Theoutputofagoodis,whereandaretheamountsoftwoinputsand0 are parameters. A government-controlled firm is directed to maximize output subject to meeting the constraint  +  = 10. Solve the firm’s problem. Use the envelope theorem to find how the maximal output changes as the parameter  or  varies.


### Problem Analysis and Formulation

The problem requires us to maximize the output function \( \text{Output} = \xi^\alpha \cdot \psi^\beta \), subject to the constraint \( \xi + \psi = 10 \), where \( \xi \) and \( \psi \) are the amounts of two inputs, and \( \alpha \) and \( \beta \) are positive parameters.

### Step 1: Set Up the Lagrangian
To solve this optimization problem, we'll use the method of Lagrange multipliers. The Lagrangian function \( \mathcal{L} \) is given by:

\[
\mathcal{L}(\xi, \psi, \lambda) = \xi^\alpha \cdot \psi^\beta + \lambda(10 - \xi - \psi)
\]

Here, \( \lambda \) is the Lagrange multiplier associated with the constraint \( \xi + \psi = 10 \).

### Step 2: Take the First-Order Conditions
To find the maximum, we take the partial derivatives of the Lagrangian with respect to \( \xi \), \( \psi \), and \( \lambda \), and set them equal to zero:

1. **Partial derivative with respect to \( \xi \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \xi} = \alpha \xi^{\alpha-1} \cdot \psi^\beta - \lambda = 0
   \]

2. **Partial derivative with respect to \( \psi \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \psi} = \beta \xi^\alpha \cdot \psi^{\beta-1} - \lambda = 0
   \]

3. **Partial derivative with respect to \( \lambda \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 10 - \xi - \psi = 0
   \]

### Step 3: Solve the First-Order Conditions

From the first two equations, we can set the two expressions for \( \lambda \) equal to each other:

\[
\alpha \xi^{\alpha-1} \psi^\beta = \beta \xi^\alpha \psi^{\beta-1}
\]

Simplifying, we get:

\[
\frac{\alpha}{\beta} = \frac{\psi}{\xi}
\]

This implies that:

\[
\psi = \frac{\alpha}{\beta} \xi
\]

Now, substitute this into the constraint \( \xi + \psi = 10 \):

\[
\xi + \frac{\alpha}{\beta} \xi = 10
\]

\[
\xi \left( 1 + \frac{\alpha}{\beta} \right) = 10
\]

\[
\xi = \frac{10 \beta}{\alpha + \beta}
\]

Using \( \psi = \frac{\alpha}{\beta} \xi \), we find:

\[
\psi = \frac{10 \alpha}{\alpha + \beta}
\]

### Step 4: Calculate the Maximal Output
Substitute \( \xi \) and \( \psi \) back into the output function:

\[
\text{Output} = \left(\frac{10 \beta}{\alpha + \beta}\right)^\alpha \cdot \left(\frac{10 \alpha}{\alpha + \beta}\right)^\beta
\]

\[
\text{Maximal Output} = 10^{\alpha + \beta} \cdot \left(\frac{\beta^\alpha \cdot \alpha^\beta}{(\alpha + \beta)^{\alpha + \beta}}\right)
\]

### Step 5: Apply the Envelope Theorem
To determine how the maximal output changes as \( \alpha \) or \( \beta \) varies, we use the envelope theorem. The envelope theorem states that the change in the value of the optimized function with respect to a parameter (holding other parameters constant) is given by the partial derivative of the Lagrangian with respect to that parameter, evaluated at the optimal solution.

1. **For \( \alpha \):**
   \[
   \frac{\partial \text{Output}}{\partial \alpha} = \text{Output} \cdot \left[\ln\left(\frac{10 \beta}{\alpha + \beta}\right) - \ln\left(\frac{\beta}{\alpha + \beta}\right)\right]
   \]

2. **For \( \beta \):**
   \[
   \frac{\partial \text{Output}}{\partial \beta} = \text{Output} \cdot \left[\ln\left(\frac{10 \alpha}{\alpha + \beta}\right) - \ln\left(\frac{\alpha}{\alpha + \beta}\right)\right]
   \]

These derivatives show how the optimal output changes with respect to changes in \( \alpha \) or \( \beta \).

### Summary
- The firm's problem was solved by maximizing the output function subject to the constraint \( \xi + \psi = 10 \).
- The optimal input quantities were found as \( \xi = \frac{10 \beta}{\alpha + \beta} \) and \( \psi = \frac{10 \alpha}{\alpha + \beta} \).
- The maximal output was calculated as \( 10^{\alpha + \beta} \cdot \frac{\beta^\alpha \cdot \alpha^\beta}{(\alpha + \beta)^{\alpha + \beta}} \).
- The envelope theorem was used to determine how the maximal output changes with respect to \( \alpha \) and \( \beta \).

Would you like more details on any of these steps?

### Follow-up Questions:
1. How does the constraint \( \xi + \psi = 10 \) affect the firm's decision on input allocation?
2. What would happen if the government increased the total amount of inputs available?
3. How would the optimal output change if \( \alpha = \beta \)?
4. What is the economic interpretation of the parameters \( \alpha \) and \( \beta \)?
5. Can you think of a real-world scenario where a firm faces a similar maximization problem?

### Tip:
When using the Lagrange multiplier method, always check the economic interpretation of the multipliers, as they often represent important marginal values like the shadow price of a constraint.

Explore how to maximize output subject to constraints using Lagrange multipliers and analyze changes with the envelope theorem. Detailed mathematical solution with applications in optimization theory.

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