Lecture notes for Macro 2

The main contents of this chapter include all of the following: Money demand (and some supply), the price of money, derivations of the pricing relations, money and sticky prices, money in models of monopolistic competition, money and price setting, monetary policy, empirical measures of the effect of money on output.

Lecture Notes for Macro 2 2001 (first year PhD

course in Stockholm)

Paul S¨oderlind1

June 2001 (some typos corrected later)

1University of St Gallen and CEPR Address: s/bf-HSG, Rosenbergstrasse 52, CH-9000 St

Gallen, Switzerland E-mail: Paul.Soderlind@unisg.ch Document name: MacAll.TeX

Contents

1.1 Money Supply 4

1.2 Overview of Money Demand 4

1.3 Money Demand: A General Equilibrium Model with Money in the Utility Function 7

1.4 The Mechanics of Money Supply∗ 16

2 The Price of Money 24 2.1 UIP, Fisher Equation, and the Expectations Hypothesis of the Yield Curve 24

2.2 The Price Level as an Asset Price: Cagan’s (1956) Model with Ratio-nal Expectations 25

2.3 A Simple Model of Exchange Rate Determination 31

A Derivations of the Pricing Relations 38 A.1 A Real Bond 40

A.2 A Nominal Bond 40

A.3 A Nominal Foreign Bond 41

A.4 Real Effects of Money? 41

A.5 Empirical Evidence on the Pricing Relations 42

3 Money and Sticky Prices: A First Look 45 3.1 Basic Models of the Effects of Monetary Policy Surprises 45

3.2 “Money and Wage Contracts in an Optimizing Model of the Business Cycle,” by Benassy 46

3.3 “Money and the Business Cycle,” by Cooley and Hansen 56

3.4 Sticky Wages or Sticky Prices? 61

4 Money in Models of Monopolistic Competition 63 4.1 Monopolistic Competition 63

5 Money and Price Setting 69 5.1 Dynamic Models of Sticky Prices 69

5.2 Aggregation of One-Sided Ss Rule: A Counter-Example to1M → 1Y∗ 80

A Summary of Solution Method for Linear RE Models 81 A.1 Summary 81

A.2 Special Case: Scalar Second Order Equation 82

A.3 An Alternative for the Scalar Second Order Equation: The Factoriza-tion Method 84

B Calvo’s Model: An Alternative Derivation 85 6 Monetary Policy 88 6.1 The IS-LM Model 88

6.2 The Barro-Gordon Model 91

6.3 Recent Models for Studying Monetary Policy 97

A Derivations of the Aggregate Demand Equation 104 7 Empirical Measures of the Effect of Money on Output 106 7.1 Some Stylized Facts about Money, Prices, and Exchange Rates 106

7.2 Early Studies of the Effect of Money on Output 107

7.3 Early Monetarist Studies of the Effect of Money on Output 108

7.4 Unanticipated or Anticipated Money∗ 115

7.5 VAR Studies 116

7.6 Structural Models of Monetary Policy 119

0 Reading List 123 0.1 Money Supply and Demand 123

0.2 Price Level and Nominal Assets 123

0.3 Money and Prices in RBC Models 124

0.4 Money and Monopolistic Competition 124

0.5 Sticky Prices 125

0.6 Monetary Policy 125

0.7 Empirical Measures of the Effect of Money on Output 126

0.8 The Transmission Mechanism from Monetary Policy to Output 126

1 Money Demand (and some Supply)

Main references: Romer (1996) (Romer), Blanchard and Fischer (1989) (BF), Obstfeldt

and Rogoff (1996) (OR), and Walsh (1998)

References: Burda and Wyplosz (1997) 9, OR 8.7.6 and Appendix 8B, and Mishkin

(1997)

The really short version: the central bank can control either some monetary aggregate

oran interest rate or the exchange rate How they do that is typically not very important

for most macroeconomic questions Still, this is discussed in Section 1.4 Why they do it,

that is, the monetary policy, is much more important—and something we will discuss at

length later

1.2.1 Money in Macroeconomics

Roles of money:medium of exchange, unit of account, and storage of value (often

domi-nated by other assets)

Money is macro model is typically identified with currency which gives no interest

The liquidity service of money ( medium of exchange) is emphasized, rather than store of

value or unit of account

1.2.2 Traditional money demand equations

References: Romer 5.2, BF 4.5, OR 8.3, Burda and Wyplosz (1997) 8

The standard money demand equation

1.2.3 Money Demand and Monetary PolicyThere are many different models for why money is used The common feature of thesemodels is that they all generate something pretty close to (1.1) But why is this broadermoney aggregate related to the monetary base, which the central bank may control? Shortanswer: the central bank creates a demand for narrow money by forcing banks to hold it(reserve requirements) and by prohibiting private substitutes to narrow money (banks arenot allowed to print bills)

The idea behind central bank interventions is to affect the money supply However,most central banks use short interest rates as their operating target In effect, the centralbank has monopoly over supply over narrow money which allows it to set the short interestrate, since short debt is a very close substitute to cash In terms of (1.1), the central bankmay set it, which for a given output and price level determines the money supply as aresidual

1.2.4 Applied money demand equationsReference: Goldfeld and Sichel (1990)

Applied money demand equations often take the form

Pt/Pt −1is thought to capture partial adjustment effects due to adjustment costs of eithernominal (b46=0) or real money balances (b4=0)

For instance, the estimate for Germany (69:1-85:4) reported by Goldfeld and Sichel(1990) is {b1, b2, b3, b4} = {0.3, −0.5, 0.7, −0.7} (They interest rate used in their esti-mation is in percentages, that is, like 7 instead of the 0.07 used here, so I have scaled their

b2= −0.005 by 100.)

In general, this type of equation worked fine until 1975, overpredicted money demand

during the late 1970s, and underpredicted money demand in the early 1980s Financial

innovations? (1.2) has been refined in various ways Various disaggregated money

mea-sures have been tried, a wealth of different interest rates and alternative costs have been

used, the income variable has been disaggregated, and fairly free adjustment models have

been tried (error correction models) Single equation estimation of (1.2) presumes that

this is a true demand function, with monetary authorities setting the interest rate, and with

the other right hand side variables being predetermined

1.2.5 Different Ways to Introduce Money in Macro Models

Reference: OR 8.3 and Walsh (1998) 2.3 and 3.3

The money in the utility function (MIU) model just postulates that real money balances

enter the utility function, so the consumer’s optimization problem is

max

{C t ,M t } ∞ 0

One motivation for having the real balances in the utility function is that having cash may

save time in transactions The correct utility function would then be uCt, ¯L − Lshoppi ng

t

,where Lshoppi ngt is a decreasing function of Mt/Pt

Cash-in-advance constraint(CIA) means that cash is needed to buy (some) goods, for

instance, consumption goods

where Mt −1was brought over from the end of period t − 1 Without uncertainty, this

restriction must hold with equality since cash pays no interest: no one would accumulate

more cash than strictly needed for consumption purposes since there are better investment

opportunities In stochastic economies, this may no longer be true

The simple CIA constraint implies that “money demand equation” does not include

the nominal interest rate If the utility function depends on consumption only, then all

rates of inflation gives the same steady state utility This stands in sharp contrast to the

MIU model, where the optimal rate of inflation is minus one times the real interest rate

(to get zero nominal interest rate) However, this is not longer true if the cash-in-advance

constraint applies only to a subset of the arguments in the utility function For instance, if

we introduce leisure or credit goods

Shopping-time modelstypically have a utility function is terms of consumption andleisure

∞

X

s=0

βsU(Ct, 1 − lt−nt) , (1.5)where ltis hours worked, and nthours spent on shopping (supposed to give disutility).The latter is typically modelled as some function which is increasing in consumption anddecreasing in cash holdings

the Utility FunctionReference: BF 4.5; OR 8.3; Walsh (1998) 2.3; and Lucas (2000)

1.3.1 Model SetupThe consumer’s optimization problem is

Production is given by a production function with constant returns to scale

straightfor-and covariances do not depend on the level of the other variables.

