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We have studied the classical theory, or markets without frictions. Discuss in 3 pages (or less) whether markets without frictions are interesting and/or important. There are no right or wrong answers – the idea is simply to tell us some things that you learned.

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Monetary Economics I:

Basic Ideas and Models

Randall Wright

Reading:

1. Lagos et al (2017) Liquidity: A New Monetarist Perspective,

JEL.

2. Rocheteau and Nosal (2017) Money, Payments, and Liquidity,

2nd ed.

3. Wright el al (2021) Directed Search and Competitive Search

Equilibria: A Guided Tour, JEL.

4. Azariadis (1993) Intertemporal Macroeconomics.

Watching: Videos on Markets With Frictions.

TA will send you more information.

Grading: Homework.

Objective:

Discuss the principles and practices of recent research on money,

credit, banking, asset markets and liquidity.

Since the
nancial crisis, economists agree liquidity is important; in

this course we aim to model it rigorously.

Goals:

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Discuss a few methodological issues.

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Study simple stylized models to make conceptual points.

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Study models more suited to policy and data analysis.

Note: Some call this research New Monetarist Economics that is

not the same as Modern Monetary Theory its the opposite!

Principle

Kareken and Wallace (1980): Progress can be made in monetary

theory and policy analysis only by modeling monetary

arrangements explicitly.

This is not the belief of many macroeconomists.

In NME agents trade with each other, as in search theory, and not

merely with budget lines, as in classical GE theory or sloppy macro.

Trading process is hindered by frictions, like spatial or temporal

separation, limited commitment, and imperfect info.

We then ask how agents trade and study institutions meant to

ameliorate frictions, like money, banks, reputation, collateral…

Methodology

From the purpose of studying the exchange process, short-cut

models like CIA or MUF are pretty much useless.

NKE models are no better (they are actually worse).

Picking a method is critical for many policy issues:

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NKE says printing money stimulates the economy.

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NKE says govt de
cits stimulate the economy.

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NKE says low nominal interest rates are bad.

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NKE says ination reduces unemployment.

NME theories generate can very di¤erent implications.

Whats in a Name?

We agree with many (not all) traditional Monetarist tenets:

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LR is more important than SR (e.g., growth matters more

than cycles);

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Phillips curves do not determine (`, ? ), except …;

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central banks control ? and i, but not r , except ….

We consider misguided the way Keynesians:

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handle microfoundations in general;

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ignore (frictions in) the exchange/payment process;

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xate on sticky prices as the key and indeed often the only

factor in all theory, empirical work and policy analysis.

A Brief History of Thought

Back in the 1960s, it was a healthy situation when Friedman and

other Monetarists constantly challenged the Keynesian consensus.

Progress in the 70s and 80s seemed to render Keynesian macro

obsolete:

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New Classical Macro

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Rational Expectations

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Real Business Cycle Theory

At some point there arose a New Keynesian consensus, more

technical but otherwise much like Old Keynesian macro.

Too much consensus is unhealthy; it should be challenged; we try

to communicate an alternative approach.

Generation 1: A Simple Model of Liquidity

The Environment (based on Kiyotaki-Wright 91,93)

1. Time is discrete and continues forever .

2. A large set of agents, say [0, 1], prod and cons specialized

goods that for now are indivisible and nonstorable .

3. Preferences: utility of cons u > 0, disutility cost of prod

c 2 [0, u ), discount factor ? = 1/ (1 + r ).

4. Agents meet bilaterally and at random at rate ?.

5. Specialization: prob (SC) = ? and prob (DC) = ?, where SC

is a single-coincidence and DC a double-coincidence meeting.

Each feature with a

merits extended discussion!

Regime 1: Barter

Let V A = 0 and V B be the value functions (life-time, expected,

discounted payo¤s) under autarky and barter.

As barter requires a DC meeting, the standard DP eqn is

h

i

V B = ? ??(u c + V B ) + (1 ??)V B

where u, c and continuation value are all discounted wlog.

Simpli
cation yields the ow DP eqn, which is nice since it actually

holds in discrete or continuous time (see Tech App below):

rV B = ?? (u

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c)

Barter captures some social gains from trade if ? > 0, but

also misses some if ? > 0.

