RDP 2011-02: Long-term Interest Rates, Risk Premia and Unconventional Monetary Policy Appendix A: The Linearised Equations

The full set of linearised equations is given by:

All variables are in log deviations from steady state. Equation (A1) gives the evolution of the aggregate marginal utility of wealth, Λt, linking it to the preference shock ât and output yt. Equations (A2) to (A4) give the restricted and unrestricted agents' intertemporal relationships. Equation (A2) is the unrestricted agents' first-order condition for long-term debt accumulation, where Inline Equation is the long-term real interest rate, Inline Equation is the nominal long-term interest rate and πt is inflation, and Inline Equation is the risk premium, where ζt is the exogenous component of the premia, mt is money demand and bL,t long-term real bond holdings. Equation (A4) gives the restricted agents' first-order condition for long-term debt accumulation. Equation (A5) combines the restricted and unrestricted agents' Lagrange multipliers, weighted by λ, the proportion of unrestricted agents. Equations (A6) and (A7) give the supply-side relations, linking inflation to marginal costs mct and technology shocks Inline Equation. Equations (A8) to (A10) govern money demand relationships, where êt is a money demand shock. Equation (A10) aggregates across agents' money holdings. Equation (A11) gives the Taylor-type rule for a central bank targeting interest rates of maturity L with money growth µt specified by Equation (A12). Equation (A13) aggregates across agents' long-term bond holdings. The exogenous processes are given by Equations (A14) to (A19).