Thom Thurston Queens College and The Graduate Center, CUNY

Transcription

1 How the Taylor Rule works in the Baseline New Keynesian Model Thom Thurston Queens College and The Graduate Center, CUNY Revised July 2012 Abstract This paper shows how to derive a Taylor rule for the New Keynesian model which is both optimal and provides a determinate equilibrium solution. The social loss function is assumed to be the standard quadratic in variances of inflation and the output gap. Four variants of the usual specification of the Taylor rule are employed: the typical one where the interest rate is a linear function of inflation to date and of the output gap, a similar one in which the inflation term is taken as forecasted future inflation rather than that just observed; the third adds the variance of the interest rate to the social loss function and in an extreme case provides a Taylor rule that will keep the interest constant. A fourth variant is a global optimal Taylor rule under commitment in which the price variable is a cumulative deviation from a fixed price level. The Taylor rule in all four versions achieves the social optimum while guaranteeing determinate equilibrium. 1

2 How Taylor Rule works in the baseline New Keynesian Model Thom Thurston Queens College and The Graduate Center, CUNY Revised July 2012 I. Introduction and bottom line The Taylor (1993) rule has assumed a major role in current monetary policy thinking and practice. In this century this rule or a variant of it has been the standard way of specifying systematic monetary policy in the New Keynesian model, which has rapidly become the dominant macro-monetary model. 1 The Taylor rule discussed here takes the discrete-time form (1) where π t is the change-in-logs inflation rate from t-1 to t, x t is the gap, measured in natural log, between actual (log of) real output and the log of full-information equilibrium output. The component reflects a reponse to IS shocks, or real interest shocks, recommended by Woodford (2001) for optimality. I will explain below why this term is necessary to get on the optimal solution path. The equations of the New Keynesian model can be written as a New Keynesian Phillips Curve, (2) and the New Keynesian version of an IS relation (3). The disturbance terms u t and g t are usually assumed to follow independent first-order processes u t=ρu t-1+η t and g t = λg t-1 + ε t. (2) and (3) are log-linear approximations of deviations from steady-state. Focusing on a Taylor rule makes irrelevant and keeps hidden two shocks to money demand and possibly to money supply 2

3 which because of interest rate targeting are automatically offset by means of endogenous money supply accommodation. Also, shocks to natural output are automatically accommodated and do not show up in the equations. 2 A well-known condition for a stable and unique solution for x t and π t is that (4). This condition is necessary and sufficient to make the solutions to the model unique and stable, which henceforth will be referred to in this paper as determinate. 3 The optimal Taylor rule requires specification of a social welfare function relating the endogenous variables. A great number of studies have derived optimal interest rate rules under the assumption that social welfare loss is specified as a time-discounted, weighted sum of the variances of π and the output gap: (5) W=. This social utility function has been rationalized by Woodford (2003a, Chapter 6) as a second-order Taylor approximation of a possibly more general utility function. The bottom line The purpose of this paper is first, to show how the optimal paths of π t and x t can be derived under several different versions of NKM. This part is not particularly original, as the optimal paths for the standard specification has been worked out by Clarida, Gali and Gertler in the famous Science of Monetary Policy paper, henceforth CGG (1999). I have some comments to add about their interpretation of the paths. Next, I illustrate how the Taylor rule can be set in order to obtain various paths including the optimal ones. This accomplishment is original, in particular my analysis of the role of ф x, and my showing that this parameter can be adjusted relative to ф π in order to obtain the optimal path while at the same time assuring the determinacy of solutions. In Sections II and III I will extend the analysis to show that the result above holds when forecasted (t to t+1) inflation is used in the Taylor rule (1) rather than the observed (t-1 to t) (Section II), and that the 3

