課程名稱︰經濟數學一 課程性質︰選修 課程教師︰周建富 開課學院：社科院 開課系所︰經濟系 考試日期（年月日）︰2012/06/20 考試時限（分鐘）：100分鐘 是否需發放獎勵金：是，謝謝! （如未明確表示，則不予發放） 試題 : 1.An individual, with utility function U(w)= -e^(-aw) , has an initial wealth W to invest, which can be divided between two assets: A safe asset with a return of $1 per $ invested and a risky asset with a random return of $z per $ invested. z has an exponential density ∞ 1 function f(z) = λe^(-λz) with E(z) = ∫ z f(z) dz = ---- > 1. 0 λ Let α and W-α denote the amounts of wealth invested in the risky and the safe asset; respectively. The wealth will be W + α(z + 1). The expected utility maximization problem is λ max E(U) = E(U(W + α(z - 1))) = [-e^(-a(W - α)) ] ------------ aα + λ (a)Calculate the FOC for an interior solution α* and find it. (b)State the Kuhn-Tucker condition for the boundary solution when α* = W . (c)Find the condition for the boundary solution to hold. (d)Show that E(U) is a concave function of α. 1 (Hint: E(U) = [-λe^(-aW)][e^(aα)] ---------- . Let x ≡aα aα + λ 1 and f(x) ≡ [e^x] ------- . x + λ 2 δ E(U) Show that f''(x)>0 so that ------------- <0.) δ α^2 PS.此處用δ代替偏微符號. a a 1 2.Let F(X,Y) = (X + Y )^(-----) , X,Y>0 , a<1. a (a)Show that F(X,Y) is homogeneous of degree 1. Y (b)Let y≡ ----- . F(X,Y) can be written as Xf(y). What is f(y)? X (c)Calculate F (X,Y) and show that it is homogeneous of degree 0. x (d)F (X,Y) can be written as g(y). What is g(y)? x 3.The utility of A (a consumer/borrower) is U(C1 ,C2) = C1 C2 , where C1 is the consumption today and C2 the consumption tomorrow. A has $ Y1 today and $ Y2 tomorrow, Y1<Y2. To simplify, assume that interest rate is r = 0 and therefore the budget constraint is C1 + C2 = Y1 + Y2. The consumer has a credit limit of $q so that C1 - Y1 <= q. The utility maximization problem is max C1C2 subject to: C1 + C2 = Y1 + Y2 , C1 - Y1 <= q. (a)State the Lagrangian of the problem and the Kuhn-Tucker conditions. (b)Find the solution(there are two cases). (c)For each case what is the marginal utility of income? (d)For each case what is the marginal utility of the credit limit q? 4.The extensive form of a 2-person game is as follows: L 1 R ／ ＼ 2／ ＼2 l ／ ＼r L ／＼ R [4] [ 2] [0] [8] [2] [10] [0] [4] (a)Use backward induction to find the sub-game perfect Nash Equilibrium of the game. (b)What are the pure strategy sets for players I and II. (c)Construct the normal(strategic) form of the game. (d)Find the other Nash Equilibrium and explain why it is not sub-game perfect. --

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