Technical Appendix : The Flow DP Eqn, page 1 of 2

One way to derive the ow DP eqn is to go from discrete to

continuous time. Let time proceed in increments dt > 0, so

t = 1, 1 + dt, 1 + 2dt… Assume meetings occur according to a

Poisson process with arrival rate ?. This means that 8t,

independent of history, the prob of 1 arrival between t and t + dt

is approximately ?dt, written ?dt + o (dt ), where o (dt ) is a

function such that limdt !0 o (dt ) /dt = 0. Hence, multple

meetings are possible, but very unlikely when dt is small. Writing

the discount rate as ? = 1/(1 + rdt ), we clearly have

VtB =

?dt? u

c + VtB+dt + (1 ?dt?)VtB+dt + o (dt )

1 + rdt

with o (dt ) capturing Poisson approximation error for dt > 0.

Technical Appendix : The Flow DP Eqn, page 2 of 2

Multiplying by 1 + rdt and subtracting VtB from BS, we get

rdtVtB = ?dt? (u

c ) + VtB+dt

VtB + o (dt ) .

Dividing BS by dt and taking dt ! 0, we get

rVtB = ?? (u

c ) + V?tB ,

where V?tB = dVtB /dt (the time derivative). Imposing stationarity,

V? B = 0, and ignoring t subscripts, we get rV B = ?? (u c ).

Continuous time is elegant, but we can get the same expression in

discrete time: simply let dt = 1 and directly assume the prob of 1

meeting per period is ? while the prob of more than 1 is 0, rather

than having that a result from Poisson arrivals.

Regime 2: A Stylized Credit System

Let V C be the value function under perfect credit, where agents

produce whenever asked. Then

rV C

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= ??(u c ) + ??u

= ? ( ? + ? ) (u c )

??c

Notice ? > 0 ) V C > V B so credit beats barter.

So if agents can commit they would commit to perfect credit.

That captures all gains from trade, given the search and

matching frictions, parameterized by ?, ? and ?.

Credit Without Commitment?

Following Kehoe-Levine, if agents can renege on promises, credit

works i¤ it satis
es the IC condition:

c +VC

µV D + (1

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V D = deviation payo¤

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µ = prob a deviation is detected.

µ) V C

Note: µ < 1 can be interpreted as imperfect memory reecting
frictions in monitoring, communication, or record keeping.
Note: µ < 1 hinders credit which may give a role to liquid assets.
__________________________
Note: In principle there are IC conditions for other trades, e.g.,
barter, but they wont bind.
Punishing Bad Behavior
To encourage good behavior (produce when asked) we must punish
deviants by either:
1. denied them future trade, V D = V A ;
2. denied them credit trade, V D = V B .
In cases 1 and 2, IC reduces to
r
µ?? (u
c
c)
r?C and r
µ?? (? + ?) (u
c
c)
r?C
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note r?C > r?C because harsher punishment is better

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but case 1 requires barter cannot be hidden

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either way credit requires (r , c ) small and (?, ?, u, µ) big.

A Role for Money

If µ is low, naturally, credit is not viable, in which case let us

consider money.

Money is a storable and transferable asset produced by society at

low cost, lets say 0, but not by individuals.

As it is useful later, we endow the asset with return ?:

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? > 0 ) the standard Lucas tree from
nancial economics;

