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MSc Finance: International Finance: Lecture 5 Monetary Policy with Forward-Looking Agents Anne Sibert Autumn 2008 1. Introduction Consider the following problem: A sophisticated scud missile passes overhead. The ob- jective is to launch a rocket to destroy the scud missile as quickly as possible, to avoid the missile taking evasive action. What you control is the angle at which you shoot o¤ the rocket. This is a typical problem that a scientist might face. An important feature of the problem is that the state (here, the location of the rocket) at a given time is a function of past actions (here, the choice of the angle). With monetary policy, the current state is in‡uenced by the future. Here is an example of how this can matter 2. Monetary Policy with Forward-Looking Agents The model described here is based on Kydland and Prescott [2] and Barro and Gordon [1]. I assume that social welfare is given by 2 2 = (n no ) ; > 0: (1) where is in‡ ation, n is employment, and no is socially optimal employment.1 All three of these variables are in logarithm form. The interpretation of this is as follows. Optimal in‡ ation is assumed to be zero. This is not important and it saves on notation. Society dislikes deviations from this socially optimal rate and welfare falls at an increasing rate as in‡ ation rises above or falls below zero. Society also dislikes deviations from the socially optimal level of emplyment and welfare falls at an increasing rate as employment rises above or falls below this optimal level. The parameter tells us how society weights losses due to in‡ ation deviations versus losses due to output deviations. The higher is , the greater the weight society places on employment deviations. We now assume a standard Keynesian model of employment determination. We sup- pose that …rms and workers enter into wage contracts. These contracts specify a …xed nominal wage W . After the wage is set, the price level is realised and …rms decide how much labour to hire. Firms maximise their pro…ts by setting the marginal product of labour equal to the real wage, W=P . Thus, the …rms have a downward sloping labour demand curve. Let N denote workers’ and …rms’ most preferred level of employment, called the natural rate. Let P e be the workers’ and …rms’ expectation of the price level. This is where our model deviates from the Keynesian model. In deciding what nominal wage to choose, the private sector forecasts the price level. We will initially assume that there is no uncertainty in the model and that the private sector has perfect foresight. Thus, it will turn out that this expectation will equal the actual price level. We assume that 1 Suppose that the country is a small open economy and that PPP holds. Then the price level equals the exchange rate, as in the monetary approach model of lectures 3 and 4. Thus, in‡ation equals the depreciation of the exchange rate. 1 MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents2 the workers and …rms choose the contractual nominal wage W such that if the price level turns out to be P e , then the …rms choose employment of N . Suppose that workers and …rms are wrong and the price level turns out to be higher than expected, say P 1 > P e . Then W=P 1 < W=P e and the …rms choose employment of N 1 > N . If he price level turns out to be lower than expected, say P 2 < P e , then W=P 2 > W=P e and the …rms choose employment of N 2 < N . In general, it is clear that employment is increasing in P=P e , with employment equal to N when P=P e = 1: We can write this idea in logarithm form as n=n + (p pe ) ; > 0: (2) To simplify the notation, we let be one. We can rewrite equation (2) as n = n + (p p 1) (pe p 1) =n + e ; (3) where p 1 is last period’ price level (in logarithm form), = p p 1 is in‡ s ation and e = e p s p 1 is the private sector’ expectation of in‡ ation. Equation (3) is an expectations-augmented Phillips Curve. It shows that there is a positive relationship between employment and unexpected in‡ ation. The di¤erence between this Phillips Curve and the conventional Philips curve is a result of the private sector anticipating the future actions of the government when it sets the contractual nominal wage. Substituting equation (3) into equation (1) yields 2 e 2 = (n + no ) : (4) o s We assume that n < n : This means that private sector’ desired level of employment is less than the