O
1 u(Ct , L t ) - - 1-~
, 1 - e)l - a , (CtLt
u(C t , L t) = ln(C;L~-~),
eE(0,1),
0 ~ 1 ;
a = 1.
(2) (3)
0 < 13< 1 is the household's subjective rate of time preference; Ct is periodt consumption and L t is leisure. The time available for leisure is subject to the constraint (4)
L t = 1 - IKt -- 1Ht -- Zt,
where lze represents hours of work supplied to the final goods sector, lnt gives the hours devoted to human capital formation, and zt is the time spent transacting. The time spent transacting is described by the following technology: zt = ~b((1 + xOPtCJMt) ~=-d~((1 + ~Ovt) ~,
qb->0,
~>1,
(5)
where Pt is the money price of consumption goods, M t are nominal money balances at the start of the period, ~c is a tax on consumption goods, and vt is the velocity of money in terms of consumption. Thus the time spent transacting is increasing in consumption expenditures gross-of-tax, 3 and deereasaThis is the standard assumptionin the literature; see Guidotti and V6gh (1993),for example. 489
Jean-Franfois W e n and David R. F. Love
ing in real money balances. ~ > 1 implies that using money for transacting has diminishing returns, while qb is a scaling parameter. Note that setting ~b to zero collapses the model to a non-monetary version (velocity goes to infinity), while in the limit as ~ approaches infinity the transactions function is identical to a cash-in-advance constraint (velocity goes to one). Guidotti and V6gh (1993) show that assuming that transactions costs are homogeneous of degree zero in consumption and real money balances is consistent with well-accepted mieroeconomie foundations, as in, for example, the Baumol-Tobin paradigm. Since our model allows for continuous growth of consumption expenditures and real balances, homogeneity of degree zero is, in any ease, required if transactions time is to remain stationary in the long run. The period-t expenditures of the household must satisfy the following budget constraint (1 + ~c~)PtCt = (1 - "~l)WtHtl~ + (1 - "ck)Rt0tKt - Pt(Kt+l - (1 - 5~)K~) -
(Mt+l - Ms) + DEt + Tt,
(6)
where zz and zk and Tt are the labor income tax rate, the capital income tax rate, and a period-t lump-sum nominal transfer from the government, respectively. Wt is the nominal wage paid to an efficiency unit of labor as determined by H t, the stock of human capital; Rt is the nominal rental rate on physical capital. Kt is the stock of physical capital, 5~ is its depreciation rate, and 0t is the fraction of that stock that is rented to firms operating in the final-goods sector. Finally, DE t represents the nominal transactions balances costlessly transferred to the household through the deposit expansion mechanism, as further discussed below. The household produces and accumulates human capital according to Ht+ 1
-~
AH((1 -- Ot)Kt)~(1HtHt)1-~ + (1 - 8H)Ht,
(7)
where 0 < 7 < 1, A H > O, and 5 H is the depreciation rate on human capital. Output in the final-goods sector, Yt, is produced by firms under perfect competition using a technology of the form Y~ = A(Ot~)'~(l~Ht) 1-`~ '
(s)
where 0 < ot < -
1, A > 0. For profit maximization, W t = Pt(1 ot)A(OtKt)~(lr, tHt)-~ and Rt = PtmA(OtKt)C~-l(lmHt)l-% Market equilib-
rium in the goods market requires that, c,
490
+ K,+~ -
(1 -
~K)K~ = r ~ .
(9)
Evaluating Tax Reforms
The government in this economy sets the tax rates, ~c, ~l and ~, and determines the rate of expansion of the monetary base, while also setting the reserve requirements relating the monetary base to the supply of transactions balances according to the relationship Mt = (1/rr)MBt,
rr -< 1,
(10)
where MBt is the monetary base and rr is the deposit reserve ratio. This use of a deposit expansion mechanism in our model is intended only as an accounting device, allowing us to distinguish between the monetary base, which is relevant for measuring the seignorage revenue that enters the government's budget constraint, and the (potentially) broader monetary aggregate that serves to reduce the time-costs of transacting. 4 The monetary base is assumed to evolve according to: MBt+ 1 = ~tMBt,
(11)
where ~t is the exogenous rate of monetary expansion. Money market clearing requires that the demand for transactions balances be equal to the righthand side of (10). Both tax and seignorage revenues are assumed to be lump-sum redistributed to the household each period so that a balanced government budget is maintained: T t = ~:cPtCt + z~HtI~ + ?RtOtKt + (p - 1)MBt.
(12)
Finally, we suppose that the additions to transactions balances in excess of additions to the monetary base are also costlessly transferred to households in each period through a deposit expansion mechanism (the "money multiplier"). Thus, DEt = (Mt+l - Mr) - (MBt+I - MBt) = (1 - rr)(~t - 1)Mr.