4 basic result holds when the variance of the interest rate is included in the social welfare function (Section III). I also explain how to derive the global optimum Taylor rule under commitment, a variant that systematically influences the price level over a long horizon (Section IV) according to a fixed plan.. The optimal paths The authorities seek to maximize the expected utility in (5) under constraint of the model (2)-(4). Note that the Taylor rule (1) is not included in the constraint. As CGG point out, the g t ( IS ) shock, which tends to push both x t and π t in the same direction away from the optimal steady state, is easily neutralized by monetary policy (whatever particular form of monetary policy it is), and there is therefore no reason that this demand shock should have impact on the optimal paths. To eliminate the demand shock requires a policy that somehow moves the interest rate inversely with g t by the amount σ. Now the supply or Phillips Curve shock u t tends to move x t and π t in opposite directions and so involves a tradeoff. How to describe optimization on this tradeoff requires the modeler to make a decision regarding time-consistency: is the monetary authority able to pre-commit to keep its policy rule the same in future periods? If pre-commitment to a path for x t and π t is made, I will call the resulting paths inflation pre-commitment paths. If not, the authorities (and public) will optimize in each current period t in the absence of any connection period t s policy choice might have for choices in t+1. If the authorities optimize on this tradeoff without consideration or influence on future policies perceived by the public (a circumstance which CGG call discretionary policy and is the only time-consistent when pre-commitment is impossible) the relevant f.o.c. is that the marginal rate of substitution for each period between inflation and the output gap in (5), dπ t /dx t = -x t Γ/π t and this applies to all future periods -, be equal to the rate of transformation between inflation and the gap on the Phillips Curve, κ. This in turn implies that x t =-[κ/γ] π t. If on the other hand it is possible to inflation pre-commit in advance to a policy rule based on x t and π t, the relevant f.o.c. is that x t =-[κ/γ(1-βρ)]π t. 4 (The notion of optimum at this level of pre-commitment has no role for the level of prices. In Section IV I expand the notion of pre-commitment to include a policy based on the future level of prices. ) 4

5 CGG derive the optimal paths of the output gap and inflation based on the f.o.c. These constraints considerably simplify finding a solution for the optimal paths of the endogenous variables. Figure 1 illustrates leaning against the wind, - an old saying evoked by CGG (1999) - with indifference curves. A positive shock to the Phillips Curve (u t ) will, in the optimal case, push π up and x down by amounts indicated by the expressions in Table 1, A and B. The flatter path (less π per unit decrease in x) is the path under the pre-commitment assumption. It reflects greater leaning against the wind, indicating a greater willingness to push down demand in response to the shock (or push up output in case of a negative shock). An interesting feature of the interest rate paths is that while the more vigorous interest response to inflation under pre-commitment tends to increase the variance of interest rates, inflation under precommitment is less sensitive (and the output gap more sensitive) to the Phillips Curve shock (u t). As a result it is uncertain whether the more vigorous policy increases the variance of the interest rate. The interest rate will have higher variance under pre-commitment, it can be shown, only if ρ. 5 The real interest rate variance will however always be higher in the pre-commitment case. The coefficients on u t in Table 1 represent period t impacts. Since u t follows an AR process (u t = ρu t-1 + η t), the gap between the actual and its long-run value (zero, since measured as deviation from equilibrium) will be reduced to the fraction ρ of its previous value each period. It might be reemphasized that the above paths of i t (as well as π t and x t ) do not reflect a particular monetary policy rule, indeed were derived without reference to any particular policy rule. There interest rate equations C1 and C2 may look like rules, such rules will inevitably produce indeterminate outcomes. 6 To provide a determinate solution an interest rule must, as a necessary condition, link the interest rate to endogenous variables (π t and/or x t ). 7 The Taylor rule is thus a good candidate for an optimal rule that also produces a determinant solution. In Table 1, interest rates equations E1 and E2 could qualify as Taylor rules provided they were explicitly specified so. However, the Taylor rule must also satisfy the condition for determinacy in (4) above, and E1 and E2 do not necessarily do so. If the Taylor rule (1) is specified as the interest rate rule governing i t, the values of ф π and ф x that meet the f.o.c.( s) for maximum welfare (recall that ф g is set as σ) can be derived from the f.o.c. and rational expectations. These turn out to be 8 : discretionary (no pre-commitment) optimal (with inflation pre-commitment) 5

6 (6a). (6b) and are illustrated in Figure 2. It should be noticed that there are no unique values of ф π and ф x, rather a requirement that they lie on a particular, linear path. The role of ф x The downward-sloping line in Figure 2 is the lower boundary of the determinacy condition (4) to this model. Essentially, optimality requires being on one anywhere on one - of the upward-sloping lines (the pre-commitment case is the higher) above the determinacy boundary. Note that provided the optimality conditions are met, the optimality paths in Table 1 will hold. In this light a very important role of ф x is clear: its existence at a high enough value can guarantee that the optimal path will be feasible (in the sense of determinate). While it may be possible - depending on the parameters ρ, κ, Γ, and σ to achieve an optimal and determinate solution with zero or even negative ф π, 9 free choice of this parameter means it will always be possible to find a Taylor rule that will provide a solution that is both optimal and determinate. It might be noted that it is impossible to find an optimal and determinate policy if ф π =0. Optimality would require ф x <0 (Figure 2) while determinacy would require ф x>0. Gali (2008, pp ) notes this problem that optimal interest rate paths are not rules - and that even when they can be written as functions of endogenous variables they may not provide determinate solutions - using the same argument. As a remedy to this problem he focuses on an interest rate rule that has only the ф π term, and notes that in that case the discretionary optimal ф x will be less than unity and therefore violate the determinacy condition - if. 10 He then proposes an interest rate rule where which he shows will lead to the discretionary optimum path and as well will be determinate for any ф x >1. Gali s solution works but as my paragraphs just above have shown, the standard Taylor Rule 6

