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Monetary EconomicsChapter 6: Monetary Policy, the Friedman rule, and the cost of in�ation

Prof. Aleksander Berentsen

University of Basel

Ed Nosal and Guillaume Rocheteau Money, Payments, and Liquidity - Chapter 6 1 / 85

Structure of this chapter

1 Introduction2 The Friedman rule

OptimalityTrading frictions

3 Distributional e¤ects of monetary policy4 The hot-potato e¤ect of in�ation5 The welfare cost of in�ation6 Conclusion


Introduction

Fiat money is useful in environments where credit arrangements arenot feasible.

The existence of �at money allows the implementation of sociallydesirable allocations that otherwise could not be achieved.

Thus far, the supply of money was constant. By changing the rate ofgrowth of money supply, the monetary authority is able to a¤ect therate of return of currency and, hence, agents�incentives to hold realbalances. This, in turn, has implications for equilibrium allocationsand society�s welfare.

Monetary policy takes here the simple form of a constant moneygrowth rate.


The Friedman rule

The Friedman ruleAssuming that the marginal cost of creating money is zero, the Friedmanrule advocates to set the nominal interest rate at zero.


The Friedman ruleOptimality

Mt is aggregate stock of money at the beginning of period t.

γ � Mt+1/Mt is the gross growth rate of the money supply.

Money is injected, or withdrawn, in a lump-sum fashion in thecompetitive market at night, CM.



Because agents have quasi-linear preferences in the CM � preferenceswhich eliminate wealth e¤ects� we can assume without loss ofgenerality that only buyers receive the monetary transfers.

We focus on steady-state equilibria, where the real value of themoney supply is constant over time, i.e., φtMt = φt+1Mt+1.

Note that the gross rate of return on currency isφt+1/φt = Mt/Mt+1 = γ�1.



DAY (DM)NIGHT (CM) NIGHT (CM)

Agent’s real balances:mz tφ= mz t 1

1+

− = φγ

Period t+ 1Period t

tM 1+tM 2+tM

Transfers Transfers

Figure 6.1: Timing of a representative period



Value function of the buyer at the beginning of the CM is

W b (z) = maxx ,y ,z 0

nx � y + βV b

�z 0�o

(6.1)

subject to

x + φtm0 = y + z + T (6.2)

z 0 = φt+1m0 =

φtm0

γ(6.3)

z : buyer�s real balances, i.e., z = φtmd : transfer from the buyer to the seller in the DMT : real value of the lump-sum transfer

Or, alternatively:

W b (z) = z + T +maxz 0�0

n�γz 0 + βV b

�z 0�o. (6.4)



Value function of the buyer at the beginning of the DM is

V b(z) = σnu [q (z)] +W b [z � d (z)]

o+ (1� σ)W b (z)

= σ fu [q (z)]� d(z)g+W b (z) , (6.5)

The buyer�s problem can be simpli�ed by substituting V b(z) from (6.5)into (6.4), i.e.,

maxz�0

f�iz + σ fu [q(z)]� d(z)gg , (6.6)

where 1+ i = (1+ r)γ and i is the nominal rate of interest.



Because it is costly to hold money: d = z .

Since the quantity traded in a match satis�es c(q) = z , the buyer�sproblem (6.6) can be rewritten as a choice of q, i.e.,

maxq2[0,q�]

f�ic(q) + σ [u(q)� c(q)]g . (6.7)

The �rst-order (necessary and su¢ cient) condition to the buyer�sproblem (6.7) is simply

u0(q)c 0(q)

= 1+iσ. (6.8)



The steady state solution qss to (6.8) is depicted in Figure 6.2.

)0(')0('

cu

σi

+1

1

*qssq

)(')('

qcqu

Figure 6.2: Stationary monetary equilibrium under a constant money growth rate



The optimal monetary policy requires i = 0, or, equivalently,γ = 1

1+r < 1.

This is the so-called Friedman rule.

