A Unified Framework for Monetary Theory and Policy Analysis

Transcription

1 A Unified Framework for Monetary Theory and Policy Analysis Ricardo Lagos NYU Randall Wright Penn 1

2 Introduction 2 We develop a framework that unifies micro and macro models of monetary exchange Why? Existing macro models are reduced-form models... Most existing micro models impose severe restrictions... Attempts to generalize micro models are very complicated... Outline for Today: 1. A brief review of the related literature 2. A basic version of our model 3. Policy implications and extensions

3 Micro Foundations of Money 3 Some things we ought not take for granted: the demand for money money should not be a primitive in monetary theory the auctioneer, mutilateral trade, price-taking behavior a model of money cannot be too Walrasian : it should build in frictions and strategic elements explicitly commitment/enforcement, monitoring/memory an essential role for money requires a double coincidence problem, imperfect commitment/enforcement, and imperfect memory/monitoring

4 4 1st Generation Models Assume m {0, 1} and q fixed Implies BE V 1 = b 1 + ασ(1 M) [u(q) + W 0 ] + [1 ασ(1 M)] W 1 V 0 = b 0 + ασm [W 1 c(q)] +(1 ασm)w 0 Note: typically, W m = βv m IR conditions: u(q) + W 0 W 1 W 1 c(q) W 0 Results: Existence of equilibrium with valued money, welfare...

5 5 2nd Generation Models Keep m {0, 1} but endogenize q Add BS: choose q to solve max q [u(q) + W 0 T 1 ] θ [ c(q) + W 1 T 0 ] 1 θ IC conditions u(q) + W 0 W 1 W 1 c(q) W 0 Results: As above (existence, welfare...), plus we can discuss price p = 1/q Example: Assumptions q < q but q q as β 1. This looks like standard monetary inefficiency (say, in CIA model)

6 6 1st and 2nd Generation Shortcomings Extreme assumptions on inventories of money: m {0, 1} Restrictive upper bound on money holdings models not useful to analyze policy experiments (e.g. changes in the money supply) Indivisibility of money often drives results: Berentsen and Rocheteau (JME 2002) No trade inefficiency (in 1st and 2nd generations) Too much trade inefficiency (in 2nd generation)

7 3rd Generation Models 7 Let m M R + and let F ( m) be the CDF of money holdings V (m) = b(m) + (1 2ασ)W (m) +ασ {u [q (m, m)] + W [m d (m, m)]} df ( m) +ασ {W [m + d ( m, m)] c [q ( m, m)]} df ( m) where q(m, m) and d(m, m) solve max q,d [u(q) + W (m d) T (m)]θ [ c(q) + W ( m + d) T ( m)] 1 θ s.t. u(q) + W (m d) W (m) W ( m + d) c(q) W ( m) d m Typically: W (m) = βv (m)

8 3rd Generation Issues 8 Model with M = {0, 1,..., m} Some analytic results: Green and Zhou (JET 1998), Camera and Corbae (IER 1999), Taber and Wallace (IER 1999), Zhou (IER 1999), Zhu (PhD thesis 2002) Model with M = R + Analytic results (even existence) very difficult Some numerical examples: Molico (PhD thesis 1997) One (big) complication: endogenous F (m) Shi (Econometrica 1997) provides a trick: the family We provide a different trick: competitive markets Potential advantages of our approach: * do not need unpalatable family * can use standard and simple bargaining theory * do not have to ignore incentive problems * having some centralized trading can be desirable

9 9 Our Model Discrete time, infinite horizon, [0, 1] continuum of agents 2 types of nonstorable, perfectly divisible goods: general and special General goods are consumed and produced by everyone U (Q) and C (Q) are utility of consumption and production U > 0, U 0 Special goods (subject to double-coincidence problem) 3 types of meetings: double-coincidence (with prob. δ) single-coincidence (with prob. σ) u (q) and c (q) are utility of consumption and production u (0) = c (0) = 0, u > 0, c > 0, u < 0, c 0, and u (q) = c (q) for some q > 0. Let q be defined by u (q ) = c (q ) Key ingredients two sub-periods (day and night) special goods can only be produced during the day general goods can only be produced at night C (Q) = Q

