Provided that there is congestion in the use of government services, the results are the following

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GOVERNMENT EXPENDITURE

AND ECONOMIC GROWTH IN

AN OPEN ECONOMY

Tapio Palokangas

Abstract

This paper examines ination and growth in the presence of internationally traded bonds and a complete public sector. Provided that there is congestion in the use of government services, the results are the following. Growth maximization is optimal policy for a benevolent monetary authority. Capital controls are never optimal and when the tax base is very elastic with respect to the tax rate, ination and currency depreciation promote growth and welfare. From this it follows that if the economies have very dierent conditions of public nance, currency unions comprise a potential risk for growth and welfare.

Journal of Economic Literature: ^O41, ^E31 Keywords: growth, ination, public investment

Introduction

This paper considers the eects of ination on growth and welfare in an open economy where the agents have some access to internationally traded bonds. To integrate dierent systems of exchange rate policy in the same framework, we assume that capital controls are used as a policy instrument.

Consequently, xed exchange rates are equivalent to the domestic and foreign ination rates being held equal, and exible exchange rates are equivalent to the domestic nominal interest rate being held xed. Because the optimal monetary policy is crucially dependent on what scal policy instruments

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are available for use in conjunction with it,¹ the whole public sector must be modeled at the same time with monetary instruments. In this unied approach, the public sector produces services to both consumers and private producers and nances this by taxes and seigniorage.

Almost all papers that consider the relationship of ination and growth assume lump-sum taxation, which eliminates the distortion in public nance.² In such a case, however, there is no explanation of ination: it is not optimal for the government to collect seigniorage at all, because the same nance is more easily collected by taxation. On the other hand, the papers that examine scal policy in the framework of economic growth usually ignore inationary nancing.³ Palokangas (1997) considers ination and growth in a model of public nance, but he ignores the government's spending on services and investment. This study can be viewed as an extension of this literature so that public services and public investment are nanced by seigniorage and distortionary taxation.

Grinols and Turnovsky (1998) consider optimal tax and monetary policies in a closed economy. They assume that money is a consumer good but not a producer good, output is produced from physical capital only subject to a linear stochastic technology, and that government spending is, for some unspecied reason, held in xed proportion to total output. Their key nding is that the real part of the economy is independent of the growth rate of the money stock (= the average ination rate). This study shows that if public expenditure is properly incorporated into the model as a consumption or production good, then this superneutrality result cannot hold: the growth rate of the money stock will increase the nominal interest rate and thereby the consumption and production costs in the economy.

Chang, Hsieh and Lai (2000) show that ination raises the long-run growth rate of the economy unambiguously. The intuition of this is as follows.

A permanent increase in the ination rate decreases real money balances and thereby increases the cost of consumption. This makes the households to substitute real capital for consumption. Because there are constants return to scale with respect to capital, the growth rate of the economy will raise. This result, however, depends on two critical assumptions that are not used in this paper: (I) real balances are held only for consumption but not for investment; and (II) productive capital as such yields utility. Consequently, in this paper, there can be an ination rate that will maximize the growth rate of the economy.

1See e.g. Woodford (1990), p. 1068-1070.

2Cf. Smith (1996), Zhang (1996), Huo (1997) Pecorino (1997), and Chang, Hsieh and Lai (2000).

3Cf. Barro (1990), Barro and Sala-i-Martin (1992) and Greiner and Hanusch (1998).

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Ferreira (1999) attempts to answer the same question as this study by a model where public spending is assumed to increase productivity. He ob- tained following results: (i) growth maximization can never be optimal, and (ii) for some interval of money creation rates a higher ination rate yields a higher growth rate for the economy. Ferreira's results, however, hinge on the following features of the model: (a) the 'young' save at money in order to spend it after retirement, and there is no other motivation for holding money;

(b) there are no taxes; and (c) there are no microfoundations of why public spending would be productivity enhancing. This study shows that when an access to internationally traded bonds (which contradicts (a)) and a complete government sector (in the place of (b) and (c)) are properly incorporated into a growth model, the results are dierent.

This study is structured as follows. Section I explains the assumptions of the model, section II constructs the national income function and in- troduces the framework of monetary policy, and section III presents the savings behaviour of the households. In section IV, optimal public policy is constructed for the case where government services are inputed to private production. This basic model is then extended for the case of government investment in section V. The optimal monetary policy is specied in section V I. It turns out that with distortions in public nance, ination plays an important role in the growth of national economies.

I. The institutional background

Our rst task is to nd a proper theory of money as a basis of the study.

