Question 1198784: Consider a perfectly competitive firm charging a price of P for each unit of output it sells (Q) and paying input price w and r for the two inputs it uses (L and K). Output is, in turn, a function of L and K via Q = Q(L, K) = L^α K^β; where Q>0, K>0, L>0 and α and β positive constant
A. Write the profit function of the firm (this function should not involve Q), and find the critical value (expressing the firm’s choice variables in terms of exogenous variables).
B. What restrictions must be improved for the sufficient conditions for a maximum to be satisfied?
C. For the case where the sufficient condition for a maximum are fulfilled, find the firm’s input demand function (demand for labor and demand for capital) the firm’s output supply function, and explain how each of these functions responds to change in each of its arguments.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! ### 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.
---
### 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 \( Q \).
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