🔬 Tutorial problems theta \theta

🔬 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:

\[\begin{split} A=\left(\begin{array}{cc} 3 & -2 \\ 6 & 4 \end{array}\right) \end{split}\]

[Haeussler Jr and Paul, 1987] Section 8.6, Example 3, Part (b)

\(\theta\).2#

Find the inverse matrix for the following matrix or show that it does not exist:

\[\begin{split} B=\left(\begin{array}{ccc} 1 & -2 & 1 \\ 2 & -1 & 5 \\ 1 & 1 & 4 \end{array}\right) \text {. } \end{split}\]

[Haeussler Jr and Paul, 1987] Section 8.6, Example 5

\(\theta\).3#

Apply Gauss-Jordan elimination to an appropriate augmented row matrix to solve the following system of equations:

\[\begin{split} \left\{\begin{array}{ccccc} 3 x+3 y+6 z & = & 12 & \cdots & (\text { Equation } 1) \\ x-3 y+5 z & = & 5 & \cdots & (\text { Equation } 2) \\ 2 x+10 y-3 z & = & 0 & \cdots & (\text { Equation } 3) \end{array}\right\} . \end{split}\]

[Bradley, 2013] Progress Exercises 9.3, Question 1

\(\theta\).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:

\[\begin{split} \left\{\begin{array}{ccccc} P_{1}+4 P_{2}+8 P_{3} & = &26 & \cdots & (\text { Equation } 1) \\ 5 P_{1}+7 P_{2} & = &38 & \cdots & (\text { Equation } 2) \\ 8 P_{1}+12 P_{2}+2 P_{3} & = & 66 & \cdots & (\text { Equation } 3) \end{array}\right\} . \end{split}\]

[Bradley, 2013] Progress Exercises 9.3, Question 4

\(\theta\).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.

\[\begin{split} \left\{\begin{array}{cccccc} Q^D_1 & = & 50 - 2P_1 + 5P_2 - 3P_3 & & & \cdots(D1) \\ Q^S_1 & = & 8P_1 - 5 & & & \cdots(S1) \\ Q^D_1 & = & Q^S_1 & = & Q_1 & \cdots(E1) \\ Q^D_2 & = & 22 + 7P_1 - 2P_2 + 5P_3 & & & \cdots(D2) \\ Q^S_2 & = & 12P_2 - 5 & & & \cdots(S2) \\ Q^D_2 & = & Q^S_2 & = & Q_2 & \cdots(E2) \\ Q^D_3 & = & 17 + P_1 + 5P_2 - 3P_3 & & & \cdots(D3) \\ Q^S_3 & = & 4P_3 - 1 & & & \cdots(S3) \\ Q^D_3 & = & Q^S_3 & = & Q_3 & \cdots(E3) \\ \end{array}\right\} . \end{split}\]

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

\(\theta\).6#

This problem is an example of the Keynesian cross model.

Use Cramer’s rule to solve the following system of linear equations:

\[\begin{split} \left\{\begin{array}{lll} Y=C+I_{0} & \cdots(E 1) \\ C=C_{0}+b Y & \cdots(C 1) \end{array}\right\} \end{split}\]

[Bradley, 2013] Progress Exercises 9.4, Question 9

\(\theta\).7#

Employ Gauss-Jordan elimination to determine the possible solutions of the following system of linear equations for different values of \(a\) and \(b\) :

\[\begin{split} \left\{\begin{array}{ccccc} x+y-z & = & 1 & \cdots & (\text { Equation } 1) \\ x-y+2 z & = & 2 & \cdots & (\text { Equation } 2) \\ x+2 y-a z & = & b & \cdots & (\text { Equation } 3) \end{array}\right\} . \end{split}\]

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Section 15.6, Problem 2

\(\theta\).8#

Consider the matrix equation \(A x=b\) where

\[\begin{split} A=\left(\begin{array}{ccc} 0 & -p_{1} & -p_{2} \\ -p_{1} & U_{11} & U_{12} \\ -p_{2} & U_{21} & U_{22} \end{array}\right) \end{split}\]

\(x=\left(\frac{\partial \lambda}{\partial p_{1}}, \frac{\partial x_{1}}{\partial p_{1}}, \frac{\partial x_{2}}{\partial p_{1}}\right)^{T}\) 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 consumer’s budget-constrained utility maximisation problem. Use Cramer’s rule to find the comparative static effects \(\frac{\partial x_{1}}{\partial p_{1}}\) and \(\frac{\partial x_{2}}{\partial p_{1}}\).

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