I try to apply a Linear-Quadratic approach to derive the optimal monetary policy in a standard New Keynesian model extended with credit frictions in the form of Gertler Kiyotaki (2011). The question is very specific, so it is only for those who are used to the Linear-Quadratic approach. I use standard notation of RBC, so everyone who is relevant will understand.
The particular problem I face is this:
In the Gertler Kiyotaki model, the price of capital is determined by the capital goods producers endogenously, that is a price, Q. Q is NOT equal to the real interest rate, r, as the latter is the deposit rate.
Therefore, in equilibrium, Q is NOT equal to the marginal revenue product of capital, but it is equal to the marginal cost of investment goods production.
My problem: When I take a second-order approximation of the welfare criterion of households, I cannot use the steady-state condition 1/beta = r = aY/K + (1-delta)
in order to iterate and eliminate the first-order terms of capital (necessary to be eliminated as following a Linear- Quadratic approach, the central bank’s objectives will be presented only in second-order.). Is there any condition for an efficient steady-state associated with flexible price equilibrium, that I can use to equate aY/K + (1-delta) with 1/beta ?
Notice that Woodford (2003, page 355) makes some notes about this but it is difficult for me to follow his intuition at the moment, so any comments are welcome.