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(Ir)relevance of moneyRelevance of money
.
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Money and Banking II, EC 4332Lecture 2: (Ir)relevance of money, Part I
Martin Bodenstein
NUS, AY 2013/14 Semester I
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(Ir)relevance of moneyRelevance of money
.. Goals of this unit
In lectures 2, 3 and 4 we set up and analyse a standardclassical monetary model.
The key economic themes of these lectures are (i) theconditions under which money matters in a general model, (ii)monetary (super-)neutrality in the and away from the steadystate.
With respect to Methodology, we cover the topics (i) dynamicoptimisation under with infinite horizon, (ii) (log-)linearisation ofdynamic models around the deterministic steady state, (iii) themethod of undetermined coefficients for systems of lineardifference equations.
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(Ir)relevance of moneyRelevance of money
.. Table of contents
...1 (Ir)relevance of moneyA classical monetary modelEquilibrium and optimality conditions
...2 Relevance of moneyChanging the standard modelSummarizing the full model
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. The Hahn problem
The Hahn problem in monetary economics is aboutconstructing a general equilibrium model in which money hasvalue.
Fiat money is money that derives its value from governmentregulation or law. It has no intrinsic output value.
Transactions of goods and services take place without the useof a medium of exchange in a general equilibrium model.
Furthermore, as money pays a zero nominal return, holdingmoney is dominated by holding other interest bearing assets.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Setup of the economy
There are three agents in this economy:a representative households,a representative firm,a monetary authority.
Time is discrete and is given by t = 0,1,2, ...,∞.
Agents act as price takers and hence to do not take intoaccount the effects of their choices on prices.
5 / 40
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Household
Preferences of the representative household are described bythe per-period utility function
U(
ct , lht)=
c1−σt
1 − σ−
lh1+φt
1 + φ. (1)
c1−σt
1−σ measures the contribution of consumption to utility, lh1+φt1+φ
measures the disutility from working.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Household
Note that for σ > 0
Uc
(ct , lht
)=
∂U(ct , lht
)∂ct
= c−σt > 0
Ucc
(ct , lht
)=
∂2U(ct , lht
)∂c2
t= −σc−σ−1
t < 0
and for φ > 0
Ul
(ct , lht
)=
∂U(ct , lht
)∂lht
= −lhφt < 0
Ull
(ct , lht
)=
∂2U(ct , lht
)∂lh2
t= −φlhφ−1
t < 0
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Household
The household maximises expected discounted lifetime utility
maxct ,lht ,bt ,Mt
E0
∞∑t=0
βtU(
ct , lht)
(2)
subject to the flow budget constraint
Ptct + Qtbt + Mht ≤ bt−1 + Wt lht + Mh
t−1 + Tt (3)
and non-negativity constraints
ct ⩾ 0, lht ⩾ 0,Mht ⩾ 0.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Household
0 < β < 1 is the discount factor for future utility contributions.
Each period, the household chooses consumption ct , laborsupply lht , the amount of bond holdings bt and nominal moneyholding Mh
t .
Pt is the nominal price of the consumption good and Wt is thenominal wage.
Bonds are in zero net supply, mature after one period, cost Qt ,and pay one unit of money at maturity.
The household also receives lump-sum transfers from firmsand the monetary authority Tt .
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Household
In addition, households are subjected to the solvencyconstraint:
limT→∞
Et (ΘT ) ≥ 0 (4)
for all t and Θt = bt−1 + Mht−1.
At every point in time, expectations on household assets far outin the future must be positive.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Firms
The representative firm hires labor l ft and produces output ytusing the following technology
yt = At l ft (5)
and maximises its expected discounted profits
maxyt ,l ft
E0
∞∑t=0
ψt
(Ptyt − Wt l ft
)(6)
subject to the constraint imposed by technology andnon-negativity of yt and l ft . ψt is the discount factor used by thefirm to value future period profits.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Firms
At is an exogenous shift to technology. Define the new variableat = ln(At)− ln(Ass) with the deterministic steady state ofAss = 1.
Following the literature, at follows an autoregressive process oforder 1
at = ρaat−1 + σaεa,t . (7)
with 0 < ρa < 1 and σa > 0.
