Economics 546 - Faculty and Staff | Faculty of...

Economics 546Lagos and Wright (2005)

Geoffrey Dunbar

UBC, Winter 2013

January 30, 2013

Geoffrey Dunbar (UBC, Winter 2013) Economics 546 January 30, 2013 1 / 51

The Workhorse Model

Probably fair to say that this is the current workhorse model ofsearch-money demand.It accomplishes two important things.

1 Money holdings are divisible - this comes at the cost of a degeneratedistribution of money holdings. So prices come at the cost ofheterogeneity (at least analytically).

2 Describes money as a vehicle to specialization. Money is essentialhere to buy general goods from the sale of specialized goods.Anonymity is important.


A Day and Night

A key mechanism used in the paper is to break a period into twosub-periods.In one sub-period there is a centralized market.In the other there is a decentralized market.


Environment)

Time is discrete.There is a continuum of consumers with unit measure.Consumers are infinitely lived and have discount factor β.Preferences are given by:

U(x , h,X ,H) = u(x)− c(h) + U(X )− H

where x (X ) is consumption during the day (night) and h (H) is laboursupplied during the day (night).Note the quasi-linearity of the preferences. This is important.Quasi-linearity in either X or H yields a degenerate distribution of moneyholdings. They need this for analytical tractability.


Environment

There are some technicalities required in terms of the functional forms.We need:

u′ > 0, c ′ > 0, U ′ > 0

u′′ < 0, U ′′ < 0

c ′′ > 0

u(0) = c(0) = 0

∃q∗ ∈ (0,∞) such that u′(q∗) = c ′(q∗)∃X ∗ ∈ (0,∞) such that U ′(X ∗) = 1 with U(X ∗) > X ∗

As usual, the reasons for these restrictions will only become apparent overtime.


Day Market

The first sub-period is the day market. This operates as a decentralizedmarket with bilateral exchange.Let α be the probability of meeting another agent.The day good, x , is a differentiated good that comes in many varieties.Each agent can produce a a good x using labour effort one-for-one.Agents cannot consume their own good.


Day Market

Consider two agents, i and j meeting. There are four possibilities.

1 Let δ be the probability of a double coincidence of wants.

2 Let σ be the probability of a single-coincidence of wants (this issymmetric).

3 Then 1− 2σ − δ is the probability of neither agent wanting theother’s good.


Night Market

In the night market, agents trade in a centralized market. In a Walrasianmarket, it doesn’t matter if goods are specialized or not as long as marketclearing can happen. So just call the night good a general good.In the night market, the general good can also be produced one-for-oneusing labor.


Goods

Both the special goods and the general good are non-storable but perfectlydivisible.There is also something called money, M, which is also divisible andstorable in any quantity.Money is essential here because anonymity in the day market rules outcredit and it is needed for exchange in the night market.Thus anonymity and storage drive the demand for money.


Equilibrium

Define Ft(m̃) be the measure of agents at time t starting the day marketwith money holdings m̃ ≥ m.Similarly, let Gt(m̃) be the measure of agents at time t starting the nightmarket.Assume the initial distributions at t = 0 are given exogenously.If the money stock is fixed, which we shall assume for now, then:∫

mdFt(m) =

∫mdGt(m) = M,∀t


Equilibrium

Let φt be the goods price of money in the centralized night market.Assume there is no uncertainty except for the random matching probability.This implies that aggregate variables are taken as given by the agents.So the only factor relevant for decision making is money holdings. It is thestate variable.


Equilibrium

As usual, we define the value functions for each possible state in which anagent may find himself.Let Vt(m) be the value of entering the day market with m money holdingsat time t.Similarly, let Wt(m) be the value of entering the night market with mmoney holdings at time t.Define m as the buyer’s money holdings and m̃ as the seller’s moneyholdings.Define Bt(m, m̃) be the payoff from a double coincidences of wantsmeeting.Let q(m, m̃) be the quantity of goods exchanged and d(m, m̃) the quantityof money.


Equilibrium

Then:

Vt(m) = ασ

∫{u(qt(m, m̃)) + Wt(m − dt(m, m̃))}dFt(m̃)︸︷︷︸

buying

+ασ

∫{−c(qt(m, m̃)) + Wt(m + dt(m̃,m))}dFt(m̃)︸︷︷︸

selling

+αδ

∫Bt(m, m̃)dFt(m̃)︸︷︷︸

bartering

+ (1− 2ασ − αδ)Wt(m)︸︷︷︸nottrading

(1)


Centralized Market

In the centralized market, agents solve:

Wt(m) = maxX ,H,m′

{U(X )− H + βVt+1(m′)}

subject to:

X = H + φtm − φtm′

X ≥ 0

0 ≥ H ≥ H̄

m′ ≥ 0

We will assume an interior solution.


