I try to replicate the paper ‘‘A New Keynesian model with heterogeneous expectations’’ written by Branch and McGough (2009).
I have difficulties to follow the steps from equation (13) to (14).
My problem is how to come up with the FOC w.r.t. P_t^i, namely E_t^\tau \sum_{k=0}^\infty(\gamma\beta)^k(\textrm{log}(P_t^i)-\textrm{log}(P_{t+k})-\zeta_1\Omega_{t+k}^\tau-\zeta_2y_{t+k}) = 0.
Additionally, I have some doubt w.r.t the claim in the paper that the consumption bundle of an agent i at some point in time, say t+k, is a function of P_t^i. In equation (13) they use C_{t+k}^i (P_t^i). Of course, one may think at first that the household decision to consume will depend on the income generated from selling goods. Due to Calvo Pricing, it may be the case that the household is applying the price P_t^i, even in period t+k. Since Branch and McGough (2009) introduce a risk-sharing mechanism to hedge against this Calvo risk, to my perception the budget constraint is independent from P_t^i.
So why should I expect that the consumption bundle of agent i is C_{t+k}^i(P_t^i)?
I managed to derive all intermediate steps from the FOC up to (and including) equation (15).
But I still have the problem to derive the FOC itself.
If we follow the yeoman farmer approach, the optimal pricing decision of an agent i (of type \tau) at time t is obtained from \operatorname*{max}_{P_t^i} \mathrm{E}_t^\tau \sum_{k = 0}^\infty (\gamma\beta)^k[u(C_{t+k}^i(P_t^i),\cdot) - v((\frac{P_t^i}{P_{t+k}})^{-\theta}Y_{t+k})].
How do we show up with the FOC, namely: \mathrm{E}_t^\tau\sum_{k=0}^\infty (\gamma\beta)^k(\log(P_t^i)-\log(P_{t+k})-\zeta_1\hat{\Omega}_{t+k}^\tau - \zeta_2 y_{t+k}) = 0
From the paper it is not obvious for me to see which functions the authors used for u(\cdot) \textrm{ and } v(\cdot).
It is obvious from equation (12) that u(\cdot)=\frac{c_t^{1-\sigma}-1}{1-\sigma}. My hunch is that the labor part is similar, i.e. -\psi\frac{N_t^{1+\kappa}}{1+\kappa} where \kappa is the inverse Frisch elasticity. But without knowing what the \zeta_i are, it is impossible to tell.