Stationary Monetary Equilibrium in a Baumol-Tobin Exchange Economy: Theory and Computation Jinhui H. Baiy December 27, 2005
Abstract I study a stationary monetary equilibrium in a Baumol-Tobin exchange economy with two assets (money and bonds) and long-lived, heterogeneous consumers who face uninsurable idiosyncratic endowment risk. Each consumer must pay a …xed cost of exchanging bonds for goods or money, but it is costless to trade money for goods. I characterize the theoretical properties and evaluate the model quantitatively by using calibrated parameters. First, increasing the exogenous growth rate of money increases the nominal interest rate and velocity of money, but decreases the real interest rate, in accordance with the Mundell-Tobin e¤ect. The presence of the Mundell-Tobin e¤ect makes clear the contrast with models with complete …nancial markets, in which I show that the steady-state real interest rate is equal to the common discount rate of consumers. Second, when the exogenous transaction cost is reduced, the nominal and real interest rates fall and the velocity of money increases. Third, I quantify the implementation and welfare consequences of Friedman’s rule. For reasonably calibrated model, the de‡ation rates consistent with Friedman’s rule are only slightly smaller than the discount rate of consumers. In addition, Friedman’s rule may reduce average welfare because changes in the lump-sum taxation induce redistributions across consumers. JEL classi…cation: E31, E40, E50 Keywords: Fixed Cost, Idiosyncratic Risk, Mundell-Tobin E¤ect, Friedman’s Rule I am grateful to John Geanakoplos and Tony Smith for helpful conversations throughout this project. I thank Ruediger Bachmann and Marek Weretka for many stimulating discussions. I also bene…ted from comments of Irasema Alonso, Truman Bewley, Bill Brainard, Bjoern Bruegemann, Eduardo Engel, Ray Fair, George Hall, Giuseppe Moscarini, Ingolf Schwarz, Gianluca Violante and the seminar participants at Yale University. Rudiger Bachmann generously provided his vectorized Nelder-Mead Simplex code to the author. I want to acknowledge the …nancial support from the Cowles Foundation in the form of a Carl A. Anderson fellowship. The usual disclaimer applies. y Department of Economics, Georgetown University, Washington, DC 20057-1036, U.S.A. Email:
[email protected]. Phone: +1-202-687-0935. Fax: +1-202-687-6102.
1
1
Introduction
I study a stationary monetary equilibrium in a Baumol-Tobin exchange economy with two assets (money and short-term bonds) and long-lived, heterogeneous consumers who face uninsurable idiosyncratic endowment risk. In the model, each consumer must pay a …xed cost of exchanging bonds for goods or money, but it is costless to trade money for goods. If the bond pays a positive nominal interest rate, consumers will still hold money in equilibrium to avoid paying the transaction cost. Because consumers optimize their inventory holdings of money, the velocity of money endogenously depends on economic fundamentals. Heterogeneous agents and uninsurable idiosyncratic income risk have long been thought to be important for asset pricing, welfare and distributional issues. A non-convex transaction cost plays a central role in the partial equilibrium Baumol-Tobin money demand theory (Baumol, 1952; Tobin, 1956). Therefore, theoretically it is interesting to investigate the interaction of the heterogeneous agents, incomplete …nancial markets and non-convex transaction costs in a general equilibrium framework. My paper, however, goes beyond pure theoretical curiosity. To a large extent, the paper is motivated by three empirical facts. First, there is strong evidence of the infrequent portfolio rebalance in the asset market (Vissing-Jorgensen, 2002). Since the frequency of portfolio rebalance gives a lower bound for the frequency of exchanging assets for money, this implies a sluggish money transfer between higher-yield brokerage accounts and checking accounts. Second, there exists a trend increase of the velocity of money in the postwar U.S. data (see Faig and Jerez, 2005). In addition, the increase in the velocity of money cannot be explained by conventional money demand variables such as income and nominal interest rate for the “missing money” period of mid-1970s (Goldfeld and Sichel, 1990) and in the 1990s (Faig and Jerez, 2005). It is widely believed that the changes in transaction technology through …nancial deregulation and …nancial innovation play an important role in explaining this fact. Third, cross-country data show a negative relation between long-run in‡ation and longrun real interest rates. This is the so-called Mundell-Tobin e¤ect (Mundell, 1963; Tobin, 1965). Monnet and Weber (2001), for example, …nd that the regression coe¢ cient of the long-run average nominal interest rate of government bonds on the long-run average in‡ation rate is positive and signi…cantly less than one. Rapach (2003) also identi…es a Mundell-Tobin e¤ect from a structural vector autoregression analysis of 14 industrialized countries.1 It is clear that a …xed transaction cost on the exchange of higher-yield bonds for money can capture the low frequency of portfolio rebalance. It is also intuitive, at least for the partial equilibrium Baumol-Tobin model, that the decrease in the transaction cost due to 1
It should be emphasized that the negative relation between long-run in‡ation and long-run real interest rates is conceptually di¤erent from the negative relation between real interest rates and expected in‡ation rate in the high-frequency data. The former (Monnet and Weber, 2001; Rapach, 2003) represents a property across steady states, whereas the latter (Barr and Campbell, 1997) applies to transition paths.
2
…nancial innovation may explain the trend increase in the velocity of money. These two observations motivate my formulation of a general equilibrium Baumol-Tobin model, which allows the velocity of money to depend on transaction costs. It is not immediately clear, however, whether the Baumol-Tobin model by itself can generate a Mundell-Tobin e¤ect. To investigate this issue, I …rst separate the friction of the …xed transaction cost from uninsurable idiosyncratic risk by studying stationary equilibrium with complete …nancial markets. I show that the steady-state real interest rate is equal to the common discount rate of consumers, which implies the absence of the Mundell-Tobin e¤ect. This conclusion leads me to study the uninsurable idiosyncratic risk and investigate mechanisms that generate the Mundell-Tobin e¤ect. In contrast to the analytical framework in related Baumol-Tobin-type models with complete …nancial markets (e.g., Jovanovic, 1982; Romer, 1986; Alvarez, Atkeson and Kehoe, 2002; Chiu, 2005), the incorporation of incomplete …nancial markets necessitates the use of computational methods to characterize the equilibrium. I extend the numerical methods developed in a large recent literature on computing stationary equilibria with one endogenous state variable (e.g., Imrohoroglu, 1992; Huggett, 1993; Aiyagari, 1994) to handle two endogenous state variables (money and bonds) and the corresponding market-clearing conditions. I also prove that the value function is continuous even in the presence of …xed costs. This result permits me to use standard value-function approximation methods to solve the consumer’s decision problem. I analyze the quantitative properties of the equilibrium by using calibrated parameters. Because this model has bonds and money, it is well-suited for studying the e¤ects of in‡ation and transaction costs on real variables in a fully-optimizing equilibrium framework. First, increasing the exogenous growth rate of money increases the steady-state in‡ation rate and nominal interest rate, and decreases the real interest rate in accordance with the Mundell-Tobin e¤ect. Moreover, higher in‡ation also increases the average frequency of bond-money transactions and the velocity of money and decreases the real amount of money held. As in Lucas (2000), the welfare cost of positive in‡ation relative to a zero-in‡ation equilibrium appears to be a concave function of the in‡ation rate, which implies a larger marginal gain from reducing lower in‡ation. As have others, I …nd that the welfare cost of moderate in‡ation is small, with a magnitude less than a quarter of a percent of income at a ten-percent annual in‡ation rate. Although a Mundell-Tobin e¤ect is also present in the previous study of an overlapping generations model (Chatterjee and Corbae, 1992), to the best of my knowledge this model is the …rst equilibrium model with fully-optimizing, long-lived consumers that features a quantitatively signi…cant Mundell-Tobin e¤ect. At the center of the driving forces is the contribution of incomplete …nancial markets to the heterogenous responses of consumers to in‡ation rate change. Given a real interest rate, an increase in the in‡ation rate makes consumers substitute money towards bonds as a means of saving, which tends to increase the 3
supply of bonds. At the same time, since the borrowing decision of each consumer depends on the realized history of endowment shocks due to incomplete insurance, the demand for borrowing does not change as much. Consequently, the imbalanced response of lenders and borrowers pushes down the real interest rate so as to clear the bond market. Notice that if the consumers are ex ante identical and asset markets are complete, then the heterogeneity is in e¤ect eliminated. This explains why the Mundell-Tobin e¤ect is absent in the stationary equilibrium with complete …nancial markets. To summarize the mechanism, the incompleteness of …nancial markets leads to ex post heterogeneity of consumers, and this heterogeneity in turn drives the Mundell-Tobin e¤ect. The e¤ect of heterogeneity on the Mundell-Tobin e¤ect has already been highlighted in an overlapping generations model with perfect foresight (Chatterjee and Corbae, 1992), where consumers have ex ante heterogeneous endowment pro…les. The novelty of this paper is to make the heterogeneity of consumers endogenous through the mechanism of uninsurable idiosyncratic endowment risk and to quantify the resulting Mundell-Tobin e¤ect. Second, when the exogenous transaction cost is reduced, nominal and real interest rates fall and the velocity of money increases. This prediction about the velocity of money is consistent with the data: …nancial innovation in the United States has reduced transaction costs in the past decades, and over the same period the velocity of money has increased. Because the change in the real interest rate causes a redistribution of wealth between borrowers and lenders, the model predicts that the change in the steady-state average utility is ambiguous. Like the welfare cost of in‡ation, the welfare cost of positive transaction costs is small in absolute magnitude. Third, I characterize and quantify the implementation and welfare consequences of Friedman’s rule. Friedman advocated a steady decrease in the supply of money so as to implement an equilibrium with a zero nominal interest rate. It is not clear how to implement such an equilibrium in my model: with positive transaction costs, consumers do not lend (because money is more liquid than bonds and yet pays the same rate of return), so that borrowing must be ruled out to ensure that the bond market clears. Despite this di¢ culty, I characterize the range of de‡ation rates to implement Friedman’s rule. Like the previous studies with a single asset (Bewley, 1983), the de‡ation rate is not equal to the discount rate, as Friedman suggests, but rather equal to the equilibrium interest rate in an incomplete-markets economy with a single asset— a risk-free bond— and a loose enough borrowing constraint. Although the de‡ation rate for the implementation of Friedman’s rule is strictly smaller than the discount rate, I show that quantitatively the di¤erence is almost negligible. Moreover, an equilibrium that implements Friedman’s rule not only fails to Pareto dominate a zero-in‡ation equilibrium, but may even reduce average welfare due to redistribution e¤ects through changes in the lump-sum taxation. It is worthwhile to make clear what I do not set out to do in the current paper. In focusing on steady states, I exclude aggregate shocks. Therefore, this paper does not answer questions 4
about the short-run ‡uctuation, e.g., liquidity e¤ects and the sluggish adjustment of price to monetary shocks. These issues are partly answered in Alvarez, Atkeson and Kehoe (2002), Chiu (2005) and Khan and Thomas (2005), using related models with complete …nancial markets. The transition path following a money growth rate shock and with incomplete …nancial markets is interesting because it highlights the distribution e¤ects of in‡ation. Technically, it may be implemented by using the non-linear method developed in Krusell and Smith (1999). I leave it to the future research. The remainder of the paper is organized as follows: Section 1.1 reviews the related literature; Section 2 outlines the model; Section 3 studies some theoretical properties of the model; Section 4 presents the computational results; Section 5 concludes the paper; and the appendix gives the mathematical proof and the details useful for replicating the computation.
