Inside money and the effects of inflation

Journal of Macroeconomics 27 (2005) 494–516

www.elsevier.com/locate/jmacro

Inside money and the effects of inflation

Dennis Powers

Department of Economics, University of St. Thomas, 2115 Summit Ave., St. Paul, MN 55105, USA

Received 23 March 2001; accepted 27 February 2004Available online 9 June 2005

Abstract

This paper studies inflation in a model of inside money, where inside money is a debt instru-ment that shares the same liquidity properties as outside money. The explicit modeling ofinside money allows for the standard Tobin effect of inflation to be combined with the trans-action costs of inflation. Thus the net effect of inflation may be positive, negative, or zero.However, it is demonstrated that if the net effect is positive, it is likely to be small in magnitudeand appear only at small inflation. Furthermore, numerical simulations reveal that the neteffect may be substantially negative for even moderate inflation. These results are consistentwith empirical observations.� 2005 Elsevier Inc. All rights reserved.

JEL classification: E0; E5

Keywords: Inflation; Inside money

1. Introduction

This paper analyzes the effects of inflation in a model of inside and outside money,where inside money is defined as a debt instrument sharing the same liquidity prop-erties as outside money. There is a large theoretical literature on the effects of infla-tion. This literature has identified three possible effects of inflation on output;

0164-0704/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jmacro.2004.02.009

E-mail address: [email protected]

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D. Powers / Journal of Macroeconomics 27 (2005) 494–516 495

positive (Tobin effect), negative (Reverse-Tobin effect), and zero (superneutrality).Though the three effects are theoretically possible, recent empirical research suggeststhat even moderate inflation can have substantial negative effects and that the effectsof inflation appear non-linear.1 That is, the effects of inflation appear larger whenstarting from higher inflation. Indeed, Ahmed and Rogers (2000) find evidence ofthe Tobin effect at low inflation. This paper is able to combine the Reverse-Tobinand Tobin effects and demonstrate that if the net effect is a Tobin effect, it will besmall in magnitude and exist only at low inflation. Furthermore, numerical simula-tions reveal there can be a large, negative net effect for even moderate inflation.Hence this paper can explain the above empirical results. A further interesting resultis that the model can capture all three possible effects of inflation, depending onassumptions to be specified below. Hence this paper identifies the conditions neces-sary for a particular effect of inflation to emerge.2

Modeling inside money has proven difficult since inside money, as defined above, isa perfect substitute for outside money. In a model of homogeneous agents only one ofthese assets, but not both, would be held in equilibrium.3 To explain the co-existence ofinside and outside money a type of heterogeneity is assumed among agents. It is as-sumed agents can meet their liquidity needs by borrowing (i.e. issuing inside money),but to borrow one must pay a cost to verify her capital holdings and agents differaccording to the ‘‘verification cost’’ they must pay. If most agents face a relatively highverification cost, at moderate levels of inflation relatively few agents choose to issue in-side money and the majority of agents hold outside and/or inside money to meet theirliquidity needs. Hence the quantity of inside money issued is not enough to meet thetotal demand for money. Thus both outside and inside money are held in equilibrium.

Consider the effect of inflation in this model. As inflation rises the return on out-side money falls. Since inside money is a perfect substitute for outside money, theinterest rate must fall so that the return on inside money equates to that of outsidemoney. There are two effects of the fall in the interest rate. First, some agents chooseto pay the verification cost so that they can borrow at low interest rates to meet theirliquidity needs and thereby escape the low returns on holding money. The verifica-tion cost represents the transaction costs of inflation. Of course, if it is assumed thatthe verification cost is prohibitively high, then agents will not pay it, and there will beno transaction costs of inflation. The second effect of the fall in the interest rate is

1 Regarding the negative effect of inflation see, for example, Kormendi and Meguire (1985), DeGregorio(1993), Fischer (1993), and Barro (1996). Regarding the non-linearities in the effect of inflation, see Fischer(1993) and Bruno and Easterly (1998).2 Using a money-in-the-utility function model Brock (1974) also finds all three effects are also possible.

Real effects are due to the labor decisions of agents and depend on the cross-partial derivatives of theutility function. Though a Tobin effect is possible in this model, it is not due to an increase in capitalaccumulation as in Tobin (1965), but to an increase in labor supply. Furthermore, the effects of inflationare numerically small and unable to explain the cited empirical results.3 This model assumes there exists an asset that shares the same liquidity properties as outside money and

identifies the conditions necessary for both this asset and outside money to be held. See Townsend (1989)and Cavalcanti and Wallace (1999) for models in which the economic environment that gives rise to anasset sharing the liquidity features of outside money is made explicit.

Table 1Summary of results

Inside money issued forliquidity needs only

Inside money issuedfor investment

Prohibitively high verification cost Superneutrality Tobin effect

Modest verification cost Transaction costs result Tobin effect andtransaction costs result

496 D. Powers / Journal of Macroeconomics 27 (2005) 494–516

that those who were already issuing inside money will issue more inside money. Thismay result in a Tobin effect. Specifically, if it is assumed that issues of inside money(i.e. borrowed funds) can be invested in capital, there will exist a Tobin effect alongwith the Reverse-Tobin effect. If, though, one assumes issues of inside money canonly be used to meet liquidity needs, there is no Tobin effect and only the Re-verse-Tobin effect. Table 1 summarizes the conditions under which a particular resultof inflation emerges.

The paper proceeds in the following manner. Section 2 presents a literature re-view. Section 3 presents the model and describes the equilibrium. Section 4 examinesthe effects of inflation on capital and output. Section 5 presents numerical simula-tions, and Section 6 offers concluding remarks.

2. Literature review

The Reverse-Tobin effect in this model is due to the fact that inside money is mod-eled as a socially costly substitute for outside money. There is a large literature onsuch ‘‘transactions cost’’ models, having their origins in Bailey (1956). In these mod-els, an increase in inflation causes a substitution toward the outside money substi-tute. Since this substitute requires real resources a Reverse-Tobin effect is present.Recent extensions of this literature model the outside money substitute as financialservices. Such models argue that an increase in inflation misallocates resources awayfrom productive activity and toward the provision of these financial services.4

Though this view of transactions costs shares much in common with the verificationcost used in this paper, there is an important difference between our model and thesemodels. The models in this literature do not allow for these financial services to inter-mediate capital. Indeed such models abstract from the lending/borrowing transac-tion underlying these financial services. The model of this paper can be viewed asan extension to this literature in that inside money is modeled explicitly as a debtcontract and is allowed to intermediate capital. It is this extension that drives themodel�s results regarding the non-linearities in inflation and, in particular, the con-ditions necessary for a Tobin effect to exist at low inflation.5

4 See Schreft (1992), Gillman (1993), English (1999), Frenkel and Gil (2000).5 There do exist other models that use transactions cost to generate a model of inside and outside money,

such as Freeman and Huffman (1991) and Lacker (1988). However, these model are not designed to studyinflation and cannot explain the empirical observations cited in the text.