εa,t is random variable that has a standard normal distribution.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Monetary authority
The monetary authority sets a path for the money supply
{Mt}∞t=0 . (8)
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Equilibrium
An equilibrium in this economy is given by paths for nominalprices {Pt ,Wt ,Qt}∞t=0 and real quantities
{ct , lht , yt , l ft
}∞t=0,
transfers {Tt}∞t=0 , and money{
Mt ,Mht}∞
t=0, such that...1 household utility is maximized,...2 firm profit is maximized,...3 market for goods (yt = ct) clear,...4 market for labor
(lt = l ft = lht
)clear,
...5 money market(Mh
t = Mt)
clears,...6 bond market (bt = 0) clears.
Changes in the money supply are engineered throughtransfers, i.e.,Tt = Mt − Mt−1.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model
To obtain the full set of equations that describe the equilibriumalgebraically, we derive the optimality conditions of the firm andhouseholds.
We then combine the optimality conditions with the marketclearing conditions to arrive at the conclusion that money isirrelevant in this economy.
15 / 40
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Firms
We write the Lagrangian Lf associated with the firms’maximization problem (equations 5 and 6) as
Lf = E0
∞∑t=0
{ψt
(Ptyt − Wt l ft
)+ ψtλ
ft
[At l ft − yt
]}(9)
ψt is the discount factor and λft is the Lagrange multiplier.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Firms
The first order necessary conditions are
∂Lf
∂yt= ψtPt − ψtλ
ft ⩽ 0 and yt ⩾ 0 (10)
∂Lf
∂l ft= −ψtWt + ψtλ
ft At ⩽ 0 and l ft ⩾ 0. (11)
Note that firms can either choose to operate, i.e., yt > 0 andl ft > 0, or not yt = 0 and l ft = 0.
The latter is optimal is ∂Lf
∂yt< 0 for yt > 0.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Firms
For an interior optimum with positive output and labor demand,i.e., yt > 0 and l ft > 0
Pt = λft (12)
Wt = λft At (13)
or
Wt
Pt= At . (14)
At the optimum, the firm chooses employment such that themarginal product of labor, here At , equals the real wage (thefactor costs of labor).
Furthermore, profits are zero in each period.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
We write the Lagrangian Lh associated with the household’smaximization problem (equations 2 and 3) as
Lh = E0
∞∑t=0
{βtU
(ct , lht
)}+E0
∞∑t=0
{βtλh
t
[bt−1 + Wt lht + Mt−1 + Tt − Ptct − Qtbt − Mt
]}(15)
λht is the Lagrange multiplier associated with the budget
constraint.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
The first order necessary conditions are
∂Lh
∂ct= βtUc
(ct , lht
)− βtλh
t Pt ⩽ 0 and ct ≥ 0 (16)
∂Lh
∂lht= βtUl
(ct , lht
)+ βtλh
t Wt ⩽ 0 and lht ≥ 0 (17)
∂Lh
∂bt= Et
[βt+1λh
t+1
]− βtλh
t Qt = 0 (18)
∂Lh
∂Mt= Et
[βt+1λh
t+1
]− βtλh
t ⩽ 0 and Mt ≥ 0. (19)
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
For an interior optimum with positive consumption and laborsupply, i.e., ct > 0 and lt > 0, equations (16) and (17) imply
Uc
(ct , lht
)= λh
t Pt (20)
Ul
(ct , lht
)= −λh
t Wt (21)
or
−Ul
(ct , lht
)Uc
(ct , lht
) =Wt
Pt. (22)
At the optimum, the marginal rate of substitution between laborand consumption equals the real wage.
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
The first order condition for bonds, equation (18), implies thatthe price Qt satisfies
Qt = Et
[βλh
t+1
λht
], (23)
or after eliminating λht by using equation (20)
Qt = Et
[βλh
t+1Pt+1
λht Pt
Pt
Pt+1
]= Et
[β
Uc(ct+1, lht+1
)Uc
(ct , lht
) Pt
Pt+1
]. (24)
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
To better interpret this expression, rearrange it as follows
Uc
(ct , lht
)= βEt
[Pt
Pt+1QtUc
(ct+1, lht+1
)]. (25)
Reducing consumption in period t by one unit, lowerscontemporaneous utility by Uc
(ct , lht
).