1.3.2 Optimal Consumption and Money Holdings

Use (1.7) in (1.6) to get the unconstrained problem for the consumer

which is the traditional Euler equation for real bonds (with uncertainty we need to take

the expected value of the right hand side, conditional on the information in t ) It would

also hold for any other financial asset

The first order condition for Mtis

If money would not enter the utility function, then this is a special case of (1.10) since the

real gross return on money is Pt/Pt +1 It is not obvious, however, that we get an interior

solution to money holdings unless money gives direct utility

The left hand side of (1.11) is the marginal utility lost because some resources are

taken from time t consumption, and the right hand side is the marginal utility gained by

having more cash today and the extra consumption this allows tomorrow (cash provides

utility and is also a form of saving, whose purchasing power depends on the inflation)

Substitute forβuC(Ct +1, Mt +1/Pt +1) from (1.10) in (1.11) and rearrange to get

where the convention is that the nominal interest rate is dated t since it is known as of t

Under perfect foresight, (1.12) can then be written

which highlights that the nominal interest rate is the relative price of the “money services”

we get by holding money one period instead of consuming it Note that (1.14) is a tion between real money balances, the nominal interest rate, and an activity level (hereconsumption), which is very similar to the LM equation

rela-Example 1 (Explicit money demand equation from Cobb-Douglas/CRRA.) Let the utilityfunction be

1 + it

it

,which is decreasing in it and increasing in Ct This is quite similar to the standardmoney demand equation (1.1) Take logs and make a first-order Taylor expansion of

Example 2 (Explicit money demand equation from Lucas (2000) Let the utility functionbe

in which case (1.14) can be written

in Example 1

1.3.3 General Equilibrium with MIU

We now add a few equations to close the MIU model in a closed economy The

govern-ment budget is assumed to be in balance in every period (not restrictive since Ricardian

equivalence holds in this model)

−Ts= Ms−Ms−1

Ps

so the seigniorage (right hand side) is distributed as lump sum transfers (negative taxes)

Note that this is taken as given by each individual agent

Competitive factor markets, constant returns to scale, and a fixed labor equal to one

(recall that is was not part of the utility function) give

In general, the price level is determined jointly with the rest of the dynamic

equilib-rium In special cases, as with log utility and complete depreciation of capital (as in the

model of B´enassy (1995)) there is a closed form solution However, in most cases, the

equilibrium must be computed with numerical methods

1.3.4 Steady StateTwo definitions:

• Neutrality of money: the real equilibrium is independent of the money stock

• Superneutrality of money: the real equilibrium is independent of the money growthrate

Let the money growth rate be σ, so Mt/Mt −1 = 1 +σ A steady state impliesthat inflation, consumption, and real money balances are constant: Pt/Pt −1 = 1 +π,

Ct/Ct −1=1, and(Mt/Pt) / (Mt −1/Pt −1) = 1 This implies that π = σ

The first order condition for K , (1.10), can then be simplified as 1 + rss = 1/β.Combining with (1.16) gives that steady state capital stock must solve

FK(Kss, 1) = rss

which depends only on the technology and the real discount rate, not on the money stock

or growth In steady state, the capital stock is constant so Css=F(Kss, 1), so tion in steady state is uniquely determined by the real side of the economy: in the steadystate of this model, money is neutral and superneutral Note that this is not true for thedynamics around the steady state unless marginal utility of consumption is independent

consump-of the real money balances (see Walsh (1998) 2.3 for a textbook treatment)

With a value for steady state consumption, we can solve for the steady state real moneybalances, Mss/Pss, by combining the first order

1.3.5 The Welfare Cost of Inflation

The welfare cost of inflation is typically analyzed for the steady state, since we can then

make use of the superneutrality of money The growth rate of money, and therefore the

inflation rate and nominal interest rate, can then be changed without affecting the real

equilibrium

The welfare loss from a higher nominal interest rate is often measured as the extra

consumption needed in order to achieve the same utility as in the case with lower interest

rate The approach is typically to find the money demand function which expresses real

money balances as a function of the consumption level and the nominal interest rateMP =

f(i, C), and calculate utility as the value of the period utility function u [C, f (i, C)]

If C = 1 in steady state, a certain interest rate i0gives the utility u1, f i0, 1
 The

welfare loss from another nominal interest rate, i1, is the value of C which solves

uh1, fi0, 1i=uhC, fi1, Ci (1.20)This value of C is the compensation that the consumers need to be as well off with the

interest rate i1as with i0 Note that C − 1 can be interpreted as the percentage change in

consumption needed to compensate for the higher nominal interest rate

Example 3 (Welfare loss with Cobb-Douglas/CRRA.) From Example 1, the utility at the

nominal interest rate i is

u[C, f (i, C)] =1 −1γ

"

CαC1 −αα

1 + ii

1−α#1−γ

.Consider a steady state where C =1 and suppose that inflation is zero, so i = r For

instance, to get the same utility as at C =1 and i = 3%, u [1, f (0.03, 1)], then

con-sumption must be

 1.03/ (1 + i)0.03/i

1−α

=C

Figure 1.1 illustrates the result forα = 0.99

Example 4 (Welfare loss from Lucas (1994).) From Example 2 we get

1.5

Money in utility function: cost of i>3%

Nominal interest rate, %

Cobb−Douglas, α=0.99

Lucas, B=0.0018

Figure 1.1: Utility loss, in terms of consumption, of inflation in two MIU models

To get the same utility as with C =1 and i = 3% consumption must be

1 + B1/2

i 1+i

1/2

1 + B1/2 0.03

1.03
1/2

=C

Figure 1.1 illustrates the result for B = 0.0018 (Lucas’ point estimate)

Also a cash-in-advance model (see, for instance, Cooley and Hansen (1989)) can erate welfare costs of inflation (more precise: of a non-zero nominal interest rate) if thecash-in-advance restriction applies only to a subset of the arguments in the utility func-tion A positive inflation acts like a tax on those goods that must be paid in cash, andthereby creates a distortion

gen-Friedman’s Rule for Optimal Money SupplyReference: Romer 9.8, BF 4.5

Friedman suggested a money rate growth which would set the nominal interest rate tozero and thereby saturate money demand The idea is that bills are (virtually) costless toprint and it has a (utility) value for agents, so why not give them as much as they wouldpossibly would like to have?

We know that the steady states of the real variables is unaffected by the inflation rate

(see above), so if we concentrate attention to the steady state, then (1.14) tells us that

consumers are satiated with real money balances if uM/P=0, that is, if i = 0

By the Fisher equation (1.13) this means that the monetary policy should set(1 + rt) Pt +1/Pt=

1, so the rate of deflation should equal the real interest rate In this way, holding cash gives

the same return as a real bond, so savers will be happy to keep large real money balances

and to get the utility out of it

In steady state, inflation equals the money growth rate, so a deflation requires a

shrink-ing money supply, which means that seigniorage is negative—see (1.15) In this settshrink-ing,

this is compensated by lump-sum taxes, which highlights the assumption that the

gov-ernment revenues from the inflation tax is either wasted or can be raised in other,

non-distortive, ways If, instead, a certain revenue must be raised and the alternative taxes are

distortive, then it may no longer be optimal with a zero inflation rate See Walsh (1998)

If the utility function is separable in consumption and real money balances, then this

result hold in general, not just in steady state

The Welfare Cost of Inflation - Other Arguments

Reference: Fischer (1996), Romer 9.8, Driffil, Mizon, and Ulph (1990), and Walsh (1998)

4.5-4.6

1 Inflation raises the effective capital income tax (subsidy), since the nominal return

(loss) is taxed (part of which is just compensation for inflation) The real net of tax

return is

rnet=(1 − τ) i − π

where the Fisher relation gives i = r + Eπ and we assume that π = E π This

distorts the savings decision Some calculation for the US (Feldstein, NBER, 1996)

suggest that this effect is large (twice as large as the effect on government revenues)

Counter-argument: a lower inflation and therefore lower government revenues from

capital income taxation is likely to bring higher tax rates

2 Costs of price adjustments and indexation

3 Some empirical evidence that really high inflation is bad for growth It is (boththeoretically and empirically) unclear if zero inflation is better for growth than 5%inflation

4 Seigniorage is low for most OECD countries (less than one percent of GDP, see OR8.2)

5 Low inflation means that it will be hard to drive down the real interest really low tostimulate output (The nominal interest rates cannot be negative since the nominalreturn on cash is zero.)

6 Variable inflation may lead to large inflation surprises which redistribute wealth,increases uncertainty (affects savings in which way?), and increases the informationcosts

1.3.6 The Relation to Traditional Macro ModelsEquation (1.14) is a money demand equation, which in many cases can be approximatedby

ln Mt−ln Pt=γ1ln Ct−γ2it, (1.22)which is a traditional LM equation

When the utility function is separable in consumption and real money balances, thenthe optimality condition for consumption (1.10) can often be approximated by

−γ ln Ct=ln(1 + rt +1) + γ ln Ct +1, (1.23)where consumption growth is related to the real interest rate From the Fisher equation,

we can replace ln(1 + rt +1) by it−Et(ln Pt +1−ln Pt) This is clearly reminiscent of an

IS equation

1.3.7 The Price LevelThe price level is determined simultaneously with all other variables, and there is typically

no closed form solution

In the special case where the utility function in (1.6) is separable, so the Euler equationfor consumption (1.10) is unaffected by real money balances, and where money supply

is exogenous is might be possible to arrive at an analytical expression for the price level.