(13)
Competitive Equilibrium A competitive equilibrium for this economy consists of the sequences {Ct, l~, lut, zt, Or, Kt, nt, MBt, Mr, P~, Wt, Rt, Tt, DEt}t%=0 satisfying: (i) household utility maximization subject to the conditions (2)-(7), the initial values for the state variables, and standard boundary conditions; (ii) profit
4Braun (1994) associatesonlythe monetarybase with transactions, whileCooleyand Hansen (1991) assume that only a fraction of M1 is relevant for transactions, and use this base also to c~dculateseignorage. 491
Jean-Franfois Wen and David R. F. Love
maximization by firms; (iii) government budget constraints; and (iv) goods and money market clearing. The first-order conditions for the household's optimization problem with respect to the variables C t, lse, lw, zt, Ot, Mr+ 1, K¢ + 1, and H t + 1, together with (5), (7), (9), and (10), implicitly define solutions for the variables Ct, Ira, lilt, zt, Or, Kt, mr, and ~tt, where m~ = Mt/P t and 1 + r~t = Pt+l/Pt are real transactions balances and gross inflation. Real wages, wt = W/Pt, and real returns, r t = Rt/Pt, are given residually. From the optimal consumption-leisure choice, one can calculate the effective tax rate ~ on an hour of labor income wtHt, which plays a critical role in the distinction between the shopping-time, cash-in-advance, and nonmonetary models. Thus, (1 - rt)/(1 + zc) (1
-
~) = 1 + ~(1 -
~)w,H~z/(1 + ~c)c,
(14)
The second term in the denominator of the right-hand side of (14) is the value of the household's time spent shopping for a marginal unit of consumption. If the parameter qb were set equal to zero, then zt = 0 and this term drops out, yielding the well-known expression for the leisureconsumption tax wedge in non-monetary models. In the extreme ease of a CIA constraint (lim~ --* ~), zt approaches zero "faster" than the rate of increase in {, so that the extra term vanishes asymptotically. Thus, the shopping-time model is distinct in that the consumption-leisure margin is endogenous. Another important feature of the model is the asset equilibrium condition, stating that transactions balances should be accumulated so as to equate the returns on money and capital. That is, (1 + (1 - "d)wt+lHt+l~z~+l/mt+l)/(1 + ~t+z) = (1 - ~K + (1 - ~k)rt+l) .
(15)
This arbitrage-based channel for the interaction of nominal and real magnitudes is the focus of the comparison between non-monetary and CIA models in the work of Smith (1996) and Gomme (1993). In the time-cost model, an additional unit of transactions balances acquired at time t provides a return at t + i of a unit of purchasing power plus the value of the marginal reduction in transactions costs, both adjusted for inflation. Note that (15) implicitly defines money demand in this economy as a function of the netof-tax wage, consumption, and the nominal interest rate. Balanced Growth Analysis It is useful to express the equilibrium in terms of a balanced-growth representation by defining the following variables: ht - - H t / K t , k t ~ Kt+ 1/Kt,
492
Evaluating Tax Reforms and (1 + gt) - Ct+JC~. Along a balanced-growth path, 1 + gt = kt = 1 + g gives the constant rate of growth of capital stocks, consumption, and real balanees; he, vt, lut, ln, and Ot are also constant. It is straightforward to shove5 that under balaneed-growth the stationary representation can be reduced to the following two equations in growth rates and total labor supply. (1 + g)(1-~(i-~))
=
~(~(1H + /K)n((1 _
,~/)(1-a)(1
_
.ck)c~)l-~l ..}_ 1 - - ~ K ) ,
(16)
where f~ = aA((1 - 7)AuA~4)n((1 - e~)7/a(1 - 7)) TM > 0, 1] = (1 - a)/ (1- a + 7)~(0,1),andle(1- ~)>0;
1, + IK = (1 - a)~(1 - (1 + (1 - ~){)qb(v(1 + ~))~) (1 -
a)e
+
(1 -
~)(1 -
~)(1
-
ag:(1 -
zk)/((1 -
3')0))(1 + f ) / ( 1
-
~z),
(17)
where O, v, and the composite term g:, are functions only of g, In + IK, and parameters, as given in the appendix. It is obvious from differentiating (16) with respect to each of the tax rates (holding the other tax rates constant) that dg/dx l, dg/dx k and dg/d~ ~ are all negative if the corresponding derivatives of the total labor supply with respect to the tax rates are negative. Devereux and Love (1995) show that these labor supply derivatives are indeed unambiguously negative in the case of the non-monetary model with 8n = fiK and a sufficiently low value for ~,6 but here the result is more complicated because of the velocity term v. In the appendix we show that in the monetary model the results do still hold unambiguously for the labor and capital income taxes; however, the growth effect of the consumption tax now becomes ambiguous, though it is negative for realistic parameter values, r Of course, the growth effects of revenueneutral tax reforms will depend on the relative magnitudes of these derivatives and on the sizes of the different tax bases.
