Assume that there is an economy populated by a continuum [0,1] of households living in two periods, 0 and 1. Each Household is endowed with income in both periods. In terms of the income streams, there are two types of households. First, any household i in the interval [0,6) is a type-A household. Type-A households are endowed with incomes ) and v in periods 0 and 1, respectively, where ;' is substantially greater than i (i.e., y' >> yi'). Second, any household i in the interval [#,1] is a type-B household. Type-B households are endowed with incomes yf and y in periods 0 and 1, respectively, where yf is substantially smaller than (i, y 00c#1l (a) Write down type-A households' optimization problem using the intertemporal budget con- straint. (In doing so, denote type-A households' consumption in periods 0 and 1 by and cf', respectively.) (b) Set up a Lagrangian equation for type-A households\" optimization problem and derive the optimality conditions. (c) Solve type-A households' optimal consumption and c{'. Derive their optimal amount of saving s and verify that they are savers. (d) Solve type-B households' optimal consumption cf and c?. (If you are obtaining type-B house- holds\" optimal consumption bundle without repeating parts (a) and (b), explain precisely the reason why you can do so.) Derive their optimal amount of saving s and verify that they are borrowers. (e) Derive the aggregate bond supply (5""*) and the aggregate bond demand (D?") in this econ- omy. (f) Determine the equilibrium interest rate r*