2 . Put the following macroeconomic model into matrix format using Y, C and T as the variables and find the determinant of the matrix of coefficients. There is no need to solve the equations.
Y = C + I
C = a + b (Y – T)
T = tY
Kindly help me with converting it into matrix form.
To convert the given macroeconomic model into matrix form, we can write it as a system of linear equations. Let's define the variables and coefficients as follows:
Variables:
X = [Y, C, T]^T
Coefficients:
A = [1, -1, 0]
B = [0, b, 0]
C = [0, 0, t]
Equations:
Y = C + I
C = a + b (Y – T)
T = tY
Now, we can write the system of equations in matrix form as:
AX = B + CY
Where:
A is the matrix of coefficients
X is the matrix of variables
B is the constant matrix on the right-hand side
C is the matrix of coefficients multiplying Y
In this case, the matrix form of the macroeconomic model is:
[1, -1, 0] [Y] [0] [0]
[0, b, 0] [C] + [a] + [0] [Y]
[0, 0, t] [T] [0] [0]
Now, to find the determinant of the matrix of coefficients, we can consider the matrix A:
A = [1, -1, 0]
[0, b, 0]
[0, 0, t]
The determinant of A can be calculated as follows:
det(A) = (1 * b * t) - (0 * -1 * 0) - (0 * 0 * t) = b * t
Therefore, the determinant of the matrix of coefficients is b * t.
To convert the given macroeconomic model into matrix form, we need to express the equations using matrices and then find the determinant of the matrix of coefficients.
Let's first define the variables as column vectors:
Y = [Y]
C = [C]
T = [T]
Now, let's rewrite the equations in matrix form:
Equation 1: Y = C + I
This equation suggests that Y is equal to the sum of C and I. Since we don't have the variable I in the given equations, we can treat it as a constant. So, the equation can be written as:
Y = C + [I]
Since [I] is a constant, we can call it matrix A, where A = [I]. Therefore, the equation can be rewritten as:
Y = C + A ---(Equation 2)
Equation 2: C = a + b(Y - T)
This equation suggests that C is equal to a constant term a plus b multiplied by the difference between Y and T. Rearranging the equation, we get:
C - b(Y - T) = a
Simplifying further:
C - bY + bT = a
Let's define the constant terms as a column vector:
B = [a]
C - bY = [C - bY]
bT = [bT]
Therefore, the equation can be written as:
[C - bY] + bT = [a] ---(Equation 3)
Equation 3: T = tY
This equation suggests that T is equal to t multiplied by Y. Rearranging the equation, we get:
Y - tY = T
Simplifying further:
(1 - t)Y = T
Now, let's define the variables in the equation:
(1 - t)Y = [1 - t] * [Y]
T = [T]
Therefore, the equation can be written as:
(1 - t)Y = T ---(Equation 4)
Now, let's summarize the equations in matrix form:
Equation 2: Y = C + A
Equation 3: [C - bY] + bT = [a]
Equation 4: (1 - t)Y = T
The matrix form of the equations can be expressed as:
| 1 -1 0 0 | | Y | | C + [I] |
| -b 1 b 0 | | C | = | a |
| 0 0 1 -t | | T | | 0 |
Finally, to find the determinant of the matrix of coefficients, we need to calculate the determinant of the left-hand side matrix:
Coefficients Matrix = | 1 -1 0 0 |
| -b 1 b 0 |
| 0 0 1 -t |
The determinant of the Coefficients Matrix will provide us with the answer.