Search-Theoretic Models of Money Jesús Fernández-Villaverde February 12, 2016 University of Pennsylvania

Search-Theoretic Models of Money

Jesús Fernández-Villaverde

University of Pennsylvania

February 12, 2016

Jesús Fernández-Villaverde (PENN)


February 12, 2016 1 / 49

Motivation

What is money?

Why do we use money as a society?

Use search theory to model existence of money.

Contrast with other two approaches:

1 Money in DSGE models (CIA, MIU).

2 Money in OLG models.



February 12, 2016 2 / 49

Essential models of money

Hahn (1965) : money is essential if it allows agents to achieve allocations they cannot achieve with other mechanisms that also respect the enforcement and information constraints in the environment.

Why do we care about essential models of money?

Three frictions that will make money essential:

1 Double-coincidence of wants problem.

2 Long-run commitment cannot be enforced.

3 Agents are anonymous: histories are not public information.

Money is a consequence of these frictions in trade: medium of exchange.



February 12, 2016 3 / 49

Three generations of models

1

1 unit of money, 1 unit of good: Kiyotaki and Wright (1993) .

2 1 unit of money, endogenous units of good: Trejos and Wright (1995) .

3

Endogenous units of money, endogenous units of good: Lagos and

Wright (2005) .



February 12, 2016 4 / 49

First generation: environment

[ 0 , 1 ] continuum of anonymous agents.

Live forever and discount future at rate r .

[ 0 , 1 ] continuum of goods. Good i is produced by agent i .

Goods are non-storable: no commodity money.

Unit cost of production c 0 .



February 12, 2016 5 / 49

Double-coincidence of wants problem

I do not produce what I like (non-restrictive: home production, specialization).

iWj : agent i likes to consume good produced by agent j :

1 utility u > c from consuming j .

2 utility 0 otherwise.

Probabilities of matching:

Jesús Fernández-Villaverde (PENN) p ( iWi ) = 0 p ( jWi ) = x p ( jWi j iWj ) = y


February 12, 2016 6 / 49

Fixed money and …xed good

Exogenously given quantity M storable good.

2 [ 0 , 1 ] of an indivisible unit of

Holding money yields zero utility

γ

: …at money.

Initial endowment: M agents are randomly endowed with one unit of money.

Agents holding money cannot produce (for example because you need to consume before you can produce again).

We eliminate (non-trivial) distributions.



February 12, 2016 7 / 49

Trades

Pairwise random matching of agents with Poisson arrival time

α

.

Bilateral trading is important, randomness is not ( Corbae, Temzelides, and Wright, 2003 ).

Upon meeting, agents decide whether to trade. Then, they part company and re-enter the process.

History of previous trades is unknown.

Exchange 1 unit of good for 1 unit of good (barter) or 1 unit of money.



February 12, 2016 8 / 49

Individual trading strategies

Agents never accept a good in trade if he does not like to consume it since it is not storable.

They will barter if they like the both agents in the pair like each other goods.

Would they accept money for goods and viceversa?

We will look at stationary and symmetric Nash equilibria.



February 12, 2016 9 / 49

Probabilities

You meet someone with arrival rate

α

.

This person can produce with probability 1 M .

With probability x you like what he produces.

With probability

π

=

π 0 π 1

(endogenous objects to be determined) both of you want to trade.

If

π

> 0, we say that money circulates.



February 12, 2016 10 / 49

Value functions

Value functions with money, V

1

: rV

1

=

α x ( 1 M )

π

( u + V

0

V

1

)

Value functions without money, V

0

.

rV

0

=

α xy ( 1 M )

π

( u c ) +

α xM

π

( V

1

V

0 c )

Renormalize

α x = 1 by picking time units: rV

1

= ( 1 M )

π

( u + V

0

V

1

) rV

0

= y ( 1 M )

π

( u c ) + M

π

( V

1

V

0 c )



February 12, 2016 11 / 49

Individual trading strategies

Net gain from trading goods for money:

∆

0

= V

1

V

0 c =

( 1 M ) (

π y ) ( u c ) rc r +

π

Net gain from trading money from goods:

∆

1

= u + V

0

V

1

=

( M

π

+ ( 1 M ) y ) ( u c ) + ru r +

π

Clearly:

8

<

π j

:

= 1

2 [ 0 , 1 ]

= 0

8

< as

∆ j

:

> 0

= 0

< 0

Plug those into the individual trading strategies, and check them.



February 12, 2016 12 / 49

Characterization

Clearly

∆

1

> 0 wants to trade.