When buyers have all of the bargaining power, the allocation of themonetary equilibrium under the Friedman rule coincides with thesocially e¢ cient allocation of the search good, q = q�.



The Friedman rule is the optimal monetary policy under many tradingmechanisms.

If terms of trade are determinied buy a Walrasian pricing mechanism ora competitive posting mechanism in the DM, the Friedman ruleimplements the e¢ cient allocation, q�.

But it does not always achieve the e¢ cient allocation.

Under Nash bargaining, the buyer will choose an ine¢ ciently low valuefor q. This ine¢ ciency is due to a non-monotonicity property of theNash solution



If the buyer obtains the marginal social return of his real balances, asis the case under buyers-take-all or competitive price posting, thenthe Friedman rule implements the e¢ cient allocation.

And even if this condition does not hold, the Friedman rule canachieve the socially e¢ cient allocation provided that the buyer�ssurplus from a trade increases with the total surplus of a match.



In many monetary environments, the Friedman rule ist the optimal policy.Why is it rarely implemented in practice?

The government may lack the enforcement power required toimplement the lump-sum tax needed to generate a de�ation.

The Friedman rule may not be needed if the mechanism thatdetermines the terms of trade during the DM is appropriatelydesigned.



In the following two models, we show that in environments with

search externalities, or

heterogenous agents,

a deviation from the Friedman rule (i.e. some in�ation) can be optimal.


The Friedman ruleTrading frictions

We assume that there is a unit measure of ex ante identical agents thatcan choose to be either buyers or sellers in the DM. The decision tobecome a buyer or seller in period t is taken at the beginning of theprevious CM, in period t � 1.

DAY (DM) NIGHT (CM)

buyers and sellersare matched bilaterally

and at random

n 1n Choice of being buyers or sellersin the next day

Choice ofreal balances

Figure 6.3: Timing of the representative period



Let n denote the fraction of buyers in the DM and 1� n the fractionof sellers.

A buyer meets a seller with probability 1� n, and a seller meets abuyer with probability n.

Therefore, the number of matches in the DM is n(1� n), and it ismaximized when n = 1

2 .



The value function at the beginning of the CM is analogous to (6.4),and satis�es

W j (z) = T + z +maxz 0�0

��γz 0 + βV j (z 0)

, (6.9)

where j 2 fb, sg.

The value of being a buyer in the DM satis�es

V b(z) = (1� n) fu [q(z)]� zg+maxhW b(z),W s (z)

i. (6.10)



Substituting (6.10) into (6.9), and using the linearity of W b (z) andW s (z), the value of a buyer with z units of real balances at the beginningof the CM must satisfy

W b(z) = T + z + maxq2[0,q�]

β f�iz(q) + (1� n)[u(q)� z(q)]g

+βmaxhW b(0),W s (0)

i. (6.11)

From (6.11), the buyer chooses the quantity to trade in the next DM,taking as given his matching probability, 1� n.



By similar reasoning, the value of being a seller with z units of realbalances satis�es

W s (z) = T + z + βn[z(q)� c(q)] + βmaxhW b(0),W s (0)

i. (6.12)

Equation (6.12) embodies the result that sellers do not carry moneybalances into the DM� since they do not need them� and that thequantity traded q , or equivalently the buyers�real balances, is taken asgiven.



Since both W b(z) and W s (z) are linear in z , the choice of being abuyer or a seller does not depend on z .

In a monetary equilibrium, agents must be indi¤erent between being aseller or a buyer, otherwise there will be no trade and �at money willnot be valued.

Consequently, we focus on active monetary equilibria where n 2 (0, 1)and W b(z) = W s (z).



From (6.11) and (6.12), n must satisfy

n[z(q)� c(q)] = (1� n) [u(q)� z(q)]� iz(q). (6.13)

The left side of (6.13) is the seller�s expected surplus in the DMThe right side is the buyer�s expected surplus, minus the cost ofholding real balances.

