Stochastic saddlepoint systems Stabilization policy and the stock market

JOURNAL OF Economic b namics

ELSEVIER Journal of Economic Dynamics and Control

19 (1995) 279-302 & Control

Stochastic saddlepoint systems Stabilization policy and the stock market

Marcus Miller”*b, Paul Weller*~“*”

= CEPR, London, UK b Department of Economics, University of Warwick, Coventry CV4 7AL. UK

‘Department of Finance, University qf Iowa, Iowa City. IA 52242, USA

(Received March 1993; final version received August 1993)

Abstract

We examine the effects of introducing stochastic shocks into a linear rational expectations model with saddlepoint dynamics generated by a forward-looking asset price. We derive the fundamental differential equation governing the path of the asset price as

a function of the ‘sluggish’ variable. The equation does not admit closed form solutions in general, but we provide a complete qualitative characterization of the solution paths which are symmetric about equilibrium. These are the relevant solutions to consider in the presence of symmetric boundary conditions.

We present an application of the analysis to fiscal stabilization policy where public expenditures are only adjusted when GNP moves outside a threshold around a target

level. The impact of such a policy on the level of the stock market is considered, and it is shown how the market will anticipate such state-contingent policy changes. Whether the expectations are stabilizing or not is shown to depend on whether the stock market is positively or negatively correlated with GNP.

Key words: Stochastic shocks; Rational expectations; Saddlepoint dynamics; Stabili- zation policy; Stock market JEL classijication: E44

* Corresponding author.

We are grateful for the research support provided to the authors by the CEPR. An earlier version of

this paper was circulated as CEPR Discussion Paper No. 308. We are grateful to Avinash Dixit, Paul

Krugman, and two referees for their comments, and to Alan Sutherland for research assistance,

funded in part by the House of Commons Treasury Committee.

0165-1889/95/$07.00 0 1995 Elsevier Science B.V. All rights reserved SSDI 016518899300782 Y

280 M. Miller, P. Weller / Journal of Economic Dynamics and Control 19 (199.5) 279-302

1. Introduction

Perfect foresight models with saddlepath dynamics have found wide application in contexts where a flexible asset price is coupled with a ‘fundamental’ whose speed of adjustment is slower. The ‘overshooting’ model of exchange rate determination (Dornbusch, 1976) and the analysis of interactions between output and stock market value (Blanchard, 1981) are just two such examples.’

One of the great attractions of these models has been the fact that their qualitative characteristics could be analyzed using the simple geometric structure of a phase diagram. What we show in this paper is that there exists an equally simple geometric characterization for stochastic saddlepath models, in which the fundamental is subject to random disturbances.

Our analysis can be viewed as a generalization of the approach pioneered by Krugman (1991) in the context of exchange rate dynamics and subsequently extended in a variety of ways. He was concerned with the specific problem of describing the movement of the exchange rate when the monetary authorities intervene intermittently to contain the rate within a predetermined band. Using continuous-time stochastic techniques in a rational expectations framework he obtained a simple explicit solution for the rate as a stationary, nonlinear function of fundamentals. This was an example of a much more general class of problems in economics and finance, namely to describe the impact of state- contingent, and therefore probabilistic, changes of policy or regime on forward-looking variables.

The results of Krugman were obtained by assuming a recursioe structure for the underlying model of exchange rate determination: the exchange rate (viewed as an asset price) is influenced by expected future fundamentals, but does not itself affect the evolution of these fundamentals. Such a recursive structure is however very restrictive for an asset price as important for the economy as the exchange rate. (This is equally true for the case of stock market value or the long-term bond rate.) Indeed, the essence of models with saddlepath dynamics is the existence of endogenous feedback effects which generate stable adjustment to equilibrium.

For this reason, we consider the more general class of nonrecursive models, which allow for endogenous dynamics in the fundamental. Our qualitative analysis is thus applicable in principle to a broad class of asset pricing models. There is an additional reason why our geometric techniques are useful. In contrast to recursive models, the nonrecursive structure we work with does not in general admit closed-form solutions. The only other alternative is to have recourse to numerical methods; but these require particular parameter values and give no clue as to the generality of the qualitative features associated with them.

1 See Blanchard and Fischer (1989, Chs. 5,10) for further examples.

M. Miller, P. Weller J Journal of Economic Dynamics and Control 19 (199.5) 279-302 281

We have already shown elsewhere (Miller and Weller, 1991a, b) that a quali-

tative approach can be fruitfully applied to the analysis of exchange rate bands. Here we provide an illustration of the use to which our analysis can be put by considering the effects on the stock market of a fiscal stabilization rule subject to thresholds. In contrast to our earlier work, we also allow explicitly for the presence of risk-averse agents. The model used is a stochastic version of that presented in Blanchard (1981).

In the next section we present the general form of the linear stochastic model and use a stochastic version of the model of Blanchard as an illustration. For

the purposes of comparison we first describe the effects of a threshold fiscal stabilization rule in the absence of shocks. Then we present a qualitative

characterization of all solutions to the stochastic model which pass through the saddlepoint of the deterministic system. We also prove the existence of

solutions asymptotic to the stable saddlepath. These solutions turn out to be important in the analysis of switches of stochastic regime.

