Economica (1998) 65, 401–27

Risk, Optimal Government Finance and Monetary Policies in a Growing Economy By EARL L. GRINOLS and STEPHEN J. TURNOVSKY University of Illinois, University of Washington Final version received 7 April 1997. Optimal tax and monetary policies in a stochastic monetary growth model are investigated. Our findings are of three general types. First, both capital income taxes and monetary growth are shown to influence the economy through effective risk-adjusted measures, expressed as a linear function of their respective means and variances. Second, two stochastic neutrality results relating to money and bonds, the two nominal assets in the economy, are identified. Third, optimal policy rules relating to taxes, bond finance and money creation are characterized. An essential component of optimal financial policy is a risk-adjusted balanced budget.

INTRODUCTION The effects of taxes and monetary growth on economic performance have been central issues in macroeconomics for many years. Early discussions focused on the effects of such policies on economic growth and inflation, and the potential trade-offs between them. The choice of the optimal rate of monetary growth, originally discussed by Friedman (1969), has subsequently been treated in an intertemporal optimizing framework; see Drazen (1979); Turnovsky and Brock (1980); Lucus and Stokey (1983); Kimbrough (1986); Abel (1987) and Woodford (1990). A parallel literature analyses the effects of distortionary taxes (see e.g. Feldstein 1978; Chamley 1986; and Lucas 1990), while the possible tradeoffs between the optimal rate of monetary growth and other distortioninducing policies have also been recognized; see e.g. Phelps (1973); Turnovsky and Brock (1980); Chamley (1985); Kimbrough (1986) and Woodford (1990). Although most of the existing literature addressing optimal policy-making employs a deterministic framework, there is a growing preference for the use of stochastic models that recognize the integral role played by financial markets in the risk-sharing process. Indeed, the effect of a specific policy tool may be quite different when its impact on the structure of risk in the economy and the consequences of this change in the stochastic environment on investment decisions, and subsequently on the growth rate, are taken into account. But the analysis of a complete stochastic intertemporal optimizing model is not easy. Consequently, progress has been slow in finding a way to incorporate in a single framework the different features required to analyse optimal monetary policy and fiscal policy satisfactorily, based on rational forward-looking behaviour. This paper addresses optimal tax and monetary policy determination in a stochastic economy consisting of representative private agents and the government. The private agent acts both as a firm, acquiring physical capital to carry out production subject to a stochastic linear technology, and as a consumer, making consumption and portfolio allocation decisions to maximize intertemporal utility. The government chooses its level of expenditure and monetary  The London School of Economics and Political Science 1998

402

ECONOMICA

[AUGUST

growth rate, though both can be controlled only stochastically. Given the tax rates that it also sets, the rate at which it issues new bonds is set residually so as to satisfy its budget constraint. There are thus three assets in the economy: equity claims to private capital, money, and government bonds. There are also three exogenous sources of risk: production risk, stochastic government expenditure and stochastic monetary growth. Financial markets price assets and determine the rates of return on the three assets consistent with these risks and the preferences of the representative agent. The model we develop is a stochastic general equilibrium growth model, in which all exogenous stochastic processes impinging on the economy are assumed to be generated by Brownian motions. The resulting macroeconomic equilibrium simultaneously determines both the means and the variances of the equilibrium growth rate, the inflation rate and other relevant economic variables. The model thus provides a comprehensive framework for assessing the effects of government policies on these variables as well as on economic welfare. Using this framework, we derive the optimal rate of taxation of capital based on maximizing the intertemporal welfare of the representative agent in the economy. The same criterion is employed to determine the optimal rate of monetary growth. Our findings can be categorized into three groups. First, for each of the two main policy variables we consider—capital income taxes and the money growth rate—we show how its effect on key aspects of the economy can be summarized in terms of a risk-adjusted measure, expressed as a linear combination of its mean and the variance of its effects on the economy. In both cases the deterministic and stochastic influences are offsetting.1 Second, we obtain two important policy neutrality results, one relating to monetary policy, the other to debt policy. The equilibrium dichotomizes in a fashion similar to that of the standard Sidrauski (1967) monetary growth model, but with the important difference that the dichotomy is now in terms of both the mean and the variance of the stochastic growth rate. The stochastic rate of monetary growth affects only the nominal part of the system—inflation and the portfolio share of money and bonds—and has no effect on the real part of the equilibrium, such as consumption and real growth. The superneutrality of money, traditionally associated with a deterministic monetary growth rate, now applies to the first two moments describing monetary policy. The neutrality result applying to bonds asserts that, within certain limits, the real equilibrium is independent of the type of government bond used to finance government borrowing. We show that neither the type of coupon payment nor the rate of taxation on bond income matters to the equilibrium. This outcome is explained if one considers that to hold bonds the investor requires a given after-tax real rate of return, determined by the real productive activities in the economy. It does not matter what the government gives in coupon and takes away in tax, as long as the capital gains to the holder of the bond adjust to leave the net holding real return at the required level. Flexible bond prices enable this adjustment to occur. Our analysis shows that this type of neutrality result, previously obtained in the absence of risk, extends to the present stochastic economy. The economy we consider is one in which optimal monetary and tax policies are interdependent. Our third set of results focuses on optimal monetary  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

403

and tax policies, doing so sequentially. We derive: (i) optimal tax policy for a given rate of monetary growth, (ii) optimal monetary policy for given tax rates and finally, (iii) the jointly optimal monetary and tax policies. The significant findings here are that the model implies a form of the Friedman (1969) optimal monetary growth rule, and that the jointly optimal policies yield a noborrowing or balanced-budget rule that requries the stock of government bonds outstanding to be set to zero (presuming that the government cannot lend to the private sector). For an arbitrary rate of monetary growth, the optimal tax rule is characterized by a simple relationship setting the share of wealth held in the form of capital to a specific level. In general, the optimal tax rate itself is somewhat complicated to compute, although it also leads to a balanced budget condition in special cases. In contrast to Chamley (1986) and Judd (1985), a long-run zero tax on capital income cannot be optimal when the government grows in size with the economy, thereby claiming larger resources in the future resulting from growth in the capital stock today, a condition we believe to be realistic. This result, obtained previously for a deterministic economy, extends with some adjustments for risk to a stochastically growing economy. The scope of the present paper is far-reaching and thus it can be viewed as integrating several strands of literature. In terms of its stochastic structure, our approach is related to previous contributions by Eaton (1981); Gertler and Grinols (1982); Stulz (1986); Grinols and Turnovsky (1993) and Benavie et al. (1996), though these papers do not undertake any welfare analysis of monetary or tax policies, the primary focus of the present analysis. In considering the effects of distortionary taxes in an intertemporal setting, the paper can be viewed as being a stochastic analogue to Brock and Turnovsky (1981); Judd (1985); Chamley (1986) and Lucas (1990), among others. However, these authors base their analyses on Ramsey-type models, in which the long-run growth rate is determined essentially by demographic factors and is independent of the usual fiscal instruments. More recently, authors such as Barro (1990); Rebelo (1991); Saint-Paul (1992); Jones et al. (1993); Barro and Sala-iMartin (1995) and Turnovsky (1996) have shown how, within an endogenous growth framework, fiscal policy does indeed influence the long-run growth performance of the economy. Being an endogenous growth model in which fiscal policy also has long-run effects, the present model can thus be viewed as a stochastic extension of this literature. Finally, in so far as it analyses optimal monetary policy, the paper is also a stochastic extension of the optimal monetary literature of Friedman (1969); Drazen (1979); Turnovsky and Brock (1980); Lucas and Stokey (1983); Kimbrough (1986) and Abel (1987).2 The remainder of the paper is structured as follows. Section I presents the model, with its equilibrium being discussed in Section II. Section III introduces the welfare criterion, while the following three sections derive optimal tax policy, optimal monetary policy and the jointly optimal tax and monetary policies, respectively. Our conclusions are reviewed in the final section. Technical details of the solution are derived in the Appendix. I. ELEMENTS OF THE ECONOMY This section describes the analytical framework and the behaviour of the relevant agents.  The London School of Economics and Political Science 1998

404

[AUGUST

ECONOMICA

Asset returns and price At each instant of time, the representative consumer chooses his rate of consumption and allocates his portfolio of wealth among three assets: money M, government bonds B, and equity claims on capital K. Asset returns in the economy are driven by the return to capital, which in turn is tied to the productivity of physical capital in the production process. The flow of new output dY is produced from capital by means of the stochastic constant-returns-to-scale technology dYGα K dtCα K dy,

(1)

where α is the (constant) marginal physical product of capital and dy is a temporally independent, normally distributed, stochastic process with mean zero and variance σ 2y dt over the instant dt. The equity investment is thus the real investment opportunity represented by this technology. Hence in the absence of adjustment costs to investment, the real rate of return on equity (capital) is (2a)

dRKGrK dtCduK ;

rK ≡ α ;

duK ≡ α dy.

