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FEDERAL RESERVE BANK OF CLEVELAND pd papers NUMBER 12 by Ben Craig and Guillaume Rocheteau POLICY DISCUSSION PAPER Inﬂation and Welfare: A Search Approach JANUARY 2006 POLICY DISCUSSION PAPERS FEDERAL RESERVE BANK OF CLEVELAND Inﬂation and Welfare: A Search Approach By Ben Craig and Guillaume Rocheteau This paper extends recent ﬁndings in the search-theoretic literature on monetary exchange regarding the welfare costs of inﬂation. We present ﬁrst some estimates of the welfare cost of inﬂation using the “welfare triangle” methodology of Bailey (1956) and Lucas (2000). We then derive a money demand function from the search-theoretic model of Lagos and Wright (2005) and we estimate it from U.S. data over the period 1900–2000. We show that the welfare cost of inﬂation predicted by the model accords with the welfare-triangle measure when pricing mechanisms are such that buyers appropriate the social marginal beneﬁt of their real balances. For other mechanisms, welfare triangles underestimate the true welfare cost of inﬂation because of a rent-sharing externality. We also point out other inefﬁciencies associated with noncompetitive pricing, which matter for estimating the cost of inﬂation. We then illustrate how endogenous participation decisions can mitigate or exacerbate the cost of inﬂation, and we provide calibrated examples in which a deviation from the Friedman rule is optimal. Finally, we discuss distributional effects of inﬂation. Ben Craig is an economic advisor at the Federal Reserve Bank of Cleveland, and Guillaume Rocheteau is an economist at the Bank.The authors have beneﬁted from the comments of Ray Batina, Chuck Carlstrom, Richard Dutu, Sebastien Lotz, Ed Nosal, and Christopher Waller. They thank Patrick Higgins for his research assistance and Monica Crabtree-Reusser for editorial assistance. Materials may be reprinted, provided that the source is credited. Please send copies of reprinted materials to the editor. POLICY DISCUSSION PAPERS Policy Discussion Papers are published by the Research Department of the Federal Reserve Bank of Cleveland.To receive copies or to be placed on the mailing list, e-mail your request to 4dsubscriptions@clev.frb.org or fax it to 216-579-3050. Please send your questions comments, and suggestions to us at editor@clev.frb.org. Policy Discussion Papers are available on the Cleveland Fed’s site on the World Wide Web: www.clevelandfed.org/Research. Views stated in Policy Discussion Papers are those of the authors and not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. ISSN 1528-4344 FEDERAL RESERVE BANK OF CLEVELAND Introduction Assessing the welfare costs of inﬂation requires a sound understanding of the beneﬁts of monetary exchange. The search theory of money, developed in the last 15 years from the pioneering works of Kiyotaki and Wright (1989, 1991, 1993), offers such a framework. However, the ﬁrst generation of search models of money were based on assumptions that were too restrictive to be able to deliver useful insights for monetary policy (goods and money were indivisible, for example, individuals’ portfolios were limited to one unit of one object, and so forth).These severe restrictions have been relaxed in several recent extensions of the theory, by Shi (1997, 1999), Molico (1999), and Lagos and Wright (2005).The extensions have opened up the perspectives for a better understanding of the costs, and also maybe beneﬁts, associated with inﬂationary ﬁnance.As an example, Lagos and Wright provide estimates for the cost of 10 percent inﬂation ranging from 1.4 percent of GDP to 4.6 percent of GDP. Interestingly, these numbers are signiﬁcantly larger than estimates based on the traditional method developed by Bailey (1956), which consists of computing the area underneath a money demand function. For instance, Lucas (2000), using Bailey’s approach, estimates the cost of 10 percent inﬂation at slightly less than 1 percent of GDP.1 In this paper we clarify and extend recent ﬁndings provided by models of monetary exchange to the evaluation of the cost of inﬂation for society. Our approach consists of relating the measures of the welfare cost of inﬂation obtained from different versions of the search-theoretic model of Lagos and Wright (2005) with the traditional measures based on the area underneath the money demand func- 1. Based on this methodology, Fischer (1981) and Lucas (1981) obtained estimates for the cost of 10 percent inﬂation ranging from 0.3 percent of GDP to 0.45 percent of GDP. tion. We show the conditions under which the two measures are consistent, and those under which they differ. We also disentangle the different effects of inﬂation in search models of monetary exchange: a real-balance effect, an effect on participation decisions, and a distributional effect.We show that the estimates for the welfare cost of inﬂation provided by the basic version of the search model of Lagos and Wright coincide with those provided by the Bailey method whenever money holders can appropriate the marginal social return of their real balances. This condition is satisﬁed when buyers have all the bargaining power to set prices in bilateral trades, or when pricing is competitive. If this condition does not hold, then the welfare cost of inﬂation is larger than what traditional estimates predict.This discrepancy arises because of a rent-sharing externality associated with noncompetitive pricing mechanisms.We establish a simple relationship between the cost of inﬂation, the area underneath the money demand function, and the buyer’s share in the surplus of a trade. We also discuss various inefﬁciencies associated with different bargaining solutions. We also extend the Lagos-Wright model by introducing participation decisions and trading frictions.We show that the measures of the cost of inﬂation based on the Bailey methodology are in general misleading since they do not take into account the effects of inﬂation on participation decisions. We also illustrate how the presence of search frictions can mitigate or exacerbate the welfare cost of inﬂation.We provide calibrated examples in which the Friedman rule, where the interest rate is set to zero, is not the optimal policy. The Lagos-Wright model is based on assumptions that yield a degenerate distribution of money balances in equilibrium.2 While these assumptions make the model tractable, they prevent an analysis of the distributional effects of inﬂation.We reintroduce such effects by considering a simple extension of the Lagos-Wright model in which agents receive idiosyncratic productivity shocks. We discuss the distributional effects of monetary policy and the insurance role of inﬂation. 2. Shi (1997, 1999) constructs a different model, which also yields a degenerate distribution of money balances. The economy is populated by households composed of a large number of members, who pool their money balances at the end of each period. See also Faig (2004). 1 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 The paper is organized as follows. First, we present the methodology Bailey (1956) developed to compute the welfare cost of inﬂation. Next, we use a search-theoretic model to derive a money demand function, and we ﬁt this function to the data. This allows us to compute an alternative measure of the cost of inﬂation and to compare it to the Bailey estimates.Then, the benchmark model is extended to discuss the importance of pricing mechanisms. Next, we consider participation decisions and search frictions. Finally, we introduce the distributional effects of inﬂation.3 The “Welfare Triangle” 3. A more detailed presentation of the model occurs in appendix 1, and the data are given in appendix 2. The traditional approach to measuring the cost of inﬂation was developed by Bailey (1956).The estimates provided by this approach will be useful in explaining some ﬁndings of the search model. Bailey (1956) measures the welfare cost of inﬂation by calculating the area underneath a money demand curve over an appropriate interval. We plot in ﬁgure 1 the (inverse) demand for real balances, where the cost of holding real balances, the nominal interest rate, is represented on the vertical axis.The demand for real balances is downward-sloping since individuals reduce their money balances and resort to alternative payment arrangements, such as credit or barter, as the interest rate increases. The area underneath the money demand relationship over the interval [m1, m*], the “triangle” ABC in ﬁgure 1, measures the welfare cost of having a positive interest rate r1 instead of zero. (In this analysis, the interest rate is assumed to vary one to one with the inﬂation rate.) Obviously, the welfare cost of inﬂation is minimized when the nominal interest rate is zero.4 This corresponds to the Friedman (1969) rule for optimal monetary policy. In the following, we will measure the cost of inﬂation as the cost of raising the interest rate from r0 = 3 percent, interpreted as the interest rate consistent with zero inﬂation, to r1, say, the interest rate associated with 10 percent inﬂation. Graphically, this cost is measured by the area ABDE. We deﬁne monetary assets according to the monetary aggregate M1, that is, currency and demand deposits.5 Money demand is then deﬁned by the aggregate money balances M1 divided by nominal gross domestic product.6 The nominal interest rate, r, is measured by the short-term commercial paper rate. In ﬁgure 2, we represent each observation (r,m) by a circle for the period 1900–2000. FIGURE 1 THE WELFARE TRIANGLE B r0 0 2 5. Alternatively, several authors, including Fischer (1981), deﬁne money as high-powered money. The operational deﬁnition of monetary assets is somewhat arbitrary. For a discussion, see Lucas (1981) and Marty (1999). 