monetary theory: cash in advance models - harris … theory: cash in advance models behzad diba...

Monetary Theory: Cash in Advance Models

Behzad Diba

University of Bern

May 2011

(Institute) Monetary Theory: Cash in Advance Models May 2011 1 / 18

Modeling Money Demand

Models with a cash-in-advance (CIA) constraint are more explicit thanother simple approaches (like MIUF) about the timing and structureof transactions that require money as a medium of exchange.

We will consider two CIA models

a closed economy with a stochastic endowment of a single perishablegoodan extension that introduces production (using labor as the only input)and a second good that can be purchased without cash (a "credit"good)

The second model (with cash and credit goods) will serve later in ourdiscussion of jointly optimal (Ramsey) fiscal and monetary policies;anticipating the normative application, the model incorporates a taxdistortion

a closed economy with a stochastic endowment of a single perishablegood

an extension that introduces production (using labor as the only input)and a second good that can be purchased without cash (a "credit"good)

An Exchange Economy

Consider an economy with a stochastic endowment of a perishablegood

there is a unit mass of identical households with endowments that aredifferentiated, in an otherwise inconsequential way, to create a need forexchange (households don’t like the "color" of their own endowment)but setting this aside, there is a single good that can be consumed byhouseholds or by the government

At the beginning of each period, households observe the state of theeconomy consisting of their endowment (yt), government purchases(Gt), a lump-sum tax (τt), and the money stock (Mt)

Next, they trade money and bonds in a financial exchange

Then, the financial exchange closes, and households buy goods fromeach other with cash, in the goods exchange

An Exchange Economy

there is a unit mass of identical households with endowments that aredifferentiated, in an otherwise inconsequential way, to create a need forexchange (households don’t like the "color" of their own endowment)

but setting this aside, there is a single good that can be consumed byhouseholds or by the government

An Exchange Economy

Goods Exchange

Each household consists of a seller and a shopper

During the goods exchange

the shopper visits other households and buys some of their endowment,using cashthe seller stays home and sells the endowment to other shoppers, forcash

In the goods exchange, shopper h faces the CIA constraint

PtC ht ≤ Mht (1)

Goods Exchange

PtC ht ≤ Mht (1)

Goods Exchange

the shopper visits other households and buys some of their endowment,using cash

the seller stays home and sells the endowment to other shoppers, forcash

PtC ht ≤ Mht (1)

Goods Exchange

PtC ht ≤ Mht (1)

Goods Exchange

PtC ht ≤ Mht (1)

Financial Exchange

Household h enters the financial exchange of time t with the proceedsof sales during the goods exchange of t − 1 and bonds acquired in thefinancial exchange of t − 1, plus any unused cash; the household facesthe budget constraint

Mht +B

ht +Ptτt ≤ Pt−1yt−1+(Mh

t−1−Pt−1C ht−1)+ (1+ it−1)Bht−1 ,(2)

where τt is a lump-sum tax

We can formally show that the CIA constraint (1) is binding if it > 0(but it is good enough if you just understand the intuition for this)and write (2) as

PtC ht + Bht + Ptτt ≤ Pt−1yt−1 + (1+ it−1)Bht−1 (3)

Financial Exchange

Household h enters the financial exchange of time t with the proceedsof sales during the goods exchange of t − 1 and bonds acquired in thefinancial exchange of t − 1, plus any unused cash; the household facesthe budget constraint

Mht +B

ht +Ptτt ≤ Pt−1yt−1+(Mh

t−1−Pt−1C ht−1)+ (1+ it−1)Bht−1 ,(2)

where τt is a lump-sum tax

We can formally show that the CIA constraint (1) is binding if it > 0(but it is good enough if you just understand the intuition for this)and write (2) as

PtC ht + Bht + Ptτt ≤ Pt−1yt−1 + (1+ it−1)Bht−1 (3)

Household FOCs

The representative household maximizes

∑j=0

βj[u(C ht+j )

]where 0 < β < 1, u′(.) > 0, and u′′(.) < 0, subject to (3)

Letting Λt denote the Lagrange multiplier on (3), the FOCs for C htand Bht are

u′(C ht ) = PtΛt

and the Euler equation

Λt = β(1+ it )EtΛt+1

These FOCs imply the Keynes-Ramsey rule

u′(C ht )Pt

= β(1+ it )Et

{u′(C ht+1)Pt+1

Household FOCs

∑j=0

βj[u(C ht+j )

]where 0 < β < 1, u′(.) > 0, and u′′(.) < 0, subject to (3)Letting Λt denote the Lagrange multiplier on (3), the FOCs for C htand Bht are

u′(C ht ) = PtΛt

u′(C ht )Pt

= β(1+ it )Et

{u′(C ht+1)Pt+1

Household FOCs

∑j=0

βj[u(C ht+j )

