Macroeconomics 2Lecture 5 - Money

Zsofia L. Barany

Sciences Po

2014 February

A brief history of money in macro 1.

1. Hume: money has a wealth effectmore money ⇒ increase in aggregate demand ⇒ Y ↑

2. Friedman - Schwartz: decline in the money supply is thesource of the great depression

3. Keynes - Hayek: IS-LM modelmoney demand is the key to transmission:Ms ↑ → r ↓ for Md ↑ → Y ↑, C ↑

4. econometricsempirical development: money leads outputvery consistent with IS-LMbig consensus: IS-LM + data fit ⇒ Tobin: macro is dead

A brief history of money in macro 2.

5. Barro: the theory that says money ⇒ expansion isobservationally equivalent to FED competence, i.e. theeconomy will do well, so FED starts pushing money into theeconomy⇒ data is not conclusive+ theory does not make sense: it assumes price rigidity,Ms ↑; Ms

P ↑6. Friedman + Phelps + Lucas: Ms ↑; Ms

P ↑, because P ↑ soMs

P does not change necessarilyskepticism of price rigidity (without is IS-LM collapses)

7. Lucas: island model with information asymmetriespeople don’t know what a price change means: is it a relativeprice change or inflation?as a producer sees the price of their product go up, they startto produce more, but if it went up due to inflation, theneverybody produces more ⇒ expansion

A brief history of money in macro 3.

8. Taylor, Fischer, Calvomicro-founded models of price and wage rigidity – restorationof old views

9. Romer and Romer: sometimes money supply change is trulyexogenous, not responding to expectations, but a new policylooking at these episodes the strong response it still there (↔Barro)

10. Sims and VAR: separation of endogenous and exogenous

11. put money into DSGE models

meanwhile: tool of mon pol is short term interest rate (not Ms)empirical models, VAR, theory is focusing on FFR

Introducing money raises a lot of questions:

I why money? what kind of money?

I can there be competing currencies?

I is money a unit of account, a medium of exchange or a storeof value?

I fiat money or commodity money?

most of the time: money is fiat money, it is used in transactions,and it is the numeraire and the medium of exchange at the sametime

Taking the above as given many other questions arise:

I how different is the economy with money?

I what determines the demand for money, the price level andthe nominal interest rate?

I does the presence of money affect the consumption/savingchoice?

I how do changes in the growth rate of money affect realactivity and inflation?

Output and money in the data

I long-held belief that money has big effects on output →money in macro

I in the data (Stock and Watson): inflation lags output, strongcorrelation

I money leads output, strong, positive correlation

I but what does this prove? Fed competence? ...

I Christiano, Eichenbaum and Evans: structural VAR estimate1 % increase in FFR : long lasting effects on output,employment, unemployment, on price level only after 6quarterswhat are exogenous monetary policy shocks?

The cash-in-advance model

benchmark model, abstract from labor/leisure choice and ignoreuncertainty

consumers solve the following problem:

∞∑i=0

βiU(Ct+i )

subject to

PtCt + Mt+1 + Bt+1 + PtKt+1 == Wt + Πt + Mt + (1 + it)Bt + (1 + rt)PtKt + Xt (BC)

and

PtCt ≤ Mt + Xt (CIA)

PtCt + Mt+1 + Bt+1 + PtKt+1 == Wt + Πt + Mt + (1 + it)Bt + (1 + rt)PtKt + Xt

I Pt the price level, i.e. the price of goods in terms of thenumeraire

I Mt ,Bt ,Kt are holdings of money, bonds and capital at thestart of period t

I Wt ,Πt are the nominal wage and the nominal profit of eachhousehold

I it is the nominal interest rate paid on bonds

I rt is the gross real rental rate paid on capital (in terms ofgoods)

I note: money pays no interest → cost of holding money

I Xt the value of a nominal transfer paid by the government →this will be the method of increasing money supply, the extraprinted money will go to the households – as a ”helicopterdrop” of money

So we saw the cost of holding money. But what is the benefit ofholding money?

I consumers only care about consumption

I i.e. they would not hold money unless they needed to, asmoney does not improve their utility, and as opposed to bondsand capital, which have a positive return, money does not

I but people must enter each period with enough nominalmoney supply to pay for consumption - cash in advanceconstraint

PtCt ≤ Mt + Xt

I why? – liquidity services of money* worker - consumer household* money balances reduce the costs of buying consumptiongoods

Solving the consumer’s problem

The Lagrangian can be written as:

L =∞∑i=0

βiU(Ct+i )

−∞∑i=0

βiλt+i [Pt+iCt+i + Mt+i+1 + Bt+i+1 + Pt+iKt+i+1

−(Wt+i + Πt+i + Mt+i + (1 + it+i )Bt+i + (1 + rt+i )Pt+iKt+i + Xt+i )]

