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Welfare Costs of Inflation, Seigniorage, and Financial Innovation

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1991
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A reduction in the costs of inflation may induce an increase in the rate of inflation. This increase may be large enough to reduce welfare. Fischer and Summers (1989) have explored this issue in the context of a monetary policy game, following Barro and Gordon (1983). In these models, inflation arises because the government is unable to commit to the optimal rate of inflation (assumed to be zero) and tries to exploit a Phillips curve relationship. In the model's simplest form, reducing the costs of inflation reduces welfare because of the more than offsetting increase in the equilibrium rate of inflation.

Fischer and Summers (1989) extended the model to consider imperfect control of inflation. They showed that mitigating the costs of inflation may be desirable only in high-inflation economies, since in low-inflation economies reducing these costs may lower welfare.

In this paper, the problem of reducing the costs of inflation is studied in the context of a simple monetary model, where inflation arises because of the need to finance government spending rather than as a result of trying to exploit a short-run trade-off between output and unexpected inflation. This approach is particularly relevant to an analysis of high-inflation experiences, where the evidence shows that the main underlying cause of inflation is a large fiscal deficit.1 Thus, seigniorage considerations have to be taken into account in determining the welfare effects of inflation mitigation.

In the model developed in this paper, money is introduced through a transactions technology, and government expenditure is assumed constant over time. Only one case of reducing the costs of inflation is considered, namely, decreasing the necessity for people to hold money. There are several potential ways of achieving this—among them, financial deepening, paying interest on deposits, lowering the reserve-de posit ratio, indexing money, or allowing foreign exchange-denominated deposits.

In all of these cases, the shift in money demand requires an increase in inflation to finance the deficit. The rise in inflation produces an additional reduction in real money balances, and results opposite to those of Fischer and Summers (1989) will be shown to hold—that is, a reduction in the social costs of inflation is more likely to increase welfare at low levels of inflation.

The transactions technology makes it possible to interpret reductions in the welfare costs of inflation as financial innovation. Thus, this paper can also be seen as an exploration of the welfare effects of financial innovation in economies that are subject to seigniorage constraints, and the main result can be reformulated as follows: in the presence of seigniorage constraints, financial innovation increases welfare in low-inflation economies, and reduces welfare in high-inflation economies.

Financial innovation per se is welfare enhancing, since it enables individuals to reduce their real money balances for the same amount of transactions, reducing the distortion caused by the inflation tax. However, financial innovation has an indirect negative welfare effect, since the reduction in real balances induces an increase in inflation to meet the seigniorage constraint. The net effect depends on the level of seigniorage required to finance the budget. In the extreme case where seigniorage equals zero, financial innovation does not affect the rate of inflation (zero in this case), and therefore, only the positive welfare effect is present. For positive seigniorage, the larger the interest rate elasticity (in absolute value) of money demand, the larger will be the required increase in inflation to offset a fall in real balances. Provided that the rate of inflation is not on the “wrong side of the Laffer curve,” the elasticity increases with the level of seigniorage. Therefore, at the other extreme, where inflation is close to the revenue-maximizing rate of inflation, the resulting shift in money demand reduces welfare—that is, the indirect effect of higher inflation dominates the positive effect of financial innovation.

The paper is organized as follows. Section I presents a simple monetary model where the only asset and medium of exchange is money. People demand money because it facilitates transactions. So, money reduces transaction costs. In Section II the question of a reduction in required money holdings, which is related on a one-to-one basis to the welfare costs of inflation, is analyzed. Section II also contains the main results of the paper: given the rate of inflation, a reduction in money holdings improves welfare. Once the indirect effect on inflation is added, the result is ambiguous. The direction of the welfare change depends, however, on the level of inflation. At low levels of inflation, the reduction in the social costs of inflation will increase welfare, while at high levels of inflation and, consequently, high seigniorage, welfare falls.

Section III extends the mode! to the case where deposits are also available for making transactions. Deposits are assumed to pay a fixed interest rate and are imperfect substitutes of money. Analogous to Section II, increasing the interest paid on deposits is shown to reduce welfare at high rates of inflation and to increase it at low rates of inflation. If the interest rate were determined competitively, a reduction in reserve requirements would have the same effect as an increase in the interest paid on deposits.

The main part of the paper treats government spending as exogenous and inflation as being the only source of revenue. In Section IV, however, both assumptions are relaxed. First, it is assumed that government spending consists of providing a public good, which is optimally chosen. It is shown that the results obtained for the case of an exogenous government spending are still valid. Second, Section IV extends the model to an optimal taxation scheme, in which the government levies a consumption tax in addition to the inflation tax. It is shown that the main results from the previous sections still hold, albeit with some qualifications. Finally, Section V provides the conclusions and discusses other possible extensions.

I. A Simple Monetary Model

The economy is populated by a constant number of identical, infinitely lived individuals. The representative consumer maximizes the present discounted value of a concave instantaneous utility of consumption u(ct):

There is no production and the individual has a constant flow endowment of y. The only asset is money, which is used because it facilitates transactions. The budget constraint is (time subscripts are omitted)

where π denotes inflation, m denotes real money balances, and g denotes government lump-sum transfers.2

Per capita government transfers are denoted by g and are financed exclusively through the inflation tax.3 Alternatively, it could be assumed that seigniorage is not returned to consumers. The model would be basically the same, because the main result is concerned with a constant level of g.

