Mathematical Annex 5 Models with Rational Expectations

Transcription

1 George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 5 Models with Rational Expectations In this mathematical annex we examine the properties and alternative solution methods for models with rational expectations. We start with simple models of exogenous stochastic processes, and susequently examine more general first and second order linear economic models with one endogenous and one exogenous variale. A5.1 The Definition of Rational Expectations Rational expectations are defined as the mathematical expectations for the future evolution of a variale, ased on a given information set. We define the rational expectation of a variale x in the period t+1, ased on availale information in period t, as,! x t+1 E(x t+1 I t ) (A5.1) where,! I t {x t i,z t i,i 0,1,2,...,} (A5.2) is the set of availale information, which consists of current and past values of the variale x itself, as well as the current and past values of a set of variales z, which affect, and thus help predict, the future values of x. It is worth noting that this definition of the information set does not involve loss of memory, as whatever is known in period t is also known in period t+1 and all future periods. More generally, we define the rational expectation of a variale x at t+s, ased on availale information at t, as,! E( I t ) (A5.3) In order to define rational expectations for a variale more precisely it is not enough to know the information set, ut also the model of how this variale is determined and evolves over time. We egin with the simplest model for determining a variale, that of a univariate stochastic process. Then we will examine more general models, according to which an endogenous variale depends

2 on future expectations for its evolution, as well as an exogenous variale. Even more general models of a system of endogenous and exogenous variales can e solved y similar methods. A5.2 Rational Expectations for Linear Autoregressive Processes Let us assume a variale x, which follows a linear autoregressive stochastic process of the form, x t (1 λ)x _ + λx t 1 + ε t (A5.4) where,! is a constant, the mean of the variale x, and ε a white noise stochastic process, with zero mean and constant variance. x _ We shall define the deviation of x from its mean as, t x t x _ (A5.5) From (A5.4) and (A5.5) we get that, t λ t 1+ ε t (A5.6) It is easo see, y repeated sustitutions, that, t+1 λ t, t+2 λ 2 t, t+3 λ 3 t,..., t+s λ s t (A5.6) The rational expectation of a first order linear autoregressive stochastic process depends only on its current value, with a coefficient that depends only on λ. If the stochastic process (A5.4) is stationary, i.e if -1<λ<1, then the impact of the current value of the variale on its rational expectation at time t+s is decreasing with s. As s approaches infinity, the limit of the rational expectation is given y, lim t+s lim λ s t 0 s s (A5.7) (A5.7) and (A5.5) implhat, lim x _ s (A5.8) In this sense, the mean of the stationary stochastic process x, which is its long run equilirium value, is also the limit to which rational expectations aout its future evolution converge over time. If the process is not stationary, ut a random walk, i.e if λ1, then (A5.6) is transformed to, t+1 t, t, t,..., t (A5.6 ) t+2 t+3 t+s!2

3 In this case, the rational expectation for the future value of x is the current value of x, independently of s, as the variale does not converge to a long run equilirium. A5.3 First Order Linear Models with Rational Expectations We now turn to the solution of a linear model in which a variale y depends on the rational expectation for its future value, and another exogenous variale x. The model is descried y a first order lineal equation of the form, a +1 + x t (A5.9) The rational expectations hypothesis implies that economic agents know that the variale y is determined y (A5.9). We also assume that all economic agents have access to the same information set. There are a numer of methods for the solution of (A5.9). All methods are ased on the law of iterated expectations, which requires that the current expectation of the future expectation of a future value of a variale, is nothing more than the current expectation of the future value of the variale. That is, that, ( +1 ) (A5.10) A5.3.1 The Method of Repeated Sustitutions The simplest method for solving (A5.9) is the method of repeated sustitutions, a method that we also used in finding the rational expectations of the simple first order autoregressive process in the previous section. From (A5.9) and (A5.10), +1 a ( ) + x t+1 a +2 + x t+1 (A5.11) Sustituting (A5.11) in (A5.9), we get, a a x t+1 + x t (A5.12) Repeatedly sustitutiong the future expectations of y, until time T, we get, a T +1 +T +1 + T a s (A5.13) In order to have convergence of the last term of (A5.13), as T tends to infinity, the asolute value of a must e less than one, and the expected value of x should not increase too quickly. If the expected value of x increases exponentially, its growth rate should not exceed (1/a)-1. Under these conditions, it follows that,!3