Terms of Trade

Let’s imagine that two agents meet and satisfy double-coincidences ofwants.Given their preferences, we ask how much will they give each other? Sincethe goods are non-storable then it must be that:

u′(q∗) = c ′(q∗)

i.e. the agents maximize by equating the marginal utility of consumptionwith the marginal disutility of production. Since these agents are otherwiseidentical then they trade q∗ goods. This implies:

Bt(m, m̃) = u(q∗)− c(q∗) + Wt(m)


Single Coincidence of Wants

In a single coincidence of wants, assume that the buyer has bargainingpower, θ, and that threat points are given by continuation values. Then,the generalized Nash solution implies that (q, d) maximizes:

(u(q) + Wt(m − d)−Wt(m))θ(−c(q) + Wt(m̃ + d) + Wt(m̃))1−θ

subject to d ≤ m and q ≥ 0.


Equilibrium Definition

This is standard. Basically, prices and quantities must satisfy individualmaximization, market clearing and aggregation.


Point of the Next Part

What we do now is to use the centralized market ‘equilibrium’ to informthe decentralized ‘equilibrium.’ In particular, Lagos and Wright show thatmoney holdings are degenerate – everyone holds the same m′.


Centralized Market

Using H = X − φtm + φtm′ into Wt(m) yields

Wt(m) = φtm + maxX ,m′{U(X )− X − φtm′ + βVt+1(m′)}

By implication, U ′(X ∗) = 1 gives the maximizing choice of X .Also, note that m′t does not depend on initial money holdings, m.Finally, dWt(m)/dm = φt . This is linear.These equations suggest that the agents leaving the centralized market arelikely to choose the same m′. This is the degeneracy.


Nash Bargaining

Since Wt(m) is part of the bargaining solution for the single-coincidence ofwants then we can rewrite that as:

maxq,d{(u(q)− φtd)θ(−c(q) + φtd)1−θ}

Ignoring for the moment the restriction d ≤ m and taking first-orderconditions yields:

θ(−c(q) + φt(d))u′(q) = (1− θ)(u(q)− φtd)c ′(q))

and

θ(−c(q) + φt(d)) = (1− θ)(u(q)− φtd)

Consequently, u′(q) = c ′(q) which by our assumptions yields q∗.


Nash Bargaining

Similarly, we can rewrite to solve for d in which case we should find:

d =θc(q∗) + (1− θ)u(q∗)

φt= m∗t

This, of course, assumed interiority but now we can relax that somewhat.As long as m ≥ m∗t then this solution holds. If m < m∗t then the solutionis d = m and:

φtm =θc(q)u′(q) + (1− θ)u(q)c ′(q)

θu′(q) + (1− θ)c ′(q)= z(q) = φtd


Equilibria

Notice that the solution does not depend on the seller’s money holdings!This is really useful because we can now concentrate only on the buyer’smoney holdings, m.Notice as well that q(m) and d(m) are independent of m̃ because m̃doesn’t affect the solution.Finally notice that φtm = z(q(m)), so by the chain rule:

φt = z ′(q)q′(m)

You can verify in the paper that z ′(q) > 0 so q(m) is increasing.


Equilibria

We can now reduce the value function since we know Bt , have anexpression for Wt , know q(m), etc. This yields:

Vt(m) = vt(m) + φtm + maxm′{−φtm′ + βVt+1(m′)},

where vt is current period payoff.Notice how similar this is to a Bellman equation. The problem withdynamic programming is that V is unbounded so traditional methods donot necessarily apply.


Equilibria

The first time I saw this paper they used the contraction mapping theoremand an assumption of stationarity to solve this via first-order conditionsand the envelope condition. Most people still do this when they look atapplications.In the published paper they pursue a more general solution by noting thatthey can use repeated substitution to eliminate the Vt+1.Now they have a sequence of vj whose payoffs they know up to anunknown constant (the distribution F ). This affects the intercept but notthe slope and so does not affect the maximal choice of m′.


Equilibria

Using the repeated substitution ‘trick’ they obtain,

Vt(mt) = vt(mt)+φtmt+∞∑j=t

βj−t maxmj+1

{−φjmj+1+β[vj+1(mj+1+φj+1mj+1]}

This is a key equation because for any m′ > m∗, it turns out thatv′(m′) = 0. This means that there is no solution to the dynamic programif φt < βφt+1 because V (m) is strictly increasing. This is a contradiction.Thus φt ≥ βφt+1 for an equilibrium. This restriction implies that thedynamic program is non-increasing in m′ for m′ > m∗.This restriction also means deflation because φ is the goods price ofmoney. At equality this is the Friedman rule.


