MONETARY ECONOMICS.

Theory and Policy.

Professor: Mikel Casares

Departamento de Economa,

Universidad Pblica de Navarra

October 8, 2007

Abstract

The object of these lectures is to provide a short course on monetary economics and essential

portions of macroeconomics. The course is intended for advanced students but presented

in a manner that emphasizes economic substance rather than mathematical technique. We

will begin with three chapters devoted to analyze the role of money in modern economies by

using a general equilibrium deterministic framework. Four models will be presented: money

in the utility function model, cash-in-advance model, shopping time model, and transaction

costs model. All of them have in common the medium of exchange role for the asset to be

regarded as money. A money demand function will be derived in each of them in order to

determine the motivations to hold money. The superneutrality property, the Chicago rule

on optimal ination, and the welfare cost of ination will be then discussed.

In the second part of the course we will learn how to solve rational expectation models in

order to obtain the Minimal State Variable solution. These techniques will allow us to solve

the dynamic models in the short run in the presence of uncertainty. As one initial example,

an IS-LM type model of the aggregate demand with constant prices will be solved. Then, we

will lift the constant price assumption by introducing exible prices and some price setting

behavior. In turn, the aggregate supply equation of the model will be obtained to complete

the model.

In the last part of the course, attention will focus on business cycle and monetary pol-

icy analysis. Monetary policy rules will be examined taking either the targeting rules or

instrument rules approach. Taylor type monetary policy rules will be studied in detail as a

representative example of instrument rules on the nominal interest rate.

PART 1. Long-run analysis.Reading list:

Casares, Miguel (2000), Long-run and short-run issues in optimizing monetary models.

Doctoral thesis. Chapter 1, pages 5-61.

McCallum, Bennett T., (1989). Monetary Economics. Theory and Policy. Macmillan

Publishing Company Eds. New York, USA. Chapter 2, pages 16-32.

McCallum, Bennett T., (1999). A Course in Macro and Monetary Economics. Lecture

Notes. GSIA, Carnegie Mellon University. Chapter 1, pages 1-12.

Walsh, Carl E., (2003). Monetary Theory and Policy, 2nd Edition. MIT Press Eds.

Cambridge, MA. Chapters 2 and 3, pages 43-134.

1 A General-Equilibrium Framework with Money

1.1 Introduction

Denition of money.

The starting point of any Monetary Economics course should be some discussion on the

nature and denition of money. Our denition of money will be M1 which roughly speaking

comprises two components: currency in circulation and checking accounts. Broader den-

itions of money like M2 or M3 would also incorporate savings deposits or money market

(short-term) assets. However we will not consider them in this course for reasons men-

tioned below. According to the M1 denition, the stock of money of one economy has two

characteristics:

1) Perfect liquidity. Money is an asset that can be used by every consumer to buy every

good without any charge.

2) Zero nominal yield. Money is an asset that pays no nominal interest rate. Its real

rate of return will be given by the rate of ination. If prices go up (positive ination) the

purchasing power of money will fall so as to have a negative real rate of return. Likewise, a

positive rate of return could be obtained if prices went down and more goods and services

could be purchased with the same amount of money.

Functions of money.

It is said that there are three functions of money:

1

1) Medium of exchange. Money facilitates transactions as it is accepted by everybody

in the exchange. This function is based on the perfect liquiditycharacteristic mentioned

above. Obviously, the medium of exchange role of money has a tremendous impact on the

volume of purchases and sales in the economy. Think for a moment of two economies with

the same resources but one with money and one without money (barter economy). What

would happen in the non-monetary economy ? Problems such as the double coincidence of

wants and non-divisible goods would arise making the exchanges very costly. Transaction

costs would be much higher in this economy without money.

2) Medium of account. Another important function provided by money is serving as a

medium of account that gives prices to all the distinct goods of the economy. This function of

money also reduces substantially the transaction costs. In particular, the information costs

are much lower in a monetary economy. The introduction of money allows us to express the

value of all goods (their price) in terms of the units of money. In an economy with N goods,

there would be N-1 prices to learn. If this economy were a barter economy shoppers would

have N(N-1)/2 relative prices to learn. As one illustrative example, I suggest you calculate

the number of prices to memorize when there are 1000 goods in the economy (N=1000).

3) Store of value. Income can be saved for the future in the form of money. As money

does not deteriorate over time (at least over quite a long period of time!), it can be used to

store value. In modern economies the store-of-value property is also found in other assets

such as bonds, shares or real state properties. Since money pays no interest, it is considered

inferior to any of the other store-of-value assets: bonds pay an interest yield, shares pay

dividends and real state pay a rental rate. This inferiority is increased when money is falling

in value because of a positive rate of ination. From a purely rational perspective, our

savings should be placed in some risk-free interest-bearing asset and money demand for this

purpose should be zero.

During this course the role of money as a medium of exchange will be emphasized so

as to be the nal motivation to demand money. The reason is as follows: the medium

of exchange function of money is the only one that distinguishes money from any other

asset because the medium-of-account role could conceivably fall to some other commodity

whereas there are many nonmonetary assets that serve as stores of value. Subsequently, our

denition of money should only include purely medium-of exchange assets like M1. Other

more inclusive denitions (M2, M3), which will not be considered here, include some interest-

bearing components that loose the full liquidity property and gain motivation to demand

money as a store of value.

2

Our models will be built from optimizing behavior of economic agents. In particular,

a discrete-time version of the neoclassical model developed by Sidrauski (1967) and Brock

(1975) will be taken here. In this framework, the economy is formed by innitely-lived

households who are also producers and maximize their intertemporal utility subject to a

sequence of constraints.

The nature of money will be analyzed in four dierent models that incorporate, more

or less explicitly, the medium-of-exchange function of money: money in the utility function

model (MIU), cash-in-advance model (CIA), transactions technology model, and shopping

time model. In this rst part of the course attention will focus on long-run analysis. The

eects of money and monetary policy in the long run will be explored in a deterministic

framework. As stochastic elements have no inuence in the steady state, the models will be

studied in a perfect foresight scenario.

1.2 Money in the utility function model (MIU)

In the money in the utility function model (MIU), the arguments of the utility function

are consumption units c, leisure time l, and real money balances m = MPbeing M nominal

money and P the price level. Thus, the representative household of the economy wants at

time t to maximize

U(ct;mt; lt) + U(ct+1;mt+1; lt+1) + 2U(ct+2;mt+2; lt+2) + :::;

where = 11+

and > 0 is the intertemporal rate of discount. Households are also producers

and obtain their output yt by employing the demand for labor ndt and the stock of capital ktwithin the existing production technology

yt = f(ndt ; kt):

Both the utility function and the production function are well-behaved functional forms in

the sense that they have positive rst-order derivatives, negative second-order derivatives,

and satisfy the Inada conditions.

The budget constraint faced by the household in period t is in nominal terms

Ptf(ndt ; kt)+Gt = Ptct+Pt (kt+1 (1 )kt)+Wt(ndtnst)+(1+Rt)1Bt+1Bt+MtMt1:

There are two sources of income: Ptf(ndt ; kt) is the nominal output produced in period t, and

Gt is the government net transfers to the household. Income is spent on consumption Ptct, on

3

capital accumulation for next periods production net of capital depreciation kt+1(1)kt,being the rate of depreciation of capital, on nominal wage spending for the labor demand net

of labor supplyWt(ndtnst), on increasing government bonds balances (1+Rt)1Bt+1Bt, andon increasing money balances for the current period Mt Mt1. Regarding the governmentbonds purchases, (1 +Rt)1Bt+1 is the amount bought in period t by the household so that

it produces a reimbursement equal to Bt+1 in period t+ 1. Thus, Rt is the nominal interest

rate over period t.

If we divide the previous expression by the price level, the household budget constraint

in real terms becomes

f(ndt ; kt) + gt = ct + kt+1 (1 )kt + wt(ndt nst) + (1 +Rt)1Bt+1Pt

bt +mt Mt1Pt

;

where lower case variables denote the real-term value of the capital case variable.

Let us dene t as the rate of ination in period t

t =PtPt1

1;

and rt as the real rate of interest in period t from

1 + rt =1 +Rt1 + t+1

Back to the household budget constraint, dividing and multiplying Bt+1Pt

by Pt+1 andMt1Ptby

Pt1, and using both the ination and the real interest rate denitions result in

f(ndt ; kt)+ gt = ct+ kt+1 (1 )kt+wt(ndt nst)+ (1+ rt)1bt+1 bt+mt (1+t)1mt1:

As for the time constraint, households divide their total time T between labor supply and

leisure time

T = nst + lt:

Taking the previous lines into consideration, the optimizing program to solve in the MIU

model is

Maxct;kt+1;ndt ;n

st ;lt;bt+1;mt

U(ct;mt; lt) + U(ct+1;mt+1; lt+1) + 2U(ct+2;mt+2; lt+2)::::::

subject to

f(ndt+j; kt+j) + gt+j ct+j kt+1+j + (1 )kt+j wt+j(ndt+j nst+j) (1 + rt+j)1bt+1+j + bt+j mt+j + (1 + t+j)1mt1+j = 0; j = 0; 1; 2; :::

4

T nst+j lt+j = 0; j = 0; 1; 2; :::Let us recall that rst order conditions are obtained by taking partial derivatives of the

Lagrangian function with respect to all the choice variables and the Lagrange multipliers,

and making them equal to zero. The Lagrangian function of the optimizing program at hand

is

Lt = U(ct;mt; lt) + U(ct+1;mt+1; lt+1) + 2U(ct+2;mt+2; lt+2) + :::

+tf(ndt ; kt) + gt ct kt+1 + (1 )kt wt(ndt nst) (1 + rt)1bt+1 + bt mt + (1 + t)1mt1

+t+1[f(n

dt+1; kt+1)+gt+1ct+1kt+2+(1)kt+1wt+1(ndt+1nst+1)(1+rt+1)1bt+2+bt+1mt+1 + (1 + t+1)1mt] + :::+ 't [T nst lt] + 't+1

