Directed Search Lecture 5: Monetary Economics. October c Shouyong Shi

Transcription

1 Directed Search Lecture 5: Monetary Economics October 2012 c Shouyong Shi

2 Main sources of this lecture: Menzio, G., Shi, S. and H. Sun, 2011, A Monetary Theory with Non-Degenerate Distributions, manuscript. Gonzalez, F.M. and S. Shi, 2010, An Equilibrium Theory of Learning, Search and Wages, ECMA 78, Topkis, D.M., 1998, Supermodularity and Complementarity. Princeton, NJ: Princeton University Press. 2

3 1. Transactions Cost and Monetary Policy 2m = py/n m/p = [y b/(2 i)] 0.5 min[bn + i m/p] ::::::::::::: 0 t 2t Nt =1 time Baumol-Tobin inventory model of money 3

4 An important feature: transactions cost = limited participation in financial markets Policy implication: limited participation = open market operations can affect real activity Examples: Grossman and Weiss (83), Rotemberg (84) Lucas (90), Alvarez, et al. (02) 4

5 But the real effect can be short-lived! 2m' 2m m'/p' = m/p t' = t N' = N ::::::::::::: 0 t0 t t+t' t+(n'-1)t' =1 money injection (in contrast to VAR evidence by Christiano et al., 99) 5

6 Staggered participation and persistent effect 2m' 2m ::::::::::::: 0 money injection 6

7 Our interpretation: A non-degenerate distribution of money holdings is critical for money injection to have persistent real effects The objective: to give a tractable characterization of a monetary eqm in which there is microfoundation of money, and money distribution is non-degenerate 7

8 Search theory is a natural framework for both: microfoundation of money: decentralized exchange with lack of double coincidence of wants; anonymity exchange generates distribution of money holdings: no match: = buyer s money: = match seller s money: 8

9 ...but challenging to characterize such an equilibrium: distribution is endogenous state with a large dimension individuals decisions, trading prob. state variable aggregation distribution of individuals over money Even numerical computation can be challenging: Molico (06), Chiu and Molico (08) 9

10 Generations of money search models: 1st generation (Kiyotaki-Wright 89) assumes: all holdings are either zero or one unit 2nd generation: Shi (95)-Trejos-Wright (95): indivisible and divisible goods Green-Zhou (98): discrete and indivisible goods 3rd generation: divisible goods and Shi (97): a larger number of members in each household Lagos-Wright (05): centralized mkt with quasi-linear pref. We characterize equilibrium without these assumptions. 10

11 2. The Model Model Environment large numbers of types of perishable goods, ; each is produced by a large number of individuals no double coincidence of wants: type consumes good but produces good +1. anonymity: no record keeping fiat money: intrinsically useless; stock = 11

12 firms: types production: of type ( 1) labor = of type goods selling: of type ( 1) labor = one trading post competitive labor market: monetary wage rate = individuals own firms through a diversified mutual fund numeraire: labor 12

13 Events in a period: chooses to be lotteries on money = aworker or a buyer = markets open; search & match; consume A worker s decision: policy function ( ) [0 1] solves ( ) =max[ ( + ) ( )] : real balance; : ex ante value function 13

14 Directed search in the goods market: Buyers and firms choose which submarket to enter. A continuum of submarkets ( ) foreachtype good: submarket ( ): real balances for units of goods. market tightness ( ): # trading posts # buyers matching probability in submarket ( ): a buyer: ( ) = ( ( )) 0 ( ) 0 apost: ( ) = ( ( )) ( ) = ( ) 14

15 A buyer s decisions: chooses which submarket ( ) toenter ( ) = ( )+max ( ( )) [ ( )+ ( ) ( )] {z } surplus from trade s.t. [0 ], 0. policy functions: quantity of goods bought: ( ) real balance spent: ( ) residual balance: ( ) ( ) trading probability: ( ) = ( ( ( ) ( ))) 15

16 A firm s decisions: demand for labor to produce and sell goods number of trading posts to be created in submarket ( ): =, if ( ( ))( ) =0, if ( ( ))( ) [0 ), if ( ( ))( ) = ( ( )): matching prob for a post in submarket ( ) 16

17 Market tightness function ( ): ( ( ))( ) and with complementary slackness for ALL ( ) R ( ) Restrictions on beliefs out of the equilibrium: some submarkets ( ) are inactive in equilibrium, but we still require ( ) to satisfy the condition above. 17

18 Choice of being a buyer or a worker: at the beginning of every period, an individual chooses: Lottery ( ): ( ) = ( ) =max{ ( ) ( )} max ( ) 1 ( 1 )+ 2 ( 2 ) s.t =, =1, 2 1, [0 1] and 0for =1 2 policy functions: ( ( ) ( )) =1 2 18

19 E D B(m) V(m) C W (m) Α 0 k m 0 m Figure 1. Lotteries and the ex ante value function 19

20 Definition of a monetary steady state: Block 1: value functions: ( ), policy functions: ( ), and market tightness function ( ) (i) and solve a worker s problem (ii) and ( ) solve a buyer s problem (iii) and ( ) solve the lottery problem (iv) is consistent with free entry of trading posts 20

21 Block 2: distribution of real balances:, andwagerate: (v) is ergodic and generated by ( ) (vi) money is valued ( ) and Z all money is held: 1 = ( ). 21

22 A monetary steady state is block recursive: block 1: value functions, policy functions, market tightness aggregation block 2: distribution wage rate block recursivity makes equilibrium tractable: state variable in block 1: agent s own money balance block 2 is easy: counting flows 22