3. Numerical Results
Calibration There are four categories of parameters requiring specification: (i) production function parameters, A, ~, ~K, An, 7, 8n; (ii) preference parameters, 5See our working paper for the fun derivation. 6The ambiguity of the theoretical result for low values of the intertemporal elasticity of substitution, which varies inverselywith ¢y,is similar to Smith's (1996) finding that the growth effect of a tax depends qualitativelyon the magnitude of ¢y. VTheambiguity of the growth effect of the consumption tax stems from the fact that, unlike in the income tax cases, a partial effect of the consumption tax is to reduce the velocity of money, which in turn raises the total supply of labor. 493
Jean-Franfois Wen and David R. F. Love
TABLE 1. e~ = 7 = [3 = qb =
0.360 0.310 0.997 183.0
Summary of Parameter Values
8K = 0.024 8H ---- 0.005 (~ = 2.000 ~ = 8.000
A AH e rr
= = = =
0.0705 0.0705 0.395 0.107
[3, e, c; (iii) transactions-cost function parameters, qb, ~; and (iv) government policy parameters, xk, xt, zc, g, and rr. Time periods are assumed to correspond to quarterly observations. The specification of the production function and preference parameters is standard, replicating long-run averages of post-World War II U.S. maeroeconomic data. s Table i summarizes the parameter values of the model. Regarding the calibration of the monetary aspects of the model, it is easy to show that Equation (15) generates the following money-demand relationship: mt = fI(NRt)-I/(I+~)((1 - "cl)wtHt)l/(l+~)((1 + "cc)Ct)~J
(18)
where NRt = (1 + gt)(1 + (1 - "~k)rt - 8K) - 1 is the net nominal interest rate, and f~ = (+{)v(l+g). ~ uniquely determines money demand elasticities in our model, providing an observable gauge for calibration of this transactions-cost parameter. Given {, the scale parameter qb can be chosen to pin down either time spent transacting, or a measure of velocity. Since velocity estimates are much more reliable, we calibrate qb according to this criterion and leave time spent transacting to be determined residually. We choose to calibrate using M2 as our measure of transactions balances for the following reasons. First, the money demand Equation (18) describes a long-run relationship, and recent empirical evidence (e.g., Haler and Jansen 1991; Mehra 1993) suggests that the existence of an equilibrium relationship for quarterly U.S. M1 is subject to doubt, but not so for M2. Second, it is well known that U.S. M1 velocities have displayed a significant trend in the past, largely attributable to new technologies, while M2 velocities are stationary. Hetzel and Mehra (1989) have suggested that as a result, M2 would be a better policy predictor. Finally, Mankiw and Summers (1986) show that 96% of M2 holdings can be related to consumption, supporting SHowever, unlike in the calibration of King and Rebelo (1990) and Gomme (1993), we do not assume identical technologies in the human-capital and final-goods sectors. In addition to being unlikely in reality, assuming identical technologies across sectors implies unrealistically fast adjustment transitions of just a single period, as well as implying that more time is allocated to human-capital production in the economy than to final goods production. Instead, our calibration assumes that human-capital production is relatively more labor-intensive.
494
Evaluating Tax Reforms the intuition that a large portion of savings balances are accumulated for discretionary consumption expenditures on things like vacations and large durables. Hafer and Jansen (1991) and Mehra (1993) report long-rnn interest rate elasticities for U.S. quarterly M2 demand in the range of ( - 0 . 0 3 , - 0.19). Thus we choose the mid-point for the interest elasticity: - 1/(1 + ~) = -0.11; note that this implies a long-run consumption elasticity of ~/(1 + ~) = 0.88, which is comparable to the income elasticity of 0.89 reported by Haler and Jansen since consumption and income are proportional in our model in the long run. Quarterly M2 income velocity (where the price level includes consumption taxes) for the post World-War II U.S. has averaged 0.41. A choice of qb = 183 yields this value for our benchmark economy. Finally, the government policy parameters were chosen as follows. In accordance with the average quarterly rate of growth of the U.S. monetary base, we set g = 1.017, yielding an annual rate of inflation of 4.9%, given the benchmark real economic growth rate of 2% per year. The reserve ratio is set equal to the average ratio of the monetary base to M2; r r = 0.107. The benchmark tax rates ~w = 0.25 and ~c = 0.06 are based on Mendoza, Razin and Tesar (1994), and @ = 0.30 was chosen residually to yield the historical average ratio of total government spending to output of 0.31.