In Section 3 we show specifically how our techniques can be applied to describe the effects of fiscal stabilization in the stochastic stock market model,

and contrast the results with those for the deterministic case. We find in particular that the stock market anticipates the effects of the threshold policy even before it is triggered; whether these anticipations help in stabilizing GNP or not depends crucially on how sensitive dividends are to GNP. Section

4 contains concluding comments.

2. Qualitative analysis of linear saddlepoint systems with white noise disturbances

2.1. The linear stochastic model

Our analysis is restricted to linear saddlepoint systems. The essential features are, first, a linear stochastic differential equation expressing the evolution of the economic fundamental as the weighted sum of three components - a white noise disturbance, the current value of the fundamental, and an asset price - and, second, the assumption that this asset price is a rational expectations forecast of future fundamentals. (Thus the asset price is random only through its dependence on fundamentals.) Formally,

dx(t) = a(x(t) - x*)dt + p(s(t) - s*)dt + cdz(t),

where

(1)

s cc

s(t) = E, - y(x(r) - x*)e-d(r-‘)dr + s*, (2) f

282 M. Miller, P. Weller / Journal of Economic Dynamics and Control I9 (199.5) 279-302

with variables and operators defined as

x(t) = the economic fundamental, s(t) = the asset price, dz(t) = increment to a standard Wiener process, E, = an expectation operator conditional on information at time t.

An asterisk denotes an equilibrium value.

On differentiating (2) with respect to t one obtains

E,(ds(t)) = 7(x(t) - x*)dt + 6@(t) - s*)dt,

where

E, (ds(t))/dt denotes lim E, s(t + E) - s(t)

E-0 E

So the evolution of the fundamental and the expected evolution of the asset price can be summarized as

where for convenience we drop the explicit dating of the variables x, s, z.

2.1.1. An illustration: Blanchard’s model of the stock market As an illustration of the general structure described above, we present

a stochastic version of the model in Blanchard (1981) relating stock market prices to the level of real activity in the economy.

The equations of the model take the following form:

m = a,x - a2i,

y = ujx + uqs + g, 0 < u3 -c 1,

dx = a,(y - x)dt + cdz,

c = a5 + a6x,

E(ds) = [(i + pB,cr2)s - c]dt,

(4)

(5)

(6)

(7)

(8)

M. Miller, P. Weller / Journal of Economic Dynamics and Control 19 (1995) 279-302 283

where

m = level of money balances, x = output, i = short-term nominal interest rate, y = level of demand, s = stock market value, CJ = index of fiscal policy t’ = stock market dividends

We assume that prices are fixed and all variables are measured in real terms. Eq. (4) gives the condition for money market equilibrium. Eq. (5) specifies the determinants of demand, namely the level of output, the value of the stock market (through a wealth effect), and an index of fiscal policy which we specify more precisely beiow.2 E q. (7) postulates a simple relationship between output and dividends, and in (8) it is assumed that arbitrage equates the rate of return on stocks to the yield on short-term bonds plus a risk premium.3 We model the risk premium in a form consistent with a simple version of the consumption-based capital asset pricing model of Breeden (1979). It takes the form p8,c?, where p is an index of risk aversion, 8, is the slope of the stable saddlepath (derived below), and o2 is the instantaneous variance of output. The term H,YaZ captures the covariance between aggregate consumption and the stock market4

If the system above is linearized about the long-run equilibrium of the deterministic system (x*,s*) we can characterize it in the following way:

(9)

where

A= - a,(1 - a3) a4a7

(al/a2 + pdsc2)s* - (16 i* I

and

i* = (at/a2)x* - (m/a2)

The associated deterministic system is obtained by setting C-J to zero.

‘There is a lag between demand and output produced as is described in Eq. (6).

3 We are grateful to a referee for the suggestion that we incorporate a risk premium into the arbitrage

equation.

4Strictly speaking, 0,~’ is the instantaneous covariance between output and the stock market in the

absence of any stabilization. But since output and consumption are perfectly correlated here. the

distinction is of no consequence. And to retain tractability we ignore the direct effect upon the risk

premium of anticipated stabilization.

284 hf. Miller, P. Weller 1 Journal of Economic Dynamics and Control 19 (1995) 279-302

2.1.2. The eflects offiscal stabilization in a deterministic framework Consider now in its deterministic formulation (a = 0) how this model can be

used to analyze the impact upon the stock market of a fiscal policy rule involving a discrete adjustment in taxation or government expenditure at some predetermined level for GNP. The rule is designed to stabilize the economy, and so will involve a tightening of fiscal stance when output is at some point above its long-run equilibrium value and a relaxation when output has fallen a given amount below equilibrium. Specifically, g = S for 5 I x I X, and g = S - a8(x - x*) otherwise.