While this paper extends transactions cost models to include inside money, therehave been many other papers that have studied inflation in a model of inside money.It is important to compare this literature with the present paper.

Cothren and Mukherji (1997) is similar in spirit to our paper in that they usecostly inside money to combine Tobin and Reverse-Tobin effects and find that theReverse-Tobin effects may dominate the Tobin effect. However, they only find sucheffects in the credit rationing equilibrium. When credit is not rationed inflation issuperneutral. Hence, the results presented in this paper would appear more general.

Gale (1983) also considers the effect of inflation in a model of inside money. De-spite the fact that inside money intermediates capital in his model, he finds a Re-verse-Tobin effect. The reason for this effect is that inside money is the only formof money that exists in his model. An increase in inflation reduces the demand for(inside) money, thereby reducing capital. In our model, and that of Tobin (1965), in-side money is a substitute for outside money. An increase in inflation causes one tosubstitute inside money for outside money, thereby having the potential to increasecapital.

A popular framework in which to study financial market imperfections and infla-tion has been the Diamond–Dybvig (DD) model of demand deposit banking. Sincedemand deposits are commonly considered inside money, it is particularly importantto compare this literature based on the DD model with the model in this paper.

Recent work has extended the DD model along a number of dimensions to ex-plain the Reverse-Tobin effects and the non-linearities mentioned above.6 Althoughthe DD model is a model of inside money, the concept of inside money in the DDmodels is very different from that being using in this paper. In the DD model agentswithdraw from their demand deposits reserves of outside money. The outside moneyis then used in transactions. Hence deposits are never used to conduct transactions,and thus do not provide the same services of outside money. They are simply theoptimal way to insure the individuals who need outside money will have access tooutside money. Hence these demand deposits do not represent inside money, as Ihave defined it.7 This is not to suggest that studying inflation in DD models is notuseful, but it does suggest that DD models offer a complementary model, rather thana competing model, in which to study the effects of inflation.

Finally, though models in this literature may have a role for inside money andallow inside money to intermediate capital, there is an important difference betweenthe model presented in this paper and those found in this literature. The model pre-sented allows for the possibility that inside money does not intermediate capital, but

6 For example, Espinosa-Vega and Yip (1999) use a DD model to study the interaction betweenmonetary and fiscal policy, and Cothren and Mukherji (1997) use the DD model to study inflation withfinancial market imperfections and costly production of demand deposits.7 If one extended the DD model to allow demand deposits to be used as money, since deposits are

produced at zero cost, the return on deposits would exceed that of outside money and this would result inzero demand for outside money. This is exactly the problem described above in which inside money is anasset providing the same services as outside money. Thus one would have a model of inside money as isbeing described in this paper.


is issued only to meet liquidity needs. The question of whether inside money is issuedto meet liquidity needs or is used for investment is shown to be crucial in determiningwhether a Tobin effect exists. To the best of my knowledge this question has beenneglected in the literature.

3. The model

3.1. Description of the model

The structure of the model is similar to that of Townsend (1980) and Bewley(1980). Infinitely-lived agents receive income every other period, but consume everyperiod. This creates a natural environment to study the demand for liquid assets. Inprinciple both money and bonds could serve as liquid assets. If one assumes there isprivate information, capital holdings cannot be verified and there is no market forbonds. Thus a monetary equilibrium emerges. If, though, capital holdings can beverified a bond market will exist and if the interest rate exceeds the return to moneythere will be no demand for money. This paper investigates a model in which bothbonds and outside money coexist. Since the definition of inside money is a debtinstrument that has the same liquidity properties of outside money, the bonds of thismodel represent inside money and these terms will be used interchangeably through-out the paper.

The coexistence of money and bonds is accomplished in the model by assumingthere are two types of capital; verifiable and non-verifiable. Furthermore, it is as-sumed that agents must specialize using either verifiable or non-verifiable capital.Those who use verifiable capital can borrow (i.e. issue inside money), while thosewho use non-verifiable capital cannot borrow. Therefore those who specialize withnon-verifiable capital will hold outside and/or inside money to provide liquidity,while those who specialize with verifiable capital can borrow or hold money to pro-vide liquidity. However, it will be shown that in the equilibrium with positive infla-tion they will only choose to borrow (i.e. issue inside money). The final assumption isthat agents have a comparative advantage in either verifiable or non-verifiable cap-ital. This comparative advantage will in part determine the capital with which theyspecialize.

The population of the economy is normalized to unity. The fraction of the pop-ulation that have a comparative advantage producing with non-verifiable capital isgiven by k and the fraction of the population that have a comparative advantageproducing with verifiable capital is given by (1 � k). It is assumed that half of thek receive their income in odd periods and the other half in even periods. Similarly,half of the (1 � k) receive their income in odd periods and the other half in evenperiods.

For simplicity, assume that if one produces with their comparative advantage cap-ital the production function is given by ka, where 0 < a < 1, while if one produceswith their non-comparative advantage capital the production function is given by(1 � j) * k

a, where 0 6 j 6 1. The value of j captures the cost of switching from


non-verifiable capital to verifiable capital and represents the cost of verifying one�scapital, or establishing one�s creditworthiness. Agents can be further differentiatedby the value of j. Let us define the function cðjÞ as the fraction of the k agents forwhom j 6 j. Hence the fraction of the k agents for whom j ¼ j is given by c0ðjÞ. Notethat if k = 1 no agent has a comparative advantage using verifiable capital, hence allagents face a cost of verifying their capital stock. Since it seems reasonable to believemost agents would face such a verification cost, k is assumed to be unity or nearunity.

3.2. The structure of trade

There are two types of heterogeneity in the model leading to four types of agents.One type of heterogeneity to the period in which income is received; odd (type Oagents) or even (type E agents). The other type of heterogeneity refers to the typeof capital used; non-verifiable (type N agents) or verifiable (type V agents). Hencethe four types of agents are NE, NO, VE, and VO. Recall some of the agents usingverifiable capital are those who have a comparative advantage using non-verifiablecapital but have switched to verifiable capital. When this distinction is importantthe agents that switch will be referred to as type S.

