Monetary Policy Effectiveness in a Dynaic AS/AD Model with Sticky Wages H J Departent of Econoics University of Copenhagen Version 1.0, April 12, 2012 Teaching note for M.Sc. course on "Monetary Econoics: Macro Aspects". c 2012 Henrik Jensen. All rights reserved. The note builds on, and extends, a odel laid out in Walsh (2010), as well as y earlier teaching aterial on this subject. I thank Christian Groth and Carl E. Walsh for coents and discussions on earlier versions. Address: Øster Fariagsgade 5, Building 26, Dk-1353 Copenhagen K, Denark. E-ail: Henrik.Jensen@econ.ku.dk. Web: hjeconoics.dk.
Preface This note deals with onetary policy effectiveness in a New-Classical odel with icrofoudations. The odel is a siple oney-in-the-utility function odel with one-period noinal wage rigidity. The aggregate supply curve of the odel is akin to the Lucas supply function, which is a defining eleent of New-Classical econoics. This research progra, focusing on rational expectations, gave rise to the faous Policy Ineffectiveness Proposition, stating that systeatic coponents in policy design had no influence on real activity. This note shows how systeatic coponents of policyaking affect the variability output, thereby ephasizing that even within early rational-expectations odels of this type, there is real ipact of systeatic policy intervention. Henrik Jensen, April 2012
1 Introduction Consider the log-linearized, siplified oney-in-the utility function odel with flexible prices as presented in Walsh (2010) p. 227. 1 All lower-case letters denote log-deviations fro steady state, except interest rates which are absolute deviations of respective rates fro their steady-state values. See Walsh (2010, Section 2.7.1) for the derivation of the full version of the odel. Output, y t, is produced by labor, n t, according to a Cobb-Douglas technology: y t = (1 α) n t + e t, 0 < α < 1, (6.1) where e t is a ean-zero supply shock assued to be serially uncorrelated (this akes a distinction between the supply shock e t and the shock ε t e t E t 1 e t defined in Walsh, 2010, irrelevant). All output is consued in equilibriu: y t = c t, (6.2) where c t is consuption. Firs hire labor up to the point where the arginal product of labor equals the real wage, such that labor deand is characterized by y t n t = w t p t, (6.3) where w t is the noinal wage and p t is the price level. Consuers choose consuption over tie so as to satisfy the conventional Keynes-Rasey rule, which in logs can be written as ΦE t (c t+1 c t ) r t = 0, Φ > 0, (6.4) where r t is the real interest rate and Φ is the coeffi cient of relative risk aversion in consuption, which in this setting equals the inverse of the interteporal rate of substitution. E t is the rational-expectations operator conditional on all inforation up until, and including, period t. Consuers choose to supply labor such that the arginal rate of substitution between leisure and consuption equals the real wage; in logs this is ( ) n ss η n 1 n ss t + Φc t = w t p t, η > 0, (6.5) where n ss is the steady-state value of labor, and η is the coeffi cient of relative risk aversion in leisure, which in this setting ultiplied by n ss / (1 n ss ) can be interpreted as the 1 In order to facilitate the coparison with Walsh (2010), I retain his equation nubering when adequate. 1
inverse (Frisch) elasticity of labor supply. With flexible wages, as well as prices, (6.1) (6.5) deterine y t, c t, n t, r t and w t p t. Monetary policy will have no role for output deterination. Money deand is given by t p t = c t ( ) 1 i bi ss t, b = Φ > 0, (6.6) where t is the noinal oney supply, i t is the noinal interest rate, i ss is the steady-state noinal interest rate, and b is a utility function paraeter deterining the interest-rate elasticity of oney deand. This version of the odel assues b = Φ, so as to focus on the siple version with separable utility of consuption, real oney holdings and leisure. (The paraeters b and Φ will, however, be used independently throughout so as to highlight their different roles.) The Fisher equation links the noinal interest rate to the real rate and expected inflation: i t = r t + E t p t+1 p t. (6.7) Finally, the odel is closed by a specification of the process for the oney supply: t = ρ t 1 + s t, 0 ρ 1, (6.8) where the shock s t has ean zero and is serially uncorrelated. To