−∞∑i=0

βiµt+i (Pt+iCt+i −Mt+i − Xt+i )

the consumer chooses Ct+i ,Mt+i+1,Kt+i+1,Bt+i+1 for every i ≥ 0

The FOCs for the consumer in period t (for i = 0) are:

Ct : U ′(Ct) = (λt + µt)Pt

Mt+1 : λt = β(λt+1 + µt+1)

Bt+1 : λt = βλt+1(1 + it+1)

Kt+1 : λtPt = βλt+1(1 + rt+1)Pt+1

Nominal and real interest rates

Combine the FOC of Bt+1 and Kt+1:

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

Pt

= (1 + rt+1)(1 + πt+1)

where we defined the inflation rate as 1 + πt+1 = Pt+1

Pt

as an approximation:

rt+1 ≈ it+1 − πt+1

The Euler equation

Combine the FOC of Ct and Mt+1:

λt = βU ′(Ct+1)

Pt+1λt+1 = β

U ′(Ct+2)

Pt+2

Use these in the FOC of Bt+1:

U ′(Ct+1)

Pt+1= β(1 + it+1)

U ′(Ct+2)

Pt+2

divide by (1 + it+1) and multiply by Pt+1:

U ′(Ct+1)

1 + it+1= β

U ′(Ct+2)Pt+2

Pt+1

= β(1 + rt+2)U ′(Ct+2)

1 + it+2

U ′(Ct+1)

1 + it+1= β(1 + rt+2)

U ′(Ct+2)

1 + it+2

I since people have to hold money one period in advance, theeffective price of consumption is (1 + i) rather than 1

I after adjusting for this price effect, it boils down to the sameas before: marginal utility today has to equal marginal utilitytomorrow, times the real interest rate and discounted

I both the value or real and nominal interest rates matter –latter mainly through the effective price

I if the nominal interest rate is constant ⇒ the equationreduces to the original Euler equation (except one periodahead - why?)

Money demand

from the FOC of Mt+1 and Bt+1:

µt+1 = it+1λt+1

if the interest rate, it+1 > 0, then it+1λt+1 > 0 holds as well ⇒µt+1 > 0 as well ⇒ the CIA constraint is binding

Mt + Xt

Pt= Ct

pure quantity theory of money: the only reason to hold money isfor transactions

note: no elasticity wrt the interest rate

Equilibrium

I firms:PtFL(Kt , Lt) = Wt FK (Kt , Lt) = Rt

where rt = Rt − δI as before, CRS and perfect competition ⇒ Πt = 0

I no labor/leisure choice ⇒ Lt = 1

I Xt is the transfer to households from the government, theextra money printed:

Mt+1 −Mt = Xt

I bonds are issued by households, closed economy, identicalagents

Bt+1 = Bt = 0

The budget constraint of the household was:

PtCt + Mt+1︸ ︷︷ ︸=Mt+Xt

+Bt+1︸︷︷︸=0

+PtKt+1 =

= Wt + Πt︸︷︷︸=0

+Mt+(1 + it) Bt︸︷︷︸=0

+(1 + rt)PtKt+Xt

which simplifies to

PtCt + PtKt+1 = Wt + (1 + rt)PtKt

= PtFL(Kt , 1) + (1 + FK (Kt , 1)− δ)PtKt

= PtF (Kt , 1) + (1− δ)PtKt

Divide by Pt and reorganize to get the usual capital accumulationequation:

Kt+1 = F (Kt , 1) + (1− δ)Kt − Ct

The equations that characterize the equilibrium:

(1 + it+1)= (1 + rt+1)(1 + πt+1)

U ′(Ct+1)

1 + it+1= β(1 + rt+2)

U ′(Ct+2)

1 + it+2

Mt + Xt

Pt= Ct

1 + rt= 1 + FK (Kt , 1)− δ

Kt+1= F (Kt , 1) + (1− δ)Kt − Ct

The steady state of the model

I assume that the growth rate of money is constant at γ, thenXt/Pt = γMt/Pt

I in the steady state all of the real variables are constant

I real money supply has to be constant:

Mt+1

Pt+1=

Mt

Pt⇒ Mt(1 + γ)

Pt(1 + πt+1)=

Mt

Pt

I the inflation rate equals the growth rate of money:πt = π = γ

I the nominal interest rate:

i = r + π = r + γ

I then from the Euler equation of the consumer and the rentalrate of the firm we get:

1 + r = 1 + FK (K , 1)− δ =1

β

this is the same as in the economy without money→modified golden rule

I from the resource constraint consumption is given by:

C = F (K , 1)− δK

superneutrality of money any change in the growth rate ofmoney has no real effects

Putting togetheri = r + π = r + γ

and

1 + r = 1 + FK (K , 1)− δ =1

β

we get the Fischer effect:

I the real interest rate is pinned down by preferences, and isindependent of any monetary measures

I any change in the growth rate of money is absorbed by aone-to-one increase in the inflation rate

Welfare costs of inflation?