socially optimal rate. A reason for this might be distortionary taxes (such as income taxes) in labour markets. Let d no n > 0 denote the deviation between the optimal and natural rates of employment. Use this notation to rewrite equation (4): 2 e 2 = ( d) : (5) We see from equation (5) that because of the distortion in the labour market, the government wants to increase employment. It can do this, but at the cost of higher in‡ation. Recall that the timing is that the private sector forms its expectation of in‡ ation and incorporates this expectation into its nominal wage contract. Then the central bank chooses monetary policy. This implies that when the central bank picks its policy, it takes expected in‡ation as given –that is, it treats it as a constant in its optimisation problem. We assume that the central bank can perfectly control in‡ ation. Thus, it chooses to maximise equation (5). To …nd the …rst-order condition of the maximisation problem di¤erentiate with respect to and set the result equal to zero e d =d = 2 2 ( d) = 0: (6) To verify that the solution to this is indeed a maximum. We …nd …nd the second-order condition; di¤erentiate (6) and verify the result is strictly negative. d2 =d 2 = 2 2 < 0: (7) MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents3 Solving equation (6) for in‡ation yields ( e + d) = : (8) 1+ Thus, we have that in‡ ation is increasing in expected in‡ ation. This is because the higher is expected in‡ ation, the higher actual in‡ation has to be to create a given amount of unexpected in‡ ation. In‡ ation is increasing in d because the greater the deviation between the optimal and the natural rates of employment, the more incentive the policy maker has to create unexpected in‡ ation. In‡ation is also rising in the parameter ; this is because the greater the weight that the policy maker puts employment deviations relative to in‡ ation deviations, the more he is willing to in‡ ate. The public knows the preferences of the policy maker; hence, the public can solve for in‡ s ation. Thus, we assume that the public’ expectation of in‡ ation equals actual in‡ ation. Substituting e = into equation (8) yields ( + d) = , = d: (9) 1+ Note from equation (3) that workers and …rms get their most preferred level of em- ployment, n . The outcome also has strictly positive in‡ation. Everyone would be better o¤ if the policy maker could commit himself to in‡ ation of zero. If zero in‡ ation were expected, then workers and …rms would still have employment of n : This is the optimal solution. But, the policy maker cannot attain the optimal solution. If zero in‡ation were expected, then by equation (8), the central bank would want to reneg and set d = : (10) 1+ Anticipating this, the public will not expect zero in‡ation. Thus, we say that the opti- mal solution of zero in‡ ation is not time consistent. This problem –that the government will choose not to follow its optimal policy is called the time inconsistency problem. It appears that a solution is to appoint a conservative central banker – that is, a central banker who puts no weight on employment deviations relative to output. Such a central banker will always choose zero in‡ ation. 3. The Time-Inconsistency Problem Suppose that a policy maker must pick a series of actions between time 0 and time T . We denote the time-t action by xt . The action might be a choice of in‡ ation or a tax or some other policy. Suppose that at time zero, the policy maker found the optimal plan: x0 ; x1 ; :::xT . If there is uncertainty, one might think of the optimal action at time t as a vector specifying what to do in each possible state of the world. There are no surprises; the policy maker knows what states of the world can occur. Thus, it seems that at time t; 0 < t T , the policy maker would …nd it optimal follow xt ; xt+1 ; :::xT . If, however, there is a t where he would not want to do this, then the optimal plan is said to be time inconsistent. In our previous example, think of the plan as covering periods zero and one. In period zero, the private sector forms its expectations, picking expected in‡ ation equal to the in‡ ation rate the government will choose in period one. There is no government policy. In period one, the government chooses in‡ ation. If the government were able to commit s to a policy in period zero, it would take into account that the private sector’ expectation MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents4 of in‡ation will equal the in‡ation rate it chooses in period one. Thus, it would attempt to in‡ uence this expectation. It would optimise by choosing zero in‡ ation. If the government were then given the opportunity to change its mind in period one, it would want to do so. This is because in period one, expectations have already been formed so the government would not try to in‡ uence them. They are now treated as a constant. 