The term θF(m) represents the transactions technology. Modeling money as an intermediate input in transactions has been used by Fischer (1983), McCallum (1983), Kimbrough (1986a, 1986b), Benhabib and Bull (1983), and, more recently, by Faig (1988) and Vegh (1989), among others. As shown by Feenstra (1986), there is a close relationship between this approach and the introduction of money into the utility function.

In contrast to the traditional formulation, it is assumed in this paper that transaction costs are independent of the level of consumption. This is a convenient shortcut and does not alter the results.4

The transactions (or shopping) technology represents real resources that are foregone in transactions; θF(m) wiil depend on institutional considerations—for example, the degree of development of credit markets, where θ is a parameter reflecting those aspects of the transactions technology.

The technology assumed has the advantage that θ can be also interpreted as a parameter of the welfare costs of inflation. In the present model inflation is a distortionary tax, whose welfare costs depend positively on θ.

The following assumptions with respect to the transactions technology are made:

The reason for the third assumption will be clear later. It is a natural requirement, since it will guarantee that, other things being equal, a reduction of θ will reduce the amount of resources spent in transactions, and it will thus increase welfare.

The necessary conditions for optimality of consumer plans are

where λt, is the current marginal utility value of an extra unit of money at time t.

The general equilibrium can now be characterized. As is well known from Brock (1974) and Calvo (1979), this economy may have multiple equilibrium paths. However, there is only one bubbleless equilibrium in which M̄ = 0. In this equilibrium, inflation is equal to the rate of money growth. This unique saddle-path stable equilibrium will be examined.5

Because there is no capital and prices are fully flexible, the model has no inherent dynamics. Hence, the economy is always at the steady state. This equilibrium is characterized by the following equations:6

A caveat: the optimal quantity of money and maximum seigniorage. Two recurrent issues in the literature on inflation and inflationary taxation can be reproduced with this model. First, the optimal rate of money growth, which equals the rate of inflation, is the Friedman (1969) rule: π = -δ, which is a zero nominal interest rate. This is checked after maximizing consumption in the steady state. This rate of inflation is optimal since it equates the marginal productivity of money with the marginal cost of production. Second, the maximum level of seigniorage is found by solving

which yields the following expression for the revenue-maximizing rate of inflation:

which, together with equation (4), determines (π*,m*).7 The revenue-maximizing rate of inflation is given by the standard rule that the money-demand inflation elasticity has to be unity. Also, it can be checked using the envelope theorem that dg* /dθ > 0.

Following the tenets of public finance, it is known that, up to a first approximation, the deadweight loss of distortionary taxation is proportional to the square of the tax rate. In this model the deadweight loss of inflation will be proportional to the square of π + δ, which by equation (4) implies that it will be proportional to the square of θ. Therefore, θ has a direct interpretation in terms of welfare.

II. Welfare Effects of Financial Innovation

This section looks at the effects of a reduction of θ on welfare in two cases. Because the economy is always in steady state and consumption is the only argument in the utility function, it is enough to analyze the effects of θ on consumption.

The fall in θ is interpreted as financial innovation, which allows people to require lower money balances to carry the same amount of transactions. A more concrete example will be discussed in Section III, where θ is related to deposits and interest rates paid on deposits. The case developed in this section shows the basis of the model and its main implications in the simplest setting. First, the welfare effect of a change in θ given the rate of money growth is examined. Then the constraint that the rate of money growth and the equilibrium real balances must satisfy the government flow budget constraint is added.

Inflation is included because it finances the budget deficit. Therefore, the purpose of the first exercise is to show that, other things being equal, a fall in θ will increase welfare.8 Thus, the welfare costs of inflation are positively related to θ.

Proposition 1. Given the rate of inflation, a decrease in θ increases welfare.

Proof. Differentiating equations (3) and (4):

and

hence

therefore

because of the third assumption (A3).

When θ falls, equilibrium money holdings will fall. This effect does not, however, offset the direct effect on welfare. When the government is constrained to raise a given amount of revenue through an inflation tax, the previous result changes. Denoting the maximum seigniorage for a given level of θ as g*(θ):

Proposition 2. For a given g, the welfare effect of a change in θ is ambiguous. For g close to and below g*, a decrease in θ will decrease welfare. For g equal to zero, a decrease in θ will increase welfare.

Corollary. IF, in addition, it is assumed that F‴≥ 0, there exists ḡ(θ) such that a decrease in θ increases welfare for all g ∈ [0, ¯g) and decreases welfare for all g ∈ (ḡ, g*).

Proof. Differentiating (3), (4), and (5):

and

where

therefore

which is negative for g = π = 0. and goes to infinity as π approaches from below to the revenue-maximizing rate of inflation.

To have monotonicity in dc/dθ, d2c/dθ2 > 0 is required, so that dc/dθ will start from infinity for the lowest feasible θ, denoted as ¯θ: this is the value that satisfies g = g*(θ¯). As θ increases, g will become small relative to g*(6), and dc/dθ will become negative. Therefore, for any g, there will be a unique θ, θ¯, such that dc /dθ = 0. Define this function as θ = θ¯(g). The inverse is g = ḡ(θ). Hence, the rest of the proof consists of showing that the function c = c(θ) is concave.