In return, the monetary amount Pt ×1 can be used to buy PtQt
×1bonds and to purchase 1
Pt+1× Pt
Qtunits of consumption in t + 1.
Thus, utility in t + 1 rises by PtPt+1Qt
Uc(ct+1, lht+1
). Since the
future is discounted and uncertain, the contemporaneousvaluation of postponing consumption by one period isβEt
[Pt
Pt+1QtUc
(ct+1, lht+1
)].
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Solving the model - FONC Households
Moving to the first order condition with respect to money,equation (19)
∂Lh
∂Mt= Et
[βt+1λh
t+1
]− βtλh
t ⩽ 0 and Mt ≥ 0. (26)
Note that using equation (23), Qt = Et
[βλh
t+1λh
t
], to rewrite the
equation
∂Lh
∂Mt= Et
[βt+1λh
t+1
]− βtλh
t
=
{Et
[βλh
t+1
λht
]− 1
}βtλh
t
= {Qt − 1}βtλht . (27)
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(Ir)relevance of moneyRelevance of money
A classical monetary modelEquilibrium and optimality conditions
.. Irrelevance of money
As marginal utility of consumption is positive for ct > 0,equation (20) implies λh
t ⩾ 0.
Define it ≡ − log (Qt), which is the nominal interest rate. Withthe price of the bond being less than 1 (to imply a positiveinterest rate), it is ∂Lh
∂Mt⩽ 0.
The value of the objective function can be increased byreducing money demand as much as possible implyinghousehold’s money demand to be non-positive.
Equilibrium requires the monetary authority to set Mt = 0 for allt . We are back in a standard real economy with nominal pricesbeing undetermined.
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Relevance of money
How to create positive money demand in a general equilibriummodel?
Many suggestions are offered in the literature. Anon-exhaustive list consists of:
money-in-the-utility-function (MIU) approach,cash-in-advance (CIA) approach (see homework),shopping time models (see homework),inventory models,money search models.
26 / 40
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Money in the utility function
Following most of the literature on applied monetary theory, wewill rely on the MIU approach. Under the MIU:
money enters through real money holdings(
Mht
Pt
)directly
as a service flow in the utility function of agents just likeconsumption,money is not used in the transaction process itself,under suitable restrictions, money demand and the valueof money will be positive.
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Money in the utility function
Augment the utility function in (1) to include real balances
U(
ct , lht ,Mh
tPt
)=
c1−σt
1 − σ−
lh1+φt
1 + φ+
(Mh
t /Pt)1−ν
1 − ν. (28)
These preferences are separable in consumption, leisure andmoney holdings.
Other than the change in the utility function, the model isunchanged, i.e., equations (2) through (8) still apply.
28 / 40
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Household problem under MIU
Write the Lagrangian Lh of the household problem as
Lh = E0
∞∑t=0
βtU(
ct , lht ,Mh
tPt
)
+E0
∞∑t=0
βtλht
[bt−1 + Wt lht + Mt−1 + Tt − Ptct − Qtbt − Mt
].
(29)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Household problem under MIU
The first order conditions are
∂Lh
∂ct= βtUc
(ct , lht ,
Mht
Pt
)− βtλh
t Pt ⩽ 0 and ct ≥ 0 (30)
∂Lh
∂lht= βtUl
(ct , lht ,
Mht
Pt
)+ βtλh
t Wt ⩽ 0 and lht ≥ 0 (31)
∂Lh
∂bt= Et
[βt+1λh
t+1
]− βtλh
t Qt = 0 (32)
∂Lh
∂Mt= βtUm
(ct , lht ,
Mht
Pt
)1Pt
+ Et
[βt+1λh
t+1
]− βtλh
t ⩽ 0
and Mt ≥ 0. (33)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Household problem under MIU
Compare the first order condition with respect to money,equation (33)
∂Lh
∂Mt= βtUm
(ct , lht ,
Mht
Pt
)1Pt
+ Et
[βt+1λh
t+1
]− βtλh
t = 0
to the first order condition for money in the original model,equation (19)
∂Lh
∂Mt= Et
[βt+1λh
t+1
]− βtλh
t = 0.