Concerning the role of money in the economy, the most rigorous analysis has been given by Bewley (1980), (1983).⁴ These papers start from the plausi- ble assumption that money does not yield utility as such but is held only as a means to achieving a more desirable consumption allocation.⁵ Bewley introduced a nite number of households, who are dierent in their preferences and initial endownments, into an intertemporal environment with uncertainty. Although this model would have been the best one from the viewpoint of the microfoundations, it is, however, too complicated to be incorporated into a model of economic growth. Therefore, in line with some other authors, we choose the best available substitute and introduce money as an intermediary good that reduces transaction costs.⁶

4A good survey of the Bewley model is presented also in Woodford (1990).

5In Bewley's analysis, cash balances are useful only insofar as they can be used in certain periods to allow consumption of a value greater than the value of a household's endowment in that period. Money is necessary for this purpose when the households' are unable to borrow against future endowment income.

6Zhang (1996), Palokangas (1997), Pecorino (1997) and Ferreira (1999) too introduce

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Second, we must nd a proper model of endogenous growth as well. The alternatives are the multi-sector models, where a specic research sector acts as the engine of growth in the economy, and the AK models, where growth results from the property that goods are produced from reproducible resources only.⁷ Because the existence of a research sector would complicate the model of a monetary economy signicantly, and because, in line with Barro (1990) and Barro and Sala-i-Martin (1992), it is easy to integrate the government sector into the AK model, we end up with the latter.

We assume that there is congestion in the use of public services (i.e. these are rival). This means that a (small) agent takes the amount of public services relative to its own input rather than the total amount of these services in the economy as given. The results will be sensitive to this assumption, but it is hard to invent any examples in which it would not hold. The owner of a car or a lorry benets from roads the more, the more he/she drives.

Defense and security benet more those who have more to defend or pro- tect, i.e. more wealthy people. Education benets a person the more, the more he uses his/her knowledge in his/her work or hobbies. Even the Royal Family benets the British economy the more, the more there is tourism. Be- cause, for technical reasons, government services and government investment cannot be examined simultaneously in the same model, we consider these successively as follows. In sections II-IV, government services are inputed directly to the utility and production functions, while in section V, the government accumulates public capital which yields services to the households and rms.

In our growth model two assets, human and physical capital, are used in production. In line with Barro, Mankiw and Sala-i-Martin (1995) and Palokangas (1996), we eliminate the dierential equation associated with the accumulation of physical capital as follows. Human and physical capital can be distinguished by whether the cumulative goods can serve as collateral for borrowing. Hence, the domestic households cannot borrow with human capital as collateral and foreigners cannot own domestic human capital. This means that the domestic households can purchase any amount of physical capital from abroad using foreign debt, but they can purchase human capital only with their savings. Consequently, if the borrowing constraint for the

the transaction-cost demand for money into a growth model. Furthermore, in a growth model, Smith (1996), Mino (1997), Huo (1997), and Chang, Hsieh and Lai (2000) use a cash-in-advance constraint, which is a special case of the transaction demand for money.

Feenstra (1986) shows that under certain regularity conditions, the approach of placing money in the utility function is equivalent to the transaction-cost approach that is used in this study.

7See e.g. Jones (1998).

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households is binding, then they face an unlimited supply of physical capital from abroad for a given price, but they must nance the accumulation of human capital by saving.

Following the second-best theory of taxation, we assume that lump-sum taxes are incentive incompatible and therefore not feasible. Otherwise, an optimal pattern of resource allocation would be possible without anyone ever holding money and monetary policy would be irrelevant. To make things simple, we assume that a household acts both as a rm, acquiring human and physical capital to carry out production, and as a consumer, making consumption and portfolio allocation decisions to maximize intertemporal utility. The government is assumed to be unable to observe the dierence between private consumption and investment in human capital. From all this and the unlimited supply of physical capital that was mentioned above, it follows that the only eective tax is that the income tax.

II. Production and foreign exchange

There are two groups of agents: households, who consume, save, invest in capital, hold money for transaction purposes, and produce goods from human capital and imported inputs; and brokers, who produce transaction services for the households from goods. Since all goods that are produced, consumed and invested by the domestic households are tradables the relative prices of which are set from abroad, we can aggregate these into a single good which can be consumed, invested, or used as an input by the brokers. This composite good is also used as an input in the government sector.

We choose the foreign price of the composite good as unity, so that the domestic price level is equal to the foreign exchange rate p. Furthermore, we assume that one unit of transaction services is produced from one unit of this composite good, so that the price of transaction services is equal to the foreign exchange rate pas well. We assume, for simplicity, that the domestic agents hold only home currency for transaction purposes.⁸

At the level of the whole economy, outputY is produced from government input G, human capital N and physical capital K through the neoclassical production functionY =F(G;N;K;) with constant returns to scale, where is a parameter that will be specied later. With congestion, a household takes government input relative to the aggregate stock of human capital, g :=G=N, as given and observes government services in the formgN, whereN is the household's own human capital. The household's production function

8This excludes cases of hyperination in which foreign currency replaces domestic money as a means of payment. The introduction of the demand for foreign currency would complicate the model without having any qualitative impact on the results.