In any monetary equilibrium:

n =u(q)� (1+ i)z(q)u(q)� c(q) . (6.14)



From (6.11), q solves

maxq2[0,q�]

f�iz(q) + (1� n)[u(q)� z(q)]g . (6.15)



De�nitionA steady-state monetary equilibrium is a pair (q, n) such that q > 0 is asolution to (6.15) and n 2 (0, 1) satis�es (6.14).



In�ation has a direct e¤ect on the equilibrium allocation by raisingthe cost of holding real balances and, therefore, by reducing q.

If the pricing mechanism delivers q = q� under the Friedman rule,then n decreases with in�ation since

dndi

��i=0

=�z(q�)

u(q�)� c(q�) < 0.

Since in�ation is a direct tax on agents who hold money, as in�ationincreases agents have less incentives to be buyers.As a result the matching probability of buyers, 1� n, increases within�ation (close to the Friedman rule).So even though there are fewer buyers, they spend their moneybalances in the DM faster.



We measure social welfare by the sum of all trade surpluses in aperiod, i.e., W = n(1� n)[u(q)� c(q)].

Welfare is maximized when the surplus of each match ismaximized� which requires q = q�� and when the number ofmatches in the DM is maximized� which requires n = 1/2.



Suppose that the trading mechanism in the DM implements q� at theFriedman rule.

The �rst condition for e¢ ciency requires that the Friedman rule isimplemented.

From (6.14), the second condition for e¢ ciency requires that

u (q�)� z (q�)u (q�)� c (q�) =

12

(6.16)



Equation (6.16) turns out to be a restatement of the so-called �Hosioscondition� for e¢ ciency in models with search externalities.

Search externalities arise when agents�decisions to participate in amarket a¤ect the trading probabilities of other agents in the market.

The buyer�s contribution in the creation of matches in the DM mustbe rewarded by giving buyers a share in the match surplus that isequal to the fraction of matches that buyers are responsible for.



The welfare e¤ect of a change in i in the neighborhood of i = 0 is

dWdi

��i=0

=u0 (q) [u0 (q)� c 0(q)][u00(q)� z 00(q)]

n2

1� n + (2n� 1) z(q). (6.17)

Assuming that q = q� at the Friedman rule� which is a valid assumptionunder proportional bargaining� we can evaluate the welfare implications ofa deviation from the Friedman rule by evaluating the second term in(6.17), (2n� 1) z(q). If n 6= 1/2, then a deviation from the Friedmanrule may actually be welfare improving.



A deviation will be optimal, i.e., dWdi��i=0 > 0, if and only if

u(q�)� z(q�)u(q�)� c(q�) = n > 1/2.

When the buyer�s share of match surplus is greater than one-half, there aretoo many buyers from a social perspective.



A deviation from the Friedman rule will be optimal wheneverθ 2 (0.5, 1):

the policy maker is willing to trade o¤ e¢ ciency on the intensivemargin (the quantities traded in each match) in order to improve theextensive margin (the number of trade matches in the DM) by raisingin�ation, which increases the number of sellers and reduces the numberof buyers.

If, on the other hand, θ 2 (0, 0.5), there will be too many sellers inequilibrium at the Friedman rule:

the Friedman rule is the optimal policy since a deviation would onlyfurther increase the number of sellers in the economy.


Distributional e¤ects of monetary policy

In�ation can have a positive e¤ect when the distribution of moneybalances across agents is not degenerate.

Indeed, a positive in�ation, which is engineered by lump-sum moneyinjections, redistributes wealth from the richest to the poorest agents.

If some agents are poor because of uninsurable idiosyncratic shocks,this redistribution can raise social welfare.



To capture a distributional e¤ect of monetary policy, we modify thebenchmark model as follows:

Buyers and sellers live for only three subperiods: they are born at thebeginning of the night and die at the end of the following period.Agents can, therefore, potentially trade three times.For simplicity, assume that agents do not discount across periods, i.e.,r = 0.