7 can be adjusted rather simply to provide optimality and determinacy through appropriate selection of ф x and ф x. How do you tell how vigorous the Taylor rule is? Figure 1 illustrates several additional and interesting features of the Taylor rule. Since the optimal paths can be reached with infinite combinations of the ф s, it will often not be obvious whether one combination involves more or less policy vigor than the other. For instance, if ф x increases for a given ф π, the result is a less vigorous policy. Note that the difference in the optimal and discretionary iso-vigor loci (Figure 2) comes from the term (1-βρ)<0 in the denominator of the former, which makes it higher and steeper. If this term is replaced by Θ then we can represent policy vigor as a family of loci having the form (7) The lower Θ gets, the higher will be the locus (and its slope) in Figure 2, and the flatter will be the π,x locus as in Figure 1. The lower the Θ, the lower will be the variance of inflation and the higher the variance of x. The two loci in Figure 2 are the special cases where Θ=(1-βρ) and and Θ = 1 for the optimal precommitment and optimal discretionary cases, respectively. These denote the highest-welfare paths which are time-consistent (the pre-commitment case of course being the highest welfare). Three loci for other Θ s lie inside and outside the ones illustrated. The impact of higher Θ it to lower the variance of nominal and real interest rates (less vigor). Thus the vigor of policy is unambiguously determined by the parameters and the value of Θ. Alternatively, the value of Θ is indicated by any combination of ф π and ф x. 11 Lower Θ is monotonically associated with higher real interest variance (although not necessarily, as mentioned, with nominal interest rate variance). The setting of Θ itself has more than one interpretation. As indicated, Θ=1 is the natural result of time consistency with no pre-commitment. If pre-commitment is not possible, there will be no other choice. Other values of Θ might be set under pre-commitment assumptions, although the motivation is for those except Θ = (1-βρ) is unclear. Finally, Θ might be the result of a difference between the central banker s perceived weight on x t in his social welfare function and that of the true or societal weight (Г). As CGG (1999) point out, choosing a central banker having personal weight (1-βρ)Γ on the output gap term in the welfare function would produce optimal paths identical to those of the pre-commitment case. What if anything does ф x have to do with how much the authorities care about the output gap relative to inflation? There is obviously no superficial link along the lines of the more we care (higher Г) the higher 7

8 we will make ф x. Rather, ф x is set in such a way relative to ф π so as to maintain the f.o.c. in the timeconsistent equilibrium. Since this requires x t and π t to have opposite signs, higher ф x must accompany a higher ф π. For instance, if the u t-shock is positive, this will push π t up and x t down. In this instance the positive ф x will tend to push the interest rate down by the right amount to maintain π t and x t on the appropriate locus of Figures 1 and 2. Section II. When the inflation term is a forecast A popular variant of the Taylor rule uses E t(π t+1) a forecast of the inflation from t to t+1 as the inflation term: (4 ) i t = ф π E t(π t+1) + ф x x t + ф g g t When this specification is put in the the model, the effects of this modification turn out to be minor. The loci of optimizing ф previously labeled 6a and 6b are the same except each term is scaled by 1/ρ. Since ρ is a fraction, this has the effect of lifting the loci and of guaranteeing that the intercepts on the фπ axis of a figure like Figure 2 will be greater than unity. There is also an effect on the determinacy conditions: in addition to the condition noted above that (1-β)ф x + κ(ф π -1) > 0 encountered earlier (4), we have an additional condition that ф x > σ(β-1) (note that 0<β<1). 12 Thus, it remains true that free selection of ф x on the part of the authorities again guarantees that the optimal and determinate solution can be obtained. Finally, when the appropriate optimality condition is imposed the optimal paths of endogenous variables is identical to those described in Table 1. It follows that outcomes will not be affected if this modified Taylor rule is used. This is not to say there is no point in employing inflation forecasts in Taylor rules in general, but if such practice improves outcomes it must be because the model employed in this paper has a misspecification: specifically, that current u- and g-shocks and their auto-regressive parameters (ρ and λ) are not perfectly perceived by agents (as this model assumes). Practical applications at the empirical level may we well reveal gains from using forecasted inflation and perhaps output. Section III. When interest variability is an issue 8