23 Why is the steady state block recursive? directed search + free entry of posts. No mixing between different : higher higher matching probability and higher spending ; higher quantity of goods obtained: A buyer with a particular only cares about ( ) and in the particular submarket he will enter; Each submarket is catered to buyers with a particular = depends on ( ) but NOT on the distribution 23

24 Eqm is NOT block recursive when search is undirected: bargaining on terms of trade: match surplus depends on in the match distribution of matters for profit ofapost= tightness depends on distribution ; value and policy functions depend on. price posting (with undirected search): whether a meeting results in a trade depends on the random buyer s distribution of again matters for profit ofapost. 24

25 3. Equilibrium Value and Policy Functions A buyer s problem: ( ) max{ ( ) : [0 ] [0 1]} ( ) ( ) = ( )+ [ ( )+ ( ) ( )] objective function is not concave in ( ) jointly standard approach in dynamic prog. does not work here But we want to use first-order and envelope conditions 25

26 C[0 ] : continuous, increasing functions on [0 ]; V[0 ] :subsetofc[0 ] with concave functions. Assume, takeany V[0 ], and prove: 3.1. worker s =,where : V[0 ] V[0 ]; 3.2. buyer s =,where : V[0 ] C[0 ]; monotone policy function, first-order and envelope conditions lottery problem = : V[0 ] V[0 ]; is monotone contraction = unique V[0 ]. prove, indeed. 26

27 Textbook approach (Stokey et al. 98) does not work here: ( ) = ( ) max{ ( ) : [0 ] [0 1]} ( ) ( ) = ( )+ [ ( )+ ( ) ( )] Objective function is not concave in ( ) jointly. why cannot we simply assume that is concave? need restrictions on endogenous that cannot be verified. some other approaches are not applicable either: e.g., equi-differentiability in (Milgrom and Segal 02). 27

28 Our approach (related to that in Gonzalez and Shi 10): ( ) =max{ ( ) : [0 ] [0 1]} ( ) (a) adapt lattice-theoretic approach (Topkis 98) to prove that policy functions are monotone (b) use (a) and first principles of calculus to prove and are differentiable at induced by optimal choices (c) (d) use (b) to validate first-order and envelope conditions prove that is differentiable at all 28

29 More specifics of step (a): ( ) isnotsupermodular! ( ) = ( )+ [ ( )+ ( ) ( )] To go around this problem: For any given, optimal solves: max ( ) ( ) is supermodular = ( ) = ( )+ ( ). (i) optimal ( ) isincreasing; (ii) ( ) =max ( ) is supermodular. Optimal solves: max ( ) ( ) =(1 ) ( )+ ( ). ( ) is supermodular = optimal ( ) isincreasing 29

30 4. Monetary Steady State A buyer s spending pattern: m x*(m) φ(m) x*(φ(m)) < x*(m) φ(φ(m)) ::::::::::::: 0 t 1 t 2 > 2t 1 t 3 >2t 2 -t 1 time 30

31 Contrast to Baumol-Tobin inventory model of money: Transactions cost lies not in getting money, but in spending money: one example is cost of trading post Endogenous features of trade: A buyer chooses how much money to spend: ( ) how much consumption to have: ( ) how quickly to get a trade: ( ) Staggered transaction pattern can be an equilibrium outcome 31

32 Existence and uniqueness of an equilibrium: A unique monetary steady state exists and is block recursive A lottery may or may not be used in the equilibrium Ifalotteryisused,itisusedonlyatthehighestbalance, ˆ : convex disutility of labor = need to smooth marginal cost by working for consecutive periods, each with low hours; lottery does the smoothing better 32

33 Does the equilibrium produce money dispersion? Yes, if individuals are sufficiently patient: 0. spend all ˆ once: in a buying sequence: pros: high current, smoothing in over periods cons: little smoothing in, discounting of future 0 is lower (money dispersion is more likely) if utility of consumption is more concave disutility of labor is less convex cost of maintaining trading posts is lower. 33

34 Equilibrium distribution of money holdings: density of money dist. trading prob. :::::::::: 0 φ n (m^) φ 3 (m^) φ 2 (m^) φ(m^) m^ Density of money distribution and trading probability 34

35 Effect of a one-time money injection: neutral in the steady state proportional injection is also neutral in the short run other types of injection are NOT neutral in the short run: a lump-sum injection tends to compress distribution and increase average real balance; open market operations are likely to have real effects how persistent the real effect is depends on how dispersed money holdings are. 35

36 What about studying money growth and inflation? equilibrium is still block recursive in the steady state: relatively simple to do comparative statics. transitional dynamics are not block recursive, but they depend on distribution only through the scalar : given, compute decisions and matching prob s flow equations = next period s distribution and +1 iterate on this procedure 36

37 We actually did compute the steady state: discount factor: =(1 05) 1 4 utility: ( ) = 1 1 = 2 3 disutility of labor: ( ) = 1 cost of post: =0 005 (markup=0.3) matching function: ( )= + 37

38 Welfare Money Growth Rate 38

39 Real Balances Money Growth Rate Z-average Z -c oe ff va r Velocity 39

40 Time Allocation Money Growth Rate Work Search&Buy Search&NotBuy 40

41 Production Money Growth Rate O utput C ons umption 41

42 5. Conclusion Formalized a monetary theory with non-degenerate distribution: directed search induces buyers to sort by into markets with different ( ) block recursive equilibrium with money dispersion some tools to do dynamic programming potentially persistent real effects of monetary policy future work on policy analysis: a lot to do 42