Generation t

Generation t+1Productivity shocksTransfers

Competitive markets

Bilateraltrades

Figure 6.4: Overlapping generations



The utility function of buyers is xy + u(q) + xo ,

xy 2 R is the utility of consumption net of the disutility of productionin the CM when young,xo is the net utility of consumption in the CM when old,u(q) is the utility of consumption in the DM.

Similarly, the utility function of sellers is xy � c(q) + xo .



We assume that newly-born buyers di¤er in terms of their productivityin the �rst period of their lives.

a fraction ρ 2 (0, 1) of newly-born buyers are productive, while theremaining fraction is unproductive.as a result, the productive buyers can participate in the CM toaccumulate money balances, while unproductive buyers cannot.

The government is not able to make di¤erentiated transfers toproductive and unproductive buyers.



The problem of a productive newly-born buyer, which is similar to (6.6), is

maxm�0

��φtm+ σ

�u[q(φt+1m)]� c [q(φt+1m)]

+ φt+1m

. (6.18)

The buyer who has access to the CM when he is born produces φtmunits of the general good in exchange for m units of money.

If he doesn�t meet a seller in the next DM, then he spends his moneybalances in the CM before he dies.

If he does meet a seller, we assume that the buyer captures the entiresurplus from the match.



Denote z = φt+1m as the choice of real balances for the next DM.

The productive buyer�s problem (6.18) can then be simpli�ed to read

maxz�0

f�(γ� 1)z + σ fu[q(z)]� c [q(z)]gg . (6.19)



Since dqdz =

1c 0(q) , the �rst-order condition for problem (6.19) is

u0(q)c 0(q)

= 1+γ� 1

σ. (6.20)

If the money supply is constant, i.e., γ = 1, newly-born productivebuyers consume q� units of the DM good.

However, unproductive newly-born buyers cannot produce in exchangefor money balances and, therefore, cannot consume in the DM.



Assume now that there is a constant (positive) in�ation, γ > 1, and thatmoney is injected into the economy through lump-sum transfers to allnewly-born buyers in the CM. Let ∆t denote a transfer at night in periodt � 1 which can be used in the DM of period t. We have

∆t = Mt �Mt�1 =γ� 1

γMt . (6.21)



Let mt represent the money balances of buyers in the DM of period twho had access to the CM when they were young. Hence, equilibriumin the money market requires that

ρmt + (1� ρ)∆t = Mt . (6.22)

The fraction ρ of productive buyers hold mt units of money while the1� ρ unproductive buyers hold ∆t . The sum of the individual moneyholdings must add up to the money supply, Mt .



Substituting ∆t from (6.21) into (6.22) and rearranging, we get

mt =Mt

ρ

�1+ ρ(γ� 1)

γ

�, (6.23)

and, from (6.21) and (6.23), we get

∆t =ρ(γ� 1)

1+ ρ(γ� 1)mt . (6.24)

Equation (6.24) implies that ∆t < mt : unproductive buyers are poorerthan productive ones.



Let q̃ denote the DM consumption of unproductive buyers.

Since productive buyers spend their mt units of money in the DM, thesame is true for unproductive buyer who hold fewer money balances,∆t < mt .From the buyer-takes-all assumption, c(qt ) = φtmt andc(q̃t ) = φt∆t . Hence, (6.24) implies

c(q̃t ) =ρ(γ� 1)

1+ ρ(γ� 1)c(qt ). (6.25)



From (6.25) q̃t < qt . As γ increases, qt decreases through a standardin�ation-tax e¤ect, see (6.20).

But in�ation also a¤ects the distribution of real balances acrossbuyers. Indeed, the dispersion of real balances, as measured byc (qt )�c (q̃t )

c (qt )= 1

1+ρ(γ�1) , decreases as γ increases.

So the policy-maker faces a trade-o¤ between smoothing consumptionacross buyers and preserving the purchasing power of real balances.



In terms of social welfare:

W = σρ[u(q)� c(q)] + σ(1� ρ)[u(q̃)� c(q̃)]. (6.26)

In the neighborhood of price stability, an increase in in�ation only hasa second-order e¤ect on the match surpluses of productive buyers,d [u(q)�c (q)]

dγ

��γ=1+

= 0.