? < 0 ) a bad asset, e.g. one with a storage cost;
? = 0 ) the theoretically pure case of
at money.
Note: The pure
at case may never have existed until recently,
with the advent of e-money, like bitcoin or CBDC.
Properties of Money
Traditionally, money is said to be a store of value, unit of account,
and medium of exchange, but clearly its salient role is as a medium
of exchange, also called a means of payment.
A good money should have (or tends to have?) properties like
storability, portability, transferability, recognizability, and divisibility.
For now it has all but the last i.e., money is indivisible and
agents can only store m 2 f0, 1g.
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even with this restriction we can derive important results.
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note it is an assumption on the storage technology, not on
behavior; also, in some models it can be endogenized.
If M is the money supply, a fraction M of agents, called buyers,
have m = 1, while 1 M, called sellers, have m = 0.
When is Monetary Exchange Viable?
Assuming ?, like the payo¤s from trade and continuation values, is
discounted, we have
rV0 = ??(u
c ) + ??M ( c + V1
rV1 = ??(u
c ) + ??(1
V0 )
M ) ( u + V0
V1 ) + ?
Assuming j?j is not too big, the key IC condition is for sellers to
produce in exchange for money, c + V1 V0 , or
r
(1
M ) ?? (u
c
c) + ?
r?M .
Consider ? = 0 (
at currency) and ask, when is money essential?
Here essential means socially useful i.e., we can support better
outcomes with money than without it.
When is Monetary Exchange Essential?
Here are versions of important results (e.g., Kocherlakota 1998):
Prop: If µ = 1
at money cannot be essential.
Proof: It is easy to check r?M < r?c . Hence, if we can can support
monetary exchange we can also support perfect credit, and the
latter is better i.e., it delivers higher payo¤s.
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Why is credit is better than money? In a monetary regime,
trade fails in single-coincidence meetings if the (potential)
consumer has m = 0 or producer has m = 1.
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In some speci
cations (e.g., directed rather than random
search), for some parameters money may do as well as credit,
but it cannot do better.
Necessary Conditions for Essentiality
Prop: Assume µ < µ , where µ 2 (0, 1); then if
at money is
viable it is essential.
Proof: If µ is low then credit is not viable. Money is viable if
r r?M , independent of µ. So for some parameters credit cannot
work, but money can, and while not as good as credit it beats
barter.
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Some people interpret this to say that money is a substitute,
albeit an imperfect substitute, for memory.
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These results are easy here because the environment is simple
but the ideas are robust.
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This can be a model of e-money although it abstracts from
mining (for Bitcoin, and also for monies like gold), and uses
m 2 f0, 1g, which we relax later.
An Aside
Is Bitcoin, or related instruments, money or credit?
They require signi
cant record keeping, like credit, but consider
the following:
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With credit you consume now and work o¤ debt later.
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With money you must work
rst and then consume.
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This is especially relevant with search and matching frictions,
because it is uncertain when you get to spend money.
A related point is that you can run out of money
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In the model above, that happens after one purchase, whence
you must produce to get more before consuming again.
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And you cannot run out of credit (in other models you might).
Summary of the Approach and Results
So far we studied the role of money in implementing good
outcomes, given incentive problems, in the spirit of mechanism
design, to endogenize institution of monetary exchange.
We learned several elements are key to making money essential:
1. a single-coincidence problem;
2. limited commitment;
3. imperfect information (memory, monitoring, etc.).
Some e.g. Wallace argue we should only analyze monetary
issues and policy in environments where money is essential.
This suggests any good model must contain elements 1-2 above,
but we can relax auxiliary assumptions, like indivisibility.
Equilibrium Analysis
Before relaxing assumptions, in the same environment consider
equilibrium outcomes, in the spirit of noncooperative game theory.
Let ? denote a trading strategy, which here is simply
? = prob (seller will produce to get m )
Taking as given others ??, an individual chooses a best response ?,
and we look for Nash equilibrium, ? = ??.
__________________________
Note: As in the previous analysis, in principle we should ask if
buyersare willing to go from m = 1 to m = 0 to get consumption,
but that is a dominant strategy if ? is not too big.
When to Accept Money?
rV0 = ??(u
c ) + ??M max ? ( c + ?)
rV1 = ??(u
c ) + ?? (1
?
M )?? (u
?) + ?
where ? V1 V0 , and: c < ? ) ? = 1; c > ? ) ? = 0; and

c = ? ) ? = [0, 1].