In this case, we can solve for the real equilibrium (consumption, real interest rates, etc)

without any reference to money supply The price level can then be found by solving

(1.11) and information about money supply This is an example of a classical dichotomy

Example 5 (Solving for the price level.) Use the approximate Fisher equation, it=Etln Pt +1−

ln Pt+rt +1, in the approximate money demand equation in Example 1

ln Mt−ln Pt=a +ln Ct− 1

iss(1 + iss)(Etln Pt +1−ln Pt+rt +1) ,and rewrite as

variablesln Ct,ln Mt, and rt +1

References: Burda and Wyplosz (1997) 9, OR 8.7.6 and Appendix 8B, and Mishkin

(1997)

The short version: the central bank can control either some monetary aggregate or an

interest rate or the exchange rate This section is about how they do that, even if this is

not particularly important for most macroeconomic issues Why they do it, that is, the

monetary policy, is much more important—and something we will return to later

1.4.1 Operating Procedures of the Central Bank

Suppose demand for the monetary base is decreasing in the nominal interest rate Suppose

the central bank does no interventions at all Shifts in the demand curve for money will

then lead to movements in the nominal interest rate Alternatively, suppose the central

bank announces a discount rate where any bank can lend/borrow unlimited amounts This

will fix the interest rate and any shifts in the demand curve leads to movements in the

monetary base (as the banks are free to borrow reserves and currency at the fixed rate)

Finally, it is possible to strike a compromise between these two extremes letting banks

M i

M i

M i

Figure 1.2: Partial equilibrium on money market

lend at increasing interest rates This effectively creates an upward sloping supply curvefor the monetary base This is illustrated in Figure 1.2

1.4.2 Money Supply and Budget AccountingMoney supply has a direct effect on government finances Consider the consolidated gov-ernment sector (here interpreted as treasury plus central bank) The real budget identityis

Assume the Fisher equation holds, so the nominal interest rate is

1 + it −1=Et −1(1 + rt) Pt

where rtis the real interest rate The convention is that the nominal interest rate is dated

t −1 since it is known as of t − 1 To simplify, assume rtis known in t − 1 We then get

from (1.25) that the real debt in t is

Consider the case where real government expenditures and tax revenues are unaffected

by monetary policy (money is neutral), and where the central bank increases money

sup-ply, Mt > Mt −1 This drives down the real value of government debt, Bt/Pt in two

ways First, the real revenues from money creation (printing), called seigniorage, is

(Mt−Mt −1)/Pt Second, the money supply increase will probably increase the price

level If this increase is unanticipated, then actual inflation exceeds expected inflation,

Pt −1/PtEt −1(Pt/Pt −1) < 1, so the real value of government debt brought over from t −1

decreases

1.4.3 Money Aggregates and the Balance Sheet of the Central Bank

The liabilities of the central bank are currency (Cu) plus banks reserves (Re) deposited

in the central bank, the assets are the foreign exchange reserve, the holding of domestic

bonds, and perhaps gold

Balance Sheet of Central Bank

Domestic bonds Currency (Cu)Foreign currency Reserve deposits (Re)Foreign bonds Net worth

Gold

1.4.4 Reserve Requirements and DepositsMoney stock, M, is currency, Cu, plus deposits, D, (also called “inside money” since it isgenerated inside the private banking system)

Suppose that private banks (because of reserve requirements or prudence) hold thefraction r of deposits, D, in reserves, Re This means that an increase in reserves,1Re,allows the bank to increase deposits with the reserve multiplier, 1/r,

1D = 1Re

These new deposits may be lent to someone (an thereby bring in profits to the bank,assuming the lending rate is above the deposit rate) Note that if r goes to zero, then thecentral bank cannot control the creation of new deposits by affecting the availability ofreserves

Reserve requirementsmean that a private bank must hold a fraction (usually a fewpercentage) of the (checkable) deposits in either cash (in the vault) or as reserves withthe central bank This fraction is often specified as an average over some period (twoweeks in the US) Suppose a bank needs to get more reserves (maybe depositors withdrewmoney during the preceding week) It can then either sell some assets, borrow from otherbanks (“federal funds” market in the US), or borrow from the central bank (at the Fed’s

“discount window” in the US) Note that borrowing from the central bank is effectively

a decrease (as long as the loan lasts) in the reserve requirement Therefore somethinghas to be done to make the reserve requirements bite in spite of the possibility to borrowfrom the central bank This is typically a penalty rate for these loans, or some kind ofadministrative rationing of loans

It is the fact that r < 1 that makes banks different from other financial institutions

To see why, suppose r = 1 Then all deposits would have to be kept as reserves andcouldn’t be used for lending Consequently, any lending has to be done from the banksown capital In this sense, the bank is not an intermediary any more and cannot “createmoney.”

The money stock deposits discussed above can be interpreted/measured in differentways The most common monetary aggregates are: M1 (currency, travellers’ checks,

checkable deposits); M2 (M1 plus small denomination time deposits, savings deposits,

money market funds, repos, Eurodollars); M3 (M2 plus large denomination time deposits,

and money market funds held by institutions)

1.4.5 Interventions and the Money Multiplier

The central bank can typically not control the reserves directly, only the sum of reserves

and currency (the private sector can always convert the currency into reserves and vice

versa) This sum is called the monetary base (B) (also called high-powered money, M0,

central bank money, or outside money since it is generated outside the private banking

system), is

The monetary base can be increased by, for instance, an open market operation where

the central bank buys government bonds and pays by either cash (increases Cu) or cheque

(increases Re) The same effect is achieved by a foreign exchange intervention; the central

bank buys a foreign asset and pays by either cash or cheque denominated in the domestic

currency

Suppose private agents wants to hold the fraction c of the money stock in currency

We can then write (1.29) and (1.27) as

c, since it acts like a “leakage” from the private banking system At c = 0, that is, whenthere is no currency, then the money multiplier is at a maximum and coincides with thereserve multiplier

Example 6 (US data 1994, Burda and Wyplosz (1997) 9) For the US 1994 c = 0.29,M1/M0 = 2.83, which by (1.32) should imply that r = 0.089 The reserve requirements

on demand deposits was (at least in 1995) virtually zero for small demand deposits Explanations: voluntary reserves (prudence or transaction purposes) and “leakages”(deposits in nonbank financial institutions)

Example 7 (The Great Depression.) Private sector decisions can lead to importantchanges in both c and r During the Great Depression, B was not changed much (counter

to the conventional wisdom about a contractionary monetary policy), while savers drew deposits from the banks and the banks increased the voluntary reserves (both c and

with-r incwith-reased substantially), with the with-result that M decwith-reased some 30% Feawith-r of banksgoing bust?

Example 8 (Money creation.) The central bank makes an open market purchase ofbonds This increases the monetary base by1B Recall Re = r D and Cu = cM,

1Re = φ1B where φ = r(1 − c)

c + r(1 − c)

c + r(1 − c).For concreteness, assume this could happen if the central bank buys the bonds from the(consolidated) private bank, which in turn buys some of them from savers The extrareserves allow the (consolidated) private bank to take extra deposits This can be done by

lending money to a customer by crediting his deposit account This extra deposit is

A sterilized foreign exchange intervention is when the effects on the money supply of

a foreign exchange intervention is nullified by an open market operation; the central bank

buys foreign assets, pays with cash, but sell domestic assets (government bonds) to get

the cash back

An intervention on the forward market is very similar to a sterilized intervention

Suppose the bank enters a forward contract to buy domestic currency tomorrow and to

sell foreign currency at the same time This decreases the supply of “domestic currency

tomorrow,” that is, of domestic bonds, while increasing the supply of foreign bonds - just

like in a sterilized intervention

It seems as if sterilized interventions can have effects on exchange rates and interest

rates, but we are not sure why Portfolio-balance effect (changing supply changes the risk

premium, but what about Ricardian equivalence?) or signalling?

Bibliography

B´enassy, J.-P., 1995, “Money and Wage Contracts in an Optimizing Model of the Business

Cycle,” Journal of Monetary Economics, 35, 303–315

Blanchard, O J., and S Fischer, 1989, Lectures on Macroeconomics, MIT Press

Burda, M., and C Wyplosz, 1997, Macroeconomics - A European Text, Oxford UniversityPress, 2nd edn

Cooley, T F., and G D Hansen, 1989, “The Inflation Tax in a Real Business CycleModel,” The American Economic Review, 79, 733–748

Driffil, J., G E Mizon, and A Ulph, 1990, “Costs of Inflation,” in Benjamin M Friedman,

andFrank H Hahn (ed.), Handbook of Monetary Economics , vol 2, North-Holland

Fischer, S., 1996, “Why Are Central Banks Pursuing Long-Run Price Stability,” inAchieving Price Stability, pp 7–34 Federal Reserve Bank of Kansas City

Goldfeld, S M., and D E Sichel, 1990, “The Demand for Money,” in Benjamin M mand,andFrank H Hahn (ed.), Handbook of Monetary Economics, vol 1, chap 8,North-Holland

Fried-Lucas, R E., 2000, “Inflation and Welfare,” Econometrica, 68, 247–274

Mishkin, F S., 1997, The Economics of Money, Banking, and Financial Markets,Addison-Wesley, Reading, Massachusetts, 5th edn

Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MITPress

Romer, D., 1996, Advanced Macroeconomics, McGraw-Hill

Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts

2 The Price of Money

Main references: Romer (1996) (Romer), Blanchard and Fischer (1989) (BF), Obstfeldt

and Rogoff (1996) (OR), and Walsh (1998)

Yield Curve

Reference: OR 8.7.1-8.7.3

2.1.1 Pricing Relations for Nominal Returns

This section gives three very important relations, which we will use in the subsequent

analysis These relations can be stated in several different forms, but here we use the

log-linear form which fits nicely into the linear models used in most of this class

Let itbe a continuously compounded (per period) nominal interest rate on a discount

bond (no coupons) traded at t and maturing at t + 1, and letπt +1 = ln(Pt +1/Pt) be

the corresponding inflation rate The relation between nominal interest rates, real interest

rates and expected inflation is

it=Etπt +1+rtr+ϕπ

where rr

t is a real interest rate andϕπ

t a risk premium (inflation risk premium) The Fisherequationassumes that the risk premium is zero or constant, and sometimes also that the

real interest rate is constant

The relation between domestic interest rates, foreign interest rates (indicated by a

star∗), and expected exchange rate depreciation is

it−it∗=Etln(St +1/St) − ϕs

where St is the exchange rate (units of domestic currency per unit of foreign currency,

for instance, 8 SEK per USD), andϕs

t is a risk premium (exchange rate risk premium)

