Eco504, Part II Spring 2002 C. SimsPOLICY GAMES1. THE SAN JOSE MODELThe policy authority believesut=θ0−θ1πt+εt. (1)In fact, though,ut= ¯u−α· (πt− Et−1πt) +ξt. (2)2. POLICYMAKERS’ BEHAVIORThey minimize∞∑t=0βt(u2t+ωπ2t) . (3)At each t, they estimateθ0andθ1— by a method that may allow for variation over time inthese parameters (the Kalman Filter). They do not controlπprecisely, but instead control gt,withπt= gt−1+νt. (4)3. EQUILIBRIUMThey do not take account of their own learning pattern, but instead just optimize at eacht as if their current estimates were true values that would remain constant forever, whichmeans they setgt=θ1θ0ω+θ21. (5)Substituting this expression into the true Phillips curve (2) and matching coefficients tells usthat OLS applied to data from this situation and to the false model (1) would deliverθ1=αθ0= ¯u+αθ1θ0ω+θ21.c°2002 by Christopher A. Sims. This document may be reproduced for educational and research purposes, solong as the copies contain this notice and are retained for personal use or distributed free.POLICY GAMES 24. DYNAMICSSolving these equations forθ0andθ1tells us the equilibrium position of the false PhillipsCurve. It implies that in steady state gt≡α¯u/ω. As we would expect, equilibrium inflationis higher the greater the natural rate, the greater the apparent effect of inflation on unemploy-ment in the Phillips Curve, and the smaller the weight on inflation in the objective function.But how do we get there? The theory so far only considers what estimation will deliverif OLS is used and g is held constant. But to progress from low inflation to the equilibrium,the policy authority will have to change g. When it does so, it will generate data in whichπis changing without producing any effect on unemployment. This will make theαappearsmaller, and reduce the apparent gains to inflation. So progress to the Kydland-Prescottequilibrium is slow.5. FIGURESThe figures that follow are from (Sims, 1988). They are not those that appeared in the originalarticle, but replacements that appear in the web version. The models and discussion of how thecharts were generated are described in the web version of the paper. All the figures show simulatedtime series from economies in which there is a natural rate Phillips curve and the Kydland-Prescottequilibrium level of inflation is 6%. Figures 1 and 2 illustrate the fact that such simulations canproduce very different results depending on the first few observations. Figure 3 shows a typicalsimulation with time variation modeled by the policy authorities as equal on constant and slope,starting from low inflation. Over the 1000 year span of the graph (assuming annual data is used inthe regression updates), inflation stays permanently low, never moving toward the Kydland-Prescottequilibrium. Figure 4, in contrast shows the economy near the Kydland-Prescott equilibrium most ofthe time, with only “brief” (100 year or so) deviations from it. This is the typical outcome when thepolicy authority attributes most time variation to shifts in the constant term — i.e. the “natural rate”.Figure 5 shows what happens when the same beliefs on the part of the policy authority as in Figure3 prevail, but the economy starts near the K-P equilibrium. This figure may be misleading, in that itshows a break away from the KP equilibrium after a few hundred years, while the model was actuallyrun for 2000 years before the start of the chart that is displayed. The simulation was also continuedfor over 10,000 years after the end of the period displayed, and never returned to the neighborhood ofKP equilibrium in that span.6. “MEAN” AND “ESCAPE” DYNAMICS, AND THEIR INTERPRETATION• Sargent describes the dynamics as a “mean dynamics” drawing the economy towardKydland-Prescott equilibrium, occasionally “punctuated” by episodes of “escape dy-namics”.• He does not use the Kalman filter, and our charts show that this affects his con-clusions. The point that “escape dynamics” prevail over brief periods in which thenature of the process changes radically is correct, but there is no necessary tendencyto drift toward K-P equilibrium.POLICY GAMES 31800 1850 1900 1950 2000 2050-4-202468No time variation, initial good luckInflation Unemployment FIGURE 11500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500-4-20246810No Time Variation, not Lucky at StartunemploymentinflationFIGURE 2POLICY GAMES 41500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500-4-202468Time variation equal on constant and slopeUnemployment Inflation FIGURE 31500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500-4-20246810More Time Variation Attributed to Constant TermInflation Unemployment FIGURE 4• Sargent’s is one of several competing stories that explain why reliance on empiricalmodels that do not embody the received wisdom of natural rate theory could lead toa temporary episode of good policy that is constantly in danger of being underminedby new, but spurious empirical results.POLICY GAMES 51500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500-4-202468Time variation equal, starting from KP equilibriumInflation Unemployment FIGURE 5• An alternate view: The natural rate theory, like any other simple orthodoxy, is at bestpartially correct and at worst can end up an albatross weighing down any attempt toarrive at understanding of new policy challenges. (How much of Japan’s problem isan effect of natural rate thinking?) Good empirical models can lead to good policyeven if they do not exactly embody the truth.7. FULL COMMITMENT, TIME CONSISTENT AND GAME-THEORETIC(BARRO-GORDON) APPROACHESFull Commitment: We set the problem up as an optimization problem of the usualform, with the private sector’s behavioral equations, which generally include expec-tational equations (Euler equations or, in our current case, the true Phillips curve(2)) among the constraints. If we maintain the assumption that the policy authoritymust choose g in advance, so it has no information advantage over the private sec-tor, we will get the uninteresting and obvious conclusion that the optimal policy isgt≡ 0. [Why is it obvious?]. So we will examine the case where the policy authoritycan pickπtdirectly at time t. This implies that the policy authority can surprise theprivate sector, or equivalently that it has an information advantage.No-Commitment: • The Full Commitment solution generally implies that