introduce a role for onetary policy, soe noinal stickiness is assued. Specifically, it is stipulated that noinal wages are fixed for one period, and are deterined the period before according to: w t = E t 1 p t + E t 1 y t E t 1 n t, (1) = E t 1 p t + E t 1 ω t. I.e., the expected real wage is set so as to atch the expected arginal product of labor, labelled E t 1 ω t. With this wage-setting rule, equation (6.5) becoes redundant. Also, equation (6.8) is replaced by the slightly ore general expression t = µ + ρ t 1 + s t. where µ is soe constant. Note that in Appendix 6.5, Walsh (2010) considers the special case of t = t 1 + s t. This siplification unintentionally hides the effectiveness of systeatic onetary policy as will be clear below. 2 We now turn to the solution of the odel under sticky wages. 2 He acknowledges this in Footnote 6 on p. 230. 2
2 Solving the odel 2.1 Fleshing out an AS/AD structure It is appropriate to ake soe siplifications of the presented syste of equations. Here, it will be easy to substitute out c t by y t and r t by r t = i t (E t p t+1 p t ). Also, one can iediately substitute the noinal wage rule, (1), into (6.3). Then, a sipler syste is: y t = (1 α) n t + e t, (2) y t n t = E t 1 p t + E t 1 y t E t 1 n t p t, (3) ΦE t (y t+1 y t ) = i t (E t p t+1 p t ), (4) ( ) 1 t p t = y t i t, (5) bi ss t = µ + ρ t 1 + s t. (6) This syste of five equations will deterine the rational-expectations solutions for the five variables y t, n t, i t, p t, and t, given the shocks e t and s t and given t 1. Now we proceed as in Walsh (2010, Appendix 6.5) by splitting up the syste into the aggregate supply and deand sides. Equations (2) and (3) constitute the supply side, and give supply of output and eployent as function of prices, whereas equations (4), (5) and (6) give output deand and the noinal interest rate (via oney deand) as functions of prices. These aggregate supply and deand schedules then provide the equilibriu output and price level. Let us start with the supply side. Fro (3) we get which inserted into (2) gives n t = E t 1 n t + y t E t 1 y t + p t E t 1 p t, y t = (1 α) (E t 1 n t + y t E t 1 y t + p t E t 1 p t ) + e t, and therefore αy t = (1 α) (E t 1 n t E t 1 y t + p t E t 1 p t ) + e t. Take period-t 1 expectations on both sides of (2) to get (reebering that E t 1 e t = 0 by assuption): E t 1 y t = (1 α) E t 1 n t, and substitute out (1 α)e t 1 n t above: αy t = E t 1 y t + (1 α) ( E t 1 y t + p t E t 1 p t ) + e t, αy t = αe t 1 y t + (1 α) (p t E t 1 p t ) + e t, 3
thereby providing the output equation y t = E t 1 y t + 1 α α (p t E t 1 p t ) + 1 α e t, or, y t = E t 1 y t + a (p t E t 1 p t ) + (1 + a) e t, a (1 α) /α > 0. As E t 1 e t = 0, it follows that E t 1 y t = 0 (as E t 1 n t = 0). 3 We therefore have: y t = a (p t E t 1 p t ) + (1 + a) e t. (7) This aggregate supply schedule is a version of the Lucas supply curve that has been central in uch acroeconoics since the 1970s. Now consider the deand side. Equations (4) and (5) can iediately be rewritten as a dynaic IS curve and a standard LM curve: y t = E t y t+1 1 Φ (i t [E t p t+1 p t ]), (8) ( ) 1 t = p t + y t i bi ss t. (9) Following the approach in Walsh, we rewrite the oney deand equation in ters of expected inflation and expected output growth using the dynaic IS curve: t = p t + y t 1 bi ss (E tp t+1 p t ) Φ bi ss (E ty t+1 y t ). Using the exogenous process for the oney supply, we get a characterization of the aggregate deand side in ters of output and prices: µ + ρ t 1 + s t = p t + y t 1 bi ss (E tp t+1 p t ) Φ bi ss (E ty t+1 y t ). (10) Equation (10) indeed characterizes, for given expectations and oney supply, a negative relationship between prices and output. The reasons are partly the standard one fro the IS/LM story: Higher prices will increase noinal oney deand. To secure oney arket equilibriu, the noinal interest rate will increase, which depresses deand. 3 This follows fro the underlying odel when the real wage target ω t fro (1) is consistent with the flexible-wage eployent level; i.e., the real wage that equates labor deand with labor supply. This results in the following expression for eployent (see Walsh, 2010, p. 228): n 1 Φ t = ( ) e t. n 1 + η ss 1 n + (1 α) (Φ 1) ss This is derived by cobining (6.1), 6.2), (6.3) and (6.5). 4