4. Credibility vs. Flexibility It appears that a solution to the time-inconsistency problem is to appoint a conservative central banker –that is, a central banker who puts no weight on employment deviations relative to output. Such a central banker will always choose zero in‡ ation.Now suppose that the …rms’production functions are subject to a shock that shifts the labour demand curve. Assume that the timing of events is as follows: 1. Workers and …rms enter into nominal wage contracts specifying W . 2. A stochastic mean-zero shock shifts the labour demand curve. 3. The central bank chooses in‡ation. Consider Figure 1 again. Suppose that the workers and …rms set W = P e and subse- quently a shock shifts the labour demand curve in from N d to N1 . d If the government sets in‡ ation equal to what the private sector expected before it learned the realiszation of the shock, then employment will be N1 < N . It is to the bene…t of the private sector if the central bank in‡ ates more than the private sector expected so that P = P1 and N = N . Because the private sector forms its expectation of in‡ ation and chooses W before the shock is realised and because the central bank chooses in‡ ation after the shock is realised, the central bank has a stabilisation role. We can analyse this algebraically by rewriting equation (5) as 2 e 2 W = ( d ) : (11) where is a shock that shifts the labour demand function in if it is positive and out if it is negative. Assume that has mean zero and variance 2 . The central bank chooses in‡ation after the shock has occurred, so it treats the shock as a constant and maximises equation (11). The …rst-order condition is e 2 2 ( d ) = 0; (12) Solving yields e ( +d+ ) = : (13) 1+ We assume that the private sector has rational expectations so that its expectation of in‡ ation is the statistical expectation: e = E . Then taking expectations of both sides of equation (13) yields (E + d + ) (E + d) E =E = : (14) 1+ 1+ Solving yields MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents5 E = d: (15) Substituting equation (15) into equation (13) yields ( d+d+ ) = = d+ : (16) 1+ 1+ In‡ ation is now equal to the in‡ation bias term d that we found last time plus a stabilisation term = (1 + ). Substituting equations (15) and (16) into equation (11) gives social welfare 2 2 2 2 = d+ d = d+ +d : 1+ 1+ 1+ 1+ (17) Taking the expected value yields 2 2 2 2 + E = + d2 2 = ( + 1) d2 : (18) (1 + ) 1+ Suppose that instead the government appointed a conservative central banker or appointed an independent monetary policy committee and ordered it to follow a zero- in‡ation rule. Then = e and welfare is 2 = (d + ) : (19) Taking the expected value yields E = d2 + 2 : (20) The conservative conservative central bank or rule is better than than the central banker who maximises welfare (this case is often called discretion) if 2 2 d2 + 2 > ( + 1) d2 , d2 + 2 < ( + 1) d2 + (21) 1+ 1+ 2 , (1 + ) d2 + (1 + ) 2 < ( + 1) d2 + 2 , 2 < (1 + ) d2 : This is sensible. A central banker who maximises social welfare tends to be bad because he produces an in‡ ation bias and tends to be good because he stabilises. The conservative central banker does not produce an in‡ ation bias or stabilise. Thus, if the variance of the labour-demand shock is su¢ ciently large, the stabilisation e¤ect dominates and discretion is better than a rule. But, if the variance of the shock is su¢ ciently small, then stabilisation is not that important and the rule is better than discrestion. Society faces a tradeo¤: If it appoints a conservative central banker or if it legislates a zero- in‡ation rule, it gains credibility (for low in‡ation), but it loses ‡exibility (in responding to shocks). A challenge is to come up with monetary institutions that confer credibility without sacri…cing too much ‡ exibility. 