State-dependent attention and pricing decisions

The derivation follows closely Alvarez and Lippi (2010). All firms share the same profit function Π(Pt, Yt, Ct) = YtP −η t (Pt − Ct). Where η > 1 represents the constant price elasticity, Yt is the intercept of the demand (i.e. it’s a demand shifter) and Ct is the marginal cost at time t. I assume that Yt and Ct are perfectly correlated, i.e. when costs are high demand is also high. In order to approximate the objective function as (1), I compute a second order approximation of Π(Pt, Yt, Ct) around its frictionless price. In the RI context, the frictionless price is the optimal price under full information P ∗ t . The second order approximation of Π(Pt, Yt, Ct)