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is then given by Y =F(gN;N;K;) =NF(g;1;k;), where k :=K=N.⁹ We choose the unit of physical capital so that the foreign price of physical capital, which is given for the economy from abroad, is unity. Because the household can use physical capital K as collateral, she can nance the purchase of this, K, by borrowing at the real interest ratei. Consequently, it can adjust capital inputK at each moment of time and domestic net wealth is equal to the stock of human capital, N. Now by duality, the tax base Y and the real rate of return on domestic human capital, , become functions of the ratiog, the tax ratex, the real interest rate iand human capital N:¹⁰

(g;x;i;)max

K

[(1^;x)Y ^;iK=N = max

k

[(1^;x)F(g;1;k;)^;ik]; Y =^;^xN >0; ^g ₌: @=@g >0; ^x ₌: @=@x <0; ⁱ ₌: @=@i <0; ^xx ₌: @²=@x² >0; ^xi₌: @²=(@x@i) >0; ^xg ₌: @²=(@x@g) <0: (1) We dene, for convenience, the elasticity of the tax base Y with respect to the tax rate x, when human capital N, government input g and the real interest rate i are held constant, as follows:

:=^;x^xx=^x >0: (2) We assume that the government budget is balanced¹¹ and that the exchange rates are exible, so that the domestic price level p is endogenous.

With capital controls, the domestic real interest rateican exceed the foreign real interest ratei^f which is xed for a small economy. At every moment, the government is precommitted to some nominal interest rate r and some real interest ratei, letting the money supply to adjust to clear the money market.

The private agents take the government instruments, i.e. the tax rate x and the nominal interest rater, as given. We denote the time derivative ofpby _p. Because the brokers transfer funds between the countries until the domestic real interest rate r^;p=p_ is equal to the foreign real interest ratei, we obtain the equilibrium conditions

p=p_ =r^;i; ii^f: (3)

9See e.g. Barro (1990).

10By duality, we obtain the revenue function = (1^;^x;^;i;^g) with the properties ^j ^>0 and ^jh^>0, where subscripts^j and^hdenote the partial derivative with respect to the^jth and ^hth argument from the left. This implies ^x =^;¹ ^<0,ⁱ =^;² ^<0,

g= ³^>0,^xx= ¹¹^>0,^xi= ¹²^>0 and^xg=^;¹³^<0.

11It is possible to introduce the nance of government decit by the issue of tradable bonds as in Palokangas (1996), section^I^V, but the results would still be the same.

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III. Consumption and investment

We assume that government input G benets both production and consumption.¹² Private consumption C and government services G then yield instantaneous utility according to the neoclassical utility function

u(C;G); u^c₌: @u=@c >0; u^g ₌: @u=@g >0; u^cc ₌: @²u=(@c²)<0; u^gg ₌: @²u=(@g²)<0; u^cg ₌: @²u=(@c@g)>0; ulinear homogenous. (4) Otherwise, the households are subject to the intertemporal utility function

Z

1

0

1^;1 ^[u(C;G)^1;^;1]e^;tdt; >0; >0; ⁶= 1;

wheretis time, constantthe subjective rate of time preference and constant the inverse of the intertemporal elasticity of substitution. With congestion, a single household takes government services relative to the aggregate stock of human capital, g :=G=N, as given and observes government consumption in the form gN. This implies that a single household's utility is given by

Z

1

0

1^;1^[u(c;g)^1;N^1; ^;1]e^;tdt; (5) where c :=C=N is the ratio of private consumption to human capital. When the borrowing constraint is binding, the household saves only for the accumulation of her human capital N.

We denote the household's nominal expenditure byE =p[C+ _N], where p is the domestic price level, C consumption, and _N =dN=dt investment in human capital. Let V be the household's demand for transaction services, and := _N=N 0 the rate of investment for human capital. The price for transaction services is equal to the price for goods, p. Nominal transaction costs pV depend on nominal money stock M and nominal expenditure on consumption and investment E (c+)Np.¹³ We can specify this transaction technology as pV =v(M=E)E or

0m := M

E ⁼ M

(c+)Np m; 0v(m) = pV

E ⁼ V

(c+)Np <¹; v⁰ 0; v⁰⁰ >0; v( m) =v⁰( m) = 0; (6)

12The assumption that the government provides services to consumers and producers separately would lead to a more complicated model without any impact on the results.