5. Introduction We consider our simple monetary approach model in continuous time. MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents6 6. A Continuous-Time Model In continuous time our model becomes mt pt = y it (22) pt = et (23) it = i + ee _t (24) ee _ = _ e (25) Equation (22) is the LM curve, equation (23) is purchasing power parity, equation (24) is uncovered interest parity and equation (25) is perfect foresight. A dot over a variable _ is the conventional way of denoting a time derivative; that is, xt = dxt =dt. Uncovered interest parity and perfect foresight together imply that the rate of depreciation of the exchange rate equals the interest rate di¤erential. Since the interest rate di¤erential is …nite, the derivative of the exchange rate with respect to time is …nite as well. Thus, the exchange rate cannot "jump". Substituting equations (23) - (25) into equation (22) yields mt et = y _ et ; where y = y i : (26) Equation (26) is a …rst-order linear di¤ erential equation with a constant coe¢ cient. The generic form for this is _ xt = a + bxt + cyt : (27) Recall that there is a trick to solving this (it is the continuous-time analogue to our lag-operator approach). Rewrite equation (27) as _ xs bxs = a + cys : (28) Notice that I have changed the dummy variable from t to s. This is harmless and will be convenient because I want to solve for xt . Now multiply both sides of equation (28) by e bs (where e is the exponential function, not the exchange rate!): bs bs e _ (xs bxs ) = e (a + cys ) : (29) bs The left-hand side is the derivative of e xs . Thus, we have d bs bs e xs = e (a + cys ) ) (30) ds bs bs d e xs = e (a + cys ) ds: Now integrate both sides. What should the limits of integration be? What if we tried integrating from s = 0 to s = t: Rt bs Rt bs 0 d e xs = 0 e (a + cys ) ds ) (31) bs t R t bs e xs 0 = e (a + cys ) ds: 0 b Recall that the de…nition of the "line" notation in the above equation is F (s)ja = F (b) F (a); b > a. Thus, equation (31) becomes MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents7 bt R t bs e xt x0 = e (a + cys ) ds ) (32) 0 bt R t bs e xt = x0 + e (a + cys ) ds =) 0 R t xt = x0 ebt + eb(t s) (a + cys ) ds: 0 This gives x in terms of its past history and the initial value of x. We can see that x will go o¤ to plus or minus in…nity if b > 0. When we solve our exchange rate model, this approach will not work because the analogue of b will be strictly positive and the model will be unstable. We will thus follow the same procedure we did before: we will solve forward, rather than backwards. In terms of the above equation, we will integrate forward from t to in…nity, rather than backward from 0 to t. Our boundary condition will not be an initial value of x, but rather that x does not go to plus or minus in…nity when this is not warranted by the fundamentals. Write equation (26) in the format of equation (27): mt et = y _ et ) (33) _ et et = y mt =) 1 y ms _ es es = : I have a small notational problem. I have let e denote the exchange rate, so I will call the exponential function "exp ". Then, multiplying both sides of equation (33) by exp( s= ) and integrating forward yields 1 y ms _ es es exp( s= ) = exp( s= ) ) (34) y ms d [es exp( s= )] = exp( s= )ds ) Z 1 Z 1 1 d [es exp( s= )] = (y ms ) exp( s= )ds ) t Zt 1 1 lim [es exp( s= )] et exp( t= ) = (y ms ) exp( s= )ds: s!1 t The …rst term on the left-hand side of equation (34) is only positive when there are bubbles. This is because exp( s= ) goes to zero as s goes to in…nity. Thus, the …rst term is zero unless the exchange rate is going to plus or minus in…nity faster than exp( s= ) is going to zero. Assuming that this term is zero, we have MSc Finance: International Finance: Lecture 5Monetary Policy with Forward-Looking Agents8 Z 1 1 et exp( t= ) = (y ms ) exp( s= )ds (35) t Z 1 Z y 1 1 = exp( s= )ds ms exp( s= )ds t Z 1t 1 1 = y exp( s= )jt ms exp( s= )ds t Z 1 1 = y lim exp( s= ) + y exp( t= ) ms exp( s= )ds s!1 t Z 1 1 = y exp( t= ) ms exp( s= )ds: t This implies Z 1 1 et exp( t= ) = y exp( t= ) ms exp( s= )ds ) (36) t Z 1 1 et exp( t= ) = ms exp( s= )ds y exp( t= ) ) t Z 1 1 s t et = ms exp( )ds y : t The outcome is similar to the the discrete-time case. The exchange rate depends on the current and all future values of the money supply. References [1] Barro, R. and D. Gordon, "A Positive Theory of Monetary Policy in a Natural Rate Model," Journal of Political Economy 91, 589-610. [2] Kydland, F. and E. Prescott (1977), "Rules Rather than Discretion: The Inconsistency of Optimal Plans," Journal of Political Economy 85, 473-91.