In…nite horizons, …nite horizons, and indeterminacy in the canonical New Keynesian Sticky - Price model. Robert Driskill Department of Economics Vanderbilt University Nashville TN 37235 Email:
[email protected] October 28, 2005
1
Introduction
Much recent analysis of monetary policy has focused on issues surrounding indeterminacy of equilibrium. In particular, in the context of the canonical New Keynesian sticky-price model as embodied in Woodford (2003), various interestrate rules generate a continuum of rational expectations equilibria all of which converge to the same steady state. This feature of the canonical New Keynesian model has been put forth as an explanation for past macroeconomic behavior (see Clarida, R., J. Gali, and M. Gertler, 2000) and as a restriction on the interest-rate rules a monetary authority ought to use. This paper will show that this indeterminacy is a result of a particular modeling choice, namely the choice of an in…nite time horizon rather than an arbitrarily large but …nite time horizon. That is, we show that only one of the equilibria in the in…nite - horizon model is the limit of the equilibria of a …nite - horizon model as the horizon approaches in…nity. We then argue that the logic behind this example, coupled with related results from other models that share key features with this model, suggest that this particular modeling choice of an in…nite time horizon lies behind the large array of other macroeconomic model indeterminacies. This result suggests that one could use the limit of a …nite - horizon solution as a selection criterion for choosing among many stable equilibria in an in…nite horizon model. The rationale for such a criterion would be based in part on the pervasive use of …nite - horizon models to help people understand the workings and implications of dynamic models. This suggests people might coordinate beliefs around the equilibria of such models. 1
Another popular suggested equilibrium selection criterion for these types of models is e - stability or its close cousin "learnability." In the case of the canonical New Keynesian model with the basic Taylor - type interest rate rule, the conditions for e - stability and determinacy are identical.1 In light of this, we also suggest that a plausible if less formal version of "learnability" could be invoked as a selection criterion that would also choose only the equilibrium that is the limit of the …nite - horizon equilibria. We might note that macroeconomists hold sharply di¤ering views on the importance of indeterminacy in these dynamic rational expectations macro models. One view, perhaps most consistently articulated by McCallum (1983a, 1983b, 1990, 1999, 2003), views the possibility of multiple equilibria as irrelevant. He has forcefully argued that there is usually a "sensible" equilibrium - the minimal state variable (MSV) solution - around which we can assume economic agents coordinate beliefs. He and others also point out that in some well - known models, other selection criterion such as e-stability choose the MSV solution among the in…nite number of other possibilities.2 A second view argues that indeterminacy captures important features of economic reality and can help explain apparent sudden "belief shifts" by economic agents. Some proponents of this view have a research program that looks for more and more models in which indeterminacy can occur and in which the "learnability" selection criterion doesn’t eliminate all but one solution.3 Regardless of to which view one subscribes, it seems useful to appreciate that the in…nite - horizon assumption is a common feature of these models. This point seems especially relevant given the sensitivity of the appearance of indeterminacy in a model to the exact speci…cs of the model. For example, in a simple interest - rate monetary - policy model and in a New Keynesian model, Carlstrom and Fuerst (2001, 2004) …nd that indeterminacy depends on the assumptions about the timing of when cash balances appear in the utility function. In another paper (2005), they show that indeterminacy depends in part on whether the model is speci…ed in discrete or continuous time, and on whether investment is included in the model. Canzoneri and Diba (2005) show that price - level determinacy in the presence of interest - rate rules depends on an interplay of types of …scal policy with assumptions about transactions services provided by bonds. These speci…cation choices are not ones about which the profession has solid evidence concerning which is most preferred or are not ones that would seem to be critical in generating a generic property such as indeterminacy. The one feature that all of these models share is the assumption of an in…nite time horizon. The results we develop here suggest that if this assumption were replaced by the assumption of an arbitrarily large but …nite time horizon, indeterminacy would disappear in all of these models. To illustrate our point, we use two versions of the canonical New Keyne1 See, for example, Bullard and Mitra (2002), Evans and Honkapohja (2003), and Woodford (op. cit.). 2 See, for example, Honkapodja and Mitra (2001) and Bullard and Mitra (2002). 3 See, for example, Carlstrom and Fuerst (2004, 2005).