The structure of trade is depicted in Figs. 1–3. Type N agents hold assets (moneyand/or bonds) to provide liquidity, while type V agents issue bonds to provide liquid-ity. Given type V agents only issue bonds, all trade in money occurs between typeNE and NO agents. In even periods type NE agents receive income and trade goodsfor money. In the next period, type NO agents receive income and acquire moneyfrom type NE agents. This is depicted in Fig. 1. Given type N agents can hold thebonds of type V agents to provide liquidity, trade in bonds occurs between agenttypes N and V. In even periods, type NE agents receive income and, in additionto holding money, hold bonds issued by type VO agents. In the next period, VOagents receive income and settle their debts with type NE agents. This is depictedin Fig. 2. Similarly, in odd periods type NO agents receive income and make loans

Period t Period t + 1

Type NE (Income = y)

m h

Type NO (Income = 0)

Type NE (Income = 0)

m h

Type NO (Income = y)

Fig. 1. Time period t is even. At t NE agents acquire fiat money, given by m, from NO agents in exchangefor goods equal to their real money demand, given by h. This is reversed at period t + 1.


Type VO (Income = 0)

bD


Type VO (Income = y)

br


Fig. 2. Time period t is even. At t NE agents acquire bonds from VO agents in exchange for goods equalto their real bond demand, given by bD. At period t + 1, VO agents pay br to NE agents to settle their debt,where r is the gross interest rate.


Type VO (Income = 0)

bD


m h

Type NO (Income = 0)

br

Type VE (Income = y)

Type VO (Income = y)

br


m h

Type NO (Income = y)

bD

Type VE (Income = 0)

Fig. 3. Figure depicts the trading patterns between the four types of agents.


to type VE agents. In the next period, type VE agents settle their debt with type NOagents. The complete structure of trade is depicted in Fig. 3.

3.3. The individuals� problem

When solving the individual agents� problems, no reference is made as to whetheran agent is type E or O. Though there are four types of agents, and potentially four


types of problems, at the steady state the behavior of type E agents and type Oagents is the same under the assumptions specified. In particular, the assumptionthat half of the population is type E and half is type O will be shown to imply thatthe steady state with a constant growth rate of money will be characterized by a timeinvariant return to money and a time invariant interest rate. This implies the steadystate bond demand, money demand, and bond supply is the same for type E and typeO agents.

There are now two types of agents to consider: those who use non-verifiable cap-ital and those who use verifiable capital. Section 3.3.1 will analyze the problem forthose who use non-verifiable capital. Section 3.3.2 analyzes the problem for usersof verifiable capital assuming inside money can only be issued to meet liquidityneeds, while Section 3.3.3 analyzes the problem for users of verifiable capital assum-ing inside money can also be issued for investment purposes. Note the problem ofthose who have switched from non-verifiable to verifiable capital is just a special caseof those who have a comparative advantage using verifiable capital and is also ana-lyzed in Sections 3.3.2 and 3.3.3.

3.3.1. Users of non-verifiable capital

Consider those who specialize with non-verifiable capital. The typical individualfaces the following problem:

maxc;h;b;n

X1t¼0

bt � UðctÞ

subject to f ðkN;tÞ ¼ ct þ ht þ bN;t þ kN;tþ2; ð1Þhtvtþ1 þ stþ1 þ bN;t � rtþ1 ¼ ctþ1; ð2Þ

where t is even for type NE and odd for type NO in Eqs. (1) and (2). b is the discountfactor and the standard assumptions apply regarding the utility function. Eq. (1) isthe budget constraint during the period in which income is received. The productionfunction is given by f(kN), where kN is the capital stock. Income can be used for con-sumption (c), to acquire real money (h), to acquire bonds (bN), and to invest (kN,t+2).For simplicity there is no depreciation. Eq. (2) is the budget constraint in the periodin which there is no income from production. Consumption is financed by the returnon assets. The return on bond holdings is given by rt+1, the gross interest rate. Thatis, r = 1 + interest rate. The gross return on money holdings is vt+1 and is equal to pt/pt+1 = (1 + pt+1)

�1, where p is the price level and p is the inflation rate. Money isintroduced into the economy via lump-sum transfers from the government, distrib-uted independent of money holdings.8 These transfers are given by st+1 and are equalto lt+1Mt/pt+1 in the aggregate, where lt+1 is the growth rate of the nominal money

8 While the transfers are independent of individual money holdings, it is assumed in the aggregate alltransfers are distributed only to current money holders. This assumption, common in the literature,assures that only the distortionary effect of inflation exists. If these transfers were also given to type Vagents, then there would be a distributional effect of inflation. While such distributional effects may beinteresting, they are not addressed in this paper.


supply between periods t and t + 1 and Mt is the nominal money supply at time t.There is no other role for government in the model.

Neither investment or liquid asset holdings are included in Eq. (2) because in anequilibrium with positive inflation, type N agents would not choose to invest or holdliquid assets in the period in which no income is received. To see this, consider firstinvestment. Given it takes at least two periods for investment to produce output, atpositive rates of inflation one would never want to invest during the period in whichincome is not received as this would first require holding an asset that depreciates forone period before investing.

To see why liquid assets would not be held at period t + 1 note that at period t

there are two methods to provide for income at period t + 2. One can provide forperiod t + 2 income by investing in capital at period t and receiving a return of(1 + f 0(kN,t+2)) in t + 2, or by investing in liquid assets at periods t and t + 1 yieldinga return of v2 at t + 2. As long as (1 + f 0(kN,t+2)) exceeds v

2, which it will at the stea-dy state with positive inflation, one would never hold liquid assets at t + 1.9

3.3.1.1. A two-period problem. The repetitive structure of the model implies the max-imization problem can be set up as a sequence of two-period problems. To begin firstconsider a two period utility function defined as

W N;t ¼ UðctÞ þ bUðctþ1Þ.Using Eqs. (1) and (2) one can substitute for ct and ct+1. Defining

yN,t f(kN,t) � kN,t+2 one may rewrite the two period utility function as

W N;t ¼ UðyN;t � ht � bN;tÞ þ bUðhtvtþ1 þ stþ1 þ bN;trtþ1Þ. ð3Þ

Note that yN,t is the income that is available for consumption in periods t and t + 1.Agents choose ht and bN,t, taking as given yN,t, vt+1, st+1, and rt+1, to determine theallocation of consumption between periods t and t + 1.