1.1
Related literature
This paper is closely related to three strands of literature. The …rst is the literature on computing stationary equilibrium with uninsurable idiosyncratic risk and incomplete …nancial markets, commonly referred to as the Bewley model. Second, my model belongs to the general equilibrium Baumol-Tobin models, also referred to as endogenous asset market segmentation models. Finally, my paper contributes to the large literature on the welfare cost of anticipated in‡ation. The novelty of this paper is to combine the …rst two approaches and quantitatively study the interaction between endogenous asset markets segmentation and incomplete …nancial markets. My paper falls into the class of stationary Bewley models with two assets. Within twoasset economies, my model stays close to that of Aiyagari and Gertler (1991), who studied the coexistence of bond and equity due to a similar transaction cost pattern. Their paper focuses on the equity premium and the low frequency of trading equities. By studying a Baumol-Tobin model, I focus on the coexistence of money and risk-free bond and the e¤ects of monetary policies. Other related two-asset Bewley models with money and an interestbearing asset are Imrohoroglu and Prescott (1991), Erosa and Ventura (2002) and Akyol (2004). Imrohoroglu and Prescott (1991) uses a per-period …xed participation cost (i.e., cost on non-zero bond holdings) to support the value of money; Erosa and Ventura (2002) studies a cash-in-advance economy with costly credit;2 Akyol (2004) uses an information friction to introduce a role for money. My paper di¤ers from theirs in that I use the friction of Baumol-Tobin to motivate the use of money. This model also belongs to the general equilibrium version of the Baumol-Tobin model, also called endogenous asset market segmentation model (Alvarez, Atkeson and Kehoe, 2
The paper of Erosa and Ventura can also be understood as partly Baumol-Tobin because they require a …xed cost to withdraw money from the asset market. However, their model assumes a zero cost to deposit the money into the asset market.
5
2002).3 In a perfect foresight model with in…nitely-lived households, Jovanovic (1982) and Chiu (2005) studied the properties of stationary equilibrium with constant in‡ation rate. Gale and Hellwig (1984) proved the existence of monetary equilibrium in a related framework. Romer (1986) and Chatterjee and Corbae (1992) studied the overlapping-generations version of the models. My paper stays in the convention of in…nite-lived consumers and extends the previous analysis to the case with uninsurable idiosyncratic endowment risk. In a recent paper, Alvarez, Atkeson and Kehoe (2002) studied a Bewley-type exchange economy with a cash-in-advance constraint and endogenous market segmentation. Khan and Thomas (2005) investigated the sluggish price adjustment with an insurable idiosyncratic transaction cost shock. My paper di¤ers from their papers both in terms of model setup and the scope of the paper. In terms of modelling strategy, I study an incomplete-market economy where money is valued only through …xed transaction costs, while in their papers both complete …nancial markets and cash-in-advance constraint play important roles. In terms of the scope of the paper, my paper concentrates on the long-run e¤ects of di¤erent in‡ation rates and transaction costs on the steady-state interest rate and the welfare consequences, with a special focus on the Mundell-Tobin e¤ect and Friedman’s rule. Their papers set out to explain the persistence of the liquidity e¤ects and sluggish price adjustment through studying the transition path following an aggregate shock. Finally, my paper contributes to the large literature on the welfare cost of in‡ation. Like previous studies in other framework (Cooley and Hansen, 1989, 1991; Imrohoroglu, 1992; Imrohoroglu and Prescott, 1991; Akyol, 2004), I …nd only small aggregate welfare costs of steady in‡ation. Therefore, it provides another evidence that the search for a larger cost of in‡ation should be directed toward the variability of the in‡ation and relative prices.
2
The model
2.1
The consumer
The economic environment is a discrete time Bewley-type exchange economy with an in…nite horizon and idiosyncratic endowment uncertainty. The economy is populated by a government and a continuum of ex ante identical consumers of unit measure. In the economy, there are two assets, …at money and a risk-free real bond, available to insure partially against the idiosyncratic endowment risk. Consumers are heterogeneous ex post because the insurance 3
There is also a highly related framework referred to as the exogenous market segmentation model. In one class of the model (Grossman and Weiss, 1983; Rotemberg, 1984; Alvarez, Atkeson and Edmond, 2003), the asset market segmentation is time-dependent, where every agent has staggered access to asset markets once in N periods. In another class of model (Alvarez, Lucas, Warren, 2001; Williamson, 2005), the asset market participation is type-dependent, where agents are divided into market-participant type and non-marketparticipant type. In contrast to this literature, the endogenous market segmentation model requires a statedependent formulation of the participation structure.
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market is incomplete. In each period t, the only risk for a consumer i is an idiosyncratic endowment shock yti 2 fy1 ; y2 ; :::; yS g, which is independently distributed across the consumers. Across time, yti foli lows an S state Markov chain with transition probability P (ys0 jys ) = Pr yt+1 = ys0 jyti = ys . I assume that the Markov chain has a unique stationary distribution P (ys ), which is also the starting distribution across consumers at time t = 0. Therefore, the endowment distribution across consumers stays in the distribution P (ys ) at any time t, which further implies a conP stant aggregate endowment y = Ss=1 ys P (ys ). Let y it = y0i ; :::; yti denote a history of the idiosyncratic shock for consumer i from time 0 to t and P y it jy0i = P y1i jy0i ::: P yti jyti 1 the probability of y it . Since there is no aggregate uncertainty in the economy, y it fully summarizes the individual i’s exogenous state at time t. Each consumer i maximizes lifetime expected utility +1 X X
t
u c y it
t=0 y it
P y it jy0i ;
(1)
where 0 < < 1 is the subjective discount factor, c y it 0 is the consumption of the it single perishable good following the individual history y , and u (c) : R+ ! [ 1; +1) is the one-period utility function. I assume that the period utility function is smooth, strictly increasing, strictly concave, and satis…es u (c) > 1 for c > 0, limc!0 u0 (c) = +1 and limc!+1 u0 (c) = 0. Following the standard terminology, the value of 1 1 is called the subjective discount rate. In the computational part of this model, the period utility function takes the form of constant relative risk aversion (CRRA), i.e., u (c) =
(c)1 1
1
;
(2)
with the chosen parameter > 1. As in Alvarez, Atkeson and Kehoe (2002), the goods and asset markets are in physically separate locations. Corresponding to this, each consumer has access to two …nancial accounts: a brokerage account for bond trading in the asset market and a bank account to hold money for goods market transaction. In each period, the trades in the asset and goods markets happen at one point of time, i.e., the asset market and goods market open at the same time. In the goods market, the consumer trades goods subject to the bank account budget constraint (3) pt c y it + M y it = M y i;t 1 pt m + pt x y it + pt yti ; where pt is price for the commodity, c y it is the consumption following history y it , M y it 0 is the money taken from period t to t + 1, M y i;t 1 is the money holding from last period, m is an anonymous commodity-market lump-sum tax from the government,4 x y it is the real value of money transfer between two accounts, and yti is current realized endowment 4
To save the notation, I have not speci…ed a linear interest income and endowment income tax. Introducing
7
income. Notice that in my notation, x y it > 0 means a withdrawal of money from the brokerage account to the bank account, and x y it < 0 is a deposit of money into the brokerage account. It is also important to point out that the commodity price (pt ) depends only on the time t and is independent of any idiosyncratic shock. To study the stationary equilibrium with non-zero in‡ation, it is convenient to express the budget constraints in terms of real variables. Denote the lowercase variables as the real M (y it ) value of nominal variables de‡ated at the current price level, i.e., m y it = pt , and de…ne the constant in‡ation rate at time t as = ptpt 1 1. The equation (3) can be equivalently written as 1 m m y i;t 1 c y it + m y it = + x y it + yti : (4) 1+ In the asset market, consumers trade the risk-free bond. Although the consumer can trade the bond in a frictionless way, to make money transfers between two accounts, the consumer has to pay a …xed brokerage fee in the brokerage account. The brokerage account budget constraint is thus 1 b y it + x y it + 1+r
x y it
= b y i;t
1
b
;
(5)
where r is the time-invariant one-period real interest rate, b y it is the real bond holding5 between period t and t + 1, (x) is a transaction cost function6 such that ( 0 if x = 0 (x) = ; (6) > 0 otherwise b y i;t 1 is the real bond payo¤ this period, and b is the real value of asset market lump-sum tax. To rule out Ponzi games, I specify a borrowing constraint as b y it
b;
(7)
where b speci…es a non-positive value of ad hoc borrowing limit which is independent of idiosyncratic shock history (y it ). To see the ad hoc nature of equation (7), it is helpful to a proportional interest income tax is equivalent to rede…ning the current interest rate to be after-tax interest rate. The e¤ect of adding linear income tax amounts only to a re-normalization of units (scale the endowment in di¤erent states by the same proportion), which has no e¤ects on the results. These e¤ects are di¤erent from those in an economy with production, where proportional interest and labor income tax have distortionary e¤ects. In matching the model with data, we should understand the interest rate and endowment as the variables after the linear tax. 5 Although the model is formulated in terms of real bonds, it is equivalent to a model with nominal bonds given that the in‡ation is fully anticipated in the economy. This can be done by transforming variables through de…ning a nominal bond holding as Bti = (1 + )bit . 6 The assumption that the transaction cost is paid in the brokerage account is consistent with the brokerage fee interpretation of the cost. Alternatively, the fee can be paid in the bank account, where the cost can be interpreted as the opportunity cost of time spent in the transfer activity. It is not crucial for the result since we can always transform one form to the other by rede…ning a new variable as x e y it = x y it + x y it .