.

Hence

π 1

= 1, i.e., the agent with money always

For

π 0

, you have

∆

0

=

( 1 M ) ( u c )

π 0 r +

π 0

( 1 M ) y ( u c ) + rc r +

π 0

Then,

∆

0 has the same sign as

π 0 rc + ( 1 M ) y ( u c )

( 1 M ) ( u c )

=

π 0 b



February 12, 2016 13 / 49

Multiple equilibria

Nonmonetary equilibrium: we have an equilibrium where

π 0

= 0 .

Monetary equilibrium: if c < r

( 1 M ) ( 1 y )

+ ( 1 M ) ( 1 y ) u then b < 1 and

π 0

= 1 is an equilibrium as well.

Mixed-monetary equilibrium:

π 0

( Schevchenko and Wrigth, 2004 ).

= b . However, not robust



February 12, 2016 14 / 49

Welfare I

De…ne welfare as the average utility:

W = MV

1

+ ( 1 M ) V

0

Then: rW = ( 1 M ) [( 1 M ) y + M

π

] ( u c )

Note that welfare is increasing in

π

.

When

π

= 1: rW = ( 1 M ) [( 1 M ) y + M ] ( u c ) = My + M

2 y

Maximize W with respect to M :

M

M

=

=

1 2 y

2 2 y

0 if y if y <

1

2

1

2

Intuition: facilitate trade versus crowding out barter.



February 12, 2016 15 / 49

Welfare II

When

π

= 0: rW = ( 1 M ) [( 1 M ) y ] ( u c )

Monotonically decreasing in M ) M = 0.

Result is a little bit silly: it depends on the absence of free disposal of money. Otherwise, welfare is independent of M .

For a general

π rW = ( 1 M ) [( 1 M ) y + M

π

] ( u c ) , monotonically increasing in M in the [ 0 , 1 ] interval.



February 12, 2016 16 / 49

Comparison with alternative arrangements

Imagine that we have the credit arrangement: “produce for anyone you meet that wants your good.”

Value function rV c

= u c

Clearly rV c

> rW

However, this arrangement is not self-enforceable: histories are not observed.



February 12, 2016 17 / 49

Second generation: endogenous prices

We make the very strong assumption that we exchanged one good for one unit of money.

What if we let prices to be endogenous?

Shi (1995) and Trejos and

Wright (1995) .

We set y = 0 and we let goods to be divisible.

When agents meet, they bargain about how much q will be exchanged, or equivalently, about price 1 / q .



February 12, 2016 18 / 49

Utility and cost functions

Utility is u ( q ) and cost of production is c ( q ) .

Assumptions: u ( 0 ) = c ( 0 ) = 0 u

0

( 0 ) > c ( 0 ) u

0

( 0 ) > 0 , u

00

( 0 ) 0 c

0

( 0 ) > 0 , c

00

( 0 ) 0

Also, b and q are such that u

0 u

0

( b ) = c

0

( q ) = c

0

( b )

( q )



February 12, 2016 19 / 49

Value functions and bargaining

Take q = Q as given. Then: rV

1

= ( 1 M ) [ u ( Q ) + V

0

V

1

] rV

0

= M [ V

1

V

0 c ( Q )]

Bargaining is the generalized Nash bargaining solution: q = arg max [ u ( q ) + V

0

( Q ) T

1

] θ u ( q ) + V

0

V

1

V

1 c ( q ) V

0

[ V

1

( Q ) c ( Q ) T

0

] θ where T j is the threat point of the agent with j units of money.

We will set T j

= 0 and

θ

= 1 / 2 .



February 12, 2016 20 / 49

Equilibria

Necessary condition taking V

0

( Q ) and V

1

( Q ) as given:

[ V

1

( Q ) c ( q )] u

0

( q ) = [ u ( q ) + V

0

( Q )] c

0

( q )

The bargaining solution de…nes a function q = e ( Q ) and we look at its …xed points.

Two …xed points:

1 q = 0: nonmonetary equilibrium.

2 q = q e

> 0: monetary equilibrium.



February 12, 2016 21 / 49

E¢ ciency

Note that the e¢ cient outcome is q , i.e.

u

0 ( q ) = c

0 ( q ) .

In the monetary equilibrium: u

0 ( q e

) = u

V

1

( q e

( q

) + V

0

( q e e ) c ( q e

)

) c

0 ( q e

) > u

0 ( q ) since u ( q e

) + V

0

( q e

) > V

1

( q e

) c ( q e

) .

