Economics 202A Lecture Outline, November 25, 2008(Version 2.0)Maurice ObstfeldIssues in Monetary PolicyIn this lecture I survey several issues important in the design and imple-mentation of monetary policy in practice. Some of these are related to thegovernment's revenue needs, as disc ussed in the last lecture, but we also gobeyond that question to consider other problems.Money and welfare: Milton Friedman's \optimum quantity ofmoney"A useful dynamic framework for thinking about monetary policy in aworld of flexible prices was provided by Miguel Sidrauski and William Brock.1The representative consumer maximizesZ1tu [c(s); m(s)] e(st)ds;where c is consumption and m  M=P the stock of real balances held. Above, is the individual's subjective rate of discount, which can dier from the realinterest rate. We are motivating a demand for money by assuming that theindividual derives a ow of utility from his/her holdings of real balances| implicitly, these help the person economize on transaction costs, provideliquidity, etc.Total real nancial assets a are the sum of real money m and real bondsb, which pay a real rate of interest r(t) at time t:a = m + b:1See Miguel Sidrauski, \Rational Choice and Patterns of growth in a Monetary Econ-omy," American Economic Review 57 (May 1967): 534-44; and William A. Brock, \Moneyand Growth: The Case of Long Run perfect Foresight," International Economic Review15 (October 1974): 750-77.1Let (t) be a transfer that the individual rec eives from the government eachinstant.2Then if we assume an endowment economy with output y(t), theevolution of wealth is given by the dierential equation_a = y + rb +   c  m= y + ra +   c  (r + )m:Since this last constraint incorporates the portfolio constraint that a = m +b, we need no longer worry about it. Under an assumption that we haveperfect foresight, so that actual  =_P =P equals expected ination, theFisher equation tells us that the nominal interest rate isi = r + ;so the last constraint b ecomes_a = y + ra +   c  im:We can analyze the individual optimum using the Maximum Principle.If  denotes the costate variable, the (current-value) Hamiltonian isH = u (c; m) +  (y + ra +   im) :In the maximization problem starting at time t, a(t) is predetermined atthe level of the individual, who chooses optimal paths for c and m. ThePontryagin necessary conditions are@H@c= uc  = 0;@H@m= um i = 0;_   = @H@a= r:2Beware: in the last lecture the same symbol  denoted taxes, or negative transfers, soall signs preceding  are reversed in this lecture compared to the last.2To make life simple, let us assume that the cross-derivative ucm= 0 |making u(c; m) additively separable in consumption and real balances3. Thenwe can rewrite the last equation asucc_c = uc(  r)or equivalently, as_cc= uccucc(r  ):Do you recognize this as the continuous-time version of the intertemporalbond Euler equation? For the isoelastic utility function with intertemporalsubstitution elasticity , we write this as_cc= (r  );an equation you will see ad nauseam in Economics 202B.43This means that the utility function takes the formu(c; m) = (c) + (m)for strictly concave functions (c) and (m). Below, we will sometimes write the marginalutilities um(c; m) and uc(c; m) in their general forms, as functions of c and m, respectively,even though that dependence is trivial when ucm= 0:4To see that this is actually a familiar equation, let us imagine that we have a timeinterval of length h, that the gross return to lending over that period is 1 + rh, and thatthe discount factor between periods is  = (1 + h)1: Then the Euler equation (in theisoelastic case, for example) would bec1t=1 + rh1 + hc1t+h:Take logs of both sides and divide by h to getlog ct+h log cth=[log(1 + rh)  log(1 + h)]h:As h gets small, the approximations log(1+rh)  rh and log(1 +h)  h bcome arbitrarilyclose, and therefore so does the approximationlog ct+h log cth (r  ):3Let us consider the model's equilibrium next. The simplest assumptionis that output is constant at level y, so that, in equilibrium _c=c = 0 andtherefore r = .What is the equilibrium rate of ination? We will assume that the gov-ernment prints money to make transfers, and in such a way as to maintaina constant growth rate  of the money supply. We therefore are assumingthat real transfers are given by =_MP= _MM!MP= m:Important point: Individuals know this rule (under rational expectations),but they interpret it as  = m, where mdenotes the economy's aggregateper capita real balances. As an individual, you are under no obligation tochoose your own real balances m to equal m: Thus, you will take mto be anexogenously given datum in solving your own optimization problem. Thatis exactly how we set up the preceding individual optimization problem |with  being exogenous to the individual. It is only in equilibrium thatthe condition m = mmust hold (because we have a representative-agenteconomy). So we are allowed to impose the government budget constraint = m only when we solve for the equilibrium after having derived theindividual's money and consumption demands. (Similarly, if we imposedthe equilibrium condition c = y prior to deriving the individual's rst orderconditions, we would never be able to conclude that r =  in an equilibriumwith constant output. The reason r =  in equilibrium is that only thatlevel of the real interest rate makes people choose c = y as their optimumconsumption level.)Since we have a representative-agent economy (and have abstracted fromgovernment debt), equilibrium bond holdings are b = 0. If we substitute theother equilibrium conditions c = y and  = m into the individual constraintLetting h ! 0; we therefore obtain_cc=  (r  )in the limit of continuous time.4_a = y + ra +   c  im; we get_m = (  )mBy combining the rst-order conditions for c and m, we obtain the money-demand relationshipum(c; m)uc(c; m)= i = r + ;so substitution yields the equilibrium relationship_m =" + r um(y; m)uc(y; m)#m:Under our assumption that ucm= 0, this equation yields unstable dynamicsunless _m = 0 and  = . Thus, it is an equilibrium for the ination rate tobe constant and equal to the constant growth rate of the money