Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions...
Transcript of Transactions and Money Demand Walsh Chapter 3bd892/Walsh3.pdf · where lhs is value of transactions...
Transactions and Money Demand
Walsh Chapter 3
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
• 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
• 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
∗ if leisure and consumption are strong substitutes vlc < 0, such thatdominates, reverse sign of umc
• 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)
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
— 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)
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
3 Cash-in-Advance models
3.1 Assumptions
• Certainty
• Representative agent with utility∞∑t=0
βtu (ct)
• 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
• 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
• 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)
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
• 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
• 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
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
— 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
— 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
— 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
• 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
)
— if the nominal interest rate is positive, then the cash in advance con-straint binds
• 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
• 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
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
• 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
• 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
• 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
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
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
— Money demand (for purchasing cash goods) falls as the nominal interestrate rises
— Velocity changes with the nominal interest rate
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
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
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
• 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
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
∗ 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