2
sian sticky-price model, one without a transactions demand for money and one with such a demand. The version without a transactions demand for money, by virtue of its simplicity, leaves no room for argument about how to specify the terminal condition of the …nite-horizon model. This terminal condition is an inescapable implication of the assumed pro…t-maximizing behavior of …rms. Imposition of this terminal condition along with the requirement of boundedness of solutions (a requirement used in all of the models within this literature) leads to …nite-horizon solutions the limits of which are the same unique member of the set of in…nite-horizon solutions (the MSV solution), with one non-generic exception.4 The version in which there is a transactions demand for money has two terminal conditions, one the inescapable implication of the assumed pro…t-maximizing behavior of …rms, and the other an implication of an assumption about the endof-horizon behavior of the monetary authority. In this case, an assumption of boundedness is no longer imposed, and all …nite-horizon solutions have as their limit the same unique MSV solution from the set of ini…nite-horizon solutions. For this case, there is no exception: the result holds for all permissible parameter values.
2 2.1
The in…nite - horizon model Structure
Stripped of what are, for our purposes, inessential features, we brie‡y describe Woodford’s (2003, chapters 2-4) in…nite - horizon model of in‡ation and output.5 Goods in this model are di¤erentiated products produced by monopolistically competitive …rms. A representative household maximizes the present discounted value of instantaneous utility:6 2 3 i=1 Z t=1 X t4 U (Ct ) v(ht (i))di5 ; 0 < < 1; t=0
i=0
where is the discount factor, and Ct is an index of household consumption de…ned as: 21 3 =( 1 Z ( 1)= Ct 4 [ct (i)] di5 ; > 1: 0
The ct (i) are consumptions of the di¤erentiated products. The variable ht (i) represents the quantity of labor of type i supplied, and v (ht (i)) represents the 4 This exception occurs when the policy parameter takes on one particular value, a value which leads to characteristic roots of the solution which are unity. Such "knife-edge" cases are ignored by Woodford as non-generic (See Woodford 2003, p. 254). 5 See Woodford (op. cit.), especially chapters 3 and 4. 6 We ignore uncertainty and money demand; neither are important for the issue of indeterminacy. We introduce money demand via money in the utility function in a later section.
3
disutitlity of supplying labor of type i. Associated with Ct is the price index 21 Z 1 Pt = 4 [pt (i)] 0
31=(1
)
di5
where pt (i) is the nominal price of good i: A …rst - order condition for constrained household utility maximization is the familiar Euler condition: 1 + it =
ex U 0(Ct+1 ) U 0(Ct )
1
Pt ex Pt+1
1
where the superscript "ex" denotes the expected value of a variable. The …rst - order condition for the household’s optimal of labor supply is: wt (i) v0(ht (i) = ; U 0(Ct ) Pt where wt (i) is the nominal wage paid for labor of type i. Output of each good is related to inputs by the production function: yt (i) = f (ht (i)); h0 > 0; h00 < 0: Assuming equilibrium in factor and goods markets, the following relationship holds between real marginal cost, st (i); quantity supplied, yt (i), and aggregate Z1 output, Yt yt (i)di : 0
st (i) =
v0 f 1 (yt (i)) u0 (Yt )
f 0 (f
1 : (yt (i)))
1
Woodford developed the concept of the natural rate of output by solving the above model under the assumption of perfect price ‡exibility. Assuming …rms know that relative demand is described by yt (i) = Yt
pt (i) Pt
;
each …rm would choose a markup over marginal cost as its optimal relative price: pt (i) = Pt 1
s(yt (i); Yt ):
Hence, the relative supply of good i would be yt (i) Yt
1=
=
1 4
s(yt (i); Yt ):
In equilibrium, yt (i) is identical for all i and thus equals Yt . Hence, equilibrium output in this ‡exible - price case, denoted as Ytn where the n is mnemonic for "natural," is implicitly de…ned by the condition 1=