4 ! lim (A5.14) T at +1 +T +1 0 Then, a solution of (A5.9) can e derived from (A5.13) as, a s (A5.15) It is worth noting that (A5.15) satisfies the condition (A5.14), and is thus a solution of (A5.9). It suggests that the current value of the endogenous variale y is the discounted sum of the expected future values of the exogenous variale x, with a discount rate equal to a. This solution is usually called the fundamental solution. It is however worth noting that (A5.15) is not the only solution of (A5.9). The fundamental solution is ased only on the minimum numer of variales (x in our case), the so-called fundamentals, and satisfies (A5.14). If (A5.14) is not satisfied, then there is a host of other, non fundamental, solutions. Suppose there is an alternative solution to (A5.9), which consists of (A5.15), plus an additional variale z. This solution takes the form, a s + z t (A5.15 ) One can easily demonstrate that if the variale z satisfies,! z t a z t+1 or equivalently! z t+1 z t a then (A5.15 ) is also a solution of (A5.9). However, it is worth noting that ecause a<1, the mathematical expectation of the future z explodes over time. This can e proven aking the limit of the mathematical expectation as time tends to infinity. This limit is given y, s! lim z t+s 1, depending on whether z is positive or negative. s a z t ± Solutions ased on non fundamental variales such as z are called ules, as opposed to solutions like (A5.15) which are ased only on the fundamentals. The rest of this Annex will focus on the fundamental solutions, ignoring ules. Besides the method of repeated sustitutions, two other methods may e used to solve (A5.9). The method of factorization and the method of undetermined coefficients. These methods, are simpler to use in more complex prolems, for which the method of repeated sustitutions soon ecomes unwieldy. However, we can also applhem to the simple case of (A5.9).!4

5 A5.3.2 The Factorization Method The factorization method requires the use of the future mathematical expectations operator F. We define the future mathematical expectations operator for a variale x, as, Fx t x t+1, F 2 x t x t+2,..., F s x t (A5.16) In addition, F 1 x t x t 1 x t 1 Lx t, F 2 x t x t 2 L 2 x t,..., F s x t x t s L s x t (A5.16 ) Thus, the future mathematical expectations operator is the inverse of the lag operator L, which we have used for the solution of difference equations. Using the future mathematical expectations operator, and assuming that -1<a<1, (A5.9) can e written as, af + x t 1 af x t a s F s x t a s E t (A5.17) (A5.17) is the same as the fundamental solution (A5.15) that we found using the method of repeated sustitutions. A5.3.3 The Method of Undetermined Coefficients The method of undetermined coefficients consists in using a presumed form of a solution with undetermined coefficients, to otain the mathematical expectation of the presumed solution, replace this in place of the expectation in (A5.9), and compare the coefficients of the resulting equation with the undetermined coefficients of the presumed solution. If the form of the presumed solution is correct, this will suffice to determine the undetermined coefficients. For example, if our guess is that the solution has the form σ µ s (A5.18) where σ and µ are undetermined coefficients, then, +1 σ µ s x t+1+s (A5.19) Sustituting (A5.19) in (A5.9), and comparing coefficients etween the resulting equation and (A5.18), we find that σ and µa. This confirms our presumption in (A5.18), and the solution is exactlhe same as with the two other methods.!5

6 The choice of method to e used to solve models with rational expectations depends on the ease of application. In simple models the method of repeated sustitutions can easily e applied, ut in more complex models, the method is not easo use, and the two other methods are preferale. A5.3.4 Two Economic Examples In order to see how these methods are applied, we shall use two simple economic models that result in equations of the form of (A5.9). Α. Stock Prices in Efficient Capital Markets In our first model we assume a capital market in which investors are risk neutral. They choose etween a stock and a safe asset with a rate of return r. In equilirium, aritrage will ensure that the expected rate of return of the stock will e equal to the rate of return of the safe asset. E! t p t+1 p t + d t r (A5.20) p t p t where p is the price of the stock, and d is the dividend. The expected rate of return of the stock is equal to the expected capital gain, plus the dividend as a proportion of the stock price. (A5.20) can e rearranged as, p t 1 ( 1+ r E p + d t t+1 t ) (A5.21) (A5.21) has the form of (A5.9), with 0<a1/(1+r)<1. The solution of (A5.21) gives us the stock price as a function of future expected dividends. p t 1 1+ r 1 1+ r d t+s s (A5.22) The stock price is the present value of future expected dividends, discounted he rate of return of the safe asset. Β. The Cagan Model of Money Demand In our second model we assume consumers and firms that decide etween holding money and goods. Money is a nominal asset whose value is affected y inflation. In this case, the demand for money is a negative function of expected inflation, and money market equilirium requires that, M! t exp α E P P t t+1 t (A5.23) P t P t!6