The consumer perceives that the nominal price of output follows the stochastic process: dPyPGπ dtCdp,

(3)

where π dt is the mean expected rate of change of P over the instant dt, and dp is a temporally independent, normally distributed random variable with zero mean and variance σ 2p dt. With P being generated by (3), and assuming that money pays a zero nominal return, the before-tax real rate of return on money is (2b)

dRMGrM dtCduM ;

rM ≡ Aπ Cσ 2p ;

duM ≡ Adp.

The before-tax real rate of return on government bonds is postulated to be (2c)

dRB ≡ rB dtCduB ,

where rB and duB , along with π , dp, (and σ 2p ), will be determined endogenously in the macroeconomic equilibrium to be derived. The bonds we shall consider have an endogenously determined variable price Q. Within the class of marketpriced bonds, its further characteristics are unimportant to the equilibrium reached, e.g. whether the bond is denominated in real or in nominal terms, or the precise nature of its coupon. Equilibrium asset-pricing considerations will determine rB and duB in terms of the real shocks to the economy, and the corresponding price of bonds (which will vary with the specific bond) will adjust to support this equilibrium. For expositional convenience, we shall assume that government bonds continuously pay a fixed coupon of c dollars per bond in perpetuity, but other types of coupon are also possible. As we will note below, however, the equilibrium will be different if instead the bonds are of infinitesimal duration with a fixed nominal price. This is the case discussed by Grinols and Turnovsky (1993) and Turnovsky (1993). As shown there, without the flexibility of the adjustment in bond prices, the corresponding equilibrium can be sustained only under a restrictive form of debt  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

405

policy, namely holding the ratio of short bonds to money constant.3 That policy leads to an equilibrium having a constant nominal interest rate. Consumer optimization The representative consumer’s asset holdings are subject to the wealth constraint (4)

M QB WG C CK, P P

where W denotes real wealth. In addition, the agent is assumed to purchase output over the instant dt at the nonstochastic rate C(t) dt out of income generated by these asset holdings. His objective is to select his portfolio of assets and the rate of consumption to maximize expected lifetime utility, taken to depend upon consumption C(t) and real money balances M(t)yP(t), as represented by the isoelastic utility function4 S

(5)

E

1

# γ3 0

C(t) θ

1Aθ

γ

1P(t) 2 4 e M(t)

−ρt

dt,

ASFγF1, 0YθY1, (1Aθ )γF1, subject to the wealth constraint (4) and the stochastic wealth accumulation equation, expressed in real terms as (6)

dWGW(nM dRMCnB dRBCnK dRK )AC dtAdS,

where portfolio shares are: nM , ≡ (MyP)yWGshare of portfolio held in money; nB ≡ (QByP)yWGshare of portfolio held in government bonds; nK ≡ KyWG share of portfolio held in equity (capital); dSGtaxes paid (described below). The form of the utility function allows for different degrees of risk aversion in the choice of γ and for a different relative importance of money in the choice of θ. The logarithmic utility function corresponds to setting γG0.5 The choice of placing money in the utility function deserves some comment. As discussed by Feenstra (1986), under certain regularity conditions a number of different approaches to the demand for money, including transaction-costbased demand and cash-in-advance constraints, are functionally equivalent. The utility function in (5), defined over the positive orthant, but omitting a neighbourhood of the origin, satisfies the necessary regularity conditions discussed by Feenstra. The demand for money in the present model can therefore be interpreted as deriving from the existence of transaction costs, φ (C, MyP), where φ increases in C and decreases in MyP. The government is assumed to tax both types of income in the economy— interest and income from capital—leading to a flow of tax revenues (7)

dSGτ K rK K dtCτ K′ K duKCτ B rB (QByP) dtCτ B′ (QByP) duB .

In addition to allowing for different tax rates for real income from bonds and capital, this specification also allows for different tax rates on the deterministic and stochastic portions of real asset returns; see also Eaton (1981). Different values for τ K , τ K′ and τ B , τ B′ reflect the possibility that taxes might include  The London School of Economics and Political Science 1998

406

ECONOMICA

[AUGUST

offset provisions, having the effect of reducing the degree of after-tax randomness of real returns. We will examine below the extent to which the availability of these different tax choices is significant. Substituting for ni into (4) and for (2a)–(2c) and (7) into (6), the stochastic optimization problem can be expressed as choosing the consumption–wealth ratio CyW, and the portfolio shares ni to maximize expected intertemporal utility (5) subject to (8a) (8b)

dWGW [nM rMCnB (1−τ B )rB CnK (1Aτ K )rKACyW] dtCW dw, nMCnBCnKG1,

where for convenience we denote the stochastic component of dWyW by (8c)

dw ≡ nM duMCnB (1Aτ B′ ) duBCnK (1Aτ K′ ) duK .

In performing the optimization, the agent takes the rates of return of the assets, the expected inflation rate, and the relevant variances and covariances as given, although ultimately these will all be determined in the equilibrium to be derived. This is a standard problem in stochastic calculus, the first-order optimality conditions to which can be written in the form6 (9a)

C [ρAβγA 2 γ (γA1)σ w] , G W 1Aγθ

(9b)

{(1Aτ B )rBA [rMC((1Aθ)yθ ) CynM W]} dt

1

2

G(1Aγ ) cov [dw, (1Aτ B′ ) duBAduM ], (9c)

{(1Aτ B )rBA(1Aτ K )rK } dtG(1Aγ ) cov [dw, (1Aτ B′ ) duBA(1Aτ K′ ) duK ],

where

β ≡ nM rMCnB (1Aτ B )rBCnK (1Aτ K )rK , σ 2w ≡ E(dw)2ydtGn2M σ 2MCn2B (1Aτ B′ )2σ 2BCn2K (1Aτ K′ )2σ 2K C2nM nB (1Aτ B′ )σ M BC2nM nK (1Aτ K′ )σ MK C2nB nK (1Aτ B′ )(1Aτ K′ )σ BK . The form of the consumption–wealth ratio implied by (9a) is a generalization of Eaton (1981), reflecting the inclusion of money. The CyW ratio will increase (decrease) with the total expected return, β, and decrease (increase) with the variance of the rate of change of wealth, depending upon whether γ l 0.7 In the case of the logarithmic utility function, γG0 and (9a) reduces to the familiar relationship CyWGθρ. Equation (9b) expresses the differential after-tax rates of return on money and bonds in terms of their relative risk differentials, as measured by the covariance of their returns with the return on the overall portfolio. Note that the return on money includes its utility return, measured by [(1Aθ )yθ ](CynM W ). Because both money and bonds are risky, money is not necessarily dominated in real return by bonds, as it would be in the absence of risk. Finally, (9c) describes the relative after-tax real rates of  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

407

return on bonds and capital in terms of their respective real risk differentials. Note that, if the investor is risk-neutral (i.e. γ → 1), all after-tax real rates of return would be required to be equal, as would be the case under certainty. Solving (9b) and (9c) in conjunction with the normalized wealth constraint (8b), one can determine the agent’s portfolio demands, nM , nB , nK . Government policy Government policy is described by the choice of government expenditure, the issuing of money and bonds and the collection of taxes, all of which must be specified subject to its flow budget constraint. This may be expressed in real terms as (10)

d

1 P 2Cd 1 P 2GdGAdTCP dR C P dR , M

QB

M

QB

M

B

where dG denotes the stochastic rate of real government expenditure. Like the consumer’s budget constraint, (6), this can be derived from the basic nominal budget constraint, as in Merton (1971). Government expenditure policy is specified by (11a)

dGGgαK dtCαK dz,

where dz is an intertemporally independent, normally distributed random variable with zero mean and variance σ 2z dt. According to this specification, the mean level of public expenditure is assumed to be a fraction g of the mean level of output, with a proportional stochastic disturbance. Taxes have been described already as being levied on interest and capital income, in accordance with (7). Monetary policy is described by a stochastic monetary growth rule: (11b)

dMyMGµ dtCdx

where dx is a temporally independent, normally distributed random variable with mean zero and variance σ 2x dt. The government independently selects the expenditure level, taxes and the level of money creation, floating as many bonds as are needed to finance the budget. While the government chooses the quantity of bonds outstanding, the market determines their real value. The endogeneity of bond prices within our system allows for an equilibrium in which portfolio shares are constant over time to prevail over a far more flexible range of debt policies than do short fixed-price bonds, where this property would obtain if and only if the ratio of bonds to money is held constant.8 Goods market equilibrium Finally, the flow of physical goods in the economy to consumption, investment and government expenditure must satisfy the resource constraint (12)

dYGdCCdKCdG,

 The London School of Economics and Political Science 1998

408

[AUGUST

ECONOMICA

which, using (1), (11a) and dCGC(t) dt, implies that the equilibrium rate of capital accumulation (rate of growth) in the economy is (13)

