Transactions and Money Demand Walsh Chapter 3

Transactions and Money Demand Walsh Chapter 3

1 Shopping time models 1.1 Assumptions Purchases require transactions services ψ = ψ (m, n s ) = c where ψ n s 0, ψ m 0, ψ n s n s 0, ψ mm 0 positive but diminishing marginal productivity for both arguments

Solve for labor required to purchase c g c > 0 g m 0 n s = g (c, m) marginal product of money in reducing shopping time is g m if greater consumption raises marginal product of money in reducing shopping time (raises g m ), then reduces g m and g mc 0 utility v (c, l) leisure l = 1 n n s

equivalent to a money in the utility function model u (c, m, n) = v [c, 1 n g (c, m)] potentially sign u cm u m = v l (c, 1 n g (c, m)) g m (c, m) > 0 u mc = (v ll g c v lc ) g m v l g mc diminishing marginal productivity of leisure v ll 0, v ll g c g m 0 if g mc 0, then v l g mc 0 if these two dominate, then u mc > 0

if leisure and consumption are strong substitutes v lc < 0, such that dominates, reverse sign of u mc

Household optimization problem maximize subject to i=0 β i u [c t+i, 1 n t+i g (c t+i, m t+i )] f (k t 1, n t )+τ t +(1 δ) k t 1 + (1 + i t 1) b t 1 + m t 1 1 + π t = c t +k t +b t +m t yields money demand function f n (k t 1, n t ) g m (c t, m t ) = 1 + i t where lhs is value of transactions time saved by holding money marginal product of money in reducing shopping time ( g m ) times the marginal value of labor (f n ) i t

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 money has positive but diminishing marginal productivity in reducing transactions costs Ψ c 0, Ψ m 0, Ψ cc, Ψ mm 0

marginal transactions costs do not increase with additional money Ψ cm 0 expansion paths have non-negative slopes so that c + Ψ increases with income lim m 0 Ψ m = assures that money is essential Add transactions costs to budget constraint f (k t 1, n t ) + τ t + (1 δ) k t 1 + (1 + i t 1) b t 1 + m t 1 1 + π t = c t + k t + b t + m t + Ψ (c t, m t )

2.2 Functional Equivalence to Shopping Time Models If redefine the consumption variable in the MIU model to be c + Ψ, then money enters utility U (c) = U [W (c + Ψ, m)] Justification for MIU model Redefinition of consumption to include transactions services allows transactions cost models to be equivalent to shopping time models

3 Cash-in-Advance models 3.1 Assumptions Certainty Representative agent with utility t=0 β t u (c t )

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 P t C t M t 1 + T t in real terms C t M t 1 P t + τ t = m t 1 P t 1 P t + τ t = m t 1 1 + π t + τ t

Nominal budget constraint P t ω t = P t f (k t 1 ) + (1 δ) P t k t 1 + M t 1 + T t + (1 + i t 1 ) B t 1 P t c t + P t k t + M t + B t Real budget constraint ω t = f (k t 1 )+(1 δ) k t 1 + M t 1 +τ t + (1 + i t 1) B t 1 P t P t c t +k t +m t +b t ω t = f (k t 1 )+(1 δ) k t 1 +τ t + m t 1 + (1 + i t 1 ) b t 1 1 + π t c t +k t +m t +b t

The nominal interest rate is the opportunity cost of money Define a t = m t + b t and 1 + r t 1 = 1 + i t 1 1 + π t ω t = f (k t 1 ) + (1 δ) k t 1 + τ t + (1 + r t 1 ) a t 1 i t 1m t 1 1 + π t c t + k t + m t + b t Present value of the opportunity cost of money is i t 1 m t 1 (1 + π t ) (1 + r t 1 ) = i t 1m t 1 (1 + i t 1 )

3.2 Optimization Maximize subject to V (ω t, m t 1 ) = max {u (c t ) + βv (ω t+1, m t )} ω t c t + m t + b t + k t with multiplier λ t m t 1 1 + π t + τ t c t with multiplier µ t ω t+1 = f (k t ) + (1 δ) k t + τ t+1 + m t + (1 + i t ) b t 1 + π t+1

First order conditions c u c (c t ) λ t µ t = 0 k βv ω (ω t+1, m t ) [f k (k t ) + (1 δ)] λ t = 0 b βv ω (ω t+1, m t ) (1 + r t ) λ t = 0 m βv ω (ω t+1, m t ) 1 + π t+1 + βv m (ω t+1, m t ) λ t = 0

Envelope conditions λ t is the marginal utility of wealth V ω (ω t, m t 1 ) = λ t µ t is the marginal value of liquidity services V m (ω t, m t 1 ) = µ t 1 + π t = µ tp t 1 P t