s(Ytn ; Ytn ):
1
The canonical New Keynesian model doesn’t assume perfect price ‡exibility, though. Rather, it assumes that only a fraction (1 ) of randomly chosen …rms get to change their price each period. It turns out that …rms that can change their price in period t will all choose the same value, denoted by pt . This implies that a log - linear approximation of the relation between the price index in period t; namely Pt ; and the optimal (identical) new price, pt ; is given by the following equation: ln Pt =
ln Pt
+ (1
1
) ln pt :
(1)
The common new price of pt must satisfy the following log - linear approximation of …rst - order condition for maximization of the present discounted value of pro…ts: TX =1 T t ( ) [ln pt ln Pt xT ] = 0 (2) T =t
where xT is the output gap, de…ned as the di¤erence between the percentage deviation of output from its steady - state value and the percentage deviation of the natural rate of output from its steady - state value.7 Letting a "hat" denote a percentage deviation form steady - state value, we have: xT
YbT
YbTn :
Equations (1) and (2) can be manipulated to create an equation representing a "New Keynesian Phillips Curve": t
where
= xt + (1
ex t+1 :
)(1
(3) )
and t ln Pt ln Pt 1 : This New Keynesian Phillips Curve is one of the three structural equations that constitute the canonical New Keynesian model. A second structural equation comes from the log - linearization of the Euler equation for the representative household and the assumption of goods - market equilibrium, i.e., the assumption that Ct = Yt . This relationship, sometimes known as the New Keynesian IS curve, is given by the equation: xt = xex t+1
(bit
ex t+1
rbtn )
(4)
7 Woodford (op. cit.) has a coe¢ cient denoted by on xT . Whether this coe¢ cient is di¤erent from unity is important for some aspects of the model, but it is not for the question of determinacy. Hence, for algebraic transparency, we set it to one (1).
5
where is the intertemporal elasticity of substitution and rbtn is the "natural" rate of interest, i.e., the rate of interest that would satisfy the Euler equation under the assumption of fully ‡exible price - setting. Fully ‡exible price - setting would in turn imply that output would be at its natural level and in‡ation would be zero. Hence, rtn is implicitly de…ned as the rate of interest that solves the n Euler equation with t+1 = 0; Yt = Ytn ; and Yt+1 = Yt+1 : 1
n U 0(Yt+1 ) U 0(Ytn )
1
1 + rtn =
:
To complete the New Keynesian model, it is usually assumed that the monetary authority controls the nominal interest rate by following a Taylor rule:8 bit = it +
t
+
(xt );
x
;
0
x
(5)
where and x are constants. For simplicity of exposition we assume through the remainder of the paper that it = rbtn and = 0.9 These three structural equations of the New Keynesian model can be rewritten as the following coupled di¤erence equations: ex t+1 ex xt+1
= a11
t
+ a12 xt
(6)
= a21
t
+ a22 xt
(7)
where a11
=
a21
2.2
1
1
; a12 = 1
=
( ); 1
; a22 = 1 +
+
x
:
Solution
10 ex By Assume perfect foresight, i.e., assume ex t+1 and xt+1 = xt+1 . t+1 = well - known methods, it is known that (with the exception of parameter values which produce repeated real roots) the solutions to the model are:
= A1 (
t 1)
+ A2 (
t 2)
(8)
xt = B1 (
t 1)
+ B2 (
t 2)
(9)
t
where 2
1
and
2
[1 + ( )
satisfy the characteristic equation 1
(1 +
)+
x =4]
+
1
f1 + (
+
x )g
=0
(10)
8 Taylor (1993) argues that central bank behavior can adequately be described by the assumption that they follow a simple policy rule that links the interest rate to output and in‡ation. 9 The assumption of = 0 makes the algebra much simpler, and is innocuous for the analysis of indeterminacy because variation in x is all that is necessary to produce both determinant and indeterminant solutions. 1 0 In the absence of uncertainty, this is the same as rational expectations. Again, the issue of indeterminacy arises under the same conditions whether or not uncertainty is put into the model.