In this interteporal odel, an additional channel is present. For given period-t + 1 price expectations, a higher price in period t reduces inflation expectations, E t p t+1 p t, and increases the real interest rate. This causes consuers to substitute deand fro now to later for given expectations about future deand, period-t deand decreases. The sae will happen if, for exaple, expectations about future prices go down. This expected inflation channel is norally not presented in standard IS/LM expositions, where there is no distinction between noinal and real interest rates in the IS relationship. The distinction, arising naturally fro the underlying icro-founded odel, is crucial for the results on policy effectiveness as will be clear below. 2.2 Applying the ethod of undeterined coeffi cients With (7) and (10) we have a two-equation rational expectations odel for y t and p t. This is solved by the ethod of undeterined coeffi cients. For this purpose we conjecture, or guess, a solution in ters of undeterined coeffi cients. In this odel, we conjecture the following solution for prices: p t = γ 0 µ + γ 1 t 1 + γ 2 s t + γ 3 e t. (11) Knowing what to conjecture is not always obvious, but in linear rational expectations odels, it is safe to ake a conjecture that is a linear function of shocks, constants and state variables. In a oent we will verify the that this for of the conjecture is indeed consistent with a solution of the odel, and we can derive the values of γ 0, γ 1, γ 2, and γ 3 as function of the odel s paraeters. Now, insert y t and E t y t+1 as given by (7) into (10) so as to obtain an expression in prices only: µ + ρ t 1 + s t = p t + a [p t E t 1 p t ] + (1 + a) e t 1 bi ss (E tp t+1 p t ) Φ bi ss (E t [a {p t+1 E t p t+1 } + {1 + a} e t+1 ] [a {p t E t 1 p t } + {1 + a} e t ]). We readily see that E t [a {p t+1 E t p t+1 } + {1 + a} e t+1 ] = E t y t+1 = 0, as E t e t+1 = 0 by assuption, and as E t [a {p t+1 E t p t+1 }] = 0. Hence, we have µ + ρ t 1 + s t = p t + a [p t E t 1 p t ] + (1 + a) e t 1 bi ss (E tp t+1 p t ) + Φ bi ss (a [p t E t 1 p t ] + [1 + a] e t ). (12) 5
as a rational-expectations equation deterining prices. This is what we need, as we have ade a conjecture for the solution for prices. We can therefore apply that the conjecture, (11), iplies p t E t 1 p t = γ 2 s t + γ 3 e t, E t p t+1 p t = γ 1 t γ 1 t 1 γ 2 s t γ 3 e t, as E t 1 e t = E t 1 s t = 0 for all t. Using this in (12) then gives µ + ρ t 1 + s t = γ 0 µ + γ 1 t 1 + γ 2 s t + γ 3 e t (13) +a [γ 2 s t + γ 3 e t ] + (1 + a) e t 1 bi ss (γ 1 t γ 1 t 1 γ 2 s t γ 3 e t ) + Φ bi ss (a [γ 2s t + γ 3 e t ] + [1 + a] e t ), Equation (13) is a linear equation involving a constant, the shocks, and past and current oney supply. The latter is eliinated by use of (6), such that we finally get: µ + ρ t 1 + s t (14) = γ 0 µ + γ 1 t 1 + γ 2 s t + γ 3 e t +a [γ 2 s t + γ 3 e t ] + (1 + a) e t 1 bi ss (γ 1 [µ + ρ t 1 + s t ] γ 1 t 1 γ 2 s t γ 3 e t ) + Φ bi ss (a [γ 2s t + γ 3 e t ] + [1 + a] e t ). Equation (14) verifies the conjecture that in a rational expectations equilibriu satisfying the odel s equations, prices are a linear function of a constant, the shocks, and the past period s oney supply. We have not solved for the price level yet, but looking closer at (14), we will (hopefully) realize that the solution is right in front of us. Equation (14) ust hold for any values of µ, s t, e t and t 1. So, differentiating the left- and right-hand sides of (14) with respect to these variables in turn, provides exactly four equations that will give the solutions for the four unknown coeffi cients. Thus, we obtain the rational expectations solution for p t of the for conjectured. The four equations are: 1 = γ 0 γ 1, (15) biss 1 = γ 2 + aγ 2 1 bi ss (γ 1 γ 2 ) + Φ bi ss aγ 2, (16) 0 = γ 3 + aγ 3 + 1 + a + 1 bi ss γ 3 + Φ bi ss (aγ 3 + 1 + a), (17) ρ = γ 1 1 bi ss (γ 1ρ γ 1 ). (18) 6