Balanced Growth Solutions. The benchmark case (~ = 0.30, rz = 0.25, xc = 0.06) is contrasted with three hypothetical revenue-neutral (unanticipated and permanent) tax reforms: (i) a lower labor income tax with a higher consumption tax; (ii) a lower capital income tax with a higher consumption tax; (iii) a lower general income tax with a higher consumption tax. In each case, the relevant income tax rate was reduced by ten percentage points and the consumption tax rate was raised to generate the same ratio of government revenues to GDP along the balanced growth path. Note that this means that the present value of total tax revenues may differ across tax regimes because of adjustments along the transition paths. 9 Thus to allow for a complete assessment of the alternative tax structures, changes in the present value of tax revenues are reported together with the changes in welfare. Although our focus is on quantifying the welfare and growth effects of 9Alternatively,conductingthe experimentsby holdingthe present value of tax revenuesconstant leads to differentbalanced growth ratios of tax revenue-to-GDPacross the tax regimes. Since we do not have a theory of the size of governmentin the model,we prefer to maintaina constant GDP-shareof tax revenue,rather than fixingits presentvalue. Our generalconclusions are independentof which definitionof revenue-neutralityis used. 495
J e a n - F r a n f o i s W e n and D a v i d R. F. L o v e
alternative tax regimes in a monetary endogenous growth model based on the "shopping-time" motive for holding money, we also present a parallel set of results for our approximations to the cash-in-advance model and the non-monetary model. This makes it easy to see to what extent and how monetary considerations can affect the results. Table 2 gives the balanced growth solutions for each tax regime. 1° Table 3 presents the welfare changes arising from each tax reform relative to the benchmark. The welfare change is measured as the constant stream of consumption required to eompensate a household for not carrying out the tax reform, n and is expressed as a percentage of the original consumption level. The welfare change is also reported as a percentage of the initial income level. The benchmark case for the shopping-time model predicts that roughly one-third of one percent of total available time 12 is spent transacting. This is one-tenth of the figure employed in D e n Haan (1990). W h e n evaluated at the effective wage rate, these transactions costs are equivalent to 1% of final-goods output, as compared to the 1% of a narrow measure of output used in Braun (1994). la Seignorage amounts to roughly 1.45% of total government revenues or 0.44% of output. 14 The main features of Table 2 are that the changes in economic growth associated with each tax reform are (almost) the same across the three types of models, but that the decline in leisure is comparatively small in the shopping-time model in response to a labor income tax cut. In the shopping1°Balanced growth paths are calculated using a Newton-Raphsonalgorithm on the balanced growth equation system. The solutions for the model's transitional dynamics are calculated by solving an appropriately specified two-point boundary problem derived from the optimality conditions of the model. This method closes the difference equation system that gives the transition paths of the economy by imposing initial values for the capital stocks, and by forcing the model to converge to its balanced growth path by some arbitrarily distant terminal date (usually 75 periods from t = 0). For a detailed and comprehensive explanation of this solution method and the algorithm we used, see Press, et al. (1994), Chapter 17. nLet {C~}~=0and {L/}t=0 be the post-tax-reform levels of consumption and leisure, and {Ct}~°=0 = {C0, (1 + g)Co, (1 + g)aCo.... } and {Lt}~'=0 = {L, L, L ...} the pre-tax-reform balanced growth paths of consumption and leisure, where g is the pre-tax-reform balanced growth rate. Then our measure of welfare change, C, is given implicitlyby E~=01~tu((1 + g)t(C0 + C), L) = E~=o~tu(C~,L~). In percentage terms, the welfare measure is C/C o. lZi.e., discretionarytime of about 16 hours per day. 13Ifwe interpret transactions time as reflecting lost leisure due to trips to the bank, as in Guidotti and V6gh (1993), then our model is generating about 3.5 minutes per day or 21 hours per year spent (per household) on banking activities, which seems reasonable. Evaluatingthis time at the U.S. average hourly wage times the civilian labor force yields one-half of a percent of GDP, using 1994 data from the Bureau of Labor Statistics. 14For a 5% inflation, Cooley and Hansen (1991) report seignorage revenues of 1.43-2.39% of government revenues. Braun (1994) reports seignoragerevenues equal to 0.47% of his narrow measure of output. 496
