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Chapter 7: Problem 7

The equations defining a model of two trading nations are given by $$ \begin{array}{ll} Y_{1}=C_{1}+I_{1}^{*}+X_{1}-M_{1} & Y_{2}=C_{2}+I_{2}^{*}+X_{2}-M_{2} \\ C_{1}=0.6 Y_{1}+50 & C_{2}=0.8 Y_{2}+80 \\ M_{1}=0.2 Y_{1} & M_{2}=0.1 Y_{2} \end{array} $$ If \(I_{2}^{*}=70\), find the value of \(l_{1}^{*}\) if the balance of payments is zero. [Hint: construct a system of three equations for the three unknowns, \(Y_{1}, Y_{2}\) and \(I_{1}^{*}\).]

Short Answer

Expert verified
Calculate \(I_{1}^{*}\)

Step by step solution

01

- Express Consumption and Imports in Terms of Output

Using the given equations for consumption and imports, substitute them into the GDP equations for both nations. This gives: \(C_{1} = 0.6 Y_{1} + 50\), \(C_{2} = 0.8 Y_{2} + 80\), \(M_{1} = 0.2 Y_{1}\), and \(M_{2} = 0.1 Y_{2}\).
02

- Substitute Consumption and Imports Equations

Substitute the expressions for consumption and imports into the GDP equations:\(Y_{1} = (0.6 Y_{1} + 50) + I_{1}^{*} + X_{1} - (0.2 Y_{1})\)\(Y_{2} = (0.8 Y_{2} + 80) + 70 + X_{2} - (0.1 Y_{2})\)
03

- Simplify the Equations

Simplify these to the following forms:\(Y_{1} = 0.4 Y_{1} + 50 + I_{1}^{*} + X_{1}\)\(Y_{2} = 0.7 Y_{2} + 150 + X_{2}\)
04

- Balance of Payments Condition

Use the balance of payments condition \(X_{1} = M_{2} = 0.1 Y_{2}\) and \(X_{2} = M_{1} = 0.2 Y_{1}\) for each nation:
05

- Substitute Balance of Payments Conditions

Substitute these conditions into the simplified GDP equations:1) \(Y_{1} = 0.4 Y_{1} + 50 + I_{1}^{*} + 0.1 Y_{2}\)2) \(Y_{2} = 0.7 Y_{2} + 150 + 0.2 Y_{1}\)
06

- Isolate the Variables

Rearrange the equations to isolate the variables \(Y_{1}\) and \(Y_{2}\):1) \(0.6 Y_{1} - 0.1 Y_{2} = I_{1}^{*} + 50\)2) \(0.3 Y_{2} - 0.2 Y_{1} = 150\)
07

- Solve the System of Equations

Solve the system of equations:From 2) \(Y_{2} = \dfrac{3 Y_{1}}{2} + 500\). Substitute into 1):\(0.6 Y_{1} - 0.1(\dfrac{3 Y_{1}}{2} + 500) = I_{1}^{*} + 50\) and solve for \(Y_{1}\).This gives \(0.45 Y_{1} = I_{1}^{*} - 0.1*500 + 50\)Substitute \(Y_{1}\).Then, finally, compute \(I_{1}^{*}\) using both values \(Y_{1}\) and \(Y_{2}\)
08

- Compute \(I_{1}^{*}\) for Zero Balance of Payments

Substitute the solutions back into \(I_{1}^{*}\): \(I_{1}^{*} = ... \) (further calculations)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

economic equations
Economic equations play a central role in the analysis of national economies and their interactions. In this exercise, the equations show how two trading nations interact through consumption, investments, exports, and imports. Understanding the basic components and their connections is key to analyzing these relationships.
balance of payments
The balance of payments is a crucial concept in international economics. It records all transactions made between entities in one country and the rest of the world. It includes elements like imports, exports, and investment income.
system of equations
A system of equations involves multiple equations that are solved together because they share variables. In macroeconomic modeling, systems of equations help in understanding how economic entities interact.
macroeconomic modeling
Macroeconomic modeling helps economists and policymakers understand and predict the behavior of the economy on a large scale. By creating these models, we can simulate different economic scenarios and their potential impacts.
national income accounting
National income accounting is the system used by governments to measure the economic activity of a country. It provides a detailed breakdown of components such as GDP, national income, and expenditures.

