๐ฌ Tutorial problems theta \(\theta\)#
Note
This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.
\(\theta\).1#
Find the inverse matrix for the following matrix or show that it does not exist:
[Haeusslerย Jr and Paul, 1987] Section 8.6, Example 3, Part (b)
Note that
Thus we know that \(A\) is non-singular, so that \(A^{-1}\) exists. We will use the adjoint matrix method to find it. Note that
and
Hence the cofactor matrix associated with \(A\) is
This means that the adjoint matrix associated with \(A\) is
Thus the inverse matrix for \(A\) is
\(\theta\).2#
Find the inverse matrix for the following matrix or show that it does not exist:
[Haeusslerย Jr and Paul, 1987] Section 8.6, Example 5
Note that
Since the determinant of the matrix \(B\) is zero, we know that \(B\) is a singular matrix. This means that it is not invertible, so that \(B^{-1}\) does not exist.
\(\theta\).3#
Apply Gauss-Jordan elimination to an appropriate augmented row matrix to solve the following system of equations:
[Bradley, 2013] Progress Exercises 9.3, Question 1
Consider the following system of three linear equations in three unknown variables:
This system of equations can be rewritten as
The augmented row matrix representation of this system of equations is
Note that
Thus we can conclude that the unique solution to this system of equations is
\(\theta\).4#
This problem is a part of an application of a linear version of a three market Marshallian cross model.
Solve the following system of linear equations:
[Bradley, 2013] Progress Exercises 9.3, Question 4
Consider the following system of three linear equations in three unknown variables:
This system of equations can be rewritten as
The augmented row matrix representation of this system of equations is
Note that
Thus we can conclude that the unique solution to this system of equations is
\(\theta\).5#
This problem is a linear version of a three market Marshallian Cross model.
Assuming that all three markets are in equilibrium, solve the following system of linear equations.
Hint: There are two ways that you might proceed here.
You could use the equilibrium (market clearing) conditions for the three markets to construct a system of six linear equations in six unknown variables. In this approach, the unknown variables are the three equilibrium quantities \(\left(Q_{1}, Q_{2}\right.\) and \(\left.Q_{3}\right)\) and the three equilibrium prices \(\left(P_{1}, P_{2}\right.\) and \(P_{3}\) ).
Alternatively, you could use use the equilibrium (market clearing) conditions for the three markets to construct a system of three linear equations in three unknown variables. In this case, the unknown variables are the three equilibrium prices \(\left(P_{1}, P_{2}\right.\) and \(\left.P_{3}\right)\). If you employ this approach, you will need to use some of the original equations and the equilibrium values for the prices to obtain the three equilibrium quantities \(\left(Q_{1}, Q_{2}\right.\) and \(\left.Q_{3}\right)\).
[Bradley, 2013] Progress Exercises 9.3, Question 8
Approach One
Consider the following three market linear Marshallian cross model:
Using the three equilibrium (market clearing) conditions, we can rewrite this model as a system of six linear equations in six unknown variables as follows:
The augmented row matrix representation of this system is
Note that
Thus we can conclude that the equilibrium quantities and prices are
Approach Two
Note that
\(Q_{1}^{D}=Q_{1}^{S} \Longleftrightarrow 50-2 P_{1}+5 P_{2}-3 P_{3}=8 P_{1}-5 \Longleftrightarrow 10 P_{1}-5 P_{2}+3 P_{3}=55\),
\(Q_{2}^{D}=Q_{2}^{S} \Longleftrightarrow 22+7 P_{1}-2 P_{2}+5 P_{3}=12 P_{2}-5 \Longleftrightarrow-7 P_{1}+14 P_{2}-5 P_{3}=27\), and
\(Q_{3}^{D}=Q_{3}^{S} \Longleftrightarrow 17+1 P_{1}+5 P_{2}-3 P_{3}=4 P_{3}-1 \Longleftrightarrow-1 P_{1}-5 P_{2}+7 P_{3}=18\).
Thus we have the following system of three linear equations in three unknown variables:
Equation (1): \(10 P_{1}-5 P_{2}+3 P_{3}=55\),
Equation (2): \(-7 P_{1}+14 P_{2}-5 P_{3}=27\),
and
Equation (3): \(-1 P_{1}-5 P_{2}+7 P_{3}=18\).
The augmented row-matrix representation of this three equation system is
We now apply the Gauss-Jordan elimination procedure to this augmented row matrix.
Thus we can conclude that the equilibrium prices are \(P_{1}^{*}=\$ 7\) per unit, \(P_{2}^{*}=\$ 9\) per unit, and \(P_{3}^{*}=\$ 10\) per unit.
Upon substituting \(P_{1}=\$ 7\) into the supply equation for commodity one, we obtain
Upon substituting \(P_{2}=\$ 9\) into the supply equation for commodity two, we obtain
Upon substituting \(P_{3}=\$ 10\) into the supply equation for commodity three, we obtain
Thus we can conclude that the equilibrium quantities are \(Q_{1}^{*}=51\) units, \(Q_{2}^{*}=103\) units, and \(Q_{3}^{*}=39\) units.
\(\theta\).6#
This problem is an example of the Keynesian cross model.
Use Cramerโs rule to solve the following system of linear equations:
[Bradley, 2013] Progress Exercises 9.4, Question 9
Consider the following Keynesian cross model of a closed economy in which there is no government sector:
The equilibrium condition is given by equation (E1). The consumption function is given by equation (C1). Note that \(b, C_{0}\) and \(I_{0}\) are exogenous parameters. Note also that \(0<b<1\). Recall that \(b\) is the marginal propensity to consume, \(C_{0}\) is autonomous consumption and \(I_{0}\) is autonomous investment. This model can rewritten as
This system of two linear equations in two unknown variables can be expressed as the matrix equation
where
and
Note that
and
We have
and
Thus we have
and
\(\theta\).7#
Employ Gauss-Jordan elimination to determine the possible solutions of the following system of linear equations for different values of \(a\) and \(b\) :
[Sydsรฆter, Hammond, Strรธm, and Carvajal, 2016] Section 15.6, Problem 2
This question is left as an exercise.
\(\theta\).8#
Consider the matrix equation \(A x=b\) where
\(x=\left(\frac{\partial \lambda}{\partial p_{1}}, \frac{\partial x_{1}}{\partial p_{1}}, \frac{\partial x_{2}}{\partial p_{1}}\right)^{T}\) and \(b=\left(x_{1}, \lambda, 0\right)^{T}\). This equation might have been derived by conducting a comparative static exercise on the first-order conditions of a consumerโs budget-constrained utility maximisation problem. Use Cramerโs rule to find the comparative static effects \(\frac{\partial x_{1}}{\partial p_{1}}\) and \(\frac{\partial x_{2}}{\partial p_{1}}\).
This question is left as an exercise.