Money and Exchange Rates under Flexible Prices

Michael Bar�

May 2, 2018

Contents

1 Introduction 2

2 Basic concepts in monetary economics 22.1 In�ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 In�ation tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Seigniorage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Relationship between Seigniorage and In�ation Tax . . . . . . . . . . . . . . 62.5 Nominal interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Real interest rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Basic questions in monetary economics . . . . . . . . . . . . . . . . . . . . . 7

3 Cagan Model (1956) 83.1 Solving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Seigniorage and In�ation Tax . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 A simple Monetary Model of Exchange Rates . . . . . . . . . . . . . . . . . 13

4 Cash-In-Advance Model (Lucas 1982) 164.1 Solving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Cobb-Doglas utility: analytical solution . . . . . . . . . . . . . . . . . . . . . 204.4 Direct derivation of the Friedman rule - steady state . . . . . . . . . . . . . . 22

5 Conclusions 23

6 Appendix 24�San Francisco State University, department of economics.

1

1 Introduction

What is money? In this course we assume that money is currency, and we abstract fromthe banking system and devices such as checks and credit cards that may be used to easetransactions. We assume that currency does not bear interest, although this is no longertrue today for some checking accounts. Money serves three purposes in the economy: (1)medium of exchange, (2) store of value, and (3) nominal unit of account. As a medium ofexchange, money reduces the costs of transactions, by eliminating the need for coincidenceof wants1. Money as a store of value has a big disadvantage in that it usually does not bearinterest, but it also has an advantage - liquidity. A liquid asset is considered to be any itemof value that can be quickly turned into cash. A house is an example of non-liquid asset.Money also serves as nominal unit of account, which is convenient for pricing goods andservices. Suppose that the price of apples was denominated in gallons of milk, and the priceof milk was denominated in units of haircuts, and the price of haircuts was denominated inunits of eggs. It would have been very di¢ cult to �gure out relative prices, i.e. which goodsare more expensive. If on the other hand all the prices are quoted in currency, and say theprice of gas is $4 per gallon while the price of milk is $3 per gallon, we can easily see that onegallon of milk buys 3/4 gallons of gas. Before we discuss any economic theory, we introducesome basic concepts monetary economics.

2 Basic concepts in monetary economics

In this section, we de�ne and brie�y discuss the 5 most fundamental concepts in monetaryeconomics: (1) in�ation, (2) in�ation tax, (3) seigniorage, (4) nominal interest rate, (5) realinterest rate.

2.1 In�ation

The �rst, and the most fundamental concept in monetary economics is in�ation. In�ationis the rise of a price level over time. Mathematically, the in�ation rate in period t is thegrowth rate in price level between period t� 1 and period t:

�t =Pt � Pt�1Pt�1

Notice that from the above we have

1 + �t =PtPt�1

, or Pt = Pt�1(1 + �t)

Taking logs,log (1 + �t) = logPt � logPt�1

1In a barter exchange, the producer of shoes who wants a haircut, needs to �nd a barber who wants tobuy shoes. This is called double coincidence of wants.

2

For small in�ation rate �t, we can approximate log (1 + �t) � �t, and therefore you oftenencounter the approximate expression for in�ation

�t � logPt � logPt�1In words, if in�ation rate is small, it is approximately equal to the di¤erence between logsof price levels.

2.2 In�ation tax

The most direct e¤ect of in�ation on our lives is the loss of purchasing power - as prices goup, our money can buy less goods and services. We can view this loss in purchasing power asa tax. Formally, suppose you hold an amount of moneyMt�1 in period t�1. The real moneybalances in period t � 1 is Mt�1

Pt�1, is the purchasing power of that money, i.e. the amount of

goods and services you can buy at time t � 1 with money stock Mt�1. If there is in�ation,i.e. Pt > Pt�1, this money will have smaller purchasing power in period t, i.e.

Mt�1Pt

< Mt�1Pt�1

.In�ation tax is the loss of purchasing power of money (loss in its real value) due to in�ation:

ITt =Mt�1

Pt�1� Mt�1

Pt

Notice that whenever there is in�ation, every individual who holds money, "pays" this in�a-tion tax. In�ation tax is one obvious cost of in�ation that makes it undesirable for consumers.If people anticipate higher in�ation, they usually want to reduce the amount of money theyhold and instead keep their wealth in some other assets (bonds for example).We can rewrite the in�ation tax in a way that conveys more intuition. Using the de�nition

of in�ation, we have Pt = Pt�1(1 + �t). Plugging in the de�nition of in�ation tax, andrearranging, gives:

ITt =Mt�1

Pt�1� Mt�1

Pt�1 (1 + �t)

=Mt�1

Pt�1

�1� 1

1 + �t

�=

Mt�1

Pt�1

��t

1 + �t

�For small rates of in�ation, we have �t

1+�t� �t, so the in�ation tax (for small values of �t)

can be written as

ITt �Mt�1

Pt�1� �t (1)

Equation (1) reveals that for small rates of in�ation, the in�ation tax is approximately equalto the tax base (Mt�1

Pt�1- the real money balances last period), times the tax rate (�t - in�ation

rate since last period). So if in�ation rate is �t = 5%, the purchasing power of money heldsince last period, declines by approximately 5%.To see that the approximation is valid, consider the limit

lim�t!0

��t

1 + �t

�=�t = lim

�t!0

�1

1 + �t

�= 1

which proves that �t1+�t

! �t as �t ! 0.

