Directed Search Lecture 5: Monetary Economics October 2012 - PowerPoint PPT Presentation

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

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, 509-537. • Topkis, D.M., 1998, Supermodularity and Complementarity. Princeton, NJ: Princeton University Press. 2

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

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

But the real e ff ect can be short-lived! 2m' m'/p' = m/p 2m 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

Staggered participation and persistent e ff ect 2m' 2m ::::::::::::: 0 money injection 6

Our interpretation: A non-degenerate distribution of money holdings is critical for money injection to have persistent real e ff ects 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

• 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

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

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

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 • fi at money: intrinsically useless; stock =  11

• fi rms:  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 fi rms through a diversi fi ed mutual fund • numeraire: labor 12

Events in a period: chooses to be markets open; lotteries a worker search & match; on money = ⇒ = ⇒ or a buyer consume A worker’s decision : policy function  ∗ (  ) ∈ [0  1] solves  (  ) = max [  (  +  ) −  (  )]  : real balance;  : ex ante value function 13

Directed search in the goods market: Buyers and fi rms choose which submarket to enter. A continuum of submarkets (   ) for each type  good: • submarket (   ):  real balances for  units of goods. # trading posts • market tightness  (   ): # buyers • matching probability in submarket (   ):  0 (  )  0  a buyer:  (   ) =  (  (   ))  a post:  (   ) =  (  (   ))   (  ) =  (  )  14

A buyer’s decisions : • chooses which submarket (   ) to enter  (  ) =  (  ) + max  (  (   )) [  (  ) +  (  −  ) −  (  )] | {z } surplus from trade s.t.  ∈ [0   ],  ≥ 0. • policy functions: quantity of goods bought:  ∗ (  ) real balance spent:  ∗ (  ) residual balance:  (  ) ≡  −  ∗ (  ) trading probability:  ∗ (  ) =  (  (  ∗ (  )   ∗ (  ))) 15

A fi rm’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

Market tightness function  (   ): ⎡ ⎤  (  (   ))(  −  ) ≤  ⎦ with complementary slackness ⎣ and for ALL (   ) ∈ R 2 + .  (   ) ≥ 0 Restrictions on beliefs out of the equilibrium: some submarkets (   ) are inactive in equilibrium, but we still require  (   ) to satisfy the condition above. 17

Choice of being a buyer or a worker : at the beginning of every period, an individual chooses: ˜  (  ) = max {  (  )   (  ) } Lottery (  1   2   1   2 ): £ ¤  1 ˜  (  1 ) +  2 ˜  (  ) = max  (  2 ) (  1  2  1  2 ) s.t.  1  1 +  2  2 =  ,  1 +  2 = 1,  2 ≥  1 ,   ∈ [0  1] and   ≥ 0 for  = 1  2  policy functions: (  ∗  (  )   ∗  (  ))  =1  2 18

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

De fi nition 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

• Block 2: distribution of real balances:  , and wage rate:  (v)  is ergodic and generated by (  ∗   ∗   ∗   ∗   ∗   ) (vi) money is valued (   ∞ ) and Z all money is held: 1  =   (  ). 21

A monetary steady state is block recursive: block 1: block 2: 888888 value functions, distribution  policy functions, − − − − − − − − − → aggregation market tightness wage rate  block recursivity makes equilibrium tractable: • state variable in block 1: agent’s own money balance • block 2 is easy: counting fl ows 22

Why is the steady state block recursive? directed search + free entry of posts. • No mixing between di ff erent  : 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

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 pro fi t of a post = ⇒ 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 pro fi t of a post. 24

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 fi rst-order and envelope conditions 25

C [0  ¯  ] : continuous, increasing functions on [0  ¯  ]; V [0  ¯  ] : subset of C [0  ¯  ] with concave functions. • Assume  ≤ ¯   ∞ , take any  ∈ 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, fi rst-order and envelope conditions. 3.3. lottery problem  =  : V [0  ¯  ] → V [0  ¯  ];  is monotone contraction = ⇒ unique  ∈ V [0  ¯  ]. • prove  ≤ ¯   ∞ , indeed. 26