6. By measuring real balances as a fraction of domestic output, the area of the money triangle can be interpreted as the fraction of income that is needed to compensate individuals for an interest rate of r1 instead of zero (Lucas, 2000). Nominal interest rate r1 4. Since the interest rate is approximately the sum of a constant real interest rate and the inﬂation rate, the Friedman rule would imply that the inﬂation rate is negative and approximately equal to the opposite of the real interest rate. D A m1 E m0 C m* Money real balances FEDERAL RESERVE BANK OF CLEVELAND To measure the welfare triangle, one estimates a curve that ﬁts the observations in ﬁgure 2, and then computes the appropriate area underneath the implied money demand curve. Lucas (2000) considers two speciﬁcations for money demand: the log–log speciﬁcation, m( r ) = Ar −η , where m is aggregate real balances divided by output, r is the interest rate, and A and η are two estimated parameters; and the semilog speciﬁcation, where m( r ) = Ae −η r. In order to estimate the parameters A and η we use nonlinear least squares.7 We also estimate the money demand curve by using a kernel regression.8 It can be seen in ﬁgure 3 that the welfare cost associated with an interest rate of 13 percent (10 percent inﬂation, approximately) is quite different across speciﬁcations for the money demand 7. This method is different from the one used by Lucas (2000), who constrains the curves to pass through the geometric means of the data and who uses a visual test to identify the best ﬁt. function, from slightly more than 0.5 percent to slightly less than 1.5 percent. These differences simply reﬂect different ways to ﬁt the data.According to the R-squared criterion, the best ﬁt is obtained for the kernel regression which evaluates the cost of 10 percent inﬂation at about 1 percent of GDP.9 This number is similar to Lucas’s measure.The estimate from the semilog speciﬁcation, about 1.5 percent of GDP, is comparable to Lagos and Wright’s smallest estimate of the welfare cost of inﬂation. Search for a Money Demand Curve The Bailey approach does not identify explicitly the beneﬁts of monetary exchange for society.10 An alternative approach consists of constructing a microfounded model economy in which money has an essential role in trades, so that there is a well-speciﬁed demand for real balances and a natural measure of welfare. A theory that emphasizes the transactional role of money is the search approach of monetary exchange pioneered by Kiyotaki and Wright (1989, 1991, 1993).The recent extension proposed by Lagos and Wright (2005) describes an economy in which trades take place under different market structures. Some trades occur in a decentralized (or search) market, where buyers and sellers are matched bilaterally, and other trades occur in a centralized market. Money is useful because of a standard double-coincidence-of-wants problem in the decentralized market:The buyer does not pro- 8. The kernel was estimated with a local bandwidth computed using plug-in techniques, modiﬁed at each boundary. See Brockman et al. (1993). 9. For the kernel regression, R2=0.6795; for the log–log speciﬁcation, R2=0.6238; for the semilog speciﬁcation, R2=0.6750. 10. The Bailey analysis is subject to the following caveats. It is a partial equilibrium analysis that assumes away externalities, general equilibrium effects, and distributional effects. Also, an underlying assumption is that the government has access to nondistorting taxes, so that a change in seigniorage revenue has no welfare consequences. duce a good that the seller wants to consume. FIGURE 2 FITTING MONEY DEMAND Real balances 0.6 0.5 Nonparametric 0.4 0.3 Log-Log 0.2 Semilog 0.1 0 2 4 6 8 10 Interest rate, percent 12 14 3 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 The instantaneous utility of an agent is u( q b ) − c ( q s ) + x, where qb is the consumption and qs is the production in a bilateral match, and x is the net consumption in the centralized market (x is negative if an agent produces more at night than he consumes).11 The probability of a single-coincidenceof-wants meeting in which an agent meets someone who produces a good he likes is σ ≤ 1 2 .There are no double-coincidence-of-wants meetings in which agents could use barter. So with probability σ an agent is a buyer in a bilateral match, with probability σ he is a seller, and with probability 1 − 2σ he is unmatched.A social planner, who would dictate the quantities to produce and consume, would choose q b = q s = q *, where q* satisﬁes u′( q *) = c ′( q *) .The socially efﬁcient level of production and 11. The linearity of the preferences is what guarantees that the distribution of wealth at the beginning of each period is degenerate. Intuitively, this linearity eliminates wealth effects in the choice of real balances and ensures that all buyers carry the same amount of real balances. consumption in the centralized market is indeterminate.The quantity of ﬁat money in the economy is growing at a constant rate, π , through lump-sum transfers in the centralized market. Denote z(q) as the real balances that an agent must hold in order to buy the quantity q ∈ [ 0, q * ] in a bilateral match. The speciﬁc form for z(q) will depend on the assumed pricing mechanism in the decentralized market.The Lagos-Wright model can be reduced to one equation that speciﬁes the quantity q = q b = q s traded in bilateral matches.This equation says that an agent chooses the quantity q to consume in the decentralized market so as to maximize the expected surplus he gets as a buyer, σ [u( q ) − z ( q ) ] minus the cost of holding real balances, rz(q), q = arg max {− rz ( q ) + σ [u( q ) − z ( q ) ]} . In order to calibrate the model, we adopt the same functional forms as the ones used in LagosWright: u( q ) = q1−η / (1 − η ), where η ≥ 0 and c ( q ) = q. Furthermore, the matching probability σ is set to 1/2, so that each agent trades with probability one. Half of the time an agent is a buyer, and half of the time he is a seller.12 The money demand function that is estimated is deﬁned as aggregate money balances divided by aggregate nominal output. It is equal to L = z / (σ z + A ), where A is the real output in the centralized market (this quantity is indeterminate in the model) and where z is a function of the nominal interest rate, r.13 FIGURE 3 13. Since agents readjust their real balances in the centralized market, the output A must be at least equal to σz. WELFARE COST OF INFLATION USING THE BAILEY METHOD Cost of inflation, percent 2.0 Semilog 1.5 Nonparametric 1.0 0.5 Log-Log 0 –0.5 0 4 2 4 6 8 Interest rate, percent 10 12. Alternatively, one could consider a model in which half of the agents are buyers in even periods and the remaining half are buyers in odd periods. This formulation would give identical results for the cost of inﬂation. 12 14 FEDERAL RESERVE BANK OF CLEVELAND One needs to take a stand on how prices (or terms of trade) are determined in decentralized markets in which buyers and sellers are matched bilaterally. We will assume here that the monetary transfer from the buyer to the seller is such that the seller is exactly compensated for his production cost, z ( q ) = c ( q ) .This bargaining solution, called the dictatorial solution, is the outcome of a game in which the buyer makes an offer which the seller can accept or reject. If the offer is rejected, no trade takes place.After some calculation, aggregate real balances satisfy −1 1 ⎡ ⎤ η ⎢σ + A ⎛1 + r ⎞ ⎥ . L= ⎜ ⎟ ⎢ ⎝ σ⎠ ⎥ ⎣ ⎦ As in the previous section, the parameters A and η can then be estimated from the data for the U.S. economy from 1900 to 2000.The parameter A could be interpreted as the extent of the tax base.The parameter η represents the sensitivity of individual real balances to changes in the interest rate. In order to measure the welfare cost associated with a given interest rate, r, relative to 3 percent (the interest rate consistent with zero inﬂation), we ask the following question. What is the percentage of total consumption that individuals would be willing to sacriﬁce in order to be in the steady state with an interest rate of 3 percent instead of the steady state associated with r? In ﬁgure 4, we compare the two measures of the welfare cost of inﬂation, namely, the compensated measure and the welfare triangle measure. The welfare triangle measure is the area underneath the money demand function as estimated from the search model. Figure 4 shows that the two measures are nearly identical. In order to understand this result, consider the individual demand for real balances given by dq ⎤ ⎡ −1 , r = σ ⎢u′( q ) dz ⎥ ⎣ ⎦ FIGURE 4 COMPENSATED WELFARE AND THE WELFARE TRIANGLE Cost of inflation, percent 2.0 1.5 Compensated welfare 1.0 Welfare triangle 0.5 0 –0.5 0 2 4 8 6 Interest rate, percent 10 12 14 5 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 where dq / dz = 1 / z ′( q ) . Compute the area underneath this money demand function over the interval [ z0 , z1 ], z1 ∫ r ( z )dz = σ {u [q ( z )] − z } − σ {u [q ( z )] − z }. 