Lumpsum taxation and the superneutrality and optimum quantity of money in life cycle growth models

Journal of Public Economics 36 (1988) 339-367. North-Holland

LUMP-SUM TAXATION AND THE SUPERNEUTRALITY AND OPTIMUM QUANTITY OF MONEY IN LIFE CYCLE GROWTH

MODELS

Firouz GAHVARI* Department of Economic.s. Virginia Pol.vtechnic Institure and Stare lJniversit_v. Blacksburg.

VA 24061. USA

Received July 1986, revised version received October 1987

The question of superneutrality and the optimum quantity of money is addressed in a model where money services enter the utility function and where individuals have finite lives. It is demonstrated that the existing result in the literature regarding a positive relationship between money creation and the steady-state capital intensity and welfare is not due to monetary policy per se. On the contrary, this is shown to arise because of the intergenerational wealth effects embedded in the manner in which money is created. Assuming that preferences are separable between consumption goods and real balances and that the consumption goods are normal, it is proved that (i) money is superneutral and (ii) any other money supply rule is Pareto inferior to the Chicago Rule provided that the government also embarks on a policy of wealth redistribution across generations through the imposition of appropriate lump-sum taxes and transfers.

1. Introduction

The effects of money creation on capital intensity and welfare have been the two major questions discussed in the money-growth-welfare literature in the past two decades. Sidrauski (1967) has shown that money is ‘superneutral’ (i.e. the steady-state capital intensity is invariant to the changes in the rate of money creation) in an economy characterized by ‘infinitely-lived’ individuals. Tobin (1965) on the other hand, had shown, within the setting of a neo-classical growth model, that money is not superneutral. On the contrary, he had found a positive relationship between money creation and capital intensity.’ This result has since been confirmed for life cycle growth models with overlapping generations by Weiss (1980) Carmichael (1980), and Drazen (198 1).

In the optimum quantity of money literature, the most well-known result

*The author wishes to thank Clive Bull, Larry KotlikoII, Keiichi Koda, and Warren Weber for discussions and comments on the lirst draft of this paper. The current version has benefited substantially from the comments of anonymous referees as well as discussions with Jacques Crtmer. I am grateful to them all.

‘This tinding has further been analyzed and elaborated on by Summers (1981).

004772727/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

340 F. Gahvari, Lump-sum taxation

has been the so-called ‘Chicago Rule’ for the optimal money supply, that is, a rate of deflation equal to the real rate of interest.2 As with superneutrality of money, this result has generally been based on the assumption of an ‘immortal family’. Weiss (1980) argues that the Chicago Rule is not attained in an economy with ‘finitely-lived’ individuals. He goes on to show that steady-state welfare maximization requires a posit& rate of money creation. Wallace (1980) denies the validity of the Chicago Rule. He writes: ‘In most models, this is not even a well-defined prescription since the equilibrium rate of return on capital depends on the monetary policy followed’ (p. 60). Summers (1981), in a Solow-Swan type growth setting, also finds that inflation can have important welfare effects.3

This paper argues that the finding of a positive relationship between money creation and the steady-state capital intensity and welfare in life cycle growth models is not due to monetary policy per se. On the contrary, it is a direct result of the intergenerational wealth effects embedded in the set-up of these models. Through imposition of lump-sum taxes and transfers, money can be made to be superneutral and the Chicago Rule a Pareto superior policy given the present stock of capital in the economy.

The intuition behind this result is quite simple. An unexpected cut in the rate of increase in money supply will create an unexpected capital gain in the money holdings of those who hold money in their portfolios. In an infinitely- lived individual framework, the question of who receives the newly created money does not arise. Similarly, when individuals have intergenerationally interdependent utilities, as in Barro (1974), no such problem arises. This is so because these two settings are essentially equivalent.4 In a life cycle model, on the other hand, the question of who receives the newly created money assumes a central role. Indeed, this is the source of the sharply different result derived for life cycle models by Weiss and others as compared to the earlier result of Sidrauski and Friedman for infinitely-lived individual settings.

‘See Bailey (1956), Friedman (1969), and Johnson (1969). These papers, however, are rather informal. Turnovsky and Brock (1980) address this issue, among others, formally in a general equilibrium one-consumer setting. They show that the Chicago Rule will not hold for a general utility function and prove that this rule ‘is obtained if and only if the marginal rate of substitution of consumption for labor is independent of the real stock of money balances’ (p. 197).

3There are other criticisms of the Chicago Rule which do not directly concern the present paper. The recent overlapping generations model of money (without money in the utility function) reject the Chicago Rule [see, for example, Wallace (1980)]. But this is to be expected considering the radically different way of modelling money in this latter literature. Moreover, the Chicago Rule has been questioned on the basis of the second-best theory of taxation. On this, see Nagatani (1981, ch. 10) and Charnley (1985). These two latter works, however, assume an immortal family.

4For a demonstration, in the present context, see Carmichael (1980).

F. Gahvari, Lump-sum taxation 341

The monetary growth models typically postulate that money is created by lump-sum distribution to those who hold monetary assets. In the common two-period life cycle formulations, as in Weiss (1980), these persons happen to be the old. With no bequest motive operating, the old will have no incentives to pass to the young some of the benefits they get, under a deflationary policy, through the capital gains they experience in their holdings of money. In this circumstance, lump-sum transfer schemes will serve to ‘connect’ the utility of the young and the old in the same manner that Barro’s bequest motive does. Consequently, one can ‘mimic’ the same result here as the one derived for infinitely-lived individual models.5

It is also of some interest to recognize here that the lump-sum transfer scheme discussed in this paper does not give the government any additional policy tools over and above what has traditionally been assumed in the money-growth literature. One only has to remember that money in these models is typically created by means of lump-sum distributions. The twist here is that the lump-sum transfers should be made to all generations rather than just the old, albeit at different rates.

The analysis is carried out within a two-period overlapping generations model. It is assumed that individuals derive utility from money services as well as consumption goods. The assumption that money services directly enter the individuals’ utility functions is controversial these days. It is not the purpose of this paper to enter into this debate. In a recent issue of this Journal, Charnley (1985) has briefly reiterated the kind of justification that one might have for modelling money this way (in the context of the problems that are addressed in this paper).6 In any event, the main purpose of this paper is to demonstrate the full range implications of lump-sum transfers in life cycle models, and particularly in life cycle monetary growth models which allow such mechanisms. It is then only natural to ‘borrow’ the set-up of these models.

