Monetary Policy, Price Dynamics, and Welfare

Monetary Policy, Price Dynamics, and Welfare Jonathan Chiu Bank of Canada jchiu@bankofcanada.ca Miguel Molico Bank of Canada mmolico@bankofcanada.ca January 2008 DRAFT - Very Preliminary and Incomplete DO NOT CIRCULATE Abstract We present a micro-founded search-theoretical model of money in which agents are subject to idiosyncratic liquidity shocks as well as aggregate productivity and monetary shocks. Monetary policy has redistributive effects and persistent effects on output and prices: aggregate shocks will propagate and diffuse gradually as the money distribution adjusts over time. The model is used to study the cyclical properties of the model, the welfare costs of unanticipated inflation, the short-run (real) effects of unanticipated monetary shocks, and to evaluate the welfare of different monetary policy rules. An algorithm based on the one proposed by Algan, Allais, and den Haan (2007) is presented and used to solve the model. J.E.L. Classification Numbers: E40, E50 Keywords: search, money, aggregate uncertainty, idiosyncratic uncertainty, monetary policy, inflation 1 Introduction Many economists agree that monetary policy has persistent nominal and real effects. Still, most of the existing economic models that generate such persistent real effects lack solid The views expressed in the paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. 1

microfoundations. Firstly, they generally lack a role for money. Secondly, they typically impose some form of ad-hoc nominal rigidities of wages and/or prices. 1 Recently, a large body of work has been dedicated to develop micro-founded monetary models. A successful class of such models has been the search-theoretical monetary literature. In these models a role for money as a medium of exchange is explicitly modeled. This literature has been very successful in addressing a large number of theoretical and monetary policy questions. However, due to technical difficulties, the short-run implications of such models have not been fully explored. Existing work has typically either focused on the long-run (steady-state analysis) (e.g. Lagos and Wright (2005) and Molico (2006)), assume some type of indivisibility (Wallace (1997)), or assume some form of perfect insurance against the idiosyncratic trading risk that naturally arises in such models (e.g. Shi(1999), Arouba and Wright( 2003)). Therefore, the distributional and propagational effects of monetary shocks are not fully understood. In this paper, we present a micro-founded search-theoretical model of money in which agents are subject to idiosyncratic liquidity shocks as well as aggregate productivity and monetary shocks. Agents face idiosyncratic uncertainty regarding their productivity, preferences, and trading opportunities. Agents are unable to insure against such risk due to market incompleteness implied by the frictions that generate the role of money. Heterogenous idiosyncratic histories imply a non-degenerate distribution of money holdings which become a state variable summarizing the past histories. Monetary policy has redistributive effects and persistent effects on output and prices: aggregate shocks will propagate and diffuse gradually as the money distribution adjusts over time. The model is used to study the cyclical properties of the model, the welfare costs of unanticipated inflation, the short-run (real) effects of unanticipated monetary shocks, and to evaluate the welfare of different monetary policy rules. 1 Other class of models that can generate real effects of monetary policy are the mis-perception (adaptiveexpectations) model of Friedman (1968) and the imperfect-information model of Lucas (1972) and Phelps (1970). Friedman s model requires that agents make systematic expectations errors and an (ad-hoc) gradual adjustment of price expectations to facts. Imperfect-information rational expectations models do not allow any systematic mis-perceptions and imply a very fast response to facts. As such, these models have difficulty in generating enough persistence. 2

As is well known, solving models with heterogeneous agents and aggregate uncertainty constitutes a technical challenge since the set of state variables contains the cross-sectional distribution of agent s characteristics, which is a time-varying infinite dimensional-object in the presence of aggregate uncertainty. Recently, algorithms have been proposed which summarize the cross-section distribution of agents characteristics by a small number of moments, therefore making the dimension of the state space tractable, and calculate the law of motion for these state variables using simulation procedures. The best known examples of this approach are those of Rios-Rull (1997) and Krusell and Smith (1998). An alternative algorithm proposed by den Haan (1997) parameterizes the cross-sectional distribution, which makes it possible to obtain a numerical solution using standard quadrature and projection techniques. A drawback of this approach is that the cross-sectional distribution is fully pinned down by the set of moments which might require a large number of moments for a decent approximation. Reiter (2002) improves this algorithm by introducing the concept of reference moments to get a better characterization of the cross-sectional distribution without increasing the number of state variables. The basic idea is to, in the process of solving the model, use simulations to obtain information regarding higher moments of the distribution which are used to obtain a better approximation of the distribution without increasing the dimension of the state space. In this paper we consider a version of the algorithm proposed by Algan, Allais, and den Haan (2007) that has both of the above features - parameterized distributions and reference moments- to solve our model. This is particularly useful in a a search-theoretical model of money since the assumption of random matching implies that the agents must form expectations regarding the potential gains from trade in order to solve their problems. The techniques above allow us to compute such expectations using quadrature techniques without introducing a significant approximation error. The rest of the paper is organized as follows. Section 2 describes the model environment. Section 3 defines a recursive equilibrium. Section 4 defines the approximate economy. Section 3