¢D -q
= = = =
= = = =
zk xk zk zk
~k ~k ~k zk
= = = =
0.25, 0.25, 0.15, 0.15,
z~ r~ zc zc
= = = =
= = = =
0.25, 0.25, 0.15, 0.15,
zc zc ~ z~
0.30, 0.20, 0.30, 0.20,
~I xz ~z ~z
= = = =
0.25, 0.25, 0.15, 0.15,
xc ~c zc zc
Non-Monetary
0.30, zl 0.20, zl 0.30,~1 0.20,~l
= = = =
= = = =
Cash-in-Advance
rl zz zl ~z
Shopping-Time
0.30, 0.20, 0.30, 0.20,
0.06 0.123 0.167 0.238
0.06 0.123 0.167 0.238
0.06 0.123 0.167 0.238
0.0054 0.0056 0.0061 0.0063
0.0053 0.0055 0.0060 0.0062
0.0050 0.0052 0.0057 0.0059
g
6.524 5.677 6.230 5.445
6.558 5.706 6.260 5.471
6.662 5.795 6.354 5.552
h
0.5775 0.6092 0.5449 0.5773
0.5778 0.6095 0.5452 0.5776
0.5788 0.6104 0.5461 0.5784
0
Balanced Growth Paths of Alternative Tax Regimes
0.6290 0.5986 0.6085 0.5782
0.6289 0.5985 0.6085 0.5781
0.6287 0.5982 0.6082 0.5745
C/Y
0.0106 0.0109 0.0116 0.0119
0.0105 0.0107 0.0114 0.0117
0.0100 0.0102 0.0110 0.0112
r
0.4658 0.4640 0.4821 0.4803
0.4601 0.4583 0.4764 0.4746
0.4434 0.4416 0.4594 0.4576
In + lK
0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
0.0036 0.0036 0.0035 0.0035
z
0.9275 0.8753 0.8424 0.7941
0.2431 0.2296 0.2204 0.2079
v
NOTES: g is the (quarterly) growth rate. The approximation to the CIA model uses ~ = 1000. The approximation to the non-monetary model uses = 10 -s. Income velocity of money equals v. (Y/C).Values for ~c differ between the models only at the fourth decimal point.
= = = =
zk zk zk zk
Tax R e g i m e
T A B L E 2.
Jean-Franfois Wen and David R. F. Love T A B L E 3.
PercentageWelfare and Revenue Changesfrom Tax Reforms
Tax Regime
Consumption Gain
Income Gain
PV Tax Revenue Change
3.43 6.46 10.03
2.16 4.06 6.31
2.94 -4.13 - 1.38
3.48 6.04 9.65
2.19 3.80 6.07
2.93 -4.14 -1.41
3.50 5.89 9.52
2.20 3.71 5.99
2.93 -4.15 - 1.41
Shopping-Time xk = 0.20, z ' = 0.25, ~ = 0.123 ~k = 0.30, rt = 0.15, xc = 0.167 zk = 0.20, zl = 0.15, ~c = 0.238
Cash-in-Advance xk = 0.20, xt = 0.25, z ~ = 0.123 ~k = 0.30, zl = 0.15, ~ = 0.167 = 0.20, rl = 0.15, z C = 0.238
Non-Monetary ~k = 0.20, "d = 0.25, xc = 0.123 zk = 0.30, z l = 0.15, ~c = 0.167 "ck = 0.20, ~l = 0.15, ~ = 0.238
time model, the reduction in zl increases the opportunity cost of time, inducing agents to hold more real money balances relative to consumption, which reduces time-costs by nearly 3% in the ease of the pure labor income tax cut, even though consumption is rising due to more labor supply. 15 Turning to Table 3, the welfare rankings of the tax reforms are consistent across the models and agree with the previous findings of Devereux and Love (1994) on the relatively high welfare gains to be achieved from reducing the labor income tax rate. The highlights in this table are that the CIA model generates welfare results that are quantitatively very similar to the non-monetary model, while the shopping-time model suggests that the gains from reducing the labor income tax rate (by 10 percentage points) are understated in the non-monetary model by about half a percent in terms of consumption. The source of this extra welfare gain in the shopping-time model is the permanent decrease in transactions time permitted by the reduction in the velocity of money. This adjustment channel is unavailable in a non-monetary framework, or in the CIA model where velocity is fixed. As a bottom hne, the simulations with the shopping-time model sug15Using the same data as in footnote (13), the reduction in transactions tame under a labor income tax cut amounts to a benefit of 0.0143 percent o£ GDP, or nearly $1 billion. 498
Evaluating Tax Reforms (Deviations from Base Case) 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
i
0
5
10
15
20
Figure 1. Interest Tax Cut: Growth
gest that reducing the general income tax rate by 10 percentage points, with revenue compensation achieved through a higher consumption tax rate, would provide a welfare gain of 10% in terms of consumption, or over 6% in terms of GDP. These figures slightly overstate the gains, since tax revenues would decrease during the transition to the new balanced growth path. The general income tax reform would also be associated with a 0.09% increase in the quarterly growth rate of GDP.