Hence q > q e

, or equivalently, the price is too high.



February 12, 2016 22 / 49

Third generation: endogenous prices and goods

Relax the assumption that agents hold 0 or 1 units of money.

Problem: endogenous distribution of money that we (and the agents!) need to keep track of.

Computational: Molico (2006) .

Theoretical:

1 Families: Shi (1997) .

2 Two markets: Lagos and Wright (2005) .



February 12, 2016 23 / 49

Environment

Discrete time.

[ 0 , 1 ] continuum of in…nitely-lived agents with

β

2 ( 0 , 1 ) .

Each period is divided between two subperiods: day and night.

Utility function: u ( x ) c ( h ) + U ( X ) H where x ( X ) is consumption during the day (night) and h ( X ) is labor during the day (night).

u

U

( 0 ) = 0, u

0

00

0.

> 0, u

00 < 0, c ( 0 ) = 0, c

0 > 0, c

00

0, U

0 > 0, and

Also, assume that

U

0

9 ( q , X ) 2 ( 0 ,

∞

)

2

( X ) = 1, U ( X ) > X .

s.t.

u

0 ( q ) = c

0 ( q ) and

Key point: linearity on H .

February 12, 2016 24 / 49

Money

Money holdings m 0, perfectly divisible and storable.

Total amount of money is M .

F t

( e ) : CDF of agents with m e at start of day market at t .

G t

( e ) : CDF of agents with m e at start of night market at t .

Therefore:

Z mdF t

( e ) =

Z mdG t

( e ) = M for all t .

φ t

: price of money in the centralized market (inverse of the price level).



February 12, 2016 25 / 49

Day trading I

Decentralized market with anonymous bilateral matching and no record-tracking technology.

Probability of meeting:

α

.

x comes in many varieties. Each agent consumes only a subset.

Each agent can transform h one for one into one of these special goods that he himself does not consume.

In any random meeting between to agents:

1

2

3

Double coincidence (both agents consume what the other can produce): probability

δ

.

Single coincidence (one agent, i.e., a “buyer”, consumes what the other can produce, i.e., a “seller”, but not the other way around): probability

σ

.

No coincidence (neither agent consumes what the other can produce): probability 1 2

σ δ

.



February 12, 2016 26 / 49

Day trading II m : dollars of the buyer.

e : dollars of the seller.

q t

( m , e ) : quantity producer by seller (bought by buyer).

d t

( m , e ) : dollars paid by the buyer to the seller.

B t

( m , e ) : payo¤ in double coincidence meetings.



February 12, 2016 27 / 49

Night trading and money

Centralized market on an homogenous good X .

Homogeneity is irrelevant with Walrasian markets.

We could also allow intertemporal claims in the night markets, but they will not trade in equilibrium.

Each agent can transform H one for one into X .

Also, x and X are perfectly divisible, but non-storable.



February 12, 2016 28 / 49

Value functions

V t

( m ) : value function for an agent with decentralized market: m dollars when he enters the

V t

Z

( m ) =

ασ

|

+

ασ

|

+

αδ

|

Z

Z u ( q t

( m , e )) + W t

( m d t

( m , e

{z

Buying

)) dF t

( e

}

) c ( q t

( e , m )) + W t

( m + d t

( e , m

{z

Selling

)) dF t

( e

}

)

B t

( m , e )

{z

Bartering dF t

( e

}

)

+( 1 2

ασ αδ

) W t

( m )

| {z

Not trading

}



February 12, 2016 29 / 49

Value functions II

W t

( m ) : value function for an agent with the centralized market: m dollars when he enters

W t

( m ) = max

X , H , m 0

U ( X ) H +

β

V t + 1 m

0 s.t.

X = H +

φ t m m

0

Substituting the budget constraint:

W t

( m ) =

φ t m + max

X , H f U ( X ) X g + max m 0

=

φ t m + U ( X ) X + max m 0

φ t m

0 +

β

V t + 1 m

0

φ t m

0 +

β

V t + 1 m

0 where we have used the optimality condition U

0 ( X ) = 1.

Therefore:

1

2

3

All agents consume the same amount X .

W t

( m ) is linear in m with slope

φ t

.

Some work is necessary to show that the solution to this problem is interior.



February 12, 2016 30 / 49

Terms of trade: double coincidence I

Nash bargaining with symmetric power and threat points W t

W t

( e ) .