Equilibria

What have we got so far?We know that there is a minimum inflation consistent with equilibriaexistence.We can characterize the choice of m as a simple program.And we know something about bargaining (doesn’t depend on the seller’smoney holdings).But we cannot yet claim uniqueness.


Uniqueness

Uniqueness is this model depends on bargaining power, θ, andφt ≥ βφt+1. As long as θ is bounded away from 1 (or we are willing tomake some assumptions regarding limiting sequences), then a sufficientcondition for uniqueness is:

u′u′′′ − (u′′)2 ≤ 0

This is a restriction on preferences. Log concavity suffices and is oftenused.This implies a unique choice of mt+1 in any equilibrium. This means adegenerate distribution F .Everyone has the same money holdings in this model.


Prices

If money holdings are degenerate, then because there is a unit mass ofidentical consumers: mt+1 = MTaking the first order condition of the value function yields:

φt = β[v′t+1(M) + φt+1]

Substituting in for φt = z(qt)/M and also for v yields:

z(qt) = βz(qt+1)[ασu′(qt+1)

z ′(qt+1)− ασ + 1]

This is a difference equation in q. So now we can find output.In a steady-state, qt = qt+1 = q so:

u′(q)

z ′(q)=ασβ + 1− β

ασβ


Liquidity Premium

We often call ασ[u′(qt+1)

z ′(qt+1) − 1] the liquidity premium which is the marginalvalue of spending a dollar instead of holding it.


What the Model Can Do

It can solve for prices and output that result from both centralized anddecentralized markets.It can allow for changes in M, so one can analyze the effects of monetarypolicy.It shows that the Friedman rule is optimal but that this does not obtainthe first-best if θ < 1.If, and this is a big if, one takes the utility structure seriously then one cancompute the welfare costs of inflation.


Where It Fits

This framework is the workhorse model of money demand because it isanalytically tractable.What generates this money demand? The fact that there is bilateralmatching, anonymity and specialization.But, what sort of world does this reflect? I think Lagos and Wright wouldlike us to imagine the decentralized market represents services.But what sort of services do we provide that require us to haveanonymity? This is perhaps my biggest complaint about this model.


Alternative Specifications

This version of the money demand model has relied upon Nash Bargaining.But we can offer a different environment. Imagine that the decentralizedmarket feature Walrasian pricing. For instance, imagine that there aresub-markets where agents meet in large groups.In this environment, agents are either buyers or sellers and this isdetermined by preference and technology shocks.


A Walrasian Specification

Now assume that after the centralized market closes that agents receive ashock such that they are either a buyer or a seller. Then:

V (m) = γV b(m) + γV s(m) + (1− 2γ)W (m)

where γ is the probability of being a buyer or a seller.


A Walrasian Specifications

Then note that:

V b(m) = maxx{u(x) + W (m − p̂x)} s.t.p̂x ≤ m

and

V s(m) = maxx{−c(x) + W (m + p̂x)}

Here, φt = 1/p̂ in the centralized market but p̂ is the decentralized price.So they are not in general the same.


A Walrasian Specifications

By non-satiation it must be that p̂x = m for the buyers.Then, recall that W ′(m) = φ in the centralized market so taking the firstorder condition of the seller’s problem yields:

dV s(m)

dx= c ′(x) + W ′(2m)p̂ = −c ′(x) + φp̂

So clearly,

c ′(x) = φp̂ = φm

x

This is very similar to the Nash bargaining solution. In fact, if γ = ασ andz(x) = xc ′(x) they are the same!


So Mechanisms?

The point of this alternative approach is that we can think of somethinglike the Lucas island model (or even a Mortensen-Pissarides labor searchmodel).So in this framework, anonymity and specialization are key.


Efficiency

I briefly mentioned this before. But we might like to think a bit moreabout inefficiencies.The Hoisios(1990) condition is a remark about how to efficiently split thesurplus from matches. In this framework it implies θ = 1Basically, there is a holdup problem in money demand since buyers mustacquire cash in the CM market prior to trading in the DM. If θ < 1 thenproducers earn some of the surplus so in expectation of this, agentsunderinvest in m.Thus bargaining power is not just a technicality – it matters here forwelfare characterizations and also matching the data.


Cash-in-Advance

Notice that the holdup problem is also a little like a cash-in-advanceconstraint. Buyers have to have money to trade in the decentralizedmarket unless they are ‘lucky’ and their match satisfies the doublecoincidence of wants.Also, notice that the cost of the constraint is not inflation due to moneygrowth but rather the surplus lost through bargaining. So this is a bitdifferent from cash-in-advance.


