Are Markets without Friction Important or Exciting Discussion

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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 – it’s 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 gov’t de…cits stimulate the economy.
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NKE says low nominal interest rates are bad.
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NKE says in‡ation reduces unemployment.
NME theories generate can very di¤erent implications.
What’s 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
I
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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I
I
= ??(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
I
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” re‡ecting 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 won’t 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 I 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, let’s 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. I even with this restriction we can derive important results. I 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. I 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. I 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. I Some people interpret this to say that money is a substitute, albeit an imperfect substitute, for memory. I These results are easy here because the environment is simple – but the ideas are robust. I 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: I With credit you consume now and work o¤ debt later. I With money you must work …rst and then consume. I 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 I In the model above, that happens after one purchase, whence you must produce to get more before consuming again. I 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 buyers’are 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: I The mixed-strategy equilibrium ? = ? is not robust; it is an artifact of indivisibility. I The two pure-strategy equilibria are more robust. I Multiple equilibria are natural in models like this. I Importantly, this is not just about …at currency; it applies to any asset that facilitates trade. Big Ideas I Existence: There are equil where an asset is valued – people work to get it – even if ? = 0, contrary to standard …nance. I Robustness: Such equilibria survive even if ? < 0, at least as long as j?j is not too big. I Tenuousness: If ? 0, including the case of …at money, there exists an equil where the asset is not valued. I Volatility: Going beyond stationary outcomes, there are also equil where the asset value varies over time. I 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. I You work to get m because you believe others will work to get m from you, just like in the real word. I 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 . I This is obviously true for ? I So we can have asset bubbles, and they are good for welfare. I 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, gov’t 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 in‡ation-like taxes raise velocity.
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hot potato e¤ect: spending money faster to avoid in‡ation
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 in‡ation-like taxes raise search e¤ort and welfare.
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It is tricky to generalize to divisible goods/money, but still it’s
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 it’s 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 + ?
I
Z
ZI
I
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 I ci = ?i + ? (n? M ) ui (ri + ?n? ri + ?n? Rearranging, ? i = 1 is a BR for i i¤ n? ?i I ? (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 I Equil is a …xed point of n? = F (n? ). I Existence follows from Tarski’s Thm. I I might not need such a powerful tool here; I but we may as well use it once we’ve done the work to describe equil in terms of a threshold n? ? i . Easy to get various types of equil and multiplicity. I I the fraction of agents who accept M depends on the fraction who accept M! And now we can experiment with parameter changes: I that is not very interesting in baseline model since the robust equil involve conrner solutions, ? = 0 or ? = 1. Endogenous Acceptability Kiyotaki-Wright (JET): Barter I Goods come in varieties, agents have favorites, and utility of a good is u (z ) where z is distance from favorite, u 0 < 0. I before trading agent produces a good he doesn’t like. I prod opportunities arrive at rate ?0 and random cost c. I In meetings the z’s are independent and U [0, 1]. I Good is only storable by its producer (not critical). I 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 Lucas’suggestion, consider u (z ) = U if z otherwise, where U and x are parameters. x and 0 Then res. trade x is …xed, instead of sol’n 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 Smith’s 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 Smith’s idea that specialization is limited by the extent of the market, and that’s 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 Fed’s “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.
It’s 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 dist’n of U (w ) induced by the the dist’n 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