1 1 0 0 z0 Using the assumed dictatorial solution, where the seller is exactly compensated for production costs, z = c ( q ), it is easy to see that the right-hand side of the previous expression is just the change in steady-state welfare. So the area underneath the individual demand for real balances coincides with the change in steady-state welfare. These two measures do not exactly coincide in ﬁgure 4 because we express real balances as a fraction of aggregate output and we do not look at the change in steadystate welfare but at a compensated measure of welfare. The welfare cost of a nominal interest rate of 13 percent relative to a 3 percent interest rate is about 1.5 percent.14 This measure is bigger than the ones of Lucas (2000) but it accords with the nonlinear least square estimate based on the semilog speciﬁcation.The difference between the numbers simply stems from different strategies of ﬁtting the points in the data. In all cases, the cost of inﬂation corresponds to the area underneath a money demand function.15 14. Assuming a competitive pricing mechanism, Rocheteau and Wright (2004, 2005) and Reed and Waller (2004) ﬁnd similar estimates for the welfare cost of inﬂation, between 1 and 1.5 percent of GDP. Pricing The estimate for the welfare cost of inﬂation in the previous section has been obtained by assuming a special pricing mechanism: Buyers are able to extract the whole surplus from trade.Alternatively, one can model prices, and the way in which the surplus from a trade is shared between trading partners, differently. In this section, we consider ﬁrst a simple bargaining solution, called the proportional solution, which will illustrate the role played by the pricing mechanism in assessing the 15. Aiyagari, Braun, and Eckstein (1998) also show, using a cash-in-advance model with a credit sector, that the welfare cost of inﬂation using an estimated money demand curve is consistent with the prediction of the model. welfare cost of inﬂation. The proportional bargaining solution assumes that the buyer obtains a constant fraction, called the buyer’s share and denoted θ , of the surplus of a match deﬁned as the difference between the buyer’s utility of consumption and the seller’s disutility of production. Formally, u( q ) − z ( q ) = θ [u( q ) − c ( q ) ], and therefore z ( q ) = θ c ( q ) + (1 − θ ) u( q ). The dictatorial solution of the previous section corresponds to θ = 1. From the agent’s choice of real balances, one can ﬁnd a simple relationship between z, individual real balances, and the nominal interest rate, r. Aggregate real balances are deﬁned as L = z / (σ z + A ) . We use the same method as before to derive the cost of inﬂation: We estimate the parameters A and η of money demand generated by the model, and we use compensated welfare to compute the cost of inﬂation. In ﬁgure 5, we report the welfare cost of inﬂation for different values for the buyer’s share (θ = 0.3, 0.5, 0.8, 1) .16 When the buyer’s share is less than 1, the cost of inﬂation is typically larger than the measure given by the money triangle. In fact, the area of the welfare triangle is approximately equal to the buyer’s share multiplied by the compensated measure of the welfare cost of inﬂation. For instance, if the buyer’s share is 50 percent, the welfare cost of inﬂation is always about twice the size of the area of the money triangle (see ﬁgure 6).To understand this result, consider the area underneath the individual demand for real balances. It satisﬁes z1 ∫ r ( z )dz = σ {u [q ( z )] − z } − σ {u [q ( z )] − z } 1 1 0 0 z0 = θσ {u [ q ( z1 ) ] − c [ q ( z1 ) ]} − θσ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} . 6 16. The R2 for the case of θ = 1.0 is 0.6757, which is higher than the log–log and semilog parametric models above. Other values of θ gave smaller values for the R2 than the semilog model. FEDERAL RESERVE BANK OF CLEVELAND It is equal to the change in steady-state welfare multiplied by θ. In ﬁgure 6, real balances are divided by aggregate output, and the welfare metric of the model is a compensated measure, which explains the slight discrepancy between the two lines. As one varies the buyer’s share from 0.3 to 1, the cost of 10 percent inﬂation varies from slightly more than 1 percent to 6 percent of GDP. In order to understand why the welfare triangle can underestimate the true welfare cost of inﬂation, consider the following example. Suppose that each unit of good produced in a bilateral match is worth $1 for the buyer and costs $0.9 to produce.The marginal surplus of a trade is then $0.1. Suppose that the price is $0.95, so that both the buyer and the seller get a surplus of $0.05.The private return of money is equal to the buyer’s surplus divided by the amount of money that the buyer must carry to buy the good, 0.05/0.95 = 5.2 percent. The social return of FIGURE 5 WELFARE COST OF INFLATION UNDER PROPORTIONAL BARGAINING Cost of inflation, percent 7.0 θ = 0.3 6.5 5.5 4.5 θ = 0.5 3.5 2.5 θ = 0.8 1.5 θ = 1.0 0.5 –0.5 –1.5 0 FIGURE 6 2 4 6 8 Interest rate, percent 10 12 14 WELFARE TRIANGLE VERSUS COMPENSATED WELFARE ( θ = 0.5) Cost of inflation, percent 3.5 3.0 2.5 Compensated welfare 2.0 Welfare triangle x2 1.5 1.0 0.5 0 –0.5 0 2 4 6 8 Interest rate 10 12 14 7 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 money is the total surplus divided by the price of the good, 0.1/0.95 = 10.5 percent. If the interest rate is 10 percent, the cost of holding $0.95 for the buyer is larger than the marginal gain of $0.05. So the buyer has no incentive to bring an additional dollar even though the return of this dollar to society is larger than the opportunity cost incurred by the buyer. This discrepancy between the private and social beneﬁts of real balances arises from a rent-sharing externality. The marginal beneﬁt of the real balances from the buyer’s point of view is smaller than the marginal beneﬁt from society’s point of view. Since the money demand estimated from the data only captures the marginal beneﬁt of money from the buyer’s side, the welfare triangle misses a fraction of the welfare cost of inﬂation.This externality arises from any pricing rule that stipulates that the buyer does not get the full marginal return of his real balances.17 This point is illustrated in ﬁgure 7. For a given stock of real balances m0, the marginal beneﬁt that money provides to the money holder, the length of segment AB, is smaller than the marginal societal beneﬁt of money, the length of segment AD. If prices are determined according to a proportional solution, the ratio BA/DA is the buyer’s share. As a consequence, when measuring the area underneath the demand for real balances, the ABC area, one underestimates the social beneﬁt of money, the ADC area, by a factor equal to the inverse of the buyer’s share. In ﬁgure 7, the two curves representing the private and social beneﬁts of real balances intersect the horizontal axis (r = 0) for the same value m* of real balances.At this point, the total beneﬁts of real balances are maximized for both buyers and society.This observation implies that the Friedman rule yields the best allocation of resources for society for all values of the buyer’s share, θ . The result in which the Friedman rule generates the ﬁrst-best allocation does not hold for all bargaining solutions. For instance, it does not hold for the Nash (1950) solution, according to which z ( q ) = Θ( q )c ( q ) + [1 − Θ( q ) ] u( q ) with Θ( q ) = θ u′( q ) , θ u′( q ) + (1 − θ )c ′( q ) FIGURE 7 THE RENT-SHARING EXTERNALITY Fraction of a dollar Private marginal benefit of money = Money demand D Social marginal benefit of money r0 B A 0 8 m0 C m* Real money balances 17. This rent-sharing externality is closely related to holdup problems noted in the investment literature. FEDERAL RESERVE BANK OF CLEVELAND where θ is now the bargaining power of the buyer.18 The buyer’s share, Θ , depends on both θ and q, and it is equal to θ when q = q*. In particular, the buyer’s share, Θ , decreases as q increases.As shown by Lagos and Wright (2005), if prices in bilateral matches are determined according to the Nash solution, then the Friedman rule is optimal but the quantities traded in the decentralized market are too low. This result is illustrated in ﬁgure 8. At the Friedman rule (r = 0) the economy’s real balances are m, while the real balances that would maximize society’s welfare are m*. In other words, if the interest rate is zero, an individual’s demand for real balances is satiated even though the marginal beneﬁt of money to society is still positive.This inefﬁciency is called a “nonmonotonicity inefﬁciency” to reﬂect the fact that the buyer’s surplus from a trade does not necessarily increase with the match surplus.19 Put differently, the buyer’s surplus u( q ) − z ( q ) reaches a maximum for a value of q smaller than q*. This inefﬁciency has two important consequences for the welfare effects of inﬂation. First, a small increase of the interest rate above r = 0 will have a larger effect on welfare than in the case under proportional bargaining solutions. Indeed, the welfare cost of a small interest rate can be approximated by the change in real balances multiplied by the social beneﬁt of real balances at r = 0. This second term, measured by the length of the segment EC in ﬁgure 8, is now positive at r = 0. Second, real balances are inefﬁciently low at r = 0, and, as a consequence, the quantities produced and consumed in bilateral matches are also too low. 18. The bargaining power θ , which varies from 0 to 1, is a measure of the buyer’s strength in the bargaining process. In an explicit bargaining game with offers and counteroffers, the bargaining power of an individual depends, among other things, on his ability to threaten to terminate the negotiation if his offer is rejected. 