[試題] 97下 周建富 經濟數學一 期中考
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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04/19 00:43, , 1F
04/19 00:43, 1F