3

2.3 Seigniorage

We know that the central banks of any country have the authority to print money of thatcountry. By printing new money, central banks generate substantial pro�t, which is paid tothe government. Seigniorage represents the real revenue a government gets from printingmoney. Formally, a government seigniorage at time t is:

SEt =Mt

Pt� Mt�1

Pt=Mt �Mt�1

Pt

Notice that the numerator is the increase in the money supply (the new money printed attime t). Dividing by the price level Pt gives the amount of goods and services that thegovernment can purchase with this new money �M = Mt �Mt�1. It might seem at �rstlook that the central bank can raise unlimited seigniorage, by simply printing large amountof money. However, printing a lot of money typically leads to higher in�ation (as we willdemonstrate in these notes), and therefore the denominator Pt will increase as well. As aresult, there is a limit to how much seigniorage can be collected by printing money. Moreover,price stability is usually a key mission of most central banks. That is why we rarely observecentral banks who try to maximize the seigniorage.To illustrate the limits to seigniorage, we �rst rewrite the de�nition as follows:

SEt =Mt

Pt� Mt�1

Pt=Mt �Mt�1

Mt

� Mt

Pt

The question we would like to address is, what are the limits to the real resources a gov-ernment can collect by printing money? We will limit the discussion to the case of constantgrowth rate in money supply, i.e.

Mt �Mt�1

Mt�1= �

In addition, suppose that the demand for real balances is

Mt

Pt=

�Pt+1Pt

���, � > 0 (2)

This function re�ects the idea that the demand for real money balances is decreasing inexpected in�ation rate (recall Pt+1

Pt= 1 + �t+1). Thus, the seigniorage can be written as

SEt =Mt �Mt�1

Mt

�Pt+1Pt

���In most economic models, and in the real world, when the economy is not growing, constantmoney growth leads to constant in�ation at the same rate as the money growth: � = �. Ina growing economy, the two rates are similar (� � �� Y , where Y is the growth rate of realGDP). Using this result, gives the seigniorage as a function of money (and in�ation) growthrate:

SE =Mt�1 (1 + �)�Mt�1

Mt�1 (1 + �)

�Pt (1 + �)

Pt

���=

�

(1 + �)(1 + �)��

= � (1 + �)���1

4

Notice that when � = 0, the seigniorage is also zero, which makes sense. However, when�!1, the seigniorage also approaches zero:

lim�!1

�

(1 + �)1+�= lim

�!1

1

(1 + �) (1 + �)�= 0

Note that we used L�Hôpital�s rule in deriving the above limit. Thus, in�ation rate that istoo high, also does not generate much revenue.What is the constant growth rate of money supply that would maximize the steady state

seigniorage? It is convenient to maximize the log of the seigniorage:

d ln (SE)

d�=

d

d�[ln�� (� + 1) ln (1 + �)] =

1

�� � + 1

1 + �= 0

1 + � = �� + �

1 = ��

� =1

�

Suppose that � = 0:5, then the rate of money growth that would maximize the seigniorageis � = 200%.Intuitively, note that � is the elasticity of money demand, equation (2), with respect to

gross in�ation rate 1+ �t. The smaller the elasticity of demand is, the smaller is the declinein the real balances held by the public due to in�ation. This allows government to collectmore seigniorage. The next �gure plots the graph of seigniorage, as a function of moneygrowth rate �, for the above example.

0 2 4 6 8 10 12 14 16 18 20

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

Money groth rate

Seigniorage

Notice that seigniorage is maximized at � = 2 = 200%, and declines for higher growth ratesof money.

5

2.4 Relationship between Seigniorage and In�ation Tax

At this point you might be wondering if In�ation Tax and Seigniorage are the same, or howthese two concepts are related. We can demonstrate that the change in real money balancesequals the di¤erence between seigniorage and in�ation tax.

Mt

Pt� Mt�1

Pt�1| {z }�(MP )

=

�Mt

Pt� Mt�1

Pt

�| {z }

SEt

��Mt�1

Pt�1� Mt�1

Pt

�| {z }

ITt

Notice that if the nominal stock of money M and the price level P grow at the same rate,then the seigniorage is equal to the in�ation tax. In a growing economy the prices usuallygrow slower than the stock of money. In this case the real money balances will increase andwe must have SE > IT .

2.5 Nominal interest rate

The nominal interest rate it+1 is the amount of additional dollars that you get at time t+ 1per $1 saved at time t. For example, suppose that nominal interest rate is it+1 = 5%. Thismeans that when you save $1 at time t, you get at time t + 1 the $1 you saved, plus $0:05additional dollars of nominal interest.

2.6 Real interest rate

The real interest rate rt+1 is the amount of additional goods and services you can buy inperiod t + 1 when you give up (save) one unit of goods and services at time t. Supposethat you give up 1 unit of goods and services at time t, which means that you save Ptdollars. In the next period, this money will become Pt (1 + it+1), and can purchase rt+1 =Pt (1 + it+1) =Pt+1 � 1 additional goods and services at time t+ 1 (in addition to the 1 unityou saved at time t). Thus, the relationship between real and nominal interest rate is:

1 + rt+1 =Pt (1 + it+1)

Pt+1(3)

Since 1+ �t+1 = Pt+1=Pt, so the relationship between real and nominal interest rates can bewritten as:

1 + rt+1 =1 + it+11 + �t+1

Taking logs, and using the approximation that log (1 + g) � g for small g, gives the approx-imate relationship between real and nominal interest rate, when all rates are "small":

log (1 + rt+1) = log (1 + it+1)� log (1 + �t+1)

rt+1 � it+1 � �t+1 (4)

It is important that you do not use the approximate formula (4) for countries with in�ationmore that 10%, and instead use the exact relationship (3).