Lemma: 9? such that: ?? > ? ) ? = 1; ?? < ? ) ? = 0; and
?? = ? ) ? = [0, 1].
Exercise: Find ? in terms of parameters, give conditions such that
0 < ? < 1, and compare them to the conditions derived above
that make monetary exchange viable.
Nash Equilibrium Acceptability
Figure: Best Response Correspondence with 3 Equilibria
Monetary Equilibrium
Prop: Consider ? = 0 and 0 < ? < 1. There are three Nash
equilibria a nonmonetary equil ? = 0; a pure monetary equil
? = 1; and a mixed monetary equilibrium ? = ? .
Homework: What happens if ? > 1 or ? < 0? What if ? 6= 0?
Remarks:
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The mixed-strategy equilibrium ? = ? is not robust; it is an
artifact of indivisibility.
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The two pure-strategy equilibria are more robust.
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Multiple equilibria are natural in models like this.
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Importantly, this is not just about
at currency; it applies to
any asset that facilitates trade.
Big Ideas
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Existence: There are equil where an asset is valued people
work to get it even if ? = 0, contrary to standard
nance.
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Robustness: Such equilibria survive even if ? < 0, at least as
long as j?j is not too big.
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Tenuousness: If ? 0, including the case of
at money, there
exists an equil where the asset is not valued.
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Volatility: Going beyond stationary outcomes, there are also
equil where the asset value varies over time.
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E¢ ciency: When there are multiple equilibria they can be
ranked in terms of welfare.
More Big Ideas
Assets can be valued even if ?
0 because they convey liquidity.
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You work to get m because you believe others will work to get
m from you, just like in the real word.
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The self-referential aspect of such beliefs explains why models
of liquidity tend to have multiple equil.
Liquidity ) asset can be valued above its fundamental ?/r .
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This is obviously true for ?
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So we can have asset bubbles, and they are good for welfare.
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It is of course bad if a bubble bursts e.g., ? goes from 1 to 0
but nothing says it must.
0, but also true for ? > 0.

Still More Big Ideas

Contrary to CIA and NKE models, here money is not a problem, its

a solution:

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It ameliorates problems related to spatial/temporal separation,

limited commitment and incomplete information.

But it is an imperfect solution:

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Credit would be better, if it were viable;

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Monetary equil can be fragile, tenuous or volatile.

There are policy implications e.g., related to ? but discussion is

postponed until we relax some assumptions.

Also, above we considered money or credit, but it is interesting to

have models with both.

Application: Inside vs Ouside Money (Cavalcanti and Wallace)

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Fraction n of agents are type B (bankers) while the rest are

type A (anonymous):

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type A are never monitored;

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type B are monitored in all meetings.

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Agents can issue notes, pieces of paper with their names.

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Notes issued by A are never accepted why produce to get a

note when you can print your own?

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Notes issued by B might be thus they resemble banks.

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For illustration, suppose B never hold outside money (not

crucial).

Inside vs Ouside Money (Cavalcanti and Wallace)

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We can support outcome where B produces whenever asked,

with autarky punishment, if

r

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? (1

n )(1

M ) (u

c ) /c.

With no inside money B only consumes when meeting another

B, so rV B = ? (nu c ), while for A

rV0A

rV1A

= ?nu + ? (1

= ?nu + ? (1

n ) (1

M ) ( u + V1

n ) M ( c + V0

V0 )

V1 ) .

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With inside money, B pay A by issuing notes so W is higher.

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Not surprising but at least we can discuss relative merits of

di¤erent arrangements.

Extension: Victor Li (IER,JME)

Add search intensity, govt taxes and transfers:

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at rate ? take away m; at rate ?M/ (1

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buyers choose ?1 = ?, so sellers get ?0 = ?M/(1

M ) give it back

M)

rV1 = ? (V0 V1 ) + ? (u + V0 V1 ) k (?).

?M

?M

( c + V1 V0 )

rV0 =

( V1 V0 ) +

1 M

1 M

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Combine FOC k 0 (?) = u + V0

T ( ? ) = [r (1

V1 with (V0 , V1 ) to get

M ) + ? + M?] u

[r (1

M?c + (1

M )k ( ? )

0

M ) + ? + ?] k (?) = 0

where reasonable conditions imply T has nice properties.

Policy Implications of Li?

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Under these conditions 9! equil 8?

that makes c + V1 V0 = 0.

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In equil ??/?? > 0, so ination-like taxes raise velocity.

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hot potato e¤ect: spending money faster to avoid ination

Optimal ? is k 0 (? ) = u

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?? where ?? is the tax

c and equil is e¢ cient i¤ ? = ??