13It is possible to dene dierent transaction technologies for consumption and investment. This would complicate the model considerably without any eect on the results.

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where the function v is assumed to be thrice dierentiable. From this we obtain the household's budget constraint

(g;x;i;)N =rMp ⁺E

p ⁺V or (g;x;i;) = [1 +rm+v](c+); (7) where N is total real income, rM=p the nominal interest expenditure on money balances, E=p real expenditure on consumption and investment, and V the purchase of transaction services. From the budget constraint (7) and the denition := _N=N, we can solve the growth rate for human capital as

N=N_ = =(g;x;i;)=[1 +rm+v(m)]^;c: (8) The household maximizes her utility (5) subject to the accumulation of human capital (8), by the choice of the consumption-capital ratio cand the money-expenditure ratio m, given the nominal interest rate r and the tax rate x. It is equivalent to maximize by c and m the Hamiltonian (constant term 1=(1^;) omitted)

H =U(C;Z)^1;=(1^;) +N =u(c;g)^1;N^1;=(1^;) +N

=u(c;g)^1;N^1;=(1^;) +(g;x;i;)=[1 +rm+v(m)]^;c N; (9) where variable evolves at each moment according to

_ =^;@H=@N = (^;)^;u(c;g)^1;N^;; lim

t!1

Ne^;t = 0; (10) The maximization of the Hamiltonian (9) by c and r yields the rst-order conditions

=u(c;g)^;N^;u^c(c;g); v⁰(m) =^;r: (11) Given relations (6) and (11), we obtain that the money-expenditure ratio, m, is a function of the nominal interest rater only:

m(r); m⁰ ₌: dm

dr ⁼^;v¹⁰⁰ <0; "(r)^;rm⁰ m ⁼

(>0 for m <m;

= 0 for m= m; ⁽¹²⁾ where"is the elasticity of the demand for money with respect to the nominal interest rate r, when nominal expenditure E is held constant. Since the Hamiltonian (9) is concave in cand m, the optimum is unique.

Because there is only one state variableN, the system jumps immediately to the steady state and there is no transitional dynamics. In the steady state, the ratios of private and public consumption to human capital,c=C=N and

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g = G=N, must be constants. Taking a logarithm of the l.h. equation in (11), dierentiating this with respect to time and noting equation (8), we then obtain the steady-state condition _= = ^;N=N^_ =^;. Given (10) and (11), this condition can be transformed into the form

+ (^;1)=^;^;=_ =u(c;g)^1;N^;= =u(c;g)=u^c(c;g): (13) Now we have a system of two equations (8) and (13), from which we can solve the ratios c and . Dierentiating this system totally and noting (7), we obtain the following functions:

= (x;r;g;i;); ^x = (u²^c^;uu^cc)^x=[(1 +rm+v))(u²^c^;uu^cc)]<0; ^r ₌: @=@r =^;(c+)m^x=^x =^;[m=(1 +rm+v)]^x=^x <0; ^g ₌: @=@g = ^x^g=^x >0; ⁱ ₌: @=@i= ^xⁱ=^x<0; ^g ₌: @=@g;

c(;g); c₌: @c=@ >^;1; c^g ₌: @c=@g: (14) Finally, given (11) and (14), the Hamiltonian (9), which can be used as a measure of welfare for the household, takes the form

H(;z;N) =u(c;g)^1;N^1;=(1^;) +N

=u(c;g)^1;N^1;^f1=(1^;) +=[+ (^;1)]^g

=u^;c(;g);z^1;N^1;=^f(1^;)[+ (^;1)]^g: (15) IV. Public policy with government services In this section, we assume that government services G are made from the numeraire good. Since government expenditure G is then nanced by taxes xY and seigniorage rM=p, given relations (1), (6), (7), (12), (14) and g =G=N, the government's surplus S reads

S :=rMp ⁺xY ^;G= [(c+)rm^;x^x^;g]N =^h rm

1 +rm+v ^;x^x^;gⁱN

=^h(g;x;i;)rm(r)

1 +rm+v(m) ^;x^x(g;x;i;)^;gⁱN = 0: (16) Government expenditure g aects the surplus S both directly and indirectly through the prot function . It is plausibly to assume that the former eect outweighs the latter. Furthermore, the economy must be on the increasing part of the Laer curve. These assumptions take the following form:

Assumption 1

Suppose that human capital N, the nominal interest rate r and the real interest rate i are held constant. Then, the increase in government services G = gN causes a decit in the government's budget, S^g ₌:

@S=@g <0, and an increase in the tax rate, x^g ₌: @S=@g >0.

