Question 1198784
### Part A: Profit Function and Critical Values

#### Profit Function
The firm's profit is defined as revenue minus costs:
\[
\Pi = P \cdot Q - w \cdot L - r \cdot K
\]
Using the production function \( Q = L^\alpha K^\beta \), the profit function becomes:
\[
\Pi = P \cdot L^\alpha K^\beta - w \cdot L - r \cdot K
\]

#### First-Order Conditions (Critical Values)
To maximize profit, the firm chooses \( L \) and \( K \) such that the marginal cost equals the marginal revenue product for each input. The first-order conditions are:
1. \( \frac{\partial \Pi}{\partial L} = P \cdot \alpha L^{\alpha-1} K^\beta - w = 0 \)
2. \( \frac{\partial \Pi}{\partial K} = P \cdot \beta L^\alpha K^{\beta-1} - r = 0 \)

From these equations, solve for \( L \) and \( K \) in terms of the exogenous variables \( P, w, r, \alpha, \beta \).

---

1. Rearrange the first condition:
\[
L^{\alpha-1} K^\beta = \frac{w}{P \alpha}
\]
Solve for \( L^{\alpha-1} \):
\[
L^{\alpha-1} = \frac{w}{P \alpha K^\beta}
\]

2. Rearrange the second condition:
\[
L^\alpha K^{\beta-1} = \frac{r}{P \beta}
\]
Solve for \( K^{\beta-1} \):
\[
K^{\beta-1} = \frac{r}{P \beta L^\alpha}
\]

Substituting \( L^{\alpha-1} \) from the first condition into the second gives the critical values for \( L \) and \( K \).

---

### Part B: Sufficient Conditions for Maximum
For a maximum, the **second-order conditions** of the profit function must be satisfied:
1. The second partial derivatives with respect to \( L \) and \( K \) must be negative (\( \frac{\partial^2 \Pi}{\partial L^2} < 0 \) and \( \frac{\partial^2 \Pi}{\partial K^2} < 0 \)).
2. The Hessian determinant (\( H \)) must be positive. For a two-variable function, the Hessian is:
\[
H = \begin{vmatrix}
\frac{\partial^2 \Pi}{\partial L^2} & \frac{\partial^2 \Pi}{\partial L \partial K} \\
\frac{\partial^2 \Pi}{\partial K \partial L} & \frac{\partial^2 \Pi}{\partial K^2}
\end{vmatrix}
\]

The sufficient conditions are met if \( H > 0 \) and the diagonal elements (\( \frac{\partial^2 \Pi}{\partial L^2} \), \( \frac{\partial^2 \Pi}{\partial K^2} \)) are negative.

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### Part C: Input Demand and Output Supply Functions

#### Input Demand Functions
From the first-order conditions, the input demand functions for \( L \) and \( K \) are:
\[
L = L(P, w, r, \alpha, \beta)
\]
\[
K = K(P, w, r, \alpha, \beta)
\]
These depend on the exogenous variables: \( P \) (output price), \( w \) (wage), \( r \) (capital price), and the production elasticities \( \alpha \) and \( \beta \).

#### Output Supply Function
The output supply function is:
\[
Q = L^\alpha K^\beta
\]
Substitute the demand functions for \( L \) and \( K \) into the production function.

#### Comparative Statics
1. **Response to Changes in \( P \)**:
   - Higher \( P \) increases the marginal revenue product, leading to higher demand for \( L \) and \( K \), which increases \( Q \).

2. **Response to Changes in \( w \)**:
   - Higher \( w \) (wage) reduces demand for \( L \) as labor becomes more expensive, potentially reducing \( Q \).

3. **Response to Changes in \( r \)**:
   - Higher \( r \) (cost of capital) reduces demand for \( K \), potentially reducing \( Q \).