To solve for liquid asset holdings one maximizes WN(yN,t, . . .) with respect to h

and bN. The first-order conditions for this problem are

U 0ðctÞ ¼ bU 0ðctþ1Þ � vtþ1;U 0ðctÞ ¼ bU 0ðctþ1Þ � rtþ1.

Since outside money and inside money (bonds) are perfect substitutes, for both out-side money and bonds to co-exist it must be that vt+1 = rt+1, implying the interestrate is negative if inflation is positive. A negative interest rate on bonds is not asimplausible as it may seem. First, if an additional transaction cost was added tousing bonds the interest rate could be positive while the return on bonds remainednegative. Second, a negative interest rate on the bonds of this model is not counter-factual. These bonds pay a negative interest rate because they are a perfect substitutefor outside money. Demand deposits are the closest real world example of a perfectsubstitute for outside money, and since demand deposits pay a small interest rate, if

9 A formal proof of these propositions is available from the author upon request.


any, the real returns on demand deposits are negative for positive inflation. How-ever, regardless of the plausibility of negative interest rates on inside money, whatis important for the purpose of this paper is that the interest rate on inside moneyis positively related to the return on outside money.

Since the returns on money and bonds are the same the individual is indifferentbetween money and bonds. However, in equilibrium the demand for bonds mustequal the supply of bonds coming from agents who use verifiable capital. Thusone can treat bN,t as given. As long as bN,t is not too large, ht is uniquely determinedand is given by ht = h*(vt+1,yN,t,bN,t, st+1). The two-period utility function can thenbe written asWN,t =WN(yN,t,bN,t,vt+1, st+1,b). It is guessed that v, s, and bN are con-stant at the steady state, which is verified in Sections 4 and 3.4. This implies the two-period utility function can be written as WN,t =WN(yN,t,bN,v, s,b).

3.3.1.2. The infinite horizon problem rewritten. To solve for steady state capital stockand output the maximization problem can now be rewritten as

maxn

X1t¼0

dtW N;tðyN;t; �Þ

subject to yN;t ¼ f ðkN;tÞ � kN;tþ1; ð4Þ

where each period in this problem represents two periods from the previous problemand d is defined as b2. This is a standard optimal growth problem that can be solvedusing the Bellman�s equation V ðkN;tÞ ¼ maxktþ1fW N;t þ dV ðkN;tþ1Þg, where V(kN,t) isthe value function. The problem has the steady state solution f 0(kN) = d�1. As inSidrauski (1967), the steady state capital stock is determined by the golden ruleand the level of output is independent of the level of inflation. Also, notice thatnon-verifiable capital pays a higher return than money at equilibrium. The reasonis simply that money provides liquidity services not provided by non-verifiablecapital.

Let the steady state level of output be given by f ðk�NÞ y�. As the steady statewith a constant growth rate of money is characterized by a constant v, r and s, utilityat the steady state for non-verifiable capital holders is given by W N;t ¼ Uðy� � h��b�N;tÞ þ bUðh�vþ sþ b�N;trÞ. One may easily show WN is increasing in y and increas-ing in v (decreasing in p).

3.3.2. Users of verifiable capital: Inside money is issued only for liquidity

If inside money can only be issued for liquidity needs, those who specialize withverifiable capital face the following problem:

maxc;b;n

X1t¼0

bt � UðctÞ

subject to bV;t�1 þ ct�1 ¼ 0; ð5Þ

f ðk Þ þ b r ¼ c þ k ; ð6Þ
V;t V;t�1 t t V;tþ2


where t is even for type VE and odd for type VO in Eqs. (5) and (6). All variables areas defined above. Eq. (5) is the budget constraint in the period in which income is notreceived. It simply states that consumption is financed by borrowing (i.e. bV < 0). Eq.(6) is the budget constraint during the period in which income from production isreceived. Income may be used for consumption (c), the repayment of debt (bv,r),and investment (kV,t+2). Again, depreciation is assumed to be zero for simplicity.No investment is allowed in period t � 1 to capture the idea that one cannot issueinside money to finance investment. Also, borrowing has not been allowed in the per-iod in which income is received. The rationale is that it is assumed borrowing takesplace prior to investment. To borrow at period t requires that period t + 2 capitalholdings be verified. However if borrowing at t occurs before investment, t + 2 cap-ital is not yet known and cannot be verified. Thus borrowing at t cannot occur.

Though it may appear in Eqs. (5) and (6) that type V agents are forced to issueinside money, since liquid asset holdings are not included in the period t budget con-straint, this is not the case. Liquid asset holdings are not included because with po-sitive inflation type V agents would never choose to hold liquid assets. The reason isthat liquid asset holdings in period t provide income in t + 1 and earn a return of r.An alternative method of providing income in period t + 1 is to invest in period t andearn a return of (1 + f 0(kV)) in t + 2. Borrowing against this income yields(1 + f 0(kV))/r in period t + 1. As long as (1 + f 0(kV))/r > r, which is true since r < 1at the steady state with positive inflation, one would not hold assets at period t.10

The fact that type V agents choose to provide liquidity by issuing inside moneyinstead of holding liquid assets establishes that inside money is endogenously sup-plied in the model with positive inflation. Indeed, if inflation were negative, thensince v = r, it must be that r > 1. Hence one could find a high enough interest rate(i.e. low enough inflation rate) that (1 + f 0(kV))/r < r, implying type V agents preferto meet liquidity needs by holding assets instead of borrowing. In such a case insidemoney is not provided. Of course, this paper is concerned with case of positive infla-tion, which is the case in which inside money is endogenously supplied.

3.3.2.1. A two-period problem. As before the problem has a repetitive structure thatcan be expressed as a sequence of two-period problems. To begin define the two-period utility function as WV,t = U(ct�1) + bU(ct). Defining yV,t f(kV,t) � kV,t+2,(3) and (4) may be expressed in a present value budget constraint as yV,t = ct�1rt + ct.One can write the two-period utility function as

W V;t ¼ Uðct�1Þ þ bUðyV;t � ct�1rtÞ. ð7Þ

Since one may borrow against future income, yV,t is the income available for con-sumption in periods t � 1 and t.

To solve for bV, ct�1, and ct one maximizes WV,t with respect to ct�1. The firstorder condition is U 0(ct�1) = bU 0(yV,t � ct�1rt)rt, which implies ct�1 = c(yV,t, rt,b).As rt = vt at an interior solution, since v is constant at the steady state, r is constant

10 A formal proof is available from the author upon request.


at the steady state. This implies the two-period utility function can be written asWV,t =WV(yV,t, r,b).