(The sign of the risk premium is just a matter of definition Hereϕt > 0 means that

an investment in foreign bonds require a positive risk premium.) Uncovered interest rateparity, UIP, assumes that the risk premium is zero or constant Both (2.1) and (2.2) havesimilar forms for multi-period investments

Finally, let it,t+sbe the continuously compounded per period nominal interest rate on

a discount bond traded at t and maturing at t + s The relation between long interest ratesand expected future short interest rates is

it,t+s=1

s(Etit+Etit +1+ · · · +Etit +s−1) + ϕi, (2.3)whereϕiis a risk premium (term premium) The expectations hypothesis of interest rates(yield curve) assumes that the risk premium is zero or constant

Derivations are given in the Appendix

Rational ExpectationsReference: BF 4.7 and 5.1, Romer 9.7, OR 8.2, Walsh (1998) 4.3

This classic model was first used to discuss hyperinflation In this case prices aredriven almost entirely by the dynamics of money supply (Phillips effects and other influ-ences of real variables are of minor importance) Many hyperinflation episodes originate

in the need to generate government revenues, why we will take a look at seigniorage Thismodel also allows us to discuss the “asset pricing” aspect of the price level, and how tosolve such models

The model is an approximation to the general equilibrium model with money in theutility function, where the real side of the economy (output, consumption, real interestrate) is kept constant

2.2.1 Determination of the Price Level under Rational ExpectationsSuppose the money demand function (LM curve) is

ln Mt−ln Pt=ψ ln Yt−ωit, with ω > 0 (2.4)

Prices are assumed to be completely flexible Assume that income and the real interest

rate are constant, so by the Fisher equation

it=Et(ln Pt +1−ln Pt) + constant, (2.5)and the money demand equation can be normalized as

ln Mt−ln Pt= −ω (Etln Pt +1−ln Pt) (2.6)These assumptions could either be motivated by that this is a steady state situation (general

equilibrium with MIU) or that we want to look at hyperinflation, where movements in the

real interest rate and output are of trivial importance compared with the movements in the

money stock

Remark 9 (The price of money) Note that if one unit of the good costs Ptunits of money,

then one unit of money costs Ft=1/Ptunits of goods We can then rewrite (2.6) as

ln Ft= −ln Mt+ω (Etln Ft +1−ln Ft) This says that the price of money equals a dividend, −ln Mt, and a discounted capital

gain To see that −ln Mtis like a dividend, suppose utility is U(C, M/P) = u1(C) +

ln M/P, then UM =1/M We could therefore think of − ln Mtas an approximation of

the marginal utility of money

Rewrite (2.6) as

ln Pt=(1 − η) ln Mt+ηEtln Pt +1, withη = ω/ (1 + ω) < 1 (2.7)

The price level today depend on the money supply, but also on the expected price level

tomorrow If the price level tomorrow is expected to be very high, then currency will

be worth little tomorrow (the value of money in terms of goods is 1/Pt) Like any other

asset, the value of money will then decrease already today, which means that Ptincreases

Remark 10 (Law of iterated expectations.) We must have

EtEt +1ln Pt +2=Etln Pt +2,since the information set in t is a subset of the information set in t +1 (Senility is not

allowed.)

Substituting for Etln Pt +1in (2.7) gives

ln Pt=(1 − η) ln Mt+ηEt

(1 − η) ln Mt +1+ηEt +1ln Pt +2

ln P t +1

=(1 − η) ln Mt+η (1 − η) Etln Mt +1+η2Etln Pt +2, (2.9)where we use the law of iterated expectations By continuing the substitution we end upwith

neu-s=0ηs=1

Example 11 (Log money supply is random walk plus drift.) Suppose the growth rate ofmoney isδ plus a (serially uncorrelated) shock, or

ln Mt +1=δ + ln Mt+εt +1.Then, the log price level in (2.11) is

ln Pt∗=ln Mt+δω or lnMt

Pt∗= −δω,

so the log real balances are a decreasing function of the growth rate of money In this

special case, real money balances are not affected by the shocks (See Romer 391-394 for

a diagrammatic description and a discussion of the case with price inertia.)

Example 12 (Log money supply is random walk plus drift, continued.) From Example

11, inflation is equal to money growth

πt=1 ln P∗

t =1 ln Mt=δ + εt

A higher growth rate of money supplyδ drives up the expected inflation and therefore the

nominal interest rate (the real interest rate is assumed to be constant) which decreases

demand for real money balances With a given money supply Mt, the price level must

increase to keep the money market in equilibrium

Example 13 (Money supply and the nominal interest rate.) Consider a money demand

equationln Mt−ln Pt =ψ ln Yt−ωit, and assume thatln Ytis constant What is the

effect on the nominal interest rate, it, of a shock to money supply? When money supply

is a random walk, then the effect is zero, since ln Pt increases as much asln Mt If

ln Mt=ρ ln Mt −1+εtwith |ρ| < 1, then ln Pt= [(1 − η)/(1 − ηρ)] ln Mtsoln Ptreacts

less thanln Mtto shock In this case, itmust decrease in response to a positive shock

to money growth To get a positive effect on it, the shock to money supply must be more

persistent than a random walk, for instance, by letting1 ln Mtbe an AR(1)

2.2.2 Bubbles and Saddle Point Properties

Equation (2.11) is not the unique solution, only the unique “fundamental solution.” Let

us call it ln P∗

t, and postulate that any solution can be written as a sum of the fundamental

solution and a “bubble” bt,

Sinceη < 1, the expected value Etbt +sexplodes

Ruling out bubbles, that is, using the solution (2.11), therefore amounts to findingthe unique stable solution of an unstable difference equation, that is, exploiting the sad-dle point property Equations (2.12)-(2.14) shows that any other solution explodes (inexpectation)

2.2.3 SeigniorageSeigniorage can be an important source of funds for the government It has historicallybeen very important during specific episodes, often in conjunction with wars (Very his-torically, it was the fee the authorities asked for the service of minting your silver or gold

To increase the demand for this service, it was often forbidden to mint your own coins or

to use foreign coins or even domestic coins older than a certain number of years.) Theneed for seigniorage is reputed to be the main cause of most hyperinflations

Real revenues from money creation, seigniorage, is

This is the real revenues the government get by printing more money We can think of

Mt/Pt as the tax base and 1 − Mt −1/Mt as the tax rate In fact, seigniorage is oftencalled “inflation tax.” The tax rate is essentially the money growth rate, which is stronglycorrelated with inflation and the nominal interest rate (the Cagan model, they are thesame) We know from the money demand equation that real balances are decreasing inthe nominal interest rate (for a given output level), so increasing the money growth ratetherefore increases the tax rate and decreases the tax base—the result is often a Laffercurve Mt in (2.15) should be interpreted as the monetary base Seigniorage is fairlyunimportant for most OECD countries; it is typically less than 1% of GDP

Example 14 From Example 11 we have

Mt=Mt −1exp(δ + εt), and Pt=Mtexp(δω)

Use this in (2.15) to get

Seignioraget= Mt −1exp(δ + εt) − Mt −1

Mt −1exp(δ + εt) exp (δω)

=exp(−δω) − exp [−δ(1 + ω) − εt].This is always increasing in the money supply surprises εt, but will typically show a

”Laffer curve” with respect toδ

Example 15 (Tax smoothing and seigniorage.) Suppose the distortionary effects of taxes

are convex functions of the tax rates The optimal way to finance government expenditures

is then to keep tax rates more or less constant over time Temporary changes in

govern-ment consumption should be met by lending/borrowing Seigniorage is a tax, so this

theory suggests that seigniorage should be relatively constant In fact, however,

seignior-age seems to much more correlated with government consumption than other taxes War

financing is a particularly clear case See, for instance, Walsh (1998) 4

2.2.4 Causality or Only an Equilibrium Condition?

In Cagan’s model, money supply is exogenous, so (2.11) shows how the expectations for

money supply determine the price level

This would no longer be true if allowed output to change and to depend on prices

(something we will see later in the course), or if we assumed that money supply was

a function of output and inflation (something we will also see later in the course) In

this case, we could still combine the money demand equation and the Fisher equation to

express the price level in terms of a discounted sum of expected money supply and output,

but it would only be an equilibrium condition

To demonstrate the last point, suppose both the real interest rate, rt, and log output,

ln Yt, vary Then (2.6) should be

ln Mt−ψ ln Yt+ωrt−ln Pt= −ωEt(ln Pt +1−Pt) (2.16)

Therefore, if we replace ln Mtin (2.11) with ln Mt−ψ ln Yt+ωrt, we have a solution of

(2.16) However, this cannot really be interpreted as the cause of the price level until we

have specified how money supply, output, and the real interest rate are determined—in

particular, if they depend on the price level

2.2.5 Empirical IllustrationsBurda and Wyplosz (1997) Fig 8.9, 8.12, Box 8.5, Table 16.3, and Boxes 16.4-5; WalshFig 4.3

Reference: OR 8.2.7, 8.4.1-4, Burda and Wyplosz (1997) 18-21, Isard (1995)

2.3.1 UIP and the Exchange Rate EquationThe traditional monetary model of exchange rates starts out from a money demand equa-tion, which is combined with an UIP condition

