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）︰2005年11月9日
考試時限（分鐘）：兩小時。
是否需發放獎勵金：是
試題 :
┌1 2 3┐ ┌x1┐ ┌y1┐
１.Let A = │4 5 6│, x = │x2│, y = │y2│
└7 8 9┘ └x3┘ └y3┘
(a) Calculate Ax.
(b) If x is in the null space of A,what condition(s) should x1,x2,x3 satisfy?
(c) What is the dimension of the null space of A?
(d) If y is in the range space of A,what condition(s) should y1,y2,y3 satisfy?
(e) What is the dimension of the range space of A?
(f) Determine the rank of A.
┌f(x) a12 a13┐ ┌b11 b12 b13┐
２.Let A = │a21 a22 a23│, B = A^(-1) = │b21 b22 b23│
└a31 a32 a33┘ └b31 b32 b33┘
where aij,ij≠11,are constants.
d|A|
(a) Calculate the derivative of |A| with respect to x,----
dx
db11
(b) Calculate the derivative of b11 with respect to x,----
dx
(c) (Extension) Let A be an n*n square matrix and F(x) be the ij-th element
of A.Assume that all the other elements of A are not functions of x.
Calculate d|A| (Hint: Use Laplace expansion theorem to expand |A| on the
---- i-th row of A.)
dx
３.In an economic model there are two endogenous variables (x1,x2) and one
exogenous variable y.The two equations are
F(x1,x2,y) = f(x1,x2)-y = 0 G(x1,x2,y) = g(x1,x2)-y = 0
┌dF/dx1 dF/dx2┐
(a) Calculate the Jacobian matrix J = │ │
└dG/dx1 dG/dx2┘
(b) Assume that |J| ≠ 0 so that the implicit function theorem is applicable
in this case.Calculate the derivatives dx1/dy and dx2/dy.
４.(Short run vs long run)
In a partial market equilibrium model,the demand and supply functions are
I
Qd = --- Qs = NP
P
where I is consumers'income and N is the number of firms in the market.
The short run equilibrium condition is Qd = Qs
(a) Calculate the short run equilibrium price P^* and quantity Q^*
as functions of I and N.
(b) In the long run the number of firms becomes an endogenous variable and
is determined by a 0-profit condition.Assume that this condition is given
by N = PQ.Calculate the long run equilibrium _,_,and _ as functions of I.
P Q N
５.Consider now the general model
Demand function: Q = D(P,I) dP/dQ < 0 dD/dI > 0
Supply function: Q = S(P,N) dS/dP > 0 dS/dN > 0
0-profit condition: N = F(P,Q) dF/dP > 0 dF/dQ > 0
(a) In the long run equilibrium model,which variables are endogenous and
which are exogenous?
(b) Find the total differential of the model.
(c) Derive the comparative statics dQ/dI,dP/dI,and dN/dI.
(d) Determine the sings of the comparative statics.
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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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