Exercise1 B

Exercise Session 2, Oct 27th ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva

Exercise 1 Suppose that we have an economy with labor, transportation, and food industries. Let $1 in labor require 40 cents in transportation and 20 cents in food; while $1 in transportation takes 50 cents in labor and 30 cents in transportation; and $1 in food production uses 50 cents in labor, use 5 cents in transportation, and 35 cents in food. Let the demand for the current production period be $10; 000 labor, $20; 000 transportation, and $10; 000 food. Find the production schedule for the economy. Exercise 2 a) Compute

D=

3 5 2 1

1 1 0 5

1 3 1 3

2 4 : 1 3

b) Calculate rank(A), jAj, tr(A), A 1 for 2 3 1 4 7 A = 4 3 2 5 5: 5 2 8

Exercise 3 Given a vector y and a matrix X, we can always write y = Xb + e where e is the di¤ erence between y and Xb. e is often call the residual, or error term. The problem is to …nd the b such that y is as close as possible to Xb. The geometric solution in Euclidean space is the b that makes e perpendicular or orthogonal to Xb. Note that two nonzero vectors a and b are orthogonal, written a ? b if and only if a0 b = b0 a = 0: Solve for the optimal b depending on the rank of X and y. Exercise 4 A very useful matrix which has been used in econometrics is 1 0 11 n where 1 is an n 1 vector of ones. It can be used to transform data to deviations from their mean. a) Show that M 0 is symmetric idempotent. Hint: it must satisfy (M 0 )0 M 0 = M 0 : n P b) To obtain the sum squared deviations about the mean ( (xi x)2 ), comM 0 = In

i=1

pute x0 (M 0 )0 M 0 x:

1

Exercise 5 For what values of the constant p and q does the following system have a unique solution, several solutions, or no solution?

2x1 3x1

x1 + x2 + x3 3x2 + 2x3 2x2 + px3

= 2q = 4q = q

APPENDIX The following rules for manipulating determinants may be useful: 1. The multiplication of any one row (or column) by a scalar k will change the value of the determinant k-fold. 2. The interchange of any two rows (columns) will alter the sign but not the numerical value of the determinant. 3. If a multiple of any row is added to (or subtracted from) any other row it will not change the determinant. The same holds for columns. 4. If two rows (or columns) are identical, the determinant will vanish. 5. The determinant of a triangular matrix is a product of its principal diagonal elements.

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