Suppose the utility that the representative consumer obtains from consumption in the current period depends also on consumption in the previous period, i.e., the period utility function in period is not just w (Cy), but it depends on both Cy and Ci1: u (Ci, Ci1). This means that the consumer displays habit persistence, as assumption that is often made in models to improve their performance relative to data. Assume that the consumer supplies one unit of labor inelastically in each period, can invest in stocks, and wants to maximize the intertemporal utility function: o0 > B u(Cs, Comy) s=t where (s is consumption in period s, # is a discount factor strictly between 0 and 1, and we are assuming perfect foresight. The budget constraint in period is: Ct + vepp1 = wy + (vp + dy) . Going right to left in this equation: The consumer begins period t with stock holdings x;, receives dividends (d) and the value of selling the stocks at the price v; and the consumer receives wage income w;. The consumer then uses these resources to buy stocks to be carried into + 1 (x41) and consumption (Cf). When a consumer arrives in period , of course Cy_; cannot be changed (because it happened in the past). e Use the Lagrangian method to find the Euler equation for the consumer's optimal stock-buying decision. (Hint: You will want to use the notation wu; (e,e) and up (e, @) for partial derivatives in this problem.) e Write this Euler equation as a stock pricing equation, i.e., with only v on the left-hand side. e Consumption in which periods affects the period-t stock price under habit persis- tence? Without habit persistence (i.e., the baseline model we studied throughout the quarter), consumption in which periods affects the stock price in period t