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)