Consider the following system of equations of four variables: , ,  and :
() = +2 +3 −=2 () = 23 = 1
Suppose that around the point (1111), the implicit function theorem applies, by which two endogenous variables can be defined as differentiable functions of two ex- ogenous variables.
(a) Suppose that  is an endogenous variable, identify the other endogenous variable. (b) Suppose that each of the two exogenous variables increases by 01. Use the implicit
function theorem to estimate how each of the endogenous variables will change?

Let's carefully address the problem using the Implicit Function Theorem.

### Given System of Equations
We have the following system involving four variables \(\xi\), \(\psi\), \(\zeta\), and \(\upsilon\):

\[
\gamma(\xi, \psi, \zeta, \upsilon) = \xi + \psi^2 + \zeta^3 - \upsilon = 2
\]
\[
\eta(\xi, \psi, \zeta, \upsilon) = \xi \psi^2 \zeta^3 \upsilon = 1
\]
We are told that at the point \((\xi, \psi, \zeta, \upsilon) = (1, 1, 1, 1)\), the implicit function theorem applies. Two of the variables will be treated as endogenous, meaning they can be expressed as functions of the other two, which are exogenous.

### Part (a): Identifying the Other Endogenous Variable

Assume \(\xi\) is an endogenous variable. To identify the other endogenous variable, we examine the partial derivatives of the functions with respect to the variables and form the Jacobian matrix.

#### Step 1: Compute Partial Derivatives

1. **For** \(\gamma(\xi, \psi, \zeta, \upsilon)\):
   \[
   \frac{\partial \gamma}{\partial \xi} = 1, \quad \frac{\partial \gamma}{\partial \psi} = 2\psi, \quad \frac{\partial \gamma}{\partial \zeta} = 3\zeta^2, \quad \frac{\partial \gamma}{\partial \upsilon} = -1
   \]

2. **For** \(\eta(\xi, \psi, \zeta, \upsilon)\):
   \[
   \frac{\partial \eta}{\partial \xi} = \psi^2 \zeta^3 \upsilon, \quad \frac{\partial \eta}{\partial \psi} = 2\xi \psi \zeta^3 \upsilon, \quad \frac{\partial \eta}{\partial \zeta} = 3\xi \psi^2 \zeta^2 \upsilon, \quad \frac{\partial \eta}{\partial \upsilon} = \xi \psi^2 \zeta^3
   \]

#### Step 2: Evaluate Partial Derivatives at (1, 1, 1, 1)

Substituting \((\xi, \psi, \zeta, \upsilon) = (1, 1, 1, 1)\):

- **For** \(\gamma\):
  \[
  \frac{\partial \gamma}{\partial \xi} = 1, \quad \frac{\partial \gamma}{\partial \psi} = 2, \quad \frac{\partial \gamma}{\partial \zeta} = 3, \quad \frac{\partial \gamma}{\partial \upsilon} = -1
  \]

- **For** \(\eta\):
  \[
  \frac{\partial \eta}{\partial \xi} = 1, \quad \frac{\partial \eta}{\partial \psi} = 2, \quad \frac{\partial \eta}{\partial \zeta} = 3, \quad \frac{\partial \eta}{\partial \upsilon} = 1
  \]

The Jacobian matrix for this system, using \(\xi\) as an endogenous variable, is:
\[
J = \begin{pmatrix}
\frac{\partial \gamma}{\partial \xi} & \frac{\partial \gamma}{\partial \zeta} \\
\frac{\partial \eta}{\partial \xi} & \frac{\partial \eta}{\partial \zeta}
\end{pmatrix}
= \begin{pmatrix}
1 & 3 \\
1 & 3
\end{pmatrix}
\]

However, notice that the determinant of this matrix is zero, indicating linear dependence, so we can't use \(\zeta\) as the endogenous variable paired with \(\xi\).

Instead, let's consider \(\upsilon\) as the other endogenous variable.

The new Jacobian matrix for \(\xi\) and \(\upsilon\) is:
\[
J = \begin{pmatrix}
\frac{\partial \gamma}{\partial \xi} & \frac{\partial \gamma}{\partial \upsilon} \\
\frac{\partial \eta}{\partial \xi} & \frac{\partial \eta}{\partial \upsilon}
\end{pmatrix}
= \begin{pmatrix}
1 & -1 \\
1 & 1
\end{pmatrix}
\]

The determinant of this matrix is non-zero (\(\det(J) = 2\)), indicating that \(\xi\) and \(\upsilon\) can indeed be treated as the endogenous variables.

### Part (b): Estimating the Change in Endogenous Variables

Suppose \(\psi\) and \(\zeta\) increase by 0.1. We estimate how \(\xi\) and \(\upsilon\) will change.

#### Step 1: Compute Changes in \(\gamma\) and \(\eta\)

Since the changes in \(\psi\) and \(\zeta\) are small, we can use linear approximations:
\[
d\gamma = \frac{\partial \gamma}{\partial \psi} d\psi + \frac{\partial \gamma}{\partial \zeta} d\zeta = 2(0.1) + 3(0.1) = 0.5
\]
\[
d\eta = \frac{\partial \eta}{\partial \psi} d\psi + \frac{\partial \eta}{\partial \zeta} d\zeta = 2(0.1) + 3(0.1) = 0.5
\]

#### Step 2: Use the Jacobian to Estimate \(d\xi\) and \(d\upsilon\)

We use the relationship:
\[
\begin{pmatrix}
d\xi \\
d\upsilon
\end{pmatrix}
= -J^{-1} \cdot \begin{pmatrix}
d\gamma \\
d\eta
\end{pmatrix}
\]

First, calculate the inverse of \(J\):
\[
J^{-1} = \frac{1}{\det(J)} \begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}
= \frac{1}{2} \begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}
\]

Now, compute the changes in \(\xi\) and \(\upsilon\):
\[
\begin{pmatrix}
d\xi \\
d\upsilon
\end{pmatrix}
= -\frac{1}{2} \begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}
\begin{pmatrix}
0.5 \\
0.5
\end{pmatrix}
= -\frac{1}{2} \begin{pmatrix}
1 \times 0.5 + 1 \times 0.5 \\
-1 \times 0.5 + 1 \times 0.5
\end{pmatrix}
= -\frac{1}{2} \begin{pmatrix}
1 \\
0
\end{pmatrix}
= \begin{pmatrix}
-0.5 \\
0
\end{pmatrix}
\]

### Final Results
- **Change in \(\xi\)**: \(d\xi = -0.5\)
- **Change in \(\upsilon\)**: \(d\upsilon = 0\)

This means that if \(\psi\) and \(\zeta\) both increase by 0.1, \(\xi\) will decrease by 0.5, while \(\upsilon\) will remain unchanged.

### Next Steps
- **Do you have any questions about this result?**
- **Would you like to see how this analysis might change under different conditions?**

### Relative Questions:
1. How does the choice of endogenous variables affect the system's behavior?
2. What is the significance of the determinant of the Jacobian matrix in the Implicit Function Theorem?
3. How would this analysis change if the exogenous variables were different?
4. Can the implicit function theorem be used for non-linear systems? If so, how?
5. What happens if the partial derivatives are evaluated at a point where the determinant is zero?

### Tip:
Always ensure the Jacobian matrix has a non-zero determinant before applying the implicit function theorem, as this condition is critical for the theorem to hold.

Explore how to apply the Implicit Function Theorem to estimate changes in endogenous variables in a system of equations. Learn about partial derivatives, Jacobian matrices, and the theorem's application around a given point. Suitable for advanced undergraduate students and mathematicians.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation