Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions...

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Transactions and Money Demand Walsh Chapter 3

Transcript of Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions...

Page 1: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

Transactions and Money Demand

Walsh Chapter 3

Page 2: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

1 Shopping time models

1.1 Assumptions

• Purchases require transactions services

— ψ = ψ (m,ns) = c

— where ψns ≥ 0, ψm ≥ 0, ψnsns ≤ 0, ψmm ≤ 0

— positive but diminishing marginal productivity for both arguments

Page 3: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• Solve for labor required to purchase c

ns = g (c,m)

— gc > 0 gm ≤ 0

— marginal product of money in reducing shopping time is −gm

— if greater consumption raises marginal product of money in reducingshopping time (raises −gm), then reduces gm and gmc ≤ 0

• utilityv (c, l)

• leisurel = 1− n− ns

Page 4: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• equivalent to a money in the utility function model

u (c,m, n) = v [c, 1− n− g (c,m)]

• potentially sign ucm

— um = −vl (c, 1− n− g (c,m)) gm (c,m) > 0

— umc = (vllgc − vlc) gm − vlgmc

∗ diminishing marginal productivity of leisure

vll ≤ 0, vllgcgm ≥ 0

∗ if gmc ≤ 0, then −vlgmc ≥ 0

∗ if these two dominate, then umc > 0

Page 5: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

∗ if leisure and consumption are strong substitutes vlc < 0, such thatdominates, reverse sign of umc

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• Household optimization problem

— maximize∞∑i=0

βiu [ct+i, 1− nt+i − g (ct+i, mt+i)]

subject to

f (kt−1, nt)+τ t+(1− δ) kt−1+(1 + it−1) bt−1 +mt−1

1 + πt= ct+kt+bt+mt

— yields money demand function

−fn (kt−1, nt) gm (ct,mt) =it

1 + itwhere lhs is value of transactions time saved by holding money —mar-ginal product of money in reducing shopping time (−gm) times themarginal value of labor (fn)

Page 7: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

2 Real Resource Costs Models (Feenstra)

2.1 Assumptions

• Transactions use up real resources Ψ (c,m)

— Ψ ≥ 0

— transactions costs are zero if there is no consumption Ψ (0,m) = 0

— transactions costs rise at an increasing rate in consumption and moneyhas positive but diminishing marginal productivity in reducing transac-tions costs

Ψc ≥ 0, Ψm ≤ 0, Ψcc,Ψmm ≥ 0

Page 8: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

— marginal transactions costs do not increase with additional moneyΨcm ≤ 0

— expansion paths have non-negative slopes so that c+ Ψ increases withincome

— limm→0 Ψm = −∞ assures that money is essential

• Add transactions costs to budget constraint

f (kt−1, nt) + τ t + (1− δ) kt−1 +(1 + it−1) bt−1 +mt−1

1 + πt= ct + kt + bt +mt + Ψ (ct,mt)

Page 9: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

2.2 Functional Equivalence to Shopping Time Models

• If redefine the consumption variable in the MIU model to be c + Ψ, thenmoney enters utility

U (c) = U [W (c+ Ψ,m)]

• Justification for MIU model

• Redefinition of consumption to include transactions services allows trans-actions cost models to be equivalent to shopping time models

Page 10: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3 Cash-in-Advance models

3.1 Assumptions

• Certainty

• Representative agent with utility∞∑t=0

βtu (ct)

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• In a given period t, goods markets open before asset markets

— purchase goods with money acquired last period

— and with current government transfers

— income from production is not available until next period

PtCt ≤Mt−1 + Tt

— in real terms

Ct ≤Mt−1

Pt+ τ t = mt−1

Pt−1

Pt+ τ t =

mt−1

1 + πt+ τ t

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• Nominal budget constraint

Ptωt = Ptf (kt−1) + (1− δ)Ptkt−1 +Mt−1 + Tt + (1 + it−1)Bt−1

≥ Ptct + Ptkt +Mt +Bt

• Real budget constraint

ωt = f (kt−1)+(1− δ) kt−1+Mt−1

Pt+τ t+

(1 + it−1)Bt−1

Pt≥ ct+kt+mt+bt

ωt = f (kt−1)+(1− δ) kt−1+τ t+mt−1 + (1 + it−1) bt−1

1 + πt≥ ct+kt+mt+bt

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• The nominal interest rate is the opportunity cost of money