The paper first demonstrates that, given the assumption that money is created by lump-sum distribution to those persons who are holding money in their portfolios (as in Weiss), steady-state welfare maximization would in general require violation of the Chicago Rule. It is proved that (assuming the real rate of interest associated with a constant money supply is greater than the population growth rate) a positive rate of money creation will enhance steady-state welfare if such a policy increases capital intensity. On the other hand, if capital intensity declines with money creation, money supply must

‘Another example of lump-sum transfers fulfilling the same function as Barre’s bequest motive is provided in Gahvari (1982) and Fane (1984). They show that lump-sum transfer schemes give rise to the same result on neutrality of land taxation as the one derived by Calvo et al. (1979) on the basis of a Barro-type utility function. This latter paper had critiqued an earlier assertion by Feldstein (1977) to the contrary.

%ee also McCallum (1982) for a defense of money-in-the-utility-function in an overlapping generation’s framework.


be cut in each period. Furthermore, Weiss’s result on the positive relationship between money creation and the steady-state capital intensity (and hence welfare) is shown to be rather robust with respect to the specification of the utility function. This result is generalized to non-additive utility functions (he assumes additive preferences across time as well as between consumption goods and real money balances).’ Sufficient conditions for validity of this result are that the consumption goods must be normal and gross substitutes for money services.

Secondly, assuming that preferences are separable between consumption goods and real balances and that the consumption goods are normal, it is shown that (i) money is superneutral and (ii) any initial money supply rule is Pareto inferior to the Chicago Rule, provided that the government takes away the unexpected capital gains in the initial old’s holdings of money and gives them to the young by means of lump-sum taxes and transfers. Moreover, this policy of redistribution from the old to the young should continue for all periods in the future.

Section 2 sets out the model. Section 3 discusses existence and stability of the steady state. Sections 4 and 5 discuss the effects of increasing the rate of monetary expansion on the steady-state capital intensity and the steady-state welfare, respectively, given the assumption that the unexpected capital gains (losses) of a deflationary (inflationary) policy are borne by the initial old. Section 6 analyzes the same issues when the government utilizes lump-sum taxes and transfers. Section 7 provides an example. Finally, a brief conclusion

ends the paper.

2. The model

It is assumed that individuals in this economy are all identical and live for two periods, while only working in the first period. Individuals derive utility from consuming a composite consumption good, c, in both periods and money services, x, only in the first period. No utility is assigned to leisure, i.e. labor is supplied inelastically. The intertemporal utility function is thus written as:

u: = 44, cP+ 1, x,), (1)

where superscripts y and o refer to young and old, respectively, and subscript t denotes time. It is further assumed that ti( .) is quasi-concave, twice differentiable in all its arguments and that the lim,,,, au/acy = lim,,, au/ax =

‘Drazen’s specification of the utility function is less restrictive than Weiss’s He assumes only intertemporal separability.


lim c0_0 au/W= cu. Furthermore, while u( .) is assumed to be strictly increasing in cy and co, the possibility of satiation in real balances is not ruled out

(i.e. lim,,, &/8x=0). This possibly must be allowed for the Chicago Rule to

be well defined. The production of consumption goods are carried out with an unchanging

neoclassical technology:

r, = WG. LA (2)

where Y,, K,, and L, denote aggregate output, aggregate capital, and aggregate labor, respectively, at time t. It is assumed that output of each period can be used for consumption in the same period, C,, or as capital, K ,+ 1, in the next period production process. It is also assumed that capital does not depreciate.

The production technology is assumed to exhibit constant returns to scale. The production function can thus be rewritten as:

Y, = f(k), (2’)

where y, and k, denote output per labor and capital per labor ratios, respectively. It is further assumed that f( .) satisfies the following regularity conditions: f(O)=O; f’(k)>0 and f”(k)<0 for 0-ck-c co; lim,_,f’(k)= cc and lim,,, f’(k) =O. Assuming a competitive setting, the real wage w, (measured in units of the composite consumption good), and the real rate of interest, rt, can be found according to:

w, = Y, - W”(k) (3)

and

rt = f’(k). (4)

Money services are assumed to be proportional to real money holdings. In fact, the unit of measurement of money services is chosen so that

where M, is the stock of money at time t, L, is the number of young persons, and pr is the money price of the consumption good.

It is assumed that the population grows at the constant rate of n. That is,


L,=(l +n)L,_,. (6)

The stock of money is also assumed to increase in each period by a constant rate equal to 8 (if 19~0, money supply is being reduced). No public consumption is assumed and the money stock is increased (decreased) through the government’s lump-sum money payments to (taxes from) the old

at the beginning of each period.* The device of distributing money to those who hold monetary assets, namely the old, ensures that expected inflation (or deflation) will not induce direct wealth transfers across generations.9

It is assumed that the old leave no bequests and there is no government ownership of capital. All savings in this economy are made by the young in order to maximize their lifetime utility. In this setting, then, the savings of the young at each period are just sufficient to buy up all the existing stock of assets, namely capital to be used next period plus the outstanding money balances.

At the beginning of period t, then, the old enter the market owning all the capital assets to be used in that period, K,, plus the stock of money used in the previous period, MtP1. In addition, each old individual receives sums of money from the government equal to sy. The young, on the other hand, are the sole suppliers of labor, L,. For the purpose of production, it is of no consequence to assume whether the young are employed by the old or the old rent out their capital assets to the young. Thus, at the end of the production process, and just before the start of consumption in this period, the old would have a total of resources for consumption equal in value to p,K,( 1 + rJ +(M, _ 1 + L,_ 1sy). And since the old leave no bequests, they would consume all their resources. That is, they set the above equal to L,_ Iptcp. It then follows that

L,-,~,cp=~,K,(l+r,)+(M,-,+L,-,sp). (7)

Turning to the young now, they have, at period t, a total of resources equal in value to L,p,w,. The young will spend L,p,c: on consumption goods. The savings of the young in period t will then be equal to L,(p,w,-p,c:) with which they buy all the assets in the economy at the end of period t, namely K ,+1 (capital to be used next period) and M, (money that was used this period). Thus:

L,(ptwt - P&‘) = PA + I+ Mt. (8)

8This assumption will be dropped in section 6 below. 9Nevertheless, unexpected inflation will continue to affect the wealth position of digerent

generations differently. Indeed, it will be shown below that the results on non-superneutrality of money and the positive relationship between inflation and steady-state welfare is due to this impact of money creation.