8
+ b, a (1+r)(ymin ) natural borrowing constraint can be de…ned as b y it + 1+1 m y it , since r this involves the maximum ability for paying back debt given zero future consumption. The sequential optimization problem of the consumer i can be summarized as maximizing utility (1) subject to the constraint (4), (5), (7), and non-negativity constraint on consumption and money holdings.
specify a natural borrowing constraint. Let ymin = minfy1 ; y2 ; :::; yS g and
2.1.1
=
m
Comparison with the literature
It would be helpful to compare my model with related monetary models with …xed transaction costs in detail. Alvarez, et al. (2002) studied a related economy with both complete asset markets and cash-in-advance constraint of the Lucas (1982) timing. To extend their setup to the incomplete markets setting, I need only to add an additional constraint, i.e., m y it
yti ;
(8)
which captures the friction that the asset markets open before the commodity markets. In Erosa and Ventura (2002), they studied a monetary growth model with a one-side …xed transaction cost and a cash-in-advance constraint of the Svensson (1985) and Cooley-Hansen (1989) timing, where the commodity market opens before the asset markets. Within my exchange economy setting, this is equivalent to the two budget equations c y it
1 m y i;t 1+
m
1
+ x y it ;
1 b y it + m y it + x y it + 1+r where (x) =
(
x y it
0 if x 0 : if x > 0
(9) = b y i;t
1
+ yti
b
;
(10)
(11)
Notice that the transaction cost is incurred only when there is withdrawal of money (x > 0). Because the cash-in-advance constraint of either Lucas (1982) timing or Svensson (1985) timing is a friction conceptually di¤erent from …xed transaction costs, I choose to support the value of money only through the …xed transaction costs. By separating these two frictions, I believe that this formulation will help to sharpen our understanding of the e¤ects of …xed transaction costs.
2.2
The government and …scal policy
Denote aggregate variables with an upper bar, and express the government budget constraint as 1 1 bt + mt + = bt 1 + mt 1 + g; (12) 1+r 1+
9
where = b + m is the total lump-sum tax, mt and bt are aggregate real money and bond supply respectively, and g is aggregate government purchase. The equation (12) says that the government expenditure must be equal to the total government resource. Throughout this paper, the government maintains a constant aggregate real money and bond holding, i.e., mt ; bt = m; b , and determines the lump-sum tax according to the rule r b = b; (13) 1+r m
m + g: (14) 1+ Notice that on the aggregate level, the change of money stock comes solely from the tax term m . In other words, positive injection of money happens only through a subsidy from the government (negative lump-sum tax), or “helicopter drop.”
2.3
=
Stationary monetary equilibrium
I will focus on a stationary monetary equilibrium where aggregate real money and bond holding are constant over time, i.e., mt ; bt = m; b . Corresponding to this, the growth rate of money is …xed at a level . For the purpose of computation, it is convenient to de…ne a recursive version of stationary monetary equilibrium. De…nition 1 A stationary (recursive) monetary equilibrium is a set of functions and correspondences, including value function v (m; b; y) and policy correspondence (c; x; m0 ; b0 ) = h (m; b; y), a set of constant prices and quantities r; ; b; m , and a constant measure of consumers over assets and endowment (m; b; y) such that (a) v (m; b; y) and h (m; b; y) solve the consumer Bellman equation: 8 9 < = X 0 0 0 0 v (m; b; y) = max u (c) + P y jy v m ; b ; y ; c;x;m0 ;b0 : 0 y
s:t:
c + m0 =
1 m+x 1+
1 0 b + x + (x) = b 1+r c 0; b0 b; m0 0: (b) Government satis…es the budget equation (12) with m = 1+ m + g.
m
+ y;
b
;
(16) (17)
=
b
+
m,
where
(c) Commodity, money and bond markets clear, i.e., Z (c (m; b; y) + (x (m; b; y))) d (m; b; y) = y; Z m; b = (m; b) d (m; b; y) : 10
(15)
b
=
r 1+r b
and
(18) (19)
(d) The distribution
(m; b; y) is constant, i.e., X X m0 ; b0 ; y 0 = y
f(m0 ;b0 )=h(m;b;y)g
P y 0 jy
(m; b; y) :
(20)
R An equilibrium has an inactive bond market if (x (m; b; y)) d (m; b; y) = 0, and has R an active bond market if (x (m; b; y)) d (m; b; y) > 0. In an equilibrium with an active bond market, the transaction cost reduces the aggregate consumption. By Walras Law, the commodity market will clear automatically given the equilibrium on the money and bond markets. Consequently, without loss of generality, we can concentrate on the money and bond markets. The setup of stationary monetary equilibrium is general enough to incorporate several existing one-asset models as special cases. If the transaction cost is zero, depending on the rate of in‡ation and economic fundamentals, the equilibrium results could be that money dominates bond, bond dominates money, or money and bond coexist. As a result, we can get one-asset incomplete-markets economies as studied in Imrohoroglu (1992), Huggett (1993), or a combination of the two economies in the case of coexistence. For the case of su¢ ciently high transaction cost, bond will be driven out of the market and consumers use only …at money to smooth their consumption, as in Bewley (1986) and Imrohoroglu (1992). By using a …nite and positive transaction cost, we get the intermediate situation when coexistence of risk-free bond and money is possible for di¤erent in‡ation rates and economic fundamentals. To complete the model, a monetary policy needs to be speci…ed. I will study two kinds of monetary policies. For most of the computational experiment, the government sets a constant real debt level of bt = b as its policy target. Under this situation, the real interest rate (r) and real money supply (m) are endogenous to maintain the equilibrium. This policy is interesting since it allows the endogenous determination of real interest rate, which seems to be a necessary environment for investigating the Mundell-Tobin e¤ect. In the second monetary policy, the government pegs the real interest rate at a predetermined level, where r = r . To attain such an equilibrium, the aggregate debt b and money m must be endogenous to maintain market clearing. This policy is less interesting in investigating the relation between interest rate and in‡ation, but it allows us to investigate the welfare e¤ects of in‡ation and transaction cost after controlling for real interest rate. This controlled experiment is helpful for understanding the contribution of redistribution e¤ects through lump-sum tax on the equilibrium welfare.
2.4
An alternative view of the model
In the previous parts, I formulated a model of two assets with possibly non-zero net supply. Following an observation from Aiyagari and McGrattan (1998, p. 454) and Ljungqvist and Sargent (2000, pp. 386-390), the model can be viewed alternatively as one economy with two assets of zero net supply and rede…ned borrowing constraints for each asset. To see this, plug 11
in the tax term ( b and budget constraint as
m)
into equation (16) and (15), and attain a rede…ned consumer
c+m e0 =
1 m e + x + ye; 1+
1 e0 b + x + (x) = eb; 1+r e0 m; c 0; eb0 b b; m
(21) (22) (23)
R1 where eb = b b, m e = m m, ye = y g. It is immediate from the de…nition that 0 ebi di = 0 and R1 i e di = 0. Since m is an endogenous variable in the equilibrium, the rede…ned economy 0 m is one with two IOUs, but with endogenous borrowing constraint for m. e Therefore, the model can be viewed as a version of Aiyagari and Gertler (1991) with endogenous borrowing constraints. This alternative formulation is helpful for understanding the failure of Ricardian equivalence with an ad hoc borrowing constraint b. For the case of a natural borrowing constraint, (1+r)(ymin g+ 1+ m) 1 i.e., eb0 + 1 m e0 m, di¤erent paths of b do not change real 1+
r
1+
equilibrium allocation, as b does not appear in the transformed economy. This is the standard Ricardian equivalence result. However, in the presence of ad hoc borrowing constraint b, a di¤erent value of b will typically change the equilibrium by e¤ectively altering the borrowing constraint. From the viewpoint of numerical analysis, this alternative formulation provides a guidance in the calibration of the model. Provided that the borrowing constraint and endowment process are calibrated in an appropriate way, without loss of generality we can assume b = 0 and g = 0.
