🔬 Tutorial problems eta

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 are for additional practice. The symbol 🍹 indicates additional problems.

\(\eta\).1

Formulation

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

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

This question comes from Haeussler and Paul (1987, pp. 281-282, Section 8.6, Example 3, Part (b))

Solution

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\(\eta\).2

Formulation

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

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

This question comes from Haeussler and Paul (1987, pp. 283-284, Section 8.6, Example 5).

Solution

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\(\eta\).3

Formulation

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

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

This question comes from Bradley (2008, p. 501, Progress Exercises 9.3, Question 1).

Solution

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\(\eta\).4

Formulation

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:

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

This example comes from Bradley (2008, p. 502, Progress Exercises 9.3, Question 4).

Solution

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\(\eta\).5

Formulation

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 \(\left.Q_{3}\right)\).

This example comes from Bradley (2008, p. 502, Progress Exercises 9.3, Question 8).

Solution

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\(\eta\).6

Formulation

This problem is an example of the Keynesian cross model.

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

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

This example comes from Bradley (2008, p. 510, Progress Exercises 9.4, Question 9).

Solution

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\(\eta\).7 🍹

Formulation

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

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

This question comes from Sydsaeter and Hammond (2006, p. 575, Section 15.6, Problem 2).

Solution

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\(\eta\).8 🍹

Formulation

This question is an example of the IS-LM macroeconomic model of an economy. The ISLM model consists of two equations. The first equation is the IS equation. It describes the locus of equilibrium points in the goods market. The “IS” term comes from one version of the goods market equilibrium condition, in which investment (\(I\)) must equal savings \((S)\). The second equation is the LM equation. It describes the locus of equilibrium points in the money market. The “LM” term comes from the money market equilibrium condition, in which money demand (\(L\)) must equal money supply (\(M\)). The letter \(L\) stands for liquidity preference, which is one potential source of the demand for money. Note that there is implicitly a third market in this model. This is the bond market. It does not need to be formally considered because Walras’ law ensures that if all but one of the markets in an economy clear, then the remaining market will also clear.

The bond market is implicitly present because there is an opportunity cost involved with holding money balances. This opportunity cost takes the form of the foregone interest which could have been earned if those money balances had been used to purchase bonds. If both the goods market and the money market are simultaneously in equilibrium, then the economy as a whole is in “general equilibrium”. (Further information on the IS-LM model at the intermediate (second year undergraduate) level can be found in Levacic and Rebman (1982), Parkin and Bade (1990), and Wells (1995).)

Assuming that the economy is in a situation of general equilibrium, obtain the equations for the IS curve and LM curve from the equations below and use Cramer’s rule to solve that system of two linear equations in two unknowns to find the equilibrium values of \(Y\) and \(r\). The equations underlying this model are:

Goods Market:

\[\begin{split} \left\{\begin{array}{cccc} Y^{D} & = & C+I & \cdots(G D) \\ C & = & 237.8+0.2 Y & \cdots(C D) \\ I & = & 10-0.4 r & \cdots(I D) \\ Y^{S} & = & Y & \text { (Keynesian GS) } \\ Y^{D} & = & Y^{S} & \text { (GMEquil) } \end{array}\right\} \text{; and} \end{split}\]

Money Market:

\[\begin{split} \left\{\begin{array}{cccc} M^{D} & = & 100+0.1 Y-0.3 r & \cdots(M D) \\ M^{S} & = & 129.225 & (M S) \\ M^{D} & = & M^{S} & (M M E q u i l) \end{array}\right\}. \end{split}\]

This example comes from Bradley (2008, p. 510, Progress Exercises 9.4, Question 10).

Solution

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\(\eta\).9 🍹

Formulation

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 \(\frac{\partial x_{2}}{\partial p_{1}}\).

Solution

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