Since

Defining DmθF″ − π, it can be shown that

Again, using the fact that DmθF′, it can be seen that the two positive terms in the brackets, -F′θ and D, are less than the absolute value of Fmθ2F″/D and -2mF″θ, respectively. Therefore, for F‴ > 0, K is strictly negative, and hence, d2c/dθ2 is negative.

In contrast to Proposition 1, a decrease in θ may be welfare reducing, especially for economies with large fiscal deficits that are financed through an inflation tax. The reason is that as θ falls, real money balances also fall. When inflation remains constant the benefit of the reduction in θ is larger than the cost of holding lower real balances, as was shown in Proposition 1. For a given g, however, the government has to increase inflation because of the drop in the tax base. This increase in inflation reduces real money balances even more. In the end, the tax revenue requirement may reduce real money balances up to a point where the increase in F outweighs the fall in θ. The above corollary provides a sufficient condition to generalize the result for all relevant g.

The results of Propositions 1 and 2 are shown in Figure 1. For θ = θ11 > θ2), the demand for money, given g, is at A. In the space (c,m) point A corresponds to A′. A reduction in θ from θ1 to θ2 can be decomposed into two parts. The first part is a reduction in m, given the rate of inflation (A to Pl in the upper panel, and A′ to P1′ in the lower panel). Proposition 1 shows that consumption at P1′ is larger than consumption at A′. However, since g is fixed, inflation wilt have to rise. Therefore, there is a further reduction in real money balances (from P1 to P2), This effect will offset the increase in welfare, since consumption falls from P1′ to P2′.

Proposition 2 shows that the total welfare effect of a reduction in θ (A to P2) is ambiguous, and its sign will depend on the size of g. For g close to g* (the value of g that produces tangency between the hyperbola πm = g and money demand for θ1), consumption decreases. However, for g = 0, there is no need to increase the rate of inflation from P1 to P2.

Figure 1.Welfare Effects of a Change in θ

That welfare increases in the case of g = 0 is a particular result of Proposition 1. The interesting outcome is the reduction of welfare for g close to g*. As detailed in equation (7). what drives the result is that dm/dθ goes to infinity as g goes to g*. The intuition for this result can be provided through an analysis of a standard multiplier effect.

Consider a small reduction in θ. which reduces m by Δ percent. Since g is constant, inflation has to rise Δ percent to offset the fall in m. This increase in inflation induces an additional fall of ∈Δ percent in m, where ∈ is the money-inflation elasticity (in absolute value). Then, a new increase in inflation of ∈Δ is required with a consequent ∈2Δ percent reduction in m. A new increase in inflation will then be required, and so on. Therefore, the total fall in m in response to a small fall in θ is equal to Δ(l + ∈ + ∈2 +,…). Hence

The elasticity of money demand varies between zero and unity on the “right side” of the Laffer inflation tax curve. For g = 0, the elasticity of money demand is zero, so that there is no indirect effect from inflation on money demand. As g goes to g*, the indirect effect also increases because ∈ is increasing. In the limit, a second-order change in θ causes a first-order change in m because of the adjustment in the inflation rate needed to finance the budget. From the welfare (consumption) point of view, the increase in F(m) caused by the reduction in real balances is larger than the direct effect of the fall in θ.

Finally, note that the result is only concerned with inflation rates on the right side of the Laffer curve—that is, on the increasing portion of the seigniorage-inflation scheduled.9 Since the economy is always in the steady state, there is no reason to assume that the government is collecting seigniorage with excessive inflation.

Numerical example. Before a banking system is explicitly introduced, a numerical example may help to clarify the nature of the results. The example is illustrative and ts not intended to replicate an actual economy. The figures are, however, consistent with empirical evidence on seigniorage and inflation (see Fischer (1982), Dornbusch and Reynoso (1989), and Giavazzi and Giovannini (1989)).

Table 1.Numerical Example
Transaction technology
θF(m)=θ(β0β1m+β2m2/2)
Consumption
c=yθ(β0β1m+β2m2/2)
Money demand
m=[β1(π+δ)/θ]/β2
Maximum seigniorage and money balances at the maximum
g*=(θβ1δ)2/4θβ2m*=(θβ1δ)/2β2
Given g, required money balances (inflation tax curve as function of m)
m=m*+12θβ2(δθβ1)24gθβ2
Parameters
y = 10
g= 0.25
β0= 0.6
β1= 1
β2= 1
δ = 0.05

Table 1 provides the structure of the example. A quadratic transactions technology defined for m > β12 is assumed. Because there is a one-to-one mapping from θto g*, the simulations can be presented in two ways. The first approach is depicted in Figure 2, where there are two curves, which represent the relationship between consumption and θ under the assumptions of Propositions 1 and 2. The hump-shaped curve considers a constant value ofg equal to 0.25; this is seigniorage equal to 2.5 percent of potential gross national product (GNP) (y is normalized to be 10). This value for seigniorage is the maximum revenue for θ = 1.1:

Therefore, θ has to be restricted to be larger than 1.1. Otherwise, it would not be possible to collect g = 0.25 through seigniorage. As θ changes, the rate of money growth and, hence, inflation will adjust to keep g = 0.25. In this simulation money over GNP is between 5 percent and 10 percent. The hump shape reflects Proposition 2: the curve increases vertically when g corresponds to the maximum revenue, g*(θ = 1.1). As θ increases, the relationship between consumption and θ is negative. In terms of Proposition 2, as g becomes low with respect to g*(θ), welfare decreases monotonically in θ.