*here none (in the steady state)*if labor-leisure choice or money in the utility function there will bewelfare costs

Dynamics

I much harder to characterizeeffects come from the consumption side: changes in theeffective price of goods: γ → π → i

I for some simple cases easy to guess the solution

I an unexpected permanent increase in the money supply (inlevels) leads to a proportional increase in the price level andnone of the real variables are affected

I an unexpected permanent increase in the growth rate ofmoney supply leads to a proportional increase in the currentprice level, and an increase in the inflation rate, which affectsthe nominal interest rate, and none of the real variables areaffected

I only anticipated future shocks can have real effects bychanging the inter-temporal consumption/saving choice

Summary so far

I introduction of money as a medium of exchange in generalequilibrium models

I does not look too promising so far: economy with money doesnot look very different

I consumption/saving choice a little bit modifiedI no real effects in the steady state - neutrality & superneutralityI some dynamic effects, but quite limited

I very simple CIA model, more sophisticated: Baumol-Tobinhouseholds decide how often to go to the bank, the higher theinterest rate, the more often they gochanges in money have distributional effects, which has(limited) real effects

I have to look further

Money in the utility function

Sidrauski’s model (1967)

Consumers maximize:

∞∑i=0

βiU(Ct+i ,Mt+1

Pt+1)

subject to

PtCt + Mt+1 + Bt+1 + PtKt+1 =Wt + Πt + Mt + (1 + it)Bt + (1 + rt)PtKt + Xt

the utility function is a reduced form (a short cut) of a morecomplicated problem, where holding money makes householdsmore efficient in shopping, and increases leisure time

what properties do you think U(·) has?Um > 0,Umc ≥ 0

The FOCs from the consumer’s problem

I still ignoring uncertainty

I and using the usual βiλt+i

Ct : UC (Ct ,MtPt

) = λtPt

Mt+1 : λt = β(λt+1 + 1

Pt+1Um(Ct+1,

Mt+1

Pt+1))

Bt+1 : λt = βλt+1(1 + it+1)

Kt+1 : λtPt = βλt+1(1 + rt+1)Pt+1

We can rearrange these as:

the inter-temporal condition:

UC

(Ct ,

Mt

Pt

)= β(1 + rt+1)UC

(Ct+1,

Mt+1

Pt+1

)modified Euler equation the intra-temporal condition:

Um

(Ct ,

MtPt

)UC

(Ct ,

MtPt

) = it

the MRS between consumption and real holdings of money has toequal the opportunity cost of holding money, which is i , thenominal interest rate

Example of log utility

U

(C ,

M

P

)= log(C ) + φ log

M

P

the first order conditions become:

1Ct

= β(1 + rt+1) 1Ct+1

MtPt

= φCtit

we can interpret these as an IS and an LM relation:

I IS: a higher interest rate implies a lower consumption todaygiven future expected consumption

I LM: the money demand depends positively on consumption(transactions) and negatively on the opportunity cost ofholding money, i

note: under separability same Euler equation as before

Steady state

Nothing has changed on the firm side:

1 + r = 1 + FK (K , 1)− δ = 1β

C = F (K , 1)− δK

Um(C ,MP )Uc(C ,MP )

= i = γ + r

I same real allocation as before

I superneutrality of money still holds

I MP inversely related to π = γ

Optimal growth rate of money supply

I money is costless to produce, utility increasing in real moneyholdings

I γ has to be such that the marginal utility of real money iszero ⇔ i = γ + r

I this implies γ = −r , negative money growth and inflation,equal to the negative of the marginal product of capital

I put it another way: private opportunity cost of holdingmoney: i , social marginal cost of producing money is zero,optimal growth rate should drive this wedge to zero

I known as: Optimum quantity of moneysee Bailey (1956), Friedman (1969), Lucas (2000)

Welfare cost from deviating from this rule

I Bailey – welfare cost is the area under the money demandcurve (as a function of i), measures consumer surplus lost dueto i > 0

I Lucas – estimates the welfare costs in the US, it is between0.85 and 3 % of real GNP per percentage rise in the nominalinterest rate above zero; loss equivalent to 88-310 billion2002$ per year

cost is larger than the cost of fluctuations

Dynamics

I in general there are some real effects, but they are quitelimited(some exceptions when no real effects: for example log utility)

I but it doesn’t look like the real effects of money in the realworld

I overshooting of inflation, decrease in employment and outputfollowing a rise in employment

Summary

I CIA, MIU model

I money often superneutral

I effect of interest rates: i affects money demand, r affectsconsumption/saving choice

I quantitative effects of monetary shocks: mostly through priceadjustment, real effects die out quickly

I qualitatively at odds with the observed fluctuations

I money as a medium of exchange, without nominal rigiditiesprovides a way of thinking about the price level, the nominalinterest rate, but does not tell us much about fluctuations

I welfare costs of reducing inflation could be very important(potential overestimation due to superneutrality)