To calculate the cost function and the conditional input demands for the given linear production function, let's break down the approach.

Production Function

The production function is: $y = \sum_{i=1}^n \alpha_i x_i$ where:

• $y$ is the output,
• $x_i$ represents the quantity of the $i$-th input,
• $\alpha_i$ is the marginal product of input $x_i$.

Objective: Cost Function

The cost function, $C(w, y)$, represents the minimum cost required to produce a given level of output $y$ when input prices are $w = (w_1, w_2, \ldots, w_n)$.

1. Define Total Cost: The total cost of using inputs $x_1, x_2, \ldots, x_n$ is given by: $C = \sum_{i=1}^n w_i x_i$ where $w_i$ is the price of the $i$-th input.

2. Minimize Cost Subject to Production Constraint: To find the cost function, we minimize $C = \sum_{i=1}^n w_i x_i$ subject to the constraint: $y = \sum_{i=1}^n \alpha_i x_i$

3. Set Up the Lagrangian: $\mathcal{L} = \sum_{i=1}^n w_i x_i + \lambda \left( y - \sum_{i=1}^n \alpha_i x_i \right)$ where $\lambda$ is the Lagrange multiplier.

4. Take Partial Derivatives and Solve: For each input $x_i$, we take the partial derivative of $\mathcal{L}$ with respect to $x_i$ and set it to zero: $\frac{\partial \mathcal{L}}{\partial x_i} = w_i - \lambda \alpha_i = 0$ which gives: $\lambda = \frac{w_i}{\alpha_i}$ Since $\lambda$ is the same for all inputs, we have: $\frac{w_1}{\alpha_1} = \frac{w_2}{\alpha_2} = \cdots = \frac{w_n}{\alpha_n} = \lambda$ This condition ensures that the ratio of the input prices to their marginal products is constant across all inputs.

5. Express $x_i$ in Terms of $y$ (Conditional Input Demand): Using the production constraint $y = \sum_{i=1}^n \alpha_i x_i$, we solve for $x_i$ to find the conditional input demands.

Summary of Results

• Cost Function: The minimum cost function, $C(w, y)$, will depend on the prices $w_i$, the marginal products $\alpha_i$, and the desired output $y$.
• Conditional Input Demands: The demand for each input $x_i$ will be a function of the price $w_i$, the marginal product $\alpha_i$, and the total output $y$.

Would you like further details or calculations?

Related Questions:

1. How does the cost function change if the production function is not linear?
2. What happens to the cost function if one of the input prices, say $w_1$, increases?
3. How does the Lagrangian method help in finding the optimal input combination?
4. How are marginal products $\alpha_i$ interpreted in a production context?
5. What is the impact of increasing returns to scale on the cost function?

Tip:

Understanding cost functions is essential for analyzing how firms minimize production costs under budget constraints.