Characterizing the dynamic path of adjustment to equilibrium in the presence of such a stabilization policy is straightforward. Absence of riskless arbitrage requires that there be no perfectly anticipated jump in stock prices at the point where the policy adjustment occurs. In other words, the path of adjustment to equilibrium must be continuous. However, it must also at all points satisfy the basic arbitrage requirement given in Eq. (8). The result is illustrated in Fig.1 where it is constructed with reference to the two stable saddlepaths labelled SS and S’S’. The former is the convergent solution for the model with no intervention at all; the latter is the stable solution if there is continuous intervention, so the system takes the form

where

A’ = - a7(1 - a3 + as) a4a7

(al/a2)s* - a6 1 i* ’

The only differences between A in (9) and A’ above are (i) in the upper left-hand element, which remains negative but increases in absolute value, and (ii) in the lower left-hand element, where the risk premium is set to zero. The coefficient a8 captures the countercyclical bias in government expenditure. This bias will, of course, increase the speed of return to equilibrium, as may readily be confirmed. Consequently, the stable saddlepath S’S’ will have a flatter slope than the saddlepath in the absence of any stabilization, shown as SS (so g = S - a8(x - x*), Vx). Intuitively, the reason for this is clear enough; with stabilization, departures from equilibrium will be less enduring, so the stock price will have less abnormal profits (or losses) to forecast.

Note that whether the stock market rises or falls as output increases (relative to its equilibrium level) depends upon whether (al/a2)s* 5 a& Both for SS and S’S’ the sign of 8,, the slope of the stable saddlepath, is determined as follows:

es50 as (al/a&* - a62 0.

M. Miller, P. Weller / Journal qf Economic Dynamics and Control 19 11995) 279-302 285

s X* X X

Fig. 1. A trigger policy for fiscal stabilization: The deterministic case.

We will adopt Blanchard’s terminology to distinguish between the two cases. The good news case corresponds to the situation where the stock market rises

with output (0, > 0) and the bad news case to the situation where it falls as output rises (0, < 0). Fig. 1 illustrates the good news case.

Note also that when the economy is at a point lying within the range [x, X] the share price is not affected by the threshold policy. The stable deterministic structure means that with this initial condition the trigger points will never be reached, so the economy adjusts smoothly to equilibrium along SS. Only when GNP is initially above or below the trigger points does one observe any effect of the threshold policy.

At a point x < x, for example, stock prices will be above SS, as the policy to

counter the recession acts to raise the flow of anticipated dividends. But it will not lie on S’S’, which describes the solution for continuous fiscal feedback. The appropriate solution which discounts the effect of the policy applied only below the threshold 3 is shown as BM, starting on SS at x = x but approaching S’S asymptotically as x falls beneath x. [BM is an integral curve of Eq. (10) where

y = S - a8(x - x*).1 Note that the share price solution shows no jump at x: as the switching off of the stabilizer will be fully anticipated. [The kink at M reflects the fact that g falls abruptly by aa(x* - x) as x hits the lower trigger.] Similar arguments apply for x > X, so the required solution takes the form AM MB meeting SS at x and X and approaching S’S’ asymptotically at both extremities.

A moment’s reflection will show that this cannot be the solution appropriate for the stochastic case, because in that case, even if output lies within the range

286 154. Miller. P. Weller / Journal of Economic Dynamics and Control 19 (1995) 279-302

[x,X], there will always be a possibility that a sequence of shocks will drive it outside that range. The anticipation of this by the market should influence stock prices even when g = S. How stock prices will be affected over the range [x, X] and the stochastic analogue to the nonlinear segment of the adjustment path in Fig. 1 will be discussed after we consider the nature of stochastic solutions.

2.2. The fundamental difSerentia1 equation of the stochastic system

We return to considering the linear stochastic model in the absence of discrete policy adjustments, as presented in Section 2.1. To simplify notation, we relabel variables so that x and s measure deviations from equilibrium. Without loss of generality we assume that the coefficients of the matrix A, c(, b, y, are negative (nonpositive) while 6, the discount factor, is positive (nonnegative). The slopes of the stable and unstable eigenvectors of A, denoted BS and fI,, respectively, are given by

8, = y (A, - 6) > 0, 6, = (/I” - ‘2)/j? < 0,

where 1, and A,, denote the two roots of the characteristic equation

A2 - (trace A)1 + det A = 0,

where det A = a6 - y/I is the determinant of A, assumed to be negative.’ In the deterministic case where cr = 0 the two eigenvectors are the only paths

connected to the origin. But when the system is subject to stochastic shocks, there turns out to be an infinity of other nonlinear solutions connected to the origin. To see this, let any particular solution be expressed as a function of x, so

s =f(x).

Then Ito’s Lemma implies that

E,[ds] =f’(x)E,[dx] + ;f”(x)dt.