9 The variance of interest rates suggested by the interest rate solutions in Table 1 (C1, C2) may, depending on circumstances, be intolerably high. A long-standing literature has discussed the desirability of suppressing interest rate volatility. Woodford (2003b) and others have employed a kind of partial stock adjustment mechanism to the Taylor rule in the form i t = μi t* + (1-μ)i t-1, 0<μ<1, where i* is the Taylor rule unconstrained optimum. I apply a different approach in the present paper. 13 I impose interest rate volatility directly into the welfare function (5) with the following revision: (5 ) W =, - that is, with an additional weight Г i on interest rate variance. It turns out that with this modification it is still possible to find the optimal Taylor rule and simultaneously guarantee a determinate solution. I will illustrate this with the time-consistent, discretionary case, which has a relatively simple first order condition, (8) This results in the following optimal paths for π t and x t : (9) (10) The interest rate is difficult to express in a compact form, but it can be shown that the impacts of of u t and g t on the variance of the interest rate rises as Γ i rises. 9

10 The Taylor rule that yields the optimal paths The paths above imply the following restrictions on the Taylor rule coefficients: (11) (12) and again for determinacy it is necessary for (1-β)ϕ x + κ(ϕ π -1) >0. I impose the condition that this expression equal an arbitrarily small constant pos: (13) (1-β)ϕ x + κ(ϕ π -1) = pos. Since again the restrictions (9) and (10) the set {ϕ π, ϕ x, ϕ g } provide a redundant Taylor rule parameter that allows the satisfaction of (11). The resulting set of ϕ s is impractical to illustrate here but is linear in u[t] and g[t]. But to illustrate with a numerical example with imposed parameter values (Г i =2, Γ=2, β=.99, σ=1, ρ=.5, λ=.5, κ=.3), the resulting Taylor rule parameters are {ϕ π = pos, ϕ g = pos, ϕ x = pos}. All values of pos (>0) are associated with the same optimal paths of π t, x t and i t, and these solutions are determinate. Constant interest rates At one extreme, Γ i =0 and the analysis reverts to the previous two sections. At the other, Γ i, which would require a constant interest rate. Interestingly, the model allows for a solution 10

11 having no interest rate variance while still providing a determinate solution. The solution is also compact. The constraints become (14) (15) which again allows one to meet the determinacy condition (11). The resulting solutions for the optimal Taylor rule are (16) (17) (18) For example, with the same numerical parameter assumptions as for the previous case, the Taylor rule parameters become {ϕ g = pos, ϕ π = pos, ϕ x = pos }. Again since there are three ф s and two constraints, it is possible to find combinations of the ф s that provide a constant interest and a determinate solution. Why do we care about constant interest rate solutions? First, the discussion provides a kind of theoretical benchmark analogous to the controversy raised by Sargent and Wallace (1975) about indeterminacies arising from central bank interest rate targeting. As with the earlier literature, we find that while strictly exogenous targeting yields indeterminate results it is still possible to maintain a fixed interest rate 11

12 provided the interest rate is an endogenous variable. The discussion above explains how this can be done in the NKM framework and with a Taylor rule. Second, Gali (2007, 2011) questions the meaningfulness of constant interest rate assumptions in central banks economic projections, in particular in the context of the NKM. Gali (2011) produces three alternative interest rate rules that will maintain constant interest rates in NKM, tending to make long-range constant interest rate assumptions plausible. However, he finds that each rule produces a different pattern of long-run projections for output and inflation, raising the question which rule to select. My way of generating a constant interest rate begins with the optimal paths resulting from a utility specification that explicitly includes interest rate variance, and it generates only the Taylor rule restricted to (14) and (15), all combinations of which yield the optimal paths for π t and x t. What it is about Gali s interest rate rules that produce different long-range projections for output and inflation deserves future study. Section IV. Price level effects and a global optimum Taylor rule The standard justifications for interest rates rules such as the Taylor rule, as opposed to money stockoriented rules, is first that it is empirically observed that monetary authorities use interest rate policies in preference over aggregate money and reserve stock policies; second, the argument that in the context of the NKM, the money supply-demand relation is behind-the-scenes but not necessary. This is due to the assumption that the shocks are fully observed making the policies equivalent. For instance, using a money demand-money supply specification m t p t y b t 1 it b (ω t and error term to the demand for t money; no error entered for the money supply here) it turns out the money path needed to put π t and x t on their optimum paths (Table 1 A1 and B1) is (19) m t (1 (1 ) ) b b u 2 t (1 ) p t 1 y b f t 1 g b t t, 12