5 discusses in detail the numerical algorithm used to compute the recursive equilibria of the approximate economy. In section 6, we parametrize and characterize the model. Section 7 concludes the paper. 2 The Model Environment We consider an environment similar to that in Molico (2006) except that the economy is subject to aggregate productivity and monetary growth shocks. Time is discrete. There is a [0,1] continuum of infinitely lived agents who specialize in the consumption and production of perfectly divisible nonstorable goods. Specialization is such that no agent consumes the good he produces; each agent consumes goods that are produced by a fraction x of the population, and each agent produces goods that are consumed by a fraction x of the population. For simplicity, no two agents consume the output of each other; that is, there is never a double coincidence and hence there is no direct barter. 2 Agents produce goods by exerting labor effort h. The production function is given by q t = z t h t, where z t denotes the level of aggregate productivity. Agents derive utility u(q) from consuming q units of a good, and disutility h from exerting labor effort h. Assume u( ) is C 2, strictly increasing, strictly concave, and satisfies u(0) = 0. Let 0 < β < 1 denote the discount factor. Let z {z L, z H } follow a Markov chain with transition probabilities given by Π zz = π LL π HL π LH π HH. In this economy, there is an additional, perfectly divisible, and costlessly storable object that cannot be produced or consumed by any private individual, called fiat money. Agents 2 For money to be valued it is only required that in some meetings there is no double coincident of wants. For simplicity, we focus on purely monetary trades and, by assumption, preclude the possibility of barter. 4

can hold any non-negative amount of money ˆm R +. The money stock at the beginning of period t is denoted M t. In what follows we express all nominal variables as fraction of the beginning of the period money supply (before any of the current period s money transfers which we will describe below), e.g. m t ˆm t M t. Let λ t : B R+ [0, 1] denote the probability measure associated with the distribution of money holdings (as a fraction of the beginning of period money supply) at the beginning of period t, where B R+ denotes the Borel subsets of R +. The money supply grows at rate t = G(z t, λ t ) + ɛ t, M t+1 = t M t, where G(z t, λ t ) : {z L, z H } Λ R + is a time-invariant monetary policy rule and ɛ t is an i.i.d. money growth shock. Let ɛ t {, 0, } with Pr(ɛ = ) = Pr(ɛ = ) = τ < 1 2. Newly printed money is injected in the economy at the beginning of the period via lump-sum transfers to all agents. Meetings Each period, agents are randomly bilaterally matched. Given the pattern of specialization, in a given meeting, an individual will be a potential buyer with probability x. Also, with probability x the agent will be a potential seller. By assumption, there is never doublecoincidence-of-wants meetings. When individuals meet in a single coincidence meeting they bargain over the amount of money and goods to be traded. Let q(m b, m s ; ɛ t, z t, λ t ) 0 denote the amount of good and d(m b, m s ; ɛ t, z t, λ t ) 0 the amount of money to be determined by the bargaining process at date t between a buyer with money holdings m b and a seller with m s, when the current state of the economy is described by the triplet (ɛ t, z t, λ t ). We assume that when the two agents meet the buyer makes a take-it-or-leave-it offer to the seller, extracting all of the seller s expected surplus. This concludes the description of the environment. In what follows, we will gradually build towards the definition of an equilibrium. 5