Rearrangement and substitution yields

;fw = -f’W(@X + Bf(4) + (YX + m4),

(11)

(12)

5 Det A negative is a necessary and sufficient condition for saddlepath dynamics.


the fundamental differential equation to be satisfied by any solution to the linear

stochastic system. The qualitative nature of those solutions which pass through the origin is

shown in Fig. 2. First we note that such solutions are strictly antisymmetric about the origin. [It follows directly from the form of the equation that if f’(x) is a solution to (12) satisfying f(0) = 0, then f(x) = - f( - x)]. The linear solution corresponding to the stable arm of the saddlepath has a well-defined

asymptotic distribution and is the natural candidate for the ‘fundamental solution, obtained by imposing a suitable transversality condition. All the nonlinear solutions can be thought of as ‘bubble’ paths, along which the asset price will with positive probability deviate arbitrarily far from its fundamental

value. However, in the presence of anticipated regime changes, it is the bubble paths which characterize the relationship of asset price to fundamental. These

solutions are no longer true bubble paths because they are truncated by the regime changes, and so do not display the divergent behavior mentioned above. There is a parallel here with the deterministic case discussed in Section 2.1.2, where an anticipated policy change causes the system to move for a period along

an unstable path in the phase diagram. The parallel is not precise because Fig. 2 is not a phase diagram, in that the system moves randomly in both directions

along a solution path. We analyze in some detail later on the relationship between the phase diagram of the deterministic system and the solutions for the stochastic case.

Fig. 2. Stochastic solutions.

288 M. Miller, P. Weller /Journal of Economic Dynamics and Control 19 (1995) 279-302

The notion that rational asset prices are deterministic functions of stochastic fundamentals is, of course, familiar from much modern finance theory, e.g., the analysis of option pricing in Merton (1973). But as suggested in the introduction, the feature of stochastic saddlepoint systems which is peculiarly macroeconomic is that in general the process determining the ‘fundamentals’ is not autonomous but depends on the asset price itself. [Thus, from (1) and (2), dx = axdt + Psdt + adz which, given a solution s = f(x), becomes dx = axdt + /If(x)dt + adz so that, unless /l = 0, the diffusion process governing x will depend on the functionf(x).] It is precisely this ‘endogeneity of the fundamentals’ which makes it impossible in general to find closed form solutions and leads one to a qualitative treatment.

We now derive the qualitative treatment implicit in Fig. 2 in detail, proceed- ing from a local to a global analysis.

2.3. Local analysis: Curvature around equilibrium

If an eigenvector of matrix A is denoted [l, 0) the slopes of the stable and unstable eigenvectors may be obtained as the roots of the quadratic equation

p(B) = y + (6 - a)0 - /Xl’ = 0, (13)

and the curvature of stochastic solutions in the neighborhood of the origin may be established as follows. Let the function f”(x) be approximated to first order at the origin, so

$‘yx) N ; (f” (0) + f”’ (0)x)

= C -f”(W.O + Bf(W -f’(ONa + Bf’(W + 7 + ~.f'(O)lx

= Pm% (14)

letting 8 =f’(O). The first equality uses the fact that f”(0) = 0, which follows from (12) when f(0) = 0, and the second follows from differentiating (12) and evaluating at x = 0.

On our assumptions, p(B) must take the form illustrated in Fig. 3a, since p(0) = y < 0, and p(8) reaches a minimum at 19* < 0. This enables us to calculate the sign off” in a neighborhood of the origin, illustrated in Fig. 3b. As one


P(e)

Fig. 3a. Local analysis: The function p(H).

U

Fig. 3b. Local analysis: The sign of j”’ in a neighborhood of the origin.

proceeds clockwise round the origin, the sign off” changes with each crossing of an eigenvector and of the vertical axis.

2.4. Global analysis

Because of the symmetry of the solutions, it is only necessary to consider half the plane. Taking the right-hand half-plane, where x 2 0, we divide this into four regions as shown in Fig. 4. Regions A and B lie on either side of the stable saddlepath and above the line XX of (expected) stationarity for x. Here the expected movement of x is towards zero. In regions C and D, which lie either

290 M. Miller. P. Weller / Journal of Economic Dynamics and Control 19 (1995) 279-302

side of the unstable saddlepath and below XX, the expected movement of x is away from equilibrium. We consider each region in turn. We first observe that, by a standard theorem on differential equations (see, for example, Birkhoff and Rota, 1969, p. 152) the two initial conditions [f(O) = 0 and f’(0) = 01 determine a unique solution to (12) in any compact convex region of the (x, s) plane; so the solution trajectories do not intersect other than at the origin.

2.4.1. Stochastic solutions and integral curves The argument we develop makes use of the relationship between stochastic

solutions and integral curves of the deterministic system. Let the ratio E,(ds)/E,(dx) at any point be denoted a(x, s) where, from (3) one

finds

Et@) yx + 6s - = a(x, s) = ~ Wdx) LYx+/?s’

recalling the redefinition of the variables as deviations from equilibrium. Ob- serve further that the integral curves for the deterministic system (i.e., where 0 = 0) have to satisfy the condition

ds/dt - = a(x, s). dxjdt

These integral curves are shown as dashed lines in Fig. 4, with turning points along the lines of stationarity XX and YY, where E,(dx) = 0 and E,(ds) = 0, respectively.

