4. (a) To derive the equilibrium interest rate "y: we should find the two agents'demands first. Given ry, an agent's total income is I = eo+ e1 co (1 + rf ) + en 1+" f and his feasible budget set is B (e , rf ) = 1( co, c 1 ) = Ritz : co + C1 To find his optimal demands, the agent solves max U ( Co, c1 ) = In Co + p In c1. ( Co .)EB (e,r. ) Obviously, the Inada condition is satisfied, there is no corner solution. Thus, we just need to solve the Lagrangean problem as follows: C = U (Co, c1) - A Co + 1 = In co + pln c, - > \\co+. C1 The first-order conditions are: ac = - A = 0 Jco ac 1 A 1+ f and C1 Co + = 1. (3) Accordingly, Go = ( 1 +rf )p (4) Substituting (4) into (3) yields 1 ( 1 + " f )p = X ( 1 + rf ) * = 1 + p I == 1+ p Thus, co( 1try ) +e co (1 + " f ) + el Co = 1+p 1+ p ( 1 + " f ) ( 1 + p ) C1 ( 1+rf )p p ( 1 + r ,) co( 1 +r , ) + e (eo ( 1 + 7 f ) + en ) p 1+ p 1 +p Thus, of = ed (1 + rf ) + el and c? = o (1 + ";) + er ( 1 + " f ) ( 1+ p ) ( 1 + " f ) ( 1 + p )