T nst+1 lt+1

+ :::;

where t and 't are respectively the Lagrange multipliers of the budget and time constraints

in period t. Thus, the rst order conditions regarding the choice variables in period t are1

Uct t = 0; (cfoct )t + t+1(1 + fkt+1 ) = 0; (kfoct+1)

t(fndt wt) = 0; (nd;foct )

twt 't = 0; (ns;foct )Ult 't = 0; (lfoct )

t(1 + rt)1 + t+1 = 0; (bfoct+1)Umt t + t+1(1 + t+1)1 = 0; (mfoct )

f(ndt ; kt) + gt ct kt+1 + (1 )kt wt(ndt nst) (1 + rt)1bt+1 + bt mt + (1 + t)1mt1 = 0;(foct )

T nst lt = 0: ('foct )

A Competitive Equilibrium (CE) in the MIU model

A CE of the economy consists of the nine rst order conditions, together with the gov-

ernment budget constraint

gt = (1 + rt)1bt+1 bt +mt (1 + t)1mt1;

1The partial derivatives take the following notation for a generic F (xt; yt) function

Fxt =@F (xt;yt)

@xtand Fyt =

@F (xt;yt)@yt

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the labor market equilibrium condition

ndt = nst ;

the production technology

yt = f(ndt ; kt);

and denitions of the rate of ination t = PtPt1 1, the real interest rate 1+rt = 1+Rt1+t+1 , andreal money balances mt = MtPt .

2 In turn, there are fteen equations in the CE system. Fiscal

and monetary policy determine exogenously paths for gt andMt. Hence, the fteen equations

of our CE economy govern the dynamic behavior of the fteen endogenous variables ct; kt+1;

bt+1; ndt ; n

st ; lt, mt; yt, t, 't, wt, Pt, rt, Rt, and t provided the predetermined values kt, bt;

mt1, and Pt1:

Obtaining the Structural Equations

Substituting t+1 = t(1 + rt)1 from (bfoct+1) into (k

foct+1) and simplifying lead to the

capital accumulation equation

fkt+1 = rt:The optimal stock of capital left for the next period is the one whose marginal productivity

minus the depreciation rate is equal to the real interest rate. This is the arbitrage condition

between the return on the physical asset (capital) and the real return on the nancial asset

(bonds).

Taking the values of the Lagrange multipliers t and t+1 respectively from (cfoct ) and the

corresponding (cfoct+1) to the bonds rst order condition will bring about the intertemporal

consumption equationUct

Uct+1= 1 + rt:

Marginal utility of consumption in period t relative to discounted marginal utility of con-

sumption in period t+ 1 is equal to their relative price.

Finally the money demand equation can be obtained by inserting t+1 = t(1 + rt)1

from (bfoct+1) into (mfoct ), rearranging and using the nominal interest rate denition

UmtUct

= Rt1+Rt

:

2Since money supply is determined exogenously the real money balances denition mt = MtPt can also be

viewed as the money market equilibrium condition.

6

Now the ratio of marginal utilities is equal to their relative income price. Notice that the

price of real money balances is the discounted nominal interest rate Rt1+Rt

, which reects that

the opportunity cost of holding real money balances takes into account both the real interest

rate and next periods rate of ination.

1.3 Cash-in-advance model (CIA)

Here we have another view of the role of money as a medium-of-exchange. Money as some-

thing necessary to conduct any purchase. It is the only counterpart to any transaction.

This is the reason why the cash-in-advance constraint emerges. The household must keep

an amount of money at least equal to the desired consumption expenditures

Mt+j Pt+jct+j j = 0; 1; 2; :::;

dividing by the price level, it becomes in real terms

mt+j ct+j j = 0; 1; 2; :::

Money is only demanded to satisfy the cash-in-advance constraint. It does not enter the

utility function which now only depends on consumption and leisure. However, the cash-in-

advance is one more constraint attached to the maximization problem3

Maxct;kt+1;ndt ;n

st ;lt;bt+1;mt

U(ct; lt) + U(ct+1; lt+1) + 2U(ct+2; lt+2) + ::::::

subject to

f(ndt+j; kt+j) + gt+j ct+j kt+1+j + (1 )kt+j wt+j(ndt+j nst+j) (1 + rt+j)1bt+1+j + bt+j mt+j + (1 + t+j)1mt1+j = 0; j = 0; 1; 2; :::

T nst+j lt+j = 0; j = 0; 1; 2; :::mt+j ct+j = 0 j = 0; 1; 2; :::

3The cash-in-advance inequality turns to be binding because the nominal interest rate is always greater

than zero.

7

Once built the Lagrangian function, the computation of the rst order conditions of the

program in period t results in

Uct t t = 0; (cfoct )t + t+1(1 + fkt+1 ) = 0; (kfoct+1)

t(fndt wt) = 0; (nd;foct )

twt 't = 0; (ns;foct )Ult 't = 0; (lfoct )

t(1 + rt)1 + t+1 = 0; (bfoct+1)t + t+1(1 + t+1)1 + t = 0; (mfoct )

f(ndt ; kt) + gt ct kt+1 + (1 )kt wt(ndt nst) (1 + rt)1bt+1 + bt mt + (1 + t)1mt1 = 0;(foct )

T nst lt = 0; ('foct )mt ct = 0; (foct )

where t, 't, and t are respectively the Lagrange multipliers attached to the budget, time,

and cash-in-advance constraints in period t.

A Competitive Equilibrium (CE) in the CIA model

A CE of the economy consists of the ten rst order conditions, together with the govern-

ment budget constraint

gt = (1 + rt)1bt+1 bt +mt (1 + t)1mt1;

the labor market equilibrium condition

ndt = nst ;

the production technology

yt = f(ndt ; kt);

and denitions of the rate of ination t = PtPt1 1, the real interest rate 1+rt = 1+Rt1+t+1 , andreal money balances mt = MtPt . In turn, there are sixteen equations in the CE system. Fiscal

and monetary policy determine exogenously paths for gt and Mt. Hence, the sixteen equa-

tions of our CE economy govern the dynamic behavior of the sixteen endogenous variables

ct; kt+1; bt+1; ndt ; n

st ; lt, mt; yt, t, 't, t; wt, Pt, rt, Rt, and t provided the predetermined

values kt, bt; mt1, and Pt1:

8

1.4 Transaction costs model

The transaction costs approach assesses that some output resources of the household are

employed in paying transaction costs when carrying out transactions. To reach the desired

level of consumption ct, the household must face some costs (transportation, searching costs,

trading costs,...) by using units of output within the available transactions technology.

Thus, transaction costs are expressed in output units whereas in the shopping time model

below they will be given in time units. This creates a new element in the household budget

constraint: a transactions cost function h(ct;mt). More consumption requires more resources

utilized to pay the transaction costs for the additional purchases. Real money holdings is

the other entry of this function. Since money is the medium of exchange, it provides services

that make transactions easier and decrease the costs associated with shopping. Hence,

the transactions cost function is aected in a positive way by the consumption level and

negatively by the amount of real money

ht = h(ct;mt);

with hct > 0; hmt < 0; hc2t > 0; hm2t > 0; hctmt < 0 and h(0;mt) = 0. The transactions-

facilitating property of money as a medium of exchange is represented through the signs

hmt < 0 and hctmt < 0 which imply that the use of more monetary services reduces the total

and marginal transactions costs. Thus, the cross derivative is negative, implying that the

marginal cost of purchasing is lower in the presence of more monetary services to facilitate

transactions. In addition consumption marginal transaction costs are increasing (hc2t > 0)

and real money marginal savings of transaction costs are decreasing (hm2t > 0).

As the medium-of-exchange function of money is included in the transaction costs func-

tion, real money balances do not appear in the utility function. Households utility only

depends on consumption. The optimizing program to solve in the transaction costs model is

Maxct;kt+1;ndt ;n

st ;lt;bt+1;mt

U(ct; lt) + U(ct+1; lt+1) + 2U(ct+2; lt+2) + :::

subject to

f(ndt+j; kt+j) + gt+j ct+j kt+1+j + (1 )kt+j wt+j(ndt+j nst+j)(1+rt+j)1bt+1+j+bt+jmt+j+(1+t+j)1mt1+jh(ct+j;mt+j) = 0; j = 0; 1; 2; :::

T nst+j lt+j = 0; j = 0; 1; 2; :::

9

that turns into the following rst order conditions

Uct t(1 + hct) = 0; (cfoct )t + t+1(1 + fkt+1 ) = 0; (kfoct+1)

t(fndt wt) = 0; (nd;foct )

twt 't = 0; (ns;foct )Ult 't = 0; (lfoct )

t(1 + rt)1 + t+1 = 0; (bfoct+1)t(1 + hmt) + t+1(1 + t+1)1 = 0; (mfoct )

f(ndt ; kt) + gt ct kt+1 + (1 )kt wt(ndt nst) (1 + rt)1bt+1 + bt mt + (1 + t)1mt1 ht = 0;(foct )

T nst lt = 0; ('foct )

where t and 't represent the Lagrange multipliers associated to the budget and time con-

straints in period t. Note the presence of the transactions technology in (cfoct ), (mfoct ), and

(foct ).

1.5 Shopping time model

In a shopping time model, there are transaction costs expressed in units of time and money

appears aecting the transactions time technology. As being the medium of exchange, money

is utilized by the household to economize the time spent on carrying out transactions. With-

out money, the household must negotiate a payment by credit or spend some time on trans-

forming part of its income to money. These alternatives to money suppose a higher amount

of time spent on shopping. Besides, it seems obvious that a greater amount of consumption

will require more shopping time. Thus, we present a generic shopping time function whose

arguments are consumption and real money balances

st = s(ct;mt)

with sct > 0; smt < 0; sc2t > 0; sm2t > 0; sctmt < 0; and s(0;mt) = 0, equivalently to the

transactions technology introduced in the transaction costs model. Again, the medium-of-

exchange function of money is represented by the signs smt < 0 and sctmt < 0 which imply

that the use of more monetary services reduces the total and marginal transactions costs.

