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in Money an RBC framework Carl E. Walsh UC Santa Cruz January 2008 Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 1 / 36 A basic real business cycle (RBC) model Key components 1 2 Optimizing agents well de ned decision problems General equilibrium perspective Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 2 / 36 A basic real business cycle (RBC) model Representative household Takes prices as given and maximizes Et subject to wt (1 Lt +i ) + rt Kt UC 1 i =0 i U (Ct +1 , Lt +i ) + (1 )Kt 1 = Ct + Kt . First order conditions: t = 0 UL U L wt t = 0 ) = wt UC t + Et (rt +1 + 1 ) t +1 = 0 Combining the rst and third conditions yields the Euler condition: UC (Ct , Lt ) = Et (rt +1 + 1 Carl E. Walsh (UC Santa Cruz) Monetary Economics ) UC (Ct +1 , Lt +1 ) January 2008 3 / 36 A basic real business cycle (RBC) model Firms Firms maximize pro ts in competitive goods and factor markets, subject to a production technology. Yt = F (Nt , Kt max [F (Nt , Kt First order conditons: F N = wt FK = rt 1 , Zt ) 1 , Zt ) w t Nt rt Kt 1] Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 4 / 36 A basic real business cycle (RBC) model Market clearing: Goods market: Yt = Ct + Kt Labor market: Nt = 1 Lt (1 )Kt 1 Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 5 / 36 A basic real business cycle (RBC) model Equilibrium Given an exogenous process for the productivity shock Z , equilibrium consists of time paths for output, consumption, capital, employment, wages and rental rates that satisfy UL (Ct , 1 UC (Ct , 1 UC (Ct , 1 Nt ) = wt Nt ) ) UC (Ct +1 , 1 1 , Zt ) Nt ) = Et (rt +1 + 1 Yt = F (Nt , Kt FN (Nt , Kt FK (Nt , Kt Yt = Ct + Kt Nt +1 ) 1 , Zt ) 1 , Zt ) = wt = rt )Kt 1 (1 Does the equilibrium display business cycle behavior? Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 6 / 36 A basic real business cycle (RBC) model Simulations using a linear approximation to the equilibrium conditions Specify functional forms U= Ct1 1 (1 Lt )1+ 1+ 1 F = Zt Nt Kt1 First order conditions become (with some simpli cation) Nt = Zt Nt Ct Ct 1 Kt1 1 = +1 1 Yt Nt Ct +1 = Et (1 ) Yt Kt 1 Yt = Zt Nt Kt1 Yt = Ct + Kt Carl E. Walsh (UC Santa Cruz) (1 )Kt 1 January 2008 7 / 36 Monetary Economics A basic real business cycle (RBC) model Steady state for Y , C , K , N, R: N = C 1 = R R Y N (1 Y N ) K N K N +1 1 = Y = C + K Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 8 / 36 A basic real business cycle (RBC) model Linearizing around a Z = 1 steady state (lower case denotes % deviation) nt + ct = zt + yt ct = Et ct +1 1 nt Et (yt +1 )kt 1 (1 ) Y K kt ) yt = zt + nt + (1 yt = Assume zt = zt 1 C Y ct + K Y [kt + et (1 )kt 1] Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 9 / 36 A basic real business cycle (RBC) model Calibration I I Parameters: , , , , , , 2 e Do not use business cycle evidence to calibrate parameters Simulations Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 10 / 36 A basic real business cycle (RBC) model Evaluation Persistence Real wages and employment Ability to match historical episodes e.g., 1979-1982 Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 11 / 36 Incorporating money To employ the neoclassical framework to analyze monetary issues, a role for money must be speci ed so that the agents will wish to hold positive quantities of money. Three general approaches to incorporating money into general equilibrium models have been followed: 1 2 Assume that money yields direct utility by incorporating money balances directly into the utility functions of the agents of the model (Sidrauski 1967). Impose transactions costs 1 2 3 4 Make asset exchanges costly (Baumol 1952, Tobin 1956) Require that money be used for certain types of transactions (Clower 1967) Assume time and money can be combined to produce transaction services that are necessary for obtaining consumption goods Assume direct barter of commodities is costly (Kiyotaki and Wright 1989). 3 Treat money like any other asset used to transfer resources intertemporally (Samuelson 1958). Monetary Economics January 2008 12 / 36 Carl E. Walsh (UC Santa Cruz) A basic MIU model First approach: MIU Sidrauski (1967) Real money balances enter directly in the utility function. Given suitable restrictions on the utility function, such an approach can guarantee that, in equilibrium, agents choose to hold positive amounts of money so that money will be positively valued. Representative household takes prices as given and maximizes Et i =0 i U (Ct +i , mt +i , Lt +i ) subject to a budget constraint. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 13 / 36 A basic MIU model Budget constraint, in nominal terms: Wt (1 Lt ) + (rt + 1 ) Pt Kt 1 = Pt Ct + Pt Kt + Bt + Mt +(1 + it 1 )Bt 1 + Mt 1 + Pt t where are real, lump-sum transfers to the household (can be negative). In real terms: wt (1 Lt ) + (rt + 1 ) Kt 1 Bt Mt = Ct + Kt + + Mt 1 Bt 1 +(1 + it 1 ) Pt + Pt + t Pt Pt or wt (1 +(1 + it Lt ) + (rt + 1 ) Kt 1 = Ct + Kt + bt + mt Pt 1 Pt 1 1 ) P t bt 1 + P t mt 1 + t Monetary Economics January 2008 14 / 36 Carl E. Walsh (UC Santa Cruz) A basic MIU model First order conditions: UC UL t = 0 UL = wt UC ) t +1 = 0 t +1 = 0 t +1 = 0 wt t = 0 ) t + Et (rt +1 + 1 t + (1 + it )Et Um t + Et Pt Pt +1 Pt Pt +1 First three look just as before (except UC and UL may depend on m). Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 15 / 36 A basic MIU model The FOC for money can be written t Um = + Et Pt Pt Solve forward to yield 1 = Pt 1 t Et t +1 Pt +1 i =0 i Um (Ct +i , mt +i , Lt +i ) Pt +i Value of money equals expected present value of discounted utility return. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 16 / 36 A basic MIU model Combining the rst and third conditions yields the Euler condition: UC (Ct , mt , Lt ) = Et (rt +1 + 1 Similarly, UC (Ct , mt , Lt ) = (1 + it )Et From fourth and fth, Um = it Et UC But 1 = (1 + it )Et so Um = it Et UC Pt Pt +1 Pt Pt +1 Pt Pt +1 UC (Ct +1 , mt +1 , Lt +1 ) UC (Ct , mt , Lt ) UC (Ct +1 , mt +1 , Lt +1 ) UC (Ct , mt , Lt ) Pt Pt +1 UC (Ct +1 , mt +1 , Lt +1 ) ) UC (Ct +1 , mt +1 , Lt +1 ) UC (Ct +1 , mt +1 , Lt +1 ) it = . UC (Ct , mt , Lt ) 1 + it January 2008 17 / 36 Carl E. Walsh (UC Santa Cruz) Monetary Economics A basic MIU model Firm side of model is same as in basic RBC model. Collecting equilibrium conditions I Given an exogenous process for the productivity shock and the nominal stock of money, equilibrium consists of time paths for output, real money balances, consumption, capital, employment, wages, the nominal rate of interest, the price level, and rental rates that satisfy Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 18 / 36 A basic MIU model UL (Ct , mt , 1 UC (Ct , mt , 1 Nt ) = wt Nt ) it Um = . UC 1 + it UC (Ct , mt , 1 Nt ) = Et (rt +1 + 1 ) UC (Ct +1 , mt +1 , 1 Nt +1 ) Pt Pt +1 UC (Ct , mt , Lt ) = (1 + it )Et UC (Ct +1 , mt +1 , Lt +1 ) Yt = F (Nt , Kt FN (Nt , Kt FK (Nt , Kt Yt = Ct + Kt 1 , Zt ) 1 , Zt ) 1 , Zt ) = wt = rt )Kt 1 (1 Pt mt = Mt Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 19 / 36 A basic MIU model Steady-state Standard RBC relationships plus 1= Um (C , m, 1 UC (C , m, 1 1+i 1+ 1+ N) i = =1 N) 1+i M0 (1 + )t m These imply the Fisher equation 1 + i = (r + 1 Pt = i r + ) (1 + ) or Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 20 / 36 Superneutrality If UL and UC independent of m, real side identical to basic RBC model. (Seady state and dynamics) I I Neutrality: One time changes in the nominal quantity of money a ect only the price level. Superneutrality: Changes in growth rate of money do not a ect real variables. Changes in the rate of growth of money a ect only the in ation rate, the nominal interest rate, and real money balances. I If F displays constant returns to scale and UL and UC depend on m, still get steady-state capital-labor ratio independence of money growth. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 21 / 36 Welfare cost of in ation Because money holdings yield direct utility and higher in ation reduces real money balances, in ation generates a welfare loss. Questions: I I How large is the welfare cost of in ation? Is there an optimal rate of in ation that maximizes the steady-state welfare of the representative household? Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 22 / 36 Welfare cost of in ation The optimal rate of in ation addressed by Bailey (1956) and M. Friedman (1969). I I I I Basic intuition: the private opportunity cost of holding money depends on the nominal rate of interest. The social marginal cost of producing money is essentially zero. The wedge that arises between the private marginal cost and the social marginal cost when the nominal rate of interest is positive generates an ine ciency. This ine ciency would be eliminated if the nominal rate of interest were zero. So the optimal rate of in ation is a rate of de ation approximately equal to the real return on capital. In the steady state, real money balances are directly related to the in ation rate, so the optimal rate in of ation is also frequently discussed under the heading of the optimal quantity of money. