a 2. Consider Calvo price-setting firms with partial indexation. That is, if a firm is not visited by the Calvo tooth fairy in period t, its price in t is the previous period's price plus Yt- 1, 0 1. The average price in period tis Y is the = axt + (1 - a)(pt- 1 + yt- 1), where Pt fraction of firms visited by the Calvo tooth fairy in any given period and Xt is the price they set. The resulting Phillips curve is a hybrid one: Tt = Et t+1 + 1+ By , where> 0 is the discount factor, K = 1% [1-3 (1- a)lo > 0 determines the slope of the Phillips curve, yt is the output gap, and Et t+1 is the expectation (taken at time t) of inflation at t+1. Y 1 Tt- 1 + 1+ By 1+ By (a) Show thatxt - pt = 1 - (t - Yt- 1). (b) Use the result in (a) and the representative firm's optimal price under Calvo pricing with partial indexation being xt = pt + (1 - (1- a)) yt +3 (1 - a)(Et(xt+1 Et t+1 Ym) to derive the hybrid Phillips curve. Pt+1 ) + (c) What value of would lead to the highest degree of inflation persistence? Why?