3.3.2.2. The infinite horizon problem rewritten. The maximization problem can nowbe rewritten as

maxn

X1t¼0

dtW V;tðyV;t; �Þ

subject to yV;t ¼ f ðkV;tÞ � kV;tþ1; ð8Þ

where again each time period represents two periods from the previous problem and dis defined as b2. Proceeding as before the Bellman�s equation for this problem isV ðkV;tÞ ¼ maxktþ1fW V;t þ dV ðkV;tþ1Þg, which has the solution f 0(kV) = d�1 at the steadystate. Hence, the capital stock and level of output is independent of the interest rate.11

Since the steady state condition is the same as that for users of non-verifiable cap-ital, the steady state output for verifiable capital holders is also y*. If one were to setup the same problem for type N agents who switch to verifiable capital the results areidentical except one would write the solution to the Bellman�s equation as(1 � j)f 0(kS) = d�1, where the subscript S denotes agents who have switched to veri-fiable capital. With f(k) = ka one can show the output of an agent who has switchedto verifiable capital is yS = (1 � j)gy*, where g = a/(1 � a) > 1. Thus for type Vagents, two-period utility at the steady state is given by W VðSÞ ¼Uðc�t�1Þ þ bUðð1� jÞgy� � c�t�1rÞ, where j = 0 if an agent has a comparative advan-tage using verifiable capital. It can be shown that WV(S) is increasing in y anddecreasing in r (increasing in p).

3.3.3. Users of verifiable capital: Inside money is issued for investment

If inside money can be issued to fund investment, those specializing with verifiablecapital face the following problem:

maxc;b;n

X1t¼0

bt � UðctÞ

subject to bV;t�1 þ ct�1 þ kV;tþ2 ¼ 0; ð5aÞf ðkV;tÞ þ bV;t�1rt ¼ ct; ð6aÞ

where t is even for type VE and odd for type VO in Eqs. (5a) and (6a). The only dif-ference between this model and the previous model is that investment occurs in per-iod t � 1 since investment can be financed by borrowing. The fact that investmentcan be financed by borrowing also implies no investment will occur in period t.

11 As in the case of users of non-verifiable capital, the capital stock is determined by the golden rule. Butsince the interest rate is negative the marginal product of capital for users of verifiable capital exceeds theinterest rate. It may seem that this violates a no-arbitrage condition since type V agents can borrow.However, even though type V agents can borrow, by assumption they cannot invest these borrowed fundsin capital, implying arbitrage profits are not available. Thus there is no reason for the marginal product ofcapital to equate to the interest rate.


The reason is when there is positive inflation the interest rate is negative, hence onealways prefers to borrow to invest. That is, one unit of income in period t can be usedto invest 1/r units in period t � 1 if one borrows. Hence the return in period t + 2 fora unit sacrificed in period t is (1 + f 0(kV,t+2))/r. If one invests one unit of income inperiod t the return in period t + 2 is (1 + f 0(kV,t+2)). Since r < 1, (1 + f 0(kV,t+2))/r > (1 + f 0(kV,t+2)) and one would always prefer to borrow to invest.12

3.3.3.1. A two-period problem. As before, the repetitive structure of the problem sug-gests it can be solved as a sequence of two-period problems. Defining the two-periodutility function as WV,t = U(ct�1) + bU(ct) and yV,t f(kV,t) � kV,t+2rt, (5a) and (6a)may be expressed in a present value budget constraint as yV,t = ct�1rt + ct. One canwrite the two-period utility function as WV,t = U(ct�1) + bU(yV,t � ct�1rt).

To solve for bV, ct�1, and ct one maximizes WV,t with respect to ct�1. The firstorder condition is U 0(ct�1) = bU 0(yV,t � ct�1rt)rt, which implies ct�1 = c(yV,t, rt,b).Thus the two-period utility function can be written at the steady state as WV,t =WV(yV,t, r,b).

3.3.3.2. The infinite horizon problem rewritten. The maximization problem can thenbe written as

maxn

X1t¼0

dtW V;tðyV;t; �Þ

subject to yV;t ¼ f ðkV;tÞ � kV;tþ2rt.

ð9Þ

The main difference between this problem and the one in which inside money can-not be issued for investment is that yV,t depends on the interest rate. The Bellman�sequation for this problem is V ðkV;tÞ ¼ maxktþ1fW V;t þ dV ðkV;tþ1Þg, which has thesolution f 0(kV) = r Æ d�1 at the steady state. Hence, the capital stock and level of out-put is dependent on the interest rate and therefore dependent on the inflation rate.This implies the golden rule does not apply for type V agents when issues of insidemoney can be invested in capital. Instead there is an overaccumulation of capital.Indeed as inflation rises, the interest rate falls and capital is increased. Thus the stan-dard Tobin effect exists for type V agents when issues of inside money can be in-vested in capital.13

12 It has been assumed that investment at time t � 1 increases kt+2 rather than kt. This assumption makesthis version of the model easily comparable to the previous version, in which agents chose kt+2 at time t.Though mainly for simplicity, one could rationalize the assumption by assuming that one is working onproduction in periods (t � 1) and t and that capital must be in place both periods to have an impact onperiod t output. Since investment at t � 1 creates new capital in period t only, it can only affect output att + 2.13 In this case since type V agents can use borrowed funds to invest there is a tendency for the marginalproduct of capital to equal the interest rate. However, the marginal product of capital does not exactlyequal the interest rate because when a bond is issued and the proceeds invested, the bond is repaid in thefollowing period, but the proceeds of the investment are not received for another three periods. Hence themarginal product of capital differs from the interest rate only by the discount factor d.


It can be shown that the steady state level of output is given by hy*, whereh = (1 + p)a/(1�a) > 1 and oh/op > 0. If one considers the same problem for thosewho switch to verifiable capital the results are identical except the solution to theBellman�s equation can be written as (1 � j)f 0(kS) = r Æ d�1 � 1. One can show theoutput of an agent that has switched to verifiable capital is given by yS = (1 � j)ghy*.Hence both yV and yS are positively related to inflation. Two-period utility at thesteady state is given by W VðSÞ ¼ Uðc�t�1Þ þ bUðð1� jÞghy� � c�t�1rÞ, where j = 0 ifan agent has a comparative advantage using verifiable capital. It can be shown thatWV(S) is increasing y and decreasing in r (increasing in p).