6

2.7 Basic questions in monetary economics

Up until now we analyzed real models, that is we did not have money as one of the goodsin the model. The neoclassical growth model for example, had two commodities: labor and�nal goods. So the price of labor was expressed in units of the �nal goods, e.g. w = 7 meansthat the worker gets 7 units of �nal good per unit of time. In these notes we introducemodels with money, thus the price of all the goods will be denominated in terms of currency(nominal prices). Introducing money into the models will help us addressing questions like:

1. What causes in�ation? In�ation is de�ned as a rise in the price level over time, so thevery de�nition of in�ation requires that prices be nominal (in units of currency).

2. What is the e¤ect of money supply on exchange rates? A nominal exchange rate is aprice of one currency in terms of another, so it is impossible to talk about exchangerates without introducing money �rst.

3. What determines the amount of seigniorage? Seigniorage represents the real revenuesa government gets by printing money to buy goods and non-monetary assets.

4. What are the real e¤ects of increasing money supply? By that we mean, can highermoney supply a¤ect the amount of goods and services produced in an economy, andcan money supply a¤ect the number of workers employed in the economy at any giventime?

5. What is the optimal monetary policy? In particular, how fast should the money sup-ply grow, and should the central bank respond to economic conditions (discretionarypolicy) or use �xed policy rule?

In the rest of these notes we discuss monetary theory. We start with a deceptively simplemodel of money and in�ation due to Cagan 1956. Later in these notes we introduce modelswith micro-foundations, i.e. models in which households explicitly decide on how much towork and how much money they want to hold.

7

3 Cagan Model (1956)

The model presented below consists of demand for real balances equation and equilibriumin the money market, as follows:

[Money demand] :Mdt

Pt= exp

���Et

�Pt+1 � Pt

Pt

��[Money Supply] : M s

t

[Money mkt. eqbm.] : Mdt =M s

t =Mt

For small growth rates, we have

Pt+1 � PtPt

�= ln(Pt+1)� ln (Pt)

In this case, the demand for real balances becomes:

Mdt

Pt= exp [��Et (lnPt+1 � lnPt)]

The �rst equation says that the demand for real balances is decreasing in expected in�ationrate. The in�ation rate between time t and t+1 is de�ned as the rate of change in prices be-tween those two periods, �t+1 = (Pt+1 � Pt) =Pt, and is approximately equal to lnPt+1� lnPtfor small rates of change. The expected in�ation can be written as �et+1 = Et (lnPt+1 � lnPt).The semi-elasticity of demand is ��, which gives the percentage change in the quantity de-manded for real balances that results from 1% change in expected in�ation. To see this,notice that

Mdt

Pt= exp

����et+1

�ln

�Mdt

Pt

�= ���et+1

d ln�Mdt

Pt

�d�et+1

= ��

For example, if � = 0:5, then the above demand implies that a 1% rise in expected in�ationleads to 0.5% decline in the demand for real balances. We assume that � > 0, so that thedemand for real balances is decreasing in expected in�ation rate.The obvious weakness of the Cagan model is the disconnect between the goods market

and the money market. In other words, this is a partial equilibrium model, as opposed togeneral equilibrium models, such as the NGM. In addition, just as the Keynesian model,the Cagan model lacks micro-foundations in the sense that it is not clear what preferencesgive rise to the above mentioned demand for real balances. The Keynesian model howeverassumes that the demand for real balances also depends on real output, while the Caganmodel abstracts from this dependence. The advantage of the Cagan model is its simplicity.The model focuses on the money market in isolation from goods markets, and helps us to

8

develop basic intuition about the relationship between prices (in�ation) and money supply.The model however rules out by assumption any relationship between the money marketand real markets.Substituting the equilibrium condition in the money market into the demand equation,

and taking logs, gives

ln (Mt)� ln (Pt) = ��Et (lnPt+1 � lnPt)

Letting the small case variables denote the logs of the original variables gives

mt � pt = ��Et (pt+1 � pt) (5)

This equation describes the relationship between money supply and prices, but prices appearon both sides of the equation. In the next section we solve the model, i.e. describe the current(log of) price level as an explicit function of (log of) money supply.

3.1 Solving the model

Rewriting equation (5) gives

mt � pt = ��Et (pt+1) + �pt

(1 + �) pt = mt + �Et (pt+1)

pt =1

1 + �mt +

�

1 + �Et (pt+1) (6)

Equation (6) says that the current (log of) price level is a weighted average of current (logof) money supply and expected (log of) future price level. In what follows, I will omit the"log of" comment when it is clear from the notation that the variables are in logs. Leadingequation (6) by one period gives

pt+1 =1

1 + �mt+1 +

�

1 + �Et+1 (pt+2)

and substituting into (6), eliminates pt+1 from the equation:

pt =1

1 + �mt +

�

1 + �Et

�1

1 + �mt+1 +

�

1 + �Et+1 (pt+2)

�=

1

1 + �mt +

1

1 + �

�

1 + �Et (mt+1) +

��

1 + �

�2Et (pt+2)

=1

1 + �

��

1 + �

�0Et (mt) +

1

1 + �

��

1 + �

�1Et (mt+1) +

��

1 + �

�2Et (pt+2)

Note that we use the law of iterated expectations, i.e. E (E (Y jX)) = E (Y ). Repeating thisprocedure to eliminate pt+2, pt+3,..., gives

pt =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms) + lim

T!1

��

1 + �

�TEt (pt+T )

9

We assume that

limT!1

��

1 + �

�TEt (pt+T ) = 0

This assumption is violated if the expected log of price level grows at a rate faster than(1 + �) =�, i.e. the price level itself is growing at least as Pt+1 = P

(1+�)=�t .

