Consider again the optimal consumption and asset holding plans of households with preferences Eu Etut) where \"(c-t) = act (1,'2)c , D 5 ct 1:: (I. The budget constraint for each period t? is ct + or\" = (1 + at + tot. Assume?" = p. But now assume that households face additional :liquidity constraints' of the form at\" E D for any :1 asset holdings can never be negative} i.e. households cannot get credit. 1. Write the Lagrangian {including also the liquidity constraints with multipliers #:L take the rst order conditions {for er and aHl] and obtain the Euler equation. How is that different from the standard Euler equation that we get in the absence of liquidity constraints'.l For the following questions} suppose now for simplicity that there are only two periods} t = D, 1. You know what [12 {the amount of assets we want to buy in period 13, which is the last] will he, don't you? So we actually only need to determine two variables} cc. and :11 {as [:1 is detenuined through time 1 budget constraint by :11 and the exogenous m1]. But we have three conditions at t? = D: the Euler equation, the budget constraint. and the liquidity constraint. While the budget constraint is always satisfied with equality {strictly increasing utility function]? the liquidity constraint may or may not be binding, i.e. it may be that we choose some [11 E D {so the constraint is not binding? since we wouldn't want to borrow anyway] or it may happen that we would like some [11 1:: I}? but since we cannot borrow we are forced to set :11 = . Remember that ifa constraint is binding, its multiplier is positive, ifthe constraint is not binding} its multiplier is zero. 2. - Solve tor the optimal co and (11 {as a Jnction ofr, on, (on, Enrol] under the conjecture that the liquidity constraint is not binding. - What is the condition {on 'r, on, ma, EU ml] for the optimal :11 to be indeed non-negative {i.e. for the conjecture to be conrmed]