]where 0 < β < 1, u′(.) > 0, and u′′(.) < 0, subject to (3)Letting Λt denote the Lagrange multiplier on (3), the FOCs for C htand Bht are

u′(C ht ) = PtΛt

u′(C ht )Pt

= β(1+ it )Et

{u′(C ht+1)Pt+1

Government

There is a fiscal authority and a central bank

CIA models, in general, may or may not impose the CIA constraint ongovernment purchases; for now, we will impose it as

PtGt = Mgt

Each period, the central bank conducts open-market operationsmonetizing part of the bonds issued by the fiscal authority

The public sector’s consolidated budget constraint is

Mt + Bt + Ptτt = Mt−1 + (1+ it−1)Bt−1 + PtGt ,

where Bt denotes bonds held outside the central bank

Government

PtGt = Mgt

Government

PtGt = Mgt

Government

PtGt = Mgt

Equilibrium

Since all households are identical, we will have Bht = Bt andC ht = Ct , ∀h, in equilibrium

The equilibrium conditions in the markets for goods and money are

Ct + Gt = yt

andMht +M

gt = Mt

Since CIA constraints are binding, with a positive nominal interestrate, we also have

Mt = Ptyt

Equilibrium

Since all households are identical, we will have Bht = Bt andC ht = Ct , ∀h, in equilibriumThe equilibrium conditions in the markets for goods and money are

Ct + Gt = yt

andMht +M

gt = Mt

Mt = Ptyt

Equilibrium

Since all households are identical, we will have Bht = Bt andC ht = Ct , ∀h, in equilibriumThe equilibrium conditions in the markets for goods and money are

Ct + Gt = yt

andMht +M

gt = Mt

Mt = Ptyt

An Unusual Implication

The particular CIA model of an exchange economy outlined above nodistortion associated with inflation

This is because households spend the cash that they choose toacquire each period, and pay the inflation tax on money balances thatwere acquired by other households

So, this model does not make a case for the Friedman Rule (nor anyother prescription about optimal monetary policy)

We will, however, see, that this is no longer the case when we add aproduction decision to the model

A Production Economy

Now suppose output is produced by competitive firms with the lineartechnology

yt = ztnt

where n is the labor input and z is the stochastic level of productivity(common to all firms)

As usual, the profit maximization problem of competitive firms leadsto equality between the equilibrium real wage and the marginalproduct of labor; with our linear technology, we get

Pt= zt

Also, as usual, firms will make zero profits in equilibrium

yt = ztnt

Pt= zt

yt = ztnt

Pt= zt

Cash and Credit Goods

The model also allows for a second good that can be purchased oncredit (consumption of this good is not subject to a CIA constraint)

Cash and credit goods will be distinct goods from the households’perspective

Firms, however, are indifferent between selling their good for cash orcredit and charge the same price Pt for either

if they sell for credit in the goods market of date t, they get Pt duringthe financial exchange of t + 1if they sell for cash in the goods market of date t, they get Pt in cashbut can’t spend the cash before the financial exchange of t + 1

In terms of understanding the notation below, keep in mind that allpayments and receipts are delayed in this way because the financialexchange is before the goods exchange of each period

if they sell for credit in the goods market of date t, they get Pt duringthe financial exchange of t + 1

if they sell for cash in the goods market of date t, they get Pt in cashbut can’t spend the cash before the financial exchange of t + 1

Households

The representative household gets utility from consumption of thecash good (C1) and the credit good (C2), and disutility from work(n); household h maximizes

∑j=0

βj[u(C h1,t+j ,C

h2,t+j , n

ht+j )

]where 0 < β < 1, and u(.) has the standard properties

As before, the household must set aside money (Mht ) acquired during

the financial exchange of time t to pay for purchasing cash goods(now, from firms) in the subsequent goods exchange; the CIAconstraint is

PtC h1,t ≤ Mht ,

The household can buy C h2,t on credit and pay firms PtCh2,t for these

purchases in the financial exchange of t + 1

Households

∑j=0

βj[u(C h1,t+j ,C

h2,t+j , n

ht+j )

PtC h1,t ≤ Mht ,

Households

∑j=0

βj[u(C h1,t+j ,C

h2,t+j , n

ht+j )

PtC h1,t ≤ Mht ,

Payments in the Financial Exchange

Wage and tax payments are also delayed by one period, due to thetiming of markets

In the financial exchange at time t firms pay wages for the labor theyhired at t − 1, and workers pay the taxes due on these wages, at anincome tax rate τw set by the government