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 23 / 36 Welfare cost of in ation The major criticism of this result is that of Phelps (1973), who pointed out that money growth generates revenue for the government the in ation tax. The implicit assumption so far has been that variations in money growth are engineered via lump-sum transfers. Any e ects on government revenue can be o set by a suitable adjustment in these lump-sum transfers (taxes). But if governments only have distortionary taxes available for nancing expenditures, then reducing in ation tax revenues to achieve the Friedman rule for optimal in ation requires that the lost revenue be replaced through increases in other distortionary taxes. To minimize the total distortions associated with raising a given amount of revenue, it may be optimal to rely to some degree on the in ation tax. More recent work has reexamined these results (see Chari, Christiano, and Kehoe 1991, 1996, Correia and Teles 1996, 1999, Mulligan and Sala-i-Martin 1997). Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 24 / 36 Cash-in-advance models Clower (1967), goods buy money and money buys goods, but goods don buy goods. t And because goods don buy goods, a medium of exchange that t serves to aid the process of transacting will have value. The demand for money is then determined by the nature of the economy transactions technology. s Rather than allowing substitutability between time and money in carrying out transactions, cash-in-advance (CIA) models simply require that money balances be held to nance certain types of purchases; without money, these purchases cannot be made. CIA models. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 25 / 36 Cash-in-advance models Timing assumptions also are important in CIA models. In Lucas (1982), agents are able to allocate their portfolio between cash and other assets at the start of each period, after observing any current shocks but prior to purchasing goods. I I The asset market opens rst and then the goods market opens. If there is a positive opportunity cost of holding money and the asset market opens rst, agents will only hold an amount of money that is just su cient to nance their desired level of consumption. This implies that agents have available for spending only the cash carried over from the previous period, and so cash balances must be chosen before agents know how much spending they will wish to undertake. If uncertainty is resolved after money balances are chosen, an agent may nd that she is holding cash balances that are too low to nance her desired spending level. Or she may be left with more cash than she needs, thereby forgoing interest income. Monetary Economics January 2008 26 / 36 In Svensson (1985), the goods market opens rst. I I Carl E. Walsh (UC Santa Cruz) Cash-in-advance models The representative agent Maximizes present discounted value of expected utility subject to a sequence of new constraints. If the goods markets open rst and the agent enters the period with money holdings Mt 1 and receives a lump-sum transfer Tt (in nominal terms), the CIA constraint takes the form Pt ct Mt 1 + Tt , where c is real consumption, P is the aggregate price level, and T is the nominal lump-sum transfer. In real terms, Mt 1 Tt mt 1 + = + t , ct Pt Pt t where mt 1 = Mt 1 /Pt 1 , t = Pt /Pt 1 is 1 plus the in ation rate, and t = Tt /Pt . Income from production during period t will not be available for consumption purchases during period t. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 27 / 36 Cash-in-advance models Budget constraint in real terms, t t f (kt 1 ) + (1 ct + mt + bt + kt where mt 1 )kt 1 + t + + (1 + it t 1 )bt 1 , and m and b are real cash and bond holdings. Let at = bt + mt . Then budget constraint implies t +1 = f (kt ) + (1 )kt + t +1 + Rt at 1 + it t +1 mt . This form highlights that there is a cost to holding money when the nominal interest rate is positive. This cost is (1 + it )/ t +1 . This is the same expression for the opportunity cost of money in an MIU model. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 28 / 36 Cash-in-advance models The decision problem De ne the value function V ( t , mt 1) = maxfu(ct ) + V ( t +1 , mt )g, where the maximization is subject to the budget constraint t ct + mt + bt + kt , the CIA constraint, and the de nition of t +1 . Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 29 / 36 Cash-in-advance models The decision problem Letting t ( t ) denote the Lagrangian multiplier associated with the budget constraint (the CIA constraint), the rst order necessary conditions for the agent choice of consumption, capital, bond, and s money holdings take the form uc (ct ) [fk (kt ) + 1 t t = 0 t = 0 t = 0 t = 0. ] V ( t +1 , mt ) Rt V ( t +1 , mt ) Rt it V ( t +1 , mt ) + Vm ( t +1 , mt ) t +1 Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 30 / 36 Cash-in-advance models From the envelope theorem, V ( t , mt Vm ( t , mt 1) 1) = t 1 t t . = t is equal to the marginal utility of wealth. The marginal utility of consumption exceeds the marginal utility of wealth by the value of liquidity services, t . The individual must hold money in order to purchase consumption, so the cost, to which the marginal utility of consumption is set equal, is the marginal utility of wealth plus the cost of the liquidity services needed to nance the transaction. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 31 / 36 Cash-in-advance models The rst order conditions imply t = t +1 + t +1 t +1 1 = Pt 1 t t +1 + t +1 Pt +1 ) . This equation can also be interpreted as an asset pricing equation for money. The price of a unit of money in terms of goods is just 1/Pt at time t; its value in utility terms is t /Pt . Solving this equation forward implies that 1 = Pt 1 t i =1 i t +i Pt +i . Value of money related to present discounted value of Lagrangian multipliers on CIA constraint. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 32 / 36 Cash-in-advance models The nominal rate of interest Since t = Rt t +1 = t +1 + t +1 / t +1 , or Rt t +1 t +1 = t +1 + t +1 and 1 + it = Rt t +1 , the nominal interest rate is given by it = t +1 + t +1 t +1 1= t +1 . t +1 The nominal rate of interest is positive if and only if money yields liquidity services ( t +1 > 0). In particular, if the nominal interest rate is positive, the CIA constraint is binding ( > 0). Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 33 / 36 Cash-in-advance models Marginal utility of consumption We can use the relationship between the nominal rate of interest and the Lagrangian multipliers to rewrite the expression for the marginal utility of consumption as uc = (1 + / ) = (1 + i ) . Since represents the marginal value of income, the marginal utility of consumption exceeds that of income whenever the nominal interest rate is positive. Even though the economy technology allows output to be directly s transformed into consumption, the price of consumption is not equal to 1; it is 1 + i since the household must hold money to nance consumption. Thus, in this CIA model, a positive nominal interest rate acts as a tax on consumption; it raises the price of consumption above its production cost. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 34 / 36 Cash-in-advance models Modi cations Cash and Credit Goods I I I Lucas and Stokey (1983, 1987) introduced the idea that the CIA constraint may only apply to a subset of consumption goods. The marginal utility of cash goods will be equated to + , while the marginal utility of credit goods will be equated to . Hence, the CIA requirement for cash goods drives a wedge between the marginal utilities of the two types of goods. It is exactly as if the consumer faces a tax of / = i on purchases of the cash good. Higher in ation, by reducing holdings of real cash balances, serves to raise the tax on cash goods and generates a substitution away from the cash good and toward the credit good. CIA and Investment Goods I The in ation tax applies to both consumption and investment goods. Higher rates of in ation will tend to discourage capital accumulation lower the steady-state capital-labor ratio. Monetary Economics January 2008 35 / 36 Carl E. Walsh (UC Santa Cruz) Welfare costs of in ation In CIA models, in ation acts as a tax on goods or activities whose purchase requires cash. This tax then introduces a distortion by creating a wedge between the marginal rates of transformation implied by the economy technology s and the marginal rates of substitution faced by consumers. Since the CIA model, like the MIU model, o ers no reason for such a distortion to be introduced (there is no ine ciency that calls for Pigovian taxes or subsidies on particular activities, and the government revenue needs can be met through lump-sum taxation), s optimality calls for setting the in ation tax equal to zero. The in ation tax is directly related to the nominal rate of interest; a zero in ation tax is achieved when the nominal rate of interest is equal to zero. Carl E. Walsh (UC Santa Cruz) Monetary Economics January 2008 36 / 36

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