C dK G α (1Ag)A dtCα(dyAdz) ≡ ψ dtCα(dyAdz). K nK W

3

4

II. MACROECONOMIC EQUILIBRIUM The solution of the model is derived in the Appendix. This is based on the assumption that the equilibrium is a recurring one, in which risks and returns on assets are unchanging through time. This implies that the consumer chooses the same allocation of portfolio wealth at each instant of time. Using the information about capital accumulation in relation (13) and stochastically differentiating the definition of portfolio shares, nM , nB , nK , assuming these to be constant over time, provides three linear equations in the three unknown stochastic components of the economy, (dp, dw, duB ). With this information, equations (2) yield the solutions for duM and duK , so that all the stochastic elements of the model are known. The variances and covariances appearing in the consumer optimality conditions can then be calculated. Solving by substitution leads to the closed-form solution summarized in (14), which characterizes the equilibrium growth path. Equilibrium growth path (14a)

θ γ (γA1) 2 C G ρAγα(1AτK )C α [(2τ K′ A1)σ 2yCσ 2z ] , W 1Aγ 2

(14b)

ψGα (1AτK )A

(14c)

CyW , nKG α (τKAg)C(1yθ )(CyW)Aα 2(1Aγ)[τ K′ σ 2yCσ 2z ]

(14d)

[(1Aθ )yθ )(CyW)] nMG µAσ 2xC(1yθ )(CyW )

(14e)

πGµAα (1AτK )C

(14f)

rKGα,

(14g)

rMG−πCσ 2xCα 2(σ 2yCσ 2z ),

(14h)

(1AτB )rB G(1AτK )αC(1Aγ )

1

2

1 C Cα 2(1Aγ)[τ K′ σ 2yCσ 2z ], θW

1 C Cα 2[1A(1Aγ )τ K′ ]σ 2yCα 2γ 2σ 2z , θW

1

nBCnK 2 α (τ K′ σ 2yCσ 2z ). nB

2

This equilibrium has the following recursive structure. First, equations (14a) and (14b) jointly determine the consumption–wealth ratio and the equilibrium growth rate. Given CyW, (14c) and (14d) determine the equilibrium portfolio shares of money and capital, with the portfolio share of government bonds being determined residually from (8b). Also, having obtained CyW, the  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

409

inflation rate is determined by (14e). The final three equations, (14f )–(14h), describe the equilibrium real rates of return on the three assets. Checking these solutions, one finds that they are indeed consistent with the assumption of a recurring equilibrium in which portfolio shares remain constant through time, thereby validating this assumption. Looking at these equations, we may state the following proposition. Proposition 1: Monetary Dichotomization. The macroeconomic equilibrium described in equations (14a)–(14h) dichotomizes in the sense that the real part of the equilibrium, described by (i) the consumption–wealth ratio CyW, (ii) the real growth rate ψ and (iii) the equilibrium portfolio share of capital nK are all independent of both the mean and variance of monetary policy. In contrast, monetary policy affects the nominal part of the system described by (i) the share of the nominal assets, nM , nB in the portfolio, and (ii) the inflation rate, π. Analogous to the deterministic Sidrauski (1967) model, money can be said to be superneutral in both the mean and variance of its growth rate. Note that, in contrast to Brock and Turnovsky (1981), monetary policy remains superneutral in the presence of distortionary taxes. This is because taxes are assumed to be levied on overall real rates of return, rather than differentially on their nominal components. But as we will show below, money is not neutral with respect to its impact on welfare. A consequence of the recursivity of the system (14) is that the equilibrium growth path is completely independent of the specific characteristics of the government bond. Whatever the nature of the coupon paid—e.g. whether it is denominated in nominal or real terms—the after-tax real return to bondholders is determined by the market equilibrium. Invariance of the real equilibrium to the bond characteristics may appear surprising at first glance. It can be explained by noting that the real holding return to bonds consists of the coupon plus capital gain. Since the investor cares only about the total holding return and not about how it is packaged, market demand, meaning investor willingness to hold the bonds, forces the flexible prices bond to move as needed to deliver the required holding return. In this case, (14h) implies that the rate of return on bonds adjusts so as to equate its risk-adjusted after-tax return to the corresponding return on the productive asset, capital. The type of coupon offered, however, will affect the bond price and the quantity offered, as these both adjust in the process of supporting the equilibrium. What matters to equilibrium is not the price or quantity, but the real value of bonds that can be floated at each moment. Any piece of paper, so to speak, can be ascribed the necessary real value by the market. The one exception to this is if the bond is infinitesimally short so that its price during the holding period remains fixed (at unity, say), so that the aggregate real value of the bonds floated by the government is determined by the quantity of new bonds offered. In this case an equivalent balanced growth equilibrium with fixed portfolio shares can be supported if and only if the government willingly chooses to restrict its bond offering so that the quantity of bonds outstanding grows at each moment at the same rate as the other  The London School of Economics and Political Science 1998

410

ECONOMICA

[AUGUST

assets, money and capital.9 Equations (14) further reveal that the growth equilibrium is independent of the tax rates on interest income, τB , τ B′ —the beforetax return on bonds simply adjusts to yield the equilibrium after-tax rate of return—while taxes on capital income operate through the linear combination, T ≡ τKAα (1Aγ )σ 2y τ K′ . The invariance of the equilibrium with respect to the government bond may be summarized by the following proposition. Proposition 2: Bond Neutrality. The economy reaches the same real equilibrium whether the government finances its deficit by offering, say, bonds paying a fixed dollar coupon, indexed bonds paying a fixed number of units of real output, or bonds paying a coupon equal to a fixed percent of bond value. Though the number of bonds offered and their prices differ between the regimes, the after-tax real holding return to a bond is the same in each case, as is the market value of bonds outstanding, being determined by the real rate of return on the productive asset. Taxes on interest income, τB , τ B′ , have no impact on the real economy.10 We may summarize the impact of capital income taxes on the equilibrium in the following proposition. Proposition 3: Capital Taxation. Taxes on real capital income influence the economy through the linear combination T ≡ τKAα (1Aγ )σ 2y τ K′ . Thus, raising the tax rate τ K′ on the stochastic component of real capital income has the opposite qualitative effect to raising the tax rate τK on the deterministic component of real capital income. The contrasting qualitative effects of taxing the deterministic and stochastic components of capital income can be explained as follows. An increase in τK reduces the after-tax return to capital, thereby inducing a reduction in the holdings of capital, and reducing growth. An increase in τ K′ , on the other hand, reduces the variance and associated risk on the return to capital, inducing investors to hold a higher fraction of their portfolios in capital, thereby increasing the growth rate.11 The qualitative effects of a uniform increase in the tax on capital dτ GdτK Gdτ K′ depends upon [1Aα (1Aγ )σ 2y ], which balances the expected-returns effect with the reduction of risk. Which effect dominates depends upon the investor’s risk aversion and the amount of risk present. Thus, the qualitative effects of a uniform tax increase under certainty continue to hold under uncertainty if and only if 1Hα (1Aγ )σ 2y . However, if this inequality is reversed, the risk effect prevails, and a tax increase in this circumstance will reverse the effects that would obtain under certainty.12 Further implications of changing the tax rates will be presented in Section IV below. For the present, we conclude our consideration of the equilibrium (14) with the following observations. 1. In general, taxes affect growth, inflation and portfolio shares (a) through the consumption–wealth ratio, CyW, and (b) directly. In the case of the logarithmic utility function (γG0), CyWGθρ and is independent of taxes. Then, only the direct channel prevails.  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

411

2. The equilibrium growth rate ψ is independent of mean government expenditure g. While an increase in g reduces the real growth rate directly by absorbing resources (see (13)), this is exactly offset by the fact that it also increases nK , thereby increasing the consumption–capital ratio and maintaining the overall rate of growth unchanged; see also Eaton (1981). By contrast, the growth rate depends (positively) upon the variance of government expenditure, as well as (negatively) upon the tax rate τK . 3. Monetary policy affects the portfolio share of money balances, nM , through the risk-adjusted monetary growth rate, Λ ≡ µAσ 2x . In the event that the nominal money stock is fixed deterministically, dM ≡ 0, implying that µ Gσ 2xG0. In this case, the equilibrium share of real money balances is fixed at nMG1Aθ, the relative importance of money in consumer utility, and is independent of all forms of government policy. 4. The inflation rate π varies pari passu with the monetary growth rate µ and is independent of the variance of the monetary growth rate, σ 2x . This contrasts with Grinols and Turnovsky (1993), where we find 0FdπydµF1, and dπydσ 2xH0. The difference in this last observation from our previous result is accounted for by the fact that in that analysis the bonds were infinitesimally short bonds issued at fixed unit prices, for which the only feasible form of debt policy consistent with maintaining constant portfolio shares was to assume that the monetary authority sets a constant ratio of short bonds to money. As a consequence, in that case a change in the monetary growth rate is accompanied by an equivalent change in the growth of government bonds, leading to real effects and therefore to only partial adjustment in the inflation rate. Initial prices and wealth effects The equilibrium growth path (14) describes a stable rational expectations equilibrium. As in any rational expectations macroeconomic system, the attainment of such an equilibrium, or the shift from one equilibrium to another resulting from a structural change, is brought about by an appropriate initial jump in prices, in this case the price of output P(0) and the price of bonds Q(0). To the extent that the representative agent holds money and bonds in his equilibrium portfolio, these jumps impose initial capital gains or losses, thereby affecting initial wealth. With M and K evolving continuously in accordance with the geometric Brownian motion processes (11b) and (13) respectively, the initial stocks at time 0, M0 and K0 , are predetermined. Given constant portfolio shares, the initial price of real output and the initial dollar value of government bonds outstanding are determined by13 (15a)

