Due Date At The Beginning Of Class On Wednesday
Due Date At The Beginning Of The Class On Wednesday 11282012
Analyze and solve two economic models: a microeconomic partial equilibrium model using matrix algebra and Cramer's rule, and a macroeconomic national income model. Demonstrate the process of setting up the models using matrices and vectors, and provide detailed solutions with step-by-step calculations to find the equilibrium values of price, quantity, and endogenous variables, respectively.
Paper For Above instruction
Introduction
Economics encompasses both microeconomic and macroeconomic analyses, often requiring algebraic and matrix-based methods to determine equilibrium states in markets and economies. This paper aims to solve two fundamental models: one microeconomic, involving the determination of price and quantity for a widget market through matrix algebra and Cramer's rule; and the other macroeconomic, focusing on the calculation of national income and consumption using a simple Keynesian cross model. Both models necessitate the construction of matrices, vectors, and the application of linear algebra techniques to derive the solutions explicitly, facilitating a deeper understanding of the underlying economic relationships.
Microeconomic Model: Price and Quantity Equilibrium
The microeconomic scenario involves two equations representing supply and demand in a widget market:
\[
Q = P \quad \text{(1)}
\]
\[
Q = 40 + 20P \quad \text{(2)}
\]
The goal is to find the equilibrium price (P) and quantity (Q) using matrix algebra and Cramer's rule, considering the general form of simultaneous equations:
\[
A \mathbf{x} = \mathbf{y}
\]
where \(A\) is the coefficients matrix, \(\mathbf{x}\) is the vector of endogenous variables, and \(\mathbf{y}\) is the constants vector.
Step 1: Rewriting the system in standard form
Bring both equations to the form:
\[
P - Q = 0 \quad \text{(from (1))}
\]
\[
-20P + Q = 40 \quad \text{(from (2))}
\]
Expressed in matrix form:
\[
\begin{bmatrix}
1 & -1 \\
-20 & 1
\end{bmatrix}
\begin{bmatrix}
P \\
Q
\end{bmatrix}
=
\begin{bmatrix}
0 \\
40
\end{bmatrix}
\]
Here:
\[
A = \begin{bmatrix} 1 & -1 \\ -20 & 1 \end{bmatrix}
\]
\[
\mathbf{x} = \begin{bmatrix} P \\ Q \end{bmatrix}
\]
\[
\mathbf{y} = \begin{bmatrix} 0 \\ 40 \end{bmatrix}
\]
Step 2: Calculate the determinant of \(A\)
\[
\det(A) = (1)(1) - (-1)(-20) = 1 - 20 = -19
\]
Step 3: Apply Cramer's rule for \(P\) and \(Q\)
- For \(P\), replace the first column of \(A\) with \(\mathbf{y}\):
\[
A_P = \begin{bmatrix} 0 & -1 \\ 40 & 1 \end{bmatrix}
\]
\[
\det(A_P) = (0)(1) - (-1)(40) = 0 + 40 = 40
\]
- For \(Q\), replace the second column of \(A\) with \(\mathbf{y}\):
\[
A_Q = \begin{bmatrix} 1 & 0 \\ -20 & 40 \end{bmatrix}
\]
\[
\det(A_Q) = (1)(40) - (0)(-20) = 40 - 0 = 40
\]
Step 4: Compute \(P^\) and \(Q^\)
\[
P^* = \frac{\det(A_P)}{\det(A)} = \frac{40}{-19} \approx -2.105
\]
\[
Q^* = \frac{\det(A_Q)}{\det(A)} = \frac{40}{-19} \approx -2.105
\]
Step 5: Interpret the solution
The negative values indicate an inconsistency with realistic market outcomes, likely due to the algebraic form. Nonetheless, mathematically, the equilibrium price and quantity are approximately \(-2.105\). For economic interpretation, the original equations should be re-examined for compatibility, but the algebraic method remains valid.
---
Macroeconomic Model: National Income and Consumption
The macroeconomic framework involves the following equations:
\[
Y = C + I + G
\]
\[
C = a + bY
\]
where \(Y\) is national income, \(C\) is consumption, and \(I + G\) is the sum of autonomous investment and government spending, considered exogenous.
Step 1: Set up the model in matrix form
Rewrite \(Y = C + I + G\) as:
\[
Y = a + bY + I + G
\]
or equivalently:
\[
Y - bY = a + I + G
\]
which simplifies to:
\[
(1 - b)Y = a + I + G
\]
Expressed as a matrix equation with two endogenous variables (\(Y\) and \(C\)), though \(C\) depends on \(Y\), the model can be set as:
\[
\begin{bmatrix}
1 & -b \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
Y \\
C
\end{bmatrix}
=
\begin{bmatrix}
a + I + G \\
a
\end{bmatrix}
\]
where:
\[
A = \begin{bmatrix} 1 & -b \\ 0 & 1 \end{bmatrix}
\]
\[
\mathbf{x} = \begin{bmatrix} Y \\ C \end{bmatrix}
\]
\[
\mathbf{y} = \begin{bmatrix} a + I + G \\ a \end{bmatrix}
\]
Step 2: Solve for \(\mathbf{x} = A^{-1} \mathbf{y}\)
Since \(A\) is upper triangular with 1's on the diagonal, its determinant:
\[
\det(A) = 1 \times 1 - 0 \times (-b) = 1
\]
The inverse of \(A\):
\[
A^{-1} = \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix}
\]
The solution:
\[
\boxed{
\begin{bmatrix}
Y \\
C
\end{bmatrix}
=
A^{-1} \mathbf{y}
=
\begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix}
\begin{bmatrix}
a + I + G \\
a
\end{bmatrix}
=
\begin{bmatrix}
(a + I + G) + b \times a \\
a
\end{bmatrix}
}
\]
Thus:
\[
Y = a + I + G + b a
\]
\[
C = a
\]
These expressions provide the equilibrium values directly: the national income \(Y\) depends on autonomous consumption, investments, government spending, and the marginal propensity to consume, while consumption \(C\) is simply the autonomous component \(a\).
Conclusion
The solution exemplifies how matrices and linear algebra tools like Cramer's rule and matrix inversion are powerful in solving simultaneous equations in economics. The microeconomic model confirms the importance of contextually consistent equations—negatives in solutions highlight the need for realistic parameters. The macroeconomic model demonstrates the straightforward application of matrix algebra in deriving equilibrium levels of income and consumption, critical for understanding fiscal policy impacts and economic stability.
References
- Begg, D., Fischer, S., & Dornbusch, R. (2014). Economics (11th ed.). McGraw-Hill Education.
- Krugman, P. R., & Wells, R. (2018). Economics (5th ed.). Worth Publishers.
- Sloman, J., & Wride, A. (2015). Economics (9th ed.). Pearson Education.
- Mankiw, N. G. (2015). Principles of Economics (7th ed.). Cengage Learning.
- Samuelson, P. A., & Nordhaus, W. D. (2010). Economics (19th ed.). McGraw-Hill Education.
- Hubbard, R. G., & O'Brien, A. P. (2018). Microeconomics (6th ed.). Pearson.
- Parkin, M., & Bade, R. (2017). Foundations of Economics (7th ed.). Pearson.
- Deaton, A., & Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge University Press.
- Blanchard, O., & Johnson, D. R. (2012). Macroeconomics (6th ed.). Pearson.
- Froyen, R. T. (2014). Macro Economics: theories and policies (10th ed.). Pearson.