The Solution of Linear Difference Models under Rational Expectations

This short note solves a general class of linear difference models under rational expectations — models in which some variables (“predetermined”) are fixed by past decisions and others (“non-predetermined,” or jump variables) can move freely in response to news, including expectations of the future. Blanchard and Kahn show that existence and uniqueness of a well-behaved (non-exploding) solution hinges entirely on a counting condition: comparing the number of eigenvalues of the model’s transition matrix that lie outside the unit circle to the number of non-predetermined variables. If the two numbers are equal, a unique, stable, “forward-looking” solution exists, in which the jump variables depend on the past only through its effect on the current predetermined state. If unstable eigenvalues outnumber jump variables, generically no non-exploding solution exists at all; if jump variables outnumber unstable eigenvalues, there are infinitely many non-exploding solutions, some of which let the jump variables depend on history directly or let variables outside the model affect the outcome. The paper gives both a general closed-form solution (via Jordan decomposition of the transition matrix) and an explicit formula for the common one-predetermined, one-jump-variable case, and notes that the eigenvalue-counting condition is simply the algebraic counterpart of the “saddle-point” stability condition long used in growth theory. The result became the standard method (and the “Blanchard-Kahn conditions”) for solving and checking determinacy in linear(ized) rational-expectations and DSGE models, though the authors flag that some models with lagged expectations of future variables cannot be put in their framework at all.

Olivier Jean Blanchard | Macro Paper Warehouse

The Solution of Linear Difference Models under Rational Expectations