The results of Section 3.3 may be summarized as

yN ¼ y�; oW N=op < 0; ð10ÞyVðSÞ ¼ ð1� jÞgy�; oW VðSÞ=op > 0; ð11ÞyVðSÞ ¼ ð1� jÞghy�; oW VðSÞ=op > 0; ð12Þ

where (11) is the case in which inside money is issued only for liquidity, and (12) isthe case in which inside money may also be issued for investment. Again, j = 0 if onehas a comparative advantage using verifiable capital.

This section allows issues of inside money to intermediate capital. For agents thatswitch to verifiable capital, this implies the productivity of capital falls. It may seemodd that intermediated capital is less productive than capital that is not intermedi-ated. However, this model abstracts from financial markets, implying the non-inter-mediated capital in the model can be thought of as capital intermediated throughfinancial markets. Hence this model assumes that capital intermediated by issuesof inside money is less productive than capital intermediated through financial mar-kets, due to the verification cost of issuing inside money.

3.4. The steady state equilibrium conditions

Equilibrium is defined as a sequence (v, s, r, kN, kV, kS, cN, cV, cS, hN, bN, bV, andbS) that is consistent with individual optimization and the clearing of the money mar-ket, bond market, and commodity market. Section 3.3 has demonstrated the steadystate equilibrium conditions consistent with individual optimization under theassumption that v, s, and r are constant at the steady state and an interior solutionexists in which both inside money and outside money are held. It remains only todemonstrate that the steady state equilibrium is characterized by a time invariantv, s, and r and that both outside and inside money may be held at the steady stateequilibrium with positive inflation. By Walras� Law one need only consider two mar-kets. The money market and bond market will be examined. The money marketequilibrium will demonstrate the time invariance of v, r, and s. The bond marketequilibrium condition establishes the conditions necessary for inside and outsidemoney to both be held.

The equilibrium condition in the money market requires money supply toequal aggregate money demand. Aggregate money demand is given by0.5kð1� �cÞht, where �c is the fraction of type N agents who have switched to verifiable


capital at equilibrium and 0.5kð1� �cÞ is the fraction of agents who hold money.Equating money supply to aggregate money demand, the price level at time t is givenby

pt ¼ Mt=0.5kð1� �cÞht.

Given vt = pt/pt+1 and substituting in for the price levels one can show

vt ¼ ð1þ lÞ�1ðhtþ1=htÞ; ð13Þ

where a constant growth rate of money has been assumed. The ratio ht+1/ht is theratio of type NE money demand to type NO money demand in odd periods (vice ver-sa in even periods). Note that if money demand is identical and constant for type NEand NO agents, then v is constant. Thus a constant demand for money, which wasfound in Section 3.3.1 under the assumption of a time invariant v and s, is consistentwith a time invariant v (and r). Furthermore since st is given by st = lht, and moneydemand is constant, s is also time invariant. In this case one can show hv + s = h,establishing the social return on outside money is unity at the steady state.

To consider whether outside money would be held in equilibrium, consider thebond market equilibrium given by

0.5kð1� �cÞbN þ 0.5ð1� kÞbV þ 0.5

Z �j

j¼0kc0ðjÞbS dj ¼ 0;

where the equilibrium j is defined as �j. Recall, j represents the cost of switching fromnon-verifiable to verifiable capital. Hence, the equilibrium j will be the j such that theutility for a non-verifiable capital holder equals the utility if she switches. It is there-fore the j that solves the equation WN = WS.

14 Recall c0ðjÞ is the fraction of the kagents associated with the value of j ¼ j and the fraction of the k agents for whoj 6 �j is given by �c.

Dividing by 0.5 the bond market equilibrium condition is more simply expressedas

kð1� �cÞbN ¼ ð1� kÞð�bVÞ þZ �j

j¼0kc0ðjÞð�bSÞdj. ð14Þ

In an equilibrium with both inside and outside money, the interest rate is not deter-mined by Eq. (14), but by the condition that r = v. This interest rate determines thesupply of bonds which are held by type N agents. Outside money is held also as longas the demand for liquid assets exceed the supply of bonds, thus for both outsidemoney and inside money to be held bond supply cannot be too large. Hence an inte-rior solution is more likely to exist the fewer the number of agents that use verifiablecapital; that is, the closer is k to unity and c to zero. Since k is assumed to be near

14 It may not seem correct to compareWN toWS sinceWN,t is based on consumption at t and t + 1 whileWS,t is based on consumption at t � 1 and t. However since WN,t and WS,t are both based on yt, it iscorrect to compare them. Intuitively, WS is based on earlier consumption since one can borrow against yt.


unity, money demand will fall to zero when there is a large amount of switching toverifiable capital; that is when c is large. If the amount of switching becomes largeand the demand for outside money falls to zero, the demand for liquid assets ismet completely be the supply of bonds and the equilibrium interest rate is deter-mined by Eq. (14). This paper is concerned with the interior solution in which bothoutside and inside money are held.

4. The output effect of inflation

This section examines the effect of inflation on average output. Since output ofboth type E and type O is the same, average output at the steady state is the samein every period and is given by

y ¼ kð1� �cÞyN þ ð1� kÞyV þZ �j

j¼0kc0ðjÞyS dj. ð15Þ

Though the welfare effects are also of interest, the heterogeneity in the modelmakes it impossible to analytically determine the effect of inflation on average wel-fare without specifying a utility function. Hence a presentation of the welfare effectsof inflation is postponed until the numerical simulations are presented in Section 5.

4.1. The model with no switching

First consider the model without switching. While there is no apparent reason tobelieve agents cannot switch at a cost, this version of the model serves as a usefulbenchmark in that it is in only in this version the superneutrality result and pureTobin effect emerges. Also, since no switching implies the cost of having one�s capitalverified is very high, this version of the model applies to economies that face a highverification cost, such as those with a poorly developed financial sector.

4.1.1. Superneutrality result: Inside money is issued only for liquidity needs

With no switching and inside money only issued for liquidity needs, yN = yV = y*

and yS = 0. Eq. (15) then becomes y = ky* + (1 � k)y* = y*. Hence output is inde-pendent of inflation.

4.1.2. Tobin result: Inside money is issued for investment

If switching is not allowed but inside money may be issued for investment, thenyN = y*, yV = hy*, yS = 0 and Eq. (15) becomes y ¼ k�

y þ ð1� kÞhy�. In this case,since oh/op > 0 average output depends positively on the rate of inflation. Thusthe Tobin result emerges where an increase in the rate of inflation leads to an in-crease in steady state output. Of course, the larger is k, the smaller will be the Tobineffect. In fact, if k = 1 the Tobin effect will not exist. Since k represents the fraction ofagents who face a cost of verifying capital it seems reasonable to think of k as beingeither unity or very near to unity. Thus the Tobin effect is likely to be small.