10

Taking into account the shopping time in the time constraint, households spend their

time on three activities: labor supply (nst), shopping (s(ct;mt)) and leisure (lt),

T = nst + s(ct;mt) + lt:

Hence, the optimizing program of the shopping time model will be

Maxct;kt+1;ndt ;n

st ;lt;bt+1;mt

U(ct; lt) + U(ct+1; lt+1) + 2U(ct+2; lt+2) + :::

subject to

f(ndt+j; kt+j) + gt+j ct+j kt+1+j + (1 )kt+j wt+j(ndt+j nst+j) (1 + rt+j)1bt+1+j + bt+j mt+j + (1 + t+j)1mt1+j = 0; j = 0; 1; 2; :::

T nst+j s(ct+j;mt+j) lt+j = 0; j = 0; 1; 2; :::The rst order conditions coming up from the optimizing program in period t are

Uct t 'tsct = 0; (cfoct )t + t+1(1 + fkt+1 ) = 0; (kfoct+1)

t(fndt wt) = 0; (nd;foct )

twt 't = 0; (ns;foct )Ult 't = 0; (lfoct )

t(1 + rt)1 + t+1 = 0; (bfoct+1)t + t+1(1 + t+1)1 'tsmt = 0; (mfoct )

Ult 't = 0 (lfoct )f(ndt ; kt) + gt ct kt+1 + (1 )kt wt(ndt nst) (1 + rt)1bt+1 + bt mt + (1 + t)1mt1 = 0;

(foct )

T nst s(ct;mt) lt ('foct )

where t and 't are the Lagrange multipliers associated with the budget and time constraints

respectively. Notice the presence of the shopping technology in (cfoct ), (mfoct ), and ('

foct ).

PROBLEMS.

1.1. Let us assume that the representative household of an MIU economy has a logarith-

mic separable utility function in consumption, leisure, and real money balances U(ct;mt; lt) =

11

log ct + a log lt + b logmt with a,b > 0; and a Cobb-Douglas production function f(ndt ; kt) =ndt1

kt with 0 < < 1.

i) Calculate the capital accumulation, consumption, and money demand equations.

ii) Take logarithms on both sides of the consumption equation, use the approximation

log(1 + xt) = xt when xt is a small number for log(1 + rt) and solve for log ct. Is real moneyaecting consumption decisions?

iii) Take logarithms on both sides of the money demand equation and calculate the

elasticities of real money with respect to consumption and the nominal interest rate (takeRt1+Rt

to represent the nominal interest rate).

iv) What would happen to capital, consumption, and real money balances if there were

an increase in b?

1.2. Now the representative household has a non-separable utility function displaying

constant relative risk aversion as in Walsh (1998) page 69

U(ct;mt; lt) =

ctm

bt

11 +

l1t1 with b; ; ; > 0, 0 < b (1 ) < 1;

and still a Cobb-Douglas production function f(ndt ; kt) =ndt1

kt with 0 < < 1.

i) Calculate the capital accumulation, consumption, and money demand equations.

ii) Take logarithms on both sides of the consumption equation, use the approximation

log(1 + xt) = xt when xt is a small number for log(1 + rt) and solve for log ct. Is real moneyaecting consumption decisions?

iii) Take logarithms on both sides of the money demand equation and calculate the

elasticities of real money with respect to consumption and the nominal interest rate (takeRt1+Rt

to represent the nominal interest rate).

iv) What would happen to capital, consumption, and real money balances if there were

an increase in b?

1.3. Calculate the capital accumulation, consumption, and money demand equations for

the CIA model described in section 1.3. Regarding the money demand equation, what is the

consumption elasticity? And what is the nominal interest rate elasticity?

1.4. Describe a Competitive Equilibrium (CE) for the transactions technology model of

section 1.4.

1.5. Calculate the capital accumulation, consumption, and money demand equations for

the transactions technology model described in section 1.4.

12

1.6. In a transaction costs model the transactions technology is given by the following

transaction costs function

h(ct;mt) =

8>:0 if ct = 0

h0 + h1ch2tmh3t

if ct > 0

9>=>;with h0; h1; h3 > 0 and h2 > 1:

Assuming a logarithmic utility function in consumption and leisure U(ct; lt) = log ct+a log ltwith a > 0; and a Cobb-Douglas production function f(ndt ; kt) =

ndt1

kt with 0 < < 1,

i) Verify that the transactions technology satisfy hct > 0; hmt < 0; hctmt < 0 and

h(0;mt) = 0:

ii) Calculate the money demand equation in logarithmic representation.

iii) Describe how an increase in h3 would aect the elasticities in the money demand

equation.

1.7. Describe a Competitive Equilibrium (CE) for the shopping time model of section

1.5.

1.8. Obtain the money demand equation for the shopping time model described in section

1.5. Discuss some intuitive interpretation.

1.9. In a shopping time model the transactions technology is given by the following

function

s(ct;mt) =

8 0

9=;with s0; s1; s3 > 0 and s2 > 1:

Assuming a logarithmic utility function in consumption and leisure U(ct; lt) = log ct+a log ltwith a > 0; and a Cobb-Douglas production function f(ndt ; kt) =

ndt1

kt with 0 < < 1,

i) Calculate the money demand equation in logarithmic representation.

ii) Describe how an increase in s3 would aect the elasticities in the money demand

equation.

13

2 Steady State and Superneutrality

The steady state is a situation in which every variable of the model grows at the same constant

rate. Moreover, this constant growth scenario will be maintained forever. In this sense the

steady state is dened as a long-run equilibrium situation. In neoclassical models the source

of long-run growth (or steady-state growth) is the growth in the technological process. Since

the technological process happens to be exogenous, the neoclassical growth model is classied

as an Exogenous Growth Model. Any growth rate given to the technological process would

become the steady-state rate of growth of the variables of the economy. Having a positive

growth would just imply that variables were growing at a positive rate over time in steady

state but this would not explain anything about the last determinants of these variables.

Throughout this course we will abstract from positive economic growth because this is not

relevant on the steady state properties of the model.

Thus, our rst task is to compute the steady state solution of our monetary models. Let

us start by showing four general results.

2.1 General steady-state results

Result 1. Steady state real interest rate.

In every monetary model of chapter 1, the bonds rst order condition (bfoct+1) is

t(1 + rt)1 t+1 = 0:

The Lagrange multiplier is constant in steady state because there is no long-run growth in

consumption. If the economy were already in steady state in period t, for the coming period

the Lagrange multiplier would be the same t = t+1 and so the real interest rate would.

Simplifying notation by dropping the time subscripts

(1 + r)1 = 0:

Eliminating from the equation and recalling = 11+

r =

The real rate of interest is equal to the households intertemporal rate of discount.

Result 2. Steady state marginal product of capital.

14

In every monetary model of chapter 1, the capital rst order condition (kfoct+1) is

t + t+1(1 + fkt+1 ) = 0;

which in steady state becomes

+ (1 + fk ) = 0:

Eliminating , and using = 11+

result in

fk = +

The marginal product of capital is equal to the households intertemporal rate of discount

plus the rate of depreciation of capital. This condition is known as the modied golden rule

in Economic Growth.

Result 3. Steady state real money balances.

The money demand functions obtained from the structural equations of the MIU model

(page 6), the CIA model (exercise 1.3), the transactions technology TT model (exercise 1.5),

and the shopping time TT model (exercise 1.8) are

UmtUct

=Rt

1 +Rt(MIU)

mt = ct (CIA)

hmt =Rt

1 +Rt(TT)

wtsmt =Rt

1 +Rt(ST)

The amount of real money balances in steady state will be constant in the four models

because the nominal interest rate, consumption, and the real wage are constant values in the

no growth steady state.

Result 4. Money growth, ination and the nominal interest rate.

From the previous result we have m = MPconstant in steady state, which means that

nominal money and prices are growing at the same rate in steady state. The growth rate of

the price level is the denition of ination. The money growth rate = 4MM

is set by the

central bank exogenously. Therefore, the long-run monetary policy will determine the rate

of ination of the economy

= ;

15

and the nominal interest rate

1 +R = (1 + ) (1 + ) ;

with = .

2.2 The steady-state solution in the MIU model

The procedure consists of taking the Competitive Equilibrium of the model described in pages

5 and 6 in steady state. Then, after some algebra to eliminate the Lagrange multipliers and

using the general results 1-4, we can reach the following subset of six equations

UlUc

= w;

fn = w;

l = T n;y = f(n; k)

y = c+ k;

fk = + ;

In principle, the six-equation system would nd steady-state values for the real sector

variables y; n; k; w; c; and l. However, a closer look would bring about the question of

consumptionmoney separability. The real sector is determined by that six-equation econ-

omy only if real money does not appear in the system. If the quantity held of real money

balances m does aect marginal utility of consumption Uc; at least another equation the

money demand equationshould be added to complete the system.4 In that case, there is no

longer consumption-money separability and the equations representing the monetary sector

of the economy must be attached to the real sector

UmUc

=R

1 +R;

1 +R = (1 + ) (1 + ) ;

= ;

in order to obtain steady state values for y; n; k; w; c; l,m,R, and .

4From an intuitive perspective, the presence of money should be expected because the medium-of-exchange

function of money would yield a positive cross derivative Ucm > 0.

16

2.3 Superneutrality in the MIU model

One model is said to have the property of superneutrality when the equilibrium of the real

sector is independent of the rate of growth of nominal money . The MIU model exhibits

superneutrality if there is consumption-money separability, i.e., if Ucm = 0.

* Example of a MIU model featuring separability, Ucm = 0.