The second-order differential Eq. (12) above can thus be written in terms of a(x,s) as follows:

f”(X) = %$J Z-f’(X) [ f 1

E, WI = ---&4x3s) -f’(x)), (15)

where a(x, s) is the slope of the integral curve at (x, s) andf’(x) is the slope of the stochastic solution at that point. So the latter will only exhibit curvature when a(x,f(x)) # f’(x). Specifically, sgn(f”) = sgn(a -f’) for E,(dx) > 0 and sgn(f”) = sgn(f’ - a) for E,(dx) < 0. At a point of inflection, therefore, where f” = 0, it must be the case that a = f’ and the integral curve and the stochastic solution will be tangent. Note also that the function a(x,s) may be written in

M. Miller, P. Weller / Journal @‘Economic Dynamics and Control 19 (1995) 279-302 291

Fig. 4. Global analysis: The relationship of stochastic solutions to integral curves

the form

y + 6r h(r) = ~

CI + fir’

where Y = s/x. One finds that

cd - BY det A h’(r) = (a + /jr)’ = (&_ + /jr)2 < 0, E,(dx) # 0.

(16)

(17)

The function h identifies the slope of an integral curve along a ray from the origin, and for a saddlepoint-stable system is declining in Y.

Now we proceed to show how these properties can be used to provide a characterization of the global properties of all symmetric stochastic solutions.

2.4.2. Properties of the solution paths

Region A. In a neighborhood of the origin, all solution paths are convex, as established in Section 2.2. If any path becomes concave, it must pass through a point wheref”(x) = 0. At such a point the solution path must be tangent to an integral curve. Using the fact that h’(r) < 0 we know that all integral curves in region A have a slope everywhere less than 19,. But since f’(0) > t?,, any point of tangency would have to occur at a point satisfyingf’ (x) > 8,. So no such points of tangency exist, and all solution paths are everywhere convex.

292 M. Miller, P. Weller / Journal of Economic Dynamics and Control 19 (1995) 27%302

Region B. An exactly parallel argument can be used to show that solution paths in region B are everywhere concave.

Region C. We show first that any path in C must pass through at least one point of inflection where f” = 0. Since any path entering C from B hasf” < 0, f’ < 0 on the boundary between the two regions, and any path lying wholly in C has f” < 0, f’ < 0 in a neighborhood of the origin, we have to rule out cases where f”(x) < 0 for all x 2 0. We can immediately rule out the case where f” is bounded away from zero, since this would imply that the graph off would eventually intersect OU, which is impossible. So we need consider only the case where f” converges to zero from below as x tends to infinity. In this case the fundamental differential equation tells us that lim,,, f’(x) = 8,, and since such a path must lie everywhere strictly above UU, we must have lim,,, f(x) = 0,x + k where k > 0. But if we substitute these limiting values into (8), we obtain, using the definition of p(Q) in (13),

j-(8,)x + (6 - @,)k = 0. (18)

But we know that

de,) = Y + (6 - ale, - Be,’ = 0. (19)

It follows that

6 - pe, = a - y/e, c 0, (20)

and Eq. (18) cannot be satisfied with k # 0. Thus we may conclude that for any path in C there exists a point where f” = 0. To show that this point must be unique we refer to Fig. 5. The first point of inflection on the solution path illustrated in region C occurs at T, which must be a point of tangency with a integral curve. Suppose another such point of tangency occurs at T’. Iff” were positive after T’, then a(x,f(x)) <f’( x , and (14) implies, since E,(dx) > 0, that ) f” is negative, and we have a contradiction. A similar argument rules out the possibility that f” is negative. So no such point T’ can exist.

This property can immediately be used to demonstrate that any path in C converges asymptotically to the unstable saddlepath. From the figure we see that the solution path is bounded above by the integral curve through T, which displays the necessary convergence. If the stochastic solution through T were to intersect the integral curve through T at some point, it must pass through a point of tangency with another integral curve after T, implying f” is zero, which we have already shown to be impossible.

Region D. Exactly parallel arguments to those used for region C may be used to demonstrate the analogous properties of solution paths in region D.

M. Miller, P. Weller / Journal qf Ectmomic Dwamics and Control 19 (199Sj 279-302 293

Fig. 5. The unique point of inflection.

2.5. ClussiJicarion of special cases

Although, as we have already observed, a general solution to (12) is not available in closed form, it is useful to classify two special cases for which closed form solutions exist,

The first is obtained by setting the top row of coefficients of the matrix

A equal to zero, i.e., CI = /? = 0. The variable x then follows a driftless Brownian motion, and the general solution takes the form

s = - (y/6)x + AlePIX + A2e”l”, (21)

pl, p2 are the roots of the equation (c2/2)p2 - 6 = 0, and the constants A, and AZ are determined by boundary conditions.

The second is obtained by setting the coefficient fl alone equal to zero. Then x follows an Ornstein-Uhlenbeck process, since the coefficient a is negative. The differential equation describing the dependence of s on x takes the form

p(x) + axf’(x) - Sf(x) - yx = 0. (22)

294 M. Miller, P. Weller / Journal of Economic Dynamics and Control 19 (1995) 279-302

The general solution takes the form

s=[y/(a-6)]x+C,M [ -$:$ -$ 1 +C2M a-6 3 ax2 &x _*_a ___ 1 ~

2a ‘2’ a2 a ’ (23)

where M (O : . : .) is the confluent hypergeometric function and Ci, C2 are constants to be determined by boundary conditions. There will of course be a saddlepoint structure associated with autoregressive fundamentals, so our qualitative analysis can be applied: the restriction that /? = 0 simply excludes regions C and D, as the unstable eigenvector coincides with the vertical axis.