Notice that (18) gives the solution for γ 1, (17) gives the solution for γ 3, (16) gives the solution for γ 2 given γ 1, and (15) gives the solution for γ 0 given γ 1. Fro (18) we get and fro (17) we recover Using (19) in (16) gives γ 1 = bi ss ρ 1 + bi ss ρ, (19) γ 3 = (1 + a) (biss + Φ) 1 + bi ss + a (bi ss + Φ). (20) 1 = γ 2 + aγ 2 + 1 bi γ ss 2 + Φ bi aγ ρ ss 2, 1 + bi ss ρ bi ss bi ss ρ + 1 + bi ss ρ = (1 + bi ss + a [bi ss + Φ]) γ 2, bi ss (1 + bi ss ) 1 + bi ss ρ = (1 + bi ss + a [bi ss + Φ]) γ 2, γ 2 = and using (19) in (15) gives Hence, the solution for p t is p t = bi ss (1 + bi ss ) (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]), (21) 1 = γ 0 γ 0 = ρ 1 + bi ss ρ, 1 + bi ss 1 + bi ss ρ. (22) 1 + bi ss bi ss ρ µ + 1 + bi ss ρ 1 + bi ss t 1 (23) ρ bi ss (1 + bi ss ) + (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + a) (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t. We can iediately see that E t 1 p t = 1 + bi ss bi ss ρ µ + 1 + bi ss ρ 1 + bi ss t 1, ρ and insert these solutions for p t and E t 1 p t into (7) so as to get the solution for y t : y t = [ bi ss (1 + bi ss ) a (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + a) ] (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t + (1 + a) e t, = abi ss (1 + bi ss ) (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t + 1 + biss + a (bi ss + Φ) a (bi ss + Φ) 1 + bi ss + a (bi ss + Φ) y t = (1 + a) e t, abi ss (1 + bi ss ) (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t + (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t. (24) 7
With the solutions for p t and y t, we can find the equilibriu noinal interest rate fro the dynaic IS curve (8), since it with use of (23) and (24) becoes y t = E t y t+1 Φ 1 (i t [E t p t+1 p t ]) abi ss (1 + bi ss ) (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t + (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t = 0 Φ 1 { i t } 1 + bi +Φ 1 ss bi ss ρ µ + 1 + bi ss ρ 1 + bi ss t ρ { 1 + bi Φ 1 ss bi ss γ µ + 1 + bi ss ρ 1 + bi ss γ t 1 bi ss (1 + bi ss ) + (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + a) } (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t, Φ 1 abi ss (1 + bi ss ) i t = (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t +Φ 1 bi ss ρ (µ + ρ 1 + bi ss ρ t 1 + s t ) { Φ 1 bi ss ρ bi ss (1 + bi ss ) 1 + bi ss t 1 + ρ (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + a) (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t }, where in the second equation, the exogenous process for t has been inserted. This equation is further siplified as Φabi ss (1 + bi ss ) i t = (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t Φ (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t = bi ss ρ + (µ + ρ 1 + bi ss ρ t 1 + s t ) bi ss ρ bi ss (1 + bi ss ) 1 + bi ss t 1 ρ (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t + (1 + a) (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t, [ bi ss ρ bi ss ρ µ + bi ss 1 + bi ss ρ 1 + bi ss ρ 1 + bi ss ρ [ bi ss (1 + bi ss ρ ) (Φ 1) (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t, ] ρ t 1 Φa (1 + bi ss ) 1 + bi ss + a (bi ss + Φ) + bi ss (1 + bi ss ) 1 + bi ss + a (bi ss + Φ) ρ ] s t 8
leading to i t = bi ss ρ µ biss (1 ρ ) ρ 1 + bi ss ρ 1 + bi ss t 1 ρ [ ] bi ss (1 + bi ss ) [1 + aφ] (1 + bi ss ρ ) 1 + bi ss + a (bi ss + Φ) ρ (Φ 1) (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t. (25) s t 3 The econoics of the solution and the effectiveness of onetary policy We now have the solutions for output, prices and the noinal interest rate fro (24), (23) and (25), respectively. These are stated again for convenience, and then the properties of the solution are thoroughly discussed with focus on the role of the oney supply process paraeter ρ for the transission of shocks. y t = abi ss (1 + bi ss ) (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t + (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t. p t = 1 + bi ss bi ss ρ µ + 1 + bi ss ρ 1 + bi ss t 1 ρ bi ss (1 + bi ss ) + (1 + bi ss ρ ) (1 + bi ss + a [bi ss + Φ]) s t (1 + a) (biss + Φ) 1 + bi ss + a (bi ss + Φ) e t. i t = bi ss ρ µ biss (1 ρ ) ρ 1 + bi ss ρ 1 + bi ss t 1 ρ [ ] bi ss (1 + bi ss ) [1 + aφ] 1 + bi ss ρ 1 + bi ss + a (bi ss + Φ) ρ s t (Φ 1) (1 + biss ) (1 + a) 1 + bi ss + a (bi ss + Φ) e t. 3.1 Supply shocks Consider a positive supply shock, e t > 0. Inspection of the AS and AD schedules, (7) and (10), reveals that insofar E t y t+1 and E t p t+1 are unaffected, these schedules deterine the output and price response directly. As the shock is white noise, we iediately have that E t y t+1 = 0. Moreover, expected future prices will also be unaffected be the supply shock. This follows as E t p t+1 = γ 0 µ + γ 1 t by (11), and because the process for t is independent of the supply shock. It is thus convenient to frae the discussion of the econoy s output and price response in ters of AS and AD schedules in a (y t,p t ) space, and then trace out the noinal interest rate effects by the IS/LM schedules underlying the AD schedule in a (y t,i t ) space. A 9