With the above assumption, we have the key equation of the Cagan model

pt =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms) (7)

Thus, the current price level is weighted average of all future expected money supplies.This result suggests that besides setting the federal funds rate at certain level, the FED canin�uence the price level through announcements and by establishing reputation (which a¤ectthe Et terms).

3.2 Examples

Example 1: suppose that money supply is constant, so that mt = �m. The intuition tells usthat the in�ation rate should be zero and the price level should be constant. Indeed, using(7) gives:

pt =1

1 + �

1Xs=t

��

1 + �

�s�t�m = �m

1

1 + �

1

1� �1+�

!= �m

Example 2: suppose that mt = �m+ �t. This means that the money supply is growingat approximately constant rate2 of �.

pt =1

1 + �

1Xs=t

��

1 + �

�s�t[mt + � (s� t)]

= mt1

1 + �

1Xs=t

��

1 + �

�s�t+

�

1 + �

1Xs=t

��

1 + �

�s�t(s� t)

= mt +�

1 + �

1Xt=0

��

1 + �

�tt

= mt +

��

1 + �

�� (1 + �) = mt + ��

The last equality uses the fact that

1Xt=0

qtt =q

(1� q)2, where jqj < 1

2Recal that the slope of the ln of a variable is approximately equal to the growth of the original variable,for small growth rates.

10

See the appendix for derivation of this sum. The in�ation rate in this case is

pt+1 � pt = (mt+1 + ��)� (mt + ��)

= mt+1 �mt

= [ �m+ � (t+ 1)]� �m+ �t

= �

We obtained the intuitive result that with constant growth rate of the money supply, pricesgrow at the same rate as the stock of money.Example 3: suppose thatmt = �mt�1+"t, 0 � � � 1, and "t are i.i.d. with Et ("t+1) = 0.

Again, using equation (7), we have

pt =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms)

=1

1 + �

1Xs=t

��

1 + �

�s�t�s�tmt

=mt

1 + �

1Xs=t

���

1 + �

�s�t=

mt

1 + �

1

1� ��1+�

=mt

1 + � � ��

We see that the price level is proportional to the money supply. For example, if � = 0:9 and� = 0:5, then pt �= 0:95 �mt. In the special case of � = 1 (the shocks are permanent), theabove reduces to

pt = mt

The price level thus follows the money supply very closely, and Corr (pt;mt) = 1.

3.3 Seigniorage and In�ation Tax

Seigniorage represents the real revenues a government gets by printing money to buy goodsand non-monetary assets. A government seigniorage at time t is

SEt =Mt

Pt� Mt�1

Pt=Mt �Mt�1

Mt

� Mt

Pt(8)

The numerator is the increase in the money supply and the denominator is the price level.The question we would like to address is, what are the limits to the real resources a gov-ernment can collect by printing money? We will limit the discussion to the case of constantgrowth rate in money supply. In particular, suppose that

Mt

Mt�1= 1 + �

11

Exponentiate (5) for the deterministic case, i.e. we drop the expectation operator and thedemand for real balances becomes

Mt

Pt=

�Pt+1Pt

���Using this model, we showed (see example 2) that prices also grow at the same rate of �, i.e.

PtPt�1

= 1 + �

Using the Cagan model together with constant growth rates of money and prices, gives thefollowing seigniorage:

SEt =Mt �Mt�1

Mt

� Mt

Pt

=Mt�1 (1 + �)�Mt�1

Mt�1 (1 + �)

�Pt+1Pt

���=

�

(1 + �)(1 + �)��

= � (1 + �)���1

What is the constant growth rate of money supply that would maximize the steady stateseigniorage? It is convenient to maximize the log of the seigniorage:

d

d�[ln�� (� + 1) ln (1 + �)] =

1

�� � + 1

1 + �= 0

1 + � = �� + �

1 = ��

� =1

�

Suppose that � = 0:5, then the rate of money growth that would maximize the seigniorageis � = 200%.In�ation tax is the loss of the value of real money balances due to in�ation:

ITt =Mt�1

Pt�1� Mt�1

Pt

=Mt�1

Pt

�PtPt�1

� 1�

=Mt�1

Pt

�Pt � Pt�1Pt�1

�| {z }In�ation rate

Notice that the change in real money balances equals the di¤erence between seigniorage andin�ation tax.

Mt

Pt� Mt�1

Pt�1| {z }�(MP )

=

�Mt

Pt� Mt�1

Pt

�| {z }

SEt

��Mt�1

Pt�1� Mt�1

Pt

�| {z }

ITt

12

Notice that if the nominal stock of money M and the price level P grow at the same rate,then the seigniorage is equal to the in�ation tax. In a growing economy the prices usuallygrow slower than the stock of money. In this case the real money balances will increase andwe must have SE > IT .

