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ECON 331 - Homework #1

due in class October 2nd at 2.30pm

Late homework will not be considered. Show detailed calculations and/or provide detailed

explanations to get full credit. Partial credit may be given.

• Exercise 1. Show by recurrence that the following holds:

for any nonnegative integer n

• Exercise 2. Consider the following matrix A . We are looking for all the

matrices B that satisfy the equation AB = BA (∗).

(a) What is the size of B?

(b) For B = (bij ) (where the ranges of i and j are deduced from (a)), calculate the products

AB and BA.

(c) Rewrite the matrix equation (∗) as an equivalent system of linear equations with 4 unknowns.

(d) Solve the above system and show that all matrices B that satisfy (∗) are given by

B = αI2 + β(0 2 3 3 )

where I2 is the identity matrix of size 2 and α and β are 2 real numbers.

• Exercise 3. IS-LM model in a closed economy

The economy is made up of two sectors: the real goods sector and the monetary sector. The

goods sector involves the following equations:

C = a + b(1 − t)Y

I = d − ei

G = G0

where the endogenous variables are Y (national income), C (consumption expenditure), I

(Investment), i (interest rate) and the exogenous variable is G0 (government expenditure); the

structural parameters are: a > 0, 0 < b < 1, d > 0, e > 0, 0 < t < 1.

1

The money market involves the following equations:

{

Money demand Md = kY − li

Money supply Ms = M0

where M0 is the exogenous stock of money; k > 0, l > 0 are parameters.

In addition, we assume that: b(1 − t) ̸= 1 + (ek/l).

(a) Write the equilibrium conditions in each sector.

(b) Use (a) to write the general system of equations satisfied at the equilibrium on both

sectors. After simplifications, you should be able to rewrite this system of equations in

matrix form where the vector of unknowns is the vector of endogenous variables, that is

the following vector (Y c I i )′.

(c) Can you use Cramer’s rule to find the equilibrium national income Y

∗ Explain. If yes,

use Cramer’s rule to find it; if no, use Gaussian elimination to find it.

Note: you are not required to completely solve the system since we are only interested in

solving for Y∗

. You should find Y

∗ as linear combination of the exogenous variables.

• Exercise 4. Solve the following system of equations (with unknowns x, y, z) for all values

of the parameters a and b. Discuss also the number of degrees of freedom in the different cases.

x + 2y + 3z = 1

−x + ay − 21z = 2

3x + 7y + az = b

Note: you should find 3 different cases.

• Exercise 5. Let α be a real number and let M .

(a) Find det(M − I3), where I3 is the identity matrix of order 3.

(b) Put α = 1 and find a column vector of size 3,

such that Mv = v and v has

length 1.1

(c) What is Mn

v for n = 1, 2, · · · ? Explain.

1The length of a vector is defined as the square-root of the sum of the square of its elements: hence, in our

case,

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