There is a MIU economy where households have a separable utility function U(ct;mt; lt) =

log ct + a log lt + b logmt with a,b > 0; and a Cobb-Douglas production function f(ndt ; kt) =ndt1

kt with 0 < < 1. For a quarterly-period calibration, let us assume the values of

the parameters = 0:36, = 0:025, = 0:005, a = 1:828, b = 0:02, T = 3, and = 0:01.

i) Compute the steady state values of y; n; k; w; c; and l.

The steady-state real sector system of equation for this economy is

al1c

= w;

(1 )nk = w;l = T n;y = n1k

y = c+ k;

n1k1 = + ;

and inserting the calibration of parameters

1:828c

l= w;

0:64

k

n

0:36= w;

l = 3 n;y = n

k

n

0:36y = c+ 0:025k;

0:36

k

n

0:64= 0:03:

Solution: k = 48:553, y = 4:046, c = 2:832, w = 2:589, n = 1, and l = 2. Hint: you

should begin by using the result kn= 48:55 obtained straightforward from the last equation

to calculate easily the real wage.

17

ii) Compute the steady state values of m,R, and .

The monetary sector is dened in steady state by the following equations

0:02m1c

=R

1 +R;

1 +R = (1 + 0:005) (1 + ) ;

= 0:01;

where inserting the steady state consumption c = 2:832 lead to m = 3:833, R = 0:015, and

= 0:01:

iii) Now let us compare this economy with a very similar one that only diers from

having a higher money growth rate 0 = 0:02. Recalculate the steady state values of the

variables of and discuss the long-run eects of having a higher nominal money growth (non-

superneutrality).

There is no eect whatsoever in the real sector of the economy since no monetary variable

enter its six-equation system. However, in the monetary sector we have

0:02m01c

=R0

1 +R0;

1 +R0 = (1 + 0:005) (1 + 0) ;

0 = 0:02;

that determines a higher rate of ination 0 = 0:02, a higher nominal interest rate R0 = 0:025,

and a lower amount of real money balances m0 = 2:322:

* Example of a MIU model featuring non-separability, Ucm > 0.

There is a MIU economy characterized by a non-separable utility function U(ct;mt; lt) =(ctmbt)

1

1 + log lt with b; ; > 0, 0 < b (1 ) < 1; and a Cobb-Douglas productionfunction f(ndt ; kt) =

ndt1

kt with 0 < < 1. For a quarterly-period calibration, let us

assume the values of the parameters = 0:36, = 0:025, = 0:005; = 0:51; = 3:087,

b = 0:02, T = 3, and = 0:01.

i) Compute the steady state values of y; n; k; w; c; l;m,R, and .

18

l

cmb(1)= w;

(1 )nk = w;l = T n;y = n1k

y = c+ k;

n1k1 = + ;bc

m=

R

1 +R;

1 +R = (1 + ) (1 + ) ;

= ;

and inserting the calibration of parameters

3:09l

c0:51m0:0098= w;

0:64

k

n

0:36= w;

l = 3 n;y = n

k

n

0:36y = c+ 0:025k;

0:36

k

n

0:64= 0:03;

0:02c

m=

R

1 +R;

1 +R = (1 + 0:005) (1 + ) ;

= 0:01:

Solution: k = 48:553, y = 4:046, c = 2:829, w = 2:59, n = 1, l = 2, m = 3:828,

R = 0:015, and = 0:01.

ii) Now let us compare this economy with a very similar one that only diers from

having a higher money growth rate 0 = 0:02. Recalculate the steady state values of the

variables of and discuss the long-run eects of having a higher nominal money growth (non-

superneutrality): Solution: k0 = 48:262, y0 = 4:021, c0 = 2:818, w = 2:589, n0 = 0:994,

l0 = 2:006, m0 = 2:31, R0 = 0:025, and 0 = 0:02.

19

PROBLEMS

2.1. Superneutrality in the CIA model?

i) Obtain the steady state set of equations for the real sector in a competitive equilibrium

of the CIA model.

ii) Does the model display superneutrality? What can we expect from an increase in the

steady state rate of nominal money growth?

2.2. Superneutrality in the CIA model? One practical example.

There is a CIA economy where households have the following utility function U(ct; lt) =

log ct + a log lt with a > 0; and a Cobb-Douglas production function f(ndt ; kt) =ndt1

kt

with 0 < < 1. Parameters are calibrated as follows = 0:36, = 0:025, = 0:005;

a = 1:802, T = 3, and = 0:01.

i) Compute the steady state solution for y; n; k; w; c; l;m,R, and .

Solution: k = 48:553, y = 4:046, c = 2:833, w = 2:589, n = 1, l = 2, m = 2:833,

R = 0:015, and = 0:01

ii) Repeat the exercise for a higher rate of nominal money growth 0 = 0:02: Compare

results.

Solution: k0 = 48:247, y0 = 4:021, c0 = 2:814, w = 2:589, n0 = 0:994, l0 = 2:006,

m0 = 2:814, R0 = 0:025, and 0 = 0:02:

2.3. Superneutrality in the shopping time model?

i) Obtain the steady state set of equations for the economy in the competitive equilibrium

of the shopping time model.

ii) Can the real sector be solved separately? Does the model display superneutrality?

iii) What eects on the real and monetary sectors can we expect from an increase in

the steady state rate of nominal money growth if scm = 0, i.e., the shopping time function

displays consumption-money separability?

2.4. Superneutrality in the shopping time model? One practical example.

In a shopping time economy households have the following utility function U(ct; lt) =

log ct + a log lt with a > 0; a Cobb-Douglas production function f(ndt ; kt) =ndt1

kt with

0 < < 1, and a shopping time function.

s(ct;mt) =

8 09=; ;

20

with s0; s1 > 0, and 0 s2 1:Parameters are calibrated as follows = 0:36, = 0:025, = 0:005; a = 1:731, T = 3,

s0 = 0:056; s1 = 0:01; s2 = 0:85, and = 0:01.

i) Compute the steady state solution for y; n; k; w; c; l; s;m,R, and .

Solution: k = 48:553, y = 4:046, c = 2:832, w = 2:589, n = 1, l = 1:94, s = 0:06;m =

3:996, R = 0:015, and = 0:01

ii) Repeat the exercise for a higher rate of nominal money growth 0 = 0:02: Compare

results.

Solution: k0 = 48:116, y0 = 4:01, c0 = 2:806, w = 2:589, n0 = 0:991, l0 = 1:947, s0 = 0:062,

m0 = 3:673, R0 = 0:025, and 0 = 0:02:

2.5. Inelastic labor supply in the MIU model. Suppose that households supply labor

inelastically. If so, their labor supply is constant and leisure time is also constant (unless

the model incorporates shopping time). Let us normalize at ns = 1 so that labor can be

eliminated from the competitive equilibrium. Analyze the property of superneutrality in

a MIU model featuring non-separability between consumption and real money balances.

Proceed by looking at the steady state equations of its competitive equilibrium.

2.6. Inelastic labor supply in the transactions technology model. Let us keep the inelastic

labor supply assumption in the transactions technology model. Let us normalize at ns = 1

so that labor can be eliminated from the competitive equilibrium. Analyze the property

of superneutrality. Proceed by looking at the steady state equations of its competitive

equilibrium.

21

3 Welfare analysis

In this chapter we will focus attention on the welfare eects of money and monetary policy

in the long-run. Thus, the decentralized competitive equilibrium analysis of the previous

chapters will be joined by the social planner program that will maximize social welfare

subject to the overall resources constraint. The MIU model will be utilized through this

chapter and the deterministic scenario will be maintained aiming at long-run analysis.

3.1 The social planner program

The objective of the social planner is to reach the maximum social welfare while satisfying

the overall resources constraint and the time constraint. The social planner takes the util-

ity function of the representative household to dene the level of welfare of the economy.

The basic question then is the denition of a long-run monetary policy consistent with the

maximum social welfare criterion and the symmetric competitive equilibrium.

The social planner optimizing program is in period t

Maxmt

U(ct;mt; lt) + U(ct+1;mt+1; lt+1) + 2U(ct+2;mt+2; lt+2)::::::

subject to

f(nt+j; kt+j) ct+j kt+1+j + (1 )kt+j = 0; j = 0; 1; 2; :::;

which gives the following rst order condition

Umt = 0:

Monetary policy will be optimal in the long-run if it leads to satiation on monetary services,

i.e., if households use up all the positive marginal returns of real money.5

3.2 The optimal rate of ination

What is the long-run optimal rate of ination then? The answer should be straightforward:

the one that makes the households exploit all the transaction facilitating services of money.

In other words, the amount of money demand will increase as long as it has some positive

5It is implicitly assumed that there exists an actual satiation point for real money balances, not only

asymptotically Um = 0 when m 1.

22

value for money holders. This is a well-known result obtained by the Nobel Prize economist

Milton Friedman (see Friedman (1969)).

Now we should say something about how to proceed when dening our long-run monetary

policy strategy. The target is Um = 0 and the instrument is the nominal money growth

as assumed to be fully controllable by the central bank. In the long-run, nominal money

growth and ination have an identical value. Therefore, we can indistinctly refer to optimal

ination, optimal money growth, or optimal monetary policywhen talking about long-

run analysis in this type of models. Let us dene this optimal gure by looking at the money

demand behavior of the households in the MIU model

UmUc

=R

1 +R:

The central bank may exert some inuence on the amount of real money balances through

the money demand dependence on the nominal interest rate.6 When setting the optimality

condition Um = 0, the money demand equation leads to the optimal nominal interest rate

R = 0:

Intuitively, if the opportunity cost of money holdings is zero households will satiate them-

selves with monetary services. This is the argument of the so-called Chicago Rule that in

terms of the optimal rate of ination is expressed as

=1 +

:

This result is easily obtained by inserting the R = 0 condition into the nominal interest rate

denition 1+R = (1+ )(1+ ): The Chicago Rule states that the optimal rate of ination

is negative and approximately equal to the rate of intertemporal discount (or the real rate of

interest). This rate makes the nominal interest rate equal to zero and real money balances

equal to its satiation level.