6
and the values of A1 ; A2 ; B1 ; and B2 satisfy A1 =
a12 B1 ; a11 1
a12 B2 a11 2 Because there is no state variable in this model, there are no initial conditions that give as exogenous the initial values of either of the two variables. The only boundary condition is an assumption of a bounded output gap xt , i.e., xt 2 [xu ; xl ], where xu and xl are …nite positive numbers.11 As Woodford shows, if 1 x >1 A2 =
then the roots of the characteristic equation satisfy the following set of inequalities: 1 < 1 < 2: In this case, the only rational expectations solution which doesn’t lead to ever - increasing absolute values of the output gap (and in‡ation) is one in which A1 = A2 = 0 (which implies B1 = B2 = 0). The bounded solution is thus t = xt = 0 8t. If 1 x < 1, then 0 < 1 < 1 < 2: With boundedness of xt as the only boundary condition, this implies that A2 = B2 = 0. But the boundedness restriction leaves possible a range of values for A1 (and of course associated values of B1 ), which in turn means there are an in…nite number of possible solutions, each indexed by a di¤erent value of A1 : t
t
xt
= A1 ( 1 ) ; t = B1 (A1 ) ( 1 ) :
Note that one solution is t = xt = 0 8t. This is also the MSV solution. Note also that all solutions in this case have trajectories of t and xt that move towards the origin of the x plane along the ray a12 xt : t = a11 1 Because of the assumption of an in…nite time horizon, any initial values 0 and x0 , so long as x0 2 [xu ; xl ], are consistent with equilibrium. We should note that when +1 = 1, and the x = 1; then 1 = 2 solution is: t
xt
= A; = B
1 1 This
boundedness is an obvious implication of a …nite labor supply. Following Blanchard and Kahn (1980), most discussions of determinacy simply impose boundedness of solutions as a condition to be met by any rational expectations equilibrium.
7
where A is an arbitrary constant and B = A(1a12a11 ) .12 As we will see, this is the only set of parameter values for which indeterminacy remains in the …nite horizon model, and is thus "non - generic."
3
A …nite - horizon analogue
3.1
Structure
Consider as an alternative a …nite-horizon version of the model, in which the world is assumed to end at time . Because the Woodford model is based upon optimizing behavior, the behavior of pro…t - maximizing …rms at time provides a natural boundary condition. We will put a tilda (e) above …nite - horizon model variables to distinguish them from their counterparts in the in…nite - horizon model. In Woodford’s model, the New Keynesian IS equation, i.e., the Euler equation relating optimal consumption choices in adjacent time periods, is una¤ected by assuming a …nite horizon. But the New Keynesian Phillips Curve equation, derived from optimal staggered pricing behavior as in Calvo (1983), di¤ers. In the …nite horizon model the equation that describes the pro…t - maximizing choice of price becomes: T = h i X T t ( ) Et log pet log PeT x eT = 0: T =t
By steps analogous to those described by Woodford, we can combine a transform of this equation with the equation describing the evolution of the price index into the following non - autonomous di¤erence equation:13 et
1
=
1 1 + r + r2 + ::: + r( 1
+
t)
r + r2 + ::: + r( t) 1 + r + r2 + ::: + r( t)
where r
x et+1
et+1 :
(11)
:
Note that as ! 1; this equation reduces to the in…nite - horizon aggregate supply relation: lim et =
(1
) (1
)
!1
lim x et + !1
lim et+1 : !1
At period , …rms that have won the Calvo lottery and can re - set their prices have no future to worry about. Thus, their optimal price is given by:
1 2 The
log pe = log Pe + x e :
general solution in the case of repeated real roots involves a term Ct in this model, it turns out that C = 0: 1 3 Details are in the appendix.
8
(12) t;
because
=1
When combined with the expression that describes the evolution of the price index, this yields the boundary condition: 1
e =
x e :
(13)
The evolution of the variables in the …nite - horizon model is thus described by a New Keynesian IS curve, a New Keynesian Phillips Curve with time varying coe¢ cients, and the above boundary condition.