F. Gahuari, Lump-sum taxation 345

Putting eqs. (7) and (8) together results in the lifetime budget constraint for each young individual:

cP+ 1 CT+-

It+ 1% ST+1 Pr+1 I l+r,+, I+&+1

+-=w,+I+r,,,, (9)

where

(1 +i,+l)-(l +r,+l)(l +74+1), (10)

P!+l= 1 +n,+,. Pi

(11)

It is now assumed that individuals in this economy have perfect foresight. Each period the young maximize the intertemporal utility function (1) subject to their lifetime budget constraint (9). The first-order conditions are given by:

aqacp, 1 1

aqacy l+r,+, 3

wax, i, + 1

aqac; l+i,+I’

(12)

(13)

Finally, it is assumed that the government sets its lump-sum money payments to each old individual equally. That is,

s;=BM,_,/L,_,. (14)

One final equation is needed to close the system. This is provided by:

M,=M,_,+L,_,$. (15)

Eqs. (2’)-(5) and (7)-(15) will then determine the current values of all the

endogenous variables of the model (c:, x,, C-F+ r, y,, k,+ 1, r,, wt, p,, n,, i,, sP, M,, and L,) under the assumption of perfect foresight.

3. The steady state and stability

The dynamic evolution of this economy can be determined as follows. First, by utilizing eqs. (6), (15), and (5), eq. (7) may be rewritten as:

cP+ 1 = (1 +n)(l +r,+l)k+l +(1+4~,+~. (7’)


Second, using eqs. (6) and (5) again, eq. (8) may be rewritten as:

c:+x,+(l +n)k,+r =w,.

Third, using eqs. (15) (6), (ll), and (14), eq. (5) may be rewritten as:

1+8 X

t+1=(1+n)(l+7r,+1)Xr.

(8’)

(5’)

Finally, from eqs. (7’), (8’), (5’) (lo), (12) (13), (3), and (4) the following pair of first-order difference equations can be derived:

k t + 1= hk,, xt, @, (16)

x,+ 1 =g(k,> x,, 0). (17)

At the beginning of some arbitrary period taken as zero, k, is predetermined by past capital accumulation. The equilibrium value of the real

balances, x0 (or equivalently the initial price of consumption goods in terms of money, p,,), will be determined on the basis of expectations about future prices. Assume that such an equilibrium exists and is unique (more will be said below on the existence and uniqueness of the equilibrium). The system of difference equations (16) and (17) will then determine the entire path of the values of (k,,x,). These values will tend to a steady state if there exists a steady-state growth solution and if it is stable.

The steady-state solution of the system, if it exists, will be determined by finding the rest point of the above two equations. The solution is characterized by the nine equations, (2’)-(5’) (7’)-(8’) (lo), (12) and (13) without the time subscripts. These equations should, in principle, be sufficient to determine the nine steady-state variables (y, k, w, r, cy, co, x, i, 7~) which remain constant over time. Although existence seems likely, it is by no means guaranteed here. The regularity conditions imposed on the preferences and the production technology ensure positive values for all the variables of the economy along a solution path. However, there is no guarantee that there exists a steady-state solution to which the system might (or might not) be driven. With this reservation in mind, the paper follows the standard practice


in this literature and simply assumes existence.iO (An example is provided in section 7 where the steady state exists and is locally stable.)

The (local) stability properties of the steady-state solution can be examined in the following manner. By linearization of the system of difference equations (16) and (17) around the steady-state solution, it is found that:

where $i, &, g,, and g, respectively denote the partial derivatives of c$( .) and g( .) with respect to k and x, while k* and x* stand for the steady-state values of k and x.

The dynamic path given by (18) converges to a steady state (i.e. k, - k* and x,-x* would tend to zero) as t tends to infinity, if at time t =0 the initial values of k, and x0 are such that the initial vector (k,- k*, x0-x*) is in the space spanned by the eigenvectors of A which are associated with eigenvalues of A with modulus smaller than one. Now, since k, is predetermined, an equilibrium value for x0 exists and is unique if and only if A possesses only one eigenvalue with a modulus less than 0ne.i’ This is the assumption employed in this paper.

4. Money creation and capital intensity

In order to evaluate the long-run effects of money creation on capital

roThe existence assumption is adopted by almost everyone in the literature dealing with the overlapping generations models with neoclassical production (with or without money). See, among others, Diamond (1965, 1970) Pestieau (1974) Kotlikoff and Summers (1979) Atkinson ans Sandmo (1980), Atkinson and Stiglitz (1980, Lecture 8) Weiss (l980), Carmichael (1980) Drazen (1981) Tirole (1985) Sandmo (1985, ch. 5) and Charnley and Wright (1987). A referee rightly pointed out that unless existence is established, a host of results that exist in this literature on the ‘long-run’ impacts of various forms of taxation, public debt, and money might all be vacuous. This is clearly a most important shortcoming of this literature because at present there are no general results regarding the existence issue. Undoubtedly, this issue must, at some stage, be addressed. It, nevertheless, remains beyond the scope of the present paper.

There is one aspect of existence in monetary overlapping generations models which has been addressed in the literature. The problem concerns the possibility of money, in the limit, being driven out of the system despite the regularity assumptions on the preferences which ensure a positive holding of real balances all along a solution path (hyperintlationary equilibrium). Scheinkman (1980) and Obstfeld and Rogoff (1983) have shown that a sufficient condition (but not a necessary one) to rule this possibility out is to impose the following restriction on preferences: lim x_o~ c’u/?~>O. This restriction makes money, in some sense, essential to the economy.

“When there are two eigenvalues with modulus smaller than one, the system will be globally stable. In that case, there will in general exist a continuum of admissible values for xc, Consequently, there will exist a continuum of perfect foresight equilibria for the economy, all of which converge to the same steady-state values. For an early discussion of this possibility in a non-maximizing monetary model, see Black (1974) and in a maximizing non-monetary framework, see Calvo (1978).


intensity, the steady-state value of capital must be found and then totally differentiated with respect to 8. First, it is noted that the general equilibrium response of the economy to any increase in the rate of money creation depends crucially on wherether the young or the old would receive the extra money created. The assumption that the old would receive all the increase in money supply (pay all the money taxes), ensures that any direct and expected wealth transfers across generations are avoided. In this manner, the resulting genera1 equilibrium changes in the economy, consequent upon a change in the rate of increase of money supply, will be due to the substitution effects induced by this change and not the direct intergenerational wealth transfers. Moreover, the results of this exercise would provide a benchmark for the analyses of compensation schemes which involve (direct) intergenerational wealth transfers.