— Define

at = mt + bt

and

1 + rt−1 =1 + it−1

1 + πt

ωt = f (kt−1) + (1− δ) kt−1 + τ t + (1 + rt−1) at−1 −it−1mt−1

1 + πt≥ ct + kt +mt + bt

— Present value of the opportunity cost of money is

it−1mt−1

(1 + πt) (1 + rt−1)=it−1mt−1

(1 + it−1)

Page 14: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3.2 Optimization

• Maximize

V (ωt,mt−1) = max {u (ct) + βV (ωt+1,mt)}subject to

ωt ≥ ct +mt + bt + kt with multiplier λt

—mt−1

1 + πt+ τ t ≥ ct with multiplier µt

ωt+1 = f (kt) + (1− δ) kt + τ t+1 +mt + (1 + it) bt

1 + πt+1

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• First order conditions

— c

uc (ct)− λt − µt = 0

— k

βVω (ωt+1,mt) [fk (kt) + (1− δ)]− λt = 0

— b

βVω (ωt+1,mt) (1 + rt)− λt = 0

— mβVω (ωt+1,mt)

1 + πt+1+ βVm (ωt+1,mt)− λt = 0

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• Envelope conditions

— λt is the marginal utility of wealth

Vω (ωt,mt−1) = λt

— µt is the marginal value of liquidity services

Vm (ωt,mt−1) =µt

1 + πt=µtPt−1

Pt

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3.3 Interpretations

• marginal utility of consumption equals the marginal value of wealth plusthe marginal value of liquidity services

uc (ct) = λt + µt

• from FO condition on bonds, write the Euler equation in terms of λ: mar-ginal cost of reducing wealth must equal the utility value of carrying thatwealth forward one period, earning a gross real return 1 + rt, discountedat rate β

λt = β (1 + rt)λt+1

• use FO condition on money to derive asset pricing equation for money

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— FO condition

λt =β(λt+1 + µt+1

)1 + πt+1

=β(λt+1 + µt+1

)Pt

Pt+1

— value of a unit of money in utility terms at time t is λtPt

— dividing through by Pt and solving forward yields

λt

Pt=∞∑i=1

βi(µt+iPt+i

)

— from envelope condition

µt+iPt+i

=Vm (ωt+i,mt+i−1)

Pt+i−1

Page 19: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

— where∂V (ωt+i,mt+i−1)

∂Mt+i−1= Vm (ωt+i,mt+i−1)

∂mt+i−1

Mt+i−1=Vm (ωt+i,mt+i−1)

Pt+i−1=µt+iPt+i

— utility value of a unit of money is given by the present value of themarginal utility of money in all future periods

λt

Pt=∞∑t=1

βi∂V (ωt+i,mt+i−1)

∂Mt+i−1

∗ in general the value of an asset is the present value of its futurereturns

∗ for money, the future returns are the liquidity services provided bymoney, giving it its marginal utility

∗ if CIA constraint not binding,(µt+i = 0

)money has no value

Page 20: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

— compare with MIU model

λt

Pt= β

λt+1

Pt+1+um (ct,mt)

Pt

Solving forward yields

λt

Pt=∞∑i=1

βium (ct+i,mt+i)

Pt+i

where um plays the role of the multiplier µt

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• nominal interest rate

— combine Euler equation in λ with FO condition on money

λt = β (1 + rt)λt+1 =β(λt+1 + µt+1

)1 + πt+1

— simplify last equality

(1 + πt+1) (1 + rt)λt+1 = λt+1 + µt+1

— use definition of nominal interest rate

1 + it = 1 +µt+1

λt+1

— nominal interest rate is positive if cash in advance constraint binds(µt+1 > 0

)

Page 22: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

— if the nominal interest rate is positive, then the cash in advance con-straint binds

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• price of consumption — interest is a tax on consumption

— from FO condition on consumption and solution for nominal interestrate

uc = λ+ µ = λ

(1 +

µ

λ

)= λ (1 + i)

— when the nominal interest rate is positive, the marginal utility of con-sumption exceeds the marginal value of income (λ)