4.2. The model with switching

To determine the effect of inflation on the decision of agents to switch from non-verifiable capital to verifiable capital, consider the effect of inflation on the equilib-rium j. Recall the equilibrium j is the j that solves the equation WN =WS. Henceto determine the effect of inflation on j one totally differentiates the equationWN =WS with respect to j and p, inflation, and solve for oj/op. One can show

ojop

¼ ðoW N=op � oW S=opÞðoW S=oyÞgð1� jÞg�1y�

.

Using the results of Section 2 that oWN/op < 0, oWS/op > 0, and oWS/oy > 0 it mustbe that oj/op > 0. Intuitively, an increase in inflation reduces WN while increasingWS. Hence agents on the margin find it optimal to switch to verifiable capital. Forsuch agents the gain in being able to issue bonds to provide liquidity exceeds the costassociated with using verifiable capital.

4.2.1. Transaction costs result: Inside money is issued only for liquidityWhen inside money can be issued only for liquidity purposes and switching is

allowed, yN = yV = y* and yS = (1 � j)gy*. Eq. (15) is then given by

y ¼ kð1� �cÞy� þ ð1� kÞy� þZ �j

j¼0kc0ðjÞð1� jÞgy� dj; ð15aÞ

and the effect of inflation is

oy=op ¼ �kðoj=opÞc0ðjÞy�ð1� ð1� �jÞgÞ. ð16ÞFor positive inflation oy/op < 0 since oj/op > 0, �j > 0 and g > 0. Intuitively, sinceyN = yV = y* the only effect of inflation is to cause agents with j 6 �j to switch fromproducing with non-verifiable capital to verifiable capital. For a given agent thatswitches, her production falls from y* to (1 � j)gy*.

4.2.2. Tobin and transaction costs result: Inside money is issued for investment

When inside money may also be issued for investment, the model will combinethe Tobin result with the Transactions Costs result. As yN = y*, yV = hy* andyS = (1 � j)ghy*, Eq. (15) becomes

y ¼ kð1� �cÞy� þ ð1� kÞhy� þZ �j

j¼0kc0ðjÞð1� jÞghy� dj; ð15bÞ

and the effect of inflation is given by

oy=op ¼ �kðoj=opÞc0ðjÞy�ð1� hð1� �jÞgÞ

þ h0y� 1� k þ kZ �j

j¼0c0ðjÞð1� jÞg dj

" #. ð17Þ


There are now two effects from inflation. The first term on the right-hand side cap-tures the fact that those who switch will produce less.15 The second term captures thefact that those who are currently producing with verifiable capital will increase pro-duction. Hence the total effect on output cannot be determined. However, at p = 0,j = 0, c = 0 and h = 1. In this case

oy=opjp¼0 ¼ h0ð1� kÞy� > 0. ð18ÞThat is, the Tobin effect is present. However, one can show

o2y=opok¼�ðoj=opÞc0ðjÞy�ð1� hð1��jÞgÞþ h0y� �1þZ �i

i¼0c0ðjÞð1� jÞgdj

" #< 0;

implying that the closer is k to unity, the smaller is the Tobin effect. Indeed, if k = 1in Eq. (18) then oy/op = 0 and the Tobin effect does not exist.

This section demonstrated the conditions necessary to generate the superneutral-ity, Tobin, and transactions cost results. While the superneutrality result remains apossibility, the assumptions necessary to generate it are restrictive in that it is onlyobserved in the model in which agents cannot switch (i.e. verify capital at a cost).Whether the Tobin result is present was shown to depend crucially on whether insidemoney may be issued to finance investment. Further analysis suggests that if the To-bin effect is present it is likely to be small in magnitude. A Reverse-Tobin effect wasalso found when switching was allowed. Moreover, from Eqs. (16) and (17) oy/op is,in general, non-linear, as the empirical literature suggests (though the exact form ofthe non-linearity cannot be analytically determined with specifying the utility func-tion and the c function). The quantitative significance of the these effects is addressedin the next section.

5. Numerical simulations

It was noted earlier that empirical evidence suggests that moderate inflation canhave significant negative effects and that these effects are non-linear. This sectiondemonstrates the model can replicate these key empirical regularities. To allow forthe possibility of negative effects of inflation only the model that allows switchingto verifiable capital will be considered.

The two measures of the effect of inflation are the output effect and the welfareeffect. The output effect is the change in output due to inflation as a percentage ofoutput at zero inflation. The standard measure of welfare reduction is the reductionin output starting from zero inflation necessary to reduce utility to the value ob-tained with positive inflation. In this model the equivalent would be to find thereduction in average output that would reduce average utility to its post inflationvalue. Since at zero inflation v = 1 = r, every agent enjoys utility equal to

15 Note that y�ð1� hð1� jÞgÞ ¼ yN � �yS, where �yS is the output of an agent for whom WS =WN. SinceWN is decreasing in p and WS is increasing in p, it must be yN > �yS, which implies y�ð1� hð1� jÞgÞ > 0.


W*(1,y*,b), thus this also is average utility. Average utility with positive inflation isgiven by W ¼ kð1� �cÞW N þ ð1� kÞW V þ

R �jj¼0 kc0ðjÞW S dj. Hence the welfare effect

of inflation is given by the x that solves W ¼ W �ð1; ð1� xÞy�; bÞ.The utility function is specified to be CRRA, with the intertemporal elasticity of

substitution denoted by r. Let c be given by the function c = q Æ [1 � (1 � j)z], whereq > 0 and z > 0. With this specification one can show at the steady state

W N ¼ 1þ b1=rvð1�rÞ=r

1� r� y� � bNð1� vÞ

1þ ðb � vÞ1=r

" #1�r

; ð19Þ

W VðSÞ ¼1þ b1=rvð1�rÞ=r

1� r� ð1� jÞghy�

vþ ðbvÞ1=r

" #1�r

; ð20Þ

and

bN ¼ hy�

vþ ðbvÞ1=r

" #�

1� k þ kqz

zþ g½1� ð1� �jÞzþg�

k½1� qð1� ð1� �jÞzÞ�

264

375.

Or by defining X as the second term in brackets, one can express bN as

bN ¼ hy�

vþ ðbvÞ1=r

" #� X. ð21Þ

Substituting bN into WN and setting WN =WS, �j is the j that solves

vþ ðbvÞ1=r � ð1� vÞX ¼ ð1� jÞgh½1þ ðbvÞ1=r�. ð22ÞGiven �j one can compute the two measures of the effect of inflation.