The undoubted benefits of working with closed form solutions are demonstrated by Delgado and Dumas (1992) and Froot and Obstfeld (1989, 1991), in connection with the recursive monetary model of exchange rates. But the necessary restriction that /3 = 0 means that these explicit solutions will not apply where there is feedback from the asset price onto the evolution of economic fundamentals, as is true in the application which follows in Section 3.

2.4. Existence of solutions asymptotic to the stable saddlepath

In the cases for which there exist closed form solutions (in terms of the exponential or confluent hypergeometric function) it is easy to confirm that from any point in the (x, s) plane there exists a unique solution which is asymptotic to the stable saddlepath as x tends to plus or minus infinity. These are examples of asymmetric solutions which do not pass through the origin, and so they are not characterized for the general case by the arguments presented above. These solutions are important for the subsequent analysis because their existence allows us to impose a ‘no bubbles’ condition on the system in the fiscal intervention regimes.

In this section we describe the geometric intuition underpinning the proof of existence of unique solutions asymptotic to right- and left-hand arms of the stable saddlepath from any point in the (x,s) plane. (The proof itself we relegate to the Appendix.) To fix ideas we consider region A in Fig. 6. We impose the arbitrary initial condition that the stochastic solution should pass through point E. Exactly as was argued above, one may show that any path satisfying f’(xo) 2 8, is globally convex, and so cannot approach SS asymptotically. In the figurefi is such a path. Iff’(xe) 5 a(xofxo)), then it follows thatf”(x) < 0 for x > x0, (x,f(x)) E A. An example is the path f2, which is tangent to II, the integral curve through E. So we are able to establish upper and lower bounds on the initial gradient of any asymptotic path which may exist. As one increases the

M. Miller. P. Weller / Journal qf Economic Dynamics and Control 19 (1995) 279-302 295

E[d.sj = 0

X

Fig. 6. Solutions asymptotic to the stable saddlepath

initial gradient from a(xo,f(xo)) one generates a family of solutions such as f3 which are first convex, pass through a point of inflection, and then remain concave so long as the path remains in A. The supremum of the set of initial gradients defining this family can be shown to be the unique solution asymptotic to ss.

The argument outlined above has the practical implication of providing initial bounds on gradient, which would prove useful in any exercise involving compu- tation of asymptotic solutions. In addition, although we do not provide a formal proof, it is evident that the initial gradient of an asymptotic solution declines monotonically as f(x,,) increases from OsxO. This property can be readily confirmed for the special cases where there exists a closed form solution, and can be used to establish uniqueness of the solution path involving a locally reversible transition from one stochastic regime to another.

3. Stabilization with thresholds and the stock market

We have described the model we intend to use (see Section 2.1.1) and the effects of fiscal stabilization in a deterministic context (see Section 2.1.2). We now show how the qualitative analysis developed in Sections 2.3 to 2.6 can be used to characterize the effects of the same stabilization rule applied to the stochastic model.

296 M. Miller, P. Weller / Journal qf Economic Dynamics and Control I9 (1995) 279-302

In applying the general characterization results for the stochastic model we need first to deal with the minor complication that one of the elements of the coefficient matrix A in (9) is a function of 0,. But it is possible to show that the stable root (2,) and the associated eigenvector (1, 0,) are identical to those for the matrix

a4a7 1 i* +pa’s* ’ Since starred variables represent long-run equilibria of the deterministic

system they are invariant to changes in p or a, and we can use this equivalence to determine the effects on the solution of changing p and a. The condition determining the sign of 0, is unchanged:

Q,~O as (al/az)s* - a6 2 0.

But the absolute value of 13, declines monotonically with increases in p or a, whereas the absolute value of A, increases.

The impact of increases in p or a on the slope of the stable saddlepath can be understood by referring back to (2). The effect is to increase the value of 6, or the rate at which expected deviations from the long-run equilibrium value of the fundamental are ‘discounted’. Thus along the stable saddlepath the stock market is less sensitive to fluctuations in the fundamental.

The stable root & is a measure of the speed of mean reversion along the stable saddlepath. The more risk-averse are agents in the economy, the more heavily damped are the effects of fundamental shocks on the stock market. As p becomes large, the stable saddlepath approaches the horizontal axis regardless of whether its slope is positive or negative and 1, approaches minus infinity. Increasing levels of risk aversion thus have the interesting effect of reducing both the variability of the stock market and, through the feedback effect, the variability of output.

