lecture8 from martin (1)

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(Ir)relevance of moneyRelevance of money

.

......

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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.. 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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.. 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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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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.. 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.

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.. 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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.. 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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.. 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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.. 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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.. 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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.. 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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.. 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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.. Monetary authority

The monetary authority sets a path for the money supply

{Mt}∞t=0 . (8)

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.. 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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.. 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.

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.. 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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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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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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.. 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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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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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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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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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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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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.. 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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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.

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.. 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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.. 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.

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.. 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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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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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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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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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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.. 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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.. 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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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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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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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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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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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)

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lecture8 from martin (1)

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