1. [50 points] Suppose a representative consumer with preferences E. Bt log( ct). t = 0 Suppose also a representative firm, which produces yt at each period t. Production yt is exogenous, and evolves according to the following stochastic process: yt = y1, with probability 7; 2, with probability 1 - 1, where y1 > y2 > 0 and T E (0, 1). There is a fixed quantity of one unit of shares of the representative firm. Each share entitles the owner to a fraction of the firm's production next period (i.e. dividends, since production costs are zero). (a) [7 points] Write down the representative consumer's budget constraint at time t. (b) [7 points] Define a competitive equilibrium. (c) [7 points] Write down the consumer's problem recursively. Explain your choice of state variables. (d) [7 points] Write down the first-order and the envelope conditions for the con- sumer