3.4 A simple Monetary Model of Exchange Rates

1. Demand for real balances. Consider a variant of the Cagan model presented above,but now the demand for real balances depends on nominal interest rate between timet and time t+ 1 and real income. In log-linear form the demand is given by

mt � pt = ��it+1 + �yt, �; � > 0 (9)

Thus, the elasticity of demand for real balances with respect to nominal interest rateis �� and with respect to real income is �.

2. Purchasing Power Parity (PPP). Let E be the nominal exchange rate, de�ned asthe price of foreign currency in terms of the home currency3, and let P and P � denotethe price level in domestic currency and foreign currency of a basket of goods. PPPholds if the purchasing power of $1 in the home country is the same as in the foreigncountry. In particular

1

Pt=

1

EtP �tPt = EtP �t

The purchasing power of $1 in the home country is 1=P units of the basket of goods.If the dollar is converted to foreign currency using the exchange rate, it gives 1=E unitsof foreign currency, which buys 1=(EP �) units of the basket of goods. In log form, thePPP relationship becomes

pt = et + p�t (10)

where small case variables denote the natural logs of the original variables.

3. Uncovered Interest Parity (UIP). Let it+1 be the interest rate on bonds in thehome country and let i�t+1 be the interest rate on foreign bonds. The uncovered interestparity holds if:

1 + it+1 =�1 + i�t+1

�Et

�Et+1Et

�In a world with perfect foresight (without uncertainty), we expect the UIP to hold viasimple arbitrage argument. The left hand side of the UIP equation is the return oninvesting $1 in domestic bonds. Suppose instead that we invest the same dollar in aforeign bond. First, we need to convert the $1 into foreign currency, which gives 1=Etunits of foreign currency. The return on investing that amount in foreign bonds gives�1 + i�t+1

�=Et units of foreign currency in period t + 1. Converting back to dollars in

3In what follows, we will always assume that the home currency is $.

13

period t + 1 gives the right hand side of the last equation. The UIP in log form isapproximately

it+1 = i�t+1 + Etet+1 � et (11)

Equation (11) uses two types of approximations. The �rst one is ln (1 + g) �= g forsmall g. The second one is lnEt (Et+1=Et) �= Et (ln (Et+1=Et)). The last approximationis pretty bad, since we know that for any strictly concave function u, we always haveu [E (X)] > E [u (x)] (Jensen�s inequality).

Plugging the PPP (10) and UIP (11) relationships in the demand for real balances (9)gives

mt � pt = ��it+1 + �yt

mt � (et + p�t ) = ���i�t+1 + Etet+1 � et

�+ �yt

mt � et � p�t = ��i�t+1 � �Etet+1 + �et + �yt�mt � �yt + �i�t+1 � p�t

�+ �Etet+1 = et + �et

et =1

1 + �~mt +

�

1 + �Etet+1

where ~mt = mt � �yt + �i�t+1 � p�t

The last equation describes the law of motion of the (log of) nominal exchange rate, and isvery similar to the law of motion of prices (6). The solution to the above is thus

et =1

1 + �

1Xs=t

��

1 + �

�s�tEt�ms � �ys + �i�s+1 � p�s

�(12)

In this model, raising the path of domestic money supply, will lead to an increase in et,the price of foreign currency in terms of home currency. This is a depreciation of the homecurrency against the foreign currency. Raising the path of domestic real income or the foreignprices, will lead to appreciation. Higher path of foreign interest rate will lead to depreciationbecause the demand will shift to the foreign currency.Example 4: suppose that ��ys + �i�s+1 � p�s = (some constant), and

mt �mt�1 = � (mt�1 �mt�2) + "t, 0 � � � 1

where "t is i.i.d. with Et�1 ("t) = 0. Thus, the rate of growth of money supply follows andAR(1) process. Using equation (12), �nd the expected depreciation rate of the domesticcurrency.Solution. Lead equation (12) gives:

et =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms � )

et+1 =1

1 + �

1Xs=t+1

��

1 + �

�s�(t+1)Et+1 (ms � )

or et+1 =1

1 + �

1Xs=t

��

1 + �

�s�tEt+1 (ms+1 � )

14

Take date t expectations of both sides

Etet+1 =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms+1 � )

Subtracting et from both sides and using the AR(1) form of mt+1 �mt:

Etet+1 � et =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms+1 � )� 1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms � )

Etet+1 � et =1

1 + �

1Xs=t

��

1 + �

�s�tEt (ms+1 �ms)

Etet+1 � et =1

1 + �

1Xs=t

��

1 + �

�s�t�s�t+1 (mt �mt�1)

Etet+1 � et =�

1 + �(mt �mt�1)

1Xs=t

���

1 + �

�s�tEtet+1 � et =

�

1 + �(mt �mt�1)

1

1� ��1+�

Etet+1 � et =�

1 + � � ��(mt �mt�1)

Suppose that � = 0:9 and � = 0:5, then

�

1 + � � ��=

0:9

1 + 0:5� 0:9 � 0:5�= 0:86

Thus, if the money supply follows the AR(1) process described above, then as a result of a1% increase in the domestic money supply, the expected depreciation of home currency willbe 0:86%.

