14.06: Section HandoutProf.: George-Marios Angeletos TA: Jose TessadaApril 28, 20051 The Model1.1 The SetupConsider a set-up like in Barro and Gordon (1983)yt y =  (t et) ;  > 0 (1)Equation (1) represents the supply side economy (a Phillips curve type of equa-tion), where ytis the actual level of output, y is the natural level of output, tis thein‡ation rate, and etis the expected in‡ation rate. The natural rate of output in thismodel is the output level we would observe is there were no surprises in in‡ation.L =122t+ (yt y)2;  > 0;  > 1: (2)Equation (2) is the policymaker’s loss function which is assumed to represent thesociety’s preferences over output and in‡ation;  represents the relative weight societyassigns to in‡ation deviations.1The term (yt y) represents the deviations of actualoutput with respect to a "bliss" level which is assumed to be higher than the naturallevel, probably re‡ecting some friction in the markets.1.2 TimingWe will consider a situation where all players know y and the set-up of the economy,given by equations (1) and (2).This is a one period model, agents choose et, and then the policymaker willchooses the tthat maximizes (2) subject to (1) and et.1.3 Discretionary Policy EquilibriumThe agents and the policymaker play a sequential move game, so we can …nd theequilibrium outcome using backwards induction. Consider …rst the policymaker’s1Notice that we have assumed that the "desired" level of in‡ation  = 0. This assumption isnot crucial and will simplify the algebra.1problemmint122t+12( (t et)  y (  1))2(P1)The FOC ist+  [ (t et)  y (  1)] = 0:The optimal in‡ation rate, as a function of etist(et) =22+ et+2+ (  1) y > 0: (3)Equation (3) corresponds to the policymaker’s best response function.2We can now analyze the agents’problem. In this game with complete and perfectinformation, the agents’take the policymaker’s best response function into accountwhen setting et. Assume that the agents’payo¤ is given byU = 12(t et)2; (4)then, the agents’optimization problem ismaxet1222+ et+2+ (  1) y  et2: (P2)Clearly, the expected in‡ation rate that maximizes this function is the one thatmakes the expression in the parenthesis equal to 0. This rate is given bybet=(  1) y: (5)In equilibrium, agents’ will set a p ositive expected in‡ation rate. The optimalexpected in‡ation rate is such that t(bet) = bet: In fact this result is exactly what wewould obtain if we assume that agents are rational; under this assumption, equilibriumwe must have t= et,3thus we can …nd etas the …xed point of equation (3).As a result, the policymaker is totally unable to increase the output level of theeconomy on a permanent base, and her inability to commit to a low in‡ation rateleads to an equilibrium where output equals its natural rate (yt= y), but the in‡ationrate is ine¢ ciently high. Agents know the government cannot commit and also knowthe government has an expansionary (in‡ationary) bias, they anticipate that settinga higher expected in‡ation rate bet.2The slope of the best response f unc tion is bounded between 0 and 1. The intercept is strictlypositive.3In general we would require E (t) = et, given that there are no stochastic shocks in this model,then E (t) = t.2tettetettFigure 1: Equilibrium in the Deterministic Game.Finally, social loss in the discretionary equilibrium isLD=12(  1) y2+12(1  )2y2=122+ 1(  1)2y2LD=12(  1)2y2; (6)where  +2< 1. The social loss is increasing in the discrepancy between thenatural and the desired ("socially optimal") output level, and in the slope of theaggregate supply (Phillips curve), because this implies a higher temptation to generatesurprises in in‡ation. It is decreasing in , re‡ecting that in‡ation is relatively morecostly for the policymaker, thus less incentives to surprise with unexpected in‡ation.Why is people anticipating the "in‡ationary bias"? You can fool some peoplesome of the time, but you cannot fool all of them, all the time. They will learn andadjust to it.31.4 Equilibrium with CommitmentImagine that the government has access to a technology that makes announcementstotally credible, and allow the government to have a …rst move where it announces thein‡ation rate that will be observed in the economy.4If the announcement is credible,the best the government can do is to announce that will set t= 0; agents will thenset et= 0 and the government will e¤ectively adjust policies to make t= 0. So, theoutcome in this case would be yt= y and t= 0.The social loss is given byLR=12(  1)2y2; (7)which is smaller than LD, because  < 1. We can conclude then that the so cietywould be better o¤ if the policymaker were able to commit not to in‡ate.52 Stochastic ModelWe can interpret the commitment solution of section 1.4 as the solution when thegovernment can self-impose a strict in‡ation rule. Our mo del also tells us, that asolution like that is preferable to a solution where the government can actually usemonetary policy. If this is true, why do we observe countries using discretionarypolicy (ex. USA).We can modify our model allowing nature to play a role in this game. Modify thetiming as follows: after the agents have set expected in‡ation, the nature moves witha random shock ztwhich is publicly observed at zero cost. The policymaker movesafter the shock is realized, and will take into account the shock when setting t.Now, the aggregate supply is given byyt y =  (t et) + zt;  > 0; E (zt) = Et1(zt) = 0; V (zt) = 2z: (8)The solution looks as follows:t(et; zt) = (1  ) [et zt= + (  1) y=] (9)et=1  (  1) y(10)t=1  (  1) y(1  )zt= et(1  )zt: (11)4An alternative motivation would be to allow modify th e game making the government move…rst, and the agents move second; as you can see, "expected" is not exactly the best description forthe agents’action.5The discussion on rules and discretionary policy was emphasized by Kydland and Prescott(1977) in