From eqs. (5) and (8) it follows that:

k t+1 =(w,-c:-xX,)/(1 +n),

which at the steady state will be rewritten as:

k=(w-cY-x)/(1 +n).

Differentiating k totally with respect to 0 and simplifying yield:

dk wpe + ax/a0 de- i + n + acy/ak+ ax/ak + kf”(k)’

(19)

(19’)

(20)

The imposition of the Hicksian static stability condition on the steady state ensures that the denominator of (20) will be positive.12 On the other hand, it is shown in appendix A that the numerator of (20) cannot in genera1 be signed. Consequently, the sign of dk/dtJ will be indeterminate. The foregoing result is summarized in Proposition 1.

Proposition 1. Given a general utility function, an increase in the rate of money creation would have (a) a real effect, and (6) the direction of the effect would be ambiguous.

The first conclusion of Proposition 1 (dk/de#O) is in direct contrast to the

t2Note that this is a static stability condition in the tradition of Hicks (1946). It is different from the dynamic stability condition discussed in section 3. This is the stability condition imposed by Weiss (1980) on his system. It has also been imposed by other economists in other contexts, e.g. Feldstein (1977) and Kotlikoff and Summers (1979). Essentially, it requires that the steady-state demand for capital, as a function of the real interest rate, to be less steep than the supply of capital, Obviously, while an upward-sloping supply of capital function (along with a downward-sloping demand curve) will be sufficient for this result, it is not necessary.


result derived by Sidrauski (1967). In Sidrauski’s world money is superneutral (i.e. steady-state real variables are invariant to the inflation rate).13 This result, of course, crucially depends on the specification of an immortal family. Given certain assumptions, individuals who live for ever would choose a level of capital intensity which corresponds to that of the modified golden rule. On the other hand, the steady-state rate of return on capital for finitely-lived individuals does not generally coincide with that of the golden rule. That rate is determined endogenously [see Diamond (1965)]. It is not then surprising that the steady-state capital intensity should vary as a result of a change in the rate of inflation (an increase in the price of money services).

The second conclusion of Proposition 1 is in contrast to the result in Weiss (1980) and Drazen (1981) as well as to that in Tobin (1965).i4 In a framework similar to that of the present paper, Weiss and Drazen have shown that with separable preferences, inflation and steady-state capital intensity are positively correlated. An intuition into the different result derived here may be gained by considering what might happen to the second-period consumption, co, as inflation increases. With an increase in the inflation rate, consumption of co would decline if co were a net complement to .x.i5 It then follows that savings per young would also have to decline (because a lower amount of savings would be required to finance a decreased level of second-period consumption). If the reduction in the required savings happens to exceed the reduction in real money balances, holdings of capital would also decline. The assumptions of separability, and normality of consumption adopted by Weiss and Drazen essentially ensure that co would be a net substitute to x, thus ruling out this possibility.

It is clear from the above discussion that the additivity restriction is not necessary to derive the result that inflation increases capital intensity. In fact, in appendix A it is also shown that, at the initial value of 8= 0, dxl~30 ~0; while if co is a net substitute to x or if cy and co are both normal goods and gross substitutes to x, then c?c”/d0>0. The following proposition emerges:

Proposition 2. The assumption that co and x are net substitutes (or cy and co are normal goods and gross substitutes to x), is sufficient to ensure that dk/ dH > 0 at the initial value of’ 0, = 0.

As a final observation, it is pointed out that the separability assumption

t3Note that superneutrality is defined with respect to the invariance of steady-state values of real variables with respect to the inflation rate. Of course, this does not mean that these variables remain unchanged on the transition path. indeed, money is not neutral in this latter sense in Sidrauski’s model.

t4Tobin’s result (the so-called Tobin effect) is derived in a non-optimizing framework. There, also exists a direct (non-portfolio) impact on asset accumulation as a result of the interaction of an ad hoc saving function and the postulated impact of inflation on disposable income.

ts Recall that money is created by giving it to individuals.


ensures that dk/dO>O for any initial value of 6 and not just c?=O as had been found by Weiss. Proposition 3 generalizes Weiss’s result in this directionI

Proposition 3. Given additive preferences, an increase in the rate of money creation will raise the steady-state capital intensity, irrespective of the initial value of the inflation rate.

5. Optimal money supply in the absence of lump-sum taxation

This section investigates the relationship between the rate of change in money supply and the steady-state utility. The steady-state utility is thus totally differentiated with respect to 0:

where ur = duJi7cY, u2 = &.Q’&-~, and uj = &ldx. From (A.3) the above can be simplified to:

du gijj=u1

where

and

(22)

(23)

(24)

Substituting from (AS) into (23) and (A.8) into (24), and evaluating at the initial value of 8 = 0, it is seen that:

A=O, (230

B= - kz-; kf “(k),

16Drazen’s fmding may also be confirmed from appendix A by setting u,~=u~~=O in (A.5), assuming normality for cy and x, and taking note of the quasi-concavity restriction imposed on

u( 9


so that

(25)

From eq. (25) it is observed that (du/d@, is of the same sign as (r - yt) dk/d0. Now since at 0 = 0, i = (r - n)/( 1 + n), one also has to impose the condition that r2n.l’ It follows that capital intensity should be increased (unless r =n which implies l3=0 is optimal and capital intensity should not change). This would require increasing the rate of change in the money supply if (and only if) dk/de> 0 and decreasing it if (and only if) dk/dt’<O (assuming r>n). Weiss’s result (based on separability) is thus generalized in the following proposition:

Proposition 4. The assumption that co and x are net substitutes (or cy and co are normal goods and gross substitutes to x,) is sufficient to ensure du/de>O at

the initial value of’ 0, = 0.18

The above findings are in agreement with the usual golden rule results. As was originally demonstrated by Samuelson (1958) and Diamond (1965), competitive economies with infinite time horizons and without complete markets for all future goods in general settle on steady-state paths where the representative individual’s utility is less than that which is feasible.19 In these circumstances, introducing ‘distortionary’ taxes [as in Diamond (1970)] may be ‘beneficial’ in that they can be used to increase the level of steady-state welfare. In the present context, increasing the opportunity cost of holding money (by raising 0) plays a similar role to taxation.