However, it has a �rst-order e¤ect on the match surpluses ofunproductive buyers. Di¤erentiating (6.25) with respect to γ, we get

dq̃tdγ

��γ=1+

=ρc(q�)c 0(0)

.



The welfare e¤ect of an increase in�ation from price stability is, from(6.26), given by

dWdγ

��γ=1+

= σ(1� ρ)

�u0(0)c 0(0)

� 1�

ρc(q�) > 0.

Hence, an increase in in�ation from γ = 1 is welfare-improving because itallows unproductive buyers to consume, while the negative e¤ect onproductive buyers�welfare is of only a second-order consequence.


The hot-potato e¤ect of in�ation

There is a commonly held view that higher in�ation makes agents spendtheir money holdings faster (hot-potato e¤ect), implying that in�ation canincrease the velocity of money and the frequency of trades.



In this section we want to give an explicit choice to buyers to spendtheir money faster: we will let buyers increase or decrease the speedat which they spend their money balances by altering the rate atwhich they search for trading opportunities.

We will investigate whether the model can generate an hot-potatoe¤ect of in�ation and we will explore the consequences of such ane¤ect for the optimal monetary policy.



Sellers are evenly distributed across several market places that worklike Walrasian markets.

Buyers are able to �nd a market place with a good they like withprobability e, where e 2 [0, 1] can be interpreted as a search e¤ort.

The cost of search e¤ort, e, is ψ(e), where ψ(0) = 0, ψ0 > 0,ψ00 > 0, ψ0(0) = 0, and ψ0(1) = ∞.



The price of a DM good in terms of the general good is p.

The value function of a buyer in the DM is

V b(z) = maxef�ψ(e) + emax

qb[u(qb)� pqb ]

+W b(z)g (6.27)

subject to e 2 [0, 1] and pqb � z .



The quantity of the search good demanded by the buyer is given by

qb (z) =�u0�1(p) if u0(z/p) � pzp otherwise. (6.28)



If we substitute V b(z) into the CM value function, then the buyer�sproblem can be expressed as

maxz ,e

n�iz � ψ(e) + e

hu[qb (z)]� pqb (z)

io(6.29)

subject to z � 0 and e 2 [0, 1] ,where qb (z) is given by (6.28).



The �rst-order conditions for the choices of real balances and search e¤ortare:

u0�qb�= p

�1+

ie

�(6.30)

ψ0(e) = u�qb�� pqb . (6.31)

If i = 0, then the solution to (6.30)-(6.31) is unique: qb = u0�1(p) ande = ψ0�1

�u�qb�� pqb

�.



We will focus our attention on monetary policies close to i = 0. Theproblem of a seller in the DM is to supply an amount of the search good,qs , taking the price, p, as given, i.e.,

maxqsfpqs � c (qs )g . (6.32)

The solution isc 0 (qs ) = p. (6.33)



The market-clearing condition in the DM is simply

eqb = qs . (6.34)

From (6.30), (6.31) and (6.34), e and qb � q will satisfy:

u0 (q)c 0(eq)

= 1+ie

(6.35)

ψ0(e) = u (q)� qc 0(eq) (6.36)



Society�s welfare is measured by the sum of the utility �ows of buyers andsellers in the DM net of the buyers�search costs,

W = eu(q)� c(eq)� ψ(e).



The �rst-order conditions that characterize the social optimum are:

u0(q) = c 0(eq), (6.37)

ψ0(e) = u(q)� qc 0(eq). (6.38)

The �rst-best allocation�qfb , e fb

�is unique and can be implemented

under the Friedman rule.



SI

Q

i↑

fbq

fbe

Figure 6.5: E¤ects of an increase in i in the neighborhood of the Friedman rule



In Figure 6.5, we represent the condition (6.35) in the space (q, e) by thecurve labelled Q and the condition (6.36) by the curve labelled SI .

Following an increase in i from i = 0, e increases but q decreases.

