Macroeconomics Theory II

Macroeconomics Theory II

Francesco Franco

FEUNL

March 2011

Francesco Franco Macroeconomics Theory II 1/34

Trading in GE

In our standard business cycle mode there is no role for money.Implicitly, trades are carried in centralized markets, with anauctioneer. Different markets organizations are possible:• Arrow-Debreu: open once, with full set of contingent

markets (No heterogeneity, no idiosyncratic shocks)• Hicks: Spot markets, open every period, based on

expectations of the future. For example, market for goods,labor, and one period bonds. A sequence of temporaryequilibria.


Trading in GE

But we use money. So need to move to an economy wheremoney plays a role. The required ingredients:• No auctioneer. Geographically decentralized trades• Then, problem of double coincidence of wants. Barter is

not convenient. Money, accepted on one side of eachtransaction, is much more so


Foundations

• Why money? What kind of money will emerge?• Can there be competing monies? silver/gold?

Dollars/domestic currency?• Fiat versus commodity money?• Numeraire versus medium of exchange? Should they be the

same, or not?• For example: Kiyotaky-Wright.


Macro short-cuts

If we take the previous questions as given, then we can askanother set of questions:• How different does a decentralized economy with money

look like?• What determines the demand for money, the equilibrium

price level, nominal interest rates?• How does the presence of money affect the

consumption/saving choice?• Steady state and dynamic effects on real activity and

inflation of changes in the rate of money growth


Macro short-cuts

Two approaches to incorporate money into general equilibriummodels are:• Cash in Advance: requiring money to be used for certain

types of transactions• Assume money yields direct utility by incorporating money

balances directly in the utility function of the agent


CIA

Ignore labor choice and assume households supply one unit oflabor inelastically each period. Ignore uncertainty.The agent problem:

U =∞�

i=0βiU(Ct+i)

subject to:

PtCt + PtKt + Mt + PtBt= PtWtN + PtRtKt−1 + Mt−1 + PtTt + (1 + it−1)PtBt−1

and

PtCt ≤ Mt−1 + PtTt


CIA

• Bt are real one period bonds, it−1 is the nominal interestrate from period t − 1 to period t• Notice we are assuming that income from production in t is

not available for consumption in t• Define:

1 + πt =Pt

Pt−1,

mt = Mt/Pt


CIA in real terms

The CIA in real terms:

Ct ≤mt−11 + πt

+ Tt

The BC in real terms:

Ct + Kt + mt + Bt

= WtN + RtKt−1 +mt−11 + πt

+ Tt +(1 + it−1)

1 + πtBt−1

Define the real interest rate (Fisher) (will be true in eq):

1 + rt−1 =(1 + it−1)

1 + πt


Opportunity cost of money

Define At = mt + Bt the agent’s holding of financial assets. Youcan write the RHS (income side) as:

WtN + RtKt−1 + Tt + (1 + rt−1)At−1 −it−1

1 + πtmt−1

This highlights that there is a cost to holding money when thenominal interest rate is positive. Actually at time t the presentvalue of the opportunity cost of holding mt is equal to:

it1+πt+1

1 + rt=

it1 + it


Foc

Lett λt and µt be the Lagrange multiplier on respectively thebudget constraint and the CIA:

C : Uc(Ct) = λt + µt

m : λt = βλt+1 + µt+1

1 + πt+1K : λt = βλt+1Rt+1

B : λt = βλt+1(1 + it)

1 + πt+1


First Order Conditions

• The existence of a liquidity constraint (if binding) drives awedge between the marginal utility of consumption and ofwealth for wealth cannot buy istantaneously consumption• Along the optimal path the marginal cost of reducing

wealth must be equal to the discounted utility value ofcarrying that wealth forward (in the form of any asset)earning a gross return


The price of Money

Define Pm = 1P the price of money (purchasing power in terms

of goods), using the foc on money:

Pm =∞�

j=1βj

�µt+jPt+j

�

λt

The price of a unit of money is the discounted sum of allperiods expected direct marginal utility of money divided bythe present MU of wealth. This says that money is just like anyother asset whose returns take the form of liquidity services


Nominal interest rate

We also can obtain an expression for the nominal interest ratecombining the second and the fourth foc:

it =µt+1λt+1

which says that the nominal interest rate is positive only ifmoney yields liquidity services (the CIA is binding)


The price of C

Using it = µt+1λt+1

expression we obtain for the marginal utility ofconsumption:

Uc(Ct) = λt(1 + it−1),

This says that even if the economy allows output to be directlytransformed into consumption, the price of consumption is notequal to 1 but to 1 + it−1 since households must hold money tofinance consumption


Quantity theory

As long as µ > 0 the CIA binds. Define M̃t = Mt−1 + Tt :

M̃t = PtCt .

Pure quantity theory. No interest rate elasticity


FOC revisited

Using the expressions we have derived, combining all the 4 foc:

Uc(Ct)(1 + it−1)

= βUc(Ct+1)(1 + it)

(1 + rt)

Once we adjust for C price effect, get the same old relation,between marginal utility this period, marginal utility nextperiod, and the real interest rate. Note the role of both thenominal and the real interest rates.