19. In order to illustrate this inefﬁciency, consider a bargaining problem where two individuals must share a prize. If one of the two individuals gets worse off as the size of the prize increases, then the bargaining solution is said to be nonmonotonic. For a detailed treatment of alternative bargaining solutions and their properties in monetary economies, see Rocheteau and Waller (2004). Figure 9 plots the welfare cost of inﬂation when prices are determined according to the Nash solution for different values of the buyer’s bargaining power, θ . The comparison of ﬁgures 5 and 9 reveals that the welfare cost of 10 percent inﬂation under the Nash solution is of same magnitude as the cost under the proportional bargaining solution. In both cases, there is a rent-sharing externality at work, which ampliﬁes the cost of inﬂation. However, under the Nash solution, the buyer’s share, Θ, gets larger for higher inﬂation rates so that the rent-sharing externality gets smaller.The most noticeable difference between ﬁgures 5 and 9 is the gain associated with a reduction of the interest rate from 3 percent to zero (which corresponds to the optimal deﬂation rate).This gain can be as high as 2.5 percent of GDP when the buyer’s bargaining power is 0.3. Under the proportional solution, this FIGURE 8 BARGAINING INEFFICIENCIES Fraction of a dollar Money demand D Social marginal benefit of money r0 B E 0 A m0 C ~ m F m* Real money balances 9 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 gain is about 0.5 percent of GDP. So the nonmonotonicity inefﬁciency is important in that it predicts large welfare gains if inﬂation is reduced from zero to the optimal deﬂation rate. To conclude this section, we present a pricing mechanism which exhibits the same type of inefﬁciencies as the Nash bargaining solution but which is based on the possibly more familiar idea that prices are set as a markup over the cost incurred by sellers.20 More precisely, the transfer of money from the buyer to the seller corresponds to the cost incurred by the seller in producing the amount asked for by the buyer multiplied by a constant factor, 1 + μ, (where μ ≥ 0 ), which we interpret as the “markup,” z ( q ) = (1 + μ )c ( q ). As shown in ﬁgure 10, the cost of inﬂation increases with the markup. When the markup is 20 percent, the cost of 10 percent inﬂation is slightly more than 3 percent of GDP, which is similar to the prediction of the model under the symmetric Nash solution. FIGURE 9 COST OF INFLATION UNDER NASH BARGAINING Cost of inflation, percent 6.0 θ = 0.3 5.0 4.0 θ = 0.5 3.0 2.0 θ = 0.8 1.0 θ = 1.0 0 –1.0 –2.0 –3.0 0 FIGURE 10 2 4 6 8 Interest rate, percent 10 12 14 12 14 COST OF INFLATION UNDER CONSTANT MARKUP Cost of inflation, percent 3.5 μ = 20% 3.0 2.5 μ = 10% 2.0 1.5 1.0 μ=0 0.5 0 –0.5 –1.0 –1.5 0 10 2 4 8 6 Interest rate, percent 10 20. For a search model with price posting by sellers, see Ennis (2004). Ennis (2004) describes an economy in which buyers have private information about their tastes, and sellers make take-it-or-leave-it offers. The annual welfare cost of a 10 percent inﬂation in this model is between 4 percent and 7 percent of GDP. FEDERAL RESERVE BANK OF CLEVELAND Figure 11 illustrates how a constant markup affects the assessment of the welfare cost of inﬂation. The constant markup, μ , shifts the curve indicating the social return of real balances up by a constant amount (BD = CE), which reﬂects the seller’s marginal beneﬁt from buyers’ real balances. The larger the markup, the larger the difference between private and social beneﬁts of real balances. Also, it is clear from ﬁgures 8 and 11 that both the Nash solution and the pricing with constant markup induce qualitatively similar effects of inﬂation. In both cases, the quantities traded at the Friedman rule (r = 0) are too low. Participation Decisions In the basic model described in the previous sections, the frequency of trade is assumed to be constant. So inﬂation affects the quantities traded in bilateral meetings, but it does not affect the number of those meetings. In order to endogenize the number of trade matches, one can let buyers and sellers choose whether or not to participate in the market, or let them choose on which side of the market to participate in, or even let them choose the resources they will invest in the search for a trading partner.21 By taking into account these participation decisions, one can introduce general equilibrium effects of inﬂation, which are not captured by the Bailey methodology. Also, in environments with search frictions, participation decisions tend to be inefﬁcient. Consequently, the welfare effects of inﬂation are ambiguous and depend on the way in which inﬂation distorts participation decisions. We consider in the following an extension of the Lagos-Wright model, which is based on an assumption in Shi (1997).22 The economy is similar to the one previously described except that at the beginning of each period, before matches are formed, individuals can choose to be buyers or sellers in the decentralized market. For instance, agents in the labor market can choose to be buyers (entrepreneurs) or sellers (workers).An agent who chooses to be a buyer cannot produce during the day, while an agent who chooses to be a seller cannot consume during the day.The composition of buyers and 21. Rocheteau and Wright (2005) consider a model with free entry of sellers. Shi (1997) describes an economy in which individuals can choose which side of the market to participate in. See also Rocheteau and Wright (2004) and Faig (2004). Li (1995, 1997) introduces endogenous search intensities. 22. Our model is similar to the one in Rocheteau and Wright (2004), except that we consider different pricing mechanisms. sellers is then endogenous. Let n denote the fraction of sellers in the economy. Assume further that the matching process is such that a buyer meets a seller with probability n, whereas a seller meets a FIGURE 11 CONSTANT MARKUP Fraction of a dollar Money demand Social marginal benefit of money D r0 B E A 0 m0 C ~ m Real money balances 11 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 buyer with probability 1 – n.23 The fraction of sellers in equilibrium is such that agents are indifferent to being buyers or sellers, −rz ( q ) + n [u( q ) − z ( q ) ] = (1 − n ) [ z ( q ) − c ( q ) ] . The left-hand side of the previous equation is the expected utility of a buyer, the right-hand side is the expected utility of a seller, and q is the equilibrium quantity traded in bilateral matches.24 We assume that prices are determined according to the proportional bargaining solution, z ( q ) = θ c ( q ) + (1 − θ )u( q ). The equilibrium allocation is socially efﬁcient if θ = 0.5 and r = 0. The second requirement corresponds to the Friedman rule, and it guarantees that q = q*. The ﬁrst requirement, θ = 0.5, is the 23. The speciﬁcation for the matching function is the same as the one used in most monetary models, including Kiyotaki and Wright (1993). Obviously, it would be desirable to estimate the transaction technology. This strategy has been pursued successfully in the labor literature. We leave this extension for future investigation. 24. The money supply is growing through lump-sum transfers. Such transfers do not affect agents’ decisions to be buyers or sellers. condition under which the number of trades is maximized; this maximization requires n = 1/2. It corresponds to the Hosios (1990) condition according to which the number of trades is efﬁcient only in the unlikely event that an agent’s share of the match surplus is equal to his marginal contribution to the creation of trade matches.25 The best outcome for a search monetary economy requires that both market, A, 25. The elasticity of the number of trades with respect to the number of buyers in the economy is n. Therefore, the Hosios condition requires θ = n. When r = 0, n = 1 − θ , so that the optimal allocation requires θ = 0.5. (1 − n ) z . L= n (1 − n ) z + A 26. For an elaboration of this idea, see Berentsen, Rocheteau, and Shi (2004). the Hosios condition and the Friedman rule hold.26 The deﬁnition of the aggregate demand for money is the money held by the 1 – n buyers divided by the sum of the output in the decentralized market, n(1 – n)z, and the output in the centralized We use the same strategy as before to estimate the parameters A and η of money demand. In ﬁgure 12, we report the welfare cost of inﬂation. When the buyer’s share is less than 50 percent, the number of buyers is too low and inﬂation lowers the fraction of buyers even further.The welfare cost of low inﬂation is then larger than what would be obtained under the assumption in which the frequency of trades is constant. Reciprocally, the welfare gains from reducing the interest rate to zero are also large.When the buyer’s share is above 50 percent, a deviation from the Friedman rule is optimal. FIGURE 12 COST OF INFLATION WITH SEARCH EXTERNALITIES (PROPORTIONAL BARGAINING) Cost of inflation, percent 6.0 5.5 θ = 0.3 5.0 4.5 4.0 3.5 3.0 θ = 0.5 2.5 2.0 1.5 1.0 θ = 0.8 0.5 0 θ = 0.9 –0.5 –1.0 0 12 2 4 8 6 Interest rate, percent 10 12 14 FEDERAL RESERVE BANK OF