Let's go step-by-step to solve Question 3.35:

Problem Statement:

We are asked to calculate the cost function and the conditional input demands for the linear production function given by:

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

where:

• $y$ is the output.
• $x_i$ are the input quantities.
• $\alpha_i$ are parameters representing the productivity or input coefficients for each input $i$.

1. Cost Minimization Problem:

To find the cost function, we need to determine how the inputs $x_i$ are allocated to minimize the total cost for a given output $y$.

The total cost function is:

$C(w, y) = \sum_{i=1}^{n} w_i x_i$

where $w_i$ is the price of input $x_i$.

2. Cost Minimization Subject to the Production Function:

The goal is to minimize the total cost $C(w, y)$ subject to the constraint:

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

Using the Lagrange method, we form the Lagrangian:

$\mathcal{L} = \sum_{i=1}^{n} w_i x_i + \lambda \left( y - \sum_{i=1}^{n} \alpha_i x_i \right).$

3. First-Order Conditions (FOCs):

We take the partial derivatives of the Lagrangian with respect to each $x_i$ and the Lagrange multiplier $\lambda$, and set them to 0:

$\frac{\partial \mathcal{L}}{\partial x_i} = w_i - \lambda \alpha_i = 0 \quad \Rightarrow \quad \lambda = \frac{w_i}{\alpha_i}.$

Since $\lambda$ must be the same for all inputs, we get:

$\frac{w_i}{\alpha_i} = \frac{w_j}{\alpha_j} \quad \forall \, i, j.$

This implies that the ratio of input prices to their productivity parameters must be equal for all inputs.

4. Solving for Conditional Input Demands:

From the production function constraint $y = \sum_{i=1}^{n} \alpha_i x_i$, we solve for the input demands $x_i$.

The optimal input allocation is:

$x_i^*(w, y) = \frac{\alpha_i y}{\sum_{j=1}^{n} \alpha_j w_j / w_i}.$

5. Cost Function:

Plugging the optimal inputs $x_i^*(w, y)$ into the total cost expression:

$C(w, y) = \sum_{i=1}^{n} w_i x_i^*(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.$

Thus, the cost function for the linear production function is:

$C(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.$

Summary:

1. Cost Function:
   $C(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.$

2. Conditional Input Demands:
   $x_i^*(w, y) = \frac{\alpha_i y}{w_i}.$

Would you like any further clarification or a breakdown of specific steps?

Related Questions:

1. How does the choice of production function affect the shape of the cost function?
2. What happens to the cost function if one of the input prices $w_i$ increases?
3. How would the results change if the production function had decreasing returns to scale?
4. What is the interpretation of the Lagrange multiplier $\lambda$ in the optimization process?
5. How do linear production functions compare to Cobb-Douglas functions in terms of input substitutability?

Tip: In linear production functions, inputs are often perfect substitutes, meaning firms can easily swap one input for another at a fixed rate.