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1月12日。
是否需發放獎勵金：是。
試題 :
１.(15) Consider the following 2-variable function:
X1^a X2^b
F(X1,X2) = ------- - --------
a b X1≧0,X2≧0,a>0,b>0.
(a) Calculate the Hessian and bordered Hessian matrices of F(X1,X2). (5分)
(b) Determine the ranges of a and b such that F(X1,X2) is concave. (5分)
(c) Determine the ranges of a and b such that F(X1,X2) is quasi-concave. (5分)
２.(25) The utility function of a consumer is U(x1, x2) = [x1．x2].
The market price is p1 = p2 = 1 and the consumer has $12.
Therefore the budget constraint is x1+x2 ≦ 12.Suppose that both products
are under rationing.Besides the money price,the consumer has to pay ρ_i
rationing points for each unit of product i consumed.Assume that ρ1 = 1
and ρ2 = 2 and the consumer has q rationing points.The rationing point
constraint is x1+2x2≦q.The utility maximization problem is given by
max U(x1,x2) = [x1．x2] subject to x1+x2 ≦ 12
x1,x2
x1+2x2 ≦ q , x1,x2 ≧ 0
(a) State the Lagrangian function and derive the K-T conditions. (5分)
(b) Assume that 12 < q < 24.Solve the utility maximzation problem. (10分)
(There are 3 cases.)
Assume now that p1 = 1 and p2 = p so that the budget constraint is
x1+px2≦12 and that the consumer has 18 rationing points.
The utility maximization problem becomes
max U(x1,x2) = [x1．x2] subject to x1+px2 ≦ 12
x1,x2
x1+px2 ≦ 12 , x1+2x2 ≦ 18 , x1,x2 ≧ 0
(c) For p < (4/3),derive the comsumer's demand function x2(p).
(Also 3 cases.) (10分)
３.(20) Consider the following 2-market dynamic model:
. .
p1 = -4p1 - 2p2 + 20 , p2 = 2p1- 4p2 + 20
(a) Find the equilibrium prices p1^* and p2^*. (2分)
(b) Let y1 = p1-p1^* and y2 = p2-p2^*.Find the differential equations for y1
and y2. (2分)
.
(c) Eliminating y2 and y2 to derive the second order DE for y1. (4分)
(d) State the characteristic equation of the DE and find the characteristic
roots. (4分) (Hint: They are conjugate complex numbers.)
(e) Solve the DE for y1(t) and derive the solution of y2(t). (4分)
(f) Find p1(t) and p2(t). (2分)
(g) Is the equilibrium stable? Why or why not? (2分)
４.(12) The difference equation of a nonlinear dynamic multiplier model is
given by 61
Y_t = aY_t-1．[1-Y_t-1], a = 3.05 = ----
20
(a) Find the equilibrium Y^* > 0. (3分)
(b) Determine whether Y^* is stable. (3分)
36
(c) Let Y0 = ----
61 Calculate Y1 and Y2.What should be (Y3,Y4,Y5....)? (3分)
(d) Draw the phase diagram and the path of Y_t for a Y0 屬於 (36/61,Y^*) (3分)
┌1 4┐
５.(12) Let A = │ │
└2 3┘
(a) State the characteristic equation of A and find A's eigenvalues. (4分)
(b) Find A's eigenvectors. (4分)
(c) Let y_t+1 = Ay_t and y0 = (2,-1)'. Find y_t. (4分)
６.(16) Consider the following discrete time optimal control problem of an
optimal growth model:
∞
max U = Σ (0.5)^t．ln C_t subject to k_t+1 = 2(k_t-c_t), k0 = 12
t=0
(a) State the Lagrangian of the problem and derive the FOCs. (5分)
(b) Solve the equation system to find the solution paths of k_t and c_t. (6分)
The budget constraint of the optimal growth model is equivalent to the
following infinite-period budget constriant:
∞
Σ (0.5)^t．c_t = 12
t=0
Therefore, the maximization problem is equivalent to the following problem:
∞ ∞
max Σ (0.5)^t．ln C_t subject to Σ (0.5)^t．c_t = 12
t=0 t=0
(c) State the Lagrangian and FOC of the optimization problem.Solve it. (5分)
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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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