3.3 Interpretations marginal utility of consumption equals the marginal value of wealth plus the marginal value of liquidity services u c (c t ) = λ t + µ t from FO condition on bonds, write the Euler equation in terms of λ: marginal cost of reducing wealth must equal the utility value of carrying that wealth forward one period, earning a gross real return 1 + r t, discounted at rate β λ t = β (1 + r t ) λ t+1 use FO condition on money to derive asset pricing equation for money

FO condition λ t = β ( λ t+1 + µ t+1 = β 1 + π t+1 ) ( λt+1 + µ t+1 ) Pt P t+1 value of a unit of money in utility terms at time t is λ t P t dividing through by P t and solving forward yields from envelope condition λ t P t = i=1 β i ( µt+i P t+i ) µ t+i P t+i = V m (ω t+i, m t+i 1 ) P t+i 1

where V (ω t+i, m t+i 1 ) M t+i 1 = V m (ω t+i, m t+i 1 ) m t+i 1 M t+i 1 = V m (ω t+i, m t+i 1 ) P t+i 1 = utility value of a unit of money is given by the present value of the marginal utility of money in all future periods λ t P t = t=1 β i V (ω t+i, m t+i 1 ) M t+i 1 in general the value of an asset is the present value of its future returns for money, the future returns are the liquidity services provided by money, giving it its marginal utility if CIA constraint not binding, ( µ t+i = 0 ) money has no value

compare with MIU model Solving forward yields λ t P t = β λ t+1 P t+1 + u m (c t, m t ) P t λ t P t = i=1 β iu m (c t+i, m t+i ) P t+i where u m plays the role of the multiplier µ t

nominal interest rate combine Euler equation in λ with FO condition on money simplify last equality λ t = β (1 + r t ) λ t+1 = β ( λ t+1 + µ t+1 1 + π t+1 (1 + π t+1 ) (1 + r t ) λ t+1 = λ t+1 + µ t+1 use definition of nominal interest rate 1 + i t = 1 + µ t+1 λ t+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 constraint binds

price of consumption interest is a tax on consumption from FO condition on consumption and solution for nominal interest rate ( u c = λ + µ = λ 1 + µ ) = λ (1 + i) λ when the nominal interest rate is positive, the marginal utility of consumption exceeds the marginal value of income (λ) price of consumption is 1 + i since an agent must hold money for one period at an opportunity cost of i before he can purchase consumption i represents a tax on consumption, raising the price of consumption above production cost

Velocity cash in advance constraint c t = M t 1 P t + τ t equilibrium value of transfers equilibrium velocity is unity τ t = M t M t 1 P t c t = M t P t 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 βv ω (ω t+1, m t ) [f k (k t ) + (1 δ)] λ t = 0 substitute using envelope condition βλ t+1 [f k (k t ) + (1 δ)] λ t = 0 dropping time subscripts f k (k ss ) = β 1 1 + δ capital stock is independent of the level and rate of growth of money

consumption using aggregated budget constraint c ss = f (k ss ) δk ss consumption is independent of the level and rate of growth of money inflation has no effect on steady state value of consumption even though acts as a tax because cannot avoid it inflation has no effect on real money balances m ss = c ss

relative price of money in terms of consumption in MIU u m u c = i 1 + i in cash in advance µ u c = µ λ (1 + i) = i 1 + i but cannot use this relationship to solve for money demand in cash in advance

3.5 Welfare Cost of Inflation Welfare is given by t=0 β t u (c ss ) = u (css ) 1 β Since steady-state consumpiton is independent of inflation, there is no welfare 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 goods are 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 of cash goods relative to credit goods and reducing their consumption

Money demand (for purchasing cash goods) falls as the nominal interest rate 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, discouraging investment, 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 in advance constraints, relative to their cost of production This wedge is ineffi cient, implying that the optimal wedge is zero, achieved when 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 is not Reduction in labor supply reduces steady-state capital stock money is not superneutral compares to ambiguous relation in MIU depending on sign of u cm Steady-state output/capital and consumption/capital ratios are unaffected

Short-run effects of changes in money growth Unexpected increase in M has only a price level effect, so money is neutral requires that money increase used for transfers get effects if used for government spending Expected increase in money growth raises i, yielding substitution out of consumption and into labor, reducing labor supply If policy has money growth react to productivity shocks, then money growth 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 therefore occurring with low probability goods for money, requiring only a single coincidence of wants and occurring with higher probability

therefore, existence of money raises the probability of mutually beneficial exchange, raising welfare Price is determined by bargaining and depends on the quantity of money