7 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 5 where M is the nominal money supply, P the price level, and α>0 the semi-elasticity of money demand with respect to expected inflation. Taking logarithms on oth sides, and denoting the logarithm of the nominal money supply y m and the logarithm of the price level y p, the model can e written as,! m t p t α( p t+1 p t ) (A5.24) Solving for p, p t α 1+ α E p + 1 t t+1 1+ α m t (A5.25) (A5.25) has the form of (A5.9), with, a α, 1 a 1+ α The solution takes the form, ' p t 1 α (A5.26) 1+ α 1+ α m t+s The current price level depends on the discounted expectations for the evolution of the money supply in the future, with a discount factor equal to 0<α/(1+α)<1. This model was first used y Cagan (1956) in order to explain hyperinflations. A5.3.5 Alternative Assumptions aout the Evolution of Endogenous Variales The solution of equation (A5.9), for example in (A5.15), is ased on the discounted rational expectations aout the future path of the exogenous variale x. In order to have a closed form solution for the variale y, we must make additional assumptions aout the evolution of the exogenous variale x. We shall use two alternative examples. s Our first assumption is that the exogenous variale x is expected to remain constant at the level x0. From (A5.15) the endogenous variale y is determined y, ' a s E (A5.27) t a s x 0 1 a x 0 (A5.27) can e used to analyze the impact of an announcement for a future change in the constant level of the exogenous variale.!7

8 Let us assume that at time t0, it is announced that from period t1 onwards, the variale x will rise from x0 to a new constant level x1. In this case, the path of the endogenous variale y will e as follows: ' for! 1 a x 0 t < t 0 ' 1 a t 1 t ( ) for! (A5.28) 1 a x + 0 at 1 t 1 a x 1 t 0 t < t 1 for 1 a x 1 t t 1 At the time of the announcement t0, variale y rises, as the expectations aout the future path of x change. Until the change is implemented at time t1, y rises gradually, as the impact of the higher x1 after period t0 rises, compared to the impact of the lower x0 in the interval t1-t. After period t1, the endogenous variale y is stailized at the higher level corresponding to the higher x1. If x is the dividend, as in our first model, the share price will rise immediately after the announcement of a future dividend increase and will continue to grow as the expected dividend increase comes closer. When the increase in the dividend materializes, the stock price stops growing and is stailized at the new higher level corresponding to the higher dividend. If x is the money supply, as in our second model, the price level will rise immediately after the announcement of a future increase in the money supply, and will continue to grow until the rise in the money supply materializes. The price level will then stailize at its new higher level. The second case that will e analyzed is ased on the assumption that the exogenous variale x follows an autoregressive stochastic process, as in (A5.4). In this case, a s E t 1 a x_ + ( aλ) s x t x _ 1 a x_ + 1 aλ x t x_ (A5.29) (A5.29) can e written as, y _, where, (A5.30) 1 aλ x t x_ y _ 1 a x_ In the stock price model, (A5.30) tells us that the price of the stock will e a function only of the current dividend. This is ecause if dividends follow a first degree autoregressive stochastic process, as we have assumed, the current dividend is the only element required to predict future dividends. Similarly, in the money demand model, (A5.30) tells us that the price level will e determined y onlhe current money supply. This is ecause, if the money supply follows a first order autoregressive stochastic process, as assumed, the current money supply is the only element required to predict the future course of the money supply.!8