The UIP (uncovered interest rate parity) condition is

Et1 ln St +1=it−i∗t, (2.17)where itand it∗are the (per period) domestic and foreign currency interest rates, respec-tively, and St is the number of units of domestic currency per unit of foreign currency(example: 8 SEK per USD) The condition says that the expected returns (measured in

a common currency) of lending in the domestic currency or in the foreign currency areequal This typically requires full capital mobility and risk neutrality

The money demand equation is

ln Mt−ln Pt=ψ ln Yt−ωit, or

it= 1

ω(ψ ln Yt−ln Mt+ln Pt) (2.18)There is a similar demand equation in the foreign country, so the interest rate differentialcan be written

=“8 kronor per dollar × 1 dollar per US hamburger”

“10 kronor per Swedish hamburger” (2.20)

You may note that the purchasing power parity (PPP) issue is about whether Qtis constant

or not The flexible-price monetary model of exchange rates would set Qtconstant We

will not impose that, so the equation for the exchange rate that we will arrive at can only

by regarded as an equilibrium condition

Use (2.20) to substitute for ln Pt−ln Pt∗= −ln Qt+ln Stin (2.19), and then combine

with the UIP condition (2.17) to get

Et1 ln St +1= 1

ω

ψ ln Yt−ln Yt∗
− ln Mt−ln Mt∗
− ln Qt +1

ωln St (2.21)Collect the “fundamental” driving variable into

dividend, vt, plus a discounted capital gain,ωEt1 ln St +1 Note that (2.23) is just an

equilibrium condition—it cannot be given a causal interpretations without making further

assumptions

2.3.2 A Fixed Exchange Rate (Stfixed, Mtvariable)

Suppose the central bank pursues a policy of a unilateral fixed exchange rate, and that it

manages to make this policy credible For instance, let ln St=Et1 ln St +1=0 in (2.23),

and note that this requiresvt=0 From (2.22), we see that this means that

ln Mt=ln Mt∗+ψ ln Yt−ln Yt∗
+ ln Pt−ln Pt∗
(2.24)

If the money stock is the monetary policy instrument, then this equation shows how the

central bank must act to keep the exchange rate fixed In this sense, the central bank has

no control over the money stock in a fixed exchange rate regime By fixing the exchange

rate (or any other financial price) the country looses its monetary policy independence

The mechanism is the following: suppose we get a temporary shock to ln Yt This

increases money demand according to (2.18); the central bank increases money supply

by either an open market operation (buying bonds, selling domestic currency) or an tervention on the foreign exchange market (buying foreign exchange, selling domesticcurrency) This restores money market equilibrium at an unchanged interest rate By UIP(2.17), this is compatible with a fixed exchange rate

in-Instead of a unilateral peg, suppose the home country and one foreign country decide

to fix their bilateral exchange rate (for instance, St=0) According to (2.24), this puts arequirement on the relative money supply, Mt/M∗

t The level of money supply (“nominalanchor”) can be set to meet some other objective (the exchange rate with a third currency,the price level, or for stabilization purposes) (This is often called the “N-1 problem.”)Example 16 (Bretton Woods.) The Bretton Woods system was originally based on theUSD being pegged to gold, and the other currencies being pegged to the USD The fixedexchange rate forced other countries to behave according to (2.24) As a consequence, theother countries more or less adopted the US inflation rate, see (2.11) The gold peg meantthat foreign central banks (not private agents) could buy gold per 35 USD per ounce

It was expected to discipline the US since too fast US money creation would lead othercentral banks to convert dollars into gold However, this did not work as expected duringthe 1960s when the US money growth rate increased Several countries, like Germanyand Japan, had strong anti-inflationary preferences but abstained from converting dollarsinto gold, possibly because of the strong political dependence of the US At the end of the1960s/beginning of the 1970s a series of speculative attacks toppled the Bretton Woodsregime

Example 17 (EMS.) Until the mid 1980s EMS was characterized by a series of smallrealignments, but thereafter it was very much a system for fixed exchange rates wheremost European currencies were effectively pegged to the DM The boom in Germany afterthe unification lead to inflationary pressure, while most other European countries were in

a fairly deep recession with almost deflationary tendencies Bundesbank wanted to keepmonetary policy tight, which forced other countries to follow (see(2.24)) In the end, thepolitical pressure for looser monetary policy (for instance, in the UK) undermined thecredibility of the system and a series of speculative attacks forced a number or currencies

to abandon the peg Why? Central banks should in most cases be able to buy back themonetary base and thereby restore the exchange rate However, by (2.18) this would

(at unchanged prices and output) lead to very high interest rates, which may disrupt the

economy (especially the banking sector which typically borrows short and lends long)

The Swedish central bank was willing to accept extreme interest rates during the first

attack on the krona in September 1992, but not during the second attack two months later

2.3.3 A Floating Exchange Rate

Suppose instead that the central bank lets the exchange rate move In the extreme case,

Mtis fixed, but that is clearly not necessary for the exchange rate to move Rearrange

(2.23) to express ln Stas a function ofvtand EtSt +1on the left hand side

ln St=vt+ωEtln St +1−ω ln St

=(1 − η) vt+ηEtln St +1, with η = ω/ (1 + ω) < 1 (2.25)becauseω > 0 The stable solution (ruling out bubbles) is then

This expresses the current exchange rate in terms of the expected values of future

“divi-dends.” This is the “asset pricing” view of exchange rate determination Since we have

not said anything about howvtis determined, this is only an equilibrium condition In

particular, there is plenty of evidence that the real exchange rate (which is invt) is affected

by the nominal exchange rate, at least in the short to medium run

Example 18 (AR(1) fundamental.) Ifvt +1 =ρvt+εt +1withεt +1iid, then Etvt +s =

inρ, since a larger ρ means that the shock has a more long lasting effect on vt +s

We have, once again, nominal neutrality in the sense that increasing Mtin all periods

by the factorγ increases the exchange rate with the same factor Also note from (2.25)

that asω becomes large, ln Stwill be very similar to a martingale (for instance, a randomwalk)

(2.26) gives a key role to the expectations formation An unanticipated increase inthe fundamental will cause a large increase in ln S1, while an anticipated increase in thefundamentalcauses the exchange rate to increase already at the date of announcement Asimple example illustrates that

Example 19 (Anticipated versus unanticipated shocks.) Suppose

Example 20 (A mean reverting shock to the fundamental.) Whenvtis iid, then (2.26)becomes

ln St=(1 − η) vt,so

1 ln St=(1 − η) (vt−vt −1) , and

Et1 ln St +1=(1 − η) Et(vt +1−vt) = − (1 − η) vt

Since it−it =Et1 ln St +1we get

Cov(1 ln St, it−it∗) = − (1 − η)2Var(vt) ,which is negative It is straightforward to show that we get Cov(1 ln St, it−i∗

t) < 0for any stationary AR(1) specification ofvt In contrast, most empirical estimates of this

covariance are positive

Example 21 (Permanent shock to the level.) Consider the case when there are permanent

shocks to the level of money supply

t) = 0, which is still too low compared with mostempirical estimates

Example 22 (A more than permanent shock.) Let the fundamental be a sum of a random

walk and the shock to the random walk

vt=ut+θ0εt, where ut=ut −1+εtandεtis iid

This means that

Et1 ln St +1= −(1 − η) θ0εt

Combine to get Cov(1 ln St, it−it∗) asCov(1 ln St, Et1 ln St +1) = E{[1 + (1 − η) θ0]εt−(1 − η) θ0εt −1}{−(1 − η) θ0εt}

= −[1 +(1 − η) θ0](1 − η) θ0Var(εt)which can have either sign Forθ0 = 0 it is zero, since this is the random walk casediscussed above Forθ0> 0 it is always negative, since εt> 0 shifts the permanent level

ofvtup, but also gives a temporary positive blip in t , so1 ln St> 0 but Et1 ln St +1< 0.Forθ0< 0, we can get a positive covariance, since εt > 0 gives a temporary negativeblip, that is, the upward permanent shift invtis not realized until t +1 For instance, for

θ0= −1, the covariance isη (1 − η)Var(εt), which is positive The basic mechanism isthat the expected effect of the shock on the fundamental grows over time

2.3.4 The Correlation Between Real and Nominal Exchange Rates

A stylized fact is that the real, Qt, and the nominal, St, exchange rates are strongly tively correlated If we assumed that Stdoes not affect Qt, then the only explanation forCov(Qt, St) > 0 in this setting is that shocks to the real exchange rate is the main drivingforce behind the nominal exchange rate As seen from (2.22) and (2.23), the nominal ex-change rate is an increasing function of the real exchange rate Monetary shocks should

posi-be small There is plenty of empirical evidence against this view The basic mechanismwould then be that monetary shocks drive St which in turn drive Qt, because nominalprices Pt∗/Pttend to be sticky

2.3.5 Capital Controls and Risk Premia

The preceding discussion shows that the country can chose either fixed exchange rate or

monetary policy independence It cannot have both, unless there is some way to break the