3
Theory
3.1
The complete markets case
To understand the e¤ects of uninsurable idiosyncratic risk, it would be helpful to study a corner case where there is a complete set of state-contingent bonds to insure against every possible idiosyncratic shock. Since yti follows a S state …rst-order Markovian process, S 2 Arrow securities are enough to complete the market. More speci…cally, for each s; s0 2 f1; 2; :::; Sg, there exist Arrow securities b (ys0 jys ). Each unit of b (ys0 jys ) is sold at the price i q (ys0 jys ) to consumer i with yti = ys and pays o¤ one unit of consumption good if yt+1 = y s0 7 and zero otherwise. Like before, the government issues only risk-free bond with real interest rate r. The requirement that consumer i with current shock ys can only trade b (ys0 jys ) for every s0 is necessary to rule out arbitrage for general Markovian process. 7
12
In an Arrow-Debreu exchange economy without aggregate shocks, it is well known that the prices of Arrow securities depend only on the risk-free rate and objective probability, i.e., 1 q (ys0 jys ) = 1+r P (ys0 jys ). This risk-neutral pricing result is independent of the number of consumer types in the economy. With …xed transaction costs, it is not clear whether this result can be achieved with a …nite number of agents. However, with a continuum of agents who face an endowment shock with the same distribution, the no-arbitrage requirement yields the same asset price as in the Arrow-Debreu economy. To see this, notice that purchasing one unit of b (ys0 jys ) at price q (ys0 jys ) from a continuum of consumers with yti = ys will produce P (ys0 jys ) unit of payments for sure, due to the Law of Large Numbers. Since the same investment on the risk-free bond can produce q (ys0 jys ) (1 + r) unit of goods, no-arbitrage 1 P (ys0 jys ).8 Notice that the assumption of continuum of agents requires that q (ys0 jys ) = 1+r plays a crucial role in achieving this result. Now I describe the consumer’s decision problem with complete …nancial markets. For an individual state y it , denote with b ys0 jy it the quantity of consumer i’s holding of Arrow security b ys0 jyti . The brokerage account budget constraint with complete Arrow securities reads S X 1 b P ys0 jyti b ys0 jy it + x y it + x y it = b yti jy i;t 1 (24) t; 1 + r 0 s =1
1 P ys0 jyti . I also impose a standard no-Ponzi-games where I use the fact that q ys0 jyti = 1+r condition as S X 1 in i in 0 lim b ys0 jy in 0: (25) n+1 P y jy0 P ys jy n!+1 (1 + r) 0 s =1
The bank account budget constraint for the consumer i is the same as equation (4). The stationary monetary equilibrium can be de…ned in a similar way as in the incomplete markets case. Given the presence of …xed transaction costs, the equilibrium consumption in stationary monetary equilibrium will typically be di¤erent from Arrow-Debreu equilibrium. However, for a complete-markets equilibrium with a positive fraction of active consumers who exchange bond for money or goods, the equilibrium real interest rate is still equal to the Arrow-Debreu interest rate 1 1. Proposition 1 In a stationary equilibrium with complete …nancial markets and an active bond market, r = 1 1. The proof of Proposition 1 uses a version of multi-period Euler equations for the active consumer who happens to exchange bonds for money or goods in two periods. Given this Euler equation, the consumption of active consumers either goes to in…nity or to zero if the real interest rate is not equal to the discount rate. This either contradicts the market clearing 8
Alvarez, et al. (2002) have shown the same result by introducing a …nancial intermediary with a special trading arrangement. The underlying no-arbitrage argument is the same as given here.
13
condition or the fact that there is a positive measure of active consumers. Consequently, the only equilibrium real interest rate in an equilibrium with an active bond market is the discount rate of consumers. An immediate implication of Proposition 1 is that the real interest rate is independent of in‡ation rate and transaction costs. Therefore, to make the Mundell-Tobin e¤ect possible, a natural step is to introduce incomplete …nancial markets as in Section 2. Since there is no known analytical solution to the current model with the incomplete markets, I need to solve the equilibrium recursively by using numerical methods. To this end, the …rst step is to understand the properties of the consumer’s dynamic programming problem.
3.2
The consumer’s dynamic programming problem
The incorporation of …xed transaction cost into the budget set brings a potential di¢ culty for the consumer’s decision problem. The …xed cost would seem to jeopardize the continuity of the budget set and the value function in the dynamic programming problem, an undesirable property for the numerical approximation of the value function. Fortunately, in Proposition 2, I prove that the value function is continuous even in the presence of …xed costs. To do this, de…ne the budget correspondence for …xed r as (m; b; y; r) =
c; x; m0 ; b0 2 R+
R
R+
Rj (15)
(17) hold ;
and de…ne an operator T (v) on the space of bounded continuous function C (m; b; y) as 8 9 < = X T v (m; b; y; r) = max u (c) + P y 0 jy v m0 ; b0 ; y 0 ; r : (26) ; (c;x;m0 ;b0 )2 (m;b;y;r) : 0 y
The main properties of the dynamic programming problems are summarized in the following proposition.
Proposition 2 If b > erties are true:
(1+r)(ymin r
)
and supc!+1 u (c) < +1, then the following prop-
(a) T (v) is a contraction mapping from the space of bounded continuous function into itself, i.e., T : C (m; b; y) ! C (m; b; y). (b) There exists a unique continuous value function v (m; b; y) for the Bellman equation. (c) The optimal choice h (m; b; y) is an upper hemi-continuous correspondence of (m; b; y). (d) v ( ; b; y) is strictly increasing in m for each (b; y). In addition, v (m; ; y) is weakly increasing in b for each (m; y) and strictly increasing in an equilibrium with an active bond market. If in addition, the transition probability matrix P (y 0 jy) is monotone,9 then v (m; b; ) is strictly increasing in y for each (m; b). 9
For the de…nition of monotone transition function, see Stokey and Lucas (1989, p. 220).
14
For most parts, the proof of Proposition 2 proceeds by applying the standard techniques in the dynamic programming problem. Two non-traditional steps are to show the continuity of budget correspondence (m; b; y; r) and deal with the possibility of unboundedness of u (c) at c = 0. Both are necessary to apply the Theorem of Maximum. It is important to point out that Proposition 2 applies to the CRRA utility with > 1. Consequently, it provides a justi…cation for the numerical steps. Speci…cally, Proposition 2 is informative about at least three points. First, the continuity property is indeed true despite the …xed cost. This permits me to use standard value-function approximation methods to solve the consumer’s decision problem. Second, the optimal choice can not be guaranteed to be a single-valued function, due to the presence of non-convex transaction cost. Consequently, the Euler-equation-based numerical methods do not apply well to this problem. Third, the concavity of the value function is not true any more, as can be expected from the …xed transaction cost. The non-concave value function prevents us from relying on the gradientbased method in the numerical optimization problem.10
3.3
Friedman’s rule
One of the most celebrated propositions in the monetary economics is Friedman’s rule, i.e., the proposal to have an equilibrium with a zero nominal interest rate. Given that there are many policy instruments available to the government, it is not surprising that there are di¤erent ways to implement Friedman’s rule. I choose to study a strong form of Friedman’s rule (Woodford, 1990, p. 1071), which requires a steady contraction of the money supply that is so low as to implement an equilibrium with a zero nominal interest rate.11 Given this goal, the task is to characterize the range of de‡ation rate for the implementation of Friedman’s rule and discuss its welfare consequences. In a monetary model without an explicit transaction cost, e.g., the cash-in-advance model, a zero nominal interest rate makes consumers indi¤erent between money and bond holdings in terms of saving. As a result, such an economy with the coexistence of money and bond is equivalent to one with only a single asset, where borrowing is performed in the bond and saving is made in terms of either bond or money (or both). The presence of a positive transaction cost, however, leads to a di¤erent form of equilibrium for Friedman’s rule. Under Friedman’s rule with a positive transaction cost, money strictly dominates bonds as a means of saving because it is more liquid. Consequently, in 10
It would be desirable to obtain a (S; s) kind of optimal choice rule in this setup. However, given the counter example of Corbae (1993) in a three-period model, it does not seem possible to show this property. 11 There is also a weak form of Friedman’s rule (Woodford, 1990, p. 1070), which allows any policy choice to implement a zero nominal interest rate. For example, the central bank can attain a zero-nominal-rate equilibrium through an interest rate peg and adjusting the bond supply (b) endogenously. Notice that in the weak form, it could be possible to implement Friedman’s rule for any given de‡ation rate. As a result, it is not clear which equilibrium to choose in the discussion of the positive and normative issues with Friedman’s rule.
15
equilibrium, every consumer holds bond in a quantity equal to the market supply (b) only for the purpose of paying b . Since consumers only use money to smooth their consumption, bond is e¤ectively driven out of the economy. Therefore, an equilibrium under Friedman’s rule is reduced to the one studied in Bewley (1986) and Imrohoroglu (1992). To emphasize this point, it is convenient to de…ne an equilibrium under Friedman’s rule. De…nition 2 A stationary recursive equilibrium under Friedman’s rule is a set of functions, including value function v (m; y) and policy function m0 = h (m; y), an in‡ation rate and money supply ( ; m ), and a distribution (m; y) such that (a) Given (
; m ), v (m; y) and m0 = h (m; y) solve the problem v (m; y) = max0 u (c) + E v m0 ; y 0 jy c;m
s:t: 1 m + (y 1+ 0; m0 0:
c + m0 = c (b) Market clearing: (c) The distribution
R
g) +
1+
m ;
h (m; y) d (m; y) = 0.
(m; y) is constant, i.e.,
(m0 ; y 0 ) =
P P y
fm e 0 =h(m;y)g P
(y 0 jy) (m; y) :
The implementation of such an equilibrium, however, requires a high enough de‡ation rate. If nobody is holding bonds, then borrowing must also be ruled out to ensure that the bond market clears. To discourage borrowing, a high enough real interest rate (or de‡ation rate) is necessary. Under incomplete-markets settings, there is no reason to expect a unique de‡ation rate to implement the Friedman’s rule. In Proposition 3, I give a su¢ cient condition for the implementation of Friedman’s rule. It should be emphasized that a complete characterization of necessary conditions for Friedman’s rule is still an open question. Proposition 3 If ( ; m ) is an equilibrium according to De…nition 2 such that < 0 and 1+ m (ymin g) + , then ; m ; R = 0; b is an equilibrium according to De…nition 1, implementing Friedman’s rule. The proof of Proposition 3 builds on the relation between m and maximum ability of paying back debt. When m satis…es the assumption in Proposition 2, the lump sum tax is so large that it reduces the household’s ability to pay back debt. As a result, no one wants to borrow in the equilibrium. This establishes the bond market clearing and hence the equilibrium under Friedman’s rule. From Proposition 3, the de‡ation rate is not equal to the discount rate, as Friedman suggests, but rather equal to the equilibrium interest rate in an incomplete-markets economy with only …at money and a high enough de‡ation rate. As shown in Ljungvist and Sargent 16
(2000), this equilibrium is also equivalent to an equilibrium in an incomplete-markets economy with a risk-free bond and a borrowing limit of m . In addition, it is well known that these values of de‡ation rates are strictly smaller than the discount rate of the consumers. Because each equilibrium under Friedman’s rule corresponds to a zero nominal interest rate, the set of real money supply m corresponds to the region of the liquidity trap in the money demand function. Di¤erent equilibrium in the liquidity trap may correspond to a different equilibrium distribution. Consequently, the welfare is di¤erent for each implementation of Friedman’s rule.
