課程名稱︰經濟數學一 課程性質︰經濟系選修 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）︰2009/4/15 考試時限（分鐘）：兩小時 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : ┌ 1/6 1/3 1/2 ┐ │ │ ┌ x1 ┐ ┌ y1 ┐ 1.Let A = │ 1/6 1/3 1/2 │，x = │ x2 │，and y = │ y2 │ │ │ └ x3 ┘ └ y3 ┘ └ 1/6 1/3 1/2 ┘ (a) If x is in the null space of A, what condition(s) should x1,x2,x3 satisfy? (b)If y is in the range space of A, what condition(s) should y1,y2,y3 satisfy? In homework problem 3.4, it is shown that if A is an idempotent matrix, then [I-A] is idempotent and that the range space of A is the null space of [I-A] and vice versa. (c) Verify that A is idempotent. (d) Find N[I-A] and R[I-A]. ┌ 1 x 1 ┐ 2. Let B= │ 1 1 y │. └ 1 1 1 ┘ (a) If Rank[A] = 1, what is(are) the values of (x,y)? (b) Find the condition(s) on (x,y) such that Rank[A] = 3. (c) Find the condition(s) on (x,y) such that Rank[A] = 2. (d) Is it possible that Rank[A] = 0? Explain. 3. In an economic model there are 3 endogenous variables (x1,x2,x3) and one exogenous variable y. The 3 equations are: x2 F1(x1,x2,x3,y) = x1 + — - 1 - y = 0, x3 x3 F2(x1,x2,x3,y) = x2 + — - 2 = 0, 　　　　　　　　　　 x1 x1 F3(x1,x2,x3,y) = x3 + — - 2 = 0. x2 We consider the case where all variables are non-negative. (a) Let y=1. Find an equilibrium (x1*,X2*,X3*). (Hint: When y=1, then 3 equations are cyclically symmetric and hence there is a solution such that x1=x2=x3 .) ┌F11 F12 F13┐ (b) Calculate the Jacobian matrix J= │F21 F22 F23│. └F31 F32 F33┘ (c) Show that│J│≠ 0 at the equilibrium when y=1 so that the implicit function theorem is applicable in this case. (d) Calculate the derivatives dx1/dy , dx2/dy ,and dx3/dy at the equilibrium when y=1. (註:題目這裡的d實際上全是用偏微符號,BBS無法打出,抱歉) 4. Consider an income determination model with import and export: _ C=C(Y), I=I, M=M(Y), X=X(Y*,W), C+I+X-M=Y. where import M is a function of domestic income and export X is a function of foreign income Y* and foreign wealth W, both are assumed here as exogenous variables. And Cy>0, My>0, 1>Cy+My>0, Xy*>0, Xw>0. Substituting consumption, import, and export functions into the equilibrium condition, we have: _ C(Y) + I + X(Y*,W) - M(Y) = Y. (a) Use implicit function rule to derive dY/dW(實際上是偏微符號) and determine its sign. Now extend the above model to analyze the case when domestic income and foreign income are interdependent: _ _ C(Y) + I +X(Y*,W) - M(Y) = Y, C*(Y*,W) + I* + X*(Y) - M*(Y*,W) = Y* With a similar assumption on the foreigner's consumption and import functions: * * * * Cy* >0, Cw* >0, 1> Cy* + My* >0. Since domestic import is the same as foreigner's export and domestic export is foreigner's import, X*(Y) = M(Y) and M*(Y*,W)= X(Y*,W) and the system decomes: _ _ C(Y) + I + X(Y*,W) - M(Y) = Y, C*(Y*,W) + I* + M(Y) - X(Y*,W) = Y*, (b) Calculate the total differential of the system. (Now Y* becomes an endogenous variable.) (c) Use Cramer's rule to derive dY/dW and dY*/dW and determine their signs. (d) Compare the magnitude of dY/dW in answers (a) and (c): dY│ dY│ —│ and —│ . 　　dW│(a) dW│(c) That is, in which case a change in the foreign wealth W has a greater impact on domestic income? Explain Why. (註:同前述，(c)(d)兩小題的d實際上也是偏微符號.) --

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