15

4 Cash-In-Advance Model (Lucas 1982)

The Cagan model gave us some insight into the possible e¤ects of money on prices of goodsand exchange rates (prices of currencies). As was mentioned before, in the Cagan model, allreal magnitudes (real output, hours worked, etc.) are exogenous. In this section we considera version of the Neoclassical Growth Model with money, in an attempt to analyze the reale¤ects of in�ation.

1. Money supply. We assume that government and the central bank are one entity(have consolidated balance sheet). When the government wants to increase the moneysupply, it gives the households a lump-sum transfer of � t. To reduce the money supply,the government collects a lump-sum tax. The money supply therefore evolves accordingto ms

t+1 = mst + � t =

�1 + �t+1

�mst , where �t+1 is the growth rate of money supply.

The required lump-sum transfer at time t is � t = �t+1mst .

2. Bonds. The household can buy or sell privately issued bonds bt that pay nominalinterest it.

3. Cash in advance constraint. In this economy the household must hold cash beforebuying goods. The cash used in purchases of consumption must be set aside in theprevious period (in advance). The CIA constraint is ptct � md

t . Notice that thisconstraint must hold with equality since by holding more than necessary cash, thehousehold gives up interest. The demand for real balances is therefore

mdt

pt= ct

4. Household. A representative household derives utility from consumption and leisure.The period utility is given by u (ct; 1� ht), where ct is consumption and ht is hoursworked. For simplicity, output is produced by the household with production functionyt = ht. The household�s problem is:

maxfct;ht;bdt+1;md

t+1g1t=0

1Xt=0

�tu (ct; 1� ht)

[B.C.] : ptct +mdt+1 + bt+1 = md

t + (1 + it) bt + ptht + � t 8t[CIA] : ptct = md

t 8t

5. Market clearing. For all t = 0; 1; 2; :::

[Labor] : yt = ht

[Goods] : yt = ct

[Money] : mst = md

t = mt

[Bonds] : bst = bdt = 0

Since we have a representative household, it has no one to borrow from or lend to, soin equilibrium, bonds will not be held. An extension of this model that has di¤erent

16

types of households, the sum of all bonds issued is still equal to the bonds bought, buthouseholds will di¤er according to their bonds holdings. For example, in a model withyoung and old households, we expect that the young would buy bonds, and sell themwhen they are retired.

4.1 Solving the model

The Lagrange function for the household�s problem is:

L =1Xt=0

�tu (ct; 1� ht)�1Xt=0

�t [ptct +mt+1 + bt+1 �mt � (1 + it) bt � ptht � � t]�1Xt=0

�t [ptct �mt]

where f�tg1t=0 is a sequence of Lagrange multipliers on the budget constraints and f�tg1t=0 is

the sequence of multipliers on the CIA constraints. Notice that we used the money marketequilibrium condition. The �rst order conditions, for all t = 0; 1; 2; :::, are:

[ct] : �tu1 (ct; 1� ht)� �tpt � �tpt = 0 (13)

[ht] : ��tu2 (ct; 1� ht) + �tpt = 0 (14)

[bt+1] : ��t + �t+1 (1 + it+1) = 0 (15)

[mt+1] : ��t + �t+1 + �t+1 = 0 (16)

Dividing (13) at time t by itself in time t+ 1 gives

�tu1 (ct; 1� ht)

�t+1u1 (ct+1; 1� ht+1)=

pt (�t + �t)

pt+1��t+1 + �t+1

�Using (16) in the above gives

u1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= �

pt�t�1pt+1�t

Now using (15) for �t�1 leads to the Euler equation for optimal investment in bonds:

u1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= �

pt (1 + it)

pt+1(17)

This Euler equation di¤ers from the one in the Neoclassical Growth Model. Remember thatin the NGM we had

u1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= � [1 + rt+1 � �]

The di¤erence between these Euler equations stems from the di¤erent channels that facilitateintertemporal tradeo¤. In the NGM, giving up one unit of ct enables increasing investment inphysical capital, which in period t+1 generates gross return, net of depreciation, of 1+rt+1��.In our version of the CIA model we do not have physical capital, so the depreciation is notpresent there. In the CIA model, giving up one unit of ct allows reducing the beginning ofperiod money, mt, by pt and buy additional bonds bt for that amount. These bonds in turn

17

allow increasing mt+1 by pt (1 + it) and in period t + 1 we can buy pt (1 + it) =pt+1 units ofct+1.An interesting result from the CIA model can be obtained when the CIA constraint is

not binding, i.e. �t = 0. The Euler equation in this case becomes

u1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= �

pt�tpt+1�t+1

Using (15) leads tou1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= �

pt (1 + it+1)

pt+1

Letting the in�ation rate between t and t+ 1 be �t+1, i.e. 1 + �t+1 = pt+1=pt, leads to

u1 (ct; 1� ht)

u1 (ct+1; 1� ht+1)= �

�1 + it+11 + �t+1

�= � [1 + rt+1]

where the last step uses the de�nition or gross real interest rate: 1+rt+1 = (1 + it+1) = (1 + �t+1).The last Euler equation is exactly the same as in the NGM, except the depreciation rate isnot present since we don�t have physical capital. This condition also arises from the solutionof the social planner�s problem, meaning that optimal consumption path would be attainedif it was not for the CIA constraint.Next, dividing (14) by (13) gives

u2 (ct; 1� ht)

u1 (ct; 1� ht)=

pt�tpt (�t + �t)

