Consider a two-period economy with a representative household who has a known endowment of y1 in the first period but no endowment in the second period. The household has the following lifetime utility function:c1 c11 +2 , (2)1 1where > 0, and (0, 1). The household finances consumption in period 2 by purchasingrisky assets. There are two possible states in period 2: boom (B) or crash (C). Asset 1 has 1a gross return of R1,B = 2 in the boom state but R1,C = 0 in the crash state. Asset 2 has a return of R2,B = 0.95 in the boom state and R2,C = 0.5 in the crash state. The household can purchase as much of either asset as they like but cannot sell either asset. Assume that booms and crashes occur with equal probability.

(a) Set up the household's decision problem and derive the first order conditions and complimentary slackness conditions.

(b) Show that the household will always hold asset 2. Give an economic explanation for why this is the case, even though expected returns on asset 2 are less than those on asset 1.

(c) When = 1 the household will hold both assets. Taking this as given, find the equilibrium values of period 1 consumption and holdings of asset 1 and asset 2. Use values = 0.95 and y1 = 1. [Hint: you can use the R function nleqslv or the MATLAB function fsolve.]

(d) Find the largest value of for which the household will hold both assets.