We assume that the upper and lower trigger points for fiscal intervention are situated equidistant from x*. Thus X - x* = x* - x. The solution to the model for the good news case in the presence of the threshold stabilization rule is illustrated in Fig, 7a. As in Fig. 1, it is made up of segments corresponding to three different regimes. Between x and X the curve corresponds to one of the symmetric solutions to the fundamental differential Eq. (12), with coefficients given by the elements of the matrix A in (9). But above X and below 5, the path solves (12) with coefficients given by the elements of A’ in (10). The boundary conditions are (i) asymptotic convergence to S’S’ and (ii) ‘value matching’ and ‘smooth pasting’ at x and 2. As in the deterministic case condition (i)

M. Miller, P. Weller / Journal of‘ Economic D_vnamics and Control 19 (1995) 275302 291

6 x* X

Fig. 7a. Fiscal stabilization in the stochastic model: The good news case.

Fig. 7b. Fiscal stabilization in the stochastic model: The bad news case

corresponds to the standard ‘no bubbles’ assumption in rational expectations models. Likewise for condition (ii) it is clear enough why the solution path should be continuous at the trigger points x and X (value matching): as the reasoning is exactly the same as in the perfect foresight case discussed in Section 2.1.2. What requires explanation is the ‘smooth pasting’ condition, namely that .ft(x) = f;(x) and f;(X) = f;(X), where the subscripts L and R denote left- and right-hand derivatives. A formal statement of this requirement is given in

298 M. Miller. P. Weller / Journal qf Economic Dynamics and Control I9 (1995) 279-302

Whittle (1983, Ch. 35, pp. 201-204). However, an intuitive justification can be provided as follows. Assume the contrary, that one has a solution which passes through the origin and satisfies (9) between 5 and X, and converges to S’S’ while satisfying (10) elsewhere, is continuous but does not smooth paste at the trigger points. The path B’OB’ in the figure illustrates this possibility. Using a discrete time approximation, consider a point X( - ) just to the left of X. Since B’OB

coincides with SS at that point, the arbitrage condition governing stock prices must be

E(ds) = [(ai/az)s* - a6](Z( - ) - x*)dt + i*(8,X( - ) - s*)dt, (25)

where s increases along SS with a positive output shock and decreases along SS for a negative shock. But with a switch of regime at X, the solution for s lies a distance of first-order magnitude below SS for a point X( + ) just to the right of X, and will violate the arbitrage condition (25) evaluated at X( + ). The solution BOB is not subject to this inconsistency since the smooth pasting condition ensures that the impact of a possible switch of regime on expectations formed at points just to the left of 2 is of second-order magnitude.

The reason why smooth pasting was not needed in the deterministic framework was essentially because the movement of the system is unidirectional, so the regime switch was (locally) irreversible. When the system is subject to stochastic shocks, however, its movement is bidirectional (shocks to the fundamental can lead it to increase or decrease) and the regime switch is locally reversible. It is only for locally reversible switches that the arbitrage condition requires smooth pasting (Whittle, 1983).

The solution path BOB shows very clearly how the market anticipates the state-contingent switch of regime induced by the particular threshold stabilization rule we consider. In contrast to the deterministic case, there are effects when output lies inside the interval [x, X] where no stabilizing adjustments to government spending are actually occurring.

What then are the implications for stabilization policy of these anticipatory movements in the market? The answer is immediately apparent from the figure. In the good news case illustrated in Fig. 7a we find that the solution is flatter than SS over the interval [5, _?I, i.e., even when the thresholds have not yet been reached. This is due to the stock market anticipating the state-contingent fiscal adjustments that will take place when x moves outside this range. These anticipations will, via their effects on consumption and investment, exert a stabilizing effect on the economy. So the stochastic solution spreads the benefit of the policy over the whole range of x.

However, when we consider the bud news case illustrated in Fig. 7b, the effects are exactly reversed. When output is above x* for example, the stock market is higher than it would have been in the absence of threshold stabilization. The effect of this on the level of output is therefore destabilizing, increasing it when it

M. Miller, P. Weller / Journal qf Economic Dvnamics and Conrrol 19 11995) 279-3017 299

is high and depressing it when it is low. In short, the bad news case turns out to be bad news for stabilization policy.

The bad news case is more likely to arise the less responsive are dividends to the level of output (a6 small). Then the effect of higher interest rates more than offsets the prospect of increased dividend payments. Alternatively, the monetary authorities may choose to introduce a countercyclical component into monetary policy as well. This would in effect reduce the magnitude of the coefficient a2, thus making the interest rate more sensitive to fluctuations in output.

The idea that anticipated fiscal policy can have perverse effects on the economy in the period before its implementation is familiar enough; see for example the discussion of the contractionary effects of the Reagan tax cuts analyzed by Branson et al. (1985). But in his 1981 paper Blanchard showed that the condition for announced fiscal expansion to prove contractionary ex ante was precisely that the stock market should be negatively correlated with output. So we find that the bad news case generates perverse results whether the fiscal changes are state-contingent or time-dependent. Evidently the earlier debate about time-dependent policy changes in deterministic models has clear implications for the effects of state contingent switches in stochastic models.

4. Conclusion

We have presented a technique for analyzing the behaviour of stochastic saddlepath models which resembles the use of phase diagrams for analyzing perfect foresight models and have demonstrated the existence of a number of close parallels between the deterministic and stochastic cases. It turns out that many of the qualitative features of the stochastic model can be inferred from the properties of the associated deterministic one.