Hosios condition: buyers equate MC of search to their MB,

which is below social MB unless they get the whole suplus.

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Key result is ination-like taxes raise search e¤ort and welfare.

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It is tricky to generalize to divisible goods/money, but still its

a nice example of a classic issue in monetary econ requiring

search theory how else do we model spending money faster ?

Extension: Equilibria for Any ?

Above we assumed j?j is not too big; here is the equil set for the

general case where we endogenize:

? 0 = pr (seller trades good for m ) and ? 1 = pr (buyer trades m for good)

Extension: Partial Acceptability

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Above ? 2 (0, 1) has partial acceptability but its not robust

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not an equil if goods or assets divisible or we allow lotteries

unstable wrt trembles, evolution, etc.

Shevchenko and Wright (ET) assume individuals are heterog

wrt (ri , ui , ci , ?i ), i 2 I.

R

Let ? i = pr (i accepts M ), mi = pr (i has M), n? = I ? i .

ri Vi 0 = B + ?

ri Vi 1 = B + ?

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Z

ZI

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mj ? i (?i

(1

ci ) dj

mj ) ? j (ui

?i ) dj + ?i .

In ss mi = M/n? 8i, assuming M < n? (this maybe can
endogenized by allowing agents to throw away money).
Equilibrium Acceptability
ri ?i = ?i + ? (1
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ci =
?i + ? (n?
M ) ui (ri + ?n?
ri + ?n?
Rearranging, ? i = 1 is a BR for i i¤ n?
?i
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? (M/n? ) ? i (?i
ci ) n?
It is a BR for i to set ? i = 1 i¤
?i
I
?i ) n?
M/n? ) (ui
?M (ui
ci ) + ri ci
? ( u i ci )
If CDF of ? i is F (? ) = prob (? i
accept M is n? = F (n? ).
?M ) ci
0
? i where
?i
.
? ), then the measure who
Endogenous Acceptability
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Equil is a
xed point of n? = F (n? ).
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Existence follows from Tarskis Thm.
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might not need such a powerful tool here;
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but we may as well use it once weve done the work to describe
equil in terms of a threshold n? ? i .
Easy to get various types of equil and multiplicity.
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the fraction of agents who accept M depends on the fraction
who accept M!
And now we can experiment with parameter changes:
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that is not very interesting in baseline model since the robust
equil involve conrner solutions, ? = 0 or ? = 1.
Endogenous Acceptability
Kiyotaki-Wright (JET): Barter
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Goods come in varieties, agents have favorites, and utility of a
good is u (z ) where z is distance from favorite, u 0 < 0.
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before trading agent produces a good he doesnt like.
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prod opportunities arrive at rate ?0 and random cost c.
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In meetings the zs are independent and U [0, 1].
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Good is only storable by its producer (not critical).
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Given X = prob(random agent wants to trade), choose res.
cost k and res. trade x, k = u (x ) = Vg V0 :
rV0 = ?0
rVg
Z k
= ?1 X
(k
c ) dF (c )
0
Z x
0
[u (z )
u (x )] dz
Kiyotaki-Wright (JET): Money
Prop: 9! barter equil where X = x, making ? = x 2 endogenous.
Small x implies very small x 2 , motivating a role for money:
rV0 = ?0
rVg
rVm
Z k
(k
c ) dF (c )
0
= ?1 (1
n? ) X
= ?1 (1
n? ) ??
Z x
Z
[u (z )
u (x )] dz + ?1 n?Y max ? (Vm
?
0
y
[u (z )
u (y )] dz
0
where in equil n? = nm / (nm + ng ), x = X , y = Y and ? = ??.
Prop: Given any M there 9 n? consistent with ss, and given n?, 9!
equil with ? = 1. In equil x 2 < y < x.
Vg )
Simpli
ed Kiyotaki-Wright (AER)
Following Lucassuggestion, consider u (z ) = U if z
otherwise, where U and x are parameters.
x and 0
Then res. trade x is
xed, instead of soln to u (x ) = Vg
V0 .
If we also make c nonrandom, and small, then
rV0 = ?0 (Vg
rVg
rVm
2
V0
c ) dF (c )
= ?1 x (1 n?) (U + V0 Vg ) + ?1 n?x ? (Vm
= ?1 (1 n?) x ?? (U + V0 Vm )
Vg )
Prop: Given M there are exactly three equil: ? = 1; ? = 0; and
? = x. Under reasonable conditions, equil can be ranked in terms
of welfare.
Specialization and E¢ ciency: Kiyotaki-Wright (AER)
Adam Smiths connection between specialization and e¢ ciency:
you choose output per unit time ?0 , but high ?0 means low prob x
that your produce is desired by others.
rV0 = max ?0 (x ) [Vg (x )
x
rVg (x ) = ?1 xX (1
rVm
= ?1 (1
V0
n? ) (U + V0
n? ) X ?? (U + V0
c ] dF (c )
Vg ) + ?1 n?x ? [Vm (x )
Vg ]
Vm )
Adam Smiths idea that specialization is limited by the extent of
the market, and thats where money really helps:
Prop: There are three equil, ? = 1, ? = 0 and ? = x, with high,
low and medium ?0 .
Prop: When ? = 1 and ?1 ! ?, we get complete specialization
x = 0, and all trade is monetary, ?1 x > 0 while ?1 x 2 = 0.