The household’s budget constraint in the financial exchange of time tis:

Mht + B

ht + Pt−1C

h2,t−1 ≤ (1− τwt−1)Wt−1nt−1 (4)

+(Mht−1 − Pt−1C h1,t−1) + (1+ it−1)Bht−1

Mht + B

ht + Pt−1C

h2,t−1 ≤ (1− τwt−1)Wt−1nt−1 (4)

+(Mht−1 − Pt−1C h1,t−1) + (1+ it−1)Bht−1

Mht + B

ht + Pt−1C

h2,t−1 ≤ (1− τwt−1)Wt−1nt−1 (4)

+(Mht−1 − Pt−1C h1,t−1) + (1+ it−1)Bht−1

Budget Constraint

With a positive interest rate, the CIA constraint is binding, and wecan write (4) as

Bht + PtCh1,t + Pt−1C

h2,t−1 ≤ (1− τwt−1)Wt−1nht−1 (5)

+(1+ it−1)Bht−1

Note that from the household’s perspective, the payment for cashgoods entails an opportunity cost of foregone interest

This is how the opportunity cost of holding money is reflected in themodel

Since the cost of producing the two goods is the same, a positiveinterest rate (i.e., a departure from the Friedman Rule) will have awelfare cost in this model (as we will see)

Budget Constraint

h2,t−1 ≤ (1− τwt−1)Wt−1nht−1 (5)

+(1+ it−1)Bht−1

Budget Constraint

h2,t−1 ≤ (1− τwt−1)Wt−1nht−1 (5)

+(1+ it−1)Bht−1

Budget Constraint

h2,t−1 ≤ (1− τwt−1)Wt−1nht−1 (5)

+(1+ it−1)Bht−1

FOCs for Consumption and Bonds

Letting Λt denote the Lagrange multiplier on (5), the FOCs for C h1,tand C h2,t are

u1(C h1,t ,Ch2,t , n

ht ) = PtΛt

andu2(C h1,t ,C

h2,t , n

ht ) = βPtΛt+1

(note that Λt+1 is not random with respect to the information set ofdate t)

The FOC for Bht gives

Λt = β(1+ it )Λt+1

FOCs for Consumption and Bonds

Letting Λt denote the Lagrange multiplier on (5), the FOCs for C h1,tand C h2,t are

u1(C h1,t ,Ch2,t , n

ht ) = PtΛt

andu2(C h1,t ,C

h2,t , n

ht ) = βPtΛt+1

(note that Λt+1 is not random with respect to the information set ofdate t)

The FOC for Bht gives

Λt = β(1+ it )Λt+1

Consumption Distortion

The FOCs for cash and credit goods and bonds imply

u1(C1,t ,C2,t , nt )u2(C1,t ,C2,t , nt )

= 1+ it

Since the marginal cost of producing the two goods is the same,having a wedge (it > 0) between their marginal utilities is ineffi cient

A departure from the Friedman Rule has a welfare cost

We will see later that the Friedman Rule is optimal in this model,assuming flexible prices, even though the government does not haveaccess to a lump-sum tax

u1(C1,t ,C2,t , nt )u2(C1,t ,C2,t , nt )

= 1+ it

u1(C1,t ,C2,t , nt )u2(C1,t ,C2,t , nt )

= 1+ it

u1(C1,t ,C2,t , nt )u2(C1,t ,C2,t , nt )

= 1+ it

Labor Distortion

The FOC for nht is

−un(C h1,t ,C h2,t , nht ) = β(1− τwt )WtΛt+1

Combining this with other FOCs, we get the equilibrium conditions

−un(C1,t ,C2,t , nt )u2(C1,t ,C2,t , nt )

= (1− τwt )

(1− τwt1+ it

)Note that the labor tax and a positive interest rate distort the marginbetween C1 and n in the same way; "leisure is a credit good"

Labor Distortion

The FOC for nht is

= (1− τwt )

(1− τwt1+ it

Note that the labor tax and a positive interest rate distort the marginbetween C1 and n in the same way; "leisure is a credit good"

Labor Distortion

The FOC for nht is

= (1− τwt )

(1− τwt1+ it

)Note that the labor tax and a positive interest rate distort the marginbetween C1 and n in the same way; "leisure is a credit good"

Government and Equilibrium

Assuming (just for concreteness) the government purchases creditgoods, the public sector’s consolidated budget constraint at time t is

Mt + Bt + τwt−1Wt−1nt−1 = Mt−1 + (1+ it−1)Bt−1 + Pt−1Gt−1 ,

and the equilibrium conditions are Bht = Bt , Mht = Mt , and

C1,t + C2,t + Gt = yt

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