Money shock ε Productivity shock z Money transfer =G(z,λ)+ε Agents meet and trade goods t t+1 λ V(m,ε,z, λ) λ =H(ε,z,λ) 3 Recursive Equilibrium In this section we define a recursive equilibrium for this economy. A key element in such definition is the law of motion of the aggregate state of the economy. The aggregate state is the triplet (ɛ, z, λ): the current aggregate productivity and monetary shocks, and the current probability measure over money holdings. The part of the law of motion that concerns ɛ and z are exogenous; it can be described by the i.i.d. process governing ɛ and the by z s transition matrix. The part that concerns updating λ is denoted H, λ = H(ɛ, z, λ), where prime denotes the future period. For the individual agent, the relevant state variable is his holdings of money and the aggregate state: (m, ɛ, z, λ). Agents take as given the law of motion of the distribution of money holdings, the process governing the productivity and monetary policy shocks, and the monetary policy rule. Let V denote the value of money and at the beginning of the period before monetary transfers but after observing the current period s aggregate shocks z and ɛ. Given our assumption that buyer s make take-it-or-leave-it offers to sellers the value function can be expressed as 3 : 3 We assume that V (m) is continuous, which is sufficient for q(m b, m s ) and d(m b, m s ) to be measurable functions and therefore the integral to be well defined. 6

V (m; ɛ, z, λ) = xβ {u[q(m, m s ; ɛ, z, λ)]+ 0 { [ ] [ ]}} m + 1 d(m, ms ; ɛ, z, λ) m + 1 E ɛ E z z V, z, λ V, z, λ λ(dm s ) [ ] m + 1 + βe ɛ E z zv, z, λ. (1) where E denotes the expectations operator and a prime superscript a next period s variables. The terms of trade d(m b, m s ; ɛ, z, λ) and q(m b, m s ; ɛ, z, λ) solve 4 : max 0 d m b + 1,q 0 u(q) + E ɛ E z zv [ mb + 1 d, ɛ, z, λ ]. (2) subject to q = z E ɛ E z z λ = H (ɛ, z, λ) { V [ ] ms + 1 + d, ɛ, z, λ V [ ]} ms + 1, z, λ = G (z, λ) + ɛ. We now describe the law of motion of the distribution of money holdings H. The law of motion of the money holdings distribution can be described as a Markov process with an associated transition function P (m, B; ɛ, z, λ). Intuitively, P (m, B; ɛ, z, λ) is the probability that an individual currently with money holdings m will have money holdings in B B R+ next period, given the current aggregate state (ɛ, z, λ). In what follows we will construct such function. Let T = {buyer, seller, neither} and define the space (T, T), where T is the σ-algebra. Define the probability measure τ : T [0, 1], with τ(buyer) = τ(seller) = x, 4 Given the assumed continuity of V, it can easily be shown by the Theorem of the Maximum that the set of solutions of the maximization problem is non-empty, compact-valued, and upper hemicontinuous. Thus, by the Measurable Selection Theorem, it admits a measurable selection. We define d(m b, m s ; ɛ, z, λ) and q(m b, m s ; ɛ, z, λ) to be that selection. 7

and τ(neither) = 1 2x. Then, (T, T, τ) is a measure space. Define an event to be a pair e = (t, m), where t T, m R +. Intuitively, t denotes an agent s trading status and m the money holdings of his current trading partner. Let (E, E) be the space of such events, where E = T R + and E = T B R+. Furthermore, let ξ : E [0, 1] be the product probability measure. Define the mapping γ(m, e) : R + E R +, where γ(m, e; ɛ mu, z, λ) = m+ 1 d(m, ;ɛ,z,λ), if e = (buyer, ); m+ 1+d(,m;ɛ,z,λ), if e = (seller, ); m+ 1, otherwise. We can now define P : R + B R+ [0, 1] to be P (m, B; ɛ, z, λ) ξ({e E γ(m, e; ɛ, z, λ) B}). P is a well defined transition function. 5 Then, λ (B) = H(ɛ, z, λ)(b) 0 P (m, B; ɛ, z, λ) λ(dm) B B R+. (3) A recursive equilibrium is then a law of motion H, a value function V, terms of trade d and q, and monetary policy rule G such that (i) V satisfies (1), (ii) d and q solve the bargaining problem (2), and (iii) H is defined by (3). 4 The Approximate Economy Approximated Problem Solving the stochastic heterogeneous agents model of the previous section with a continuum of agents constitutes an impossible technical problem given that the state vector includes the 5 By construction, for each m, P (m, ; ɛ, z, λ) is a probability measure on (R +, B R+ ). Furthermore, given the measurability of d(, ; ɛ, z, λ), P (, B; ɛ, z, λ) is a B R+ -measurable function. 8