4 4.1
Computation and the results Calibration and computation
Since the decision to transfer fund between the brokerage account and the bank account is a high-frequency phenomenon, it requires a short time period. To incorporate this consideration and also to compare with the existing literature (e.g., Imrohoroglu, 1992), I pick the time period as six weeks. The calibration is summarized in Table 1. Table 1: Parameter values y 0:995
1:5
(1; 0:25)
(P (y1 jy1 ) ; P (y2 jy2 )) (0:9565; 0:5)
g=y 0
b=y 2:6
b=y
=y
0:0
0:2%=0:6%=1%
The parameters of ( ; ; y; P (y1 jy1 ) ; P (y2 jy2 )) are picked directly from Imrohoroglu (1992). The discount rate = 0:995 corresponds approximately to an annual discount rate of 0:96. The elasticity of intertemporal substitution ( ) is also consistent with the literature. Since the model is scale-free in the level of endowment income, it is without loss of generality to normalize the high endowment shock to be 1. The low endowment shock corresponds to an unemployment subsidy of twenty-…ve percent of employed income, which seems to be reasonable. The transition probability matrix implies an average unemployment persistence of one quarter and the average unemployment rate of eight percent. Given the discussion in Section 2.4, it is without loss of generality to choose the aggregate bond supply and government expenditure equal to zero, i.e., b = 0 and g = 0. It remains to calibrate the borrowing constraint (b). The borrowing constraint is important since it has a direct in‡uence on the equilibrium real interest rate. I calibrate the borrowing constraint (b) to match the long-run average real interest rate in the U.S. economy. Based on the U.S. data, the annual average in‡ation rate is approximately equal to 4 percent and the real interest rate is equal to 1 percent. This can be calibrated with b=y = 2:6 at the transaction cost =y = 1%. Since a systematic estimation of the …xed transaction cost is not available from the empirical work, I experiment with several seemingly plausible transaction costs, which are chosen to be 0:2%, 0:6% and 1% of the period endowment. According to the annual statistics in the 17
United States, 1% of six week’s income of an average U.S. household is approximately equal to 50 dollars. With these calibrated parameters, I solve the equilibrium real interest rate and real money supply by using a two-dimensional …xed-point-iteration method. Within each step of iteration, I solve the consumer’s dynamic programming problem through value-function iteration. One additional complication in the dynamic programming problem involves the instability due to extrapolation errors associated with two-dimensional rectangular grid. To solve this problem, I reformulate the consumer’s dynamic programming problem in terms of a rede…ned …nancial wealth and a portfolio composition variable. This reformulation step could be helpful in other applications with two-dimensional endogenous state variables as well.
4.2
Welfare measure
To evaluate the welfare consequences of di¤erent experiments, a well-de…ned social welfare function is needed. I follow the standard practice to study a equal-weighted utilitarian social R welfare function v (m; b; y) d (m; b; y), where v (m; b; y) and (m; b; y) are the value function and stationary measure over pairs (m; b; y), respectively.12 Given this measurement of social welfare, the welfare in di¤erent economies can be compared by using the compensating variation (CV). Following Lucas (1987), the compensating variation (CV) can be de…ned as the percentage decrease13 of endowment (or consumption) in all states in the economy after policy change to maintain the same welfare as in the benchmark economy. For the CRRA utility function, it is easy to derive CV14 as CV
R 0 v (m; b; y) d R v 1 (m; b; y) d
=1
0
(m; b; y) 1 (m; b; y)
1 1
;
where a superscript zero (one) means the benchmark economy (economy after change). Notice that the welfare increases relative to the benchmark economy if and only if CV is positive. In accordance to the common practice, I report the welfare cost as the negative of CV .
4.3
The e¤ects of in‡ation and transaction costs
In this section, I report the steady-state e¤ects of changing long-run in‡ation and transaction costs on the real variables of the economy. Towards this end, I compute the stationary equilibria for di¤erent combination of transaction costs and money growth rates, which range 12
For di¤erent interpretation and justi…cation of this approach, see Aiyagari and McGrattan (1998, p. 455). Another possibility is to measure the welfare change in terms of absolute magnitude, as in Mas-Colell et al. (1995, p. 82). The advantage of percentage change is that it is scale-free. 14 Similarly, the equivalent variation (EV) can be de…ned as the percentage increase of income (or consumption) in all states in the economy studied to maintain the same welfare as in the benchmark economy. This 13
R
1
v (m;b;y)d
(m;b;y)
1
measure can be calculated as EV = R v0 (m;b;y)d 0 (m;b;y) 1: EV is used in Akyol (2004). Since EV is quantitatively close to CV, we only report the result based on CV.
18
from the de‡ation rate required by Friedman’s rule to a medium annual in‡ation rate of 10%. The e¤ects of changing in‡ation and transaction costs can be summarized as follows. First, changing in‡ation and transaction costs changes the equilibrium nominal and real interest rate. Given a …xed transaction cost, when the exogenous money growth rate increases, the nominal interest rate increases, but the real interest rate decreases. In Figure 1, I plot the equilibrium real interest rates and in‡ation rates for three di¤erent values of transaction costs. For each given transaction cost, it is easy to identify a quantitatively signi…cant MundellTobin e¤ect, i.e., a negative relation between the in‡ation rate and real interest rate. The presence of Mundell-Tobin e¤ect is consistent with the empirical study of Monnet and Weber (2001). In addition, the relation appears to be concave: the marginal e¤ect of in‡ation on the real interest rate is most signi…cant when the in‡ation rate is negative, but becomes smaller as the in‡ation rate increases. It is also important to point out that the highest real interest rate in Figure 1 is associated with Friedman’s rule.15 Although it is still smaller than the discount rate, the quantitative di¤erence is negligible. (insert Figure 1 around here) Although there exists the Mundell-Tobin e¤ect, the change in the real interest rate cannot fully compensate the change of the in‡ation rate. This leads to a positive relation between nominal interest rates and in‡ation rate, as presented in Figure 2. Consistent with Figure 1, the positive comovement between the in‡ation and nominal interest rate is smaller than one to one. Therefore, even in the long run, the Fisher equation does not hold exactly, although it still dominates the Mundell-Tobin e¤ect. (insert Figure 2 around here) Both Figure 1 and Figure 2 show positive relations between transaction costs and interest rates for …xed in‡ation rates. An increase in the transaction cost reduces the relative liquidity of the bond and hence lowers the demand for bonds. To make the bond market clear, the real interest rate has to rise. In other words, an increase in the transaction cost results in a liquidity premium for the interest rate. In the extreme case with an in…nite transaction cost, the real interest rate is always equal to 1+ , which achieves the most signi…cant Mundell-Tobin e¤ect. By varying the magnitude of transaction costs, we can achieve the Mundell-Tobin e¤ects with di¤erent shapes between real interest rates and the in‡ation rates. This explains the insigni…cant Mundell-Tobin e¤ects for positive in‡ation rates with current calibration of small transaction costs, as shown in the Figure 1 and Figure 2. Second, changing in‡ation and transaction costs also changes the equilibrium velocity of money and real money balance. Given a transaction cost, increasing the in‡ation increases the 15
In Proposition 3, I characterize a range of de‡ation rate consistent with Friedman’s rule. However, quantitatively the range is too small to be interesting. As a result, I only report the equilibrium with the lowest money supply within the range.
19
frequency of money transfer between asset markets and goods markets, increases the velocity of money and decreases the money holdings. These results are quite intuitive. Given a positive relation between in‡ation and nominal interest rate, a higher in‡ation rate increases the opportunity cost of holding money, which leads to more frequent bank trips, higher velocity of money and lower money demand. Figure 3 gives the positive comovement between the velocity of money and in‡ation rate. This positive relation is consistent with the cross-country study of Rodriguez Mendizabal (2004). To see the same e¤ect through a di¤erent perspective, Figure 4 plots the equilibrium money-income ratio together with di¤erent equilibrium nominal interest rates. As explained in Lucas (2000, p. 258), we can view this across-steady-state relation between equilibrium real money balance and nominal interest rate as a long-run money demand function. Interpreted this way, Figure 4 produces a money demand function in a similar shape as empirical studies.16 (insert Figure 3 and 4 around here) Figures 3 and 4 also tell the e¤ects of a change in the transaction cost on the velocity of money and money demand. A decrease in the transaction cost causes an increase in the velocity of money, as shown in Figure 3. This is consistent with the U.S. data: the …nancial innovation has decreased transaction cost in recent decades, and at the same time the velocity of money increases. Consistent with this, Figure 4 shows a positive relation between transaction costs and money demand. All these results are in accordance with simple economic intuition. Since higher transaction cost makes money transfer between asset markets and goods markets more costly, consumers tend to reduce the frequency of transfer, increase the inventory of money and decrease the velocity of money. Third, the welfare cost of in‡ation and transaction cost is reported in Figure 5 and Figure 6. In Figure 5, I draw the in‡ation and its welfare cost relative to a zero-in‡ation rate. For the in‡ation rate larger than a lower bound, the welfare cost of in‡ation appears to be a concave function of in‡ation, which implies a larger marginal gain at a low in‡ation rate. Within this range of in‡ation rates, the welfare cost appears to be a small number, with a magnitude less than a quarter of a percentage point at a ten-percent annual in‡ation rate. However, at the lowest de‡ation rate, i.e., the value that implements a zero nominal interest rate as claimed in Friedman’s rule, the average welfare is decreased relative to a zero-in‡ation economy. In addition, the welfare loss is around 1:5%, a much larger number than the 10% annual in‡ation rate. This result is in sharp contrast to an economy with representative agents, where Friedman’s rule typically increases the welfare. The central driving force is the redistribution between the debtors and creditors caused by the change in the real interest rate. A large de‡ation in the implementation of Friedman’s rule results 16
Figure 4 presents a semi-log form of money demand function. Indeed, the log-log form of money demand can …t the data point better. I choose to report the semi-log form because the logarithm of nominal interest rate is not de…ned at the zero nominal interest rate, the equilibrium under Friedman’s rule.