Transitional Dynamics Figures 1-8 depict the transition paths (expressed as deviations from the base case) for key variables following the tax reforms. The figures reveal important differences in the dynamics associated with reductions in the two factor taxes. They also show that the differences in the welfare effects across the three types of models are attributable to "level effects," rather than growth rate effects. Figures 1-2 depict the growth rate. Both labor and capital income tax cuts generate large short-run increases in growth, as physical capital is accumulated. However, the growth rate response is nearly identical in the monetary and non-monetary models. It is apparent from Figures 3-4, which plot the value of discounted utility, ~tu(Ct, L t ) , for each period, that the transitional dynamics resulting from the capital income tax cut generates welfare gains in every period, but 499
(Deviations ~ o m B a s e Case) ,
0.016
i
0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
5
Shopping Time Model
I
I
10
15
Cash in Advance . . . . . . Model
20 Non-Monetary Model
Figure 2. Wage Tax Cut: Growth
(Deviations from Base Case) , ~
0.06
,
0.05 0.04 0.03 0.02 0.01 0
0
5
10
Figure 3. Interest Tax Cut: Discounted Utility
500
15
2O
[
1
Deviations from Base Case) 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 0 __
5
Shopping Time Model
10 . . . . . . Cash in Advance Model
15 ~
20 Non-Monetary Model
I
I
Figure 4. Wage Tax Cut: Discounted Utility
(Deviations from Base Case) 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0
5
10
15
20
Figure 5. Interest Tax Cut: Leisure
501
(Deviations ~ o m B a s e Case) I
~
I
I 5
I 10
15
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 -0.018
--
0
Shopping Time Model .....
Cash in Advance " . . . . . Model
20
Non-Monetary ~Model
Figure 6. Wage Tax Cut: Leisure
(Deviations from Base Case)
0
I
I
-0.0002 -0.0004 -0.0006 -0.0008 -0.001 [
-0.0012 0
5
.....
I
I
10
15
Figure 7. Interest Tax Cut: Consumption 502
20
[
Evaluating Tax Reforms (Deviations from Base Case)
0
I
I
I
5
10
15
-0.0002 -0.0004 -0.0006 -0.0008 -0.001 -0.0012 -0.0014 0 _ _ S h o p p i n g Time Model
. . . . . . Cash in Advance Model
20
~Non-Monetary Model
[
Figure 8. Wage Tax Cut: Consumption
that there are substantial losses initially from a cut in the labor income tax. 16 However, the comparatively higher long-run economic growth rate associated with the labor income tax cut more than offsets the transitory losses, so that the labor income tax reform generates the greatest total improvement in welfare. Also note that there are differences in the adjustment paths for discounted utihty across the three types of models. The dynamics of leisure and consumption, the ultimate sources of welfare gains, are given in Figures 5-8. The paths of leisure and consumption in the shopping-time model lay above the corresponding paths in the non-monetary and CIA models, especially in the case of the labor income tax reform.
4, Conclusions Using a calibrated endogenous growth monetary model, where money serves to reduce the time-costs of transacting, we find that a ten percentage point reduction in the income tax rate, with revenues replaced by consump16In fact, the path of discounted utility lies below the base case balanced growth path in each of the periods depicted in Figure 4. Of course, eventuallythe welfare change turns positive.
503
Jean-Francois Wen and David R. F. Love tion taxes, yields a welfare gain of about 10% of consumption, and a 0.36 percentage point increase in the annual growth rate. The parametric form of the transactions technology used in this paper allows for a comparison of effects of tax reforms across broad classes of macroeconomic models: time-cost, cash-in-advance, and non-monetary. We find that the growth effects of taxation are almost unaffected by monetary considerations relative to the traditional non-monetary analysis. However, the benefit from reducing the labor income tax is (slightly) understated in both the non-monetary and cash-in-advance models, compared to the shopping-time specification. This extra welfare gain in the shopping-time model arises because of a decrease in transactions costs associated with a fall in the velocity of money. On the whole, we conclude that the policy evaluations stemming from non-monetary tax models are robust to the inclusion of a role for money at moderate levels of inflation. However, the shopping-time specification could be modified to assume that investment in capital also involves a time-cost. Stockman (1981) shows that the superneutrality of money depends on whether a cash-in-advance constraint applies to consumption or investment. Thus the differences between the evaluation of tax policies conducted in monetary and non-monetary settings could be more significant in that case. Received: October 1996 Final version: June 1997