Equilibrium

• Firm rent capital and labor:

FN = Wt FK = rt + δ

• There is a monetary authority that controls the growthrate of nominal money M according to

Mt = Mt−1(1 + xt)

where xt is the growth rate of money. The authorityrebates the increase in money to Households with a lumpssum transfer PtTt = xtMt−1• As bonds are issued by agents (not firms, renting capital

and labor services) zero net supply• Replacing all these conditions in the accumulation equation

of consumers-workers gives:Ct + Kt = F (Kt ,N) + (1− δ)Kt−1


Equilibrium

Ct + Kt = F (Kt ,N) + (1− δ)Kt−1

Uc(Ct)(1 + it−1)

= βUc(Ct+1)(1 + it)

(1 + rt)

(1 + rt) = (1− δ) + FK

Ct =(1 + xt) Mt−1

Pt

1 + rt−1 =(1 + it−1)

1 + πt

πt =Pt+1Pt− 1


Steady State

The steady state implies that

1 + r =1β

andFK = r + δ

Same as without money: Modified golden rule. You also get

C = F (K ,N)− δK

None is dependent on the inflation rate : the model exhibitssuperneutrality: on the real side, the economy looks the same aswithout money


Steady State

From the CIA constraint

C =m

1 + π̄+ T

we get C = m( 1+x1+π̄ ) so that to have a constant real consumption

π̄ = x : inflation is purely a monetary phenomena. Finally:

i =1 + π̄

β− 1

which is the Fisher relationship where the nominal interest ratemoves one to one with inflation


Welfare costs of inflation

• having introduced inflation we can answer our first policyquestion: how costly is inflation?• In this frame work because of superneutrality the steady

state welfare:W =

U(C)

1− βis invariant to inflation


Dynamics

• An unexpected permanent increase in money leads to anequal proportional increase in the price level, and nochange in the real variables. (ρ = 0)• An unexpected permanent increase in money growth, from

x to x �. In this case, this leads to a proportional increase inthe price level today, and inflation at rate x � from then on.Nominal interest rates increase by x � − x . Real variablesare unaffected


Dynamics

• A transitory increase in x . Not trivial. Inflation goes uptransitory. Nominal interest increase transitory. Price ofconsumption goes up transitory. Savings increasetransitory. Capital stock increases. Output increases• Might have positive effects on output but not on

consumption• Quantitatively very small effects


Labor supply

• inflation: wedge of inefficiency cannot compute the paretoallocation and get the decentralized equilibrium• Leisure can be purchased without money: variation in

inflation will affect the MRS between consumption andleisure (cash and credit goods)• Implications for welfare: inflation will have an effect• Dynamics change: higher money growth induce higher

inflation which now increases the demand for leisure andhence reduces output


MIU

CIA models can become very cumbersome. Thus, we often useshort cuts. The most popular one is to introduce money in theutility function. Consider the following optimization problem(known as the Sidrauski):

∞�

i=0βi

U(Ct+i ,Mt+iPt+i� �� mt+i

)

subject to:

Ct + Kt + mt + Bt

= WtN + (1 + rt−1)Kt−1 +mt−11 + πt

+ Tt +(1 + it−1)

1 + πtBt−1

Think of the utility function as a reduced form of a morecomplex problem in which by holding more money, householdscan shop more efficiently, increase leisure time, and so on


Foc

C : Uc(Ct ,mt) = λt

m : Um(Ct ,mt) + βλt+1

1 + πt+1= λt

B : β(1 + it)1 + πt+1

λt+1 = λt

K : β(1 + rt)λt+1 = λt


Foc

Combining the first three you obtain:

Um(Ct ,mt)Uc(Ct ,mt)

=it

(1 + it)

which is the opportunity cost of holding money as in the CIAmodel. You can also derive an asset price equation for the priceof money using the second foc


Equilibrium

Ct + Kt = F (Kt ,N) + (1− δ)Kt−1

Uc(Ct) = βUc(Ct+1)(1 + rt)

(1 + rt) = (1− δ) + FK

1 + rt−1 =(1 + it−1)

1 + πt

Um(Ct ,mt)Uc(Ct ,mt)

=it

(1 + it)

πt =Pt+1Pt− 1


Steady State

The steady state implies that

1 + r =1β

andFK = r + δ

Same as without money: Modified golden rule. You also get

C = F (K ,N)− δK

None is dependent on the inflation rate : the model exhibitssuperneutrality: on the real side, the economy looks the same aswithout money


Steady State

i =1 + π

β− 1

Um(C ,m)

Uc(C ,m)=

1 + π − β1 + π

The level of real money balances inversely proportional to therate of inflation, itself equal to the rate of money growth


Optimality

Here there is an optimal rate of inflation: as money is costlessto produce, the optimal rate is such as to drive the marginalutility of real money to zero, so to drive i = 0:

1 + r =1

1 + π,

π � −r ,x � −r .

This result is known as the Optimum Quantity of Money(Friedman)


Dynamics

• Dynamic effects of changes in money on real activity? Ingeneral, yes, but limited• And nothing which looks like the real effects of money in

the real world• Money as a medium of exchange, without nominal rigidities

gives us a way of thinking about the economy, the pricelevel, the nominal interest rate, but not much in the way ofexplaining fluctuations


References

Jordi Gali, “Monetary Policy. Inflation and the BusinessCycle” chap 2. Princeton 2010

Carl Walsh “Monetary Theory and Policy”, 2nd EditionMIT 2003 chap 2 and chap 3.

Cooley, Thomas and Gary Hansen. “ The inflation tax in aReal Business Cycle model.” AER 1989 (September)


Documents

Macroeconomics Theory II