The problem asks to calculate the cost function and conditional input demands for the linear production function defined by:

$y = \sum_{i=1}^n \alpha_i x_i$

where:

• $y$ is the output,
• $x_i$ represents the input quantities,
• $\alpha_i$ are positive parameters indicating the marginal products of each input $x_i$.

Solution Outline

1. Define the Cost Function: The cost function $C(w, y)$ represents the minimum cost of producing a given output level $y$ when input prices are given by $w = (w_1, w_2, \dots, w_n)$.

2. Formulate the Minimization Problem: To find the cost function, we solve a cost-minimization problem: $\min_{x_1, x_2, \dots, x_n} \sum_{i=1}^n w_i x_i$ subject to the production constraint: $\sum_{i=1}^n \alpha_i x_i = y.$

3. Apply the Lagrange Multiplier Method: Set up the Lagrangian $\mathcal{L}$: $\mathcal{L} = \sum_{i=1}^n w_i x_i + \lambda \left( y - \sum_{i=1}^n \alpha_i x_i \right).$

   Differentiate with respect to each $x_i$ and $\lambda$, and solve the resulting system of equations.

Step-by-Step Solution

1. Differentiate the Lagrangian with respect to $x_i$: $\frac{\partial \mathcal{L}}{\partial x_i} = w_i - \lambda \alpha_i = 0 \Rightarrow \lambda = \frac{w_i}{\alpha_i}.$ Since $\lambda$ is the same for all $i$, we equate $\frac{w_i}{\alpha_i} = \frac{w_j}{\alpha_j}$ for all $i, j$.

2. Solve for $x_i$ in terms of $y$: Using the production constraint, we can express $x_i$ in terms of $y$ and $w_i$.

3. Substitute $x_i$ back into the cost function to obtain $C(w, y)$.

Would you like a detailed calculation for each step?

Follow-Up Questions

1. What are the properties of a cost function for a linear production function?
2. How does the linearity of the production function impact the form of the cost function?
3. Can you derive the marginal cost from the cost function obtained?
4. How would changes in input prices $w_i$ affect the conditional input demands?
5. What does the term "conditional input demand" mean in the context of this production function?

Tip

For linear production functions, conditional input demands often reveal insights about substitution between inputs under cost minimization constraints.