3.2
Solution
Using the boundary condition (13), the …nite - horizon New Keynesian Phiillips Curve (11), the IS curve (4), and the Taylor rule (5), we can use backward induction from time to establish a time - varying proportional relationship between in‡ation and output: et = at x et ; t
1
; a
:
(14)
Using this in conjunction with the IS curve implies: x et =
[1 + at+1 ] x et+1 ; t < [1 + x ]
:
(15)
Using the time - varying proportionality relationship between in‡ation and the output gap in conjunction with the …nite - horizon New Keynesian Phillips Curve (11), we also have h i 1 r+r 2 +:::+r ( t) at+1 1+r+r 2 +:::+r ( t) ix x et = h et+1 : (16) 1 1 at ( t) 2 1+r+r +:::+r Equating coe¢ cients on the above two di¤erence equations yields the following recursive relationship that holds for any t < : at =
t
+
(1 + x ) at+1 1 + at+1
t
(17)
where 1 t
1+r+ 1
t
1 + ::: + r(
r + r2 + ::: + r( t) 1 + r + r2 + ::: + r( t)
Hence, the solution for x et is: "n= x et =
r2
Yt
n=0
1
(1 + a (1 + 9
n) x)
#
x e
t)
;
:
(18)
where at is generated by the recursive relation (17). The solution for et is found by multiplying the solution for x et by at : # "n= t 1 Y (1 + a n ) x e (19) et = at (1 + x ) n=0
4
The limit of the …nite - horizon solutions as ! 1:
First we establish that at converges as (
t) goes to in…nity:
Lemma 1 For non - negative values of the policy parameter ), x = 1 lim at = a+ (
x
(except for
t)!1
where a+ is the unique positive solution to the quadratic equation: a2 + a[1
(1 +
x)
]
= 0:
Proof. See Appendix. Knowing that at converges to the positive number a+ , we see from (18) that a+ the limiting behavior of x et and et depend critically on whether 1+ is greater 1+ x
a+ or less than one (1). If 1+ 1+ x > 1, the limit of the solution described by (18) and (19) will involve multiplication of (at most) a …nite string of positive fractions times an in…nite string of numbers greater than one (1). Hence, in this case, je xt j gets greater and greater than je x j as t gets farther and farther away from . This implies that for any non - zero value of x e , there is a value of t su¢ ciently large such that x et violates the boundedness assumption. That is, unless x e = 0; lim !1 x et = 1. Because et = at x et , this also implies that lim !1 et = 1. This implies that it must be that x e = 0, which in turn implies that x et = et = 08t. Put another way, in this case, the limit of the …nite - horizon solution as the horizon goes to in…nity is the in…nite - horizon solution xt = t = 0 8t: a+ On the other hand, if 1+ 1+ x < 1; the limit of the solution described by (18) and (19) will involve multiplication of (at most) a …nite string of positive numbers greater than one (1) times an in…nite string of numbers less than one (1). Hence, in this case, no matter the value of x e , lim( t)!1 x et = lim( t) et = 0: Put another way, in this case as well as in the previous one, the limit of the …nite - horizon solution as the horizon goes to in…nity is the in…nite - horizon solution xt = t = 0 8t: a+ Only in the case in which 1+ 1+ x = 1 would the limit of the …nite - horizon solution be indeterminate. The following proposition states that in fact this only occurs for the single policy parameter value x = 1 :
10
Proposition 2 For non - negative values of the policy parameter x such that ; the limit of solutions to the above …nite - horizon canonical model is x 6= 1 the solution xt = t = 0 8t of the analogous in…nite - horizon canonical model. a+ Proof. What we will show is that 1+ 7 x . That is, for 1+ x 7 1 as 1 parameter values that give rise to indeterminacy in the in…nite - horizon model, 1+ a+ < 1, and for parameter values that give rise to a determinate solution in 1+ x
a+ the in…nite - horizon model, 1+ 1+ x > 1: Consider …rst solutions to the in…nite - horizon model which lie along the parametric path: t = axt :
Substituting this into equations (6) and (7), we get equations of motion for xt along this path: 1 xt+1 = (1 )xt : a xt+1 = 1 + x + 1 ( a) xt : Equating the coe¢ cients on xt from the two equations implies that there is one positive and one negative value of a that solves the quadratic equation: a2 + a[1
(1 +
x)
]
= 0:
This quadratic equation is identical to that which describes the stationary value of the recursion relationship in the …nite - horizon model for at . Thus, the positive root is a+ . First consider the case in which x < 1 . The only solutions that converge to (0; 0) are t