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Household problem under MIU
Note that for an interior equilibrium, equations (30) and (33)imply
βtUm
(ct , lht ,
Mht
Pt
)1Pt
βtUc
(ct , lht ,
Mht
Pt
) =−Et
[βt+1λh
t+1]+ βtλh
t
βtλht Pt
(34)
or using equation (32), Qt = Et
[βλh
t+1λh
t
],
Um
(ct , lht ,
Mht
Pt
)Uc
(ct , lht ,
Mht
Pt
) = −Et
[βλh
t+1
λht
]+ 1 = 1 − Qt = 1 − exp (−it) .
(35)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Household problem under MIU
The marginal rate of substitution between real balances andconsumption equals the opportunity cost (price) of holdingmoney.
The opportunity cost of holding money is directly related to thenominal interest rate.
The household could reduce nominal money holdings in periodt by one unit, purchase one unit of the bond at price Qt < 1. Asthe bond pays one unit in t + 1, this transaction allows keepingt + 1 assets unchanged while increasing time t consumption by1 − Qt > 0.
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Equilibrium
Again, an equilibrium in this economy is given by paths fornominal prices {Pt ,Wt ,Qt}∞t=0 and real quantities{
ct , lht , yt , l ft}∞
t=0, transfers {Tt}∞t=0 , and money{
Mt ,Mht}∞
t=0,such that
...1 household utility is maximized,
...2 firm profit is maximized,
...3 market for goods (yt = ct) clear,
...4 market for labor(lt = l ft = lht
)clear,
...5 money market(Mh
t = Mt)
clears,...6 bond market (bt = 0) clears.
Changes in the money supply are engineered throughtransfers, i.e.,Tt = Mt − Mt−1.
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
Collecting the relationships that characterize the equilibrium
lφt cσt = wt (36)
exp (−it) = βEt
[(ct+1
ct
)−σ 1exp(πt+1)
](37)
m−νt cσ
t = 1 − exp (−it) (38)yt = exp(at)lt (39)wt = exp(at) (40)at = ρaat−1 + σaεa,t (41)yt = ct (42)
Mt
Mt−1=
mt
mt−1exp(πt) (43)
{Mt}∞t=0 . (44)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
We have used the definitions for the real wage
wt =Wt
Pt, (45)
the nominal interest rate
Qt = exp(−it), (46)
real money holdingsMt/Pt = mt , (47)
the inflation ratePt/Pt−1 = exp(πt), (48)
and technologyat = log(At). (49)
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.
(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
In stating the model equations, we have used the utility functionin (28)
U(
ct , lht ,Mh
tPt
)=
c1−σt
1 − σ−
lh1+φt
1 + φ+
(Mh
t /Pt)1−ν
1 − ν
to replace the generic notations of marginal utilities by
Uc
(ct , lht ,
Mht
Pt
)= c−σ
t (50)
Ul
(ct , lht ,
Mht
Pt
)= −
(lht)φ
(51)
Um
(ct , lht ,
Mht
Pt
)=
(Mh
t /Pt
)−ν=
(mh
t
)−ν. (52)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
Equations (36 to 38) are derived from the first order conditionsof the household, compare to (22, 24 and 35).
Equation (39) stems from the firm’s production technology,compare to (5).
Equation (40) is derived from the first order condition of thefirm, compare to (14).
Equation (41) is the exogenous technology shock in (7).
Equation (42) is the goods market clearing condition(production equals consumption).
Equation (43) connects money supply (growth) and realbalances.
Equation (44) is the behaviour of the central bank stated in (8).
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
Notice, that the budget constraint of the household (3) alwaysholds in equilibrium, for
Ptct + Qtbt + Mht = bt−1 + Wt lht + Mh
t−1 + Tt (53)
implies using bt = 0 and setting the transfer Tt = Mt − Mt−1 weobtain
ct = exp(at)lt = yt . (54)
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(Ir)relevance of moneyRelevance of money
Changing the standard modelSummarizing the full model
.. Summarizing the full model
Note that equation (43) is derived from the condition that themoney supply equals money demand in equilibrium:
Mt
Mt−1=
MtPt−1
PtMt−1
Pt
Pt−1=
mt
mt−1exp(πt). (55)
40 / 40