3.3 The welfare cost of ination

We have just seen that the optimal rate of ination is a negative small number. In modern

economies real rates of interest are at around 2% per year so that the optimal price evolution

should be near 2% decline per year. However, it is quite rare to see continuous deation

6The transmission mechanism channel in the CIA model is by a consumption/leisure substitution (to be

worked out in Problem 3.1).

23

episodes in the real world.7 Therefore if the rate of ination of one economy is positive in

a long-run perspective, there should be some welfare cost according to the Chicago Rule.

Moreover, this would be a permanent welfare cost since referred to a steady state analysis.

How large is this welfare cost? Is a 2% for example close enough to the optimal rate of

ination in the sense of having a low welfare cost?

Let us try to give an answer to these question with our monetary models. Our approach

follows the analysis of the Nobel Prize economist Robert Lucas in Lucas (2000), and also

appearing in Walsh (1998), pages 61-64. The welfare cost of one given positive rate of

ination is measured as the amount of consumption necessary to gain utility in the utility

function in order to be indierent to the optimal rate of ination economy. This consumption-

equivalent value is reported as a percentage of steady state output.

Here comes one exercise as an example of a welfare cost of ination calculation:

There is a MIU economy where households have a separable utility function U(ct;mt; lt) =

log ct + a log lt + b (logmt hmt) with a,b; h > 0; and a Cobb-Douglas production functionf(ndt ; kt) =

ndt1

kt with 0 < < 1. For a quarterly-period calibration, let us assume

the values of the parameters = 0:36, = 0:025, = 0:005, a = 1:828, b = 0:02, h = 0:1,

T = 3, and = 0:01.

i) What is the optimal rate of ination?

=1 +

= 0:004975;

which is -1.99% per year.

ii) How much is the welfare cost of the current steady state 4% annual rate of ination?

The solution for the real economy sector is k = 48:553, y = 4:046, c = 2:832, w = 2:589,

n = 1, and l = 2 (see page 17). The monetary sector is dened in steady state by the

following equations

0:021m 0:1

1c

=R

1 +R;

1 +R = (1 + 0:005) (1 + ) ;

= 0:01;

where inserting the steady state consumption c = 2:832 leads to m = 3:609, R = 0:015, and

= 0:01:

7Here we need to mention the Japanese deationary experience from the mid-90s to present time.

24

Plugging c = 2:832, m = 3:609, and l = 2 in the utility function the level of utility

reached is U(2:832; 3:609; 2) = 2:3265:

Now we recalculate the steady state solution at the optimal rate of ination =

0:004975. This rate leads to a zero nominal interest rate that changes the monetary sectorto m = 10, R = 0, and = 0:004975: The real economy sector is not aected as the utilityfunction features money-consumption separability. In turn the steady state values entering

the utility function under the optimal rate of ination are c = 2:832, m = 10, and l = 2:

The level of utility goes up to U(2:832; 10; 2) = 2:3341:

The amount of consumption needed to compensate the loss of welfare at = 0:01 is

c = 2:8536 because U(2:8536; 3:609; 2) = 2:3341: Subsequently, if consumption is raised by

2:8536 2:832 = 0:0216 units, households would be indierent between the current andthe optimal ination scenario. The increase in consumption required is in terms of output

0:0216=4:046 = 0:0053 which means that the welfare cost of a steady state 4% annual ination

is a permanent 0.53% of output.

PROBLEMS

3.1. Verify that the long-run optimal rate of ination is provided by the Chicago Rule

in:

i) The CIA model.

ii) The transactions technology model.

iii) The shopping time model.

3.2. In the shopping time economy described by problem 2.4, calculate the welfare gains

of obtaining price stability in the long-run by moving down the steady state rate of ination

from = 0:01 to = 0:0:

3.3. Steady-state analysis in a transactions technology model.

There is a monetary economy in which all households have the same logarithmic utility

function whose arguments are consumption ct and leisure lt

U(ct; lt) = log ct + a log lt;

with a > 0:0. They maximize current and future utility values discounted at the rate

= (1 + )1 with > 0:0. These households also use the Cobb-Douglas production

technology

yt = f(kt; ndt ) = k

t

ndt1

;

25

with 0:0 < < 1:0, in order to obtain units of output yt that sell in a perfect competi-

tion market. There is also a labor perfect competition market where households demand

labor ndt and place their labor supply at the real wage wt. The process of capital accumu-

lation kt involves a constant rate of depreciation per period equal to . Finally, households

need to take some output resources to pay the transaction costs due to the purchases of

consumption goods. In particular there is a transactions-technology function (entering the

budget constraint) that provides the amount of transaction costs ht depending positively on

the level of consumption ct, and negatively (up to some satiation point) on the amount of

transactions-facilitating real money balances mt

ht = h(ct;mt) = h1ct +mh2t + h3mt

with h1; h2; h3 > 0:0.

i) Represent the steady-state equations of the competitive equilibrium of this economy.

Can the real sector be solved separately from the monetary sector? Does the model display

superneutrality?

ii) The parameters of this economy (calibrated for quarterly observations) take the fol-

lowing values: a = 1:8, = 0:005, = 0:36, h1 = 0:05, h2 = 2:0, h3 = 0:006, = 0:025, the

total number of units of time available is T = 3:0. The monetary authorities print money

at a 1% per quarter growth rate, = 0:01. Solve the model in steady state for the (ten)

endogenous variables y, w, k, n, c, l, m, h, R, and .

iii) Optimal long-run monetary policy. Solve again the model in steady state now as-

suming that the central bank runs the optimal long-run policy. Therefore, the monetary

authorities apply the Chicago rule and set the average nominal interest rate at 0.0: Compare

the results with the case of ii).

iv) Calculate the welfare cost of a 4% annualized rate of ination in this economy.

3.4. Optimal monetary policy in a shopping-time model.

A monetary general-equilibrium economy is formed by identical households whose utility

function is logarithmic in both consumption ct and leisure lt

U(ct; lt) = log ct + a log lt;

with a > 0:0. As usual, they maximize current and future utility values discounted at the

rate = (1 + )1 with > 0:0. These households obtain income from producing units of

output, yt, with the following Cobb-Douglas production technology

yt = f(kt; ndt ) = k

t

ndt1

;

26

where kt is the stock of capital, ndt is the labor demand, and 0:0 < < 1:0. The labor

market operates in perfect competition and households place the amounts of labor demand

labor and labor supply that maximize their intertemporal utility at the (market-clearing)

real wage wt. Households spend their income on consumption, capital accumulation, net

purchases of government bonds and net increases on money balances. The process of capital

accumulation involves a constant rate of depreciation per period equal to . Finally, pur-

chases of consumption goods are time consuming. Thus, a shopping time function provides

the amount of time devoted to purchases depending positively on the level of consumption

ct, and negatively (up to some satiation point) on the amount of transactions-facilitating

real money balances mt

st = s(ct;mt) = s1c2tmt+ s2mt

with s1; s2 > 0:0.

i) Represent the steady-state equations of the competitive equilibrium of this economy.

Can the real sector be solved separately from the monetary sector? Does the model display

superneutrality?

ii) The parameters of this economy (calibrated for quarterly observations) take the fol-

lowing values: a = 1:672, = 0:005, = 0:36, s1 = 0:0156, s2 = 0:001, = 0:025, the

total number of units of time available is T = 3:0. The monetary authorities print money

at a 2.5% per quarter growth rate, = 0:025. Solve the model in steady state for the (ten)

endogenous variables y, w, k, n, c, l, m, s, R, and .

iii) Optimal long-run monetary policy. Solve again the model in steady state now assum-

ing that the central bank runs the optimal long-run policy. Compare the results with the

case of ii).

iv) Calculate the welfare cost of the 10% annualized rate of ination held in the economy

described in ii).

27

REFERENCES

Brock, W.A. (1974), Money and growth: the case of long-run perfect foresight, Interna-

tional Economic Review, 15, 750-777.

Friedman, M. (1969), Optimum quantity of money and other essays, Aldine Press.

Lucas, R.E. Jr (2000), Ination and welfare, Econometrica, 68 (2), 247-274.

McCallum, Bennett T. (1989), Monetary Economics. Theory and Policy, Macmillan

Publishing Company Eds. New York, USA.

McCallum, Bennett T. (1998), Solutions to linear rational expectations models: a compact

exposition, Economic Letters 61, 143-147.

McCallum, Bennett T., (1999), A Course in Macro and Monetary Economics, Lecture

Notes. GSIA, Carnegie Mellon University.

Sidrauski, M. (1967), Rational choice and patterns of growth in a monetary economy,

American Economic Association Papers and Proceedings, 57, 534-544.

Walsh, Carl E. (2003), Monetary Theory and Policy, 2nd edition. MIT Press Eds. Cam-

bridge, MA.

28

PART 2. Short-run analysis.

Reading list:

McCallum, Bennett T., (1989). Monetary Economics. Theory and Policy. Macmillan

Publishing Company Eds. New York, USA. Chapter 8, pages 145-173, and Chapters 11-12,

pages 221-248.

McCallum, Bennett T., (1999). A Course in Macro and Monetary Economics. Lecture

Notes. GSIA, Carnegie Mellon University. Chapters 3-6, pages 26-106.

Walsh, Carl E., (2003). Monetary Theory and Policy. 2nd edition. MIT Press Eds.

Cambridge, MA. Chapter 5, pages 199-269; and Chapter 11, pages 517-558.

Woodford, M., (2003). Interest and prices: Foundations of a theory of monetary policy.

Princeton University Press.

4 Solutions to Linear Rational Expectations Models

In this second part of the course attention is placed on the dynamic period-to-period evolution

of the models, the so-called short-run analysis. The deterministic world of the steady state

analysis of the rst lectures is abandoned for an economy with uncertainty. Two new elements

will be introduced: stochastic terms (shocks) and rational expectations operators. Prior to

getting into the monetary analysis in the short-run, we must take some time to learn how

to solve linear rational expectations models.