CLEVELAND In this case, if r = 0, then the number of buyers is too high and the number of trades is too low. Since inﬂation has a direct negative effect on buyers’ expected utility, the number of buyers falls, while the number of sellers increases.27 The composition in the market becomes more even and the number of trades increases.28 When the buyer’s share is 90 percent, then the cost of implementing the Friedman rule is about 0.5 percent of GDP while the cost of 10 percent inﬂation is close to 0. The results reported in ﬁgure 12 have been derived by assuming that terms of trade are set according to a proportional bargaining solution.As in the previous section, different bargaining solutions can have very different properties in terms of the efﬁciency of the equilibrium allocation at the Friedman rule. For example, in the basic Lagos-Wright model, the equilibrium allocation is efﬁcient at the Friedman rule under the proportional bargaining solution, but it is inefﬁcient under the Nash solution. Not too surprisingly, the choice of the pricing mechanism will also matter considerably for the welfare cost of inﬂation when participation decisions are endogenous. Figure 13 reports the cost of inﬂation when the terms of trade are determined according to the Nash solution. In sharp contrast to the results obtained under the proportional solution, there is no positive effect of inﬂation on welfare. The optimal monetary policy is the Friedman rule. The “nonmonotonicity inefﬁciency” of the Nash solution changes the nature of the trade-off between the negative effect of inﬂation on individual real balances and the effect of inﬂation on the composition of the market. Indeed, since real balances are inefﬁciently low at the Friedman rule under the Nash solution, a small increase of the interest rate reduces real balances, which has a ﬁrst-order negative effect on welfare. For the calibrated version of the model, this negative real balance effect dominates any positive effect of inﬂation on the composition of the market.29 As a consequence, a deviation from the Friedman rule is not optimal under the Nash solution. The comparison of ﬁgures 9 and 13 reveals that the presence of search externalities exacerbates the cost of inﬂation for large values of the buyer’s bargaining power. If the buyer’s bargaining power is 0.5, results are largely similar to those obtained in the absence of search externalities, because at this value of θ , the composition of the market is similar to the market in which σ = 1 2. If the buyer’s bargaining power is 0.9, the cost of 10 percent inﬂation is about 4.5 percent of GDP in the presence of search frictions, whereas it is less FIGURE 13 COST OF INFLATION WITH SEARCH EXTERNALITIES (NASH BARGAINING) Cost of inflation, percent 6.0 5.5 27. This effect is sensitive to the choice of the pricing mechanism. For instance, under the Nash solution the number of buyers can increase with inﬂation because the buyer’s share in the surplus of a match gets bigger. 28. The result in which the Friedman rule is not always optimal in the presence of search externalities could raise the following objection. If the government was able to make transfers contingent on agents’ participation decisions, then it could take care of the search externalities by an appropriate transfer scheme, while the Friedman rule would be used to reduce the monetary wedge associated with the inﬂation tax. In practice, however, it may be difﬁcult to implement transfers contingent on agents’ participation decisions since these decisions may not be observable. Inﬂation, which is a tax on market activities, may be an effective instrument for correcting inefﬁcient participation in the market. 29. For other parameterizations, a deviation from the Friedman rule could raise society’s welfare. However, the result according to which a deviation from the Friedman rule is less likely under Nash bargaining than under proportional bargaining is robust. θ = 0.3 5.0 4.5 θ = 0.9 4.0 3.5 θ = 0.8 3.0 θ = 0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1.0 –1.5 –2.0 –2.5 0 2 4 8 6 Interest rate, percent 10 12 14 13 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 than 2 percent of GDP when the frequency of trades is exogenous. Also, the welfare cost of inﬂation is not a monotonic function of the buyer’s bargaining power. An increase in the buyer’s bargaining power makes the rent-sharing externality less severe, but it also distorts the composition of the market toward too many buyers. To summarize, the introduction of endogenous participation decisions has several implications. First, the welfare triangle is a misleading measure since it does not capture the distortionary effects of inﬂation on individuals’ participation decisions. Second, the presence of search externalities can mitigate or exacerbate the welfare cost of inﬂation.Third, the Friedman rule may no longer be optimal, as the positive effect of inﬂation on the composition of the market and the frequency of trades can dominate the negative effect of inﬂation on real balances. Distributional Effects The Lagos-Wright model described in the previous sections has been designed in such a way that all agents, despite different trading histories in the decentralized market, start each period with the same money balances.This property of the model is what makes it tractable. However, because the distribution of money balances is degenerate, inﬂation does not have any distributional effect. Levine (1991), Molico (1999), and Deviatov and Wallace (2001), among others, have provided examples in which a policy that consists of increasing the money supply through lump-sum transfers induces some redistribution across individuals. For individuals with large stocks of nominal assets, the burden associated with the inﬂation tax is greater than the beneﬁt of receiving a lump-sum transfer. On the other hand, individuals with few nominal assets enjoy a net beneﬁt from the lump-sum transfer. In some circumstances this redistribution can be beneﬁcial to society. For example, if agents are subject to idiosyncratic shocks (on endowments, productivity, and so forth) that cannot be insured against, money growth can provide an insurance mechanism.30 We capture the distributional effects of inﬂation through a simple extension of the Lagos- Wright model. Recall that in the Lagos-Wright model individuals live forever, and, in each period of their lives, they trade sequentially in a centralized market and a decentralized market. We depart from these assumptions by assuming that individuals live only two periods.They are born at the beginning of the centralized market, and they die at the end of the following period after the centralized market has closed. Also, agents do not discount utility across periods. When agents are born, they have access to the centralized market in order to make their choice of money balances. In the second period of their lives, they trade in the decentralized market and then have access to the centralized market before they die. Only a fraction p of the newly-born agents are able to produce in the ﬁrst period of their lives. One can interpret this assumption as individuals receiving productivity shocks. In the absence of money growth, agents who cannot produce cannot accumulate money and, therefore, cannot consume in the decentralized market in the second period of their lives. By inﬂating the money supply the government can transfer money to all individuals, irrespective of their productivities, and therefore smooth consumption across all agents. The demand for real balances from productive agents is the same as the one described in the previous sections.The main difference with respect to the previous models is the fact that the distribution of real balances has two points, z, the real balances of productive agents, and z , the real balances of unproductive agents.The aggregate demand for money balances is then 14 30. In Molico’s (1999) version of the search-theoretic model, individuals trade only in a decentralized market with bilateral random matching. Since the matching process is random, trading opportunities arrive according to a stochastic process. Some individuals are lucky and can sell their output often: They then have a large stock of money balances. Other individuals are less lucky and have a low stock of real balances. Molico shows that for low inﬂation rates the redistributive effect of inﬂation, according to which inﬂation acts as a subsidy for the poor and a tax on the rich, can dominate the real balance effect of inﬂation, according to which inﬂation reduces aggregate real balances. In Craig and Waller’s (2004) version of a decentralized market with two currencies, one of which inﬂates, low rates of inﬂation can induce a strong redistributive effect in the inﬂating currency where buyers are likely to hold either large amounts of the currency or none at all. FEDERAL RESERVE BANK OF CLEVELAND L= pz + (1 − p ) z . σ ⎡ pz + (1 − p ) z ⎤ + A ⎣ ⎦ The strategy to estimate the model is the same as before. Society’s welfare is measured by the sum of the trade surpluses in all matches.The welfare cost corresponding to an interest rate, r, is the fraction by which total consumption at the steady state with a 3 percent interest rate must be reduced in order to achieve the same welfare as the one that prevails at the steady state with r.31 In ﬁgure 14, we plot the welfare cost of inﬂation when prices are determined