From the above equations, it can also be verified that steady-state utility maximization neither requires r=n (unless it so happens that at the initial value of 8 =O, r = n), nor the Chicago Rule of i =O. One question then remains: What, if anything, is wrong with this rule?

The key to this apparent paradox must be sought in the changes that would occur in the wealth position of the individuals who live on the transitional path from one steady state to the next when the rate of monetary expansion is altered. In the original specification of the problem, individuals were assumed to live indefinitely. Thus, no distinction could arise

“Note that this assumption immediately puts the economy outside the ‘inefficient’ range of the golden rule results.

180L course, the results of this paper imply that normality plus additivity of the utility function also ensure that (du/dH), will be positive.

19Note that this is not to say that these paths are necessarily Pareto inefficient. For that it is not just the steady-state utility which is at stake, but the utility of all the existing and future generations.


between individuals on the transitional path and those at the steady state. Here such a distinction becomes of central importance as the following discussion clarifies.

Assume that the economy is initially at a steady state with 8=0. The government unexpectedly starts a policy of deflation by continually reducing the money supply at a rate equal to 8 through lump-sum money taxes levied on the old and ‘burning’ the proceeds. This policy of deflation would reduce the nominal rate of interest and thus increases the demand for real balances. At least in the first period after the announcement of the new policy, then, the rate of deflation would have to be higher than the rate of decrease in the money supply (adjusted for population growth). Hence, the initial old, who hold all the stock of money, would definitely be better off as a result of this policy. The future generations on the transitional path may also be better off because of such positive wealth effects. However, their welfare will also be affected by the decline in the capital intensity of the economy. Finally, when the new steady state is reached the wealth effect completely disappears and individuals will only be faced with a smaller stock of capital. It is not surprising then that steady-state utility will have gone down.20

One may now ask whether it would be possible to transfer the benefits of deflation from individuals on the transitional path to all future generations in the new steady state. This question is discussed next.

6. Optimal money supply in the presence of lump-sum taxation

Assume that initially at time t - 1 the economy is at a steady state. Denote the value of the existing real balances of a young person by X. Further assume that at time t the government unexpectedly cuts the rate of increase in money supply to a new and lower level. It is now observed from eq. (7’) that an unexpected increase in the real purchasing power of money from t - 1 to t would make each old person wealthier by (1 + n)(x, - 2).

Now suppose that in every period the government would impose a lump- sum tax on each old person equal to:

TP=(l +n)(x,-x). (26)

The proceeds of this tax are distributed to the young, again in a lump-sum fashion.21 Thus, each young person at time t will receive:

R:=x,-2. (27)

20This is closely related to Fried’s (1980) discussion of an ‘income increasing innovation’ which may cause all future generations to be worse off. But, as will be argued below, there already exists a mechanism in the present model to avoid such an outcome.

21 Note that the rate of change in money supply must nevertheless remain at 0.


Further assume that this transfer policy will continue indefinitely into the future. Thus, the lifetime budget constraint for an individual born at time t + ~(t 2 0) will be qua1 to [from eqs. (9) (14), (5), (1 l), (26), and (27)]:

c+,+ 1 ++r+1xt+r t?X c,Y+7+ ~~ + ~~.

1 +rt+7+1 1+4+,+1 =w,+r+ t+r

1 +ir+r+1

+ R;+,- _1;,,,1 1+1,+,+1’

The above equation can further be simplified into:

c:+,+ cP+z+ 1 (rl+r+l -42 1 +r,+r+l =Wt+r- l+r,+,+1

(28)

(29)

Furthermore, capital intensity per young at time t + t + 1 will now be equal to:

k W t+r+R;+,-G-x,+, Wt+,--CtY+r-2

r+c+1=- l&i l+n -.

(30)

In order to find the effects of a change in money creation on long-run capital intensity in this case, the steady-state version of eq. (30) is totally differentiated with respect to 8. This will result in:

(31)

The system of equations (B.2) of appendix B indicates that the above expression cannot in general be signed.22

Now assume that the preferences are separable between consumption

22Moreover, it may now be the case that restricting cy and co to be normal and gross substitutes to X, would be inconsistent with existence of an equilibrium. To see this note that from (B.2):

L’cy WJdtI z+I+r- =O.

Consequently, normality of both cy and co, and their gross substitutability to .x, would require (:c~/~B=&~/~O=O. It then follows from (31) that dk/d@=O. Thus

&/c’P 1

l?uIScY 1 +r should also remain constant. But since x has increased, this cannot occur unless the marginal rate of substitution between cY and co is independent of x.


goods on the one hand and real balances on the other.23 It is then immediately observed from (B.2) that dcY/aQ = &“/ZJ =O, and as a consequence:

dk de=o. (32)

Hence, money will presently be superneutral. This result is summarized in the following proposition:

Proposition 5. Imposition of appropriate lump-sum taxes and transfers would make money superneutral in life cycle growth models (assuming that preferences are separable between consumption goods and real balances).

Turning to steady-state utility, it is observed from eqs. (22)324), (B.2), and (B.3) that:

(33)

It is again the case that unless the separability constraint is imposed, du/d8 will be ambiguous in sign. While with the separability of cy and co from x, eq. (33) will be reduced to [by substituting from (32)]:

du iu, 2x

d0 l+i &I’ (34)

It is shown in appendix C that separability of preferences between consumption goods and real balances in addition to the normality of consumption goods will be sufficient to ensure that ax/# ~0. Consequently, du/d8 <O (assuming i>O). A cut in 0, it is thus seen, always (for i> 0) enhances steady-state utility. Furthermore, u( .) attains a maximum at i=O as

23Thi~ is clearly less restrictive than Weiss’s formulation in which preferences are assumed to be separable in all goods. Consequently, the discussion in this section does apply to Weiss’s conclusions. However, whether separability is a ‘realistic’ assumption or not remains an open question. On this, see McCallum (1982) and Charnley (1985).


prescribed by the Chicago Rule24 assuming that there exists a satiation level of real ba1ances.*5v?6 This result is summarized in the following proposition:

Proposition 6. In the presence of lump-sum taxation mechanisms, steady-state

utility is maximized when the opportunity cost of holding money is pushed down to zero (assuming that preferences are separable between consumption goods

and real balances and that the consumption goods are normal).