10 10 Trading Feasible trades: day: special for special goods and money for special goods night: general for general goods and money for general goods Assume: day: decentralized trading night: centralized trading Note: Agents cannot commit to future actions; and no memory role for money in decentralized trading

11 Value Function V (m) = ασ {u [q (m, m)] + W [m d (m, m)]} df ( m) +ασ {W [m + d ( m, m)] c [q ( m, m)]} df ( m) +αδ B(m, m)df ( m) + (1 2ασ αδ)w (m) 11 V (m) : value of entering the search market with m dollars W (m) : value of entering the centralized market with m dollars F (m) : CDF of money holdings (endogenous) M : total stock of money; mdf (m) = M q (m, m) : quantity of special good exchanged if the buyer has m and the seller m dollars d (m, m) : dollars exchanged if the buyer has m and the seller m dollars B (m, m) : expected net payoff from a barter trade if the buyer has m and the seller m dollars Next: look at W (m); determine single-coincidence terms of trade q (m, m) and d (m, m); and value of barter B (m, m)

12 Centralized Market 12 β : discount factor φ : price of money in terms of general goods In the centralized market agents solve: W (m) = max X,Y,m U(X) Y + βv (m ) s.t. X + φm = Y + φm W (m) = max X,m U(X) X + φm φm + βv (m ) W (m) = U(X ) X + φm + max m {βv (m ) φm } where U (X ) = 1 Observation: m is independent of m Corollary 1: V (m) strictly concave F (m) degenerate Corollary 2: W (m) is affine, W (m) = W (0) + φm

13 Decentralized Market: Terms of Trade 13 Double-coincidence meetings Symmetric Nash solution (i) each agent produces q for the other, and (ii) no money changes hands B (m, m) = b + W (m); where b u (q ) c (q ) Single-coincidence meetings In general BS is subject to max q,d [u (q) + W (m d) T (m)]θ [ c (q) + W ( m + d) T ( m)] 1 θ u (q) + W (m d) W (m) c (q) + W ( m + d) W ( m) d m T (m) : the threat point of an agent with m units of money Linear W and T (m) = W (m) BS becomes:

14 14 max q,d [u (q) φd]θ [ c (q) + φd] 1 θ s.t d m q = { q(m) m < m q m m d = { m m < m m m m where φm = θc(q ) + (1 θ)u(q ) and q(m) solves FOC: φm = θu (q)c(q) + (1 θ)c (q)u(q) θu (q) + (1 θ)c (q) f (q)

15 Observation 15 q(m) solves FOC: φm = θu (q)c(q) + (1 θ)c (q)u(q) f (q) θu (q) + (1 θ)c (q) q φ[θu + (1 θ)c ] 2 (m) = u c [θu + (1 θ)c ] + θ(1 θ)(u c)(u c c u ) lim (m) = 1 φ m m q u (q ) 1 + θ(1 θ) (u c ) c (q ) u (q ) u (q ) 2 u (q ) q (m ) < φ if θ < 1 u (q ) q (m ) = φ if θ = 1

16 Value Function 16 V (m) = αδ [b + W (m)] + ασe F { c [q ( m)] + W [m + d ( m)]} +ασ {u [q (m)] + W [m d (m)]} +(1 2ασ αδ)w (m) = αδb + ασe F { c [q ( m)] + φd ( m)} +ασ {u [q (m)] φd (m)} + W (m) Recall: W (m) = U(X ) X +φm+max m {βv (m ) φm } let: κ = αδb + ασe F { c [q ( m)] + φd ( m)} + U (X ) X and let: v (m) = κ + ασ {u [q (m)] φd (m)} Then the value function can be written as V (m) = v(m) + φm + max m {βv (m ) φm }

17 Value Function: Existence and Uniqueness V (m) = max m {v(m) + φm φm + βv (m )} 17 Note: The RHS defines a contraction on a complete metric space B φ = {f : R R f(x) = g(x) + φx, g(x) B} where: B = {g : R R g is cont. and bounded in the sup norm} Hence! V (m) in B φ solving BE Remark: Our methods also work for V (m, φ, F )