13 where p t-1 is the lagged level of prices and is the full-information output which defines the output gap x t. Although the level of money in this policy rule is explicit, the determinacy conditions for the corresponding money supply rule, which take the form (20) mt t x xt p 1 pt 1 y yt g gt t, has similar restrictions that I found above for the Taylor rule(including that of sufficient parameter freedom to obtain optimality and determinacy as in Section I). 14 Since the interest rate policy avoids having to consider specification of money demand and supply, on this ground it still seems to be the better policy choice. f On the other hand, there is reason to expect that money stock anchors entered into the policy rule will improve welfare. CGG (1999) and Gali (2008) have discussed paths that involve a commitment to control the level of future prices in particular make the future price level stationary. CGG (1999) and Gali (2008) show that such multi-period optimization results in a set of f.o.c. conditions (21) where that =π t ). = p t+i - p -1 and p -1 is the price level the period before the commitment begins (note The optimal path of price level p t and of x t level follows (22) (23) where represents the deviation of the (log) price level at t from the (log) level at period to the log level at period t-1 (p -1). Note that at the start of the policy and. δ, which must be < 1, turns out to be 13

14 A standard Taylor rule as in (1) based only on the output gap and inflation cannot produce the optimum, but a slight modification that will suffice is of the four-parameter form: (24) where refers to p[t]-p[-1] and where p[-1] refers to a fixed price level for the period before the beginning of the optimal policy period. This form releases a restriction from the earlier ( 1) in which implicitly. The optimal settings (which meet the optimality conditions) are 15 (25) (26) (27) It should be noted that this Taylor rule, while not based on an explicit, exogenous money supply nevertheless has a nominal anchor in the form of p -1 and an implicit money supply measure need to achieve the optimal paths of endogenous variables. Work still in progress, but I can report that Luc Marest has found 14

15 that the simple rule i t= will inevitably meet the determinacy conditions. Mr. Marest also finds that a rule such as i t = p t (p t the log level of price) will also inevitably meet the determinacy conditions. The anchor appears to be critical. Section IV. Concluding statement Probably in order of importance, the contribution of this paper has been to show: 1. The role of ф x in the Taylor rule may be crucial, as it guarantees that the authorities can reach optimality and determinacy simultaneously. 2. The vigor or forcefulness of the rule is defined by a linear locus of the ф π and ф x. It can be summarized in a single parameter in the formula for this locus (I have used Θ). Increasing vigor of policy is indicated by the variance of the real interest rate, which moves inversely with Θ above. 3. The same type of conclusion holds if the variance of interest rates enters the quadratic welfare function. The Taylor rule coefficients can be manipulated to provide both the optimal paths of endogenous variables and to meet the determinacy condition. 4. It is even possible to have a constant interest rate and determinate solution of the model with the simple Taylor rule. 5. A modification of the Taylor rule can be made connecting the interest rate it to the (log) level of prices in such a way as to create stationary and optimal projects for price levels. This globally optimal plan improves social welfare beyond what can be achieved with rules that govern projections of output and inflation alone (and exclude reactions to price levels). 15

16 unit u shock on pre-commitment path π unit u shock on no-commitment path x Figure 1 Leaning against the wind Pre-commitment solutions involve smaller inflation movements and larger output gap movements. Social welfare is however improved with pre-commitment. 16

17 Table 1. Optimal Paths discretionary (without pre-commitment) A1 optimal (with inflation pre-commitment, no commitment to price level) A2 B1 B2 C1 C2 D1 D2 written as a linear combination of and E1 E2 Note: The absolute value of (D2) is smaller (welfare is higher) than in (D1); that is, pre-commitment increases welfare. 17

18 ф π Inflation precommitment optimal no pre-commitment time-consistent Figure 2 ф x Loci of ф π and ф x having the same paths of endogenous variables 18

