→ neverneverfu:多謝版友指正，已修正手誤^^" 01/26 23:45
課程名稱︰經濟數學一
課程性質︰系訂選修
課程教師︰周建富
開課學院：社會科學院
開課系所︰經濟系
考試日期（年月日）︰2006年1月11日
考試時限（分鐘）：兩小時。
是否需發放獎勵金：是。
試題 :
１.(Timing problem)
Let A(t,γ) = K[e^(t-t^2)][e^(-γt)] be the present value of an asset,
where γ is market interest rate.
(a) (5) Find the FOC of the maximization problem max_t A(t,γ).
(b) (5) Solve the FOC to derive the optimal value of t as a function of γ,
t^* = T(γ).
(c) (5) Check that the SOC is satisfied.
In general,let A(t,γ) = K[e^f(t)][e^(-γt)] and T(γ) the optimal value of
t that maximizes A(t,γ).
(d) (5) Determine the sign of T'(γ) and explain its meaning.
(e) (5) Let W(γ) = A(T(γ),γ),use envelop theory to find W'(γ) and
explain its meaning.
２.Consider the following 2-variable function:
F(X1.X2) = X1^(1-a) - 1 X2^(1-a) - 1
-------------- ＋ -------------
1-a 1-a
X1,X2,a 皆大於等於零且a不等於1
(a) (10) Use L'Hopital to find lim_a→1 F(X1,X2).
(b) (5) Calculate the Hessian and bordered Hessian matrices of F(X1,X2).
(c) (5) Determine the range of a such that F(X1,X2) is concave.
(d) (5) Determine the range of a such that F(X1,X2) is quasi-concave.
３.(Monopsonist (獨買) utility maximization)
The utility function of consumer A is U(x1,x2) = x1．x2.
He has $300 to spend on X1 and X2.He is a monopsonist in X1 market and the
supply function of X1 is x1 = p1.Therefore,Consumer A's budget constraint
is p1x1 + p2x2 = x1^2 + p2x2 = 300.The utility maximization problem is
max x1x2 Subject to x1^2 + p2x2 = 300.
(a) (5) State the Lagrangian and FOC for the maximization problem.
(b) (10) Find the solution of the utility maximization problem.
(c) (10) Use the bordered Hessian test to verify that the SOC is satisfied.
４.Consider the following nonlinear programming problem:
max U(X,Y) = lnX + lnY s.t. 2X + Y ≦ 24
X + Y ≦ q
X,Y皆大於等於零
Assume that 12 < q < 24 so that the intersection of the two constraints,
(x,y) = (24-q,2q-24) >> 0.
(a) (5) State the Lagrangian and Kuhn-Tucker conditions for the maximization
problem.
(b) (10) Assume that (x,y) = (24-q,2q-24) >> 0. Solve the Kuhn-Tucker
conditions to find the values of the two Lagrangian multipliers.
(c) (10) For what range of q the solution is (x,y) = (24-q,2q-24)?
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陷自己於忙碌之中 也許不寂寞但卻很孤獨......
陪伴的人也許很多 不孤獨卻不代表不寂寞......
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◆ From: 140.112.215.14
※ 編輯: neverneverfu 來自: 140.112.215.14 (01/26 23:42)