according to the proportional bargaining solution with θ = 1/2. As in Molico (1999), society’s welfare is maximized for a positive inﬂation rate. The welfare gains associated with a positive interest rate are rather small. When calibrating the FIGURE 14 31. Note that we kept the benchmark for the interest rate at 3 percent. Since agents do not discount future utility, this interest rate corresponds to a 3 percent inﬂation. DISTRIBUTIONAL EFFECTS OF INFLATION Cost of inflation, percent 3.5 3.0 2.5 2.0 1.5 1.0 p = 0.8 0.5 p = 0.5 p = 0.3 0 p = 1.0 –0.5 0 FIGURE 15 2 4 6 8 Interest rate, percent 10 12 14 DISTRIBUTIONAL EFFECTS OF INFLATION (HIGHER RISK AVERSION) Cost of inflation, percent 3.5 p=1.0 3.0 p=0.8 2.5 p=0.5 2.0 p=0.3 1.5 1.0 0.5 0 –0.5 0 2 4 6 8 10 12 14 Interest rate, percent 15 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 model, the coefﬁcient that describes the aversion of individuals toward risk is low (the coefﬁcient of relative risk aversion is close to 0.2), and it is, in fact, lower than what is usually thought realistic. If one increases this coefﬁcient to make it a little bit more realistic, the welfare gain corresponding to the redistribution effect of inﬂation gets larger. For example, in ﬁgure 15 we plot the cost of inﬂation for the same value of the parameter A as the one used to draw ﬁgure 14 but for a coefﬁcient of relative risk aversion raised to 0.5.The beneﬁts of inﬂation get signiﬁcantly bigger and increase with the probability that an agent receives a negative productivity shock. Conclusion We have presented some insights provided by the search theory of monetary exchange for the understanding of the welfare cost of inﬂation. Using different extensions of the model of Lagos and Wright (2005), we have identiﬁed and quantiﬁed various effects of inﬂation on welfare. First, inﬂation has a negative real balance effect.The inﬂation tax introduces a wedge in the decision to invest in real balances. The extent of this distortion depends crucially on the assumed pricing mechanism. If buyers receive the full marginal beneﬁt of their money balances, the cost of inﬂation is essentially the Bailey measure, given by the area underneath the money demand function. In all other cases, the Bailey measure has to be scaled up by a function that depends on the sellers’ share in the surplus of a trade. We have also provided examples of pricing mechanisms under which the Friedman rule fails to generate the ﬁrst-best allocation. In such cases, the beneﬁt of implementing the optimal deﬂation is large. Second, inﬂation affects agents’ decisions to participate in the market and therefore it affects the number of trades. However, since participation decisions tend to be inefﬁcient in search environments, the effect of inﬂation on participation choices can be welfare-enhancing or welfare-worsening. We have provided calibrated examples in which a deviation from the Friedman rule is optimal.We have shown once again that the welfare effects of inﬂation depend crucially on the pricing mechanism.Third, inﬂation can generate a redistribution across agents. If agents are subject to idiosyncratic shocks, this redistribution can prove useful to society. Additional extensions would be worth considering. As emphasized by Cooley and Hansen (1989, 1991) and Dotsey and Ireland (1996), the inﬂation tax can distort a variety of marginal decisions: among others, the leisure-consumption choice and the accumulation of capital. Aruoba and Wright (2003) and Aruoba, Waller, and Wright (2004) have extended the Lagos-Wright model to allow for capital accumulation and have considered various assumptions regarding how capital enters the economy. Also, as Kocherlakota (2004) pointed out, it is important to extend search models in order to incorporate additional assets beside money, such as government bonds, and to take into account distortionary taxes. Finally, one can introduce realistic nominal rigidities in the search model of money along the lines suggested by Craig and Rocheteau (2004). 16 FEDERAL RESERVE BANK OF CLEVELAND Appendix 1:The Model The Lagos-Wright Model of Monetary Exchange We present brieﬂy the search-theoretic model of monetary exchange proposed by Lagos and Wright (2005). Time is discrete and each period of time is divided into two subperiods: the day and night. During the day trades take place in a decentralized market where agents are matched bilaterally. There is a lack of double coincidence of wants in bilateral matches. The probability for an agent to ﬁnd someone who produces a good he likes is σ , and the probability to ﬁnd someone who likes the good he produces is σ . With probability 1 − 2σ an agent is unmatched. Night trades take place in a competitive market. Money is introduced in the economy through lump-sum transfers in the centralized market and the supply of money is growing at the rate π . We only focus on steady-state equilibria where real balances are constant. An agent’s utility function is u( q b ) − c ( q s ) + x , where qb is the consumption and q s the production in a bilateral match and x is the net consumption in the centralized market (x is negative if an agent produces more at night than he consumes). When we calibrate the money demand function we assume u( q ) = q1−η /(1 − η ) , where η ∈ ( 0,1) and c ( q ) = q .The discount factor is β = (1 + ρ ) −1 ∈ ( 0,1) . Let pt denote the price in the centralized market.The utility of an agent holding mt units of money during the night of period t, denoted by W ( mt / pt ) , satisﬁes ⎧ ⎛m ⎞ ⎛ m ⎞⎫ ˆ ⎪ ⎪ W ⎜ t ⎟ = max ⎨ xt + β V ⎜ t +1 ⎟ ⎬ , ˆ xt mt +1 ⎪ ⎝ pt ⎠ ⎝ pt +1 ⎠ ⎪ ⎩ ⎭ ˆ s.t. pt xt + mt +1 = mt + Tt where V (⋅) is the expected utility of the individual in the decentralized market. The individual receives a lump-sum transfer Tt and chooses his net consumption xt and his money balances mt +1 in the next period. It is straightforward to check that the value function is linear and ˆ that the choice of mt +1 is independent of mt. Note that the absolute value of consumption and producˆ tion in the centralized market is not determined within the model. The aggregate production in the centralized market will be determined by calibration. Terms of trade (q,d) in a bilateral match, where q is the output and d the real transfer of money, are determined by a bargaining solution. In general, the terms of trade maximize a monotonic function of the surpluses of the buyer and the seller in the match.Those surpluses are independent of the money balances of the buyer and the seller.Also, for standard bargaining solutions, (q,d) only depends on the real balances zt = mt / pt of the buyer. The utility of an agent in the decentralized market satisﬁes ⎡ ⎡ ⎛m ⎞ ⎛m ⎞ ⎛m ⎞⎤ ⎛m ⎞⎤ V ⎜ t ⎟ = σ ⎢u( q b ) + W ⎜ t − d b ⎟ ⎥ + σ ⎢ −c ( q s ) + W ⎜ t − d s ⎟ ⎥ + (1 − 2σ )W ⎜ t ⎟ , pt pt pt ⎠ ⎥ ⎥ ⎢ ⎢ ⎝ pt ⎠ ⎝ ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ⎣ 17 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 where (q b ,d b ) are the terms of trade when the agent is the buyer and (q s ,d s ) are the terms of trade when the agent is the seller. In equilibrium, ( q b , d b ) = ( q s , d s ) since all agents hold the same money balances. Substitute V(m t / pt) into the Bellman equation for W ( mt / pt ) and rearrange in order to obtain the following problem max {−r z ( q ) + σ [u( q ) − z ( q )]} , t t t t qt where the nominal interest rate rt satisﬁes 1 + rt = (1 + π )(1 + ρ ) and where z(q) relates the buyer’s real balances to the quantity of goods he can purchase from the seller. So the agent essentially maximizes the expected surplus he gets when he is a buyer minus the cost of holding real balances. Pricing Mechanisms The form of the function z(q) depends on the bargaining solution. If buyers have all the bargaining power, then z(q) = c(q) and the ﬁrst-order condition is u′( q ) r = 1+ . c ′( q ) σ Using the speciﬁcations for the utility and cost functions, this gives −1 r ⎞η ⎛ q = z = ⎜1 + ⎟ . ⎝ σ⎠ If terms of trade are determined according to the proportional bargaining solution, then the relationship between individual balances and the quantity traded in a match is given by z ( q ) = θ c ( q ) + (1 − θ )u( q ) , where θ is the buyer’s share in the match surplus. Using the linear speciﬁcation for c(q) and the CRRA speciﬁcation for u(q), and the ﬁrst-order condition for the choice of real balances, one obtains ⎛ 1 − θ ⎞ ⎛ ( r + σ )θ ⎞ z =⎜ ⎟⎜ ⎟ ⎝ 1 − η ⎠ ⎝ σθ − r (1 − θ ) ⎠ − ( 1−η ) η −1 ⎛ ( r + σ )θ ⎞ η +θ ⎜ ⎟ . ⎝ σθ − r (1 − θ ) ⎠ If terms of trade are determined according to the generalized Nash bargaining solution where the buyer’s bargaining power is θ , the pair (z,q) satisﬁes θ u′( q )c ( q ) + (1 − θ )c ′( q )u( q ) , θ u′( q ) + (1 − θ )c ′( q ) r u′( q ) = 1+ . z ′( q ) σ z (q ) = Finally, if prices are set as a constant markup over the cost incurred by sellers, the function z(q) satisﬁes z ( q ) = (1 + μ )c ( q ). 