References Auerbach, Alan J., and Laurence J. Kotlikoff. Dynamic Fiscal Policy. Cambridge: Cambridge University Press, 1987. Braun, Anton R. "Another Attempt to Quantify the Benefits of Reducing Inflation." Federal Reserve Bank of Minneapolis Quarterly Review Fall (1994): 17-25. Chamley, Christophe. "The Welfare Cost of Capital Income Taxation in a Growing Economy." Journal of Political Economy 89 (1981): 468-95. Cooley, Thomas, and Gary Hansen. "The Welfare Costs of Moderate Inflations." Journal of Money, Credit, and Banking 23 (1991): 483-503. • "Tax Distortions in a Neoclassical Monetary Economy." Journal of Economic Theory 58 (1992): 290-316. Den Haan, Wouter J. "The Optimal Inflation Path in a Sidranski-Type Model with Uncertainty." Journal of Monetary Economics 25 (1990): 389-409. Devereux, Michael B., and David R. F. Love. "The Effects of Factor Tax-
504
Evaluating Tax Reforms ation in a Two-Sector Model of Endogenous Growth." CanadianJournal of Economics 28 (1994): 509-335. --. "The Dynamic Effects of Government Spending Policies in a TwoSector Endogenous Growth Model."Journalof Money, Credit, and Banking 27 (1995): 167-82. Dotsey, Michael, and Peter Ireland. "The Welfare Costs of Inflation in General Equilibrium." Journal of Monetary Economics 37 (1996): 29-47. Dowd, Kevin. "The Value of Time and the Transactions Demand for Money." Journal of Money, Credit, and Banking 22 (1990): 51-64. Gillman, Max. "The Welfare Cost of Inflation in a Cash-in-Advance Economy with Costly Credit." Journal of Monetary Economics 31 (1993): 97115. Gomme, Paul. "Money and Growth Revisited: Measuring the Costs of Inflation in an Endogenous Growth Model." Journal of Monetary Economics 32 (1993): 51-78. Guidotti, Pablo E., and Carlos A. V6gh. "The Optimal Inflation Tax When Money Reduces Transactions Costs: A Reconsideration."Journal of Monetary Economics 31 (1993): 189-205. Hafer, R. W., and Dennis W. Jansen. "The Demand for Money in the U.S.: Evidence from Cointegration Tests."Journal of Money, Credit, and Banking 23 (1991): 155-68. Hetzel, Robert L., and Yash P. Mehra. "The Behavior of Money Demand in the 1980's." Journal of Money, Credit, and Banking 21 (1989): 45563. Judd, Kenneth L. "The Welfare Cost of Factor Taxation in a Perfect Foresight Model." Journal of Political Economy 95 (1987): 675-709. Kimborough, Kent P. "Inflation, Employment, and Welfare in the Presence of Transactions Costs." Journal of Money, Credit, and Banking 18 (1986): 129-39. King, Robert G., and Sergio T. Rebelo. "Public Policy and Economic Growth: Developing Neoclassical Implications."Journalof PoliticalEconomy 98 (1990): $126-50. Lacker, Jeffrey M., and Stacey L. Schreft. "Money and Credit as a Means of Payment." Journal of Monetary Economics 38 (1996): 3-23. Love, David R. F. "The Transactions Demand for Money and the Value of Time: Some Empirical Evidence." Brock University discussion paper no. 1995-02. Love, David R. F., and Jean-Frangois Wen. "Inflation, Welfare, and the Time-Costs of Transacting." Brock University discussion paper no. 199505. Lucas, Robert E. "Equilibrium in a Pure Currency Economy." In Models of
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Jean-Franfois Wen and David R. F. Love Monetary Economies, edited by J. H. Karaken and N. Wallace. Minneapolis Minn: Federal Reserve Bank of Minneapolis, 1980. . "Supply-Side Economics: An Analytical Review." Oxford Economic Papers 42 (1990): 293-315. --. "On the Welfare Cost of Inflation." University of Chicago, Department of Economics, 1993. Mimeo. Mankiw, Gregory N., and Lawrence H. Summers. "Money Demand and the Effects of Fiscal Policies." Journal of Money, Credit, and Banking 18 (1986): 415-29. Marquis, Milton H., and Kevin L. Reffett. "New Technology Spillovers into the Payment System." The Economic Journal (September 1994): 112338. McCallum, Bennett T. Monetary Policy: Theory and Practice. New York: Macmillan Publishing Company, 1989. Mehra, Yash P. "The Stability of the M2 Demand Function: Evidence from an Error-Correction Model." Journal of Money, Credit, and Banking 25 (1993): 455-60. Mendoza, Enrique G., Assif Razin, and Linda L. Tesar. "Effective Tax Rates in Macroeconomics, Cross-Country Estimates of Tax Rates on Factor Incomes and Consumption." Journal of Monetary Economics 34 (1994): 297-323. Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing (C + + version). Cambridge: Cambridge University Press, 1992. Schreft, Stacey L. "Welfare-Improving Credit Controls." Journal of Monetary Economics 30 (1992): 57-72. Smith, Todd R. "Money, Taxes, and Endogenous Growth."Journal of Macroeconomics 18 (1996): 449-62. Stockman, Alan C. "Anticipated Inflation and the Capital Stock in a CashIn-Advance Economy." Journal of Monetary Economics 8 (1981): 38793. Summers, Lawrence. "Capital Taxation and Accumulation in a Life-Cycle Growth Model." American Economic Review 71 (1981): 533-44. Wang, P., and C. K. Yip. "Real Effects of Money and Welfare Costs of Inflation in an Endogenously Growing Economy with Transactions Costs." Research paper 9311, Research Department, Federal Reserve Bank of Dallas.