4.1 The Minimal State Variables (MSV) solution

Let us denoteE as the rational expectation operator. Then, Etyt+1 is the rational expectation

of yt+1 in period t and Etyt+1 yt+1 is its expectational error. A rational expectation (RE)is dened as the expectation over a future variable whose expectational error is unrelated to

the set of available information.8 In this regard it is considered as the bestexpectation

for the economic agent because it exploits all the available information.9

Nonlinear models with RE are quite di cult to solve. Thus, the original non-linear

equations of an optimizing model with RE are usually approximated by log-linear equations

8See McCallum (1989), pages 146-147, for the proof.9By contrast, adaptative expectations may have systematic expectational errors that could be avoided by

using correctly the available information.

29

representing the variables in deviations from the steady state solution. Loglinearizing tech-

niques are well described in Uhlig (1999). In this chapter, we will directly work on linear

equations.

One model formed by a linear system of equations that incorporates rational expectation

operators is said to be a Linear RE Model. The Minimal State Variable solution is the

one written as a linear function with the minimal set of state variables. The minimal set

of state variables contains two types of variables: predetermined and shocks. So, the way

of proceeding is by selecting the minimal set of state variables and nding the coe cients

attached to these state variables.

Sometimes Linear RE Models have multiple solutions. In those cases, the MSV solution

has the desirable property of being the bubble-free (fundamental) solution, i.e., it rules out

explosive dynamic patterns.

4.2 Solution procedure: The Undetermined Coe cients technique

One univariate example will serve to explain the procedure to nd the MSV solution suing

the undetermined coe cients technique in simple models.10 The dynamic evolution of yt is

governed by the following linear equation

yt = a+ bEtyt+1 + cyt1 + ut; (1)

with ut being a white noise random perturbance, ut N(0; 2u). Since yt1 and ut are thetwo state variables, the MSV solution for yt should take the form

yt = 0 + 1yt1 + 2ut: (2)

Now we need to gure out the values of the coe cients 0, 1, and 2: According to our

conjecture for yt the value of Etyt+1 would be determined by

Etyt+1 = 0 + 1yt: (3)

Inserting (3) into (1) yields

yt = a+ b0 + b1yt + cyt1 + ut;

10Some computer routine is required to solve more sophisticated models. The paper by McCallum (1998)

provides the guidelines for programming such a routine. One example to be used on MatLab is written at

Michael Woodfords website (http://www.columbia.edu/~mw2230/)

30

and moving b1yt to the left hand side of the equation

(1 b1)yt = a+ b0 + cyt1 + ut: (4)

By substituting yt from equation (2) in (4), it is obtained

(1 b1)0 + (1 b1)1yt1 + (1 b1)2ut = a+ b0 + cyt1 + ut: (5)

Now the undetermined coe cients technique consists of making the coe cients attached to

the left hand side of (5) equal to their value on the right hand side. It turns out into the

three-equation system

(1 b1)0 = a+ b0; (6a)(1 b1)1 = c; (6b)(1 b1)2 = 1: (6c)

Notice that (6b) determines the value of coe cient 1 from the second order equation

b21 1 + c = 0;

which happens to have two roots, +1 =1+p14bc2b

and 1 =1p14bc

2b. Therefore, there are

two solutions for the triplet of coe cients, one coming from +1 and the other one from 1 .

It is crucial then to determine which of the two is the bubble-free MSV solution. The MSV

solution will be found by looking at the implications of the solution form (2) in some relevant

particular case. Hence, if c = 0 the evolution of yt is not dependant of its lagged value yt1.

In such situation, the value of the coe cient attached to yt1 in (2) should be 1 = 0 and

the minimal number of state variables would drop from two to one: Now we can verify that

1 = 0 when c = 0 whereas +1 =

1b. Consequently, the MSV solution is provided by 1 and

the bubble solution will be given by +1 .

Taking 1 =1p14bc

2bin (6a) and rearranging terms result in

0 =a

12 b+

q14bc2

:

Similarly, taking 1 =1p14bc

2bin (6c) and rearranging terms result in

2 =2

1 +p1 4bc:

31

The MSV solution for yt is 0 =a

12b+p

14bc2

, 1 =1p14bc

2b, and 2 =

21+p14bc . In turn, the

dynamic evolution of yt can be expressed as linearly dependant of a constant term and the

state variables as follows

yt =a

12b+p

14bc2

+ 1p14bc2b

yt1 + 21+p14bcut

PROBLEMS.

4.1. This problem represents the Cagan model and has been taken from McCallum

(1999), pages 26-28. A bivariate money market model consists of the money demand and

money supply functions

logMt logPt = (Et logPt+1 logPt) + ut;logMt = 0 + 1 logMt1 + "t;

where , , 0 > 0, 0 < 1 < 1, and both ut and "t are white-noise shocks.

i) Find the MSV solution:

ii) How would a money supply shock (monetary surprise) would aect the price level?

4.2. Consider a market in which the quantity supplied is

qt = 0 + 1pt + 2qt1, with 0; 1 > 0 and 0 < 2 < 1,

and the quantity demanded is

qt = 0 + 1pt + 2 (Etpt+1 pt) + vt,

where 0; 2 > 0, 1 < 0, and vt is a white-noise disturbance. This linear RE model has two

solutions. Discriminate between the fundamental MSV solution and the bubble solution.

32

5 The Optimizing IS-LM Model

In this chapter an IS-LM optimizing model will be described and found its MSV solution.

Two scenarios regarding price behavior will be considered. First, the price level will be

assumed to be constant. This assumption is literally taken from the standard textbook-style

IS-LM model. In these models the constant-price case is meant to reect the extreme case

of a Keynesian economy with a horizontal Aggregate Supply curve. Under our second price

scenario, the price level will be endogenous. Moreover, the price level will adjust to clear

the perfect competition goods market. Thus, our second scenario corresponds to a standard

Classical economy with a vertical Aggregate Supply curve.

5.1 Constant-Price Economy

In this second part of the course, we will set apart the production sector from the consumption

sector. Households raise income from the earnings obtained for the labor services supplied to

rms. Therefore, the budget constraint of the representative household is written as follows

wtnst + r

kt kt + gt = ct + kt+1 (1 )kt (1 + rt)1bt+1 + bt mt + (1 + t)1mt1: (1)

For simplicity, our utility function is separable in consumption, real money, and work hours

as follows

U(t; ct;mt; nst) = exp(t)

c1t1 +

m1t1

(nst)1+

1 + with ; ;;; > 0; (2)

where there is a white-noise consumption preference shock t N(0; 2).Taking the previous lines into consideration and recalling the constant-capital assump-

tion, the optimizing program for the representative household becomes

Maxct;bt+1;mt;nst

Et

1Xj=0

j

"exp(t+j)

c1t+j1 +

m1t+j1

(nst)1+

1 +

#subject to

Et[wt+jnst+j+r

kt k+gt+jct+jk(1+rt+j)1bt+1+j+bt+jmt+j+(1+t+j)1mt1+j] = 0; j = 0; 1; 2; :::;

33

which results in the rst order conditions

exp(t)ct t = 0; (cfoct )

t(1 + rt)1 + Ett+1 = 0; (bfoct+1)mt t + Et

t+1(1 + t+1)

1 = 0; (mfoct ) (nst) + twt = 0; (ns;foct )

wtnst + r

kt k + gt ct k (1 + rt)1bt+1 + bt mt + (1 + t)1mt1 = 0: (foct )

Inserting the Lagrange multipliers from the consumption rst order condition in (bfoct+1)

yields the consumption structural equation

exp(t)ct

1 + rt= Et exp(t+1)c

t+1;

which happens to be a non-linear relation. Taking logs on both sides and using log(1+rt) ' rtlead to the following log-linear approximation

log ct = Et log ct+1 1(rt ) + 1

t:

Let us denote hatvariables as per-unit deviations from steady state. For example, bct =logctcss

where the ss superscript represents the steady-state gure. Then, the log-linear

consumption equation becomes

bct = Etbct+1 1(rt rss) + 1

t; (3)

where rt rss represents the deviation with respect of the steady-state real interest rate.The overall resources constraint for this economy can be obtained by plugging both the

government budget constraint and the labor market equilibrium condition into the household

budget constraint. It results in the equation

yt = ct + k;

where yt = f(zt; nt). The overall resources constraint in logarithmic deviations from steady

state is11 byt = cssyssbct: (4)11Notice that yt = yss exp(byt). Then, by using the approximation exp(byt) ' 1 + byt, it is obtained yt '

yss + yssbyt. The same procedure applied to ct brings about equation (4). See Uhlig (1999) for details onloglinearizing techniques.

34

Equations (3) and (4) imply the following dynamic behavior of output

byt = Etbyt+1 1css

yss(rt rss) + 1

css

ysst: (5)

This is the so-called optimizing IS curve because its negative relationship between output

and the real interest rate resembles the ad hoc IS setup from traditional Macroeconomic

textbooks. Notice the presence of Etbyt+1 in the equation, which gives a forward-lookingpattern on output dynamic evolution.

In a constant-price economy, the nominal and real interest rate are identical because the

rate of ination is zero at all times.12 Therefore, equation (5) can also be written as

byt = Etbyt+1 1css

yss(Rt Rss) + 1

css

ysst: (6)

Our next step is to nd the MSV solution for byt. In order to do that, we need to specifysome dynamic behavior of the nominal interest rate due to its presence in (6). Hence, there

will be a two-equation system to be solved for byt and Rt. It is assumed that the central bankimplements a monetary policy rule looking for stabilizing output around its steady state

value

Rt = Rss + 2byt with 2 > 0: (7)

If byt is greater than zero the central bank would raise the interest rate so as to make outputfall. Likewise, if the economy is in the downturn with byt < 0 the central bank would cut theinterest rate to expand aggregate demand as implied by the IS curve (6).

