Question 1169746: 4) The equations defining a model of two trading nations are given by
๐ ๐ ๐ถ1 = 0.7๐1 + 50 ๐ถ2 = 0.8๐2 + 100 ๐ผ ๐ผ ๐1 = 0.3๐1 ๐2 = 0.1๐2
a. Express this system in matrix form and hence find the values of ๐1 and ๐2.
b.Calculate the balance of payments between these countries
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**Understanding the Model**
We are given a model of two trading nations, with the following equations:
* **Country 1:**
* Y1 = C1 + I1 + X1 - M1
* C1 = 0.7Y1 + 50
* M1 = 0.3Y1
* **Country 2:**
* Y2 = C2 + I2 + X2 - M2
* C2 = 0.8Y2 + 100
* M2 = 0.1Y2
Where:
* Y = National Income
* C = Consumption
* I = Investment (we'll assume I1 and I2 are exogenous, so we'll treat them as constants)
* X = Exports
* M = Imports
We also know that:
* X1 = M2 (Exports of country 1 are imports of country 2)
* X2 = M1 (Exports of country 2 are imports of country 1)
**a) Express the System in Matrix Form and Find Y1 and Y2**
1. **Substitute the Equations:**
* For Country 1:
* Y1 = (0.7Y1 + 50) + I1 + M2 - 0.3Y1
* Y1 = 0.4Y1 + 50 + I1 + 0.1Y2
* 0.6Y1 - 0.1Y2 = 50 + I1
* For Country 2:
* Y2 = (0.8Y2 + 100) + I2 + M1 - 0.1Y2
* Y2 = 0.7Y2 + 100 + I2 + 0.3Y1
* -0.3Y1 + 0.3Y2 = 100 + I2
2. **Matrix Form:**
We can write this system of equations in matrix form as:
```
[ 0.6 -0.1 ] [ Y1 ] = [ 50 + I1 ]
[ -0.3 0.3 ] [ Y2 ] = [ 100 + I2 ]
```
Let's represent this as:
```
A * Y = B
```
Where:
```
A = [ 0.6 -0.1 ]
[ -0.3 0.3 ]
Y = [ Y1 ]
[ Y2 ]
B = [ 50 + I1 ]
[ 100 + I2 ]
```
3. **Solve for Y (Y1 and Y2):**
To find Y, we need to calculate Aโปยน and multiply it by B:
```
Y = Aโปยน * B
```
First, find the determinant of A:
* det(A) = (0.6 * 0.3) - (-0.1 * -0.3) = 0.18 - 0.03 = 0.15
Now, find the inverse of A:
```
Aโปยน = (1 / det(A)) * [ 0.3 0.1 ]
[ 0.3 0.6 ]
Aโปยน = (1 / 0.15) * [ 0.3 0.1 ]
[ 0.3 0.6 ]
Aโปยน = [ 2 2/3 ]
[ 2 4 ]
```
Now, multiply Aโปยน by B:
```
Y = [ 2 2/3 ] [ 50 + I1 ]
[ 2 4 ] [ 100 + I2 ]
Y1 = 2(50 + I1) + (2/3)(100 + I2)
Y1 = 100 + 2I1 + 200/3 + (2/3)I2
Y1 = 500/3 + 2I1 + (2/3)I2
Y2 = 2(50 + I1) + 4(100 + I2)
Y2 = 100 + 2I1 + 400 + 4I2
Y2 = 500 + 2I1 + 4I2
```
**b) Calculate the Balance of Payments Between These Countries**
The balance of payments is the difference between exports and imports.
* Balance of Payments for Country 1 = X1 - M1 = M2 - M1 = 0.1Y2 - 0.3Y1
* Balance of Payments for Country 2 = X2 - M2 = M1 - M2 = 0.3Y1 - 0.1Y2
Let's calculate the balance of payments for Country 1:
* Balance of Payments 1 = 0.1Y2 - 0.3Y1
* Balance of Payments 1 = 0.1(500 + 2I1 + 4I2) - 0.3(500/3 + 2I1 + (2/3)I2)
* Balance of Payments 1 = 50 + 0.2I1 + 0.4I2 - 50 - 0.6I1 - 0.2I2
* Balance of Payments 1 = -0.4I1 + 0.2I2
Let's calculate the balance of payments for Country 2:
* Balance of Payments 2 = 0.3Y1 - 0.1Y2
* Balance of Payments 2 = 0.3(500/3 + 2I1 + (2/3)I2) - 0.1(500 + 2I1 + 4I2)
* Balance of Payments 2 = 50 + 0.6I1 + 0.2I2 - 50 - 0.2I1 - 0.4I2
* Balance of Payments 2 = 0.4I1 - 0.2I2
**Conclusion**
* Y1 = 500/3 + 2I1 + (2/3)I2
* Y2 = 500 + 2I1 + 4I2
* Balance of Payments for Country 1 = -0.4I1 + 0.2I2
* Balance of Payments for Country 2 = 0.4I1 - 0.2I2
The balance of payments for Country 2 is the negative of the balance of payments for Country 1, meaning that if one country has a surplus, the other has a deficit.
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