Figure 2.Numerical Example: Consumption and θ

Forθ = 1.1, the rate of inflation that maximizes revenue is 52 percent. Any inflation rate above that value will be on the “wrong side” of the Laffer curve. For the range of θ depicted in Figure 2, the rate of inflation changes from 52 percent for θ = 1.1 to 30 percent for θ = 2.1.

The monotonically decreasing curve corresponds to consumption, keeping the rate of money growth and, hence, inflation constant. Given this rate of inflation (assumed to be zero), m can be found from the money demand for any value of θ. Therefore, this curve illustrates the result of Proposition 1: that the relationship between consumption and θ is negative everywhere when inflation is constant—at zero in this example. Note that the optimal rate of inflation should be -5 percent; therefore, in both exercises there is a welfare loss due to inflation tax. This loss varies between 1 percent and 2.5 percent of potential GNP.10

In Figure 3, Propositions 1 and 2 are illustrated in terms of the slope of the consumption-θ schedule. The curves are drawn for a fixed value of θ (1.1), and g is allowed to vary. Proposition 1 says that for any constant rate of inflation the slope is negative. This is the curve at the bottom of Figure 3. For any initial g, a change in θ without adjusting inflation will cause consumption to move in the opposite direction. Once the restriction that g has to be constant is incorporated and inflation is adjusted accordingly, the welfare effect of a change in θ is ambiguous. This is reflected in the other two curves, drawn for different values of β2. As g approaches g*(θ = 1.1) = 0.25, the marginal change in welfare goes to infinity; that is, c(θ) becomes vertical as in Figure 2.

Figure 3.Numerical Example: Slope of the Consumption-θ Schedule

(θ = 1.1)

The two curves are drawn for different assumptions about β2 (1 for the curve above, and 0.85 for the other) to show that the value of g, at which dc/dθ changes sign (ḡ in Proposition 2), is very sensitive to the parameters. For β2 = 1, welfare falls when θ falls for any g larger than 2 percent of potential GNP; for β2 = 0.85, welfare falls when θ falls for anyg larger than 0.8 percent of potential GNP.

The gap between the curve for the inflation constant and the one for g constant can be interpreted as the welfare cost arising from the adjustment of inflation to raise the required revenue. In terms of Figure 1, this is the change in c from P1 to P2′.

III. Financial Innovation with a Banking System

In this section the monetary model is extended to incorporate deposits. Deposits are imperfect substitutes of money, but they also provide transaction services.11 In this case what is called money is more precisely a monetary base, which, under the absence of reserves, corresponds to currency.

Money, Deposits, and Interest Rates

The transactions technology is now F(m, e), where e denotes deposits. The three assumptions set out in Section I become

The third assumption simply says that money, as well as deposits, is a normal good.12

Financial intermediation is made through banks. Deposits are the banks' liabilities. On the assets side, because there is no capital in this model, it is assumed that banks have exclusive access to a storage technology with a return equal to δ.13 The marginal cost of providing deposits is zero and banks pay an interest rate equal to i. Further, it is assumed that the interest rate paid on deposits is fixed by the government and there are no reserves. The interest rate is kept below its competitive level, representing a financially repressed economy.

Now the consumer's problem is to maximize equation (1), subject to the following budget constraint:

where f is banks' profits. The nominal return of banks is π + δ. Therefore, the competitive interest rate is equal to π + δ.14 If i is less than π + δ,f is equal to (π + δ − i)e.

At the competitive interest rate, and for any i < δ + π, people do not want to save, although they still maintain deposits, since they, as well as money, facilitate transactions.

The steady state is now characterized by

and the government budget constraint, equation (5).

Note that again the optimal rate of inflation (first best) is equal to Friedman's rule. At that optimum, the interest rate is i = 0. In this case, the marginal productivity of both—money and deposits—is equated to their marginal cost of production. This optimum is attained by setting the rate of deflation equal to δ and allowing banks to pay the competitive interest rate.

Note also that by (A1′), the problem can have an interior solution only for i ≤ δ + π. In the rest of this section the focus will be on the interest rate on deposits between zero and δ + π. The question is, what are the welfare effects of an increase in the interest rate toward its competitive level? By making the financial system more competitive, financial repression is alleviated.

Using equations (13) and (14), it is possible to show that the rate of inflation that maximizes government revenue is given by15

The results from Propositions 1 and 2 can now be extended to the following proposition:

Proposition 3. (i) For a given inflation rate an increase in the interest rate will increase welfare, (ii) For a given g. an increase in the interest rate will increase welfare for g around zero and will decrease welfare for g around, but below, g*.