Money Demand

Let’s now specialize this model a little bit. Let

U(X ) = log(X )

u(x) = Ax1−a/(1− a)

c(x) = x

β = 1/(1 + r)

α = 1

σ = 0.5

We still need to determine θ.


Money Demand

If U(X ) = logX then in the centralized market agents produce X ∗ = 1(keeping our assumption that U ′(X ∗) = 1).Thus nominal output in the centralized market is PX = 1/φ.In the decentralized market output is M/2 since ασ = 0.5 and so half theagents produce and in every single coincidence of wants meeting M dollarschange hands.Total output in both markets is 1/φ+ M/2


Money Demand

Now recall that z(x) = φM. So,

M

PY=

z(x)

1 + z(x)/2

This is the money demand equation (technically the inverse of velocity).Now define the nominal interest rate as 1 + i = (1 + r)(1 + φt/φt+1) sinceφt/φt+1 is the (steady-state) inflation rate.


Money Demand

What we need, which we do not have yet, is a steady-state condition forhow money growth (inflation) affects x . We can get this from the solutionto the general model from above including money growth (so the M nolonger cancels from both sides):

z(qt)

Mt= β

z(qt+1)

Mt+1[ασ

u′(qt+1)

z ′(qt+1)− ασ + 1]

In the steady-state this becomes:

u′(q)

z ′(q)= 1 +

1 + φt/φt+1 − βασβ

Thus rewriting we should get:

u′(q)

z ′(q)= 1 + i/ασ

By our assumptions on functional forms this implies that dx/di < 0.


Money Demand

Since this is true for our specialized example then:

M

PY=

z(x)

1 + z(x)/2

is well defined, given θ, to estimate preferences because z(x) is a functionof preferences.Given time series data on M/PY , one can uncover A and a. But whichmoney series to use?Also, given preferences and holding P and M constant, one gets a LMcurve straight from Hick’s IS-LM.


Money Demand


Extensions to Lagos and Wright

There are too many to list. Some important ones are Aruoba, Waller andWright (2009) on the coexistence of money and capital.A very nice overview of the search money literature can be found inWilliamson and Wright (2010) “New Monetarist Economics: Models”which is a chapter from the most recent Handbook of MonetaryEconomics.In fact, they extend this setting to include nominal price rigidity either asan assumption or an equilibrium outcome. The key difference from thisliterature and the New Keynesian literature is in the search frameworkmoney is neutral. Thus policy prescriptions are, in general, quite different.


Molico (2006)

Since no PhD student wishes to discuss this paper, I will do so ratherbriefly.The point of this paper is that the distributions of money holdings are, inthe real world, not degenerate and this has implications for both policyand welfare.In simplest terms, imagine a central bank that generated an inflation. Ifmoney holdings are not identical, then inflation implies a distribution ofreal resources across agents. This cannot be Pareto-efficient and alsoimplies welfare costs.


Molico (2006)

I don’t think that the usefulness of this paper is really to do with the exactcomputation of the welfare costs. Welfare measures are, in my mind, notreally very interesting in a world with migration, death, birth and aging.This is the world of macro.So, what can we learn? Maybe something about magnitudes. Maybe.


Molico

Numerically, computing heterogeneous money holdings is very difficultbecause it is the entire distribution that is of interest to agents in a worldwith random matching.Other types of hetergeneous agent models often have centralized marketsin which a price summarizes the relevant part of the distribution and inwhich a subset of well-defined moments can forecast these prices forward.


Molico

The main findings of this paper are that money is neutral if monetaryexpansions are proportionate to agents money holdings. So moneyneutrality is fragile. This matters viz a viz the price rigidity discussionabove.Second, dispersion matters. More dispersed money holdings imply moredispersion in prices and lower output (and therefore welfare).Finally, in environments of low inflation lump-sum transfers of money canimprove welfare by decreasing dispersion of money holdings. The same isnot true in high inflation environments.


Molico

Because this is a computational paper it is difficult to understand theintuition underlying the inflation results.Nevertheless they are interesting and suggest that degeneracy may be animportant restriction to overcome.


Kocherlakota (1998): Money as Memory

This paper is also interesting because it claims that money performs anadditional role: it summarizes our history of trades.This is somewhat obvious in the Lagos and Wright model over a shorthorizon because whether an agent exits the decentralized market or notimplies that he or she has traded or not.But equally clearly, over a longer-horizon the degeneracy of moneyholdings implies that there is no long-run memory since all agents hold thesame amount of money exiting the centralized market.Thus degeneracy and memory are linked.


Economics 546 - Faculty and Staff | Faculty of...

Documents

Transcript of Economics 546 - Faculty and Staff | Faculty of...