UIP condition (which here serves as the equilibrium condition for the capital market) One

possibility of doing that is to let the central bank affect risk premia by changing the relative

supplies of different assets For instance, the portfolio-balance approach emphasizes risk

premia and discusses how sterilized interventions change the risk characteristics of the

private-sector portfolio and thereby asset prices Another possibility is to impose capital

controls, which make it costly to move capital

2.3.6 Other Candidate Exchange Rate Equations

There are several other candidate exchange rate equations Monetary sticky-price models

relaxes the assumption of instantaneous price flexibility, which often adds inflation terms

tovtand makes the real exchange rate endogenous The Dornbusch model (see OR 9.2)

is one example

Structural exchange rate equations, which try to explain exchange rates in terms of

macro variables often fail to improve upon a simple random walk—at least for relatively

short horizons

2.3.7 Empirical Illustrations

Burda and Wyplosz (1997) Figs 8.9, 13.6, and 19.1; OR Figs 9.1; Isard (1995) Figs

3.2, 4.2, and 11.1

Consider an agent who chooses consumption and asset holdings optimally Let Rt +1j be

the one-period gross return from investing in asset j in t The first order condition is

1 = EtRt +1i exp(qt +1) , (A.1)where qt +1is the log real discount factor between t and t + 1 For instance, if the utility

function is time separable, Et6∞

t +1

εj

t +1

#with" εq

#," σqq σq j

σq j σj j

#!.(A.3)Example 23 (CRRA utility.) The real discount factor, q, would be normally distributed,for instance, ifln Ct +1andln(Mt +1/Pt +1) are normally distributed and the utility func-tion is isoelastic, so U(Ct, Mt/Pt) = Cα

Remark 24 If x ∼ N(Ex, Var(x)), then

Eexp(x) = exp [Ex + Var (x) /2] The distribution could be interpreted as a conditional or an unconditional distribution.Take logs of (A.2), and apply the rule for Eexp(x) when x is normally distributed

0 = Etrt +1j +Etqt +1+Vart



rt +1j +qt +1 /2 (A.4)This can be written as

The variance terms are due to the non-linear transformation (recall Jensen’s inequality)

and typically not very interesting The covariance is more important, since it captures risk

aversion

A real bond has a known real interest rate (it is safe), rt +1j = rtr In this case (A.5)

which shows how the expected return in excess of the safe return depends on a Jensen’s

inequality term and the covariance of the return with the stochastic discount factor A

negative covariance means that the asset tends to have an unexpectedly low return when

marginal utility is unexpectedly high This is like a negative insurance, so investors will

require a positive risk premium,ϕj

A nominal bond has an uncertain real return, rt +1j =it−πt +1, since inflation is uncertain

In this case (A.7) is

it−Etπt +1−rtr= −1

2Vart(−πt +1) − Covt(−πt +1, qt +1) , (A.9)which has the same form as (2.1) Note that itis known, so only −πt +1enter the variance

and covariance terms If the covariance is negative, then inflation tend to be unexpectedly

high (the real return on the nominal bond is unexpectedly low), when marginal utility is

unexpectedly high Investors will then require a positive inflation risk premium The sign

of the inflation risk premium can clearly be different in different economies It can also

change over time if the conditional covariance does

A nominal foreign bond has also an uncertain real return, it∗−πt +1+ln(St +1/St), sinceboth inflation and the exchange rate depreciation are uncertain To see that this is thereal return, note that giving up one unit of goods today gives Ptunits of domestic cur-rency, and therefore Pt/Stunits of foreign currency In t + 1, this gives exp i∗

t
Pt/St

units of foreign currency, or exp i∗

t
PtSt +1/Stunits of domestic currency, and thereforeexp i∗

t

(Pt/Pt +1) (St +1/St) units of goods

In this case (A.7) is

The derivation of the pricing relations does not rule out real effects of money In fact, ing has been said about how the conditional distribution (A.3) is determined: it could bethe case that money supply changes affect output and therefore the optimal consumptiondecision Alternatively, it could be the case that monetary policy cannot affect the averagelevel of output, but it may have an effect on the volatilities In this case, monetary policyaffects the risk premia for financial assets, which could affect the consumption/savingstrade-off

noth-A.5 Empirical Evidence on the Pricing Relations

Reference: S¨oderlind and Svensson (1997)

A.5.1 UIP

Several of these relations have been studied by running “ex post” regressions For

in-stance, to test the UIP, we could add the innovation in the log exchange rate, ut +1 =

ln St +1−Etln St +1, to both sides of (2.2) to get

ln St +1−ln St=it−it∗+ut +1, (A.13)provided UIP holds, that is, ifϕs

t =0 We would test this relation by running the sion

regres-ln St +1−ln St=a + b it−it∗
+εt +1, (A.14)and test the null hypothesis that a = 0 and b = 1 Under the null hypothesis that UIP

holds, this regression should give consistent estimates of a and b, since the innovation

ut +1must (by definition) be uncorrelated with the regressors, or for that matter, everything

else in period t or earlier

This testing approach is a special case of the more general implication of UIP: ln St +1−

ln St−it+i∗

t should be unforecastable, and therefore uncorrelated with all information

in period t

The test of the null hypothesis is, of course, a joint test of rational expectations (that

it−it∗equals Etln St +1−ln Strather than some other expectation of the exchange rate

depreciation) and of no risk premia If a 6= 0 but b = 1, then this might (under RE)

be interpreted as constant risk premia, which in (A.11)-(A.12) corresponds to constant

second moments

The typical result from a large number of studies is that a 6= 0 but b 6= 1 (often

b < 0) This could be due to time-varying risk premia (which requires time-varying

variances and covariances in the theory for risk premia presented above) An alternative

explanation is that the sample is not long enough for ex post data to produce all the

jumps (devaluations) and other features that seems to be part of market expectations of

exchange rates (The Mexican peso is the classic case, where the interest rate differential

to USD interest rates was positive for many years—and it took a very long time before the

realignment eventually came In many studies, the sample ended before the realignment:the “peso problem.”) Evidence from survey data suggest that this might be the case

A.5.2 Fisher EquationThe Fisher equation is typically tested in much the same way as UIP: inflation is related

to the nominal interest rate and the null hypothesis is that the sum of the real interest rateand the inflation risk premium is a constant Most empirical evidence suggests that this isnot true, in particularly not for short maturities

A.5.3 Expectations Hypothesis of Interest RatesThe expectations hypothesis of interest rates is also tested in a similar way: future longinterest rates are related to current long interest rates Most evidence suggest that theexpectations hypothesis of interest rates works fairly well for very short maturities, butperhaps less well for maturities of 6 months up to a couple of years The empirical evi-dence for really long maturities is very mixed

A.5.4 Empirical IllustrationsMcCallum (1996) Fig 9.1; S¨oderlind and Svensson (1997) Fig 5; S¨oderlind (1998) Fig 2

Bibliography

Blanchard, O J., and S Fischer, 1989, Lectures on Macroeconomics, MIT Press.Burda, M., and C Wyplosz, 1997, Macroeconomics - A European Text, Oxford UniversityPress, 2nd edn

Isard, P., 1995, Exchange Rate Economics, Cambridge University Press

McCallum, B T., 1996, International Monetary Economics, Oxford University Press,Oxford

Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MITPress

Romer, D., 1996, Advanced Macroeconomics, McGraw-Hill.

S¨oderlind, P., 1998, “Nominal Interest Rates as Indicators of Inflation Expectations,”

Scandinavian Journal of Economics, 100, 457–472

S¨oderlind, P., and L E O Svensson, 1997, “New Techniques to Extract Market

Expecta-tions from Financial Instruments,” Journal of Monetary Economics, 40, 383–420

Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts

Main references: Romer (1996) (Romer), Blanchard and Fischer (1989) (BF), Obstfeldtand Rogoff (1996) (OR), and Walsh (1998)

3.1.1 One-Period Wage ContractsReferences: BF 8.2, Romer 6.8, Walsh 5.3

The firm has a Cobb-Douglas production function so log output is

ln Yt=ln Zt+α ln Kt+(1 − α) ln ht, (3.1)where Ztis the productivity level, Ktcapital stock, and htemployment

On a competitive labor market, the log nominal wage would be

lnwt=ln Pt+ln(1 − α) + ln Yt−ln ht (3.2)Instead, nominal wages are here set, one period in advance, equal to the expected value

of the market clearing wage (in logs, to simplify)

Now, use (3.4) to substitute for the labor input innovation in (3.5) After rearrangement

In this model monetary policy surprises can have affect on output by causing a price

surprise, while anticipated monetary policy cannot For instance, if monetary policy could

react to innovations in the productivity level, then it might be possible to stabilize output

3.1.2 Lucas’s Model of the Phillips Curve

Reference: BF 356-361, Romer 6.1-6.4,

The Lucas model is another way to get one-period effects of monetary supply shocks

The basic mechanism is that firm i has the supply curve

yi= 1

where piis the firm’s price, and Eipthe expectation about the general price level based

on the information set of firm i It is assumed that firm i observes only piand yiand uses

these to infer the general price level The firm cannot distinguish between real shocks

(to the relative price, pi−p) and nominal shocks It therefore reacts, to some extent,

to all observed movements in pi: we get real effects of monetary policy as long as the

policy surprise lasts - the macro implication is very similar to the sticky wage model (3.6)

However, the coefficient in front of the innovation in prices here depends on the volatility

of prices, rather than on the production function parameter

A major criticism against the Lucas model is that the misperception story is weak: the

consumer price index is published with a very short lag

Busi-ness Cycle,” by Benassy

3.2.1 Baseline Model

Reference: B´enassy (1995), Walsh 5.3

This model is a general equilibrium model with money in the utility function, butsticky nominal wages