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Now we can, without losing any generality, solve the budget constraint (16) in terms of the tax ratex. Dierentiating (16) totally, and noting^x <0 from (1), denitions (2) and (12) as well as assumption 1, we obtain

x(r;g;i;) with x^g ₌: @x

@g ⁼^; S^g ^xN

h+ rm

1 +rm+v ^;¹

i>0; + rm

1 +rm+v <¹; x^r ₌: @x

@r ⁼ m

(1 +rm+v)^x

h"+ rm

1 +rm+v ^;¹

i

h+ rm

1 +rm+v ^;¹

i; xⁱ ₌: @x

@i ^{= 1}^x

hx^xi^; rmⁱ 1 +rm+v

i

h+ rm

1 +rm+v ^;¹

i >0: (17) Given the functions (14) and (17), the growth rate can be specied as a function of policy instruments g and r:

(r;g;i;)_{= (}: x(r;g;i;);r;g;i;); @=@i= ⁱ+ ^xxⁱ <0: (18) In the model, all domestic households are similar. Therefore, it is plausi- ble to assume that the policy makers are benevolent, i.e. that they maximize the welfare of the representative household. We call the one deciding on taxation and public expenditure as the scal authority, and the one deciding on the interest ratesrandias the monetary authority, for convenience. Because all policy makers have the same objective function, the result would be the same if scal and monetary policy were exercised by a single authority.

The maximization of the welfare of the household is equivalent to the maximization of the Hamiltonian of the household (15) at each moment of time. It is also equivalent to maximize the following increasing transforma- tion of the Hamiltonian (15) taking the state variable N as given:

(r;i;g;N) = (;z;N)_{= log[(1}:

;)H]=(1^;)

= log u^;c(;g);z+ log[+ (^;1)]=(^;1) + ; (19) where is given by the function (18) and term consists of terms that are independent of g and . Given (4), (14) and (19), we obtain that when g is held constant, welfare is an increasing function of the growth rate:

@ =@= (u^c=u)c+ 1=[+ (^;1)] = (c+ 1)u^c=u >0: (20) The scal authority maximizes the function (19) by government services g. This means that the monetary authority takes g as given and maximizes the function (19) at each moment of time by the interest ratesr and i, given the inequalityii^f from (3). It is equivalent to this to maximize the growth rate by r andisubject to ii^f at each moment of time takingg as given, which leads to the following result:

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Proposition 1

Growth maximization is optimal monetary policy.

This result can be contrasted with Ferreira (1999), who obtains that growth maximization can never be the optimal policy. The reason for this outcome is that Ferreira does not specify any production function: he only assumes that, for some unspecied reason, government spending increases private sector productivity. When both a proper production function and the congestion of public services is incorporated into the model, the result will be reversed. The existence of Ramsey preferences for a household is the necessary condition for the existence of an equilibrium rate of growth in the economy. If the scal authority determines government consumption optimally { or if there is no government consumption at all { then, with Ramsey preferences, the maximization of the private utilities is equivalent to the maximization of the growth rate of the economy.

V. Public policy with government investment In this section, we assume that one unit of public capital produces one unit of services G. Hence, the stock of public capital, which is accumulated from the numeraire good, can be denoted by G. We dene the ratio of public investment to human capital by s = _G=N. Given this and (8), the relative capital stock g :=G=N evolves according to

g_ =d[G=N]=dt= _G=N ^;(G=N) _N=N =s^;g: (21) The government's budget constraint is given by xY + rM=p = _G = sN;

wherexY is tax revenue, rM=pseigniorage and _Gpublic investment. Noting relations (6), (7), (12) and (14), we can write this constraint as follows:

srMp ⁺xYN¹ ⁼ (g;x;i;)rm(r)

1 +rm+v(m) ^;x^x(g;x;i;): (22) Dierentiating (22) with respect to x; r and s, and using (1), (11) and (12), we can express the budget constraint in terms of the tax rate:

x(r;s;g;i;) with x^s ₌: @x=@s= (^x)^;1[+rm=(1 +rm+v)^;1]^;1; x^r ₌: @x=@r =mx^s["+rm=(1 +rm+v)^;1]=(1 +rm+v);

xⁱ ₌: @x=@i=x^s[x^xi^;rmⁱ=(1 +rm+v)]: (23) We assume that the economy is on the increasing part of the Laer curve:

Assumption 2

When the nominal interest rate r and the real interest rate i are held constant, the increase in total government investment s increases the tax rate x, x^s>0.