From (6.31), for e to increase, p must decrease, which implies from(6.33) that aggregate production eq decreases.


The hot-potato e¤ect of in�ationSummary

The model where buyers can choose their search e¤ort in the DMgenerates a hot-potato e¤ect: as in�ation increases (away from theFriedman rule), agents spend their cash faster.

! This is because the price of the DM good decreases, which raisesbuyers�surplus.

! Still, aggregate output and welfare fall.


The hot-potato e¤ect of in�ationSummary

The optimal monetary policy is the Friedman rule, and it achieves the�rst-best allocation.

So the presence of an hot-potato e¤ect does not imply that thepolicy-maker should deviate from the Friedman rule.

This result di¤ers from the before, where the Friedman rule wassometimes suboptimal, because here buyers�search e¤orts do notgenerate a trading externality.


The welfare cost of in�ation

Let us now turn on the welfare cost of in�ation.

In�ation distorts the allocations by inducing agents to reduce theirreal balances, and hence the quantities they trade in the DM.

If the costs associated with a moderate level of in�ation are verysmall, then in�ation will not be an important policy concern.

Search-theoretic monetary models predict a welfare cost of in�ationwhich can be signi�cantly higher than traditional estimates.



A typical calibration procedure adopts a representative-agent versionof the model studied so far.

The night-time utility function: B ln x � h, where x is consumption, his the hours of work, and h hours produces h units of the generalgood.

Given these preferences, the production in the CM is B. One caninterpret B as the quantity of goods that do not require money to betraded.



The functional forms for utility during the day areu(q) = q1�η/(1� η) and c(q) = q. The parameters (η,B) arechosen to �t money demand as described in the model to the data.

The cost of holding real balances, i , is measured by the commercialpaper rate and M is measured by M1, which is cash plus liquiddeposits.

The typical measure of the cost of in�ation is the fraction of totalconsumption that agents would be willing to give up to have zeroin�ation instead of 10 percent in�ation.



The results of existing studies are summarized in Table 1.

Trading mechanism Cost of in�ation (% of GDP)Buyers-take-all 1.2-1.4Nash solution 3.2-3.3

Generalized Nash up to 5.2Egalitarian 3.2

Price-posting (private info) 6.1-7.2

Price-taking 1-1.5

Gen. Nash w/ ext. margin 3.2-5.4Proportional w/ ext. margin 0.2-5.5Comp. search w/ ext. margin 1.1

Table 1: Summary of studies on the cost of in�ation



Under the buyer-takes-all bargaining solution, the welfare cost of 10per cent in�ation is typically between 1 percent and 1.5 per cent ofGDP per year.

One �nds a similar magnitude for the welfare cost of in�ation underprice-taking or competitive price posting. This is a sizeable number.

Graphically, this number is approximately equal to the areaunderneath the money demand curve.



To see this, integrate the inverse (individual) money demand function,

which is given by i(z) = σnu 0[q(z )]c 0[q(z )] � 1

o, see (6.8), to obtain

Z z1

z0i(z)dz = σ fu [q(z1)]� c [q(z1)]g�σ fu [q(z1.1)]� c [q(z1.1)]g ,

where z1 represents real balances when γ = 1 and z1.1 represents realbalances when γ = 1.1.

The left side of the above expression is the area underneath theindividual money demand curve while the right side is the change insociety�s welfare.



In Figure 6.6 we represent the individual money demand function,i(z). As the nominal interest rate approaches to 0, real balancesapproach their maximum level, z�.

Under buyers-take-all, z� = c(q�).

Consider two nominal interest rates, i > 0 and i 0 > i . The welfarecost from raising the nominal interest rate from i to i 0 corresponds tothe area underneath money demand, ABDE .

The welfare cost from raising the interest rate from its optimal levelgiven by the Friedman rule to i 0 is the area ABC .



i

i

0 *zA

B

CE

D

'i

'z

[ ]

−= 1)(')(')(

zqzzquzi σ

Figure 6.6: Welfare cost of in�ation and the area underneath money demand



If sellers have some bargaining power, then the welfare cost ofin�ation is bigger.