18 FEDERAL RESERVE BANK OF CLEVELAND Calibration The money demand function that is confronted with the data is deﬁned as aggregate money balances divided by aggregate nominal output. It is equal to M , σ M + pA where M is aggregate money balances, σ is the frequency of trades in the decentralized market, p is the price in the centralized market, and A is the real output in the centralized market. Since z = M/p, the money demand function is z . σz+ A If the pricing mechanism is “buyers take all” then the expression for money demand is z = σZ + A 1 1 r ⎞η ⎛ σ + A ⎜1 + ⎟ ⎝ σ⎠ . The frequency of trades is set to σ = 0.5 so that each individual is matched and is either a buyer or a seller.The parameters A and η are chosen to ﬁt money demand in the United States over the period 1900–2000.The same procedure is used for other pricing mechanisms. Social and Private Marginal Returns of Real Balances A marginal unit of real balances allows a buyer to buy ∂q / ∂z units of goods in the event a match occurs. The expected private marginal return of real balances, or equivalently, the expected increase of the buyer’s utility from holding an additional unit of real balances, is then σ [u′( q ) − z ′( q ) ] ∂q / ∂z = σ [u′( q ) / z ′( q ) − 1] , which is precisely r from the ﬁrst-order condition for the choice of real balances. The expected social marginal return of real balances is σ [u′( q ) − c ′( q ) ] ∂q / ∂z = σ [u′( q ) − c ′( q ) ] / z ′( q ). Note that the private and social returns of real balances are equal when buyers have all the bargaining power since z(q) = c(q). If prices are determined by the proportional bargaining solution then the social return of real balances is ⎡ u′( q ) − c ′( q ) ⎤ σ ⎡ u′( q ) − z ′( q ) ⎤ r σ⎢ ⎥= . ⎥= ⎢ z ′( q ) z ′( q ) ⎦ θ ⎦ θ ⎣ ⎣ So the social return of real balances is equal to the interest rate divided by the buyer’s bargaining power. Finally, if prices are determined according to a constant markup μ over the cost incurred by the seller then the social return of real balances is ⎡ u′( q ) − z ′( q ) /(1 + μ ) ⎤ ⎡ u′( q ) − c ′( q ) ⎤ μ σ⎢ . ⎥ = r +σ ⎥ =σ ⎢ z ′( q ) z ′( q ) 1+ μ ⎦ ⎣ ⎦ ⎣ So the social return of real balances is equal to the interest rate plus a constant term that is increasing in μ . 19 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 Accurateness of the Welfare Triangle Measure The inverse money demand function of the search model is given by r = σ {u′ [ q ( x ) ] q′( x ) − 1} , where q(x) is the function that speciﬁes the output traded in bilateral matches as a function of real balances. Denote z0(zr) the real balances at the steady-state equilibrium with an interest rate of 3 percent (r percent). Integrate the inverse money demand function from zr to z0 in order to obtain the welfare triangle measure (denoted WT), z0 WT = ∫ σ {u′ [ q ( x ) ] q′( x ) − 1} dx zr = σ {u [ q ( z0 ) ] − z0 } − σ {u [ q ( z r ) ] − z r } . When buyers make take-it-or-leave-it offers, WT = σ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} − σ {u [ q ( z r ) ] − c [ q ( z r ) ]} . In this case, he welfare triangle coincides exactly with the change in society’s welfare. When prices are determined according to the proportional bargaining solution, WT = σθ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} − σθ {u [ q ( z r ) ] − c [ q ( z r ) ]} . Now the welfare triangle measure is equal to the buyer’s share times the change in society’s welfare. Measuring the Welfare Cost of Inﬂation Consider two steady states, one associated with an interest rate of 3 percent (the nominal interest rate that is consistent with 0 inﬂation) and one associated with an interest r. The welfare cost of an interest rate r is measured by the rate Δ , at which consumption in the steady state with an interest rate of 3 percent must be decreased to make agents indifferent between this steady state and the steady state with a nominal interest rate r. Let qr denote the quantities traded in a steady state when the interest rate is r.The cost of inﬂation Δ solves σ {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ = σ [u( qr ) − c ( qr ) ] . ⎣ ⎦ Using the functional forms for u(q) and c(q) this gives ( q0.03 )1−η ( q )1−η A (1 − Δ )1−η − Δ = r + q0.03 − qr . 1 −η σ 1 −η Endogenous Composition of Buyers and Sellers We introduce an assumption used by Shi (1997), and subsequently by Rocheteau and Wright (2004), Rocheteau and Waller (2005) and Faig (2004), to endogenize the frequency of trades. In the LagosWright model, each individual can be a buyer or a seller in the decentralized market depending on whom he meets.The frequency of trades is then given by an exogenous matching probability σ . We 20 FEDERAL RESERVE BANK OF CLEVELAND now allow each agent to choose on which side of the market to participate in: Each agent can choose to be a buyer or a seller in the decentralized market. Let n denote the fraction of sellers.The matching technology is such that a buyer meets a seller with probability n and a seller meets a buyer with probability 1 – n. So the aggregate number of trades is n(1 – n) and it is maximum when the composition of the market is symmetric, n = 1 – n = 1/2.The number of sellers in equilibrium is such that an agent is indifferent between being a buyer or a seller. Consequently, n satisﬁes −rz ( q ) + n [u( q ) − z ( q ) ] = (1 − n ) [ z ( q ) − c ( q ) ] , where z(q) is the buyer’s real balances as a function of q. (The growth of money supply occurs through lump-sum transfers. These transfers do not affect agents’ decisions to be buyers or sellers.) The right hand-side is the expected surplus of a buyer in the decentralized market net of the cost of holding real balances. The right hand-side is the expected utility of a seller in the decentralized market.The previous equation can be solved for n, n= (1 + r ) z ( q ) − c ( q ) . u( q ) − c ( q ) The form taken by z(q) depends on the pricing mechanism that is assumed. If prices are determined by the proportional bargaining solution then z ( q ) = θ c ( q ) + (1 − θ )u( q ). The choice of real balances satisﬁes an equation similar to the one in the basic Lagos-Wright model, that is, u′( q ) r = 1+ . z ′( q ) n Substituting n by its expression as a function of q gives r [u( q ) − c ( q ) ] u′( q ) = 1+ . z ′( q ) (1 + r ) z ( q ) − c ( q ) The aggregate money demand function that is ﬁtted to the data is (1 − n ) z , n(1 − n ) z + A where both n and z are functions of the interest rate r. Since the number of trades is n(1 – n), steadystate welfare is measured by n(1 – n)[u(q) – c(q)]. The welfare cost of inﬂation is the value Δ that solves n0.03 (1 − n0.03 ) {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ = nr (1 − nr ) [u( qr ) − c ( qr ) ] , ⎣ ⎦ where nr and qr are the values for n and q at the steady state corresponding to the interest rate r. 21 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 Distributional Effects of Monetary Policy We now consider a variant of the Lagos-Wright model where individuals only live for three periods.All agents are born at the beginning of night before the centralized market opens, and they die at the end of the subsequent period after the centralized market closes. So agents can trade in the ﬁrst period of their life in the centralized market and in the second period of their life in both the decentralized market and the centralized market. For simplicity, we assume that agents do not discount future utility, ρ = 0, so that r = π . In the second period of their lives, agents are identical to the agents in the LagosWright model. In order to introduce distributional effects of monetary policy we add the following assumption. Only a fraction p of the newly-born agents are able to produce in the ﬁrst period of their lives.The remaining 1 – p cannot produce and therefore cannot obtain money balances except from lump-sum transfers by the government. If p = 1 the model is analogous to the Lagos-Wright model. The choice q (the quantity consumed in the decentralized market) by newly-born agents satisﬁes max {−rz ( q ) + σ [u( q ) − z ( q )]} , q where z(q) is given by the bargaining solution. Let Tt denote the lump-sum transfer at night in period t. By deﬁnition Tt = Mt +1 − Mt = π Mt , where Mt is the quantity of money in the decentralized market in period t. Let mt denote the nominal money balances held by agents who can produce.Agents who cannot produce hold Tt–1.Then, pm t + (1 – p)T t–1 = M t and Tt −1 = pπ mt . 1 + pπ Let qt ( qt ) denote the quantity consumed in the decentralized market by those agents who can (cannot) produce in the ﬁrst period of their lives. Since z ( qt ) = Tt −1 / pt and z ( qt ) = mt / pt then z (q ) = pπ z ( q ). 1 + pπ If π = 0, then q = 0 ; and if π = ∞ , then q = q . The aggregate money demand function that has to be ﬁtted to the data is pz ( q ) + (1 − p ) z ( q ) z (q ) = . σ [ pz ( q ) + (1 − p ) z ( q ) ] + A σ z ( q ) + (1 + pπ ) A p(1 + π ) Social welfare is measured simply by the sum of utilities across agents, that is, σ p [u( q ) − c ( q ) ] + σ (1 − p ) [u( q ) − c ( q ) ] . The welfare cost of inﬂation is measured by Δ that solves σ p {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} + σ (1 − p ) {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ ⎣ ⎦ ⎣ ⎦ = σ p [u( qr ) − c ( qr ) ] + σ (1 − p ) [u( qr ) − c ( qr ) ] . 