The utility of our approach stems not only from the inherent simplicity of geometric treatment, but also from the fact that for the class of models we consider no explicit solutions are usually available. (This is in contrast to the more restricted class of model analyzed in the context of the exchange rate band literature.)

The stochastic solution we obtained for Blanchard’s stock market model showed how expectations of state-contingent policy changes can be system- atically incorporated into asset values. The debate as to whether such anticipations may not have perverse effects on output has a close parallel here. Whether the anticipation of threshold fiscal policy was stabilizing or not depended on the same conditions that determined whether future anticipated fiscal expansion was good news or bad news for the economy in a deterministic framework.

The use of stochastic rational expectations models with endogenous dynamics for the fundamental has surely been hampered by the absence of closed form

300 M. Miller, P. Weller / Journal of Economic Dynamics and Control 19 (1995) 279-302

solutions. Our qualitative analysis indicates that an effective alternative to the use of numerical methods is available.

One would expect the solutions we obtain here to be relevant for determining optimal feedback in linear quadratic Gaussian control problems. This is one area for further investigation. Another is to see whether the approach we have pursued is necessarily restricted to bivariate models. It would be useful to learn whether this qualitative analysis could be extended to cases where there are more cumulative shocks, or additional state variables.

Appendix

Existence of solutions asymptotic to the stable saddlepath

Consider a family of solutions f(x : b) to the stochastic system, where b = (b,, b2) is a vector specifying initial conditions. Thus b = (f(x,& f’ (x0)). We confine attention to solutions passing through the interior of region A. Thus bl =f(xo) > &x,, and x0 > 0. Note first that from Eq. (ll), iff’(xo) 2 8,, then f”(x) > 0 for all x 2 x0 and f’(x) could not approach 8, as x goes to infinity. So 8, must provide an upper bound on the initial gradient of an asymptotic solution.

Now suppose that f’(xo) = a(xo,f(xo)), This implies from (11) that f”(xo) = 0. Since the integral curve is convex at (xo,~(xo)), for small positive E we have j’(xo + E) < a(xo + &,f(xO + E)), from which it follows that f”(xo + E) < 0. It is immediate that there cannot be a point x1 > x0 at whichf”(x,) = 0, so long as the solution path remains in region A.

To rule out the possibility that a path exiting from region A becomes asymptotic to SS, we subdivide into two cases. First, if a path enters B with a negative slope, it would have to pass through a point xi at which f’(xi) = 0 and f”(xi) > 0. But in region B, a(x, y) > 0 and E(dx) < 0, which shows from (11) that, if ever f’(x) = 0, then f”(x) < 0. Similarly in regions C and D, a(x, y) < 0 and E(dx) > 0, and we reach the same conclusion. So such a path never has a turning point, and so cannot approach SS asymptotically. The second case to consider is where the path enters B with positive slope less than 8,. For such a path to become asymptotic to SS, it would either have to have a turning point, which we have already ruled out, or it would have to have a point of inflection in B. Such a point of inflection would have to occur at a point where f’ < 8,. But at a point of inflection f’(x) = a(x,f(x)), and in region B we know that a(x, y) > 8,. So such a path cannot approach SS asymptotically.

Now consider the set B defined as

B(b,) = (6, 1 3x* s.t. f”(x* : bl, b,) = 0, x* 2 x0).

M. Milkr. P. Weller i Journal qf Ecwmnic Dxnawks and Control 19 (1995) 279-302 301

We know that for (x,, 1(x0)) E A, this set is bounded above by O,, and is nonempty.

So it must have a supremum, b$(b,). We wish to examine the properties of the

solution f(x : hl, bt(b,)). Suppose there exists xf such that .f“‘(x: : hl, bz(b,)) = 0. Such a point must occur in the interior of region A, since on the boundary defined by SS, the linear saddlepath is the unique solution satisfying this condition.

We have already established that J” is declining in x in the neighborhood of a point of inflection. Choose a &neighborhood of XT such that the graph of j’over the interval [xf, x: + 61 is in the interior of A. Then we know first that

limf”‘(x: + B : h,, h;(h,) + E) 2 0. (A.11 c-o

This follows from the fact that .f”(x : hl, b2) must be bounded away from zero by definition, for all b, > h:(b,), x 2 x0.

We also know that there must exist q > 0 such that

,f”‘(x: + 6 : h,, b;(b,)) < il. (A.21

But (A.l) and (A.2) imply that ,f” (x : b,, b,) is discontinuous in b, at XT + 6, which is not possible.

So it must be the case that f“‘(x : h,, bf(b,)) > 0 for s 2 x0, and that

lim,, X .f’(x : b,, bf (b,)) = O,yx + k, and we have already established in Eqs.

(14)-(16) that (8) is not consistent with k # 0. Therefore ,f(x : bl, b:(b,)) is the unique solution through point (xc), b,) E A asymptotic to the right-hand arm of the stable saddlepath.

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Stochastic saddlepoint systems Stabilization policy and the stock market

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