xt
= A1 ( =
1
t 1)
;
a11 a12
A1 (
t 1)
:
Hence, any solution lies along the parametric path a12 xt t = a11 1 1 Now, a11 = > 1 and 1 < 1 so the denominator in the above expression is 1 negative. Because a12 = < 0, this means a12a11 > 0. Hence, along 1 this path, t = a+ xt : Substituting this back into equation (4), the IS equation, we have (1 + a+ ) xt = xt+1 : 1+ x Because we know that along this path xt+1 is closer to zero than xt , it must be a+ ) et = 0: that (1+ t)!1 x 1+ x > 1. Hence, lim( Now consider the case in which x > 1 . In this case, no in…nite horizon solution can converge to (0; 0); so along the path t = a+ xt it must be a+ ) < 1: Hence, lim( t)!1 x et = 0: that (1+ 1+ x
11
5
Learning
One response to indeterminacy in rational expectations models has been to invoke a selection criterion based on some form of "learnability." Most of the learning literature assumes economic agents know something about the model: perhaps that it is linear, but with unknown parameters or with unknown solution values for variables. A key feature of all these schemes is that agents’ beliefs iterate on the basis of past observations and whatever knowledge of the model they are assumed to have, e.g, linearity of functions. Much of the literature focuses on least - squares learning, which has been shown by Evans and Honkapohja (2001) to be frequently isomorphic to the more user-friendly concept of e - stability. For the canonical New Keynesian model, Woodford (2003 p. 269) shows that none of the equilibria in the in…nite - horizon indeterminacy case are iteratively e - stable.14 This means that in such cases if agents start with arbitrary ex exogenous beliefs about values of ex t+1 and xt+1 , say 0 and x0 , a "natural" iterative updating scheme fails to lead to a convergence of beliefs to any of the rational expectations equilibria. In fact, beliefs in this case explode. Thus, for people who view learnability as the key selection criteriwhether or not indeterminacy is an artifact of the choice of an in…nite horizon might be viewed as less interesting than The iterative learning scheme used can be viewed as an "eductive" learning process in which agents reason about what must be the behavior of expectations, given what they know about the model and how they assume agents form expectations.15 To see how agents are assumed to update these beliefs, we can rewrite the structural model embodied in equations (6) and (7) as = b11
ex t+1
+ b12 xex t+1
(20)
xt = b21
ex t+1
+ b22 xex t+1
(21)
t
where b11 b21
1 b12 = b22
a22 a21
a12 ; a11
a11 a21
a12 : a22
Substituting the exogenous expectations 0 and x0 into this structural model would generate "actual" values of t and xt : = b11
0
+ b12 x0
(22)
xt = b21
0
+ b22 x0 :
(23)
t
The "beliefs updating scheme" assumes agents use these values of t and xt as their beliefs about next - period values. Hence the evolutions of beliefs is generated by the following coupled pair of di¤erence equations: N +1
= b11
1 4 See
N
+ b12 xN
(24)
Evans and Honkapohja (2001) for the complete discussion of the relation between e stability (iterative and otherwise) and learnability. 1 5 See Evans and Honkapohja 2001 p. 372.
12
xN +1 = b21 with
0
N
+ b22 xN ; N = 0; 1; 2; :::
(25)
and x0 given. The solution of this coupled system is given by N
N
N
= A1 (b1 ) + A2 (b2 ) N
N
xN = B1 (b1 ) + B2 (b2 ) ; N = 0; 1; 2; :::; where b1 =
1 1
and b2 =
1 2
0
(27)
, and the values of A1 ; A2 ; B1 ; and B2 satisfy b1 b11 ; b12 B1
q1 A1 = B1 ; q1 =
b2 b11 : b12 B2
q2 A2 = B2 ; q2 = With mined as
(26)
and x0 given as exogenous, this implies that A1 and A2 are deterA1 =
0 q2
x0
q2 x0 A2 = q2
q1 0 q1 : q1
Because the roots b1 and b2 are inverses of the roots of the characteristic equation of the in…nite - horizon model, the above system is unstable in the indeterminate case in which 1 < 1. That is, in this case b1 > 1 and b2 < 1: As a consequence, beliefs explode unless A1 = 0. The eductive learners in this model would thus conclude that the only permissible beliefs about initial values of 0 and x0 would be those which made A1 = 0, namely, initial beliefs that lie along the ray: x0 = q2
0:
With this constraint on initial beliefs, limN !1 N = limN !1 xN = 0. Hence, beliefs would converge to the MSV solution. In the determinate case, 0 < 1 1 < 1 and 0 < 2 1 < 1, and beliefs converge to (0; 0) regardless of initial beliefs.