In the end, the model consists of the pair of equations

byt = Etbyt+1 b (Rt Rss) + bt; (8a)Rt = R

ss + 2byt; (8b)where b = 1

css

yss. Since the only state variable is the demand shock t, the MSV solution will

take the form

byt = 1t;Rt Rss = 2t:

Using the undetermined coe cients technique, the solution for the coe cients is 1 =b

1+b2

and 2 =b21+b2

. Both gures are strictly positive reecting that output and the nominal

interest rate will increase if there is a positive demand shock t. This can be represented in

a (R; by) diagram as a shift of the IS curve to the right.12Taking logs in the denition of the nominal interest rate leads to the Fisher equation that relates nominal

and real interest rates as follows Rt = rt Ett+1.

35

5.2 Flexible-Price Economy

Let us have the IS-LMmodel with endogenous price adjustments to guarantee a general equi-

librium. Thus, the price level is assumed to be fully exible to restore the full-employment

level of output after any demand or technology shock. Firms decide on labor demand as

they seek to maximize prots by producing output with a given technology. For simplicity,

we will assume that the stock of capital is constant and the only variable input for the rm

is labor. Subsequently, the production technology can be represented by this Cobb-Douglas

production function that incorporates a labor-augmenting technology shock, zt,

f(zt; nt) =exp(zt)n

dt

1with 0 < < 1, (9)

and where the constant capital term has been normalized to the value k = 1:0. The tech-

nology shock follows the AR(1) stochastic process

zt = zt1 + "t;

where "t is a white-noise technology innovation. A representative rm would maximize this

intertemporal prot function

Et

1Xj=0

jf(zt+j; n

dt+j) wt+jndt+j

;

subject to the production technology available (9) in period t and future periods. The rst

order condition for the optimal demand of labor in period t implies

fndt wt = 0; (10)

which leads to the standard micro-founded prescription of equation the marginal product of

labor to the real wage.

Turning to the household sector, the labor supply curve is reached by combiningns;foct

with

cfoct

from above

(nst)

exp(t)ct

= wt; (11)

which implies that the marginal rate of substitution between work hours and consumption

must be equal to the real wage. Combining (9) and (11), it yields

(nst)

exp(t)ct

= (1 ) ndt (exp(zt))1 ;36

that implies that the labor market is in equilibrium when the householdsmarginal rate of

substitution and the rmsmarginal product of labor coincide. The price level will move

as necessary to adjust the real wage at the value that makes these two variables identical.

Inserting the labor market equilibrium condition (nst = ndt = nt), taking logs, and dropping

constant terms, we obtain

bnt t + bct = bnt + (1 ) zt, (12)where bnt and bct are the (market-clearing) levels of employment and consumption. TheCobb-Douglas production technology in loglinear terms for full-employment labor is

yt = (1 ) zt + (1 ) bnt,which implies bnt = (1 )1 yt zt (13)The substitution of (13) into (12) results in

+

1 yt = t bct + (1 + ) zt;

where, using (4), we can replace bct for ysscssbyt to reach this equation for full-employment outputbyt = 1 + + b1(1 ) (t + (1 + ) zt) , (14)At this point, we are able to present the exible-price equation as the market-clearing

condition byt = byt; (15)that will determine endogenously the dynamic behavior of the price level. A variable price

level makes the rate of ination also variable and the nominal and real interest rate no

longer coincide. Monetary policy will now probably aim to stabilize both ination and

output around its steady state values. Thus, the monetary policy rule sets the nominal

interest rate in response to changes in both byt and t as followsRt = R

ss + 1 (t ss) + 2byt; (16)with 1 > 1 and 2 > 0: This type of monetary policy rule is called a Taylor rule because it

was rst proposed by John Taylor (see Taylor, 1993). The imposed condition 1 > 1 reects

the Taylor principle.13

13The Taylor principle argues that responses of the nominal interest rate to ination deviations must be

greater than one-by-one so that the real interest rate moves in the same direction as the nominal interest

rate.

37

Hence, taking the IS curve derived above with variable ination and equations (14)-(16)

into account, the exible-price economy is characterized by the following three equations

byt = Etbyt+1 b(Rt Ett+1 rss) + bt; (17a)Rt Rss = 1 (t ss) + 2byt; (17b)byt = 1

+ + b1(1 ) (t + (1 + ) zt) ; (17c)

that can be solved for the three endogenous variables byt, t, and Rt. There are three statevariables in the system: the (predetermined) lag of the AR(1) technology shock, zt1, the

innovation on technolgy, "t, and the consumption preference white-noise shock, t, and the

AR(1) technology shock, zt. In turn, our conjecture of the MSV solution is

byt = 11zt1 + 12"t + 13t;t ss = 21zt1 + 22"t + 23t;Rt Rss = 31zt1 + 32"t + 33t:

PROBLEMS

5.1. Obtain the money demand LM curve from the model described in the rst section

of this chapter.

i) Express the LM curve depending on byt; t, and Rt (use rst order conditions, and theapproximations Rt ' Rt1+Rt and Rt ' Rss(1 + log

RtRss

)

ii) Find the MSV solution for bmt:iii) Discuss what may happen to the LM curve if there is a positive demand shock.

5.2. The constant-price model is calibrated at = 2, css

yss= 0:7, and 2 = 0:5:

i) Calculate the impulse response functions for byt; and Rt Rss of a unit consumptionpreference (demand) shock. Take up to 4 periods after the shock (1 year).

5.3. Find the MSV solution for the exible-price model described in the text.

i) What is the impact of a positive demand shock t on byt; t ss, and Rt Rss?ii) What is the impact of a positive technology shock "t on byt; t ss, and Rt Rss?5.4. The exible-price model can be represented in a (; by) diagram with an Aggregate

Supply (AS) function and Aggregate Demand (AD) function. The AS function displays the

supply-side behavior to clear the labor market as collected by (17c), whereas the AD curve

38

is the result of combining the IS demand curve (17a) with the Taylor-type monetary policy

rule (17b).

i) Discuss the shifts on either the (AD) or the (AS) fucntions after a positive technology

shock and its implications on and by.ii) Discuss the shifts on either the (AD) or the (AS) fucntions after a positive demand

shock and its implications on and by.5.5. The exible-price model is calibrated at = 2, c

ss

yss= 0:7, 1 = 1:5, 2 = 0:5,

= 0:36, = 4:0, and = 0:90.

i) Calculate the impulse response functions for the three endogenous variables of a unit

innovation on the technology shock. Take up to 4 periods after the shock (1 year).

ii) Calculate the impulse response functions for the three endogenous variables of a unit

innovation on the consumption preference (demand) shock. Take up to 4 periods after the

shock (1 year).

39

6 Sticky prices and the New Keynesian model

Over the past ten years or so, structural monetary models with sticky prices have become

increasingly popular for dynamic macroeconomic analysis. There are two major reasons

for the uprising of this approach that also justify its use in this course. First, the Phillips

curve dening the dynamic evolution of ination is assumed to be independent from the

monetary/scal policy regime because it is obtained from rational (optimizing) decisions

under any implemented policy. In other words, the Phillips curve is not subject to the Lucas

critique. Secondly, nominal rigidities can help capturing the short-run real eects of these

policies observed in actual data.

Price stickiness arises when setting the selling price. Here we follow the xed probabilities

scheme introduced in Calvo (1983) and quite popular today.14 Firms are not allowed to

set the optimal price with an exogenous constant probability. Interestingly, the ination

dynamics generated by the Calvo constant probability of optimal pricing are equivalent to

those obtained when assuming the presence of menu costs la Rotemberg (1982) with a

quadratic adjustment cost function for price changes (see Roberts, 1995; or Khan, 2005).

Other alternatives of price stickiness are Taylors (1980) staggered prices, and the Fuhrer

and Moores (1995) price contracts specication which are both described in Walsh (2003,

chapter 5).

6.1 The New Keynesian Phillips Curve

Price setting takes place in a monopolistic competition framework as in Dixit and Stiglitz

(1977). Firms are no longer alike as producing a dierentiated consumption good. They set

the selling price while the amount produced meets the monopolistic competition demand

function. Thus, output produced by the ith household in period t is15

yt(i) =

Pt(i)

Pt

yt; (1)

14As several examples, see Yun (1996), Erceg, Henderson, and Levin (2000), Smets and Wouters (2003),

Christiano, Eichenbaum, and Evans (2005), and Casares (2007a, 2007b).15It is implicitly assumed that a competitive rm could produce the consumption bundle (by using the

Dixit-Stiglitz output aggregator technology dened in the text to assemble dierentiated goods). If so,

the prot-maximizing criterion leads to demand equation (1), and the zero-prot condition leads to the

Dixit-Stiglitz aggregate price level denition.

40

in which is the elasticity of substitution between dierentiated goods, Pt =hR 10[Pt(i)]

1 dii1=(1)

is the aggregate Dixit-Stiglitz price level, and yt =hR 10[yt(i)]

(1)= dii=(1)

is the Dixit-

Stiglitz aggregate output.

The introduction of market power for setting prices implies that the producer-specic

labor demand ndt (i) is the one needed to obtain the amount of dierentiated output yt(i).

The decision on the selling price Pt(i) determines the amount to produce yt(i). This can be

obtained with the production technology yt(i) = f(zt; ndt (i)) by employing some certain units

of labor ndt (i) provided the state of technology zt. How do we calculate the (market-clearing)

real wage with this inelastic labor demand? It can be given by the labor supply function at

the level of labor demanded by the rms. For this reason, we need to drop the inelastic labor

supply assumption. Thus, the optimizing program for a rm that sets the price optimally

in period t is

MaxPt(i)

Et

1Xj=0

jh

Pt+j(i)

Pt+j

i1yt+j wt+jndt+j(i)

;

whose rst order condition is

(1 )hPt(i)Pt

iyt wt@n

dt (i)

@yt(i)

@yt(i)

@Pt(i)= 0: (2)

Notice that the partial derivatives that appear in the second term of (2) are the inverse of

the marginal product of labor (@ndt (i)

@yt(i)= 1

fndt (i)

), and the marginal response of the amount of

output produced by the i rm to a change in her selling price which using (1) results as

follows@yt(i)

@Pt(i)=

hPt(i)Pt

i1 ytPt= yt(i)

Pt(i)Pt

1

Pt= yt(i)

Pt(i): (3)

Let us denote with t(i) the real marginal cost for the i rm computed as the ratio between

the real wage and the marginal product of labor16

t(i) =wtfndt (i)

: (4)

Using (3) and (4), we can simplify terms of the rst order condition (2) in order to reach

Pt(i) =

1Pt t(i); (5)

16Total real cost can be written as TCt(i) = wtnt(i). The real marginal cost is therefore@TCt(i)@yt(i)

=

wt@nt(i)@yt(i)

= wtfndt (i)

.

