Elementary Mathematical Techniques for Economics

Instructor: Ram Sewak Dubey ECON 317: Problem Set 4 October 23, 2018

1. [Determinant value and unique solution to system of linear equations]

Use determinants to find out if a unique solution exists in each of the following system of linear equations.

(a)

12x1 +7x2 = 147 15x1 +19x2 = 168.

(b)

2x1 +3x2 = 27 6x1 +9x2 = 81.

(c)

4x1 +3x2 +5x3 = 27 x1 +6x2 +2x3 = 19 3x1 + x2 +3x3 = 15.

(d)

4x1 +2x2 +6x3 = 28 3x1 + x2 +2x3 = 20

10x1 +5x2 +15x3 = 70.

2. [Matrix Inversion in Market Equilibrium]

Find the equilibrium price for the following related markets for two goods by using the technique of matrix inversion.

(a)

18P1 −P2 = 87 −2P1 +36P2 = 98.

Submission deadline: October 31, 2018 Page 1 of 3

Instructor: Ram Sewak Dubey ECON 317: Problem Set 4 October 23, 2018

(b)

5P1 −2P2 = 15 −P1 +8P2 = 16.

(c)

11P1 −P2 −P3 = 31 −P1 +6P2 −2P3 = 26 −P1 −2P2 +7P3 = 24.

Please use the information that the inverse of 3×3 matrix

A =

11 −1 −1−1 6 −2 −1 −2 7

 is

A−1 = 1

401

38 9 89 76 23 8 23 65

 . 3. [Matrix Inversion in IS-LM Model]

(a) Given the IS equation 0.3Y +100r−252 = 0,

and the LM equation 0.25Y −200r−176 = 0,

find the equilibrium level of national income Y and the interest rate r by using the technique of matrix inversion.

(b) Given Y = C+ I0, where C = C0 + bY , find the equilibrium level of national income Y and the aggregate household consumption C by using the technique of matrix inver- sion.

Submission deadline: October 31, 2018 Page 2 of 3

Instructor: Ram Sewak Dubey ECON 317: Problem Set 4 October 23, 2018

4. [Cramer’s Rule]

Use Cramer’s rule to solve for the unknowns in each of the following.

(a) Equilibrium Prices in related markets for two goods

7P1 +2P2 = 60 P1 +8P2 = 78.

Solve for P1 and P2.

(b) Equilibrium national income Y and the interest rate r

0.4Y +150r = 209 0.1Y −250r = 35.

Solve for Y and r.

(c) Equilibrium Quantities in related markets for three goods

5q1 −2q2 +3q3 = 16 2q1 +3q2 −5q3 = 2 4q1 −5q2 +6q3 = 7.

Solve for q1, q2, and q3.

Submission deadline: October 31, 2018 Page 3 of 3