Proof. For part (i), differentiating (12):

Now, differentiating (13) and (14):

Denoting as H the hessian of F(.,.) and considering dπ = 0:

(ii) Using dπ = -πdm /m from equation (5), and substituting into equation (17), the system can be written as

Let us call H′ the square matrix on the left-hand side. Note that for π = 0, H′ = H; hence, the first part of (ii) is true from equation (18).

After solving the system

For π < π*: det(H′) > 0; and for π = π*: det(H′) = 0. So, provided that the expression in square brackets is nonzero, dc/di will diverge to plus or minus infinity, depending on whether K is larger or less than zero, respectively. Replacing in K the value of π* from (15):

Therefore, close to g*, dc/di goes to minus infinity, and welfare falls.

The intuition for part (i) of Proposition 3 is simply that an increase in the interest rate will increase the productivity of holding deposits. The negative effect of economizing in money holdings is smaller, given the assumption that money and deposits are normal goods.

Part (ii) of Proposition 3 follows from Proposition 2 in Section II, which says that an increase in the interest rate paid on deposits will decrease welfare when there is a high budget deficit and. hence, a high rate of inflation. In contrast, increasing interest on deposits will increase welfare at low rates of inflation.

As in the previous section, the increase in the interest rate paid on deposits requires a further increase in inflation to collect the required revenue. This effect reduces money balances even more, leading to an unambiguous welfare loss at high rates of inflation. As was already shown, the welfare effect depends on the elasticity of money demand, which determines the change in inflation and its implied further reduction in real balances required to finance the budget.

Reserve-Deposit Ratio in a Competitive Banking System

The critical element for the results of this paper is the unitary elasticity of money demand at the revenue-maximizing rate of inflation. The results may not be robust under some specifications that do not satisfy the unitary elasticity rule, which may be the case with the positive reserve-deposit ratio.

Brock (1989) showed that for a positive reserve-deposit ratio the elasticity of money demand (currency) that maximizes revenue is not unity. The reason is that the tax base is high-powered money. If ρ is the actual reserve-deposit ratio, seigniorage, which by assumption is equal to g, will be given by

Therefore, the rate of inflation that maximizes revenue is that at which the elasticity of h (in absolute value) with respect to π (∈h) is equal to unity. Hence, the elasticity of currency, m. will, in general, be different from unity. In what follows it will be shown that the result of Proposition 3 can be extended to this case, and reinterpreted in the context of a competitive banking system where the reserve-deposit ratio falls.

The increase in the interest rate paid on deposits can be envisioned as a reduction in reserve requirements in a competitive banking system (Calvo and Fernández (1983)). If banks were allowed to compete and required to hold a fraction ρ of their deposits as non-interest-bearing reserves, the competitive interest rate would be (δ + π)(1 − ρ).

In this case, (14) would become

The effect of a change in ρ on consumption is

and the effect on high-powered money is

Substituting equations (24), (22). and (13) in (23), the following expression for the welfare effects of a change in ρ is obtained:

The same intuitive argument explaining Proposition 2 can be used to show that at high levels of seigniorage welfare falls when ρ falls. It is enough to note that dh/dρ goes to infinity as the elasticity of high-powered money (∈h) goes to unity.

A change in ρ that causes a Δ percent change in h will require an increase of Δ percent in inflation to offset it. The increase in inflation will reduce high-powered money by an additional ∈h Δ. Inflation will then be required to increase further by ∈h Δ percent, where h will consequently fall by ϵh2Δ, and so on. Therefore, the total fall in h will be Δ(1+ϵh+ϵh2+ϵh3+,). which goes to infinity when ∈h goes to unity. Then, according to equation (25), welfare will unambiguously fall “close” to the maximum inflation-tax revenue because of the excessive increase in the inflation rate and the fall in real balances.

IV. Endogenous Government Spending and Optimal Taxation

In previous sections the government has been assigned a passive role through the imposition of an exogenous tax structure to finance a given budget. This section examines government behavior by analyzing two separate cases. First, government spending is assumed to be a public good that is optimally provided, and, hence, one that will be modified when financial innovation occurs. Second, an optimal taxation approach is followed by assuming that, in addition to the inflation tax, the government can resort to a consumption tax in order to finance a given level of government spending.

Endogenous Government Spending

In this section g is considered to be a public good that enters into the individual utility function. For simplicity, the utility function is assumed to be separable into a consumption good and a public good. In this case there will also be an optimal response of g to a reduction in θ (financial innovation). The response will be a reduction in the provision of the public good as the welfare cost of providing it increases.

It is shown below that the main results of previous sections hold: in high-inflation economies financial innovation will reduce welfare, whereas in low-inflation economies welfare increases.

