Economics 202A Lecture Outline, November 27, 2007(Version 1.2)Maurice ObstfeldGovernment Revenue from Money CreationIn general the central government has a monopoly right to issue money,and that privilege is a source of revenue. This lecture shows how to integratemoney creation by the central government into national budgetary accounts.They relate the discussion to the notion of the revenue-maximizing rate ofmonetary growth. For a classic application, see the paper on \UnpleasantMonetarist Arithmetic" by Sargent and Wallace.What is seigniorage?If the private sector is willing to hold paper money that the governmentsupplies, the government can buy real goods and services that the privatesector produces with money that is (virtually) costless for the governmentto print.1The real resources that the government acquires in this way equalits seigniorage revenue. To dene seigniorage we need not know how or whythe private sector is willing to accept the government's at money; all thatmatters is that there is a demand for it.In a discrete time mode, seigniorage in period t is given byMt Mt1Pt;that is, it is the real resources the government acquires through increases inthe nominal money balances the public is willing to hold. A useful way torewrite this expression is asMt Mt1Pt= tmt1+ (mt mt1); (1)1Money that is not backed by a real commodity (such as gold) is called at money.1where t (Pt Pt1)=Ptand m  M=P . This expression emphasizes twodistinct sources of seigniorage. First is the ination tax, the amount peoplemust give to the government to hold their real money balances constantin that face of rising prices.2Second is the public's desire to alter its realmoney holdings, given the ination rate. The same decomposition applies incontinuous time. Seigniorage at time t is_M(t)P (t)= (t)m(t) + _m(t);as you can easily check. Observe that seigniorage need not equal inationtax revenue, which is m only.The revenue-maximizing steady-state ination rateAn important theoretical concept is the revenue-maximizing steady-stateination rate: what is the highest rate at which we can squeeze golden eggsout of the proverbial goose? It turns out that the concept is slightly am-biguous, and for a reason that lies at the heart of discussions over timeinconsistency in monetary policy.One approach to the problem is to simply maximize_M=P , which doesequal m in a steady state (since _m = 0.) Thus, if r is the real interest rateand money demand is a declining function of i = r + , we solveddm(r + ) = 0;which yieldsm + m0(i) = 0;orm0(i)m= 1: (2)2You probably are used to dening ination as (PtPt1)=Pt1. However, the inationconcept that concerns us here is the fraction of an agent's real balances that is \conscated"through a rise in the price level, and that equals (Pt Pt1)=Pt= t. Notice that as therate of price level increase becomes arbitrarily big, t! 1, meaning that an \innite" rateof price increase reduces the value of real balances by 100 percent.2This formula instructs us to look for the point on the money demand curvewhere the ination elasticity is 1. It is a standard monopoly pricing formula,which equates the marginal cost of producing money (zero) to the marginalrevenue from creating it (zero, at the point where condition (2) holds).3Conceptually this approach is a bit unsatisfactory because it apparentlyfails to answer the following dynamic question. Suppose we are in a steadystate with ination rate . When will it be the case that we cannot raisepresent and future seigniorage revenue by raising ination? The fundamentaldierence between this question and the one answered in the last paragraphis that now we must worry about how the initial ination change, whichoccasions a price level jump and a jump in real money demand impactsseigniorage revenue. Suppose that the economy always jumps to its steadystate in response to an unexpected change in ination. Then, according toeq. (1)|which is appropriate because there will b e discrete changes in Pand in m at the moment of the change|the present discounted value of thechange in seigniorage revenue resulting from a small change in ination is1PdPdm + m0(i)| {z }+R10ert[m + m0(i)] dt;| {z }initial stock adjustment ongoing ination tax ow(3)where dP denotes the initial equilibrium price level change due to the changein the ination rate.4Observe that since the nominal money supply M is3I assume the second-order condition that the function m(i) is concave where thepreceding condition holds:2m0(i) + m00(i) < 0:4It may be helpful to note that the path of the price level P may not be dierentiable.Strictly speaking, I am thinking of (t) here as the expected rate of ination going forward| in technical terms, the right-hand derivative of the price level, orlimh!0Pt+h PthPt+h:The eect we are considering is the impact of this change in the expected future inationrate on the price level and real money demand today.3not changed at time 0 when  rises,m0(i) =dmd= MP2dPd= 1PdPdm:Thus, the equation for total discounted seigniorage revenue reduces toZ10ert[m + m0(i)] dt:(Intuition for the cancellation of initial eects: The initial nominal moneysupply does not jump, so the government cannot gain any seigniorage at theinitial instant.) Plainly, maximizing this with respect to  simply leads tothe same answer we found before, eq. (2).This solution is still somewhat problematic, however, because it entailsan unexpected expropriation of private sector real wealth equal to1PdPdm:This surprise ination tax receipt osets the decline in seigniorage revenuecaused by the initial fall in real money demand when  is raised.We could well imagine, however, that the government has promised toavoid surprise changes in the value of real balances; perhaps, like Brazil in1998, it is allowing prices to rise gradually over time but has pledged not toengage in a \maxi-devaluation" that reduces the value of the currency by adiscrete amount. In the presence of such an \honest government" constraint,5a small rise in ination would raise government seigniorage revenue by onlym0(i) +Z10ert[m + m0(i)] dt;and not by the amount in eq. (3). The reason: to ensure that dP = 0when ination rises (say), the government must reduce the nominal moneysupply sharply; it might nance this