( m ) and

Bargaining on q i

, q j

0 and

∆ m : q i max

, q j

, ∆

[ u ( q j

) c ( q i

) + ( W t

( m

∆

) W t

[ u ( q i

) c ( q j

) + ( W t

( e + ∆ ) W t

( m ))]

1 / 2

( e ))]

1 / 2 s .

t .

∆ m i and

∆ m j

Given the linearity of W t

( m ) on m , we can rewrite the problem: q i max

, q j

,

∆ log [ u ( q j

) c ( q i

)

φ t

∆ ] + log [ u ( q i

) c ( q j

) +

φ t

∆ ] s .

t .

∆ m i and

∆ m j



February 12, 2016 31 / 49

Terms of trade: double coincidence II

Optimality conditions: q i q j

: c

0 ( q i

)

:

∆

: u u u

( q j

) c ( q i

)

φ t u

0 ( q j

)

∆

(

( q j

) c ( q i

)

φ t q j

) c ( q i

)

φ t

∆

∆

= u u

0 ( q i

)

( q i

) c ( q j

) +

φ t

∆ u c

0 ( q j

)

=

= u (

( q j

) c ( q i

) +

φ t q j

) c ( q i

) +

φ t

∆

∆

Thus: q = q i

= q j

, s.t.

u

0

∆

= 0

( q ) = c

0

( q ) and

B t

( m , e ) = u ( q ) c ( q ) + W t

( m )



February 12, 2016 32 / 49

Terms of trade: single coincidence I

Nash bargaining with power

θ

W t

( m ) and W t

( e ) .

> 0 for buyer and threat points

Bargaining on q 0 and d m :

( max q , d [

[ u ( q ) + W t

( m d ) W t

( m )] θ c ( q ) + W t

( e + d ) W t

( e )]

1

θ

)

Given the linearity of W t

( m ) on m , we can rewrite the problem:

θ max q , d

1

θ log [ u ( q )

φ t d ] + log [ c ( q ) +

φ t d ]



February 12, 2016 33 / 49

Terms of trade: single coincidence II

De…ne: m t

=

1

[( 1

θ

) u ( q

φ t

Claim: solution to bargaining problem is

) +

θ c ( q )] q t

( m , ˜ ) = q t

( m ) if m < m t q if m m t and d t

( m , ˜ ) = m if m m t

< m t if m m t where ˆ t

( m ) solves: m =

1

φ t

( 1

θ

) u ( q ) c

0 ( q ) +

θ c ( q ) u

0

( 1

θ

) c 0 ( q ) +

θ u 0 ( q )

( q )

=

1

φ t z ( q ) .

Note that this solution does not depend on ˜ .



February 12, 2016 34 / 49

Proof I

Optimality conditions if d m : q : d :

θ u

0 ( q )

1

θ u

θ

( q )

φ t d

1

1

θ u ( q )

φ t d

=

=

Thus: c c c

0 ( q )

( q ) +

φ t d

1

( q ) +

φ t d

θ

1

θ u c ( q ) +

φ t d

( q )

φ t d

= 1 ) u

0

( q ) = c

0

( q ) and: d =

1

φ t

[( 1

θ

) u ( q ) +

θ c ( q )] = m t



February 12, 2016 35 / 49

Proof II

Optimality conditions if d = m , we still have q :

θ u

0 ( q )

1

θ u ( q )

φ t d

= c

0 ( q ) c ( q ) +

φ t d but now evaluated at:

θ u

0 ( q )

1

θ u ( q )

φ t m

= or c c

0 ( q )

( q ) +

φ t m d t

( m ) = m =

1

φ t |

( 1

θ

) u ( q ) c

0 ( q ) +

θ c ( q ) u

0 ( q )

( 1

θ

) c 0 ( q ) +

θ u 0 ( q )

{z z ( q )

}

=

1

φ t z ( q )



February 12, 2016 36 / 49

Some properties I

Remember m =

Then, for any m that:

1 z ( q ) .

φ t

< m t

, it follows from the implicit function theorem q t

0

( m ) = ˆ t

0

( m ) = z 0

φ t

( q t

( m ))

Also, remember that: z ( q ) =

( 1

θ

) u ( q ) c

0 ( q ) +

θ c ( q ) u

0 ( q )

θ u 0 ( q ) + ( 1

θ

) c 0 ( q ) and then (where all functions are evaluated at q ): z

0

= u

0 c

0 [

θ u

0 + ( 1

θ

) c

0 ] +

θ

( 1

θ

) ( u c ) ( u

0 c

00 c

0 u

00 )

[

θ u 0 + ( 1

θ

) c 0 ]

2

> 0 > 0

ˆ t t

( m ) !

q as m !

m .

