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

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

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] ::::::::::::: t 2t Nt =1 time

Baumol-Tobin inventory model of money

3

An important feature: transactions cost = ⇒ limited participation in financial markets Policy implication: limited participation = ⇒

• pen market operations can affect real activity

Examples: Grossman and Weiss (83), Rotemberg (84) Lucas (90), Alvarez, et al. (02)

4

But the real effect can be short-lived!

2m' m'/p' = m/p 2m t' = t N' = N ::::::::::::: 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 effect

2m' 2m ::::::::::::: money injection

6

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

• 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:  match ⎧ ⎨ ⎩ buyer’s money:  −  seller’s money:  = ⇒ · · · · · ·

8

...but challenging to characterize such an equilibrium: distribution is endogenous state with a large dimension individuals’ decisions, trading prob. state variable ← − − − − − − − − − − − − − − − − − − − − → aggregation distribution

• f individuals
• ver money

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
• fiat money: intrinsically useless;

stock = 

11

• 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

Events in a period: lotteries

• n money =

⇒ chooses to be a worker

• r a buyer

= ⇒ markets open; search & match; 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 firms choose which submarket to enter. A continuum of submarkets ( ) for each type  good:

• submarket ( ):

 real balances for  units of goods.

• market tightness ( ):

# trading posts # buyers

• matching probability in submarket ( ):

a buyer: ( ) = (( )) 0()  0 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 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

Market tightness function ( ): ⎡ ⎣ (( ))( − ) ≤  and ( ) ≥ 0 ⎤ ⎦ with complementary slackness for ALL ( ) ∈ R2

+.

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):  () = max

(1212)

£ 1 ˜  (1) + 2 ˜  (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 k m0 m

Figure 1. Lotteries and the ex ante value function

19

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

• Block 2:

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

21

A monetary steady state is block recursive: block 1: value functions, policy functions, market tightness 888888 − − − − − − − − − → 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

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

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 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

• n the random buyer’s 

— distribution of  again matters for profit 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 first-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, first-order and envelope conditions. 3.3. lottery problem  =  : V[0 ¯ ] → V[0 ¯ ];  is monotone contraction = ⇒ unique  ∈ V[0 ¯ ].

• prove  ≤ ¯

  ∞, indeed.

26

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

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) use (b) to validate first-order and envelope conditions (d) prove that  is differentiable at all 

28

More specifics of step (a): (  ) is NOT supermodular! (  ) =  () +  × [( ) +  ( − ) −  ()] To go around this problem:

• For any given , optimal  solves: max (  )

(  ) = ( ) +  ( − ). (  ) is supermodular = ⇒ (i) optimal ˜ ( ) is increasing; (ii) ˜ ( ) = max (  ) is supermodular.

• Optimal  solves: max ˜

( ) ˜ ( ) = (1 − ) () +  ˜ ( ). ˜ ( ) is supermodular = ⇒ optimal () is increasing

29

• 4. Monetary Steady State

A buyer’s spending pattern:

m x*(m) φ(m) x*(φ(m)) < x*(m) φ(φ(m)) ::::::::::::: t1 t2 > 2t1 t3>2t2 -t1 time

30

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

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
• If a lottery is used, it is used only at the highest balance, 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

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

Equilibrium distribution of money holdings:

density trading prob.

• f money dist.

:::::::::: φn(m^) φ3(m^) φ2(m^) φ(m^) m^

Density of money distribution and trading probability

34

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

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

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

We lfare

0.99 0.992 0.994 0.996 0.998 1 1.002 0.00 0.02 0.04 0.06 0.08 0.10 Mon e y Growth Rate 38

Real Balances

0.75 0.825 0.9 0.975 1.05 1.125 1.2 0.00 0.02 0.04 0.06 0.08 0.10 Mone y Growth Rate

Z-average Z-c oeff var V eloc ity

39

Time Allocation

0.9 0.925 0.95 0.975 1 1.025 1.05 1.075 1.1 0.00 0.02 0.04 0.06 0.08 0.10 Mon e y Growth Rate

W ork S earc h&B uy S earc h&NotB uy

40

Production

0.9 0.925 0.95 0.975 1 1.025 1.05 0.00 0.02 0.04 0.06 0.08 0.10 Mon e y Growth Rate

O utput C ons um ption

41

• 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