P(0)G

1 n 21 K 2 , nK

M0 0

M

(15b)

Q(0)B0G

1n 2 M . nB

0

M

Thus, given M0 and K0 , any policy that generates a change in the relative portfolio shares, nK ynM , will lead to a jump in the initial price level P(0). As  The London School of Economics and Political Science 1998

412

[AUGUST

ECONOMICA

seen in (15b), the equilibrium determines only the initial market value of bonds, Q(0)B0 . The initial price itself, Q(0), will depend (inversely) upon the initial quantity B0 , which in turn will depend upon the specific type of bond offered. This is a manifestation of the invariance of the growth equilibrium with respect to the specific government bond. The corresponding initial wealth, W(0), of the agent is (15c)

M0 Q(0)B0 (nMCnB ) M0 K0 W(0)G C CK0 G CK0 G . P(0) P(0) nM P(0) nK

Taking the differential of this expression, for given initial stocks M0 and K0 , the change in initial wealth generated by the capital gains or losses on money and bonds is14 (15c′)

dW(0) M0 G d W(0) W(0)

nMCnB 1 dnK G− . nM P(0) nK

31

2 4

This initial wealth effect must be taken into account in assessing the effects of any structural or policy change on economic welfare; see Section III. Feasibility of equilibrium Finally, the equilibrium must satisfy certain feasibility conditions. First is the transversality condition, which for each asset requires lim E

t→S

31P 2 X M

W

4

exp (−ρt) G lim E t→S

31 P 2 X QB

W

4

exp (−ρt)

G lim E [KXW exp (−ρt)]G0, t→S

where XW denotes the marginal utility of wealth which, in the case of the constant elasticity utility function, is proportional to W γA1.15 For constant portfolio shares, these conditions all reduce to (16)

lim E [W γ exp (−ρt]G0.

t→S

Using (13), condition (16) can be shown to be equivalent to the condition Cy WH0, as originally shown by Merton (1969).16 With the equilibrium being one of balanced real growth, in which all real assets grow at the same rate, (16) also implies that the intertemporal government budget constraint is met. Using (14a), condition (16) implies the following constraint on the composite tax rate T, and other parameters: (16′)

γ (γA1) 2 2 ρAγα(1AT)C α (σ z Aσ 2y )H0. 2

Note that (16′) is automatically met in the case of the logarithmic utility function (γ G0). In other cases, this condition may impose restrictions in order for the tax rate to remain feasible.17 But provided (16′) holds, the equilibrium is viable in the sense of being consistent with the intertemporal solvency of the government. Second, with non-negative stocks of money and capital in existence, the equilibrium portfolio shares nM X0, nK X0. These inequalities impose further  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

413

restrictions on T and the monetary growth parameter Λ. If the government is permitted to borrow and lend, then no restriction on nB is imposed. However, if such lending to the private sector is ruled out, the additional restriction nB X0 is required to be met.18 Equilibrium bond pricing This completes the characterization of the equilibrium. Given the real part of the equilibrium as determined by (14) and the initial prices, P(0), Q(0), it is perfectly consistent and sustainable as long as the feasibility conditions set out in the previous subsection are met. The characterization of the equilibrium is completed by considering the intertemporal pricing behaviour of government bonds. This is determined residually given its real rate of return as determined by (14) and the specific coupon assumed. Since the pricing behaviour of a specific bond is of no particular significance in so far as the basic equilibrium is concerned, there is no need to pursue this aspect further at this time.19. III. WELFARE To assess the consequences of tax and monetary policy on economic welfare, we consider the welfare of the representative agent as specified by the intertemporal utility function (5) evaluated along the equilibrium path. By definition, this equals the value function used to solve the intertemporal optimization problem. It can be shown that for the constant elasticity utility function the optimized level of utility, starting from an initial stock of wealth, W(0), and computed in this way, is given by (17)

θ C X(W(0))G nγM(1Aθ) γ W

1 2

θγA1

W(0)γ,

where nW , CyW are the equilibrium values given in (14). Using the relationship (15c), the welfare criterion (17) can be expressed as (18)

θ C X(K0 )G n γM(1Aθ) n−Kγ γ W

1 2

θγA1

K 0γ ,

where CyW, nK , nM are obtained from (14a), (14c), and (14d) respectively. Assuming that these solutions are all positive implies that γX(K0 )H0, as well. Taking the differential of (18) yields (19)

dX dnK dnM d(CyW) Aγ C(θγA1) Gγ (1Aθ ) , X nM nK CyW

and thus the effect of any policy change on welfare can be assessed in terms of its impact on: (i) the portfolio share of money balances, nM ; (ii) the consumption–wealth ratio, CyW and (iii) the portfolio share of capital, nK . To the extent that money provides utility (1Hθ ), an increase in the portfolio share of money balances raises utility. In addition, given (1Hθγ), a decline in the CyW ratio leads to an increase in the growth rate, leading to greater expected future consumption, and this too is welfare-improving. On the other hand, as noted  The London School of Economics and Political Science 1998

414

[AUGUST

ECONOMICA

in (15c′), given nM , the increase in the portfolio share nK reflects capital losses on money and bond holdings, resulting from jumps in the initial price change causing a welfare deteriorating decline in wealth.20 Optimal welfare policy involves trading off these three effects. From the structure of the equilibrium, (14), we see that tax policy influences welfare through all three channels while monetary policy does so only through its effect on money balance holdings. IV. TAX POLICY To analyse the effects of a change in the tax on capital, as represented by the composite tax rate, T, we differentiate the expressions in (14) and utilize (19) to obtain (20a)

d(CyW ) θαγ G ; dT 1Aγ

(20b)

α dψ G− Y0; dT 1Aγ

(20c)

1 dnK α G (θγAnK ); nK dT (1Aγ )(CyW)

(20d)

1 dnM θαγ G (1AθAnM ); nM dT (1Aθ )(1Aγ )(CyW ) i.e. sgn

(20e) (20f)

i.e. sgn

1

d(CyW) Gsgn (γ ); dT

2

dnM

1 dT 2Gsgn [γ ( µAσ )]; 2 x

dπ α X0; G dT 1Aγ dX α (γX) G [nKAθ (1Cγ nM )]. dT (1Aγ )(CyW)

An increase in T therefore lowers the growth rate and raises the inflation rate. All other effects contain some ambiguity, depending in part upon γ , the elasticity of the utility function. As a benchmark case, it is instructive to focus on the logarithmic utility function when γ G0. In this case, both CyW and nM are independent of the tax rate. An increase in T, by lowering the return to capital, will cause agents to switch their portfolio away from capital towards bonds. The net effect on welfare as measured by (19′) in this case depends upon (nKAθ ). On the one hand, the decrease in growth caused by the higher tax rate (dψydTG−α) reduces the expected future consumption flow, thereby reducing welfare by the amount −αyρ2 and this is welfare-deteriorating. At the same time, the reduction in nK reduces the negative wealth effect resulting from the price rise (measured by A(1yρ) d ln nK ydTGα nK yθρ2 and this is welfare improving. The net effect thus depends upon whether nK m θ. If γ ≠ 0, most of these effects are adjusted by the impact of T on CyW. If γH0, an increase in T will raise CyW, while the net effect on nM depends upon  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

415

whether µ m σ 2x . The rise in the equilibrium CyW ratio will tend to induce investors to increase the share of capital in their portfolios, so that on balance nK may either rise or fall. The effects on the rate of growth and inflation remain unchanged, although they increase in magnitude with γ . As noted, the net effects on welfare involve taking account of the adjustments in nM , nK , and CyW as described in (19), and the optimal tax is attained where these are balanced off optimally. The main results can thus be summarized as follows. Proposition 4. Optimal Capital Taxation. An increase in the tax on capital, T ≡ τKAα (1Aγ )σ 2y τ K′ (i.e. reduced bond financing) lowers the rate of growth and raises inflation. This raises welfare if nK Hθ (1Cγ nM ) and lowers it if the inequality is reversed. Optimal tax policy is therefore characterized by portfolio shares that satisfy the condition: nK Gθ (1Cγ nM ).21 The optimal tax itself is obtained by substituting for (14a), (14c) and (14d) into the condition of Proposition 4. The resulting expression may be conveniently expressed in the form ˆ ˆ γ nˆM C (21) T GΓA , α nˆK W ˆ ˆ ˆ where Γ ≡ gCα (1Aγ )σ 2z and nM , nK , (CyW ) are obtained from the equilibrium (14), substituting for (14a), (14c) and (14d) and recognizing that (CyW )G[θy (1Aγ )(aCγ α T)], where a ≡ ρAγ α C12γ (γ A1)α 2(σ 2z Aσ 2y ). It is thusˆ seen that in general (21) defines ˆ a quadratic equation in the optimal tax rate T that can be solved to express T in terms of: (i) the exogenous parameters, and variances of the exogenous shocks, (ii) the risk-adjusted rate of monetary growth, paramˆ eter Λ, (through nM ) and (iii) the risk-adjusted rate of government expenditure, 22 Γ. One of the two roots can always be shown to satisfy ˆ the second-order condition so that in general an interior optimal tax rate, T, always exists. One further observation is that any combination of the deterministic tax rate, τ K , and the stochastic tax rate, τ K′ , consistent with (21) will generate the optimal risk-adjusted tax rate. As a special case, we may note that, if both the deterministic and stochastic components of income are taxed uniformly, i.e. if ˆ ˆ ˆ τ K Gτ K′ Gτ , then the optimal such uniform tax is ˆ γ nˆM C 1 ˆ ΓA . (21′) τ G (1Aασ 2y ) α nˆK W