Consider the same model as that in Section I. but now the instantaneous utility function is u(c) + v(g), where both u(·) and v(·) are increasing and strictly concave functions. The government will maximize this utility function, subject to consumer behavior and its budget constraint; that is, the government problem is to maximize U(c,g), subject to equations (3), (4), and (5). This problem can be conveniently written as

Thus, the government implicitly chooses m by setting the rate of inflation accordingly. The first-order condition to this problem is

The effects of financial innovation can be determined using the envelope theorem (arguments of the functions are omitted):

The first term on the right-hand side represents the standard direct effect of a reduction in transactions costs, which increases welfare when θ falls. The second term is the welfare-reducing effect of financial innovation, by which the provision of the public good will be reduced because of the increasing costs of raising revenue. In this case, dW/dθ will not necessarily diverge to infinity as inflation approaches the level that maximizes government revenue; but the same results from Propositions 1 and 2 hold. To see this, replace equation (27) in equation (28) to obtain

It is clear from equations (29) and (30) that the direct effect (first term within square brackets) is of a second order when compared to the indirect effect of rates of inflation close to the revenue-maximizing rate of inflation (equation (6)). When inflation is low. because g is low, the direct effect will dominate. Therefore, the results from Propositions 1 and 2 apply to the case of endogenous government spending.

Optimal Taxation

In addition to the inflation tax, governments use several instruments to raise revenue. Throughout this paper, g has been financed via inflation only. Therefore, g can be interpreted as the fraction of government spending that cannot be financed through nondistortionary taxes. In high-inflation countries, the heavy reliance on seigniorage may be due to the large size of the underground economy or the inefficiency of the tax system. It may also be caused by political factors, as was discussed in Cukierman, Edwards, and Tabellini (1989). who argue that political instability and the degree of polarization are key determinants of seigniorage.

Under a nonsingle tax system, when the base of one tax falls the government will, in general, adjust several other tax rates. Hence, a permanent shift in money demand will be accommodated not only by an increase in inflation, but also through an increase in other taxes.

Since Phelps (1973), a iarge body of literature has focused on inflation as part of an optimal tax system. In this case the optimal structure consists of equating the social marginal costs of different distortions.16

It is assumed for the model in Section I that money is the only asset and is used in transactions. Government optimally sets taxes on money holdings and consumption to finance g of government spending, which is then returned to consumers as a lump-sum transfer.

The consumer problem is the same as before, but now consumption is taxed at a rate, τ:

subject to

Note that despite having taxes on only the two goods—consumption and real balances—lump-sum taxation cannot be reproduced. The reason is that θF(m) plays the role of a third good, which is not taxed.

Individual behavior is characterized by the following equations:

Although taxes and g are related through the government budget constraint, the individual takes g as given. These equations describe consumption and real balances as functions of the two tax rates, which have to be considered in solving the optimal tax problem. Note that m depends only on the inflation tax.

As will be shown later, in the present model an optimal tax structure involves no tax on money, so the whole tax burden should fall on the consumption tax.17 Therefore, the model has to be amended to make an inflation tax positive. For this purpose, real costs in collecting consumption tax are assumed, whereas the cost of collecting an inflation tax is zero (Aizenman (1983) and Vegh (1989)). Collection costs are increasing in the amount of revenue collected. They are described by a function, ϕ(τc), where 0 <ϕ′(·)< 1 and ϕ″(·) > 0. The case ϕ′(·) = 0 is equivalent to the case of no collection costs. Since utility depends only on consumption, which is constant in equilibrium, the optimal tax problem consists of

subject to

The lagrangian of this problem is

It can be shown that the first-order conditions from the government problem are18

and

Since ϕ′ ∈ (0,1), μ > 1. In the case where ϕ′ = 0—that is, there are no collection costs—μ = 1. Solving equation (35) for the case of no collection costs, the following equation is obtained for the optimal tax problem:

Then, optimal taxation calls for δ + π = 0, which recovers Friedman's optimal money rule as in Kimbrough (1986b). In the presence of collection costs, however, the optimal inflation tax departs from the zero nominal interest rate rule. In this case, combining equations(35)and(36) and substituting ∂m/∂π with − 1/θF″(m), the optimal tax problem is reduced to

In this case, both money and consumption goods have to be taxed.

The question now is what is the welfare effect of a change in θ, considering that all taxes will be optimally adjusted. The total effect can be written as

The direct effect (∂c/∂θ) will be the same as the total effect computed in Proposition 1, which by the assumptions made on F(·) guarantee this to be negative (financial innovation increases welfare). 19 The other effects will have the opposite sign since taxes will have to be risen to offset the fall in the inflation-tax base. After some algebra it can be shown that

where α is the share of inflation tax revenue on total spending (G = g + ϕ), and ̂x denotes the percentage increase in the tax rate x (x = π,τ) on account of the fall in θ.

When the tax rates are not adjusted, allowing a fall tn revenue, a fall in θ will increase welfare. Therefore, only the direct effect matters. This result is merely a generalization of Proposition 1; thus, given the tax rates, a fall in θ is welfare improving.

Nevertheless, once the government budget constraint is taken into consideration, there is a negative effect because of the increase in the tax rates. A reduction in the inflation tax base requires an adjustment in tax rates. Then, a fall in θ implies that the term within square brackets is positive, offsetting the beneficial direct effect.

It is not possible to generalize Proposition 2 without making additional assumptions, although it can be argued that under general conditions the results still hold. Equation (39) provides support for this presumption on two counts. First, the negative effect of a fall in θ is positively related to g, although the change in tax rates and their share in revenue will also depend on g. Second, the increase in the rate of inflation required to raise a given amount of seigniorage is increasing in the elasticity of money demand. Therefore, the larger the elasticity of money the larger the increase in the rate of inflation and the more distortionary commodity taxation will become to offset the distortion in money holdings.