18 Value Function: Properties 18 V (m) = v(m) + φm + max m {βv (m ) φm } where v (m) = κ + ασ {u [q (m)] φd (m)} If u and c are C n then V is C n 1 a.e. and V (m) = { ασφe (q) + (1 ασ) φ m < m φ m m where e (q) = u (q)q (m) φ is the gain from having an additional unit of real balances when bargaining Lemma: for θ (0, 1) (i) lim m m V (m) < φ (ii) V (m) < 0 for m < m if u is log concave Note: if θ = 1 then: (i) lim m m V (m) = φ (ii) V (m) < 0 for m < m

19 19 Solution to the Agent s Problem in the Centralized Market max m +1 {βv (m +1 ) φm +1 } Derivative is: βφ +1 φ for m +1 m +1 β {ασu [q (m +1 )] q (m +1 ) + (1 ασ) φ +1 } φ for m +1 < m +1 where m +1 [θc(q ) + (1 θ)u(q )] /φ +1 Corollary. In any equilibrium: (i) βφ +1 φ, and (ii) m +1 < m +1 m +1 is characterized by the FOC βv (m +1 ) φm +1 = 0 Recall: under mild conditions, V (m) < 0 for m < m Thus the FOC has a unique solution, which is independent of m F (m) degenerate t > 0 in any equilibrium

20 Equilibrium 20 Definition. Given M an equilibrium is a list (V, q, d, φ, F ) satisfying: 1. the BE V (m) = max m {v(m) + φm φm + βv (m )} 2. the BS q = { q(m) m < m q m m d = { m m < m m m m where m = θc(q )+(1 θ)u(q ) φ, and q (m) solves θ [ c (q) + φm] u (q) = (1 θ) [u (q) φm] c (q) 3. the FOC βv (m ) = φ 4. F degenerate at m = M

21 21 Analysis Substitute V into FOC: φ t = β [ασu (q t+1 )q t+1(m t+1 ) + (1 ασ)φ t+1 ] Use BS to eliminate φ and q : e θ (q t+1 ) = 1 + f θ(q t ) βf θ (q t+1 ) ασβf θ (q t+1 ) f θ (q) θcu +(1 θ)uc θu +(1 θ)c and e θ (q) [θu +(1 θ)c ] 2 u u c [θu +(1 θ)c ]+θ(1 θ)(u c)(u c c u ) A monetary equilibrium is simply a sequence {q t } with q t (0, q ] that solves this difference equation Given q t, the rest of the allocation is given by General Results: (i) Classical Neutrality d t = M φ t = f θ (q t )/M m t+1 = M with prob. 1 (ii) Inefficiency: q t < q for all t in any equilibrium

22 22 Results (Steady State) MSS solves e θ (q) = β ασβ A simple case. If θ = 1, then equilibrium condition is Results: u (q) c (q) = β ασβ (i) If a MSS exists, it is unique (ii) MSS q 1 provided u (0) c (0) > β ασβ (iii) q 1 < q (iv) q 1 q as β 1; q 1 0 as ασβ 0 General case. If 0 < θ < 1, then MSS q θ and q θ < q 1. (Existence and uniqueness under mild conditions.) Result: θ < 1 q θ is bounded away from q even as β 1 Intuition. In general there are two distortions: β-wedge (standard in monetary models) θ-wedge (hold up problem)

23 23 Monetary Policy Suppose M t+1 = (1 + τ) M t Form of injections: lump-sum transfers Generalized steady state condition: e θ (q) = τ β βασ Results: (i) Inflation reduces welfare (ii) MSS exists iff τ β 1 (iii) Friedman Rule is optimal (iv) Frieman Rule achieves q iff θ = 1 Intuition: Monetary policy can correct the β-wedge but not the θ-wedge

24 Welfare Cost of Inflation 24 Implication: Welfare costs of inflation can be much higher than predicted by standard reduced form models Intuition: Envelope Theorem

25 25 Extensions and Applications Dynamics nonstationary, cyclic, sunspot and chaotic equilibria Real shocks can have d < m with positive prob. (endogenous velocity) Monetary shocks only persistent inflation affects φ (negatively) Endogenous search or specialization also makes velocity = ασ endogenous Heterogeneity (e.g. through limited participation ) F nondegenerate yet tractable inflation may increase welfare (through redistribution) Empirical implementation (e.g. use the model to quantify welfare cost of inflation)