1 21 21 2

3 1 21 21 24

Interest in the question of the optimal tax on capital was stimulated by Judd (1985) and Chamley (1986), who showed that asymptotically the optimal tax on capital should converge to zero (see also Chari, et al. 1991; Lucas 1990). That initial result was obtained in a Ramsey-type neoclassical growth model in which the government sets the level of its expenditures exogenously. Subsequent authors have revisited this issue in the context of simple deterministic models of endogenous growth, in which, as in the present analysis, government expenditure is specified to be a fixed fraction of output so that its level is no longer exogenous, but instead is proportional to the size of the growing capital stock. In that case the decision to accummulate capital by the private sector enlarges the economy and leads to an increase in the supply of public goods  The London School of Economics and Political Science 1998

416

ECONOMICA

[AUGUST

in the future. Since the private sector treats government expenditure as being independent of its own investment decision (when in fact it is not), a tax on capital can correct this distortion, and thereby internalize the externality. The magnitude of the optimal tax depends upon the nature of the government expenditure and the externalities it generates. In the basic model, in which government expenditure yields no direct benefits, the optimal tax is to finance ˆ expenditure fully with a tax on capital, i.e. to set τ Gg. This is modified in the presence of externalities, such as congestion, and for deviations in expenditure from its optimum; see Barro (1990); Barro and Sala-i-Martin (1995) and Turnovsky (1996).23 The present model also points to a balanced-budget or no-borrowing rule for government finance as a benchmark, though in this case the distortions are brought about by the endogenously generated risk. To understand ˆ these, we consider a number of special cases when the solution for T simplifies dramatically. (i) Logarithmic utility function (γ G0) In this case government expenditure generates no externalities and the optimality condition in Proposition 4 reduces to nKGθ ; i.e. the tax rate should be set such that the portfolio share of capital equals the relative importance of consumption in the utility function. The corresponding optimal tax rate is to set (22a)

ˆ T ≡ τ KAαα 2y τ K′ GgCασ 2z ≡ Γ.

This is a stochastic generalization of the deterministic balanced budget condition in that it requires the risk-adjusted tax rate T to be set equal to the riskadjusted share of government expenditure in output, Γ. Stochastic variations in government expenditure raise the risk associated with government bonds, inducing agents to hold a higher fraction of their portfolio in capital. This raises the equilibrium growth rate, thereby increasing the absolute level of government expenditure, which requires a higher risk-adjusted tax rate to finance. A larger variance in output increases the risk associated with real capital and requires a compensating increase in the mean rate of return. This also raises the equilibrium growth rate and the size of government and, given τ K′ , necessitates a larger deterministic tax rate, τ K . (ii) No direct utility to money (θ G1) With money providing no utility and being dominated in yield by incomeearning assets, the equilibrium holdings of money will be reduced to zero. That is, nMG0 and the optimum requires nKG1; i.e. all wealth should be held in the form of physical capital. Real money balances and bonds are driven to zero by ensuring that P(0) becomes infinite. Again, this equilibrium satisfies the second-order condition. In this case the optimal tax policy is (22b)

ˆ TGgCα (1Aγ )σ 2z GΓ,

which is similar to (22a), except for the modification to the risk term.24  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

417

(iii) Deterministically fixed nominal money stock (ΛG0) In the case where ΛG0, the equilibrium stock of money is nMG(1Aθ ), the relative importance of money in the utility function. The optimality condition thus requires nK Gθ [1Cγ (1Aθ )]. In this case, the quadratic equation (21) determining the optimal tax rate T reduces to a linear equation. It can be shown that the second-order condition is satisfied if and only if the optimal nK H0. For this condition to hold and be consistent with the concavity of the utility function, we require that 1Hγ (1Aθ )H−1. Note also that, for each of the above cases, the transversality condition (16′) needs to be met. This is clearly satisfied in case (i). However, in case (ii) it requires that

ρAαγ (1Ag)A 12 γ (γ A1)α 2(σ 2z Cσ 2y )X0. If γ F0, this condition imposes a constraint on the fraction of output spent by the government. If this condition is violated, the optimum cannot be attained. A similar, but more complex, constraint applies in case (iii). As long as γ ≠ 0, nM ≠ 0, risk introduces an extra externality into consumption and money holdings, and this also needs to be taken into account in determining the optimal tax rate. Example (iii), noted above, is an illustration of this. For example, suppose γ H0. In this case, the presence of risk will induce private agents to increase consumption at the expense of saving; see (9a). In order to correct for this, the return to saving should be increased and this can be achieved by reducing the tax on capital income. This is reflected in the second term of (21). Except in special cases such as those we have been considering, the choice of the optimal tax depends upon the rate of monetary growth, Λ. Differentiating (21) with respect to Λ, we derive the trade-off relationship. (23)

ˆ dT d∆

G−

γ α

1 21

dnˆM yd∆ nˆK

ˆ C . W

21 2

ˆ This relationship describes the adjustment in T in response to a change in Λ, consistent with maintaining welfare at its optimum. It is analogous to the type of trade-off relationship between monetary growth and income tax originally discussed by Phelps (1973) and later pursued by Turnovsky and Brock (1980). Since dnM ydΛF0, it follows that the trade-off will be positive or negative depending upon whether the elasticity γ m0.

V. MONETARY POLICY As discussed in Proposition 1, monetary policy has no effect on the real part of the equilibrium, affecting only the expected rate of inflation π , the portfolio share of money nM and therefore bonds. In particular, its impact on economic welfare occurs through its effect on nM , doing so through the risk-adjusted growth rate Λ ≡ µAσ 2x . Thus, differentiating (14d) with respect to Λ, we obtain  The London School of Economics and Political Science 1998

418

ECONOMICA

[AUGUST

for θF1 (24a)

θ nM 1 dnM G− F0, nM dΛ (1Aθ )(CyW)

(24b)

dX nM θ ( γ X) G− F0. dΛ CyW

An increase in the mean monetary growth rate increases inflation, reduces the return to holding money and thus reduces the fraction of money held in the representative agent’s portfolio, thereby lowering welfare. An increase in the variance has the opposite effect. The fact that an increase in money adds to real utility according to the benefits from real money balances suggests that money creation should be raised to the point where the marginal benefit of additional real money balances is zero. However, for the constant elasticity utility function, an interior solution of this type does not exist, because the marginal expected utility of money balances can never be reduced to zero. In this case, the welfare-maximizing monetary policy is to make real money holdings as large as feasible by making the return to holding money as attractive as possible. From (14d), this is achieved by letting Λ→A(1yθ )(CyW), thereby driving nM →S. Noting (14e), this condition implies the optimal inflation rate ˆ (25) π G−α (1Aτ K )Cσ 2xCα 2 [1A(1Aγ )τ K′ ]σ 2yCα 2γ 2σ 2z . In the absence of distortionary taxes and risk, this condition reduces to α Cπ G 0, which is precisely the well-known Friedman (1969) full liquidity rule, setting the nominal interest rate to zero. Condition (25) is therefore the generalization of that condition, appropriate to the present stochastic growth context. Since the equilibrium portfolio of capital nK is independent of Λ, the increase in nM required to attain this optimum is at the expense of a reduction in the portfolio share of government bonds, nB . In the absence of infinite government borrowing and lending, bonds should be reduced to their lowest feasible level, thereby driving π as close as possible to the optimal value defined by (25). Thus, we derive the following proposition. Proposition 5: Optimal Monetary Policy. Provided θF1, optimal monetary policy is to reduce the stock of bonds to its lowest feasible level. If the government is unable to lend to the private sector, then nBG0, and this optimum is characterized by portfolio shares which satisfy the condition: nB G0; nMG 1AnK . The constraint imposed on nB in Proposition 5 is viewed as representing a restriction on government policy rather than as a constraint on the representative consumers’ ability to borrow or lend, which individuals are free to do among themselves. The proposition requires the government to set its policy in such a way that the private sector in the aggregate chooses to hold the minimally acceptable portfolio share of government bonds. The result in Proposition 5 requiring nBG0 might be characterized as being a balanced budget proposition, because it asserts that government spending should not be financed by the issuance of debt. Apart from money creation (or reduction), optimal policy requires that taxes cover all public spending.25  The London School of Economics and Political Science 1998

1998]

419

MONETARY POLICIES IN A GROWING ECONOMY

The optimal monetary rule itself can be determined by substituting for (14c) and (14d) into the equality appearing in Proposition 5. This may be conveniently expressed in the form ˆ ˆ (1yθ )(CyW )(nˆKAθ ) (26) ΛG . ˆ 1AnK This equation indicates a trade-off betweenˆ a given tax rate, T, and the corresponding optimal monetary growth rate, Λ, that occurs through the optimal consumption–wealth ratio and the optimal portfolio share of capital.26 Whether the risk-adjusted monetary growth rate is positive or negative depends ˆ upon whether nˆKm θ. For the logarithmic utility function, nˆK Gθ and Λ G0. Also, in the case where θ G1, so that money yields no utility, nMG0, and the monetary growth rate is irrelevant.

