Linearing the BC in Justiniano and Preston (2010)

General DSGE Modeling
1

I have a great difficulty to linearizing the budget constraint in a paper of Justiniano and Preston (2010), titled: " Monetary policy and uncertainty in an empirical small open‐economy model" at

https://onlinelibrary.wiley.com/doi/full/10.1002/jae.1153

In particular, the level of the BC on page 96 of this paper is

P(t)C(t) + D(t) + e(t)B(t) =D(t-1)*R(t-1) + e(t)*B(t-1)*R_star(t-1)*Phi(A_t) + W(t)*N(t) + Profit_H(t) + Profit_F(t) + T(t)

and its linearized form on pape 101 is

c(t) + a(t) = (1/beta)a(t-1) - alpha[s(t) + Phi*s(t)] + y(t)

Well, I solved all other equations in this paper successfully, excepting this BC. Can someone helps me to do that since I have done so many time, it has not succeeded yet.

Thank you guys so much

2

Equation (22) is not a direct linearization of the BC on page 96. Rather, the distribution of output to factor payments and profits has been used to reduce the right-hand side of the equation.