22 FEDERAL RESERVE BANK OF CLEVELAND Appendix 2: Data Description The interest rate is the short-term commercial paper rate. From 1900 to 1975, it is taken from Friedman and Schwartz (1982), Table 4.8, Column 6. From 1976 to 1994, it is from the Economic Report of the President (1996),Table B-69. From 1995 to 1997, it is from Economic Report of the President (2003), Table B-73. From 1998 to 2000 it is the short-term 90-day AA credit rate from the Federal Reserve Board, www.federalreserve.gov/releases/h15/data/m/fp3m.txt. Money supply is M1, as of December of each year, and is not seasonally adjusted. From 1900 to 1914, it is from the Historical Statistics of the United States (1960), Series X-267. From 1915 to 1960, it is from Friedman and Schwartz (1963), pp. 708–744, col. 7. From 1961 to 2000, it is from the FRED II database of the St. Louis Fed. Nominal GDP from 1900 to 1928 is taken from the Historical Statistics of the United States, Colonial Times to Present (1970, F-1 p. 224). From 1929 to 2000, it is from the GDPA series from the Citibase database. DATA SET Year Interest rate Money demand 1900 4.38 0.307540107 1901 4.28 0.318792271 1902 4.92 0.331018519 1903 5.47 0.327729258 1904 4.2 0.34139738 1905 4.4 0.346533865 1906 5.68 0.322752613 1907 6.34 0.314868421 1908 4.37 0.32833935 1909 3.98 0.283203593 1910 5.01 0.282691218 1911 4.03 0.289860335 1912 4.74 0.277106599 1913 5.58 0.277727273 1914 4.79 0.300906736 1915 3.45 0.334925 1916 3.42 0.323043478 1917 4.74 0.303228477 1918 5.87 0.272015707 1919 5.42 0.279345238 1920 7.37 0.252983607 1921 6.53 0.296623563 1922 4.42 0.308205128 1923 4.97 0.269529965 1924 3.9 0.288099174 1925 4 0.280204082 23 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 DATA SET (CONTINUED) Year 1926 4.23 0.262257732 1927 4.02 0.271243414 1928 4.84 0.272536082 1929 5.78 0.25515444 1930 3.55 0.273267544 1931 2.63 0.286196078 1932 2.72 0.346524702 1933 1.67 0.350336879 1934 0.88 0.345060606 1935 0.75 0.368785812 1936 0.75 0.368162291 1937 0.94 0.316528836 1938 0.86 0.368513357 1939 0.72 0.390574837 1940 0.81 0.413323471 1941 0.7 0.380102605 1942 0.69 0.386689314 1943 0.72 0.402129909 1944 0.75 0.412843494 1945 0.75 0.459058718 1946 0.81 0.484030589 1947 1.03 0.459459459 1948 1.44 0.411218425 1949 1.49 0.411896745 1950 24 Interest rate Money demand 1.45 0.392443839 FEDERAL RESERVE BANK OF CLEVELAND DATA SET (CONTINUED) Year Interest rate Money demand 1951 2.16 0.359563808 1952 2.33 0.352777003 1953 2.52 0.33790195 1954 1.58 0.346477392 1955 2.18 0.324493732 1956 3.31 0.312 1957 3.81 0.293862503 1958 2.46 0.301369863 1959 3.97 0.279313068 1960 3.85 0.268047112 1961 2.97 0.266935928 1962 3.26 0.259393784 1963 3.55 0.25504452 1964 3.97 0.248426763 1965 4.38 0.240084828 1966 5.55 0.224554455 1967 5.1 0.22630675 1968 5.9 0.222897802 1969 7.83 0.212624416 1970 7.71 0.211929706 1971 5.11 0.208094224 1972 4.69 0.206849713 1973 8.15 0.195440804 1974 9.84 0.18788 1975 6.32 0.180258805 25 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 DATA SET (CONTINUED) Year 1976 5.34 0.172301539 1977 5.61 0.167401152 1978 7.99 0.160335556 1979 10.91 0.153380408 1980 12.29 0.150375336 1981 14.76 0.142873993 1982 11.89 0.149237481 1983 8.89 0.150777278 1984 10.16 0.143542408 1985 8.01 0.150059711 1986 6.39 0.165774178 1987 6.85 0.161484123 1988 7.68 0.157358047 1989 8.8 0.147793378 1990 7.95 0.14521859 1991 5.85 0.152707684 1992 3.8 0.164985089 1993 3.3 0.173235347 1994 4.925 0.166032211 1995 5.925 0.155740433 1996 5.425 0.141290153 1997 5.625 0.132086389 1998 5.05 0.128062879 1999 5.925 0.123845971 2000 26 Interest rate Money demand 6.325 0.113277783 FEDERAL RESERVE BANK OF CLEVELAND Appendix 2: Parameter Values REGRESSIONS A η R2 Log–Log: m( r ) = A−η r 0.097835 0.29953 0.6238 Semilog: m( r ) = A−η r 0.43056 0.6750 11.027 Nonparametric 0.6795 ESTIMATION OF THE SEARCH MODEL (ii) Nash bargaining (i) Proportional bargaining θ A η θ η A 1.0 1.8248 0.14421 1.0 1.8249 0.14423 0.8 1.9096 0.17601 0.8 1.8167 0.17676 0.5 2.1876 0.26441 0.5 1.7722 0.26453 0.3 2.8112 0.40346 0.3 1.6200 0.37878 (iv) Endogenous participation and proportional bargaining (iii) Constant markup μ A η θ η A 0 1.8249 0.14422 0.9 1.8166 0.1 1.0367 0.14423 0.8 1.5574 0.33898 0.2 0.61861 0.14423 0.5 0.85057 0.2855 0.3 0.44921 0.35956 (v) Endogenous participation and Nash bargaining θ A η 0.50674 (vi) Distributional effects p A η 0.9 1.5699 0.53586 1.0 2.1876 0.26441 0.8 1.6757 0.47177 0.8 1.7461 0.26128 0.5 0.88848 0.27391 0.5 1.0875 0.25658 0.3 0.48668 0.29534 0.3 0.65116 0.25358 27 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 References Aiyagari, Rao, Anton Braun, and Zvi Eckstein.“Transaction Services, Inﬂation, and Welfare,” Journal of Political Economy, vol. 106 (1998), pp. 1274–1301. Aruoba, Boragan, and Randall Wright.“Search, Money and Capital:A Neoclassical Dichotomy,” Journal of Money, Credit, and Banking, vol. 35 (2003), pp. 1085–1106. Aruoba, Boragan, Christopher Waller, and Randall Wright. “Money and Capital,” unpublished manuscript (2004). Bailey, Martin. “The Welfare Cost of Inﬂationary Finance,” Journal of Political Economy, vol. 64 (1956), pp. 93–110. Berentsen,Aleksander, Guillaume Rocheteau, and Shouyong Shi.“Friedman Meets Hosios: Efﬁciency in Search Models of Money,” Federal Reserve Bank of Cleveland,Working Paper no. 04-08, 2004. Brockman, Michael,Theo Glasser, and Eva Herrmann.“Locally Adaptive Bandwidth Choice for Kernel Estimators,” Journal of the American Statistical Association, vol. 88 (1993), pp. 1302–09. Craig, Ben, and Guillaume Rocheteau.“State-Dependent Pricing, Inﬂation, and Welfare in Search Economies,” Federal Reserve Bank of Cleveland,Working Paper no. 05-04, 2005. Craig, Ben, and Christopher Waller. “Dollarization and Currency Exchange,” Journal of Monetary Economics, vol. 51 (2004), pp. 671–89. Cooley, Thomas, and Gary Hansen. “The Inﬂation Tax in a Real Business Cycle Model,” American Economic Review 79 (1989), pp. 733–48. Cooley, Thomas, and Gary Hansen. “The Welfare Costs of Moderate Inﬂations,” Journal of Money, Credit and Banking, vol. 23 (1991), pp. 483–503. Deviatov, Alexei, and Neil Wallace.“Another Example in which Lump-sum Money Creation Is Beneﬁcial,” Advances in Macroeconomics, vol. 1 (2001). Dotsey, Michael, and Peter Ireland.“The Welfare Cost of Inﬂation in General Equilibrium,” Journal of Monetary Economics, vol. 37 (1996), pp. 29–47. Ennis, Huberto. “Search, Money, and Inﬂation under Private Information,” Federal Reserve Bank of Richmond, Discussion Paper no. 142, 2004. Faig, Miquel.“Divisible Money in an Economy with Villages,” unpublished manuscript, 2004. Fischer, Stanley.“Towards an Understanding of the Costs of Inﬂation: II,” Carnegie-Rochester Conference Series on Public Policy, vol. 15 (1981), pp. 5–42. 28 FEDERAL RESERVE BANK OF CLEVELAND Friedman, Milton.“The Optimum Quantity of Money,” in The Optimum Quantity of Money and Other Essays, Chicago:Aldine Publishing Company, 1969. Hosios, Arthur. “On the Efﬁciency of Matching and Related Models of Search and Unemployment,” Review of Economic Studies, vol. 57 (1990), pp. 279–98. Kiyotaki, Nobuhiro, and Randall Wright. “On Money as a Medium of Exchange,” Journal of Political Economy, vol. 97 (1989), pp. 927–54. Kiyotaki, Nobuhiro, and Randall Wright.“A Contribution to the Pure Theory of Money,” Journal of Economic Theory, vol. 53 (1991), pp. 215–35. Kiyotaki, Nobuhiro, and Randall Wright. “A Search-Theoretic Approach to Monetary Economics,” American Economic Review, vol. 83 (1993), pp. 63–77. Kocherlakota, Narayana. “Optimal Monetary Policy: What We Know and What We Don’t Know,” International Economic Review, forthcoming. Lagos, Ricardo, and Randall Wright. “A Uniﬁed Framework for Monetary Theory and Policy Analysis,” Journal of Political Economy, forthcoming. Levine, David.“Asset Trading Mechanisms and Expansionary Policy,” Journal of Economic Theory, vol. 54 (1991), pp. 148–64. Li, Victor. “The Optimal Taxation of Fiat Money in Search Equilibrium,” International Economic Review, vol. 36 (1995), pp. 927–42. Li,Victor.“The Efﬁciency of Monetary Exchange in Search Equilibrium,” Journal of Money, Credit and Banking 29 (1997), pp. 61–72. Lucas, Robert.“Discussion of: Stanley Fischer, Towards an Understanding of the Costs of Inﬂation: II,” Carnegie-Rochester Conference Series on Public Policy, vol. 15 (1981), pp. 43–52. Lucas, Robert.“Inﬂation and Welfare,” Econometrica, vol. 68 (2000), pp. 247–74. Marty,Alvin.“The Welfare Cost of Inﬂation:A Critique of Lucas and Bailey,” Federal Reserve Bank of St. Louis, Review, vol. 81 (1999), pp. 41–6. Molico, Miguel. “The Distribution of Money and Prices in Search Equilibrium,” unpublished manuscript (1999). Nash, John.“The Bargaining Problem,” Econometrica, vol. 18 (1950), pp. 155–62. Reed, Robert, and Christopher Waller.“Money and Risk Sharing,” unpublished manuscript (2004). Rocheteau, Guillaume, and Christopher Waller.“Bargaining and the Value of Money,” Federal Reserve Bank of Cleveland,Working Paper no. 05-01 (2005). 29 POLICY DISCUSSION PAPERS NUMBER 12, JANUARY 2006 Rocheteau, Guillaume, and Randall Wright.“Inﬂation and Welfare in Models with Trading Frictions,” in Monetary Policy in Low-Inﬂation Economies, edited by David E.Altig and Ed Nosal, Cambridge: Cambridge University Press, forthcoming. Rocheteau, Guillaume, and Randall Wright.“Money in Search Equilibrium, in Competitive Equilibrium, and in Competitive Search Equilibrium,” Econometrica, vol. 73 (2005), pp. 175–202. Siefert, Burkhardt, Michael Brockman, Joachim Engel, and Theo Gasser. “Fast Algorithms for Nonparametric Curve Estimation,” Journal of Computational and Graphical Statistics, vol. 3 (1994), pp. 192–213. Shi, Shouyong.“A Divisible Search Model of Fiat Money,” Econometrica, vol. 65 (1997), pp. 75–102. Shi, Shouyong.“Search, Inﬂation, and Capital Accumulation,” Journal of Monetary Economics, vol. 44 (1999), pp. 81–103. 30 pd papers Presorted Standard FEDERAL RESERVE BANK OF CLEVELAND RESEARCH DEPARTMENT P.O. BOX 6387 U.S. Postage Paid Cleveland, OH Permit no. 385 CLEVELAND, OHIO 44101 Return Service Requested: Please send corrected mailing label to the Federal Reserve Bank of Cleveland, Research Department, P.O. Box 6387, Cleveland, Ohio 44101.