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This assumption and relations (1) and (23) imply xⁱ > 0. Given this, (14) and (23), we can specify the growth rate in the form of the function

(r;s;g;i;)_{= (}: x(r;s;g;i;);r;g;i;) with ^r ₌: @=@r;

^s₌: @=@s; ^g ₌: @=@g and ⁱ ₌: @=@i = ⁱ+ ^xxⁱ <0: (24) The policy makers maximize the household's welfare (5) subject to the household's equilibrium conditions (14), the accumulation of assets, _N =N and (21). The Hamiltonian of this is (constant term 1(1^;) omitted)

H

= (: (r;s;g;i;);s;g;N;;')

=: u^;c(;g);z^1;N^1;=(1^;) +N +'[s^;g]; (25) where the co-state variables and 'evolve according to

_ =^;@^H=@N = (^;)^;u^1;N^;; lim

t!1

Ne^;t = 0; '_ ='^;@^H=@g

= (+)'^;(@=@)^g^;u^;N^1;[u^g+u^cc^g]; lim

t!1

'ze^;t = 0; (26) and where

@=@ =N^1;u(c;g)^;u^c(c;g)c+N ^;'g: (27) In the steady state, the growth rateand the ratios of private and public consumption to human capital, c and g, are constant. Then, from equation (27) it follows that in the steady state terms N^1; and N must grow at the same rate. Given this, the denition (8), the household's equilibrium condition (13) and equations (26), we obtain

(1^;)= (1^;) _N=N = _N=N + _==^;u^1;N^;=;

u^1;N^;= =+ (^;1)=u=u^c; =u^;N^;u^c: Substituting this into (27) and noting (14) yield

@=@ =u^;N^1;u^c[1 +c]^;'g > ^;'g: (28) The scal authority maximizes the Hamiltonian (25) by public investment s, which leads to the rst-order condition

@^H=@s = (@=@)^s+'= 0: (29) 12

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This behaviour means that the monetary authority takes s as xed and maximizes the Hamiltonian (25) by the interest rates, i and r, subject to inequality ii^f. The Lagrangean of this problem is

L=^H+#[i^;i^f]; (30) where the multiplier # satises the condition

#[i^;i^f] = 0; #0: (31) Using (24) and (25), the maximization of the Langrangean (30) yields the rst-order conditions (31) and

@^L=@r= (@=@)^r = 0; @^L=@i= (@=@)ⁱ +# = 0: (32) If @=@ = 0, then equations (28) and (29) imply '= 0 and @=@ =

;'g > 0, which cannot be true. So there must be @=@ ⁶= 0. On the other hand, if @=@ < 0, then from (24), (31) and (32) it follows that 0 = (@=@)ⁱ + # (@=@)ⁱ > 0, which cannot be true either. So

@=@ is strictly positive, and the maximization of the Lagrangean^Lby the interest rates r and i subject to i i^f is equivalent to the maximization of the growth rate by r and i subject toii^f. We conclude as follows:

Proposition 2

Growth maximization is optimal monetary policy.

VI. Optimal monetary policy

Given propositions 1 and 2, the monetary authority should maximize economic growth. Because from (18) we see that is a decreasing function of the real interest rate i and from (3) that ii^f, growth maximization by i subject to inequality ii^f in yields the following result:

Proposition 3

It is never optimal to use capital controls, so that the domestic real interest rate must always be given from abroad, i=i^f.

From relations (2), (14), (17), (23) andi=i^f, and from the maximization of the growth rate (18) by the nominal interest rate r, we obtain

@=@r= ^xx^r+ ^r = ^x^fx^r^;m=[(1 +rm+v)^x]^g

= ["(r)^;(r;)]^h+ rm

1 +rm+v ^;¹

i m^x

(1 +rm+v)^x ^{= 0}; (33) 13

(14)

where the elasticity of the tax base is given by (r;)₌:

;x(r;g;i^f;)^xx(g;x(r;g;i^f;);i^f;)=^x(g;x(r;g;i^f;);i^f;): We specify now that at all levels of the nominal interest rater, the parameter increases the elasticity of the tax base,@=@ >0. With this specication, can be used as a measure of the distortion in the government sector. Given (17), result (33) implies the Ramsey rule for optimal taxation: the elasticity of money holdings with respect to the interest rate, ", must be equal to the elasticity of the tax base with respect to the tax rate, . The higher are the elasticities =", the higher is the deathweight loss in public nance.

Given (1), (14), (17), (33) and @=@ >0, there must be

@²

@r@ ⁼

h1^;^; rm 1 +rm+v

i m^x

(1 +rm+v)^x @

@ >⁰:

From this, i = i^f, conditions (3) and the rst-order condition (33) and the second-order condition of the government's optimization, @²=@r² < 0, it follows that the ination rate _p=p (= the rate of the depreciation of the domestic currency) increases with :

d( _p=p)=d=dr=d=^;[@²=(@r@)]=[@²=(@r²)]>0: Now the ination rate _p=p is strictly positive for high enough and .