graphical representation is provided by Figure 1. 4 The supply shock oves the AS curve to the right, i.e., fro AS 0 to AS 1, such that the new equilibriu will be at point A. This involves an increase in output fro y 0 to y 1 and lower prices. This can also be confired by inspection of (24) and (23), respectively. The lower price level will atter for the IS and LM curves. For a given noinal interest rate, a lower price level increases the real oney supply and oves the LM curve to the right, fro LM 0 to LM 1, which puts downward pressure on the noinal interest rate so as to clear the oney arket. This would be the only effect of lower prices in a standard IS/LM odel. But in this dynaic odel, a lower price level also plays a role by lowering the real interest rate since expected inflation, E t p t+1 p t, increases. This increases deand, i.e., oves the IS curve fro IS 0 to IS 1, and puts upward pressure on the noinal interest rate. This expected inflation effect thus ake the ipact of the shock stronger point C, which corresponds to point A in the AS/AD graph of Figure 1, involves higher output than point B, which is the intersection point of the IS and LM curves when the IS curve does not ove. A further iplication is that the effect on the noinal interest rate is abiguous. I will depend on whether the IS- or the LM- effect is the strongest. When Φ > 1, the real interest-rate channel is rather weak (the elasticity of interteporal substitution is relatively sall), and the IS curve oves less than the LM curve. This is the case depicted in Figure 1. In that case, the noinal interest rate will fall when a positive supply shock hits. This can be seen atheatically by inspection of (25); the opposite, of course, holds for Φ < 1. Note that the oney supply process paraeter ρ, a systeatic part of onetary policyaking in the odel, has no effect on how the supply shock is transitted. Any change in this paraeter will therefore not influence output fluctuations arising fro supply shocks. The crucial factor behind this result is that E t p t+1 is unaffected, when t is. This will be evident, when we turn to the response of the econoy to the shock that does affect the process for the oney supply, and thereby E t p t+1, naely s t. 3.2 Monetary policy shocks Consider a positive noinal oney supply shock, s t > 0. In a standard, static IS/LM odel with fixed prices, this will increase output and lower the noinal interest rate as the real oney supply is increased. In this odel, the equilibriu response will be different 4 In this, and the other figures, it should be noted that the subscripts do not represent periods, but are erely used to distinguish different values. 10
as p t is endogenous, and of particular interest in this dynaic context because E t p t+1 is affected as E t p t+1 = γ 0 µ + γ 1 t. I.e., the expected inflation effect working through the real interest rate will play a crucial role. Is is instructive to split up the analysis up two parts. First, consider the special case of ρ = 0. This case iplies that the oney supply shock will not affect the next period s oney supply or expected prices. The latter can be confired analytically, as γ 1 = 0 applies when ρ = 0; cf. (19). With the shock, the LM curve oves to the right. This is depicted in Figure 2 as the ove fro LM 0 to LM 1. In absence of any effects on the real interest rate, i.e., any price effects, this would increase output fro y 0 to y 1, and the econoy would end in point A. In the associated AS/AD graph, the AD curve would ove fro AD 0 to AD 1. The increase in aggregate deand will increase output and put upward pressure on prices, and the increase in p t will dapen the increase in the real oney supply oving the LM curve back to LM 2 as well as put upward pressure on the real interest rate oving the IS curve fro IS 0 to IS 2. Both oveents serve to dapen the output increase, and output becoes y 2 (and the associated deand curve is AD 2 ). So, for a given E t p t+1 the effect of s t > 0 is an increase in output and prices and a fall in the noinal interest rate. The positive ipact on prices, not present in standard IS/LM analysis, dapens the effect on output partly through the