Question

a. a person has an original amount (a) of a good. by sacrificing (x) of it, they can produce (y = \theta x) of another good where (\theta > 0) is a parameter that measures the marginal productivity of the process. the persons utility, (u(x,y)) depends on both goods, and is given by (u(x,y) = \sqrt{x} + \sqrt{y}) so that they solve the problem
\\v(\theta) = \max_{x \ge 0} \sqrt{a - x} + \sqrt{\theta x}.\\

1. give the focs for the problem (v(\theta)).

Explanation:

🆕 New Concept Discovered: First-Order Conditions (FOC)
Finding optimal choices by setting derivatives to zero.

Step 1: Identify the objective function

The optimization problem is to maximize the utility function with respect to the decision variable \( x \):

$$ f(x) = \sqrt{a - x} + \sqrt{\theta x} $$

Here, \( x \) represents the amount of the original good sacrificed, which must satisfy the constraints \( 0 \leq x \leq a \).

Step 2: Differentiate with respect to \( x \)

To find the First-Order Condition (FOC), we take the first derivative of the objective function with respect to \( x \) and set it equal to zero.

Using the chain rule:

• The derivative of \(\sqrt{a - x}\) with respect to \( x \) is:

$$ \frac{d}{dx}(a - x)^{1/2} = \frac{1}{2}(a - x)^{-1/2} \cdot (-1) = -\frac{1}{2\sqrt{a - x}} $$

• The derivative of \(\sqrt{\theta x}\) with respect to \( x \) is:

$$ \frac{d}{dx}(\theta x)^{1/2} = \frac{1}{2}(\theta x)^{-1/2} \cdot \theta = \frac{\theta}{2\sqrt{\theta x}} = \frac{\sqrt{\theta}}{2\sqrt{x}} $$

Step 3: Set the derivative to zero

Setting the sum of these derivatives to zero gives the first-order condition for an interior optimum:

$$ -\frac{1}{2\sqrt{a - x}} + \frac{\sqrt{\theta}}{2\sqrt{x}} = 0 $$

This can be simplified by multiplying the entire equation by 2:

$$ \frac{\sqrt{\theta}}{\sqrt{x}} = \frac{1}{\sqrt{a - x}} $$

Answer:

The First-Order Condition (FOC) for an interior solution is:

$$ -\frac{1}{2\sqrt{a - x}} + \frac{\sqrt{\theta}}{2\sqrt{x}} = 0 $$

which simplifies to:

$$ \frac{\sqrt{\theta}}{\sqrt{x}} = \frac{1}{\sqrt{a - x}} $$