Lectures on

Markets with Frictions

by

Randall Wright

Part I: Labor Markets

Introduction

Search theory provide a novel, compared to classical economics,

way to study markets with frictions.

Labor markets are just one application, but a very important one:

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the economic fortunes of most people are largely determined

by labor market outcomes;

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this includes employment-unemployment experiences and

wages

Also, many policy makers focus a lot (perhaps too much) on

unemployment e.g., the Feds “dual mandate.”

Motivation

The classical tools of supply and demand analysis are very useful

for understanding some aspects of labor markets e.g., the e¤ects

of taxation.

But they are limited:

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not consistent with coexistence of unemployment and

vacancies;

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not consistent with long and variable durations of

unemployment and vacancy spells

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not consistent with wage dispersion for similar workers/jobs.

Its doubtful classic S and D analysis has anything at all to say

about equilibrium unemployment or vacancies.

Even it we allow w to be too high and call S-D unemployment, or

too low and call D-S vacancies, we cannot get both at once.

And certainly we cannot get wage dispersion.

Big Ideas

Search theory is designed to study random processes of workers

nding and losing jobs, and
rms
nding and losing employees.

It builds on the notion that it takes time and other resources to

nd most things, including a desirable house, spouse, car, … and,

of course, a job if you are a worker, or a worker if you are a
rm

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Early search models were concerned with individual activities

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Modern search models integrate individual activities into

general equilibrium settings.

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Importantly, frictions allow deviations from the “law of one

price” in standard GE theory.

More Big Ideas:

Search theory is also one of the few ways in which economists

study agents trading with each other:

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in standard GE theory agents simply trade along their budget

lines at prices they take as given.

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there is no notion of who works for whom there is only

aggregate S and D of labor,

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prices are determined by the theory i.e., by us as economists

to clear the market, not by the individuals in the model

Search is an very natural way to study decentralized markets

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frictions make things interesting!

A Decision Problem

Consider an unemployed individual looking for a job.

Assume time is discrete time his horizon is in
nite, t = 1, 2, …

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this is a good simpli
cation for many purposes.

Assume he wants to maximize the PV of life-time earnings with

discount factor ? = 1/(1 + r ), where r > 0.

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income is w if employed at w (generally, this can include

wages, plus bene
ts, commuting distance, etc.)

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income is b if unemployed (generally, this can include UI

bene
ts, plus home production, value of leisure, etc.)

One interpretation is risk neutrality; there are others, including

complete insurance markets.

A Digression on Probability Theory

Consider a random variable x. It is called a discrete random

variable if it takes values in a set X = fx1 , x2 , …g, called the

support, each with some probability. We write this as x = x1 with

prob ? 1 , x = x2 with prob ? 2 , … and, in general, x = xj with prob

? j . An example is shown in the left panel of the diagram below.

The probabilities are such that ? j

0 for all j, and typically

? j > 0 for all j (if ? j = 0 for some j we can eliminate xj from X ).