( m ) is increasing for any m < m t and continuous.



February 12, 2016 37 / 49

Equilibrium value function I

Then:

V t

( m ) =

ασ

[ u ( q t

( m ))

φ

| t d

{z t

Buying

( m )] +

ασ

W t

( m )

}

Z

+

ασ

|

[ c ( q t

( ˜ )) +

φ t d t

(

{z

Selling

˜ )] dF t

( ˜ ) +

ασ

W t

( m )

}

+

αδ

([ u ( q ) c ( q

{z

Bartering

)] + W t

( m ))

}

+( 1 2

ασ αδ

) W t

( m )

| {z

Not trading

}



February 12, 2016 38 / 49

Equilibrium value function II

Grouping W t

( m ) and substituting its value: or

V t

( m ) =

ασ

[ u ( q t

( m ))

φ

Z t

+

ασ

[ c ( q t

( ˜ d t

(

)) + m

φ t

)] + d t

( ˜ )] dF t

( ˜ )

+

αδ

[ u ( q ) c ( q )]

+

φ t m + U ( X ) X + max m 0

φ t m

0

+

β

V t + 1 m

0

V t

( m ) =

ν t

( m ) +

φ t m + max m 0

φ t m

0

+

β

V t + 1 m

0 where v t

( m )

ασ

[ u ( q t

( m ))

φ

Z t

+

ασ

[ c ( q t

( ˜ d t

(

)) + m

φ t

)] d t

( ˜ )] dF t

( ˜ )

+

αδ

[ u ( q ) c ( q )] + U ( X ) X .



February 12, 2016 39 / 49

Equilibrium value function III

By repeated substitution:

V t

( m ) = v t

( m t

) +

φ t m t

+ max m t + 1

[

φ t m t + 1

+

β

V t + 1

( m t + 1

)]

= v

+ t j

( m

∞

∑

= t t

) +

β j t

φ t m t n max m j + 1

φ j m j + 1

+

β h v j + 1

( m j + 1

) +

φ j + 1 m j + 1 io

Let us focus on the maximization problem: n max m j + 1

φ j m j + 1

+

β h v j + 1

( m j + 1

) +

φ j + 1 m j + 1 io

The optimality condition is:

φ t

+

βφ t + 1

+

β v t

0

+ 1

( m t + 1

) = 0



February 12, 2016 40 / 49

De…nition (Monetary Equilibrium)

A monetary equilibrium is a list such that: f V t

, W t

, X t

, H t

, m t

0

, q t

, d t

,

φ t

, F t

, G t g t

∞

= 0

1 the value functions V t and W t and the decision rules satisfy the equations above for every t given f q t

, d t

,

φ

( t

X t

,

, F t

H t

,

, G t m g

0 )

∞ t = 0

;

2 the terms of trade ( q t

, d t

) in the decentralized market solve the problems above for every t given f V t

, W t g

∞ t = 0

;

3

4

φ t

R

> 0 for every t ; m t

0 ( m ) dG t

( m ) = M for all t ;

5 f F t

, G t g

∞ t = 0 is consistent with initial conditions and the evolution of money holdings implied by trades in the centralized and decentralized markets.



February 12, 2016 41 / 49

Some properties of the equilibrium I

Note that, for any m m t v t

( m )

α

(

σθ

+

δ

) [ u ( q ) c ( q )]

Z

+

ασ

[ c ( q t

( ˜ )) +

φ t d t

( ˜ )] dF t

( ˜ )

+ U ( X ) X and: v t

0 ( m ) = 0

In equilibrium,

φ t m t + 1 m t

βφ t + 1

. Otherwise, the agent can pick and the optimality condition

φ t

+

βφ t + 1

> 0 implies that the maximization problem does not have a solution.

This restriction allows the Friedman rule


φ t

=

β

.

φ t + 1


February 12, 2016 42 / 49

Some properties of the equilibrium II

Note that, for any m < m t v t

0 ( m ) =

ασ u

0

=

ασ u

0

( q t

( m )) q t

0

( q t

( m )) q t

0

( m )

φ t d t

0 ( m )

( m )

φ t

Then, coming back to the optimality condition:

φ t

βφ t + 1

=

β v t

0

+ 1

(

=

βασ u

0 m

( t q

+ t

1

+

)

1

( m t + 1

)) q t

0

+ 1

( m t + 1

)

φ t + 1 u

0 ( q t + 1

( m t + 1

))

=

βασφ t + 1

2 z 0 ( t + 1

( m t + 1

))

1

=

βασφ t + 1 u

0 [

θ u

0 + ( 1

θ

) c

0 ]

2 u

0 c

0 [

θ u

0 + ( 1

θ

) c

0 ]

+

θ

( 1

θ

) ( u c ) ( u

0 c

00 c

0 u

00 ) where in the last line we evaluate the u and c functions at q t + 1

( m t + 1

) .