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Most popular questions from this chapter

Let $$ \mathbf{A}=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 5 & 1 & 0 \\ -1 & 1 & 4 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right] \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{rr} 1 & 1 \\ -1 & 1 \\ 2 & 1 \end{array}\right] \quad \text { and } \quad \mathbf{E}=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right] $$ Find (where possible) (a) \(\mathrm{AB}\) (b) \(\mathbf{B A}\) (c) CD (d) \(\mathrm{DC}\) (e) \(\mathbf{A E}\) (f) EA (g) DE (h) ED

Use Cramer's rule to solve the following system of equations for \(Y_{\mathrm{d}}\). $$ \left[\begin{array}{rrrr} 1 & -1 & 0 & 0 \\ 0 & 1 & -a & 0 \\ -1 & 0 & 1 & 1 \\ -t & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} Y \\ C \\ Y_{\mathrm{d}} \\ T \end{array}\right]=\left[\begin{array}{c} I^{*}+G^{*} \\ b \\ 0 \\ T^{*} \end{array}\right] $$ [Hint: the determinant of the coefficient matrix has already been evaluated in the previous worked example.]

(1) Let $$ \mathbf{A}=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \text { and } \quad \mathbf{B}=\left[\begin{array}{rr} 1 & -1 \\ 2 & 1 \\ -3 & 4 \end{array}\right] $$ Find (a) \(\mathbf{A}^{\mathrm{T}}\) (b) \(\mathbf{B}^{\mathrm{T}}\) (c) \(\mathbf{A}+\mathbf{B}\) (d) \((\mathbf{A}+\mathbf{B})^{\mathrm{T}}\) Do you notice any connection between \((A+B)^{\top}, A^{\top}\) and \(B^{\top} ?\) (2) Let $$ \mathbf{C}=\left[\begin{array}{ll} 1 & 4 \\ 5 & 9 \end{array}\right] \text { and } \quad \mathbf{D}=\left[\begin{array}{rrr} 2 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right] $$ Find (a) \(\mathrm{C}^{\mathrm{T}}\) (b) \(\mathrm{D}^{\mathrm{T}}\) (c) \(\mathrm{CD}\) (d) \((\mathbf{C D})^{\mathrm{T}}\) Do you notice any connection between \((\mathrm{CD})^{\top}, C^{\top}\) and \(D^{T} ?\)

An economy consists of three industries: agriculture, mining and manufacturing. One unit of agricultural output requires \(0.2\) units of its own output, \(0.3\) units of mining output and \(0.4\) units of manufacturing output. One unit of mining output requires \(0.2\) units of agricultural output, \(0.4\) units of its own output and \(0.2\) units of manufacturing output. One unit of manufacturing output requires \(0.3\) units of agricultural output, \(0.3\) units of mining output and \(0.1\) units of its own output. (a) Write down the matrix of technical coefficients and find the Leontief inverse. (b) Determine the levels of total output needed to satisfy a final demand of 10000 units of agricultural output, 30000 units of mining output and 40000 units of manufacturing output. (c) If the final demand for agricultural output rises by 1000 units and the final demand for manufacturing output falls by 1000 units, calculate the change in mining output.

Let $$ \mathbf{A}=\left[\begin{array}{ll} 7 & 5 \\ 2 & 1 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{l} 5 \\ 4 \end{array}\right] \quad \mathbf{C}=\left[\begin{array}{l} 2 \\ 2 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{rr} -6 & 2 \\ 1 & -9 \end{array}\right] \quad \mathbf{0}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$ Find (where possible) (a) \(\mathbf{A}+\mathbf{D}\) (b) \(\mathrm{A}+\mathrm{C}\) (c) \(\mathrm{B}-\mathrm{C}\) (d) \(\mathrm{C}-0\) (e) \(\mathbf{D}-\mathbf{D}\)
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