I keep the pt instead of canceling it out on purpose. Now using (16) gives:

u2 (ct; 1� ht)

u1 (ct; 1� ht)=ptpt

�t�t�1

Finally, using (15) for �t�1 leads to the condition for optimal leisure:

u2 (ct; 1� ht)

u1 (ct; 1� ht)=

ptpt (1 + it)

(18)

Recall the the analogous condition in the NGM was

u2 (ct; 1� ht)

u1 (ct; 1� ht)= wt

which equates the marginal rate of substitution between leisure and consumption, with therelative price of leisure wt. Recall also that the price of consumption was normalized to 1because in the NGMmoney didn�t exist and all prices were expressed in units of consumption.In the CIA model prices are quoted in units of currency, so the price of consumption is pt.Because of our assumption that one unit of labor produces one unit of consumption (seeabove), the wage is also pt. Thus, the relative price of leisure in the CIA model is supposedto be pt=pt = 1. However, the optimality condition (18) is distorted by the CIA constraint,

18

i.e. by the constraint that the individual must hold cash before period t. Thus, if thehousehold wants to buy extra units of consumption at time t, it needs to bring pt extraunits of money mt, which means it needs to reduce bonds bt by that amount and give upinterest. Essentially, the cost of current period consumption becomes pt (1 + it) because ofthe CIA constraint. The CIA constraint therefore distorts the optimal leisure choice bymaking consumption relatively more expensive (and the leisure relatively cheaper). Thisdistortion would disappear however if the nominal interest rate would be zero, i.e. it = 0.The condition of it = 0 is called the Friedman Rule, and we will say more about it below.To summarize, for any exogenous sequence of money supply fmtg1t=0, the equilibrium

time paths of fct; pt; itg1t=0 must solve for all t = 0; 1; 2; :::

[ht] :u2 (ct; 1� ct)

u1 (ct; 1� ct)=

1

1 + it(19)

[EE] :u1 (ct; 1� ct)

u1 (ct+1; 1� ct+1)= �

pt (1 + it)

pt+1(20)

[CIA] : ptct = mt (21)

Notice that I already plugged the feasibility condition that ct = ht in the above.

4.2 Steady state

In this section we analyze a special case of the CIA model with constant growth rate ofmoney supply:

mt+1 = (1 + �)mt

Thus, the transfers in each period are � t = �mt. In the long-run in this model, consumption,output, labor, and nominal interest rate will be constant, i.e. the model converges to asteady state. Intuitively, this model is similar to the NGM without technological progress.Note also that in�ation rate at steady state will be constant and equal to �. To see this,divide two consecutive CIA constraints:

pt+1c

ptc=

mt+1

mtpt+1pt

= 1 + �

1 + � = 1 + �

Form the condition for ht we get that the nominal interest rate must be constant at steadystate. Thus, using the steady state notation ct = yt = ht = c, the equilibrium conditionsare:

[ht] :u2 (c; 1� c)

u1 (c; 1� c)=

1

1 + i

[EE] : 1 = �1 + i

1 + �[CIA] : ptc = mt

[mt] : mt+1 = (1 + �)mt

19

From the Euler equation, the de�nition of � = 11+�

(recall that � is the discount rate), andthe de�nition of real interest rate, we get

[EE] : 1 = �1 + i

1 + �1 + � = 1 + r

r = �

Thus, the real interest rate in steady state must be equal to the discount rate. Intuitively,lower � (or higher �) means that households cares more about the current consumptionrather than the future consumption. Thus, we expect that the relative price of currentconsumption, in terms of future consumption, will be higher.

4.3 Cobb-Doglas utility: analytical solution

Now suppose that u (ct; 1� ht) = � ln ct + (1� �) ln (1� ht). In this special case, we cansolve the model analytically; not only for steady state, but the entire sequence fct; pt; itg1t=0which satis�es equations (19)-(21). With Cobb-Douglas

[ht] :1� �

�

ct1� ct

=1

1 + it

[EE] :ct+1ct

= �pt (1 + it)

pt+1[CIA] : ptct = mt

The Euler equation can be written asct+1pt+1ctpt

= � (1 + it)

Using the CIA constraint for periods t and t+ 1 gives:ct+1pt+1ctpt

=mt+1

mt

� 1 + �t+1

Thus, the Euler equation becomes

1 + �t+1 = � (1 + it) ;

and the nominal interest rate can be solved from the growth rate of money:

1 + it =1 + �t+1

�

Next, using the intratemporal optimality condition [ht], gives the solution for ct as a functionof it:

1� �

�

ct1� ct

=1

1 + it1� �

�(1 + it) ct = 1� ct

ct +1� �

�(1 + it) ct = 1

ct =1

1 + 1���(1 + it)

20

Using the solution for it gives

ct =1

1 + 1���

�1+�t+1�

� = �

� + 1���

�1 + �t+1

�Finally, solving for prices from the CIA constraint:

pt =mt

ct=mt

�

�� +

1� �

�

�1 + �t+1

��We can also solve for the in�ation rate between periods t and t+ 1:

1 + �t+1 =pt+1pt

=

mt+1

�

�� + 1��

�

�1 + �t+2

��mt

�

�� + 1��

�

�1 + �t+1

��=

�1 + �t+1

� � + 1���

�1 + �t+2

�� + 1��

�

�1 + �t+1

�Notice that in�ation rate is not equal to the growth rate of money, unless money grows atconstant rate, i.e. �t+2 = �t+1.In summary, the solution to the model, for given sequence of exogenous money supply

fmtg1t=0 and money growth rate��t+1

1t=0, is given by

it =1 + �t+1

�� 1 (22)

ct = yt = ht =1

1 + 1���(1 + it)

