V'(>) = - (1+ β + δ) φF'(l), (3.8b)
rn(f) = г* + а (п) = β
δa — F (ŋ = rn(fι)fι = [г* + а (—)] —,
(3.8с)
(3.8d)
where С = δa and μ = (1 + β + δ) φ. The Hrst two steady-state conditions are quite
straightforward: equation (3.8a) describes the long-run Hrst order condition for own
durable consumption, while equation (3.8b) is the long-run optimality condition for em-
ployment if it is the sole factor of production. In turn, equation (3.8c) describes the
steady-state maximum condition for foreign debt: the real return on debt in steady-state
equilibrium equals the given consumer-producer rate of time preference. Correspondingly,
this condition determines the steady-state stock of debt —, which is a function of the
world interest rate r*, the domestic rate of time preference β, and the curvature of the
“risk premium” function α (∙). Finally, equation (3.8d) is the steady-state version of the
current account balance in which the difference between long-run durable consumption
spending and output equals steady-state interest payments on the outstanding stock of
international debt.
Linearizing (3.7a)-(3.7d) about the steady-state equilibrium described by (3.8a)-
(3.8d), we obtain the following matrix differential equation
Z = Jsoez =
|
φ ■ ʌ |
( (1 + β + δ) O —1 O ʌ |
^ φ — φ ^ | |||
|
2 |
- cμ - (1 + δ) cμ 0 |
a — a | |||
|
μ |
OOO —fra' |
μ — μ | |||
|
(3.9) | |||||
|
∖ — ' |
— cμ — 1 - (F'lμ — cμ} β + α'f ) |
— — — — у |
where z = (φ, a, μ, —, /and Jsoe denotes the Jacobian matrix in the small open economy
(soe) case. To determine the stability properties of the equilibrium, we Hrst consider the
19
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