[size=10.000000pt]Consider a two-period small open economy model with a representative agent. The agent takes the world interestrate as given, and assume it is zero (that is, [size=10.000000pt]r [size=10.000000pt]= 0 [size=10.000000pt]and [size=10.000000pt]R [size=10.000000pt]= 1 [size=10.000000pt].)
  [size=10.000000pt]The home agent receives an endowment of [size=10.000000pt]Y[size=7.000000pt]1 [size=10.000000pt]in period1. In period 2, the endowment [size=10.000000pt]Y[size=7.000000pt]2 [size=10.000000pt]is stochastic:
[size=10.000000pt]Y[size=7.000000pt]2 [size=10.000000pt]= [size=10.000000pt]Y[size=10.000000pt] ̄ [size=10.000000pt]− [size=10.000000pt]ε [size=10.000000pt]with probability [size=10.000000pt]1[size=10.000000pt]/[size=10.000000pt]2[size=10.000000pt]Y[size=7.000000pt]2 [size=10.000000pt]= [size=10.000000pt]Y[size=10.000000pt] ̄ [size=10.000000pt]+ [size=10.000000pt]ε [size=10.000000pt]with probability [size=10.000000pt]1[size=10.000000pt]/[size=10.000000pt]2
[size=10.000000pt]Assume [size=10.000000pt]Y[size=7.000000pt]1 [size=10.000000pt]< Y[size=10.000000pt] ̄ [size=10.000000pt], and [size=10.000000pt]0 [size=10.000000pt]≤ [size=10.000000pt]ε [size=10.000000pt]≤ [size=10.000000pt]Y[size=10.000000pt] ̄[size=10.000000pt]. Assume that the country has zero initial net claims on the rest of theworld. Given the assumption of zero interest rate, the budget constraint of the representative agent is then[size=10.000000pt]C[size=7.000000pt]1 [size=10.000000pt]+ [size=10.000000pt]C[size=7.000000pt]2 [size=10.000000pt]= [size=10.000000pt]Y[size=7.000000pt]1 [size=10.000000pt]+ [size=10.000000pt]Y[size=7.000000pt]2[size=10.000000pt].

[size=10.000000pt]In this problem, ignore the non-negativity constraint on consumption. That is, do not impose the constraints[size=10.000000pt]C[size=7.000000pt]1 [size=10.000000pt]≥[size=10.000000pt]0[size=10.000000pt]and[size=10.000000pt]C[size=7.000000pt]2 [size=10.000000pt]≥[size=10.000000pt]0[size=10.000000pt].
  [size=10.000000pt]Assume utility of the representative agent is given by [size=10.000000pt]U [size=10.000000pt]= [size=10.000000pt]ln[size=10.000000pt]([size=10.000000pt]C[size=7.000000pt]1[size=10.000000pt]) + [size=10.000000pt]E[size=10.000000pt][[size=10.000000pt]ln[size=10.000000pt]([size=10.000000pt]C[size=7.000000pt]2[size=10.000000pt])][size=10.000000pt],where [size=10.000000pt]E [size=10.000000pt]means "expected value."

求 the optimal value of C1 (in terms of Y1, Y ̄ and ε).


注：Y ̄ 就是
的意思

                                                                                                                                                [size=10.000000pt]