=�

� + 1���

�1 + �t+1

� (23)

pt =mt

�

�� +

1� �

�

�1 + �t+1

��(24)

Observe that consumption (and in this model output and hours worked) is decreasing innominal interest rate it and in the growth rate of money �t+1. Once again, the intuitionis that nominal interest rate is the opportunity cost of holding money, and due to the CIAconstraint, the household is forced to hold money in order to buy consumption. The intuitionbehind the negative e¤ect of money growth on ht is that in�ation reduces the incentive towork, since the proceeds of work can be used only in the next period. The income from workin the current period, loses its purchasing power due to in�ation.Also notice that under the Friedman rule, it = 0 8t, the optimal growth rate of money

can be found:

it = 0 =1 + �t+1

�� 1

1 + �t+1 = �

��t+1 = � � 1

21

Since 0 < � < 1, the optimal growth rate of money supply is negative, i.e., the money supplyshould decrease over time. Alternatively, using � = 1

1+�

1 + �t+1 =1

1 + �

ln�1 + �t+1

�= � ln (1 + �)

��t+1 � ��

Thus, the Friedman rule can be stated as "nominal interest rate on bonds is it = 0", orthat "the growth rate of money supply should be negative, roughly the negative of the timediscount rate: �t+1 � ��".

Example 1 Solve for the equilibrium consumption, output and labor, under the Friedmanrule in this model (Cobb-Douglas utility).

Solution 2 Using equation (23) with it = 0, gives

ct = yt = ht =1

1 + 1���

=1

�+1���

= �

4.4 Direct derivation of the Friedman rule - steady state

Consider the problem of maximizing the lifetime utility at steady state, when utility functionis u (c; 1� h) = � ln c+ (1� �) ln (1� h):

maxc

1Xt=0

�t [� ln c+ (1� �) ln (1� h)] =1

1� �[� ln c+ (1� �) ln (1� c)]

In the last step we used the market clearing condition c = h. The �rst order condition is:

�

c� 1� �

1� c= 0

=) c� = �

To solve for optimal growth rate of money, �, combine the above with the solution for ct asa function of � from equation (23):

� =�

� + 1���(1 + �)

�� + (1� �) (1 + �) = �

1 + � =� (1� �)

1� �= �

�� = � � 1 � ��

This is the same result we obtained in the last section. Once again, since 0 < � < 1, theoptimal growth rate of money supply is negative, i.e., the money supply should decreaseover time. It implies that the optimal in�ation is also negative in this model, since in�ation

22

rate is the same as the growth rate of money. Since at the steady state in�ation is equal tothe growth rate of money supply, � = �, and real interest rate is equal to the time discountfactor, r = �, we have:

�� = �� � �� = �rThus, under the Friedman rule, the in�ation rate should be equal to the negative of thereal interest rate, which is approximately the same as saying that the nominal interest rateshould be zero. The intuition for this result is simple. Since there is a delay in using moneyin this model (it is earned in advance, before it can be used for purchases), the money losesvalue and imposes a distorting tax on work whenever in�ation is positive. The ine¢ ciencyin this model comes from the CIA constraint, which forces households to hold an inferiorasset that does not pay interest. However, if the nominal interest rate on bonds was zero,then money and bonds would earn the same return.

Exercise 3 Derive the Friedman rule by comparing the equilibrium conditions of the CIAmodel, to the solution of the Social Planner�s problem:

maxfct;htg1t=0

1Xt=0

�tu (ct; 1� ht)

[Feasibility] : ct = ht 8t

Observe that the social planner does not utilize markets, which require the use of money andtherefore distorted by the CIA constraint. The social planner simply tells the household howmuch to work and consume, given the feasibility constraint.

5 Conclusions

Using the Cagan model and the CIA model, we found that under certain assumptions: (1)in�ation rate at steady state (long-run) is the same as the rate of growth of money, (2)currency depreciation is positively related to the growth rate of money supply, (3) nominalinterest rates move in the same direction as money supply, and (4) consumption is negativelycorrelated with the growth rate of money supply. The last result means that increasing therate of growth of money supply, in CIA model, cannot stimulate the economy. Monetaryexpansion, which is often used to stimulate an economy in a recession, does not work inthis model. Some other features are needed for positive impact of money expansion on realconsumption, output and employment. For example, the most widely used modi�cation tothe above model is introduction of nominal wage contracts, which are �xed in the short-run.Under this condition, faster money growth and higher in�ation, may (in the short-run) lowerthe real wage and increase employment - i.e. boost the economy.

23

6 Appendix

We show how to evaluate the summationP1

t=0 qtt, where jqj < 1.

1Xt=0

qtt = 0 + q1 + 2q2 + 3q3 + :::

=

1Xt=1

qt +

1Xt=2

qt +

1Xt=3

qt + :::

=�q + q2 + q3 + :::

� 1Xt=0

qt

=q

1� q� 1

1� q=

q

(1� q)2

In the case of q = �=(1 + �) we have

1� q = 1� �

1 + �=1 + � � �

1 + �=

1

1 + �

q

(1� q)2=

�1+��11+�

�2 = � (1 + �)

24