Dear all,

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)?

Thanks in advance.