The support X may contain a
nite or an in
nite number of

elements, but in any case the law of total probability is ?j ? j = 1.

We are often interested in certain statistics, such as the mean, or

expected value, denoted Ex and de
ned by

Ex = ? ? j xj .

j =1

Figure: A Discrete (left) and a Continuous (right) Random Variable

A continuous random variable has support with a continuum of

values, e.g. X = [x, x ], where x and x > x are the lower and

upper bounds of an interval. We are not so interested in the

probability of each potential realization x, but instead in the

probability density function, or pfd, f (x ). It tells us the probability

of certain events, e.g., the probability that x is in a subinterval

(x1 , x2 ) X is given by the area under the curve between x1 and

x2 , as shown in the right panel. That area is given by the integral,

prob (x1

x

x2 ) =

Z x2

f (x ) dx.

x1

An integral is a generalization of a sum: partition (x1 , x2 ) into

subintervals; take a value of x in each subinterval and compute an

area, f (x ) times the length of the subinterval; sum up these areas;

repeat taking smaller subintervals; then continuing in this way, in

the limit we get the integral.

A special case is an interval [x, x? ], starting at lower bound x and

ending at x? > x, so prob (x x x? ) = prob (x x? ) = F (x? ),

where F ( ) is called the called the cumulative distribution

function, or cdf. It is the area under f ( ) to the left of x?:

Z x?

F (x? ) =

f (x ) dx.

x

So at any x the pdf is the derivative of the cdf, f (x ) = F 0 (x ).The

law of total probability is now

F (x ) =

Z x

f (x ) dx = 1,

Z x

xf (x ) dx.

x

and the mean is given by

Ex =

x

Some random variables are neither discrete nor continuous e.g.,

x can take values in a set fx1 , x2 , …g with some probability, and

values in an interval [x, x ] with complimentary probability. In any

case, we still have a cdf F (x? ) = prob (x x? ). Indeed, in general a

cdf describes any random variable x, including the discrete case,

the continuous case, and combination cases. In general we write

the expectation as

Z

Ex =

x

xdF (x ) .

x

If F ( ) is di¤erentiable then dF (x ) = F 0 (x ) dx = f (x ) dx, so the

expectation of a continuous random variable is a special case. With

a little more e¤ort, one can see that the expectation of a discrete

random variable is a special case, too.

For present purposes, we do not need all the power of formal

probability theory, but we will use some of the notation and idea,

in particular the idea of the mean. In fact, we are interested not

only in the mean or expected value of x, but in the expected value

of functions of x. For instance, if x is consumption and U (x ) is a

utility function, when x is random expected utility EU (x ), or more

concisely just EU, is given by

EU =

Z x

U (x ) dF (x ) .

x

Note that we sometimes write the above as Ex U to make it clear

that x is the random variable.

Also note that the expectation operator is linear i.e., for any

constants a and b, E(a + bx ) = a + bEx.

Important Concept: An agent is said to be risk neutral when his

utility function U (x ) is linear, meaning he only cares about the

mean: EU (x ) = U (Ex ). He is strictly risk averse when U (x ) is

strictly concave, in which case EU (x ) < U (Ex ) for any genuinely
random x (i.e., it has more than one value in the support, so where
there is actually some risk). And he is strictly risk loving when
U (x ) is strictly convex, in which case EU (x ) > U (Ex ) for any

genuinely random x.

In economics and
nance it is often assumed that agents are risk

averse, or, sometimes, risk neutral. But even if their direct utility

function is concave, their indirect utility (value) function can be

convex, as we will see below, making them prefer more risk.