Next, the transitional path to the new steady state is examined. Consider the initial period (time t) in which the change in the rate of money creation takes place. The capita1 intensity at t + 1 is equal to [from (30)]:

w,--CT--x k -~~ 1+1-

l+n ’ (30’)

Differentiating eq. (30’) totally with respect to 9 results in:

It can further be checked, by differentiating the system of eqs. (12), (13), and (29) (where the latter equation is rewritten for t+ 1) partially with respect to n,, 1 that, given additivity and normality assumptions,

(36)

This rather surprising result can be explained as follows. It is seen from eq. (29) that the ongoing transfers from the old to the young ensure that the total expenditure by the young on present and future consumption goods would remain unchanged, notwithstanding their higher level of real money holdings. But since, as a result of our assumptions, the consumption of both

24The Chicago Rule is sometimes specified as creating a deflation rate, equal to the real rate of interest in the economy. This is an approximation. The main idea is to push the opportunity cost of holding money, i.e. i, to zero. This implies, from the Fisher equation, that (1 + r)( 1+ II)= 1 + i= 1, or x= -r/l + r. However, the Fisher equation is sometimes written, ap

P roximately, as i = r + I[. This formulation results in z = -r.

-5Note that if one does not get satiated from the consumption of money services, it still follows that a cut in i will increase steady-state utility. However, in this case, setting i close to zero will push the consumption of real balances all the way to infinity.

26Note that because money is superneutral here, r will be independent of 0 and the Chicago Rule of setting s= -r/l +r is well defined. Wallace’s misgiving that this rule may not be ‘a well- defined prescription since the equilibrium rate of return on capital depends on the monetary policy followed’, does not apply here.


goods would have to change in the same direction, if at all, there can be no change in either.

Substituting (36) into (35) yields:

From eq. (4), eq. (37) can be further simplified into:

with the general solution of:

dk 1+1 ~ m=o.

d8

(37)

(38)

(39)

This is a most interesting result. It indicates that the government could, by imposition of the above policy, move the economy instantaneously from one steady state to a new one. That is, at period t when the rate of monetary expansion is cut, the real stock of money jumps to its new steady-state value while all other variables remain unaffected (their values would be equal under the old and the new steady states). No further changes occur as of that time and there would be no transition path.*’ This result is summarized in the following proposition:

Proposition 7. Given appropriate lump-sum taxes and transfers, a change in

the rate of money creation will instantaneously move the economy from one steady state to another.

Two implications of this result are of particular importance. First, the comparison of steady states is a useful and meaningful exercise here. Nothing of substance is being left out. This is in contrast to such undertakings in the literature, where there exists a transition path but it is ignored. This omission generally makes such comparisons to be of limited value. Second, because the move to a new steady state is instantaneous, the complexity of the calculation required on the part of the government, regarding the size of the lump-sum taxes and transfers, is greatly reduced. There would be only one such value to compute.

*‘IT the proposed policy is initiated at a time when the economy is not on a steady-state path, the dynamic path of the real variables in the economy will continue as before while the real balances evolve at a higher level.


The welfare of all existing and future generations can now be examined. In the first place, the initial old would neither lose nor gain from the policy discussed above. This can be seen by considering their budget constraint as given by eq. (7’). Clearly, the consumption of the initial old would remain constant and, consequently, so would their welfare.

The welfare of all future generations, on the other hand, can be examined by simply investigating the steady-state utility. This follows because the move to a new steady state is instantaneous. But it was seen earlier that the steady- state utility will always be enhanced as a result of a deflationary policy accompanied by the above-specified taxes and compensations. Moreover, the steady-state utility was shown to attain a maximum when i=O as prescribed by the Chicago Rule. indeed, since the welfare of the initial young and all subsequent generations would improve without hurting the initial old, any deflationary policy as prescribed above will be Pareto improving. The Chicago Rule will then be a Pareto superior policy to follow.** This result is summarized in the following proposition:

Proposition 8. Given lump-sum taxation, the Chicago Rule Pareto dominates

any other money supply rule.

The intuition behind the results of this section is quite simple. As a result of the deflation, the current young and future generations would want to hold more real balances. In the absence of the lump-sum tax and transfer scheme, this can be done only if the consumption of other goods is reduced (recall that the mechanism via which money is injected into this economy eliminates the income effects). Hence, holding of real capital would also be altered and there would be real effects. With the lump-sum tax and transfer scheme, on the other hand, the situation would be radically different. The (unexpected) gains accrued to the initial old (the higher value of their money holdings) are taxed away from them and given to the current young. This would enable the young to increase their holdings of real balances without having to reduce their consumption of other goodsz9 or having to change

‘*Levhari and Patinkin (1968) point out, in the context of a Solow-Swan type growth model, that if the government has another policy tool in addition to the rate of inflation, the steady- state utility maximization will be achieved by following the Chicago Rule. Note, however, that in the present set-up no additional policy tools are required. The model assumes that the government is capable of engaging in lump-sum transfers. The twist here is that the lump-sum transfers should be made to all generations rather than just the old. Furthermore, the discussion here is not limited to the maximization of steady-state utility which may be achieved by transfers from generations on a transitional path to those in the steady state. The Chicago Rule is shown to be Pareto superior to any other money supply rule, given the historically determined capital stock of the economy.

29Strictly speaking, their aggregate expenditures on other goods remain the same. It is the separability of preferences assumption that further ensures that the composition of total expenditures does not change either.


their holdings of real capital. This process, then, would get to be repeated every period. Consequently, there would never be a change in the real sector and money would be superneutral.

Regarding welfare effects, the initial young, and all future generations, would end up receiving a sum of money from the government in the first period of their lives and paying it back in the second period. Thus, they would all gain by an amount equal to the present value of the return to this sum. (This is accrued in terms of carrying more real balances around.) Furthermore, because the initial capital gains are created at no costs to the existing old generation, the change would be Pareto improving.30

7. An example

Assume that the preferences and the technology are represented by the following utility and production functions, respectively:

u=lncr+lncp+,-x,lnx, (O<x,sl/e), (40)

y,=k;.25. (41)

These functions satisfy all the regularity conditions imposed on u( .) and f( .). Furthermore, u( .) exhibits satiation with real balances at x = l/e.

