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Econ2125\6012, Semester-1 2023 Assignment-2
发布时间：2023-05-29
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Assignment-2
Econ2125\6012, Semester-1 2023
The eigenvalues, eigenvectors in Question-1 needs to be calculated by hand (not by computer).
Questions-1: (9 mark)
A car rental company has three locations. Every customer can rent from and return their car to any of these locations. Let x(n) = [x1 (n), x2 (n), x3 (n)]T be the percentage of cars available in the i-th location at time n and A be the matrix whose i,j-entry is the probability that a customer renting a car from location j at time n-1 returns it to location i at time n.
( 0.3 0.4 0.5)
| |
A = | 0.3 0.4 0.3 |
|\0.4 0.2 0.2)|
(i) Write down the equation for dynamics of the system. (ii) Solve the equation for x(n) .
(iii) What is the long-run percentage of the cars at each location?
Questions-2: (8 marks)
Determine the positive or negative definiteness of the following quadratic form.
2 2 2 (x1 + x2 + x3 = 0
Questions-3: (9 marks)
Consider the following unit cost function
1 n n
ln C(p) = 2 xxaij lnpi ln pj with aij = aji
where pi is the price of i-th input and aij s are the coefficients .
(i) C(p) is expected to be homogenous of degree-1, what conditions needs to be imposed on the coefficients to achieve the homogeneity property.
(ii) The equation for share of each input in costs can be obtained using si = ? ln pi .
Find the equation for si .
「 ?2 ln C ]
|L? ln pi ? ln pj 」|
Questions-4: (8 marks)
The economy of Victoria is in equilibrium when the system of equations
2xz + xy + z − 2 
= 11
xyz = 6
are at x=3; y=2 and z=1. Suppose the premier can change variable z by a simple decree.
(i) If premier raises z to 1.1, use calculus to estimate the change in x and y.
(ii) If x were in the control of premier and not y or z, can this method be used to
estimate the effect of reducing x from 3 to 2.95? Explain your answer.
Question-5: (8 marks)
Consider the minimisation problem
Minq Q = (yn
1 − Xn
k qk
1)T Ω
(yn
1 − Xn
k qk
1)
where the subscripts are the dimensions of the matrices. y, X and Ω are known matrices, Ω
?Q
?q
.
(i) Using matrix derivatives show that q* = (XT Ω−1X)−1 (XT Ω−1y )
(ii) Let y be the only stochastic term with Var(y) = Ω , show that Var(q* ) = (XT Ω −1X)−1 where Var means variance.
Question-6 (challenge question): (8 marks)
Let A be a K 
K symmetric positive definite matrix and x and y be two K 
1vectors. Knowing that je−x2
2adx = 
when a > 0 , prove the following
R
xT A −1x+yT A −1y
j e 2 dxdy =| A |