Finally, assume that the tax base becomes perfectly inelastic to the tax rate, ^!0. Given relations=", (6), (11) and (12), we then obtain" ^!0, r ^!0,v⁰ ^!0 and m^!m. These results can be summarized as follows:

Proposition 4

If the elasticity of the tax base with respect of the tax rate, , is high enough for all relevant values of the nominal interest rate r, then ination and the depreciation of the home currency is optimal public policy.

On the other hand, if the elasticity of the tax base is low enough, ^! 0, then Friedman's (1969) rule holds in the limit: the money supply should be extended to the bliss point where the nominal interest rate r is zero.

Propositions 3 and 4 are explained in the last section.

VII. Conclusions

This paper examines optimal public policy in an open economy where the private agents have some access to traded bonds and government services or the services of public capital are needed in production and consumption.

Following Barro (1990), the use of public services are assumed to be subject to 14

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congestion. Because all inputs are reproducible, persistent growth is possible.

Furthermore, because human capital cannot be used as a collateral, this must be nanced by saving. This paper shows that (a) the growth maximization is optimal policy for a benevolent monetary authority, (b) capital controls are never optimal public policy, and that (c) ination and the depreciation of the home currency is optimal for an economy with a very distorted public sector. These results can be explained as follows.

Ination has two eects on economic growth. First, an increase in the ination rate increases seigniorage, the supply of government services, private output, private saving and ultimately the accumulation of human capital. On the other hand, the increase in the ination rate also increases transaction costs in the private sector and thereby reduces the rate of return on human capital and the accumulation of human capital. Given these two opposite eects, there is an optimal ination rate that maximizes the accumulation of human capital and economic growth. Because the scal authority determines the scal policy instruments, welfare maximization by monetary policy is equivalent to growth maximization by monetary policy.

If capital controls are imposed, then the domestic real interest rate will be above the foreign real interest rate. In such a case, the domestic households pay more for investment in physical capital, which decreases the rate of return on human capital. This decreases the accumulation of human capital, economic growth and social welfare, so that capital controls should never be imposed. In an open economy with free capital movements, a positive ination rate is associated with the depreciation of the home currency. If the tax base is very elastic to the tax rate, then there is a large distortion in the public sector. In such a case, the positive eect of inationary nance on growth through government services is so strong that the optimal ination rate is positive. Consequently, it is optimal to let the currency to depreciate.

Correspondingly, if the tax base is very inelastic, the disctortion in the public sector is small and one should let the currency to appreciate.

The results above mean that if the economies have very dierent conditions of public nance, then a currency union involves a potential risk for growth and welfare. For an economy with a relatively large distortion in public nance, the optimal ination rate is very high. In a currency union, this economy will be subject to a lower than optimal ination rate and therefore, it will lose seigniorage and end up with lower government spending and slower economic growth. On the other hand, for an economy with a relatively small distortion in public nance, the ination rate is above the optimal level and therefore it will be subject to higher transaction costs and slower economic growth. Hence, from the viewpoint of economic growth and welfare, the economies must be similar for not to lose in monetary integration. Although

15

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we admit that monetary integration can be welfare enhancing for some reasons (e.g. stabilization policy in an uncertain environment) that are ignored in our model, our results show that growth considerations can nevertheless be important in the establishment and enlargement of a currency union.

The nal conclusion of this paper is that if a stylized mathematical model is used to analyse the eects of public policy on economic growth, then one should pay attention to the proper microfoundations of the model. In particular, the eects of monetary policy should examined in conjunction with the behaviour of the scal authority.

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17

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In this Appendix, we refer by equations (1)-(28) those in the paper.

The functions (14)

Dierentiating (13) totally and noting (4), we obtain (^;1)d= [1^;uu^cc=u²^c]dc+ [u^g=u^c^;uu^cg=u²^c]dg and

c(;g); @c=@ = (^;1)u²^c=(u²^c^;uu^cc)>^;u²^c=(u²^c^;uu^cc)>^;1: (34) Substituting this into the r.h. equation in (7), we obtain

[1 +rm+v(m)][c(;g) +] =(g;x;i;); (35) where, by the r.h. equation in (11), we can take the optimal value m of m as given. Dierentiating equation (35) totally and noting (7) and (34), we obtain

^gdg+^xdx+ⁱdi= (c+)mdr+ (1 +rm+v)[(c+ 1)d+c^gdg]