contractive effects of the associated decrease in expected inflation. Now consider the case of ρ > 0. The shock to noinal oney in period t under consideration then affects the noinal oney supply in period t + 1 positively, and thus period-t+1 prices and the period t expectations of these. Indeed, as γ 1 > 0 when ρ > 0, E t p t+1 will increase, which exerts upward pressure on expected inflation. The effect of the shock will therefore we very different fro the case of ρ = 0. The case is shown in Figure 3. Point A is the sae point as in Figure 2. I.e., the point where output has increased to y 1 due to the increase in the oney supply for given prices. As in the previous case, this increases deand and puts upward pressure on current prices. This effect will, as in the case of ρ = 0, reduce the real oney stock and reduce inflation expectations oving both the IS and LM curves inwards. Iportantly, when ρ > 0, expectations about the future feed back and affect current variables in this rational expectations fraework. And the higher is ρ, the stronger is the current ipact of the expected future prices. Indeed, if ρ is suffi ciently high, the increase E t p t+1 stiulates deand through the downward pressure on the real interest rate, pushing the IS curve outwards. In Figure 3, the IS curve settles at IS 2 i.e., the expansive effects of higher E t p t+1 doinates the contractive effects of higher p t. The 11
equilibriu will be characterized by point B, with an output level y 2 which is higher than the output response when ρ = 0. Hence, the size of ρ, a systeatic coponent of onetary policy, has iplications for the transission of the shock onto real variables in the odel. Specifically, a higher ρ iplies that the effects of the oney-supply shock on output and prices becoe stronger (analytically it is seen fro the coeffi cients to s t in (24) and (23), which are increasing in ρ ). In addition, the agnitude of expected future price effects ay be so strong that the associated upward pressure on the noinal interest rate ay doinate the fall in the noinal rate found in conventional static odels. See (25), where it is seen that a high ρ ay turn the coeffi cient on s t positive. 5 While this odel is erely an exaple of policy effectiveness under rational expectations, it has generality for the role of stabilization policies in such settings. A shock that persists into the future (either by itself of through a policy response), affects expectations about future variables, which ay have current effects. Policy ay then be designed so as to affect expectations about future variables appropriately. In this odel, for exaple, a policy rule that stabilizes output as uch as possible would be (6) with ρ 1; i.e., any positive oney supply shock should be reversed in the following period. In that case, E t p t+1 would fall, putting upward pressure on the real interest rate, and thus downward pressure on aggregate deand. See (24) where the coeffi cient on s t is iniized when ρ is a sall as possible (while still retaining stationarity of the oney supply). 4 General policy iplications and discussion of soe early literature The iplication of the previous discussion is that under rational expectations, onetary policy is generally not ineffective. Systeatic coponents of the onetary policy rule do not affect average output, but it can affect its variability. This odel is just one exaple of this violation of the policy ineffectiveness proposition. Interestingly, the early and very influential literature on the policy liitations under rational expectations generally did not acknowledge this at all. Papers like Sargent and Wallace (1975), Barro (1976), Woglo (1979), McCallu (1980), Canzoneri et al. (1983) were very focused on presenting Lucas-style aggregate supply equations satisfying the natural-rate hypothesis. They therefore siplified the specification of the deand side such that in their odels, systeatic onetary policy rule paraeters had no influence 5 When deand is highly real interest-rate elastic, Φ < 1, this occurs when ρ 1. 12