20
in a high real interest rate in the equilibrium. As a result, the lenders gain at the loss of borrowers. Because the equilibrium under Friedman’s rule typically features a large measure of borrowers, implementing Friedman’s rule decreases the average welfare. (insert Figure 5 around here) I also investigate the welfare cost of positive transaction costs as compared to an equilibrium with a zero cost. A higher transaction cost can have at least two e¤ects on the welfare. First, a higher transaction cost tends to use up more resources and crowd out the average consumption. This resource cost e¤ ect contributes to a decrease in average welfare. Second, increasing transaction cost leads to an increase in the real interest rate. The increase in the real interest rate will redistribute wealth from borrowers to lenders, which causes a redistribution e¤ ect. Depending on the shape of the distribution of asset holdings, the change in the average welfare could be ambiguous. Figure 6 presents the welfare e¤ects of positive transaction costs. We can see at least two important points. First, the absolute magnitude of welfare e¤ect is small. This is intuitive given that the magnitude of the transaction cost is small relative to the output. Second, the sign of welfare change depends on the given in‡ation rate. For given positive in‡ation rate, the average utility decreases with the increasing transaction cost. However, for the zero in‡ation rate, the average welfare increases with the transaction costs. These facts represent the relative magnitude of resource cost e¤ect and redistribution e¤ect. (insert Figure 6 around here)
4.4
Welfare cost under interest rate peg
Since one of the goals of this paper is to investigate e¤ects of changing in‡ation and transaction costs on the real interest rate, the previous analysis assumes a …xed bond supply and an endogenous interest rate. These assumptions sharpen the analysis on the real interest rate by focusing on the e¤ects of demand change. However, for the investigation of the welfare e¤ects of in‡ation and transaction costs, an alternative approach is to study the interest rate peg policy. Although the interest rate peg is less interesting for studying the e¤ects on the real interest rate, it gives another robustness check to the magnitude of the welfare cost of in‡ation and transaction costs in an alternative policy environment. In the experiments with the interest rate peg, I let the real interest rate stay …xed at an annual level of 1% and vary the in‡ation rates. Figure 7 reports the results for three di¤erent values of transaction cost. Even with the real interest rate …xed, a positive in‡ation still decreases the average welfare, with the magnitude and shape similar to the one with endogenous interest rates. This result leads to the conclusion that in the environment of the Baumol-Tobin model, the real interest rate is not the only driving force in the welfare cost
21
of in‡ation. Consumers will view two in‡ation regimes di¤erently even if the two in‡ation rates are accompanied by the same real interest rate.
5
Concluding remarks
In keeping the model simple, I have excluded several issues potentially important for constructing a more complete model. First, by focusing on an exchange economy, I assume away the endogeneity of the production process. The incorporation of production is potentially important in investigating the cost of disin‡ation in the presence of the Phillips curve e¤ects. Second, the absence of aggregate shocks makes the current model inappropriate for questions on economic ‡uctuations, among which are the full investigation of the liquidity e¤ect of open-market operations, the short-run sluggish response of prices to monetary shocks, and the welfare cost of in‡ation uncertainty and variability. By using current model as a benchmark, it is promising to solve a fully stochastic equilibrium with aggregate shocks through the approximation methods of Krusell and Smith (1998, 1999). Finally, the assumption of a one-good economy precludes the e¤ects of relative price variability on real variables. Recent research has already shown that such variability may create a large cost of in‡ation (see Williamson, 2005). Relative price variability might also give rise to a large welfare cost in the current model.
6 6.1
Appendix Proof of Proposition 1
The proof follows closely that given in Alvarez, Atkeson and Kehoe (2002). To prove the result, it is convenient to use the intertemporal version of the sequential budget constraint of the brokerage account. Iterating on the sequential budget constraint and using the transversality condition, we can get the intertemporal brokerage account budget constraint as +1 X t=0
1 P y it (1 + r)t
x y it +
22
x y it
+
b t
= b y0i ;
(27)
where b y0i is the start-of-period wealth in the asset markets. To solve the sequential problem, we can set up a Lagrangian as L (c; b; m; x; ) +1 X X t = u c y it
P y it
t=0 y it
+
+1 X X
t
c
1 m y i;t 1+
y it
t=0 y it
+
b
b y0i
+1 X t=0
1 P y it (1 + r)t
m t
1
+ x y it + yti
x y it +
x y it
+
c y it
b t
!
+
m y it
m
y it m y it ;
where c y it , b and m y it are the Lagrangian multipliers for the bank account budget constraint, equation (27), and non-negative money holdings. Notice that b does not depend on the idiosyncratic shock. The FOC for this problem is c y it m y it x y it 6= 0
: u0 c y it :
:
c
t
y it =
c
P y it = X
1+
y it =
b
c
y it ;
c
y i;t+1 +
(28) m
y it ;
(29)
i yt+1
1 it : tP y (1 + r)
(30)
Notice that equation (30) is valid only when x y it 6= 0. Combine equation (28) and (30) to get 1 t 0 : (31) u c y it = b (1 + r)t From equation (31), it is easy to see that c y it does not depend on individual history y it whenever x y it 6= 0. To emphasize this point, write c y it = cit if x y it 6= 0. For another history x y i;t+n 6= 0, we have n 0 u
cit+n 1 = : i 0 (1 + r)n u ct
(32)
Equation (32) says that the real interest rate is determined by the ratio of intertemporal substitution of active consumers. Given this, we will argue that 1 + r = 1 . If 1 + r > 1 , we must have lim u0 cit+n = 0 and hence cit+n ! +1. This contradicts the market clearing n!+1 condition since there is a positive measure of consumers to transfer. On the other hand, if 1+r < 1 , it follows that u0 cit+n ! +1 and hence cit+n ! 0. Since u (0) = 1, this implies a zero measure of consumers to conduct money transfer, a contradiction to the assumption of positive measure of money transfer.
23
6.2
Proof of Proposition 2
We start by de…ning the budget correspondence problem as (m; b; y) =
c; x; m0 ; b0 2 R+
R
(m; b; y) in the dynamic programming
R+
Rj (15)
(17) hold ;
where we should view the equation (15) and (16) as inequality “ ”. For …xed x such that (c; x; m0 ; b0 ) 2 (m; b; y), de…ne the x section of (m; b; y) as x (m; b; y)
=
c; m0 ; b0 2 R+
R+
Rj c; x; m0 ; b0 2
(m; b; y) :
Denote the interior of (m; b; y) and x (m; b; y) as int f (m; b; y)g and int f x (m; b; y)g. It is clear that (c; x; m0 ; b0 ) 2 int f (m; b; y)g if and only if (c; m0 ; b0 ) 2 int f x (m; b; y)g. Given the standard techniques in the dynamic programming problem (see Stokey and Lucas, 1989, chapter 9.2), the only non-trivial step is to show the continuity of the budget correspondence (m; b; y). To show this, we need to deal with the indicator function (x) involved in budget correspondence carefully.17 For upper hemi-continuity, start with a sequence (mn ; bn ) ! (m; b) and a sequence (cn ; xn ; m0n ; b0n ) ! (c; x; m0 ; b0 ) such that (cn ; xn ; m0n ; b0n ) 2 (mn ; bn ; y) for every n, we want to show (c; x; m0 ; b0 ) 2 (m; b; y). It is easy to see that (c; x; m0 ; b0 ) will satisfy the equations (15) and (17) at (m; b; y). Therefore, it only remains to show that equation (16) holds. For this purpose, it is su¢ cient to check that (x) (xn ) for n large enough. If x = 0, we have (x) = 0 (xn ); if x 6= 0, we must have xn 6= 0 for n large enough, which means (x) = 1 = (xn ). The result then follows. The proof for lower hemi-continuity adapts the steps in Hildenbrand (1974) by dealing with the function (x) carefully. The proof proceeds in three steps. First, int f (m; b; y)g is (1+r)(ymin ) . Second, int f (m; b; y)g non-empty. This is true given the assumption of b > r n n 0 0 is lower hemi-continuous, i.e., if (m ; b ) ! (m; b) and (c; x; m ; b ) 2 int f (m; b; y)g, there exists (cn ; xn ; m0n ; b0n ) 2 int f (mn ; bn ; y)g such that (cn ; xn ; m0n ; b0n ) ! (c; x; m0 ; b0 ). To show this, …rst notice (c; x; m0 ; b0 ) 2 int f (m; b; y)g implies (c; m0 ; b0 ) 2 int f x (m; b; y)g. If int f x (m; b; y)g is lower hemi-continuous, then for (mn ; bn ) ! (m; b), we can …nd (cn ; m0n ; b0n ) 2 int f x (mn ; bn ; y)g such that (cn ; m0n ; b0n ) ! (c; m0 ; b0 ), which immediately implies that (cn ; x; m0n ; b0n ) 2 int f (mn ; bn ; y)g and (cn ; x; m0n ; b0n ) ! (c; x; m0 ; b0 ). Given the non-empty interior of int f x (m; b; y)g, i.e., (c; m0 ; b0 ) 2 int f x (m; b; y)g, the lower hemi-continuity of int f x (m; b; y)g follows from standard argument. Finally, the lower hemi-continuity of (m; b; y) follows from the fact that the closure of a lower hemi-continuous correspondence is also lower hemi-continuous. (1+r)(ymin ) , the optimal consumption c (m; b; y) will be Given the assumption b > r bounded away from zero. Therefore, in the optimal choice, u (c (m; b; y)) is bounded below. 17
The proof is similar to the steps in a two-period economy in Bai and Schwarz (2005).
24
This fact together with the theorem of maximum and Blackwell’s su¢ cient condition implies that part (a) is true. Part (b) directly follows from the contraction mapping theorem. Part (c) follows from the theorem of maximum. Finally, part (d) follows from standard argument (see Theorem 9.7 and 9.11 in Stokey and Lucas, 1989).