= (c+)mdr+ (1 +rm+v)(c+ 1)d+ (1 +rm+v)c^gdg

=⁽³⁴⁾ (c+)mdr+ (1 +rm+v)^h(^;1)u²^c u²^c^;uu^cc ^{+ 1}

id+ (1 +rm+v)c^gdg

= (c+)mdr+ (1 +rm+v)u²^c^;uu^cc

u²^c^;uu^cc d+ (1 +rm+v)c^gdg

=⁽⁷⁾ m

1 +rm+v dr^{+ (1 +}rm+v)u²^c^;uu^cc

u²^c^;uu^cc d+ (1 +rm+v)c^gdg and

@@x ⁼ ⁽u²^c^;uu^cc)^x

(1 +rm+v)(u²^c^;uu^cc) <0; @@i ⁼ ⁱ ^x @

@x <⁰;

@@r ⁼^;⁽c+)m ^x @

@x <⁰; @@g ⁼ ^g ^x @

@x >⁰:

1

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The function (17)

Dividing (16) by N, we obtain

S=N =(g;x;i;)rm(r)=[1 +rm+v(m)]^;x^x(g;x;i;)^;g = 0; where, by the r.h. equation in (11), we can take the optimal value m of m as given. Dierentiating this with respect tog,x,rand iand noting (2) and (12), we obtain

;

S^g N dg⁺

idi

=^h rm^x

1 +rm+v ^;^x^;x^xxⁱdx+^h(rm⁰ +m)

1 +rm+v ^; rm² (1 +rm+v)²

idr

=^h rm

1 +rm+v ^;¹^; x^xx ^x

i^xdx+^hrm⁰

m ^{+ 1}^; rm 1 +rm+v

i m

1 +rm+v dr

=^{(2) ;(12)} ^h rm

1 +rm+v ^;^{1 +}ⁱ^xdx+^h1^;"^; rm 1 +rm+v

i m

1 +rm+v dr;

from which we solve

@x@r ⁼ m (1 +rm+v)^x

h"+ rm

1 +rm+v ^;¹

i

h+ rm

1 +rm+v ^;¹

i;

@x@g ⁼^; S^g ^xN

h+ rm

1 +rm+v ^;¹

i and

@x@i ^{= 1}^x

i

h+ rm

1 +rm+v ^;¹

i:

Because @x=@g > 0 and S^g < 0 by assumption 1, and ^x < 0 by (1), we obtain +rm=(1 +rm+v)<1.

2

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The function (23)

Dierentiating (22) with respect to (x;r;i;s) and noting (2), (12) and r =^;v⁰ from (11) yields

ds=^h rmⁱ

1 +rm+v ^;x^xiⁱdi+^h m+rm⁰

1 +rm+v ^; rm² (1 +rm+v)²

i dr +^h rm^x

1 +rm+v ^;^x^;x^xxⁱdx

=^h rmⁱ

1 +rm+v ^;x^xiⁱdi+^h1 + rm⁰

m ^; rm 1 +rm+v

i m dr 1 +rm+v

+^h rm

1 +rm+v ^;¹^; x^xx ^x

i^xdx

=^{(2) ;}⁽¹²⁾ ^h rmⁱ

1 +rm+v ^;x^xiⁱdi+^h1^;"^; rm 1 +rm+v

i m dr 1 +rm+v

+^h rm

1 +rm+v ^;^{1 +}ⁱ^xdx;

which implies

@x@s ⁼

h rm

1 +rm+v ^;^{1 +}ⁱ^;1¹^x;

@x@r ⁼^;

h1^;"^; rm 1 +rm+v

i m

1 +rm+v

h rm

1 +rm+v ^;^{1 +}ⁱ^;1¹^x

=^h"^;1 + rm 1 +rm+v

i m

1 +rm+v @x

@x @s;

@i ⁼^;

h rmⁱ

1 +rm+v ^;x^xi^ih rm

1 +rm+v ^;^{1 +}ⁱ^;1¹^x

=^hx^xi^; rmⁱ 1 +rm+v

i@x

@s:

Equation (33)

Given relations (2), (14), (17), (23) and i =i^f, the maximization of the growth rate (18) by the interest rate r yields the rst-order condition

@=@r=^{(14) ;(17)} ^xx^r+ ^r =⁽¹⁴⁾ ^x^fx^r^;m=[(1 +rm+v)^x]^g

=⁽¹⁷⁾ ^xⁿ m (1 +rm+v)^x

h"+ rm

1 +rm+v ^;¹

i

h+ rm

1 +rm+v ^;¹

i

;

(1 +rmm+v)^x

o

= m^x

(1 +rm+v)^x

n h"+ rm

1 +rm+v ^;¹

i

h+ rm

1 +rm+v ^;¹

i

;1^o

=^h+ rm

1 +rm+v ^;¹

i m^x

(1 +rm+v)^x^["(r)^;(r;)] = 0: 3