19 References Clarida, R.; Gali, J. & Gertler, M. (2000), 'Monetary Policy Rules and Macroeconomic ility: Evidence and Some Theory', The Quarterly Journal of Economics 115(1), Clarida, R.; Galí, J. & Gertler, M. (1999), 'The Science of Monetary Policy: A New Keynesian Perspective', Journal of Economic Literature 37(4), Gali, J. (2007), Constant Interest Rate Projection without the Curve of Indeterminacy: A Note, forthcoming International Journal of Economic Theory. Gali, J. (2008), Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Princeton Univ Press. Gali, J. (2011), Are central banks projections meaningful?, Journal of Monetary Economics 58, pp McCallum, B. (2009), Indeterminacy from Inflation Forecast Targeting: Problem or Pseudo-Problem?, Federal Reserve Bank of Richmond Economic Quarterly 95(1), Sargent, T. & Wallace,N. Rational Expectations, the Optimal Monetary Instrument and the Optimal Money Supply Rule. Journal of Political Economy 83(2): Taylor, J. (1993), Discretion Versus Policy Rules in Practice, Carnegie-Rochester Conference Series on Public Policy 39: Walsh, C. E. (2010), Monetary Theory and Policy, 3rd Edition, the MIT Press. Woodford, M. (2003a). Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton Univ Press. Woodford, M. (2003b), Optimal Interest Rate Smoothing, Review of Economic Studies 70, Woodford, M. (2001), 'The Taylor Rule and Optimal Monetary Policy', The American Economic Review 91(2),

20 1 According to the recent brochure for the Walsh (2010) text book (3rd edition), Much of the material on policy analysis has been reorganized to reflect the dominance of the New Keynesian approach. 2 The models solutions are for π t, i t, and the output gap (log output minus log natural output). If log natural output were to increase, restoration of equilibrium would require log output to increase by an equal amount. 3 This model can be written in the form y=ax + Be where, B is a coefficients matrix, and and Be is a matrix of error terms. The condition for determinacy is that both eigenvalues lie inside the unit circle. Condition (4) can be shown to result from the this eigenvalue condition. Violation of this condition implies either explosives solutions or indeterminate stable solutions arrived at through sunspots. Both are referred to in this paper as indeterminate solutions. 4 The marginal rate of substitution for period t is still dπ t/dx t = -x t/γπ t = -π t /(Γx t), but in this case the authorities take into account the feedback between the current choice if π t, expected inflation in the next period, E(π t+1), and the latter s impact on π t as in the Phillips Curve (2). Since in this model E(π t+1)=ρπ t the Phillips Curve constraint can be written π t(1-βρ)=κx t+u t where the marginal tradeoff dπ t/dx t is now = κ/(1-βρ). Hence, the pre-commitment f.o.c. for period t is that x t=-[κ/γ(1-βρ)] π t. 5 More generally, a decrease in Г will increase the variance of the interest rate under the above condition. 6 It can be shown that these rules will violate one eigenvalue condition, implying that the resulting solutions for π t, x t, and i t are not determinate. 7 CGG (1999) refer to their interest rate equations as interest rate policies. Gali (2008) points out that interest rate rules based only on exogenous shocks will inevitable produce eigenvalues that indicate indeterminacy condition (see endnote 3). Gali (2008) does not refer to the statements by CGG (1999) in which he was a co-author. 8 To solve, use the appropriate f.o.c. as a constraint, use (1)-(4) and rational expectations. Assume ф g=σ to eliminate the g t shock. The result will be solutions for ф π and ф x which are not unique but follow (6a) and (6b). 9 A negative ф x may sometimes be found consistent with determinacy and optimality, but a negative value is never required for determinacy and optimality. 10 Gali was not discussing the pre-commitment optimum in that section, but the corresponding optimal ф π under precommitment is which can also be less than unity. This information is the same as suggested in Figure 2. 20

21 I am indebted to Osman Dogan, a doctoral student at the CUNY Graduate Center, for pointing out to me this second condition. 13 I avoid the partial adjustment approach here for three reasons. First, including a lagged interest term considerably complicates the mathematics. It appears that conclusions may have to be reached by means of numerical examples or calibrations. Second, putting the variance of the interest rate directly into utility seems more analytically defensible. If there is a preference for reduced volatility of interest rates, it should appear in utility. Third, Woodford finds in calibration studies that the presence of the lagged interest rate appears to improve outcomes in simulations involving calibrated parameters even when interest rate volatility does not enter the welfare function. I suspect this is related to the improvement noted in Section IV of this paper, where lagged prices enters the mathematics, but have at present no way to sort this out. 14 I am grateful to Luc Marest, a dissertation student at the CUNY Graduate Center, for making available this derivation from his dissertation and a joint paper with me. 15 These solutions were also worked out recently by Luc Marest. 21