Consider an infinitely-lived agent who receives an exogenous income stream. The agent derives utility from non-durables and durables, and maximizes the following object: 00 Vo = Bu(c, de), 0 0, where a denotes the amount of non-durables consumed by the agent in period t and d denotes the stock of durables. Consumption of durables and non-durables may never be negative. The durable stock evolves according to: de = (1 - 6)du_1 + It, for t = 0, 1, 2, ..where o is the depreciation rate of durables and a denotes purchases of durables. The agent can freely borrow and lend. The budget constraint of the agent is given by C+ 9.It the = yi + (1 + r_1)by_1, for t = 0, 1, 2, ... where q is the price of durables relative to non-durables, b, denotes bond holdings, ~ is the net interest rate and y is income (all expressed in units of non-durables). The agent decides on c, de, It and b, for every period t = 0, 1, 2, .. to maximize Vo subject to the constraints mentioned above, taking as given prices, and interest rates. The agent also takes as given an initial level of bond holdings b_1 = 0, and an initial level of durables d_1 3 0. a) What is the marginal rate of substitution of durables (d) for non-durables (c.)?2 b) Derive the first-order conditions for the agent's optimization problem. You may ignore the constraint that de 2 0 and c 2 0. c) What is the intuition behind the first-order condition for durables? Relate your answer clearly to the equation. Suppose that there is a unit-mass continuum of agents exactly like the one described above. Also, suppose that the aggregate supply of durables is fixed and equal to one and the depre- ciation rate o equals zero. d) Formulate the market-clearing conditions and define a competitive equilibrium. e) How are the interest rate r, and durable-price growth q+1/qt related