whole distribution of money, which is an infinite-dimensional object. Recently, algorithms have been proposed which summarize the cross-section distribution of agents characteristics by a small number of moments, therefore making the dimension of the state space tractable, and calculate the law of motion for these state variables using simulation procedures. The best known examples of this approach are those of den Hann (1997), Rios-Rull (1997), and Krusell and Smith (1998). An added difficulty of computing equilibria in our model is that it requires us to compute the expected gain of being a buyer - the integral in equation (1) - which requires us to integrate over the money holdings of the sellers. To exactly compute that expectation might require us to keep track of a large number of moments. To overcome this difficulty we follow Algan, Allais and den Haan (2007) by parameterizing the cross-sectional distribution and using reference moments as in Reiter(2002). Parameterizing the cross-sectional distribution allows us to obtain a numerical solution using standard quadrature and projection techniques. Using reference moments allows us to get a better characterization of the cross-sectional distribution without increasing the number of state variables. Although we will provide a detailed description of our algorithm we direct the reader to Algan, Allais, and den Haan for further advantages of their algorithm. We approximate λ by a finite set of moments s = [s 1,..., s N ]. Given that one doesn t have a complete description of the cross-sectional distribution it may be worthwhile adding some lagged values of the aggregate shocks as a state variable. Here we add z 1. As mentioned above, we follow Algan, Allais, and den Hann and parameterize the crosssectional distribution by a polynomial P (m; ρ) of order N with coefficients ρ = [ρ 0, ρ 1,..., ρ N] to match not only the moments s = [s 1,..., s N ] but also a set of extended moments s = [s N+1,..., s N]. We will describe in detail the procedure to obtained the extended moments below. By adopting a particular class of approximating polynomials one can reduce this problem to a convex optimization problem. 6 In particular, the polynomial of order N is 6 Without the convexity of the problem obtaining convergence can be problematic without finding a good initial condition. 9

written as: P (m, ρ) = ρ 0 exp{ρ 1 [m s 1 ] + ρ 2 [(m s 1 ) 2 s 2 ] +... + ρ N[(m s 1 ) N s N]}. When the density is constructed in this particular way, the coefficients, except for ρ 0, can be found with the following minimization routine: min ρ 1,ρ 2,...,ρ N 0 P (m, ρ) dm. The parameter ρ 0 can be determined by the condition that the density integrates to 1, 0 P (m, ρ) dm = 1. Finally, since we only use a limited set of moments as state variables, s = [s 1,..., s N ], the transition law only needs to specify how this limited set of moments evolves over time. Thus, instead of computing H, we now calculate s = Γ n (ɛ, z 1, z, s; γ n ) where Γ n is an n th -order polynomial with coefficients γ n. The approximate problem can then be written as: V (m; ɛ, z 1, z, s) = xβ {u[q(m, m s ; ɛ, z 1, z, s)]+ 0 { [ ] m + 1 d(m, ms ; ɛ, z 1, z, s) E ɛ E z z V, z, z, s [ ]}} [ ] m + 1 m + 1 V, z, z, s P (dm s ; ρ) + βe ɛ E z zv, z, z, s. s = Γ n (ɛ, z 1, z, s; γ n ). = Φ(z 1, z, s) + ɛ. 10

Where d(m b, m s ; ɛ, z 1, z, s) and q(m b, m s ; ɛ, z 1, z, s) solve: max 0 d m b + 1,q 0 u(q) + E ɛ E z zv [ mb + 1 d, ɛ, z, z, s ]. subject to q = z E ɛ E z z { V s = Γ n (ɛ, z 1, z, s; γ n ) [ ] ms + 1 + d, ɛ, z, z, s V [ ]} ms + 1, z, z, s = Φ (z 1, z, s) + ɛ. In the implementation of the algorithm we keep only track of the second moment s = s 2 as a state variable, that is, we set N = 2. Note that in this model the first moment is constant at s 1 = 1, since the average money holdings as a fraction of the begin of the period money supply is 1. For each triplet (ɛ, z 1, z) we approximate the value function of a grid of money holdings,m, and second moments,s 2. In the m direction we use an uneven grid and a shape-preserving interpolation routine. In the s 2 direction we use an evenly spaced grid and piecewise-linear interpolation. This is done to guarantee that the algorithm preserves the concavity of V with respect to m. Computing the Reference Moments In what follows we describe the procedure used to compute the reference moments at a point in the algorithm. For a given guess for the value functions, the law of motion, and the policy function, the idea is to simulate the model to obtain more information regarding the shape of the distribution. The procedure works as follows: 1. Use a random number generator to draw a time series for aggregate productivity and monetary shocks {ɛ t, z t } T t=0. 11