VI. JOINTLY OPTIMAL TAX AND MONETARY POLICY Combining the results of Proposition 4 and Proposition 5 implies that, in the case where the government is unable to lend to the public, the overall optimal fiscal–monetary policy package is to set the composite tax rate T and the monetary growth rate Λ such that the three conditions nK Gθ (1Cγ nM ), nB G0 and nMG1AnK are met. This leads to Proposition 6. Proposition 6: Jointly Optimal Tax and Monetary Policy. The jointly optimal combination of capital tax and monetary growth is characterized by setting the composite tax rate T and the monetary growth rate Λ so that nM ≡

1Aθ X0; 1Cθγ

nBG0;

nK ≡

θ (1Cγ ) X0. 1Cθγ

That is, the government uses only tax and money creation to finance its expenditures. Note that the portfolio allocation required for Proposition 6 is feasible only if γ C1H0, i.e. only if the coefficient of relative risk aversion, ω F2. If this condition is not met, the optimum portfolio allocation will be nM G1, nBGnKG0. Recalling the solutions for nM and nK reported in (14), the optimal settings for T and Λ may be explicitly solved in terms of the various exogenous sources of risk as follows (for θF1): (27a)

ˆ TG

1

α (1Aγ

γ (γ A1) 2 2 α (σ z Aσ 2y ) 2

−γ (1Aθ ) ρAγ α C 3 θ) 5

2

6

C(1Aγ 2 )α [gCα (1Aγ )σ 2z ] ,

(27b)

ˆ γθ (1Cγ ) γ (γ A1) 2 2 ΛG ρAγ α (1Ag)A α (σ z Cσ 2y ) . 2 1Aγ θ 2

3

 The London School of Economics and Political Science 1998

4

4

420

ECONOMICA

[AUGUST

In the case of the logarithmic utility function (γ G0), these expressions simplify to ˆ T ≡ τ KAασ 2y τ K′ GgCασ 2z , ˆ Λ ≡ µAσ 2xG0. In the event that money yields no direct utility (θ G1), the equilibrium portfolio share of money is reduced to zero (nMG0). In this case, the optimal tax policy is ˆ T ≡ τ KA(1Aγ )ασ 2y τ K′ GgCα (1Aγ )σ 2z , while monetary policy is irrelevant, so that Λ can be set arbitrarily. As a final point, it is of interest to determine the effects of government expenditure on real growth, when the policy-maker adopts the optimal tax and monetary policies. Differentiating the expression forˆ ψ reported in (14b) with ˆ respect to g and σ 2z respectively, and evaluating dTydg, dTydσ 2z from (27a), yields ˆ α (1Cγ ) dψ dψ dT F0, G G− (28a) dg dT dg 1Aγ 2θ ˆ dψ dψ dT α 2(1Aγ 2) α 2(1A2γ ) 2 (28b) C . G C α (1A γ )G− dσ 2z dT dσ 2z 2(1Aγ 2θ ) 2 Thus, with optimal financing we find that an increase in mean government expenditure g, being tied to an increase in the tax rate τ K , will reduce the equilibrium rate of growth. This is in contrast with (14b), where we found that with bond financing the mean growth rate is independent of g. Second, whereas in (14b) an increase in the variance of government expenditure, σ 2z will raise the growth rate, with optimal financing this is no longer necessarily the case. VII. CONCLUSIONS This paper has investigated optimal tax and monetary policies in a stochastic monetary growth model. Our findings are of three types. We have identified measures of effective risk-adjusted policy instruments; we have highlighted two stochastic neutrality results relating to money and bonds, the two nominal assets in the economy; and we have derived optimal government policy rules relating to taxes, bond finance and money creation. We briefly summarize each in turn. Risk-adjusted measures of effective policy An important contribution of the model is to identify risk-adjusted measures showing how policy variables effectively matter to the economy. For example, taxes on the income from capital are shown to influence the equilibrium through the linear combination T ≡ τ KAα (1Aγ )σ 2y τ K′ . Thus, raising the tax rate on the stochastic component of capital income has the opposite qualitative effect to raising the tax on the deterministic component. This is because the former reduces the riskiness of holding equity, making it a more attractive asset, whereas the latter lowers the mean return, making it less desirable. A  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

421

uniform tax imposed in the presence of productivity shocks will continue to have the same qualitative effects as a tax increase under certainty if and only if 1Hα (1Aγ )σ 2y . The model highlights the possibility that large productivity shocks can cause tax increases to have seemingly perverse impacts on the economy. Similarly, monetary policy influences the economy through the linear combination Λ ≡ µAσ 2x . Thus, an increase in the mean growth rate of the money supply has the opposite effect to that of an increase in its variance. Accordingly, if an increase in the monetary growth rate is accompanied by a larger variance (for example), the direction of the response depends upon which effect dominates, a point that was first noted for investment by Gertler and Grinols (1982). Neutrality of financial assets The model yields important policy neutrality results, in spite of our view that the demand for money in this economy derives from its friction-reducing transaction demand properties. Thus, even with a real role for money, it is notable that the superneutrality of the monetary growth rate associated with the standard Sidrauski (1967) model extends to this stochastic economy in the sense that the real part of the equilibrium—the real growth rate, the consumption– wealth ratio and the portfolio share of capital—is independent of both the mean and the variance of the stochastic monetary growth rate. Optimal policy recognizes, however, that money has real effects on welfare and seeks to maintain a proper balance between money holdings and the impact that inflation has on the relative real returns to the economy’s assets. The second neutrality result we found was that the real equilibrium is independent of the nature of the bond used by the government to finance its borrowing, or of the taxes that the government levies on real bond returns. The explanation is that the private sector requires a given real return to be willing to hold bonds. From the investor’s perspective, what the government gives in terms of coupon payment and takes away in tax or bond income is irrelevant as long as the after-tax holding return is the same. With flexible bond prices, equilibrium and the willingness of the private sector to hold bonds causes the holding capital gain from owning a bond to adjust as needed to yield the required real return on bonds. Optimal policy Macroeconomic policy analysis frequently focuses on the behaviour of growth and inflation. These measures are of interest in so far as they proxy economic welfare. Being based on intertemporal utility optimization of a representative agent, the model focuses on the welfare effects of tax and monetary policies directly, as measured by their impact on this intertemporal utility measure. These operate through their effects on (i) the optimal consumption–wealth ratio CyW, (ii) the optimal portfolio share of capital nK and (iii) the optimal portfolio share of money nM . Because of the superneutrality of money, the first two channels are influenced by tax policy only, while the third is affected by both taxes and monetary policy. Consequently, optimal tax and monetary policies are in general interdependent. The optimal tax policy depends in part upon  The London School of Economics and Political Science 1998

422

ECONOMICA

[AUGUST

how monetary policy is set, and vice versa. We have studied a series of optimal policy issues. First, we have derived the optimal tax on capital (for a set monetary policy) and characterized it in terms of a simple relationship between the optimal portfolio shares of capital and money. In general, the optimal tax is a function of the means and variances of the exogenous stochastic processes impinging on the economy, and the degree of risk aversion of the representative agent. In special important cases the rule for the optimal tax rate simplifies and can become independent of the arbitrarily set monetary policy. Second, we have derived the optimal monetary policy for arbitrarily set tax rates. This involves raising the portfolio share of money at the expense of government bonds as far as is feasibly possible. This generates a form of the well-known Friedman optimal monetary rule. Finally, we have combined these two partial optimizations and derived an overall joint optimal tax–monetary policy combination. This too can be characterized simply in terms of optimal portfolio shares. A particularly interesting feature of the optimal policy is that it implies a noborrowing finance rule or balanced-budget type rule for government finance. That is, the optimal policy involves paying for government expenditure only through taxes and money creation, the latter being needed to satisfy the transaction demand for money in the economy. To establish the robustness of this rule would appear to be a valuable topic for further research. Our results suggest that a more complete model of the type developed here can provide a useful and tractable framework for assessing the effects of tax and monetary policy on macroeconomic equilibrium and welfare in an environment of risk. The tractability of the analysis in the presence of an arbitrarily set government debt policy is due to the assumption being made that the government uses long rather than short bonds, an issue we explained in terms of the long bond’s price variability. With the price of such bonds being determined endogenously, the government budget acquires a degree of freedom, thereby enabling the attainment of an equilibrium in which portfolio shares remain constant through time. By contrast, Grinols and Turnovsky (1993) and Turnovsky (1993) assume that government bonds are short bonds, implying a fixed price. In this case, the equilibrium will be one of constant portfolio shares if and only if the ratio of bonds to money is kept constant. Relaxing this assumption leads to an equilibrium in which portfolio shares evolve stochastically over time. Such equilibria are of interest, though considerably more difficult to analyse. However, it will be necessary to consider such equilibria if one wishes to introduce short bonds into the present framework. This will enable one to address important issues pertaining to the term structure of interest rates, as well as, in debt policy, relating to the mix of long and short bonds in government finance.