February 12, 2016 43 / 49

3

1

7

Some properties of the equilibrium III

As m t + 1

!

m t + 1 from below, the slope of the objective function is:

"

φ t

βφ t + 1

=

[ u

0 ( q )]

2

βασφ t + 1

|

[ u 0 ( q )]

2

+

θ

( 1

θ

) [ u ( q ) c ( q )] [ c 00 ( q ) u 00 ( q )]

{z

Σ

Σ

0.

Σ

= 0 if and only if

φ t

Since

φ t

βφ t + 1

,

=

βφ t + 1 and

θ

= 1.

φ t

+

βφ t + 1

+

βασφ t + 1

Σ < 0 which means that any solution must satisfy to m t + 1 m t + 1

< m t + 1 by the left, the return function is already falling).

(i.e., close



February 12, 2016 44 / 49

1

#

}

Some properties of the equilibrium IV

By imposing conditions either on the bargaining power

θ or on preferences, one can also prove that v t

00

+ 1

< 0.

This ensures that the choice of m t + 1

= M is unique in equilibrium.

Therefore, we have:

φ t

=

β v t

0

+ 1

( M ) +

φ t + 1 or

φ t

=

β ασ u

0 ( q t + 1

( M )) q t

0

+ 1

( M ) + ( 1

ασ

)

φ t + 1



February 12, 2016 45 / 49

Some properties of the equilibrium V

In any monetary equilibrium, we have q t

2 ( 0 , q ) : for every t = 0 , 1 , 2 , ...

Then:

M = z ( q t

)

φ t

< z ( q )

φ t q t

0

( M ) = z 0

φ t

( q t

) and: m t

φ t

=

β v t

0

+ 1

=

βφ t + 1

( m t + 1

) +

φ t + 1 u

0

ασ z 0

(

( q q t t +

+

1

1

(

( m m t t

+

+

1

1

))

))

+ 1

ασ or z ( q t

) =

β z ( q t + 1

)

ασ u

0 z 0

( q t + 1

)

( q t + 1

)

+ 1

ασ



February 12, 2016 46 / 49

Stationary equilibria

A necessary and su¢ cient condition for a stationary equilibrium is:

1 =

β ασ u

0 z 0

( q )

( q )

+ 1

ασ or u

0 ( q ) z 0 ( q )

= 1 +

1

β

.

ασβ

When

θ

< uniqueness.

1, a stationary solution exists, but we cannot guarantee

In this case, we always have q e¢ cient.

< q and the steady state is not



February 12, 2016 47 / 49

Monetary policy I

New money injected as lump-sum transfers in the centralized market:

M t + 1

= ( 1 +

τ

) M t

, with a constant

τ

0.

New equilibrium condition: u

0

( 1 +

τ

) z ( q t

) =

β z ( q t + 1

)

ασ z 0

( q t + 1

)

( q t + 1

)

+ 1

ασ

Consider only stationary equilibria in which aggregate real balances are constant over time. Then

φ t

φ t + 1

= 1 +

τ and u

0 ( q ) z 0 ( q )

= 1 +

1 +

τ β

ασβ with

∂ q /

∂τ

< 0.



February 12, 2016 48 / 49

Monetary policy II

Thus, a higher steady state in‡ation rate implies a lower level of output in the decentralized market: a higher in‡ation rate raises the opportunity cost of holding money for transaction purposes, which reduces the demand for real money balances.

The Friedman rule is the optimal monetary policy. However, it delivers the e¢ cient outcome q only in the case

θ

= 1.

The quantity traded in the decentralized market is smaller than the socially e¢ cient level owing to a hold-up problem. An additional unit of money carried into the decentralized market will not reap its full return when the seller retains some of the bargaining power, creating an additional ine¢ ciency in the model.



February 12, 2016 49 / 49

Search-Theoretic Models of Money Jesús Fernández-Villaverde February 12, 2016 University of Pennsylvania

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Search-Theoretic Models of Money Jesús Fernández-Villaverde February 12, 2016 University of Pennsylvania

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