41

which implies that the optimal price is set at a constant mark-up, 1 , over the nominal

marginal cost.

Following Calvo (1983), we introduce nominal rigidities on price setting by assuming that

rms have a chance to price optimally only when receiving a market signal that comes up

with a constant probability 1 . By contrast, there is a probability that the price cannotbe changed. The Calvo pricing behavior gives rise to the following Dixit-Stiglitz aggregate

price level17

Pt =h(1 ) [Pt(i)]1 + [Pt1]1

i1=(1); (6)

where Pt(i) is the optimal selling price during period t.

With staggered prices and xed probabilities, the (conditional) rst order equation for

the selling price (2) derived above changes to

(1 )Et1Xj=0

jjhPt(i)Pt+j

iyt+j + Et

1Xj=0

jjwt+j@ndt+j(i)

@yt+j(i)

@yt+j(i)

@Pt(i)= 0: (2)

which can be rearranged to be solved out for Pt(i) as follows

Pt(i) =

1

"EtP1

j=0 jj t+j(i) (Pt+j)

yt+j

EtP1

j=0 jj (Pt+j)

1 yt+j

#; (7)

After loglinearizing, the optimal price given by equation (7) that conveys Calvo staggered

pricing is approximated by the linear expression

logPt(i) = (1 )Et1Pj=0

jjlogPt+j + b t+j(i) ; (8)

where rm-specic variables represent gures obtained under optimal price setting. The

resulting expression implies that the log of the optimal price set depends positively on the

expected future evolution of both the log of the aggregate price level and the log of the

rm-specic real marginal cost. The rate of growth of one variable over one unit of time can

17If we aggregate prices by the period when the price was set, the Dixit-Stiglitz aggregate price level yields

Pt =

"1Pj=0

(1 )j [Ptj(i)]1# 11

;

in which (1)j is the fraction of households that set a new a price j periods ago. Note that for the laggedvalue of the price level

[Pt1]1

=1Pj=0

(1 )j [Pt1j(i)]1

Inserting the value of [Pt1]1 from the second expression into the rst one leads to equation (6).

42

be computed using the rst dierence on its natural logaritmth (for example, given a time

variable xt its growth rate in period t can be obtained asxtxt1

xt= log(xt) log(xt1)).18

therefore, the rate of ination observed in period t is t = logPt logPt1, and future valuesof the log of price level can be obtained as logPt+j = logPt+

Pkk=1 t+k. Taking into account

the last expression for logPt+j in (8), we can derive (after some algebra) an expression for

the relative price

logPt(i) logPt = (1 )Et1Pj=0

jjb t+j(i) + Et 1Pj=1

jjt+j; (9)

The denition of the Dixit-Stiglitz aggregate price level provided above can be log-

linearized to yield

logPt = (1 ) logPt(i) + logPt1;where considering the denition of the rate of ination, t = logPt logPt1, leads to

logPt logPt(i) = 1

t: (10)

Combining equations (8) and (9) results in the following formulation for ination quarter-

to-quarter changes

t ss = (Ett+1 ss) + (1 )(1 )

b t(i) (11)The ination equation obtained implies that current ination depends upon its expected

value for the next period and the current real marginal cost under optimal pricing. Ination

is purely forward looking as there is no lagged ination term in (11). As a consequence,

current ination depends positively on the present and all discounted future real marginal

cost under optimal pricing.

There is a nal step to get to the New Keynesian Phillips curve. We need to express the

ination equation in terms of aggregate variables. Following Sbordone (2002) we can nd

a linear relationship between the optimal-price real marginal cost b t(i) and the aggregatereal marginal cost b t for our Cobb-Douglas production technology.19 Substituting it out inequation (11) and, doing some algebra, yields

t ss = (Ett+1 ss) + (1 )(1 )1 +

1 b t: (12)

18The proof is straightforward in continuous time: @ log xt@t =@ log xt@xt

@xt@t =

1@xt

@xt@t .

19Concretely, b t+j(i) = b t+j 1 1t Pjk=1 t+k :43

For practical purposes, it would be even more convenient to have the Phillips curve relating

ination and some cyclical measure output. To achieve this, we must nd the link between

the aggregate real marginal cost and the output gap. The output gap is commonly dened

as the fractional deviation of current output from the exible-price level of output (see

Woodford, 2003, pages 247-249). Denoting eyt for the output gap in period t, this denitionimplies eyt = byt byt where the term byt is called either natural-rate output or potentialoutput. Natural-rate output is calculated assuming that the economy is free of nominal

rigidities when setting prices and/or wages. In the price setting scheme la Calvo, this

is the extreme case = 0:0 in which all producers receive the market signal to adjust the

price optimally. It results in a constant mark-up of prices over the marginal costs and, as

a consequence, we have b t = 0:0 and the log of the marginal product of labor coincideswith the real wage every period. Meanwhile, (optimizing) households supply the amount of

labor services that makes the marginal rate of substitution between hours and consumption

equal to the real wage (equation 11 of the previous chapter) Therefore, potential output

uctuations are driven by this equation presented in the exible-price competitive economy

of the previous chapter and rewritten herebyt = 1 + + b1(1 ) (t + (1 + ) zt) (13)A positive relationship may be expected since an increase in current output about potential

output will imply more labor employed in the production function and a lower marginal

productivity. In turn, marginal costs would rise and price ination will move upwards. Let

us check this expected sign next.

The log-linearized equations for the economy-wide real marginal cost and the market-

clearing real wage are b t = bwt byt + bnt; (14)bwt = bnt t + b1byt; (15)where (15) can be obtained by taking logs on both sides of the labor supply curve derived from

the householdsoptimizing behavior (see equation (11) from previous chapter). Combining

(14) and (15), it is obtained b t = (1 + )bnt t (1 + b1)byt: (16)The Cobb-Douglas production technology featuring a technology shock, zt, can be loglin-

earized to yield byt = (1 ) zt + (1 ) bnt. (17)44

Inserting the value of log-uctuations of labor implied by (17) into (16), we nd (after

rearranging terms) b t = + + b1(1 )1 byt (1 + )zt t: (18)Fluctuations on potential output given in (equation 14 from previous chapter) allows us to

rewrite (18) as follows

b t = + + b1(1 )1 byt + + b1(1 )1 byt;where inserting the denition of the output gap, eyt = byt byt, yields

b t = + + b1(1 )1 eyt: (19)Using (19) in the ination equation (12) leads to the so-called New Keynesian Phillips curve

t ss = (Ett+1 ss) + eyt; (20)with = (1)(1)

(1+ 1)++b1(1)

1 : The New Keynesian Phillips curve (20) will be used in the

upcoming chapter for our business cycle and monetary policy analysis.

6.2 The output gap in the IS curve and the Taylor monetary policy

rule

Let us write the optimizing IS curve (see equation 17a from the previous chapter) in terms

of the output gap. Thus, using the denition eyt = byt byt; we haveeyt = Eteyt+1 b(Rt Ett+1 rss) + bt (byt Etbyt+1);

where the process determining potential output uctuations, byt = 1++b1(1) (t + (1 + ) zt),can be inserted to obtain the IS curve20

eyt = Eteyt+1 b(Rt Ett+1 rss) + ct dzt; (21)with c = b 1

++b1(1) and d =(1)(1+)(1)++b1(1) . It is important to realize that since c > 0

a positive demand-side shock, t, always gives rise to a positive output gap. By contrast, a

positive supply-side shock zt always gives rise to a negative output gap. If the central bank

20It was also assumed that the technology shock follows the AR(1) process zt = zt1 + t with t being

white noise.

45

wishes to stabilize both ination and the output gap in a systematic way, the movements of

the nominal interest rate could be governed by the Taylor-type policy rule

Rt Rss = 1 (t ss) + 2eyt (22)The three-equation system (20), (21), and (22) gives solution paths for the three endogenous

variables t, eyt, and Rt:PROBLEMS

6.1. What is the steady-state value for Pt

Ptin the sticky-price model of the text? Hint:

take the denition of the Dixit-Stiglitz price level in steady state.

6.2. Use equation (2) to nd the steady-state value of the real marginal cost wfnwith

staggered prices la Calvo. Since the real marginal cost is the inverse of the monopolistic

competition mark-up of prices over marginal costs.

i) Is there a constant monopolistic competition mark-up term in steady state?

ii) What are the determinants of it?

6.3. Represent the New Keynesian Phillips Curve (20) in the (; ey) diagram as the AScurve of the economy together with the AD curve of the economy implied by (21) and (22).

i) How is the position of the AS curve aected by an increase in the slope coe cient on

the output gap, ?

ii) How is the position of the AS curve aected by a decrease in expected future ination

Ett+1?

iii) How is the position of the AD curve aected by a positive technology shock zt?

iv) How is the AS-AD equilibrium aected by a positive technology shock, zt?

v) How is the AS-AD equilibrium aected by a positive preference shock, t?

6.4. (For algebra lovers only). Find the New Keynesian Phillips curve if the following

stochastic indexation rule is applied for adjusting the non-optimal prices

Pt(i) = (1 + ss + & t)Pt1(i);

where the stochastic element is an AR(1) & t = & t1 + t and t N0; 2t