6.3
Proof of Proposition 3
From the discussion in Section 2.4, it su¢ ces to prove the result for b = 0. Let us start with a sequential economy with in‡ation rate < 0 under Friedman’s rule as in De…nition 2, which has the initial distribution over real money and endowment pair as (m; y) and decision rule (c; m0 ) = h (m; y). At date t = 0, introduce a bond on the asset market with r = 1+ > 0 , and the initial distribution as bi 1 = 0 for every i. I need to show that if m yminr g 1+r r i i i i i i i i the optimal decision rule is ct ; mt ; bt ; xt = hc mt 1 ; yt ; hm mt 1 ; yt ; 0; 0 for every t 0 and i. The proof proceeds by proving …ve claims. 1 First, the optimal choice requires xi0 0. Suppose not, i.e., cit ; mit ; bit ; xit t=0 involves xi0 < 0. Consider an alternative consumption sequence such that ci0 = ci0 + for t = 0 and 1 cit = cit for t 1. It is not di¢ cult to see that cit t=0 is feasible in the budget set and leads to a higher utility, a contradiction. g rm) (1+r)(ymin y g > 0, or equivalently m < min r , it must be true that Second, if r (1+r) y (1+r)(ymin g rm g ( ) min (1 + r) mi0 +bi0 > . Suppose not, i.e., (1 + r) mi0 +bi0 = r r 0. Such a choice will correspond to ci1 = 0 if y1i = ymin , which leads to an expected utility of 1 due to the Inada condition on u (c), a contradiction to the optimal choice. g rm) (1+r)(ymin y g ymin g Third, if 0 < (1 + r) , or equivalently min r >m r r 1+r i i , each consumer will choose x0 = 0. Suppose not, i.e., x0 > 0. From the second r g rm) (ymin 1 i 1 i step, mi0 + 1+r b0 > b0 , or equivalently mi0 > 1+r = xi0 > 0. r 1 Construct an alternative sequence cit ; mit ; bit ; xit t=0 such that ci0 = ci0 , mi0 = mi0 xi0 > 0, xi0 = bi0 = 0, mi1 = mi1 , bi1 = bi1 , and ci1 xi1 cit
; mit
; bit
; xit
= ci1 + (1 + r) =
xi1
=
cit
+ (1 + r) ; mit
; bit
xi1
+ xi0 ; xit
+ ,t
xi1
+
xi1
; xi1
;
2:
It can be checked that such a choice is feasible and yields a higher utility, a contradiction to the optimal choice. g ymin , in the optimal choice, xi0 = 0. Suppose not, i.e., xi0 > 0. Since Fourth, if m r ymin g m , it must be true that (1 + r) mi0 + bi0 . The same logic as in the third r step follows, which again contradicts the optimal choice. ymin g 1+r Fifth, if m , cit ; mit ; bit ; xit = hc mit 1 ; yti ; hm mit 1 ; yti ; 0; 0 for r r every t 0 and i. From step 1 to step 4, the optimal choice must involve xi0 = bi0 = 0. At t = 1, we are going back to the same situation as in step 1 to 4. Therefore, by induction the 25
rm)
<
optimal choice must involve xit = bit = 0 for every t 0. Given this fact, it must be true that cit ; mit = hc mit 1 ; yti ; hm mit 1 ; yti . This …nishes the proof.
6.4
Computational appendix: solving stationary equilibrium
I solve the stationary equilibrium in three steps. First, I transform the consumer problem into an equivalent form, which is designed to overcome the extrapolation problem associated with a two-dimensional rectangular grid. Second, I solve the transformed problem with a parametric value function iteration method. Third, I solve for the stationary equilibrium using a two-dimensional …xed point iteration method. 6.4.1
The transformation of the consumer problem
In the main text, I formulate the dynamic programming problem in terms of the state variables (m; b; y). Although intuitive, this formulation causes instability due to extrapolation with rectangular array on the (m; b) space. To avoid this problem, I transform the original problem into a di¤erent form. This reformulation could be useful in other applications as well. De…ning the new variables as R = (1 + r)(1 + ) 1, bb = (1 + ) (b b), ! = m + bb and bb = ! , I have v (!; ; y) =
max
fc 0;! 0 0;0
s:t: 1 !0 1+R
0
0
u (c) + E v ! 0 ; 0 ; y 0 jy
1;xg
+ x + (x) =
c + !0 1
0
=
1 ! 1+
1 ! (1 1+
)
b m
+
r b; 1+R
+ x + y:
To get around the non-convex budget set, I transform the problem into two parts. In the …rst step, I assume that the consumer does not transfer money, i.e., x = 0. Plug in x = 0 and after a little bit of manipulation, I get the choice problem as v 1 (!; ; y) =
max
fc 0;! 0 0;0
0
1g
u (c) + E v ! 0 ; 0 ; y 0 jy
s:t:
= (1 + r)! + rb (1 + R) b ; 1+R m c + !0 = ! + rb (1 + R) 1+ ! 0 (1 + r)! + rb (1 + R) b 0: !0
0
26
b
+ y;
In the second step, I assume that x 6= 0. Therefore, the maximization problem becomes v 2 (!; y) =
max
u (c) + E v ! 0 ; 0 ; y 0 jy
fc 0;! 0 0;0
0
c + !0 1
R 1+R
1g
s:t:
0
+
=
1 r !+ b 1+ 1+R
+ y:
Notice that v 2 (!; y) does not depend on , which brings a big saving in the computation. Given the two step problem, I can formulate an equivalent problem as v (!; ; y) = maxfv 1 (!; ; y) ; v 2 (!; y)g: It is worthwhile to point out that the results in Proposition 2 apply to the transformed problem. Speci…cally the Bellman equation de…nes a contraction mapping and the resulting value function v (!; ; y) is continuous. 6.4.2
Solving the consumer dynamic programming problem
Since the non-convexity of the transaction cost function occurs only at one point (x = 0), the discrete state method can almost never …nd the right solution. As a result, I use a continuous state value function iteration method to solve the consumer dynamic programming problem, which is described in Judd (1998, pp. 433-436). The algorithm can be summarized as follows: Algorithm 1 Parametric Dynamic Programming with Value Function Iteration Step 1: Pick a three-dimensional rectangular grid (!; ; y) and initial value of value function v0 on the grid. Set up a cubic spline to evaluate the o¤ -grid points. Step 2: Solve for the …rst step maximization problem for each grid point. Store resulting value function vn1 and policy function gn1 : Step 3: Solve for the second step optimization problem for each grid point. Store resulting value function vn2 and policy function gn2 : Step 4: Set vn = max vn1 ; vn2 and the corresponding choice gn as the new values on the grid. Step 5: Iterate until convergence. Now I give the details helpful for replicating the results. In choosing the rectangular grid on the (!; ), I use 101 points in the interval [0; 6] on the ! dimension and 51 points in the interval [0; 1] on the dimension. I typically start from a constant value v0 = 1 1 u (y) for all grid points. Since the problem involves non-convexity, the value function is possibly non-smooth and non-concave at some points. This explains the choice of cubic spline instead of polynomial approximation of value function (e.g., the Chebyshev polynomial) in step 1. In the setup of the cubic spline, the “not-a-knot” is used as the boundary condition.
27
Due to the possible non-concave regions of value function, the gradient-based optimization method can perform badly when the solution is far from the optimum. Therefore, the golden section search and Nelder-Mead simplex method are used to solve the problem in step 2 and step 3, respectively. To increase the accuracy of the optimization problem in the simplex step, I choose to re…ne the computation using a quasi-Newton method after the solution is close to the optimum. Typically I keep the tolerance level for the optimization as 10 7 , the percentage deviation of value function as 10 7 , and the absolute deviation of policy function as 10 5 .18 6.4.3
Solve for the stationary equilibrium
To solve for the nominal interest rate (R) and real money supply (m), I use a two-dimensional …xed point iteration method with dampening. The algorithm can be summarized as follows: Algorithm 2 Two-dimensional Fixed Point Iteration Step 1: Start from an initial guess of m0 and R0 < + 1 1. Set up a convergence criterion ". Step 2: Solve for the consumer’s dynamic programming problem. Step 3: Calculate long-run ergodic distribution over (!; ; y) and the corresponding real bond and money holdings bn and mn . Step 4: If bn b < " and jmn mn j < "; stop; else go to step 5. Step 5: Set
with
R;
Rn+1 =
R
mn+1 =
m
m
bn
b + Rn + (1
mn + (1
m)
R)
Rn =
R
bn
b + Rn
mn
2 (0; 1]; and go to step 2.
To approximate the ergodic distribution over (!; ; y), I discretize the continuous state space (!; ) using a large number of grids and calculate the ergodic distribution of the discrete state Markov chain.19 More speci…cally, I choose 301 uniform grid points in the interval [0; 6] on the ! dimension and 101 points in the interval [0; 1] on the dimension. To compute the invariant distribution of the corresponding discrete Markov chain, I need only to know about the probability transition matrix. The latter is achieved by applying a randomization procedure over the continuous policy function g (!; ; y). Given any grid point (!; ; y), I 18
We also …nd that the Howard improvement method often makes the iteration diverge. This is probably due to the non-convexity involved. 19 My method over two dimensional continuous choice variables is a simple generalization of the discretization and randomization procedure over one continuous-state variable, taught by Professor Tony Smith at Yale University. To check the robustness of this procedure, I also used a method based on the simulation of an agent over a long period (one million periods in my choice). The result based on simulation is quite close to this procedure.