The key equations are:

of money isµtMt −1, whereµt may be different from unity Benassy refers toµtas a

“multiplicative monetary shock.” One possible interpretation is a stochastic tax/subsidy

on cash holdings, where the government prints(µt−1)Mt −1new money at the beginning

of period t and distributes it as “interest rate payments” on cash holdings The exactdetails might not be too important The essential feature is that there is a stochastic moneysupply

3.2.2 Flexible PricesThe Lagrangian is

The first order conditions are

Note that we have used som aggregate relations (equilibrium rental rate and aggregate

resource constraint), which makes a lot of sense once we realize that tthe behaviour of a

representative agent must be compatible with equilibrium

Solve (3.17) recursively forward

de-ln Yt +1=ln Zt +1+α ln Kt +1+(1 − α) ln h (3.23)

=ln Zt +1+α (ln α + ln β + ln Yt) + (1 − α) ln h /*Kt +1=αβYt*/ (3.24)Solving this equation recursively backwards shows that only shocks to Ztdrive output:monetary shocks do not matter for real variables The reason is that the marginal utility

of consumption/leisure does not depend on the real money balances: the utility function

is separable Therefore, money is neutral both in the steady state (as in most models withmoney in the utility function) and along the adjustment path to steady state

Since labor supply is constant, it is clear from the production function (3.10) that thereal wage and output have a correlation of one

Finally, to determine the price level, use (3.13) in (3.16)

µt +1Mt Solving recursively forward shows that a stable solution must be

Mt

PtCt

Equation (3.26) shows that, as long as money supply is exogenous, prices and

con-sumption are negatively correlated The reason is that money has no effect on real

vari-ables in this model with flexible prices Formally, we have

Cov(ln Pt, ln Ct) = Cov (ln Mt−ln Ct, ln Ct) /*from (3.26)*/

=Cov(ln Mt, ln Ct) − Var (ln Ct) , (3.27)which is negative since Cov(ln Mt, ln Ct) = 0 as long as Mtis uncorrelated with Zt(and

hence output and consumption) The intuition for this result is that ln Ptis affected by

both nominal and real shocks, while ln Ctis affected by real shocks only It is clear from

(3.26) that real shocks must drive prices and consumption in different directions, since

the money stock is unaffected by these shocks

3.2.3 Asset Pricing

Add a real bond to the budget constraint: add 1 + rt −1r
Br

t to the revenues and Brt +1tothe expenditures Since the real interest rate rrtis known in t , the first order condition with

respect to Bt +1r is

−λt+β 1 + rr

t
Etλt +1=0, or 1

1 + rr t

=βEtλt +1/λt=βEtCt/Ct +1 (3.28)

We could also add a nominal bond to the budget constraint: add(1 + it −1) Bt/Pt

to the revenues and Bt +1/Ptto the expenditures Since itis known in t , the first order

condition with respect to Bt +1is

3.2.4 Discussion of the Money Demand EquationThe money demand equation (3.26) looks a bit odd, since it does not include the nominalinterest rate This section shows that this comes from the “dividends” payed to holders

of cash To see this, add a lump sum transfer to the budget constraint (3.9), and assumethat money supply evolves as, Mt +1=σt +1Mt Ifσt +1 6=µt +1, then the central bankbalances its budget by changing the lump sum transfer

The first order condition for money holdings is then still characterized by (3.13) and(3.16) The first line in (3.25) still holds, but the second does not To see what we getinstead, supposeµt +1andσt +1are known in t In this case, by multiplying the first line

This shows that whenµt +1is proportional to 1 + it, then we get a quantity equation This

is satisfied when Mt +1=µt +1M, since by combining (3.26) and (3.29) we also see that

1 + it=Mt +1/(Mtβ) = µt +1/β It can be noted that the nominal interest rate must behigher than the direct dividends on cash to compensate for the fact that cash has a direct

effect on utility Otherwise no one would like to buy nominal bonds.

In most monetary macro models, the dividend on money is not proportional to the

nominal interest rate To demonstrate this, consider the very simple case where (i) cash

has do dividends (µt +1= 1 in all periods); (ii) consumption and the real interest rates

are constant (more generally, they follow an exogenous process since they are unaffected

by money supply); (iii) the Fisher equation holds; (iv) and money supply follows some

stochastic process This is Cagan’s model, where we easily can generate movements in

the nominal interest rates by making money supply something other than a random walk

Even if the nominal interest rate cancels from the money demand equation, it is not

obvious it should simplify to a quantity equation It is straightforward to show that this

happens only when the utility function has a Cobb-Douglas form or logarithmic form in

terms of consumption and real money balances

3.2.5 Nominal Wage Contracts

Assume now that nominal wages for t are set in t − 1, with the agreement that households

will supply any labor actually demanded by firms in t (compare with the “right to manage”

in the wage literature) The log nominal wage is set equal to the log expected (as of t − 1)

nominal value of the marginal product of labor in t in the case of no stickiness This

assumption is, of course, ad hoc but not too unreasonable—and very convenient

The household still optimizes (3.12), but it now takes htas given Since the utility

function is separable, the first order conditions for Ct, Kt +1, and Mtare unchanged

To find the wage, note that from the production function we havewt=Pt(1 − α) Yt/ht,

lnwt=Et −1ln Mt+ constant (3.34)

Use (3.34) in (3.33) to solve for ht

ln ht=ln Mt−Et −1ln Mt+ constant (3.35)Combining this with (3.24), but replacing h with ht +1gives

ln Yt +1=ln Zt +1+α ln Yt+(1 − α) (ln Mt +1−Etln Mt +1) + constant, (3.36)which shows that money supply surprises affect current output It also affects futureoutput via capital accumulation

Since labor supply is now positively correlated with output, real wages are no longerperfectly correlated with output

Since consumption and output are positively correlated with money supply, the lation between prices and output does not have to be negative, see (3.27) Note that ln Mt

corre-equals ln Pt+ln Ctplus a constant Since real shocks have no effect on ln Mtthey mustmove ln Ctand ln Ptin opposite directions In contrast, nominal shocks move both ln Mt

and ln Ctin the same directions, but ln Ctmoves less (see (3.36)), so ln Ptalso moves inthe same direction The covariance of ln Ctand ln Ptis an average of these two forces -and depends therefore on the relative volatility of real and nominal shocks

Example 25 (Correlation of output and real wage in a special case.) Supposeα = 0 (orthat Ktis very stable so it doesn’t matter for the business cycle movements) By using thedefinitions in (3.10), Yt=Zthtandwt/Pt=Yt/ht=Zt, we then get

be uncorrelated with the productivity level The correlation is therefore

Std(ln Zt) ∗ Var (ln Zt) + Var (ln Mt−Et −1ln Mt)1/2

1 + Var(ln Mt−Et −1ln Mt) /Var (ln Zt)1/2which decreases towards zero as the volatility of the money supply shocks increase, and

increases towards one as the volatility of the Solow residual increases

3.2.6 Extensions of the Model: Demand Shocks

This model has two shocks: to productivity and to money supply It is easy to add demand

shocks, which could useful for discussing monetary policy The easiest way to do that is

to change the utility function (3.8) by multiplying ln Ctwith a random taste parameter,

At In this case, the first order condition for optimal consumption (3.13) becomes

investing in capital for future consumption For instance, if the current taste parameter is

higher than expected future taste parameters, At> EtAt +1+s, then Kt +1/Ctis lower than

otherwise This means that consumption is higher

Will a temporary shock to Ataffect output? Yes, it is likely to affect Yt if labour

supply increases (it typically will)—and yes, it will also affect Yt +sunless the increased

labour supply in t exactly offsets the increased propensity to consume so investment isunaffected by the shock to At

It is also possible to think of direct shock to labour supply (add a stochastic parameter

in the V function) or to money demand (letθ be stochastic) Shock to labour supply willcertainly affect output and prices, but shocks to money demand will (under flexible prices)probably not affect the real side of the economy since the utility function is separable

3.2.7 Extensions of the Model: Monetary Policy

So far, monetary policy has been described as some type of random process for moneysupply In reality, monetary policy is typically pursued with a purpose: to stabilize infla-tion or perhaps output, or even to maximize welfare

With flexible prices, monetary policy can clearly not affect the real variables likeoutput and consumption It can, however, affect prices We see from (3.26) that anychange in Mtmakes Ptchange to keep Mt/Ptunchanged (recall that Ctis unaffected).This shows that the central bank can affect prices, and even control them if it can set Mt

after the productivity shock is realized, but that this does not affect utility (none of thearguments in the utility function—Ct, Mt/Pt, and ht—is affected)

This is no longer true when there is nominal stickiness, provided monetary policy cansurprise private agents Note from (3.36) that monetary policy surprise have real effects

If money supply can be set after the productivity (or demand) shock is observed, thenmonetary policy can clearly stabilize output Too see this, decompose log productivity in

t +1 into its expectation in t and the shock: ln Zt +1=Etln Zt +1+εz

t +1and do the samefor money supply ln Mt +1=Etln Mt +1+εm

t +1 We can then write (3.36) as

cer-3.3 “Money and the Business Cycle,” by Cooley and Hansen

Reference: Cooley and Hansen (1995)

3.3.1 Stylized Facts

Hodrick-Prescott filtered data (data minus a moving average: cycles longer than 8 years

are virtually eliminated)

Variable (xt): Corr(xt −2, ln GDPt) Corr(xt, ln GDPt) Corr(xt +2, ln GDPt)

1 Log money and log velocity (ln V = ln Y + ln P − ln M) are procyclical

(a) Log money peaks before log output (Corr(ln M1t −1,ln Yt)> Corr(ln M1t,ln Yt)

> Corr(ln M1t +2,ln Yt))

(b) Log velocity lags output

2 The nominal interest rate (short) and inflation rate are also procyclical, but peaks

after output

3 The log price level is counter cyclical

3.3.2 Inflation Tax Model

This is a fairly standard real business cycle model, with some additional features A

stochastic money supply interacts with a cash-in-advance transaction technology to

cre-ate some real effects of money supply shocks The key equations are listed below (Lower

case letters denote values for a representative household, whereas upper case letters note aggregates.)