APPENDIX: DERIVATION OF EQUILIBRIUM The stochastic terms dx, dy and dz are errors (unplanned deviations) from planned policy and production choices. There is no reason to believe that these should be related to one another, and thus we take them to be uncorrelated. Alternatively, if one believes that there is a common mechanism explaining the errors, then one can easily introduce some pattern of correlation among them.  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

423

Assuming a recurring equilibrium where asset returns are constant through time implies fixed portfolio shares nM , nB , nK . The intertemporal constancy of these portfolio shares implies (A1)

d(MyP) d(QByP) dK dW G G G Gψ dtCdw, MyP QByP K W

where from (13) we have (A2)

dwGα (dyAdz).

Using the rules of stochastic calculus, (A1) implies (A3)

d(MyP) dM dP dM dP dP 2 G A A C . MyP M P M P P

1 2

Substituting for dMyM from (11b) and for dPyP from (3) into (A3), retaining terms to order dt, and using (A1) to equate the resulting expression to (13) gives the following expression for the stochastic evolution of the price level: (A4)

3 1

π dtCdpG µA α (1Ag)A

2

4

C − σxpCσ 2p dtCdxAα (dyAdz). nK W

Equating the deterministic and stochastic components of this equation implies the following expressions for the corresponding components of the inflation rate:

1

(A5a) πGµA α (1Ag)A

2

C − σxpCσ 2p , nK W

(A5b) dpGdxAα (dyAdz). Next, dividing both sides of the government budget constraint (10) by W and using (A1) implies (nMCnB )

d(MyP) dG dT G A CnM dRMCnB dRB , MyP W W

which upon substitution for dMyM, dPyP, dG, dT, dRM and dRB can be written as (A6)

(nMCnB )[( µAπAσxpCσ 2p ) dtCdxAdp] G[gα nKCnM rMCnB rB (1AτB )AnK rK τK ] dt Cα nK dzAnM dpCnB (1Aτ ′B ) duBAnK ατ ′K dy.

Equating separately the stochastic and deterministic components of (A6), while noting (2b), enables us to express the corresponding components of the after-tax real rate of return on bonds as (A7a) nB (1AτB )rBG(nMCnB )( µAπAσxpCσ 2p )Aα nK (gAτK )AnM (−πCσ 2p ), (A7b) nB (1Aτ ′B ) duBG(nMCnB )(dxAdp)Aα nK dzCnM dpCα nK τ ′K dy. Substituting for dp from (A5b) in (A7b), leads to (A8)

nB (1Aτ ′B ) duBGnM dxCα (nBCnK τ ′K ) dyAα (nBCnK ) dz.

Since duK Gα dy from (2a) and duMG−dp, (A2) and (A8) allow us to obtain

α 2nK (τ ′K σ 2yCσ 2z ) dt, (A9a) cov [dw, (1Aτ ′B ) duBAduM ]G nB α 2 (nBCnK ) (τ ′K σ 2yCσ 2z ) dt, (A9b) cov [dw, (1Aτ ′B ) duBA(1Aτ ′K ) duK ]G nB (A9c) σ 2wGα 2(σ 2yCσ 2z );

σ 2pGσ 2xCα 2 (σ 2yCσ 2z );

 The London School of Economics and Political Science 1998

σxpGσ 2x .

424

ECONOMICA

[AUGUST

Using (A9a)–(A9c), (A7a), (2) and (9), we can write the expressions for the real rates of return given in equations (14f )–(14h). Using (14f )–(14h) and (A9a)–(A9c) to eliminate rM , rB , rK , the covariances in (9a)–(9c) and (8b) lead to equations (14a)–(14d). ACKNOWLEDGMENTS This paper has benefited from comments received at a seminar presented at the University of Washington. The constructive comments of an anonymous referee are also gratefully acknowledged. NOTES 1. The effects of government expenditure do not decompose quite as neatly into a mean–variance measure since, for example, the equilibrium growth rate depends upon the variance but not the mean level of government expenditure. However, we do find that the risk-adjusted measure Γ≡gCα(1Aγ )σ 2z is useful in some instances; for example, it is through this measure that government expenditure affects the portfolio share of capital. More importantly, Γ is a crucial determinant of the optimal tax and monetary growth rates. 2. But there is also a growing literature discussing optimal policy in stochastic economies; see e.g. Den Haan (1990); Chari et al. (1991); Zhu (1992); Corsetti (1991) and Turnovsky (1993). Of these studies, the paper is most closely related to the latter two, although it differs from both in fundamental ways: Corsetti abstracts from monetary issues; Turnovsky employs a restrictive debt policy, compounding the effects of monetary and fiscal policy. 3. The present model generates naturally a recurring equilibrium in which the investor chooses from assets whose stochastic properties are the same through time, leading him to devote unchanging shares of his wealth to the various assets at different moments of time. The value of a particular asset held by the investor therefore grows with his overall wealth. Short-term bonds having a fixed nominal price are compatible with such a recurring equilibrium if and only if the government selects its supply of bonds to grow at the rate of other assets. 4. The restrictions on the elasticities γ , θ introduced in (5) are to ensure that the utility function is concave in C and MyP. For simplicity, we assume that consumers do not assign direct utility to government expenditure. Since issues of optimal government expenditure are not addressed, this assumption is inessential. An extension of the present analysis to consider optimal utility or productivity-enhancing government expenditure is worthwhile. 5. Strictly speaking, the logarithmic utility function emerges as limγ → 0 [{[C θ (MyP)1Aθ ]γA1}yγ ]. This function differs from (5) by the subtraction of the term −1 in the numerator; the two forms of utility function have identical implications. 6. See e.g. Merton (1971); Malliaris and Brock (1982); Turnovsky (1995). Details of the optimization in this specific case are provided in an expanded version of this paper. 7. The intuition for this result can be provided in terms of offsetting income and substitution effects of changes in β or σ 2w ; see Sandmo (1970). Take the former. (The latter is precisely the opposite.) An increase in the expected return on assets dβ generates a positive income effect, thereby increasing consumption. This is offset by a negative substitution effect, measured by Adβy(1Aγ ), reflecting the fact that the higher return on investment induces a switch away from consumption. The net effect on the equilibrium consumption–wealth ratio is given by the sum −γ dβy(1Aγ ), with the two effects being precisely offsetting for the logarithmic utility function. 8. If portfolio shares nM and nB are constant, then the ratio nB ynM GQByM is also constant. A fixed price of bonds, Q, would then require B and M to grow at the same rate, placing a restriction on the government’s choice of B and M; see Grinols and Turnovsky (1993). 9. A similar question can be asked about bond maturities. In this model we describe long bonds as effectively consols that pay a coupon based on a predetermined rule. Short bonds would be bonds that have a life over the planning instant and then mature, requiring the government to roll over outstanding debt at the start of the succeeding instant by issuing new debt. If this can be done costlessly, there is no reason for the government to do otherwise. It is a message of the model that different bond maturities simply become accounting conventions without real consequences. To introduce bond maturities that have real consequences requires the introduction of roll-over costs and frictions, the nature of which would have to be described and would drive the real consequences. 10. While the assumption that the coupon and nominal capital gains on bond returns are taxed uniformly is important in ensuring the neutrality of monetary policy, it does not affect the neutrality of the interest income tax. Since the after-tax real return on bonds is determined endogenously by the after-tax return on physical capital, the same neutrality would still occur if there were differential taxes applied to the nominal return and capital gains on bonds. In  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

425

that case, the nominal rate would adjust, so that, given the specific tax regime, the net return on bonds differed from that on capital by the appropriate adjustment for risk. 11. The conventional measure of financial risk in mean–variance finance is the asset’s beta coefficient, the covariance of the asset’s after-tax real return with the real return on the market portfolio, divided by the variance of the market return. In this model, the beta coefficient for equity is

βKGcov [α (1Aτ K′ ) dy, α (dyAdz)]yvar [α (dyAdz)]G(1Aτ K′ )[σ 2y y(σ 2yCσ 2z )], which declines with an increase in τ K′ . 12. The result that under uncertainty taxes can increase efficiency is similar to that previously due to Gordon (1985) who argues that by taxing the stochastic component of capital income, the government absorbs a certain fraction of the risk in the return. 13. These equations hold at all points of time, including 0. Given the constancy of portfolio shares, they are the source of the proportionality of the stochastic growth rates summarized in (A1). 14. The second equality in (15c′) can be obtained directly from the relationship W(0)GK0 ynK . It can also be derived by combining the differential of the first equality with that of (15a). 15. The utility function (as a function of wealth) can be shown to be of the form δW γ. 16. This can be established as follows. Real wealth evolves according to dWGψ W dtCαW(dyAdz), from which it follows that E[W γ exp(−ρt)]GW(0)exp ({γ [ψC 12 (γA1)α 2(σ 2y Cσ 2z )]−ρ}t).

