The Solution of Linear Difference Models under Rational Expectations

The paper responds directly to a gap identified by Shiller’s survey on rational expectations, which “indicates that the difficulty of obtaining explicit solutions for linear difference models under rational expectations may have hindered their use” — Blanchard and Kahn “attempt to remedy that problem by giving the explicit solution for an important subclass of the model Shiller refers to as the general linear difference model” (Introduction, p. 1305). The paper is explicitly a methods note: Section 1 sets up the class of models the method covers, and Section 2 “gives the solution together with the conditions for existence and uniqueness” (p. 1305).

The model is written as [X_{t+1}; tP{t+1}] = A[X_t; P_t] + γZ_t, where X is an (n×1) vector of variables “predetermined” at t, P is an (m×1) vector of “non-predetermined” variables, Z is a (k×1) vector of exogenous variables, and tP{t+1} is agents’ rational-expectations forecast of P_{t+1} formed at t (Section 1, eq. 1a-1b, p. 1306). The distinction is defined precisely: “a predetermined variable is a function only of variables known at time t…so that X_{t+1}=tX{t+1} whatever the realization of the variables in Ω_{t+1},” whereas “a non-predetermined variable P_{t+1} can be a function of any variable in Ω_{t+1},” so P_{t+1} equals its forecast tP{t+1} only if realized shocks happen to match their conditional expectations (p. 1306). A non-explosion condition on the exogenous variables Z (eq. 1c) “simply requires that the exogenous variables Z do not ’explode too fast’” (p. 1306).

The paper works through four examples to show the form’s reach: Example A reduces a linearized Sidrauski-style monetary growth model (capital predetermined, the price level non-predetermined) to the canonical form; Example B shows that models with lagged variables or expectations more than one period ahead “present no particular problem” because they can be stacked into first-order form; and Example D shows a multiplier-accelerator model reduces to the form even when the transition matrix A is singular, which “shows the absence of necessary connection between ‘real,’ ’nominal’ and ‘predetermined,’ ’non-predetermined’” (Section 1, pp. 1306-1307). But Example C is a genuine limitation: certain models — including a “zeroth order” difference equation and an asset-pricing-type equilibrium condition involving a once-lagged expectation of a future variable — “cannot be reduced to form (1)” at all, which the Conclusion flags as an open problem: “there exist models which cannot be reduced to this form,” leaving “the characterization of the class of models not reducible to (1)” as future work (p. 1307, 1309).

A solution is defined as a sequence of functions of variables in the current information set Ω_t that satisfies the model equations for all possible realizations, subject to a boundedness condition analogous to (1c) applied to the forecasts of X and P themselves — a condition the authors note “in particular rules out ‘bubbles’ of the sort considered by Flood and Garber [1979]” (Section 2, p. 1307). This restriction to non-exploding paths is what makes the existence/uniqueness question nontrivial: without it, there would typically be a continuum of mathematically valid but economically unreasonable explosive paths.

The authors transform A into Jordan canonical form, A=C⁻¹JC, with the diagonal blocks of J ordered by increasing absolute value and partitioned into J₁ (eigenvalues on or inside the unit circle) and J₂ (eigenvalues outside the unit circle); C, C⁻¹, and γ are partitioned conformably (Section 2, p. 1307). Rewriting the system in the transformed (Y,Q) coordinates splits it into a stable subsystem governed by J₁, which evolves freely from given initial conditions, and a subsystem governed by the unstable block J₂, which “will, by construction of J2, explode and violate the non-explosion condition unless” the unstable component Q_t is pinned down exactly by the discounted sum of expected future exogenous shocks (Appendix, eq. A4) — i.e., unstable dynamics are not allowed to evolve freely; they are “solved for” by imposing that they equal the forward-looking expectation term, which is precisely what prevents explosion.

Letting m̃ denote the number of eigenvalues of A outside the unit circle: Proposition 1 states that “if m̃=m, i.e., if the number of eigenvalues of A outside the unit circle is equal to the number of non-predetermined variables, then there exists a unique solution”; Proposition 2 states that “if m̃>m…there is no solution satisfying both (1) and the non-explosion condition”; and Proposition 3 states that “if m̃<m…there is an infinity of solutions” (Section 2, pp. 1308-1309, Propositions 1-3). In the Proposition-3 case, “a solution may in particular be such that the non-predetermined variables depend on the past directly and not only through its effect on the currently predetermined variables,” and a solution “may also be such that a variable not belonging to Z directly affects X and P” — i.e., extraneous or “sunspot”-like variables can enter the equilibrium when the model is underdetermined in this sense (p. 1309).

The unique solution under Proposition 1 “is ‘forward looking’ (Sargent and Wallace [1973], Shiller [1978], Blanchard [1979]) in the following sense: the non-predetermined variables depend on the past only through its effect on the current predetermined variables” (Section 2, eq. 2-3, p. 1308). The recursive form (eq. 2-3) expresses X_t and P_t in terms of lagged X, lagged Z, and a discounted sum of expected future Z’s weighted by the inverse powers of the unstable eigenvalues (J₂); solving this recursion forward gives an explicit “final form” solution (eq. 4-5) expressing current X_t and P_t directly as a function of the initial condition, the history of Z, and expected future Z’s, with no free unstable components. The paper notes for computation that “simulations should use the recursive formulas (2) and (3) rather than the final form solution (4) and (5)” because the closed form, while explicit, is less tractable (p. 1309).

The authors state that “the condition m̃=m is also clearly related to the strict saddle point property discussed in the context of growth models: the above proposition states in effect that a unique solution will exist if and only if A has the strict saddle point property,” and note that when the system in fact represents the first-order conditions of a quadratic optimization problem subject to linear constraints, “the matrix A will have more structure than we have imposed here, and the condition m̃=m will always hold” (citing Hansen and Sargent) (Section 2, p. 1308). They report that “most recent macroeconomic models also satisfy m̃=m,” citing Sargent and Wallace [1973], and larger models by Hall [1978], Taylor [1978], and Blanchard [1980], while noting that models violating this condition (m̃<m) “have been constructed by…Blanchard [1979], Shiller [1978], Taylor [1977], Calvo [1977], and Burmeister, Caton, Dobell, and Ross [1973],” with Taylor in particular having “pointed out the non-uniqueness and the possible presence of irrelevant variables in the solution” in that case (p. 1308).

For the scalar case n=m=1 — “the case where there is one predetermined and one non-predetermined variable,” which the authors note “appears often (Fischer [1979], for example)” — the paper gives a fully explicit closed-form solution in terms of the two eigenvalues λ₁ (inside the unit circle) and λ₂ (outside), avoiding the need to compute a general Jordan decomposition (Section 2, p. 1309). This is offered “for convenience” as the practically most relevant special case, since checking existence/uniqueness in the general case “require[s] computing the roots of an n+m-th order polynomial” and is “in general analytically intractable” except in this low-dimensional case (p. 1309).