28
have ! 0 ; 0 = g (!; ; y). To allocate the measure, I locate the position of ! 0 ; 0 inside the four closest points f(! l ; l ) ; (! l ; h ) ; (! h ; l ) ; (! h ; h )g such that ! l !0 ! h and 0 l h . In the transition matrix, I give the measure to the eight corresponding points, i.e., the combination of the four points and two discrete shocks y. The measure is calculated based on the transition of discrete shocks P (y; y 0 ) and the percentage of the area of the four rectangles, i.e., prob ! l ; l ; y 0 j!; ; y prob ! l ;
0
j!; ; y
=
prob ! h ; l ; y 0 j!; ; y
=
prob ! h ;
h; y
=
h; y
0
j!; ; y
=
(! h (! h (! h (! h (! 0 (! h (! 0 (! h
!0) !l ) ( !0) !l ) ( !l ) !l ) ( !l ) !l ) (
0
h
l)
h 0
l
h
l)
h
0
h 0 h
P y; y 0 ; P y; y 0 ;
l)
P y; y 0 ;
l)
P y; y 0 :
l
Notice that I randomize the measure in the “o¤-diagonal”way, which captures the idea that the closer one point is, the larger the probability is. Similar to other heterogeneous agent models, there is no guarantee on the convergence of the algorithm. Therefore, a good starting value of (R0 ; m0 ) is crucial for the convergence. The convergence is sensitive to the initial choice of nominal interest rate (R0 ), but less sensitive to the starting value of money (m0 ). Finally, the choice of relaxation parameters is important for the stability of the convergence. In my experience, typically R = 0:001 and m = 0:5 work well for most of the cases.
29
References [1] Aiyagari, R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109, 659-684. [2] Aiyagari, R., Gertler, M., 1991. Asset returns with transaction costs and uninsured individual risk. Journal of Monetary Economics 27, 311-331. [3] Aiyagari, R., McGrattan, E., 1998. The optimum quantity of debt. Journal of Monetary Economics 42, 447-469. [4] Akyol, A., 2004. Optimal monetary policy in an economy with incomplete markets and idiosyncratic risk. Journal of Monetary Economics 51, 1245-1269. [5] Alvarez, F., Atkeson, A., Kehoe, P.J., 2002. Money, interest rates, and exchange rates with endogenously segmented markets. Journal of Political Economy 110, 73-112. [6] Alvarez, F., Atkeson, A., Edmond, C., 2003. On the sluggish response of prices to money in an inventory-theoretic model of money demand. NBER Working Paper 10016. [7] Alvarez, F., Lucas, R.E., Jr., Weber, W.E., 2001. Interest rates and in‡ation. American Economic Review 91 (2), 219-225. [8] Avery, R., Elliehausen, G., Kennickell, A., 1987. Changes in the use of transaction accounts and cash from 1984 to 1986. Federal Reserve Bulletin 73 (3), 179-195. [9] Bai, J.H., Schwarz, I., 2005. Existence of monetary equilibria in a Baumol-Tobin economy: The two period case. Mimeo. [10] Barr, D.G., Campbell, J.Y., 1997. In‡ation, real interest rates, and the bond market: A study of UK nominal and index-linked government bond prices. Journal of Monetary Economics 39, 361-383. [11] Baumol, W.J., 1952. The transactions demand for cash: An inventory theoretic approach. Quarterly Journal of Economics 66, 545-556. [12] Bewley, T., 1983. A di¢ culty with the optimum quantity of money. Econometrica 51, 1485-1504. [13] Bewley, T., 1986. Stationary monetary equilibrium with a continuum of independently ‡uctuating consumers, in: Hildenbrand, W., Mas-Colell, A. (Eds.), Contributions to Mathematical Economics in Honor of Gerard Debreu. North-Holland, pp. 79-102. [14] Chatterjee, S., Corbae, D., 1992. Endogenous market participation and the general equilibrium value of money. Journal of Political Economy 100, 615-646.
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[15] Chiu, J., 2005. Endogenously segmented asset market in an inventory theoretic model of money demand. University of Western Ontario Working Paper. [16] Cooley, T.F., Hansen, G., 1989. The in‡ation tax in a real business cycle model. American Economic Review 79, 733-748. [17] Cooley, T.F., Hansen, G., 1991. The welfare costs of moderate in‡ations. Journal of Money, Credit and Banking 23 (3), 483-503. [18] Corbae, D., 1993. Relaxing the cash-in-advance constraint at a …xed cost: Are simple trigger-target portfolio rules optimal? Journal of Economic Dynamics and Control, 17, 51-64. [19] Erosa, A., Ventura, G., 2002. On in‡ation as a regressive consumption tax. Journal of Monetary Economics 49, 761-795. [20] Faig, M., Jerez, B., 2005, Precautionary balances and the velocity of circulation of money. Mimeo, University of Toronto. [21] Gale, D., Hellwig, M., 1984. A general-equilibrium model of the transactions demand for money. CARESS Working Paper 85-07. [22] Goldfeld, S.M., Sichel, D.E., 1990. The demand for money, in: Friedman, B.M., Hahn, F.H. (Eds.), Handbook of Monetary Economics, Vol. 1. North Holland, Amsterdam, pp. 300-356. [23] Grossman, S., Weiss, L., 1983. A transaction-based model of the monetary transmission mechanism. American Economic Review 73, 871-880. [24] Hildenbrand, W., 1974. Core and Equilibria of a Large Economy. Princeton University Press, Princeton, New Jersey. [25] Huggett, M., 1993. The risk-free rate in heterogeneous-agent incomplete-insurance economies. Journal of Economic Dynamics and Control 17, 953-969. [26] Imrohoroglu, A., 1992. The welfare cost of in‡ation under imperfect insurance. Journal of Economic Dynamics and Control 16, 79-91. [27] Imrohoroglu, A., Prescott, E., 1991. Seigniorage as a tax: A quantitative evaluation. Journal of Money, Credit and Banking 23 (3), 462-475. [28] Jovanovic, B., 1982. In‡ation and welfare in the steady state. Journal of Political Economy 90, 561-577. [29] Judd, K., 1998. Numerical Methods in Economics. MIT Press, Cambridge, Massachusetts. 31
[30] Kahn, A., Thomas, J.K., 2005. In‡ation and interest rates with endogenous market segmentation. Mimeo, University of Minnesota. [31] Krusell, P., Smith, A.A., Jr., 1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106, 867-896. [32] Krusell, P., Smith, A.A., Jr., 1999. On the welfare e¤ects of eliminating business cycles. Review of Economic Dynamics 2, 245–272. [33] Ljungqvist, L., Sargent, T.J., 2000. Recursive Macroeconomic Theory. MIT Press, Cambridge, Massachusette. [34] Lucas, R.E., Jr., 1982. Interest Rates and Currency Prices in a Two-Country World. Journal of Monetary Economics, 10, 335-359. [35] Lucas, R.E., Jr., 1987. Models of Business Cycles. Basil Blackwell, New York. [36] Lucas, R.E., Jr., 2000. In‡ation and welfare. Econometrica 68, 247-274. [37] Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University Press, New York. [38] Monnet, C., Weber, W.E., 2001. Money and interest rates. Federal Reserve Bank of Minneapolis Quarterly Review 25 (4), 2-13. [39] Mulligan, C., Sala-i-Martin, X., 2000. Extensive margins and the demand for money at low interest rates. Journal of Political Economy 108, 961-991. [40] Mundell, R., 1963. In‡ation and real interest. Journal of Political Economy 71, 280-283. [41] Rapach, D. E., 2003, International evidence on the long-run impact of in‡ation. Journal of Money, Credit, and Banking 35 (1), 23-47. [42] Rodriguez Mendizabal, H., 2004. The behavior of money velocity in low and high in‡ation countries. forthcoming Journal of Money, Credit and Banking. [43] Rotemberg, J.J., 1984. A monetary equilibrium model with transactions costs. Journal of Political Economy 92, 40-58. [44] Stokey, N.L., Lucas, R.E., Jr., with Prescott, E., 1989. Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, Massachusetts. [45] Svensson, L., 1985. Money and asset prices in a cash-in-advance economy. Journal of Political Economy 93, 919-944. [46] Tobin, J., 1956. The interest elasticity of transactions demand for cash. Review of Economics and Statistics 38, 241-247. 32
[47] Tobin, J., 1965. Money and economic growth. Econometrica 33, 671-684. [48] Vissing-Jorgensen, A., 2002. Towards an explanation of household portfolio choice heterogeneity: Non…nancial income and participation cost structures. NBER Working Paper W8884. [49] Williamson, S.D., 2005. Monetary policy and distribution. Mimeo. [50] Woodford, M., 1990. The optimum quantity of money, in: Friedman, B.M., Hahn, F.H. (Eds.), Handbook of Monetary Economics, Vol. 2. North Holland, Amsterdam, pp. 10681152.
33
Figure 1: Inf lation and real interest rate 4.5 γ = 0.2% γ = 0.6% γ = 1%
4
Annual real interest rate (%)
3.5
3
2.5
2
1.5
1
0.5 -4
-2
0
2 4 Annual inflation rate π (%)
6
8
10
8
10
Figure 2: Inf lation and nominal interest rate 12 γ = 0.2% γ = 0.6% γ = 1%
Annual nominal interest rate (%)
10
8
6
4
2
0 -4
-2
0
2 4 Annual inflation rate π (%)
6
Figure 3: Inf lation and v elocity of money 4
3
Velocity of money (log)
2
1
0
-1 γ = 0.2% γ = 0.6% γ = 1%
-2
-3
-4 -4
-2
0
2 4 Annual inflation rate π (%)
6
8
10
Figure 4: Nominal interest rate and money demand 4 γ = 0.2% γ = 0.6% γ = 1%
3
Real balances-income ratio (log)
2
1
0
-1
-2
-3
-4
0
2
4
6 Annual nominal interest rate (%)
8
10
12
Figure 5: Inf lation and welf are 1.6 γ = 0.2% γ = 0.6% γ = 1%
←Friedman
1.4
1.2
Welfare cost of inflation (%)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4 -4
-2
0
2 4 Annual inflation rate π (%)
6
8
10
Figure 6: Transaction cost and welf are 0.3 π=0 π = 4% π = 10%
0.25
Welfare cost of transaction cost (%)
0.2
0.15
0.1
0.05
0
-0.05
0
0.1
0.2
0.3
0.4 0.5 0.6 Transaction cost (%)
0.7
0.8
0.9
1
Figure 7: Inf lation and welf are with interest rate peg 0.2 γ = 0.2% γ = 0.6% γ = 1%
0.18
0.16
Welfare cost of inflation (%)
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4 5 6 Annual inflation rate (%)
37
7
8
9
10