It is possible, however, that a fall in θ will always increase welfare. It is not a general proposition under the assumptions of this section that the indirect effect of tax rate changes is of a first order when compared to the direct effect of financial innovation—which resolved the ambiguity. The reason is that when the elasticity of money demand is close to unity, the required revenue will be raised mainly through commodity taxation. Thus, the welfare effect will also depend on the elasticity of the demand for consumption goods. Nevertheless, the direction of the result is the same: the larger the seigniorage, the larger will be the distortion introduced in consumption because of the increase in the tax rate.

V. Conclusions

Only two interpretations have been fully developed for the change in the marginal productivity of money and, consequently, for the reduction in the welfare costs of inflation. The results of Section III can, however, be easily extended to include other sources of reduction in the welfare costs of inflation.

Deposits can be interpreted as indexed money, which is an imperfect substitute for nonindexed money. More important, they represent an alternative asset to money. This asset can be used in transactions and it yields higher interest than money. Policies that make the use of this asset more attractive will increase the velocity of money. Consequently, the inflation tax base will fall. This is the case of “dollarization.” where there is a shift from domestic to foreign money. Dollarizations are usually observed when a country's inflation rate increases (Fischer (1982)).20,21

In the model presented here, for a constant rate of money growth an increase in velocity is welfare improving. When a given revenue has to be raised through an inflation tax. however, the result can be reversed. The main conclusion is that negative effects on welfare will occur in large seigniorage economies. In contrast, the lower the inflation tax, the more beneficial is the reduction in inflation costs.

Faced with high inflation, people seek institutional changes that will protect them from inflation. For example, wage indexation and short-term financial instruments become very important. Usually, there are demands for government to introduce changes that reduce the costs associated with transactions, such as a reduction in the reserve-deposit ratios. The perverse effect that some of these changes have when the deficit is financed mainly by seigniorage may, however, explain why governments are reluctant to accept changes that may look, other things being equal, positive.

Note that the results of this paper could easily be extended to any form of taxation and its relationship to technical progress. The parameter θ is equivalent to a Hicks neutral parameter of technical progress, and m is equivalent to an input. Therefore, technical progress can be immiserizing when it produces a fall in the demand for taxed inputs. The required increase in the tax rate and, hence, in the degree of distortion may end up reducing welfare. For example, imagine an economy where the only available tax is a tax on gasoline. Technical progress that saves on inputs will reduce the use of gasoline. Therefore, the required increase in the gasoline tax rate may end up reducing welfare.

Inflation and capital flight are frequently observed in inflationary economies. In this paper, Propositions 1 and 2 assume that θ is exogenous. Making this variable endogenous may explain inflation and capital flight as a coordination failure. If θ is interpreted as a parameter of the structure of credit markets, it can be related to the extent of capital flight. Proposition 1 shows the private incentive to reduce θ because it takes inflation as given; individual decisions have no effect on inflation. Under the assumptions of the model, the private incentive to reduce θ is always positive. Instead. Proposition 2 shows that the total effect is uncertain and depends on the current level of inflation (through seigniorage). Therefore, a welfare-reducing increase in inflation may be the result of spillover effects from capital flight to real balances and, hence, to inflation.

Finally, this paper connects the degree of inflationary finance with developments in financial markets, an issue recently addressed by Dornbusch and Reynoso (1989). Financial deepening may lead to a deterioration in welfare through an increase in the rate of inflation. Therefore, a prerequisite for the removal of financial repression is fiscal discipline. Relying heavily on seigniorage to finance the budget may outweigh the advantage of a more developed financial market.

In summarizing the results of the original question—what are the welfare effects of a reduction in the costs of inflation—this paper concludes that they are positive (negative) in low (high) seigniorage economies. In a broader interpretation, concerning the welfare effects of financial innovation, it can be concluded that the benefits of improved financial intermediation may be offset by the negative effects of a higher rate of inflation.

APPENDIX

The Optimal Tax Problem

Consumer behavior is given by the following equations:

Note that as the optimum, c will be a function of both taxes, τ and π, while m is only a function of π. Although in equilibrium g equals πm + cτ. the individual takes g as a given transfer. The following partial derivatives can be obtained from equations (40) and (41):

The effect on consumption, then welfare, of a change in θ is given by

The direct effect is the one total derivative from Proposition 1—that is, assuming all tax rates as constants:

The government optimal tax problem consists of max c(τ. π;θ),πτ

subject to

The lagrangian of this problem is

where m is a lagrange multiplier. The first-order conditions of this problem are (subscripts denote partial derivatives)

Substituting equation (44) in (47), this first-order condition becomes

Since ϕ′ ∈ (0,1), μ > 1. In the case that ϕ′ = 0—that is, there is no collection cost—μ = 1.

Introducing equations (43) and (44) into equation (49) yields

It can be seen that when ϕ′ = 0 and, hence, μ = 1, equation (51) collapses to

So, optimal taxation calls for δ + π = 0, which recovers Friedman“s optimal money rule as in Kimbrough (1986b). But in the presence of collection costs the optimal inflation tax departs from the zero nominal interest rate rule. Replacing equations (50) and (42) in (51) yields the following expression characterizing the optimal tax scheme:22

Substituting equations (42), (43), and (44) in (45) (consumer behavior in equation for the total effect of θ in consumption) yields

which, after using the optimal tax rule (equation (53)), becomes equation (39) in the text:

Note that this equation can be further reduced to find an expression for the percentage change of the tax rate by totally differentiating the government budget constraint.