Production function : Yt=eztKθ

tH1−θ

t Capital accumulation : kt +1=(1 − δ) kt+xt.Government budget constraint : Tt=1Mt +1

Money supply : 1 ln Mt +1=0.491 ln Mt+ξt +1, ln ξt +1∼N, known at t Log productivity : zt +1=0.95zt+t +1, t +1∼N0, 4.9 × 10− 5

(Note: it should be Tt/Ptin the real budget constraint; there is a typo in the book.) Thenotation is: capital stock (K ), money stock (M), price level (P), wage rate (W ), hoursworked (H ), output (Y ), investment (X ), and productivity (z) Note that the notationdiffers somewhat from the model in B´enassy (1995): the money stock held at the end ofperiod t is denoted Mt +1(Mtin Benassy)

Private consumption consists of a “cash good,” c1t, and a “credit good,” c2t Oneinterpretation of the trading sequence within a time period t is the following

1 In the beginning of the period, the household carries over mtfrom t − 1, and gets

Ttis cash transfers from the government Households also own all physical capital(kt) Firms hold no cash or physical capital The government finances the transfers

by printing new money

2 Firms rent capital and labor (the rent and wages are paid somewhat later in theperiod), and produce goods

3 The household buys the cash good with the available cash, where the advance restriction Ptc1t ≤ mt+Ttmust hold (The log-normal distribution ofthe money supply shockξtmeans that the money stock can never decrease, which

cash-in-is enough to ensure that the CIA constraint always binds: positive nominal interestrate with probability one.) Firms now hold mt+Ttin cash

4 The household receives nominal factor paymentswtht+Ptrtktfrom the firms

(ex-hausts all profits), and buys credit goods (Ptc2t) and investment goods (Ptxt) The

firms now hold no cash; households own the physical capital kt +1=(1 − δ) kt+xt,

and the cash mt +1=wtht+Ptrtkt−Ptc2t−Ptxt

5 In equilibrium, the money stock held by the households (mt +1) must equal money

supply by the central bank (mt+Tt=Mt +1)

Calibration

The parameters in the production function, depreciation, Solow residual, and time

pref-erence are chosen as in standard RBC models The money supply process (for M1) is

estimated with least squares The a parameter is estimated from the first-order condition

from how they estimate the AR(1) for money supply, where they use all of M1)

Iden-tifying a from the intercept, they get a = 0.85 (If they had identified α from the slope

instead, then they would have gotα = 0.9.)

To sum up, they useθ = 0.4, δ = 0.019, β = 0.989, γ = 2.53, and a = 0.84

Solving the Model

The inflation tax means that the competitive solution will not coincide with the social

planners’s solution The solution algorithm is therefore a based on the concept of recursive

competitive equilibrium Solving a quadratic approximation (in logs) of the model results

in a set of linear decision rules in terms of the state of the economy Productivity is

stationary (|ρ| < 1), but the money supply is not, so prices will also be non-stationary It

is therefore very convenient to “detrend” all nominal variables by dividing by Mtbefore

the solution algorithm is applied

Results

The model is simulated 100 times to generate artificial samples of 150 quarters, and the

simulated data is subsequently filtered with the Hodrick-Prescott filter (Note: Tables

7.3-7.5 have the x(±s) columns in the reverse order compared with Tables 7.1-7.2, even if

this is not reflected by the headers: they should read (x(+5), , x, , x(−5).)Output is virtually neutral with respect to money supply shocks, even if the composi-tion of aggregate demand is affected by the inflation tax Intuition:1 ln Mt⇒expectedinflation⇒expensive to hold money so households substitute credit goods (and saving,that is future consumption) for cash goods

The following table summarizes some properties of data and simulations (simulationresults in parentheses):

Variable (xt): Corr(xt −2, ln G DPt) Corr(xt, ln G DPt) Corr(xt +2, ln G DPt)

2 Correlation output and interest rate/inflation is completely wrong

3 Correlation output and velocity is much too strong

4 For impulse response function, see Cooley Fig 7.6

3.3.3 A Model with Nominal Wage StickinessThe wage contract is based on the one-period ahead expectation of the marginal product

of labor The first order condition for profit maximization is

t −1, while Pt, Ht, and ztare not The deterministic steady state of the economy with

the this type of wage contracts is the same as in the economy without wage contracts

(simplifies a lot)

The nominal wage is fixed in t − 1, and the price level is observed in t Money

supply shocks may therefore affect the real wage by affecting the price level Workers are

assumed to supply inelastically at the going real wage (firms are on their labor demand

schedules) A positive money supply shock will decrease the real wage and therefore

increase labor demand and output As usual, this effect lasts as long as some prices

remain fixed: here it is one period since we have one-period labor contracts Consumers

(which own both the firms and the labor resources and therefore get all output) choose

to consume only a fraction of the temporary income increase, so most of output increase

spills over to investment (saving)

Results:

The following table summarizes some properties of data and simulations (with the

simulation results in the parentheses)

Variable (xt): Corr(xt −2, ln G DPt) Corr(xt, ln G DPt) Corr(xt +2, ln G DPt)

1 Much stronger correlation of1 ln Mtand real variables than in inflation tax model

(at odds with data)

2 Better fit of the contemporaneous correlation of nominal interest rate/inflation and

ln G D Pt, but worsens the correlation of the price level and ln Yt But the fit for

leads and lags still poor

3 For impulse response function, see Cooley Fig 7.7

Reference: Romer 5.4, Mankiw’s comment to Rotemberg (1987)

The distinction between wages and prices is often blurred in models of nominal iness It is still interesting to take a quick look at the different possibilities

stick-3.4.1 Sticky Wages and Flexible PricesThe combination of sticky wages, flexible prices, and an additional assumption about thatthe firms are on their labor demand curve means that workers have a completely elasticlabor supply at the given real wage (see Keynes, B´enassy (1995), Cooley and Hansen(1995)): unemployment/overemployment Firms hire labor until FL(L) = W/P holds,

so demand shocks lead to counter cyclical real wages (a demand shock increases the pricelevel and therefore decreases the real wage which makes it profitable to hire more workers;both P and L increases so FL(L) decreases if F (L) is a concave production function).However, this implication can be overturned by assuming that the markup of pricesover marginal costs,µ, is counter-cyclical Since nominal marginal costs is W/FL(L),

we have FL(L) /µ = W/P Consider a fixed nominal wage and an increase in demand

If the markup is constant, then we have the previous case However, ifµ is lower inbooms (higher demand elasticities in booms because shopping around or more difficult tosustain collusion in booms?), then prices need not increase and the real wage is constant

3.4.2 Flexible Wages, Sticky Prices with Monopolistic Competition

In this case, prices are fixed, but MC > P so firms are willing to supply demand (up

to where MC = P, assume this never happens), so we here assume a completely elasticsupply of goods Labor demand is then found by inverting the production function Ld=

F− 1(Y ) Workers are on their labor supply curve (no unemployment/overemployment),

so the real wage is given by the condition that the labor market clears Ld = Ls, or

F− 1(Y ) = L (W/P) A demand shock increases Y and L which requires that W/Pincreases: demand shocks lead to pro-cyclical real wages

However, the implication of no unemployment can easily be overturned by addingreal labor market frictions where there is equilibrium unemployment (efficiency wages,insider-outsiders) A common specification is that there is a real wage function W/P =

w (L) which together with labor demand determine a real wage rate, where workers would

be willing to supply more labor

Bibliography

B´enassy, J.-P., 1995, “Money and Wage Contracts in an Optimizing Model of the Business

Cycle,” Journal of Monetary Economics, 35, 303–315

Blanchard, O J., and S Fischer, 1989, Lectures on Macroeconomics, MIT Press

Cooley, T F., and G D Hansen, 1995, “Money and the Business Cycle,” in Thomas F

Cooley (ed.), Frontiers of Business Cycle Research, Princeton University Press,

Prince-ton, New Jersey

Long, J B., and C I Plosser, 1983, “Real Business Cycles,” Journal of Political Economy,

91, 39–69

Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MIT

Press

Romer, D., 1996, Advanced Macroeconomics, McGraw-Hill

Rotemberg, J J., 1987, “New Keynesian Microfoundations,” in Stanley Fischer (ed.),

NBER Macroeconomics Annual pp 69–104, NBER

Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts

4.1.1 Producers of the Final GoodThe final good (the good that enters the utility function of agents) is produced by compet-itive firms The production function is a CES function (with constant returns to scale) of

a continuum of intermediate goods Y(i), indexed by i ∈ [0, 1]

Yt=

Z 1 0

Y(i)qdi

!1 q

Y(i)qdi

!1 q

−

Z1 0