NEED MATRICES HELP (economics style matrix question)

[Baxter_Slade]

I am currently completing my undergraduate degree in economics. I have been given problem sheets on matrices for study and have no idea how to do the following question:

Consider the following Keynesian macroeconomic model:
Y = C + I + G
C = 200 + 0.8Y
I = 1000 − 2000R

where the endogenous variables are national income (Y), consumption (C) and investment (I), and where the exogenous variables are government spending (G) and the interest rate (R).

i. Write the above equations in a matrix form:

[Y]
A [C] = B (Note, that's 1 vertical matrix (Vector) [Y,C,I])
[I ]
where A is a 3 × 3 matrix and B is a 3 × 1 matrix with entries involving the exogenous variables.


ii. Find the inverse of A and solve for Y, C and I


Any help that you could give me would be most appreciated

 

[stapel]

I...have no idea how to do the following question:

Was this material (the terms, concepts, techniques, etc) not covered in class?

Consider the following Keynesian macroeconomic model:
Y = C + I + G
C = 200 + 0.8Y
I = 1000 − 2000R....

i. Write the above equations in a matrix form:

[Y]
A [C] = B (Note, that's 1 vertical matrix (Vector) [Y,C,I])
[I ]
where A is a 3 × 3 matrix and B is a 3 × 1 matrix with entries involving the exogenous variables.

Would it be correct to read the desired equation to be of the following form?

. . . . .\(\displaystyle \left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]\, \left[\begin{array}{c}Y\\C\\I\end{array}\right]\, =\, \left[\begin{array}{c}b_1\\b_2\\b_3\end{array}\right]\)

...with:

. . . . .\(\displaystyle A\, =\, \left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]\, \mbox{ and }\, B\, =\, \left[\begin{array}{c}b_1\\b_2\\b_3\end{array}\right]\)

ii. Find the inverse of A and solve for Y, C and I

If you were given the completed matrix equation, would you be able to find the inverse?

Thank you!

 

[Ishuda]

I am currently completing my undergraduate degree in economics. I have been given problem sheets on matrices for study and have no idea how to do the following question:

Consider the following Keynesian macroeconomic model:
Y = C + I + G
C = 200 + 0.8Y
I = 1000 − 2000R

where the endogenous variables are national income (Y), consumption (C) and investment (I), and where the exogenous variables are government spending (G) and the interest rate (R).

i. Write the above equations in a matrix form:

[Y]
A [C] = B (Note, that's 1 vertical matrix (Vector) [Y,C,I])
[I ]
where A is a 3 × 3 matrix and B is a 3 × 1 matrix with entries involving the exogenous variables.


ii. Find the inverse of A and solve for Y, C and I


Any help that you could give me would be most appreciated

Generally in a matrix equation you like to get all of one kind of variable [the 'unknowns'] on one side and the other variables [the 'knowns'] on the other. The number of unknowns is usually the same number as the rows of equations you have. In this case you are told which variables the unknowns are in part ii (variables Y, C, and I). The next thing to do is to always put the unknowns in the same order for each equation (row). Doing this we have

[FONT=courier new]   1 Y  - 1 C - 1 I =  G
-0.8 Y  + 1 C + 0 I = 200
   0 Y  + 0 C + 1 I = 1000 − 2000R
[/FONT]

Now just "read off" the matrices A and B. For example, the first row of A is (1, -1, -1).