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José De Gregorio, an Economist in the European Department, was an Economist in the Research Department when this work was completed. This paper is a revised version of the first chapter of his doctoral dissertation for the Massachusetts Institute of Technology. The author is grateful to Guillermo Calvo, Christophe Chamley, Rudiger Dornbusch, Robert Flood, Pablo Guidotti, Larry Kotlikoff, Jeff Miron. Federico Sturzenegger, Carlos Végh, Peter Wick-ham, and seminar participants at Boston University, the Massachusetts Institute of Technology, the World Bank, and the University of California (Los Angeles and San Diego).

Money is the only asset in this economy, although in general equilibrium an interest-bearing bond can be introduced at the margin, paying a real interest rate equal to δ. The case of debt financing is not considered. Since g is constant, the constant rate of inflation can be interpreted as the result of optimal tax smoothing. See Mankiw (1987) and Barro (1988) for recent applications.

Seigniorage is πm, which implicitly assumes that individuals own the initial stock of money and the government allows the price level to jump when the equilibrium changes, contrary to the “honest government” of Auenheimer (1974). The result does not change if (δ + π)m is used instead of πm.

This is equivalent to assuming that money provides “shopping services,” as suggested in Dornbusch and Frenkel (1973). The individual budget constraint would be c + M0304; + πm = y(l + v(m)) + g, so -yv(m) ≡ θF(m). Hence, the transactions technology would depend on y rather than c. But. since y is constant, it is omitted as an argument of F.

Bubbles can be ruled out assuming that the transactions technology satisfies

see Blanchard and Fischer (1989, chap. 4 and references therein).

To save in notation and given that the economy is always in the steady state, superscripts or subscripts to denote equilibrium values are omitted.

It is assumed that the second-order condition holds: 2F″(m*) + F‴“(m*)m* >0. A sufficient condition for a unique interior solution is that F′(m) goes to minus infinity when m goes to zero.

This exercise assumes that the government budget constraint, equation (5), does not hold.

Eckstein and Leiderman (1989) estimated the relationship between seigniorage and inflation for Israel. In their model they found that the Laffer curve is always increasing, becoming almost flat for races of inflation above 5 percent a quarter.

Assuming an inflation rate of 7 percent and θ = 1.1, the welfare loss is 1.2 percent of potential GNP. Fischer (1981) computed a welfare loss (“inflation triangle”) of 0.3 percent of GNP due to a 12 percent deviation of inflation from its social optimum.

Fischer (1983) and, more recently, Brock (1989) introduced deposits in the transactions technology. Romer (1985) included deposits in the utility function, and Walsh (1984) included them in a cash-in-advance constraint. In all cases, deposits are assumed to be imperfect substitutes of money. See also Calvo (1986) for a related discussion.

This assumption is equivalent to (A3) in Section I, noting that an increase in θ is equivalent to a fall in e.

This is equivalent to assuming that banks are the only ones allowed to hold foreign assets. The world interest rate is δ. In addition, it should be assumed that purchasing power parity (PPP) holds, and the nominal exchange rate grows at the same rate as money and domestic prices.

The first version of this paper assumed a zero return on this storage technology; therefore, banks were the only providers of an alternative asset to make transactions. The competitive interest rate in this case is π, so they provide “indexed money.”

It is also assumed that the second-order condition holds and the solution is interior.

Poterba and Rotemberg (1990) and Grilli (1989) cast serious doubts on the empirical validity of the optimal seigniorage theory when tested in industrialized countries.

The specification of the problem in this section is consistent with the result of Kimbrough (1986b), who extended the inflation tax problem in Diamond and Mirrlees' (1971) principle that intermediate inputs should not be taxed. As pointed out in Guidotti and Végh (1990), however, these results hold true only under particular characteristics of the transactions technology, which apply in the above case.

For a full derivation of the remaining equations, see the Appendix.

One might suspect by the envelope theorem that only the direct effect of θ matters. Equations (28) and (33) show why this intuition is wrong. The government does not set ∂c/∂ (tax rate) equal to zero, but rather ∂L/∂ (tax rate) equal to zero. In fact, both taxes are distortionary, so a decrease in θ will cause both tax rates to increase such that at the margin the distortions are equated, but the indirect effects through the tax rates cannot be eliminated.

For the case of Mexico, see Ortiz (1983). In November 1989, in the middle of an unsuccessful stabilization program in Argentina, the Government allowed deposits in foreign currency, which were to be guaranteed.

Arrau and De Gregorio (1990), in a framework similar to the one in this paper, estimated empirically the role of financial innovation in money demand equations for Chile and Mexico. The results showed that an important component of money demand fluctuations corresponds to financial innovation.

The final solution could be obtained by replacing m as a function of π from equation (41) in (52) to have a single equation for π. Then, substituting c from equation (40) as a function of π and τ in the government budget constraint, the solution for τ would be obtained.

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