9 A5.4 Second Order Dynamic Models with Rational Expectations We next turn to the methods of solving a second order, linear, dynamic model with rational expectations. In this model, we assume that an endogenous variale y depends on the rational expectation of its future value, its lagged value, and an exogenous variale x. This model comines rational expectations aout the future value of the variale, with the impact of lagged values of the variale. Our model is linear and takes the form, a cx t (A5.31) where, a, >0, and a+<1. This model can e solved either using the factorization method, or the method of undetermined coefficients. Oviously, oth methods result in the same fundamental solution. A5.5.1 The Factorization Method Using the future expectations operator F, (A5.31) can e written as, af + F 1 + cx t (A5.32) where F -1, the inverse of the future expectations operator, is the same as the lag operator. Moving all the terms that contain y on the left hand side, we get, ( 1 af F 1 ) cx t (A5.33) (A5.33) can e multiplied y -F/a on oth sides, which results in, F 2 1 a F + a c a Fx t (A5.34) (A5.34), can e factorized as, ( ) c a Fx t F 2 1 a F + a (F λ)(f µ) F 2 (λ + µ)f + λµ (A5.35) where λ and µ are the two roots of the characteristic polynomial of, F 2 1 a F + a (A5.36) From (A5.35), we know that, λ+µ1/a, λµ/a. It is simple to show that one root is smaller than unity and the other is higher than unity. We shall assume that λ<1, and that µ>1.!9

10 !! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 5 The characteristic polynomial of (A5.36) is given y, Φ(φ) φ 2 1 a φ + a (A5.37) In order to show that one root is smaller than unity, we shall calculate the characteristic polynomial for φ0 and φ1. We get,! Φ(0) και! a > 0, 1 a Φ(1) < 0 a Hence, there is a root λ, etween zero and one, for which Φ(λ)0. The second root µ is determined y, µ aλ We shall have µ>1, if λ</a. This is actuallrue, since, Φ a (1 a ) a 2 < 0 Therefore we shall have that λ</a<1 και µ>1. Dividing oth sides of (A5.35) y F(F-µ), we get, ( 1 λf 1 ) c 1 a µ F x t c 1 aµ 1 µ 1 F x t λc 1 1 µ 1 F x t (A5.38) where in the last part of (A5.38) we have made use of the properthat aµ/λ. This follows from (A5.35). From (A5.38), it follows that, λ 1 + λc 1 1 µ 1 F x t λ 1 + λc s 1 µ (A5.39) (A5.39) is the fundamental solution of (A5.31). As in the case of (A5.15), (A5.39) suggests that the current value of the endogenous variale y depends on the discounted sum of the expected future values of the exogenous variale x, with a discount factor equal to 1/µ<1. It also depends on its own lagged value, with a coefficient equal to λ<1. The factorization method is perhaps the most effective method of solving equations of the form (A5.31). An alternative method is the method of undetermined coefficients.!10

11 A5.5.2 The Method of Undetermined Coefficients In order to applhe method of undetermined coefficients, as in the case of the first order model, we presume that the solution of (A5.31) takes the form, φ 1 +ψ ω s (A5.40) with undetermined coefficients φ,ψ,ω. From (A5.40), the rational expectation of the future y is given y, +1 φ +ψ s1 ω s 1 (A5.41) Sustituting (A5.41) in (A5.31), we get, 1 aφ y + c t 1 1 aφ x + aψ t 1 aφ s1 ω s 1 (A5.42) Comparing coefficients etween (A5.42) and (A5.40) we can solve for the undetermined coefficients, as, φ ψ ω 1 aφ c 1 aφ a 1 aφ (A5.43a) (A5.43) (A5.43c) From (A5.43a) φ will e the smaller root of the polynomial, Φ(φ) φ 2 1 a φ + a (A5.44) This is none other than (A5.37), analyzed in the factorization method. There will e two roots, λ and µ, where 0<λ<1 and µ>1. The roots will satisfy, λ + µ 1 a,λµ a (A5.45) From (A5.43)-(A5.45) we get,!11

12 φ λ < 1, ψ λc, ω 1 µ < 1 (A5.46) As a result, the fundamental solution is given y, λ 1 + λc s 1 µ (A5.47) which has exactlhe same form as (A5.39). For solution methods to more general multivariate linear models with rational expectations, see Blanchard and Kahn (1980), Klein (2000), or Sims (2001).!12

13 References Blanchard O.J. and Kahn C. (1980), The Solution of Linear Difference Equations under Rational Expectations, Econometrica, 48, pp Klein P. (2000), Using the Generalized Schur Form to Solve a Multivariate Linear Rational Expectations Model, Journal of Economic Dynamics and Control, 24, pp Sims C.A. (2001), Solving Linear Rational Expectations Models, Computational Economics, 20, pp !13