Appendix Derivatives with Respect to Taxes Proof That d(l~ + dlK)/d~k < 0 and d(l~ + l~)/d~l) < 0 (and therefore that dg/dxk < 0 and dg/dxz < 0) when 6~ = 5K = ~ < 1: 506
Evaluating Tax Reforms W r i t e (17) as l H + lK = W A w h e r e , ~7 A = FI(~, 0, .~k, .el, .cc)
(1 - c~)e --=
[-
(1
a)e + (1
-
e)(1
-
-
ar{(1 - xk)] (1 + ~ ) ;
r{)[1
(1
-
T = F2(v, "if) -- 1 - (1 + (1 - e)~)qb(v(1 + xc))~ ;
13(1 - 7)(g + 5) r{ = Fz(g) =- (1 + g)1-~(1-~) _ it(1 _ 8 ) , 0 = F4(e, ~ , ~)---
(19)
7) 0 /j (1-- zz)
-
~(1 - ~)(1 - ~?)(1 - ~)
( 1 - a)7-~(1- ~) + a(1 - 7)(1- ~ ) ( 1 - ~k),
(20)
(21)
(22)
v = F~((A" W), ~, g, 0, x ~, ~1, .f)
I( ---
or(l--_ "gk)~.l{(1--~)(ZH-l-iK)(~t(1-b 1
0(1 - 3') ] \
g)~(a--1) __ ~))]1](~--1)
(1 - :Z)l~b(1 + "f)~
(23)
A > 0 since the t e r m 1 - otr{(1 - xk)/(1 - 7)0 = Ct/Yt is positive, and t h e r e f o r e qJ > 0 for positive l a b o r supply. It is easy to show that r{ = 1H/(lH + IK) a n d so 1 - r{ > 0. N o w w e n e e d to s h o w that d(~P • A)/dz k < 0 and d ( q ~. A ) d z z < 0. Totally differentiating lH + 1K = T A yields:
d(W.a) = ~P ((OF1/Or{)dr{ + (OF1/O0) dO + (OFJO~) dx k (+)
(+)
(?)
(-)
(?)
(-)
(+)
+ (OFa/O~) dx 1 + (OF1/O~c) dz ~ )
(-)
(+)
(-)
(+)
+ A ((OFJOv)dv + (OFJz c) dx ~ ) ~-~ ~ ~.~ ~,. ~.~ ;
(+)
(-)
(?)
(-)
(+)
(24)
w h e r e the sign o f the t e r m is p r o v i d e d u n d e r n e a t h each expression. E x c e p t
17The full derivation of expressions (16), (17), and the solutions for g:, 0, and v are provided in Love and Wen (1995). 507
Jean-Francois Wen and David R. F. Love in the cases of the three terms with (?) below them, the signs are self-evident from partially differentiating the functions F1 • • • F5 Suppose for now that dg < O, d¢: < 0 and dz c = 0. Then it is easy to see that dO > 0 (by totally differentiating F4(¢~, zl, zk)) and dv > 0 (by totally differentiating Fs((A" qJ), g:, g, 0, ,ok, rl, xc)). Filling these signs in for (?) in expression (24), one sees that the sign for d(1H + lK) would be negative, which would in turn be consistent with dg < O. Given dg < O, it is clear from (21) that dR < 0 if a~3g > 0. It is straightforward from differentiating ~ = F3(g) to show that the latter inequality holds for r~ -< i as well as for larger values for ¢r within some upper bound. Finally, the growth derivative with respect to the consumption tax is theoretically ambiguous because of one term in the total derivative of v; that is, 3v/az ~ < 0 (see Equation [23]). In fact, substituting v = F~(.) into the expression z = qb(v(1 + zc))~ reveals that transactions time couldfaU as z~ rises, if i < ~ < 2. In that case, labor supply would tend to increase, raising the marginal product of capital, investment, and growth. This positive growth effect resulting from an increase in z~ does not arise in our simulations, as we set ~ = 8.
508