17. 18. 19. 20.

The transversality condition (16) will be met if and only if γ [ψC12(γA1)α 2(σ 2yCσ 2z )]AρF0. Substituting for ψ from (14b), this reduces to CyWH0. Such a restriction arises if γF0, i.e. if the agent’s degree of relative risk aversion exceeds that corresponding to the logarithmic utility function. The equilibrium in which the portfolio share of government bonds, nBG0, is one in which there are no outside bonds. Such an equilibrium still allows for the presence of inside bonds that are perfect substitutes for government bonds. The implications for the pricing of specific bonds is available in an expanded version of this paper. The expression for X(K0 ) becomes more intuitive in the case of the logarithmic utility function ( γ→ ), when it reduces to X(K0 )G(θyρ) ln (θρ) C[(1Aθ )yρ] ln nMC(ψyρ 2 )A(α 2y2ρ 2 )(σ 2yCσ 2z )A(1yρ) ln nKC(1yρ) ln K0 .

21. The condition in Proposition 4 is only a first-order condition. To ensure an interior optimum, the second-order condition d 2XydT 2F0 must also be met. Evaluating this quantity at the optimum, this is given by sgn [(dnK ydT )Aθγ (dnM ydT )]Gsgn {[θy(1Aθ )]γ 2n 2MA2γ nMC(γA1)}F0, where nM is given by (14d). ˆ 22. The explicit quadratic equation determining T is given by [(γαTCa)y(1Aγ )] ΛC[1Cγ (1Aθ )][(γαTCa)y(1Aγ )] G . [(αTCa)y(1Aγ )]AE ΛC[(γαTCa)y(1Aγ )] 23. As Jones et al. (1993) point out, one can still derive a Chamley–Judd result of a zero asymptotic tax on capital in an endogenous growth framework. If the government continues to maintain its level of expenditures fixed in absolute terms, then, if the economy follows a path of steady endogenous growth, the share of government expenditure as a fraction of output g→0, implying that the corresponding optimal tax on capital τK will tend to zero as well. But this would be an economy with a negligibly small government sector, and therefore of only limited interest in a world of ongoing growth. A more interesting case arises if the government sets its expenditure level optimally. In this case, in the absence of other externalities such as congestion, there are no spillovers from the expenditure decision to financial markets, and in general capital income should remain untaxed; see Turnovsky (1996). 24. This result is similar to that obtained by Corsetti (1991). 25. The finding that bonds have no welfare-enhancing role is also obtained in an overlappinggenerations model by Saint-Paul (1992). In the stochastic equilibrium, where bonds are absent and all expenditures must be financed by taxes, the stochastic component of the government budget constraint (A8) reduces to nM dxCαnK τ K′ dyAαnK dzG0. In this case, it is no longer  The London School of Economics and Political Science 1998

426

ECONOMICA

[AUGUST

possible for dx, dy and dz all to be mutually uncorrelated, as we have been assuming. Now, one of them, presumably dx or dz, must adjust to finance the government budget. It is straightforward to resolve the model under this assumption. 26. The explicit solution for Λ can now be written as

θα (TAΓ)( γαTCa) . ΛG− (1Aγ )[α(TAΓ)C((1Aθ )y(1Aγ ))(γαTCa)]

REFERENCES ABEL, A. B. (1987). Optimal monetary growth. Journal of Monetary Economics, 19, 437–50. BARRO, R. J. (1990). Government spending in a simple model of endogenous growth. Journal of Political Economy, 98, S103–S125. —— and SALA-I-MARTIN, X. (1995). Economic Growth. New York: McGraw-Hill. BENAVIE, A., GRINOLS, E. and TURNOVSKY, S. J. (1996). Adjustment costs and investment in a stochastic endogenous growth model. Journal of Monetary Economics, 38, 77–100. BROCK, W. A. and TURNOVSKY, S. J. (1981). The analysis of macroeconomic policies in perfect foresight equilibrium. International Economic Review, 22, 179–209. CHAMLEY, C. (1985). On a simple rule for the optimal inflation rate in second best taxation. Journal of Public Economics, 26, 35–50. —— (1986). Optimal taxation of capital income in general equilibrium with infinite lives. Econometrica, 54, 607–22. CHARI, V. V., CHRISTIANO, L. J. and KEHOE, P. J. (1991). Optimal fiscal and monetary policy: some recent results. Journal of Money, Credit, and Banking, 23, 519–39. CORSETTI, G. (1991). Fiscal policy and endogenous growth: a mean-variance approach. Unpublished paper, Yale University. DEN HAAN, W. J. (1990). The optimal inflation path in a Sidrauski-type model with uncertainty. Journal of Monetary Economics, 25, 389–409. DRAZEN, A. (1979). The optimal rate of inflation revisited. Journal of Monetary Economics, 5, 231–48. EATON, J. (1981). Fiscal policy, inflation, and the accumulation of risky capital. Review of Economic Studies, 48, 435–45. FEENSTRA, R. (1986). Functional equivalence between liquidity costs and the utility of money. Journal of Monetary Economics, 17, 271–92. FELDSTEIN, M. (1978). The welfare cost of income taxation. Journal of Political Economy, 86, S29–S51. FRIEDMAN, M. (1969). The optimum quantity of money. In M. Friedman (ed.), The Optimum Quantity of Money and Other Essays. Chicago: Aldine. GERTLER, M. and GRINOLS, E. (1982). Monetary randomness and investment. Journal of Monetary Economics, 10, 239–58. GORDON, R. H. (1985). Taxation of corporate income: tax revenues vs. tax distortions. Quarterly Journal of Economics, 100, 1–27. GRINOLS, E. and TURNOVSKY, S. J. (1993). Risk, the financial market, and macroeconomic equilibrium. Journal of Economic Dynamics and Control, 17, 1–36. JONES, L. E., MANUELLI, R. E. and ROSSI, P. E. (1993). Optimal taxation in models of endogenous growth. Journal of Political Economy, 101, 485–517. JUDD, K. L. (1985). Debt and distortionary taxes in a simple perfect foresight model. Journal of Monetary Economics, 20, 51–72. KIMBROUGH, K. P. (1986). The optimum quantity of money rule in the theory of public finance. Journal of Monetary Economics, 18, 277–84. LUCAS, R. E. (1990). Supply-side economics: an analytical review. Oxford Economic Papers, 42, 293–316. —— and STOKEY, N. L. (1983). Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics, 12, 55–93. MALLIARIS, A. G. and BROCK, W. A. (1982). Stochastic Methods in Economics and Finance. Amsterdam: North-Holland. MERTON, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. Review of Economics and Statistics, 51, 247–57. —— (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373–413.  The London School of Economics and Political Science 1998

1998]

MONETARY POLICIES IN A GROWING ECONOMY

427

PHELPS, E. S. (1973). Inflation in the theory of public finance. Swedish Journal of Economics, 75, 67–82. REBELO, S. (1991). Long-run policy analysis and long-run growth. Journal of Political Economy, 99, 1500–21. SAINT-PAUL, G. (1992). Fiscal policy in an endogenous growth model. Quarterly Journal of Economics, 107, 1243–59. SANDMO, A. (1970). The effect of uncertainty on savings decisions. Review of Economic Studies, 37, 353–60. SIDRAUSKI, M. (1967). Rational choice and patterns of growth in a monetary economy. American Economic Review, 57, 534–44. STULZ, R. (1986). Interest rates and monetary policy uncertainty. Journal of Monetary Economics, 17, 331–47. TURNOVSKY, S. J. (1993). Macroeconomic policies, growth, and welfare in a stochastic economy. International Economic Review, 35, 953–81. —— (1995). Methods of Macroeconomic Dynamics. Cambridge, Mass.: MIT Press. —— (1996). Optimal tax, debt, and expenditure policies in a growing economy. Journal of Public Economics, 60, 21–44. —— and BROCK, W. A. (1980). Time consistency and optimal government policies in perfect foresight equilibrium. Journal of Public Economics, 13, 83–212. WOODFORD, M. (1990). The optimum quantity of money. In B. M. Friedman and F. H. Han (eds.), Handbook of Monetary Economics, Vol. 2. Amsterdam: Elsevier Science. ZHU, X. (1992). Optimal fiscal policy in a stochastic growth model. Journal of Economic Theory, 58, 250–89.

 The London School of Economics and Political Science 1998