2. In period 1, the procedure starts with a given set of moments. To start the algorithm, we start with the moments of the stationary equilibrium of a model where the shocks are set to their unconditional mean. We set N = 6, that is we use 4 additional reference moments. Given the set of moments we use the procedure described above to compute the coefficients of the polynomial used to approximate the density, ρ 1. 3. Given the value function, the law of motion, and the policy rule we compute the decision rules d(m b, m s ; ɛ, z 1, z, s) (on a grid of money holdings and use interpolation). Given d, the monetary transfers, P (m, ρ 1 ) and the fraction of agents that are buyers and sellers we can then compute the new moments of the distribution using quadrature. Note that in this step we are using the fact that there is a continuum of agents and computing the moments without sampling variation. 4. Given these new moments we use again the procedure to compute the coefficients of the density ρ 2. 5. Repeat the process for the next period until t = T. To ensure that the sample used to obtain information about the cross-sectional distribution has reached (or is at least close to) its ergodic distribution one should disregard an initial set of observations. We set T = 2000 and discard 500 observation. We then compute the average of the reference moments conditional on z 1. Given that this procedure to compute the reference moments is computationally costly we only update the reference moments every 10 iterations of the algorithm which we describe in more detail in the next section. 5 Numerical Algorithm We now present a brief description of the algorithm used to compute the recursive equilibrium. The basic strategy of the algorithm is to iterate on the mapping defined by equation 12

(??) starting from an initial guess for V (m, ɛ, z 1, z, s 2 ) and for the law of motion Γ n. The algorithm works as follows: Compute Reference Moments - In iteration 1, we begin by computing the set of reference moments using the procedure described in the last section. As mentioned before, we only update the reference moments every 10 iterations given that the procedure is computationally time-costly. Computing the Cross-sectional Distributions - For each state (z 1, s 2 ) on the grid and the associated reference moments (conditional on z 1 ) previously obtained, we compute the coefficients of the associated densities P (m, ρ). Updating the Value Function - Given V (m, ɛ, z 1, z, s 2 ), the law of motion Γ n and the densities P (m, ρ) obtained in the last step, for each quintuplet (m, ɛ, z 1, z, s 2 ) we use equation (??) to update the V on the state space grid using quadrature methods. For the integration, we define d(m b, m s ; ɛ, z 1, z, s 2 ) and q(m b, m s ; ɛ, z 1, z, s 2 ) as functions obtained by solving the bargaining problem when needed. We do not support the decision rules on a grid to avoid introducing approximation errors, in particular violations of the participation constraints of the agents, that could lead to the loss of concavity of the value function. Updating the Law of Motion - Given P (m, ρ), the monetary policy rule, and the decision rules d(m b, m s ; ɛ, z 1, z, s 2 ), for each quadruplet (ɛ, z 1, z, s) on the state space grid, we compute a new set of second moments s 2. Note that, conditional on the realizations of the aggregate shocks and the current period s cross-sectional distribution, next period s cross sectional distribution is known with certainty. We then use standard projection methods to update the coefficients of Γ n. Note that this step does not require any kind of simulation procedure and thus avoids introducing cross-sectional sampling variation. We repeat the steps above until the law of motion has converged. 13

6 A Numerical Illustration 7 Discussion and Conclusion 14

References Algan, Yann, Oliver Allais, and Wouter den Hann. (2007). Solving Heterogeneous- Agent Models with Parameterized Cross-Sectional Distributions, C.E.P.R. Discussion Papers 6062. Berentsen, Aleksander, Gabriele Camera and Christopher Waller. (2005). The Distribution Of Money Balances And The Nonneutrality Of Money, International Economic Review, 46, 465-487. den Hann, Wouter J. (1997). Solving dynamic models with aggregate shocks and heterogeneous agents, Macroeconomic Dynamics, 1, 355-86. Krusell, Per and Anthony A. Smith, Jr. (1998). Income and Wealth Heterogeneity in the Macroeconomy, Journal of Political Economy, 106, 867-896. Lucas, Robert E., Jr. (1972). Expectations and the Neutrality of Money, Journal of Economic Theory, 4, 103-124. Molico, Miguel. (2006). The Distribution of Money and Prices in Search Equilibrium, International Economic Review, 47, 701-722. Reiter, Michael. (2002). Recursive Computation of Heterogeneous Agent Models, manuscript, Universitat Pompeu Fabra. Rios-Rull, J. Victor. (1997). Computation of Equilibria in heterogeneous agent models, Federal Reserve Bank of Minneapolis Staff Report 231. 15