Assume a zero population growth (n=O) and initially no growth in money supply (0=0). Going through the steps discussed in section 3 results in the following two difference equations which describe the dynamic evolution of

this economy:

k (0.75k~~25-~t)[x,(l+lnx,)-l]+~~

t+1= x,(1 +lnx,)-2

(42)

(0.75k~~25-xt)[x1(1+lnx,)-l]+x, X 0.75k;.25-xt-2---

x,(l+lnx,)-2

(0.75k~~25-x,)[x,(l+lnxt)-l]+xt -O.”

x,(l+lnx,)-2 ] }. (43)

30A succinct intuitive explanation of the result of this section is provided by the following remarks of a referee: ‘The government says: I am going to pay an interest on money (starting next period), the young will demand and have more real money balances now (this creates a welfare gain). How are these balances created? By an increase of the value of the existing nominal quantity of money. Will it reduce capital accumulation? No, because all the increase occurs as a capital gain for the old, and I (government) take it from the old and give it to the young so that they have the resources to buy this increment of cash balances without reducing the savings in capital.’

It is easily checked steady-state solution:


that eqs. (42) and (43) give rise to the following

k* = 0.2090 [r* =0.8088]; x* = 0.0574; u* = - 2.092.

Linearization of the pair of difference equations, (42) and (43), around this steady state yields:

k f + i - 0.2090

= x,,i -0.0574 I[ 0.3186 -W~O][~k$.?_O2O]~

-0.0848 (44)

The 2 x 2 matrix in (44) has two eigenvalues given by 1, =0.2588 and i,,= 1.597. Consequently, the above system is locally stable and the steady- state path is unique. The (unique) initial value of real balances, x0, for the historically determined value of k, can then be found by determining the eigenvector associated with i, =0.2588. This is found from:

x0 - O.O663k, -0.0435 = 0. (45)

Now assume that the economy is on its steady-state path associated with 0=0. At some arbitary time period t, the government sets 6= -r*/l +r* = -0.8088/1.8088, which is the Chicago Rule yielding i= 0. This policy is accompanied by a lump-sum tax on the old and a lump-sum transfer to the young equal to 0.3105. (This tax-transfer scheme will continue indefinitely into the future.) As a consequence, the economy instantaneously moves to a new steady-state characterized by:

&=0.2090; i=O.3679; ir= - 1.888,

where c= k*, 1 is the satiation level of x, and ir>u*.

8. Conclusion

This paper re-examines the question of superneutrality and the optimum quantity of money in an overlapping generations model in which money services directly enter individuals’ utility functions. It argues that the money- growth-welfare literature has failed to realize the significance of its allowing lump-sum transfer mechanisms as government policy tools. In particular, it criticizes the widely held belief of a positive relationship between money creation and steady-state capital intensity and welfare in life cycle models. This result is demonstrated to be due to a particular mode of money creation, namely its lump-sum distribution to those who hold money in their


portfolios. This form of money creation is shown to result in outcomes that are Pareto inferior to the ones which can be achieved if certain lump-sum tax and transfer schemes are used.

It is argued that in life cycle models, lump-sum tax and transfer mechanisms may be used to connect the utility of the young and the old just like the bequest motive a la Barro. Of course, bequests are voluntary, while tax payments are not. But without a bequest motive to entice individuals to care about future generations, there is no decentralized mechanism present in these models (where the time horizon of the economy is infinite but the individuals live for a finite period only3r) to permit a first-best optimum to be derived.‘* This is the source of the possible inefficiency in the overlapping generations models and a central message of Samuelson (1958) and Diamond (1965). In these settings, lump-sum taxation gives the government the required tool to force the economy to achieve a first-best optimum.33 In this fashion, then, one can mimic the results derived for infinitely-lived individual models in life cycle frameworks.

The first major finding of the paper is that, given the assumption that money is created by lump-sum distribution to those persons who are holding money in their portfolios, in addition to the assumptions of the normality of the consumption good and its gross substitutability with money services, a positive relationship will emerge between money creation and the steady- state capital intensity, and hence welfare.

Secondly, when preferences are separable between consumption goods and real balances, and consumption goods are normal, it is shown that (i) money is superneutral and (ii) any other money supply rule is Pareto inferior to the Chicago Rule is followed, provided that the government follows the ‘right’ tax and transfer policy. This is to say, if it takes away the unexpected capital gains in the initial old’s holdings of money and gives them to the young by means of lump-sum taxation, and continues with such a policy of redistribution from the old to the young for all periods in the future.

Appendix A

A.1. Utilizing eqs. (14), (5), (ll), and (lo), the young’s lifetime budget constraint may, at the steady state, be written as:

3’See Shell (1971). 32 Recall that the initial old have no incentive not to consume their windfall gains. 331t would be most interesting to investigate what welfare gains could be obtained via the

imposition of distortionary taxes, if lump-sum taxation is not available and there is no bequest motive. This issue needs a careful study and should be addressed in future research.


64.1)

where I =9x/l + i is the present value of the lump-sum money distribution when old. Moreover, from the steady-state condition xf=x,_ , and eq. (5), it follows that:

1+e 1+71=------

l+n’

Now utilizing eqs. (A.2) and (lo), one can rewrite (A.l) and the optimality conditions [eqs. (12) and (13)] in the steady state as:

CO cy+p

r-n

1+r+1+r px=w,

Ul U2 = Fr’

l+n l-(l+e)(l+r) Ul. 1

Partially differentiating this system of equations with respect to k constant, yields:

1 1 r-n

l+r l+r

Ull U

U12 U13

21-l+r ~ u22- Fr “23-~+~

‘Ull U

lU12 ZUl.

31-1si u32- 7 u33- ~

1+1 l+

(A.3

first-order

(A.3)

8, holding

(A.4)

where Uij denotes the second partial derivative of u with respect to goods i and j (i and j stand for cy, co, and x).