Under the (symmetric) Nash solution or the egalitarian solution (i.e.,proportional with θ = 0.5), the welfare cost of 10 per cent in�ation isbetween 3 and 4 per cent of GDP.



Whenever θ < 1 and i > 0 any bargaining solution generates aholdup problem on money holdings: buyers incur a cost frominvesting in real balances in the CM that they cannot fully recoveronce they are matched in the DM.

The severity of this holdup problem depends on the seller�s bargainingpower, 1� θ, and the average cost of holding real balances, i/σ.

As in�ation increases, the holdup problem is more severe, whichinduces buyers to underinvest in real balances.



Once again, this argument can be illustrated using the areaunderneath the money demand function.

The inverse (individual) money demand function is

i(z) = σnu 0[q(z )]z 0[q(z )] � 1

o.

The area underneath money demand isZ z1

z0i(z)dz = σ fu [q(z1)]� z1g � σ fu [q(z1.1)]� z1.1g .



Under proportional bargaining, u [q(z)]� z = θ fu [q(z)]� c [q(z)]g,the area underneath the money demand function isZ z1

z0i(z)dz = θσ fu [q(z1)]� c [q(z1)]g� θσ fu [q(z1.1)]� c [q(z1.1)]g

The private loss due to an increase in the in�ation rate corresponds toleft side of the above expression.

It is equal to a fraction θ of the welfare loss for society, the right sideof the above expression.



In Figure 6.7 we represent the individual demand for real balances aswell as the social return of those real balances (the dashed curve).

The welfare cost from raising the nominal interest rate from 0 to i isgiven by the area ADC , while the welfare cost to the buyer is the areaunderneath money demand, ABC .



So the individual money demand does not accurately capture thesocial value of holding money since it ignores the surplus that theseller enjoys when the buyer increases his real balances.

If, for example, θ = 1/2 (the egalitarian solution), then the socialwelfare cost of in�ation is approximately twice the private cost formoney holders. This private cost has been estimated to be about 1.5percent of GDP, so the total welfare cost of in�ation for society isthen about 3 percent of GDP.



A

B

D

C0

i

i

*z

[ ]

−= 1)(')(')(

zqzzquzi σ

[ ] θσ

)()('

)(')(' zizqz

zqczqu=

−

Figure 6.7: Holdup problem and the cost of in�ation



In the case where θ = 0, there is no monetary equilibrium sincebuyers get no surplus from holding money.

If buyers are heterogenous and the buyer�s willingness to trade isprivate information, then a monetary equilibrium can be restored.

In this case, the welfare cost of in�ation can be as high as 7 percentof GDP per year.



The introduction of an endogenous participation decision, see"Trading frictions", can either mitigate or exacerbate the cost ofin�ation, depending on agents�bargaining powers.

As we saw earlier, in some instances, the cost of small in�ation can benegative.

Under competitive posting (i.e., competitive search equilibrium),search externalities are internalized and the cost of in�ation isapproximately the same as the one obtained without extensive marginand competitive pricing.


Conclusion

Under standard pricing mechanisms, the optimal monetary policycorresponds to the Friedman rule. The money growth rate must benegative so that the rate of return of money balances is approximatelyequal to the rate of time preference.

Under this policy the cost of holding money is driven to zero andagents�demand for money balances are satiated.

To be implemented, the Friedman rule requires that money iswithdrawn from the economy through lump-sum taxation.


Conclusion

If the government�s coercive power is limited, the Friedman rule is notalways incentive feasible.

The optimality of the Friedman rule seems at odds with the usualpractise of central banking.

Some extensions of the model show where a deviation from theFriedman rule is optimal.

Finally, in�ation leads to welfare costs, but those depend crucially onthe assumed pricing mechanism.


monetary economics - sfu.cadandolfa/chapter 6.pdf · structure of this chapter 1 introduction 2 the...

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