6
Adding money demand
To this point, our analysis has ignored the possibiblity of a transactions - based demand for money. We did this in part to keep focus on the basic New Keynesian model with Taylor rules in which, even in the presence of money demand, the behavior of the key endogenous variables xt and t are una¤ected by the presence of money demand. We also did this because such a model is consistent with a "cashless society" (see Woodford, op. cit., p. 31) which is perhaps on the technological horizon. We now introduce money demand in a …nite - horizon model. This amendment adds a di¤erent boundary condition to the model, one that replaces the 13
assumption of boundedness. We show that addition of this new boundary condition leads to limiting behavior of solutions that converge to the unique …nite - horizon MSV solution even in the non - generic case in which the Taylor rule was such that the roots of the characteristic equation of the in…nite - horizon solution are repeated and equal to unity. That is, even when x = 1 , the unique limit of the …nite - horizon solution is the MSV solution xt = t = 0 8t.16 Consider the intertemporal household maximization problem with real balances in the utility function and with separable utility. The ‡ow budget constraint at time t < is Pet Yet + (1 + it
e
ft +M
1 )Bt 1
et + M ft + B et : = Pet C
1
e = 0. At , though, B First - order conditions for t < are the familiar Euler equation and the marginal condition that determines the demand for money function: ( ) 1 e ex ) U 0(C Pet t+1 1 1 + it = et ) U 0(C Peex t+1
Um (m e t) =
it 1 + it
et ): UC (C
where m is real balances and Um (m) and UC (C) are the partial derivatives of the utility function. At ; though, we have e ): Um (m e ) = UC (C
(28)
Log - linear approximations of the money demand function would thus be ln Pet ln Pe
ln Mt ln M
et xx
=
e xx
=
b
i it ;
:
t< ;
At time 1, we assume the monetary authority is …nished: it has set its last interest rate. This means that f =M f M
1:
(29)
This implies the following boundary condition: et =
et xx
+(
x i x )e
x
1:
(30)
The other boundary condition that e = a x e can be combined with this to derive the following relation between x e 1 and x e : x e =
(
i x)
x
(a +
x)
x e
1:
(31)
1 6 In fact, the result is stronger in some sense: every …nite - horizon solution is x et = et = 0 8t < :
14
Now, it is also the case that x e
1
=
[1 + at+1 ] x e : [1 + x ]
(32)
There are only two ways in which both of these equations can be true: either x = 0 (which implies that xt = t = 0 8t ), or a + (
x
x i x)
=
1+ a : 1+ x
(33)
If this equality is satis…ed, then the behavior of x et and et and at is identical to what it would be in the non - monetary …nite - horizon model, and the proposition that the limit of the …nite - horizon solution being the in…nite horizon solution of xt = t = 0 8t applies except if x = 1 . But if x = 1 , could the above equality be satis…ed? The following proposition states that the answer is no. Proposition 3 For non - negative values of the policy parameter x ; regardless of whether x + 1 ; the limit of solutions to the above money - in -the - utility - function …nite - horizon canonical model is the solution xt = t = 0 8t of the analogous in…nite - horizon canonical model. Proof. See appendix. The addition of a transactions technology embodied in money in the utility function of the representative agent gives rise to a "sharper" boundary condition than does just the assumption of boundedness, and hence eliminates the possibility of indeterminacy even in the "non - generic" case for which the in…nite - horizon model has equal real roots (equal to unity) to the characteristic equation.
7
Conclusion
In the context of the canonical New Keynesian model as developed by Woodford (op. cit.), we have shown that for almost all feasible parameter values, some of which give rise to indeterminacy in the in…nite - horizon model, the limit of the …nite - horizon equilibria is the unique MSV equilibrium of the in…nite - horizon model. For two reasons, we think this result will extend to the other models in this literature for which indeterminacy is an issue. First, we believe multiple equilibria arise in these models because of the conjunction of the assumption of rational expectations and an in…nite time horizon. As noted by McCallum (2003, p.1157), all these models are dynamic and feature expectations of future endogenous variables in their structural equations. As argued in Driskill (2005), such a combination generically leads to solutions that need one more boundary condition in order to have a unique solution. In brief, these models have a structure in which at every time period one equation links current variables to 15