Solving (A.4) results in:

aCy (1 +nh (r-4u12 u13 -= u23 (r-n)u22 -_____ ae d(1 +r)(l +Q2 (1 +r)2 (l+r)‘+l+r 1 l+r ’


ace (1 + +1 (r-nb12 u13 (~--n)u11

ae=Ll(l +r)(l +e)’ l+r +---

l+r -‘23-- (~+~)z 1 ’ (A3 ax (1 +nbJ,

[

2u12 Ull

%=d(l +r)(l +f3)2 1 +r+(l +r)’ +u22 5 1

where

A=w33-u:3 -----------++22U33-U;3+

i(r--n)(ullu22-G2)

(1 +r)2 (1 +i)(l+r)

+2(“13u23-u12u33)

l+r

(‘4.6)

The quasi-concavity of u( .) would ensure that the last bracket in ax/a0 is negative. The same condition would also imply that, at the initial value of 8=0, A >O. This follows because at 8=0, (r-n)/( 1 +I) will be equal to and may, in (A.6), be replaced by i/( 1 + i). Consequently, at 8 = 0, ax/a0 < 0.

A.2. At 8=0, the inflation (tax) compensated partial derivatives, ax/%, acy/8, and &?/%I yield the Hicksian compensated terms. This is because money is created in this model by giving it to individuals. It thus follows that if co and x are net substitutes, W’/ae>O.

Alternatively, from (A.l) and (A.3), co is found to be a function of i/l +i, l/l + r, and w + I. Differentiating it partially with respect to f3 yields:

ace ace a(i/i +i) ace ar

ZT-a(i/l + i) ae +are’

But from (lo), (A.l), and (A.2):

a(i/l + i) l+n

ae (1 +r)(l +d)‘“’

(A.7)

Clearly, at 0 =O, aI/ae> 0. Consequently, normality of co and its gross


substitutability to x imply that &“/aO>O. The same result obviously also holds for cy.

Now given either of the above assumptions (which ensure that &“/atI>O), and rewriting the first equation in (A.4) yield:

acy ax I ace i+nax de+%= - --~ -+ ~ -<o.

I+~ a0 i+r ae (A.7)

On the other hand, if u( .) is additive and all goods are normal, ax/a0 < 0, &Y/a0 > 0, &?‘/a0 ~0 for any initial value of 8 (assuming r > n). This follows because normality of all goods and additivity (in addition to the usual quasi- concavity) of the utility function imply diminishing marginal utility for each good.j4 Hence, in this case, (A.7) will also be true for any initial value of 8.

A.3. Similarly, differentiating (A.3) partially with respect to k and solving the resulting system of equations will yield:

u,,u33_u~3+u’3u2;I~12u33+ ~(~12U23-U13U22) l+i 1

(r-n)u 23-uj3+i(u13-(r--II)uIz) l+n u~~-(I-n)u*~ __-__ l+r ( 1 + i)( 1 + r) 1+e l+r

1 +n (r-n)u,,-uu,,

1st) II (l+r)’ ’

ace f"(k)

dk A

-43 i(u12u13-ullu23) u13uz3-ulzu33+u11u33 l+r

+- l+i 1

(1 ?r,1 L”

r-n i (r-n)u,, --u,3+-- ~-~--u

33-l+r l+i ( l+r l3 >

1 +n (r-n)u12 l+nu 13-(r-4ulll(l +I)

l+% ---i+ - u23 1+e l+r

ax f”(k)

k- A u12u,3-u,3u,,+U12U1;~~r1U23+

44 lU22 -42)

l+i 3

64.8)

34 For a proof see Gahvari (1986). For another application see Gahvari (1984)

364 F. Gahuari, Lump-sum taxation

1 +n 2u12 +------- 11 l+Bl+r .

Appendix B

Replace the first equation in (A.3) by:

Co r-n CY+p --z=w.

1+r+1+r

Taking the same steps as in appendix A, it is now found that:

acy (1 +nh

[

U13

a0 d’(l+r)(l +8)2 (1 +r)Z+ l”f-‘r 1 ’

03.1)

ax (1 +nh [

2u12 Ull

%=A’(1 +r)(l+8)2 +---_i+u22 9

l+r (l+r) 1 and

acy Q-‘W -u33

[

iu13 (1 +n)u2, (1 +4u13

ak=- Ll’(1 +r)2 l+r +(l+i)(l+r) -(1+0)(1+r)+(l+O)(l+r)2 1 ’ ace

ak=-

ax

ak=-

4, (1 +4u23 (1 +nh3 u33p1+ 1+8 -(l+O)(l+r) ’ 1

-u”+&(“,- s)- G(&+‘i)

03.3)

2(1+442

+(l+e)(l+r) ’ 1

where

+ u22”33 -43 + 2(u13~23-~12~33)

l+r

F. Gakrari. Lump-sum ta.uation 365

03.4)

Appendix C

Assume that the quasi-concave utility function u=u(cy, co, x) is additively separable between (cy,cO) and X. That is:

u=~(cY,cO)+h(x). (C.1)

Maximization of u( .) subject to the budget constraint,

1 cy+m ~ p +

1 fr l- .y = I,

I +i (C.2)

where I denotes income, yields the following first-order conditions:

u2 1 u3 i

uY=im+r’ u1 l+i’ (C.3)

Partially differentiating (C.l) and (C.2) with respect to I (income) and solving the resulting system of equations yield, among other things:

where D denotes the following determinant:

D=

1 1

I+r

Ull u21

u12

l+r ‘“-T+r

i i u31- -u11

I+i u32-m ~-IA12

l+i

i

l+i

u13 u23 - ~-.~

I+r

i u33- i+<“l”

From quasi-concavity of u( .) it follows that D>0. If it is now cy and co are normal goods, it will follow from (C.4) that either

(C.4)

(C.5)

assumed that

366 F‘. Gahcari, Lump-sum tararion

(i) u,,>o,(u22- fl::)>Oand ($ _,,,),,: or

(ii) u,,cO,(u,,- ~)<Oand(~r -n,,)iO.

(C.6)

(C.7)

But if (i) holds, then:

(u22- $)+llr(;(;(;lr -u,*)=u2*+~~;)I - ;5;>0_

which will violate the quasi-concavity conditions. Hence, only (ii) can hold. That is:

u33 <o. (C.8)

Now consider the expression for A’ from (B.4) in appendix B. It can easily be checked that if u( .) is additively separable between consumption goods and real balances, A’ will be reduced to:

A’=u,, -!!!L+u

242 t1 +rJ2 22- i+r 1 (C.9)

Substituting in (B.2) of appendix B results in:

Px (1 +r)ui

Z-(1 +r)(l +8)2u,,’

It thus follows from (C.8) and (C.10) that di/M<O.

(C.10)

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Lumpsum taxation and the superneutrality and optimum quantity of money in life cycle growth models

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