on output deterination. These siplifications included either an ad hoc definition of the real interest rate that failed to account for period-t expectations of future prices, or a specification of aggregate deand that eliinated any real interest rate effects by replacing the IS/LM equations by a quantity equation t = p t + y t. As evident by this odel, such siplifications are way too strong when it coes to aking stateents about policy ineffectiveness. Ineffectiveness fails when there are oney supply shocks. Moreover, if the process for the oney supply rule is further aended by, e.g., inclusion of the supply shock and/or proper responses to output, the policy ineffectiveness proposition fails ore generally. In this odel, any policy rule that will affect price expectations, i.e., E t p t+1, will affect the real interest rate and thereby aggregate deand and output in the short run. Sargent (1973) odels the real interest rate as a function of expected prices, and his rational-expectations solution derived in the article s appendix reveal upon inspection that policy rule paraeters do affect output deterination. To the best of y reading, he does not ake coent of that fact. Dotsey and King (1983), on the other hand, did note how a feedback policy could affect future prices and thereby current prices and output volatility. Their odel is a Lucas-style iperfect inforation odel, and the coputation of rational expectations equilibria under different feedback policies is coplicated as the coeffi cients of the odel are functions of the variances of underlying shocks and thus to what extent they are stabilized. They are able to show, however, that a oney supply feedback rule that abstains fro responding to past period s oney supply reduces output variance (in this odel, it would correspond to setting ρ = 0), as does a countercyclical response to past period s output. 6 Groth (1997) also deonstrates how a countercyclical oney supply rule stabilizes output. He lets t be a negative function of y t 1 in a odel very siilar to the one of this exercise. This eans that a shock (e.g., a supply shock) that increases output in period t, is expected to be followed by a lower oney stock in period t + 1; E t p t+1 go down, the real interest rate increases and the expansive effects of the shock are reduced. So, by appropriate policy design, fluctuations due to supply shocks can also be reduced by systeatic onetary policy. The introduction of rational expectations into acroeconoics was by any seen as paving the death for deand-oriented public policies (and still is for soe), as the 6 As they can only deonstrate it for a particular case, they write in their Footnote 12 (p. 370): We do not attach a great deal of iportance to the particular exaple chosen, as it was selected for analytical convenience. 13
policy ineffectiveness proposition was considered central. The upshot of this analysis is, ironically, that it is indeed the rational expectations about future prices that create a scope for policy effectiveness in an otherwise standard New-Classical odel fraework. References [1] Barro, R. J., 1976, Rational Expectations and the Roles of Monetary Policy, Journal of Monetary Econoics 85, 1 32. [2] Canzoneri, M. B., D. W. Henderson, and K. S. Rogoff, 1983, The Inforation Content of the Interest Rate and Optial Monetary Policy, The Quarterly Journal of Econoics, 546 566. [3] Dotsey, M. and R. G. King, 1983, Monetary Instruents and Policy Rules in a Rational Expectations Environent, Journal of Monetary Econoics 12, 357 382. [4] Groth, C., 1997, Nyklassisk akroteori. In: C. Groth, L. H. Pedersen and P. B. Sørensen, Noter til Makroøkonoi, Undervisningsnoter nr. 75, Departent of Econoics, University of Copenhagen, 147 194. [5] McCallu, B. T., 1980, Rational Expectations and Macroeconoic Stabilization Policy, Journal of Money, Credit, and Banking 12, 716-746. [6] Sargent, T. J., 1973, Rational Expectations, the Real Rate of Interest, and the Natural Rate of Uneployent, Brookings Papers on Econoic Activity 2, 429 472. [7] Sargent, T. J. and N. Wallace, 1975, Rational Expectations, the Optial Monetary Policy Instruent, and the Optial Money Supply Rule, Journal of Political Econoy 83, 241 254. [8] Walsh, C. E., 2010, Monetary Theory and Policy. Third Edition, The MIT Press. [9] Woglo, G., 1979, Rational Expectations and Monetary Policy in a Siple Macroeconoic Model, The Quarterly Journal of Econoics 93, 91 105. 14
Figures Figure 1: Effects of a positive supply shock, e > 0 15
Figure 2: Effects of a positive oney supply shock, s > 0. The case of ρ = 0. 16
Figure 3: Effects of a positive oney supply shock, s > 0. The case of ρ > 0. 17