— price of consumption is 1 + i since an agent must hold money for oneperiod at an opportunity cost of i before he can purchase consumption

— i represents a tax on consumption, raising the price of consumptionabove production cost

Page 24: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• Velocity

— cash in advance constraint

ct =Mt−1

Pt+ τ t

— equilibrium value of transfers

τ t =Mt −Mt−1

Pt

— equilibrium velocity is unity

ct =Mt

Pt

— money demand does not depend on the interest rate

Page 25: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3.4 Steady State Equilibrium

• using the λ Euler equation

β (1 + r) = 1

1 + r =1

β

• using the definition of the nominal interest rate (Fisher relation) yields

1 + i = (1 + π) (1 + r) =1 + π

β

Page 26: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• capital stock

— first order condition on capital

βVω (ωt+1,mt) [fk (kt) + (1− δ)]− λt = 0

— substitute using envelope condition

βλt+1 [fk (kt) + (1− δ)]− λt = 0

— dropping time subscripts

fk (kss) = β−1 − 1 + δ

— capital stock is independent of the level and rate of growth of money

Page 27: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• consumption

— using aggregated budget constraint

css = f (kss)− δkss

— consumption is independent of the level and rate of growth of money

— inflation has no effect on steady state value of consumption even thoughacts as a tax because cannot avoid it

— inflation has no effect on real money balances

mss = css

Page 28: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• relative price of money in terms of consumption

— in MIUum

uc=

i

1 + i

— in cash in advanceµ

uc=

µ

λ (1 + i)=

i

1 + i

— but cannot use this relationship to solve for money demand in cash inadvance

Page 29: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3.5 Welfare Cost of Inflation

• Welfare is given by∞∑t=0

βtu (css) =u (css)

1− β

• Since steady-state consumpiton is independent of inflation, there is nowelfare cost of inflation and no optimal rate of inflation

Page 30: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3.6 Modifications of the Basic Model

3.6.1 Cash and Credit Goods

• Cash goods are subject to a cash in advance constraint, but credit goodsare not

• In equilibrium, real money balances will equal consumption of cash goods

— velocity is not unity

• An increase in inflation raises the nominal interest rate, raising the cost ofcash goods relative to credit goods and reducing their consumption

Page 31: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

— Money demand (for purchasing cash goods) falls as the nominal interestrate rises

— Velocity changes with the nominal interest rate

Page 32: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

3.6.2 Investment

• Cash in advance constraint applies to investment expenditures

• Increase in inflation acts as a tax on capital accumulation, discouraginginvestment, and having real effects

3.6.3 Optimal Rate of Inflation in Multi-good Models

• Inflation drives a wedge between the price of goods with different cash inadvance constraints, relative to their cost of production

• This wedge is ineffi cient, implying that the optimal wedge is zero, achievedwhen the nominal interest rate is zero

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3.7 Stochastic CIA Model

3.7.1 Assumptions

• Add capital with an investment decision

• Add a labor-leisure choice

• Consumption is a cash good, while investment and labor are credit goods

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Effects of an Increase in Inflation

• Agents shift away from consumption, which is taxed, to leisure, which isnot

— Reduction in labor supply reduces steady-state capital stock

∗ money is not superneutral

∗ compares to ambiguous relation in MIU depending on sign of ucm

— Steady-state output/capital and consumption/capital ratios are unaf-fected

Page 35: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

• Short-run effects of changes in money growth

— Unexpected increase in M has only a price level effect, so money isneutral

∗ requires that money increase used for transfers

∗ get effects if used for government spending

— Expected increase in money growth raises i, yielding substitution outof consumption and into labor, reducing labor supply

— If policy has money growth react to productivity shocks, then moneygrowth can have real effects

Page 36: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

4 Search Models

• Focus on money’s fundamental role as a medium of exchange

• Agents engage in search to make trades

— A trade occurs only if both parties agree

— A trade can exchange

∗ goods for goods, requiring a double coincidence of wants and there-fore occurring with low probability

∗ goods for money, requiring only a single coincidence of wants andoccurring with higher probability

Page 37: Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions time saved by holding money Œmar- ... Œ utility value of a unit of money is given

∗ therefore, existence of money raises the probability of mutually ben-eficial exchange, raising welfare

— Price is determined by bargaining and depends on the quantity of money