From the graph this is clear: EU (x ) < U (Ex ) (i.e., risk aversion)
is the same as U (x ) concave.
As a limiting case, EU (x ) = U (Ex ) (i.e., risk neutrality) is the
same as U (x ) linear.
Back to the Decision Problem
Assume one job o¤er w arrives each period, which come as i.i.d.
random draws from a known cdf F (w ).
I
if multiple o¤er arrive in a period, simply interpret F (w ) as
the cdf of a new random variable, the best o¤er.
I
of course learning about F (w ) is important, but that is a
complication best ignored for now.
Also assume:
I
rejected o¤ers are lost forever (no recall);
I
accepted o¤ers entail permanent employment (no quits);
I
in fact, these restrictions are not binding, given our other
assumptions.
A Dynamic Programming Formulation
Let W (w ) be the (present) value of accepting w .
Let U the value of rejecting it, which does not depend on w given
our assumptions.
Due to stationarity, we can write these recursively as follows:
W (w ) = w + ?W (w )
U
= b + ?Ew? max fW (w? ), U g
Here Ew? is the expectation of the value of the next o¤er w? , and
max captures the idea that w? can be accepted or rejected.
Hint: to remember notation, W and U stand for "working" and
"unemployed."
Optimal Search Behavior
First it is clear that
W (w ) = w + ?W (w ) ) W (w ) =
w
1
?
,
which actually should be obvious, given accepted o¤ers entail
permanent employment.
As W (w ) is strictly increasing and U is independent of w , there is
a unique w , called the reservation wage, where W (w ) = U.
Then the optimal search strategy is clearly:
w < w ) reject and w
w ) accept
Hence, while the problem may seem di¢ cult, the solution is
conceptually simple!
The optimal search strategy has a simple representation in terms of
w , although we still have to solve for w .
An Aside on Risk
Before solving for w , notice something interesting in the above
diagram: the value of the next o¤er maxfW (w ), U g is convex.
This means that agents like risk!
The reason is simple: they have the option to reject low w while
accepting high w , and a more disperse o¤er distribution lets them
better take advantage of that.
Now if they actually get utility U (w ) and U (b ) while employed
and while unemployed, this e¤ect is tempered by risk aversion
captured by assuming U ( ) is concave.
But we can still say they like risk in they distn of U (w ) induced
by the the distn of w .
Finding w
Notice b + ?Ew? max fW (w? ), U g = w / (1
w
= (1
? ) b + (1
= (1
? ) b + (1
?), and so
?) ?Ew? max fW (w? ), U g
w?
w
?) ?Ew? max
,
1 ? 1 ?
Cancelling 1 ? in the last term, we have w = T (w ), where T
is a nice function i.e., a contraction given by
T (w ) = (1
?) b + ?Ew? max fw? , w g .
In other words, w is the solution to T (w ) = w .
One might ask, is there a solution to T (w ) = w ? If so, is it
unique? And, how can we
nd it?
Since T (w ) is a contraction, w = T (w ) has a unique solution,
and it can be found it by iterating on wn +1 = T (wn ) for any w1 .
In fact, if w1 = b then wn is the reservation wage when there are n
periods left to search, and wn ! w as n ! ?.
Another Version of the Solution
Subtracting ?w from BS of w = T (w ) and simplifying, we get
w =b+
?
1
?
Ew? max fw?
w , 0g ,
R w?
Using ?/ (1 ?) = 1/r and Ew? g (w? ) = 0 g (w? )dF (w? ) for any
function g ( ), we get an eqn often seen in the literature
w =b+
1
r
Z w?
(w?
w )dF (w? ),
w
where max is taken into account by the lower limit of the integral,
and w? denotes the upper bound.
Intuitively, the LHS is the opportunity cost of rejecting w , while
the RHS is the bene
t, b plus the value of further search, which is
the discounted expected gain from seeing another o¤er.
Some Simple Experiments
From the reservation wage eqn, it is easy (if you know calculus, as
discussed below in a Technical Digression) to derive:
?w
?w
> 0 and

> 0.

?b

??

Intuitively, when you
nd unemployment less painful (higher b) or

you are more patient (higher ?) you become more selective.

In particular, higher UI bene
ts make agents less inclined to accept

o¤ers, so they have longer unemployment spells on average.

But of course this does not mean they are worse o¤!

Can one even contemplate such questions using S and D analysis?

Generalizations

Now suppose o¤ers arrive with probability ?

unemployed.

1 each period while

And workers lose their job (e.g., get laid o¤) each period with

probability ? 1 while employed.

The previous methods imply there is again a reservation wage w .

If we also assume, mainly to ease notation, that w and b accrue at

the end of each period, the re