5.1. The model with inside money issued only for liquidity needs

Table 2 presents the results of numerical simulations for the case in which insidemoney can only be issued for liquidity needs. The non-linearity in the effects of infla-tion are clearly demonstrated as an increase in inflation of 5% starting from zeroinflation causes a negligible effect on output, while a 5% increase in inflation starting

Table 2h 1: The model with inside money issued only for liquidity needs

p c Change in average output (%) Change in average welfare (%)

5 0.08 �0.11 �0.1310 0.17 �0.45 �0.5915 0.26 �1.07 �1.4620 0.35 �2.04 �2.9025 0.44 �3.49 �5.27k = 0.9, z = 2, q = 0.75, a = 0.33, b = 0.98, r = 5.


from 15% causes a substantial fall in output. This non-linearity is due to the fact thatas inflation rises the fraction of agents who switch (i.e. c) increases and the cost atwhich they switch (given by j) also increases.

Regarding the magnitude of the effect of inflation, the effects of a moderate infla-tion eventually become quite large as the amount of switching becomes large. Itshould be noted that the inflation at which the effects become large is sensitive tothe parameter values. If z or q were reduced, then there would be little switchingat moderate inflation rates. In this case the effect of inflation would not be large untilinflation was relatively high. Similarly, if k were increased near unity, then the effectsof inflation would be bigger at lower inflation. Hence it is not possible to identify theinflation rate at which the effect becomes large without having estimates of k, z, andq. However, regardless of the values of these parameters there would be some infla-tion rate at which the costs would become large and there would still exist non-linearity in the effects of inflation.

5.2. The model with inside money issued for investment

The numerical simulations are repeated in Table 3, except that the Tobin effect isallowed for by allowing inside money to be issued for investment. Clearly the effectsare mitigated, particularly at low inflation where a 5% inflation increases both outputand welfare. However, as the theoretical results of Section 3 suggested, the positiveeffects of inflation are never large at any inflation rate. Also, once again there arestrong non-linearities. A 5% increase in inflation starting from 10% has a small neg-ative effect on output, but a significant negative effect on welfare. As before, the levelof inflation that produces large negative effects will be sensitive to the values chosenfor k, z, and q.

The effects of inflation on output and welfare are significantly different. This is be-cause much of the effect on average welfare is driven by the welfare of money holdersand the welfare of money holders falls by a greater amount when inside money maybe issued for investment. The reason is an increase in inflation causes a larger in-crease in issues of inside money when inside money is issued for investment com-pared to when it is only issued for liquidity purposes. Hence moneyholders holdrelatively more inside money, and since inside money earns a return of r < 1, whileoutside money earns a social return of unity, the welfare of money holders is lower.

Table 3h = (1 + p)a/1�a: The model with inside money issued for investment

p c Change in average output (%) Change in average welfare (%)

5 0.15 +0.20 +0.1110 0.29 +0.26 �0.2615 0.42 +0.02 �1.5420 – – –

k = 0.9, z = 2, q = 0.75, a = 0.33, b = 0.98, r = 5.


Finally, note that no numbers are reported for the 20% inflation rate in Table 3.This is because the demand for outside money has fallen to zero by this inflation rate.Hence when the effects of inflation become large, implying substantial switching toverifiable capital, outside money demand falls to zero and an inside money equilib-rium exists. While this appears troublesome in that positive money demand is ob-served at high rates of inflation, it presents no problem if one assumed issuers ofinside money had to hold reserves of outside money, or if some transactions werecash-only. In this case money demand would never go to zero.

6. Concluding remarks

This paper has presented a model of inside and outside money to examine the ef-fects of inflation. Recent research which has modeled financial services as a costlysubstitute for outside money. This paper has extended such models by allowingthe outside money substitute to represent a debt contract and intermediate capital.It was shown that moderate inflation can have substantial negative effects and theeffects are non-linear. Moreover, it is possible at low inflation for the Tobin effectto exist. These results are consistent with recent empirical evidence.

It should also be noted that the results are also consistent with empirical resultsregarding inflation, financial activity, and the capital stock. In the model of thispaper, increases in inflation cause an increase in the amount of inside money. English(1999) and Aiyagari et al. (1998) find empirical evidence supporting this result. Also,in this model when there is a net negative impact of inflation it is a direct result of asmaller amount of capital held by agents who have switched, suggesting as inflationrises, the capital stock falls. There is a large literature, such as Boyd et al. (2001) andFischer (1993), which finds evidence of this effect.

In addition to explaining the empirical evidence, an important feature of themodel is the ability to include all three possible effects of inflation (Super-neutrality,Reverse-Tobin effect, and Tobin effect), allowing one to determine the conditionsnecessary for a particular effect to emerge. The effect that emerges depends onassumptions made regarding the ability of agents to switch for non-verifiable capitalto verifiable capital and whether issues of inside money could be used for investmentpurposes.

While all three effects are possible in the model, superneutrality and a pure Tobineffect both seem unlikely from a theoretical perspective. There are two ways forsuperneutrality to emerge. First, every agent could face a prohibitively high costof issuing inside money (i.e. k = 0 and no switching). As this would imply thenon-existence of inside money, this is counterfactual. Superneutrality can also occurif most agents face a prohibitively high cost of issuing inside money, while some canissue inside money at zero cost (i.e. k > 0 and no switching) but issues of insidemoney cannot be used for investment. If, though, the issues of inside money canbe used for investment, then a pure Tobin effect exists. However, this case seemsimplausible as it assumes some agents have a prohibitively high cost of issuing insidemoney, other agents have a zero cost, but no agents have a modest cost.


Hence superneutrality and a pure Tobin effect appear to be special cases. That is,it would seem that at least some agents face a modest cost to producing insidemoney, and thus there would always be a tendency for a Reverse-Tobin effect.The only question is whether issues of inside money are used for investment, whichwould create a tendency for the Tobin effect to exist simultaneously with the Re-verse-Tobin effect. As was shown in Section 4 of the paper, the net effect will be aReverse-Tobin effect if it is costly to produce inside money for all agents (i.e.k = 0). If some agents face a zero cost of producing inside money (i.e. k > 0) thenthe net effect will be the Tobin effect only when starting from low inflation. Sincethe question of how issues of inside money are used appears important, future re-search should address this question.

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Inside money and the effects of inflation