expectations of a future variable. The rational expectation of the future variable (because agents are assumed to "know the model"), is found by scrolling this equation forward one period. Thus, over any n periods, there are n equations st and n + 1 endogenous variables, the (n + 1) variable being the expectation in period n of the value of the variable in the n + 1 period. To have as many equations as unknowns, we need one more equation. In many models, the other equation is provided by a boundary condition that requires a bounded solution. In these models, only one of the solutions is bounded (see. for example, the original Muth (1961) inventory model). More problematic, though, are those models in which there are multiple bounded solutions. In …nite - horizon models, though, economic analysis usually provides a boundary condition based on optimal behavior at the end of the horizon that provides the necessary equation for a determinate solution. New Keynesian models are built on optimizing behavior, and thus satisfy this criterion. A second reason we think use of a …nite - horizon model will eliminate indeterminacy is the existence of other examples in which this is the case. In Driskill (2005), a variety of in…nite - horizon dynamic rational - expectations models in which there were multiple stable equilibria were shown to have only one equilibrium that was the limit of an analogous …nite - horizon solution. These results suggest that one can use the limit as the horizon gets arbitrarily large of …nite - horizon solutions as a selection criterion for choosing among the many in…nite - horizon solutions. Acknowledgement 4 I thank Michael Woodford for useful comments. Appendix Derivation of the …nite - horizon AS curve The pro…t - maximizing condition is T = X
(
T
)
t
Et [log pt
T =t
16
log PT
xT ] = 0:
Rewrite as ln pt
+::: +(
)
0
De…ne: (ln pt pbt
t
t+1
xt+1 5
t+2
}| { z }| { z ln Pt + ln Pt ln Pt+1 + ln Pt+1 ln Pt+2
(A.1) 3
xt+2 5
3 t+2 t+1 z }| { z }| { ln Pt + ln Pt ln Pt+1 + ln Pt+1 ln Pt+2 + ::: 7 7 t+T 5 z }| { ln Pt+T 1 ln Pt+T xt+T
2
6 ln pt 6 4
ln Pt )
3
t+1
}| { z ln Pt + ln Pt ln Pt+1
4ln pt
xt +
2
2 ) 4ln pt
+(
=
ln Pt
2
pbt : Then,
1 + r + ::: + r
t
xt + rxt+1 + r2 xt+2 + ::: + r
=
2
t
+ r + r + ::: + r 2
3
+ r + r + ::: + r t
+::: + r
t
x
t+1 t
t+2
:
So, pbt+1
1 + r + ::: + r
(t+1)
h
xt+1 + rxt+2 + r2 xt+3 + ::: + r (t+1) x h i + r + r2 + ::: + r (t+1) t+2 h i + r2 + r3 + ::: + r (t+1) t+3 h i +::: + r (t+1) :
=
Hence, r
pbt+1
1 + r + ::: + r
(t+1)
=
rxt+1 + r2 xt+2 + ::: + r + r2 + r3 + ::: + r 3
t
+ r + ::: + r +::: + r
t
t
t
x
t+2 t+3
:
Comparing coe¢ cients, we see that pbt
1 + r + ::: + r
t
= xt + r + r2 + ::: + r +r
17
pbt+1
t
1 + r + ::: + r
t+1 (t+1)
:
i
Now, pbt =
1
t:
Hence, (let n = t
t): 1 r 1 rn+1 r rn : 1 rn+1
1
= xt
1
+
t+1
Proof of Lemma 1 Note …rst that at is bounded: value of t is and the maximum i h the minimum x )at+1 is an increasing concave function of value of t is ; and t + t (1+ 1+ at+1 h 1+ x at+1 . Hence, at 2 ; + . Furthermore, as ( t) ! 1; t ! i h x )at+1 and t ! , so the function t + t (1+ becomes arbitrarily close 1+ at+1 i h (1+ x )at+1 , which is an increasing concave function of at+1 with to + 1+ at+1
1+ x : The phase diagram minimum value and maximum value + depicting the behavior of at as a function of at+1 with this limiting function of at+1 is depicted in Figure 1.Setting at = at+1 = a in this limiting recusive relationship gives us the quadratic equation for which a+ is the unique positive root.
Proof of Proposition 2 If
a + (
x
x i
x)
=
1+ a ; 1+ x
then, upon rearrangement, we have + + x i
a =
x
1
x x (1
+
i)
:
Now, indeterminacy in the …nite horizon model occurs only when 1 x
= 1:
Remember, =
1
(1
);
so that this condition can be re - written as x
or
1
=
1
=
1 1
x
18
1 1
:
(P2.1)
Remember, a = 1 : So, if x 6= 0, upon substitution of into (P2.1), we have: z
a
x
}| 1 1
{
+ + x i
=
x
1
x x (1
Cancelling the x on both sides, and noting that 0 < this could only be true if 0
