Quantitative Macroeconomics equation

And, assuming a steady-state scenario or debt levels approximated by and substituted into equation 1, the new equation becomes,

The equation remains essentially unchanged, but the stationary switches from dynamic to static. The relationship between the variables remains constant, but their movement does. In this case, the sign makes sense since it represents the constant displacement around which the debt level swings. The bond price-schedule is represented by the term ( in equation I. Bond price schedule is dependent on the coupon rate and never on the value of v. assuming that we have a coupon rate of , the bond price-schedule is found to be estimated by equation II below;

…. II

In this scenario, the stochastic component v is irrelevant since the schedule is deterministic.

Assuming a static response to the debt schedule, and assuming that equation II represent the debt response,

The maximum bond price schedule is fixed on a contractually agreed timeline, and the optimum debt level is determined by the maximum amount of stochastic component v* whereby Vt> V* where V* represents the upper stochastic ceiling or level.

The debt level at cut off therefore becomes,

In order to derive the equation for the bond price schedule, we must start from the debt amount formula;

Differentiation this equation with respect to T and taking maximum limits.

…

Then

We find the equation

The condition for alpha such that a steady state debt level is observed at is when the stochastic component is assumed to be showing since of movements. Therefore, the state is approximated by equation I