Calculate explicitly the profit function for the technology y = x^a，forpoint that yields a positive profit.
0< a < 1 and verify that it is homogeneous and convex in (p,w).

To calculate the profit function and verify its properties, let's follow the steps carefully. We are given the technology:

\[
y = x^a \quad \text{where} \quad 0 < a < 1
\]

where:
- \( y \) is the output.
- \( x \) is the input.
- \( a \) is the output elasticity with respect to input \( x \), and it is between 0 and 1.

### Step 1: Define the Profit Function

The profit function is defined as the difference between total revenue and total cost:

\[
\pi(p, w) = \max_{x} \left( p \cdot y - w \cdot x \right)
\]

where:
- \( p \) is the price of output.
- \( w \) is the price (cost) of input.
- \( x \) is the amount of input used in production.
- \( y = x^a \) is the output given the input.

Substitute \( y = x^a \) into the profit function:

\[
\pi(p, w) = \max_{x} \left( p \cdot x^a - w \cdot x \right)
\]

### Step 2: First-Order Condition (FOC)

To maximize the profit with respect to \( x \), we take the derivative of the profit function with respect to \( x \) and set it equal to zero:

\[
\frac{d}{dx} \left( p \cdot x^a - w \cdot x \right) = 0
\]

The derivative is:

\[
a \cdot p \cdot x^{a-1} - w = 0
\]

Solving for \( x \), we get:

\[
a \cdot p \cdot x^{a-1} = w
\]
\[
x^{a-1} = \frac{w}{a \cdot p}
\]
\[
x = \left( \frac{w}{a \cdot p} \right)^{\frac{1}{a-1}} = \left( \frac{a \cdot p}{w} \right)^{\frac{1}{1-a}}
\]

### Step 3: Calculate the Profit Function

Now that we have the optimal input \( x^* \), substitute this back into the original profit function:

\[
\pi(p, w) = p \cdot \left( x^* \right)^a - w \cdot x^*
\]

Substitute \( x^* = \left( \frac{a \cdot p}{w} \right)^{\frac{1}{1-a}} \):

\[
\pi(p, w) = p \cdot \left( \left( \frac{a \cdot p}{w} \right)^{\frac{1}{1-a}} \right)^a - w \cdot \left( \frac{a \cdot p}{w} \right)^{\frac{1}{1-a}}
\]

Simplify both terms:

1. **First term:**

\[
p \cdot \left( \frac{a \cdot p}{w} \right)^{\frac{a}{1-a}} = p \cdot \left( \frac{a^a \cdot p^a}{w^a} \right)^{\frac{1}{1-a}} = \frac{a^a \cdot p^{1+a}}{w^{a}}
\]

2. **Second term:**



Calculate explicitly the profit function for the technology y = x^a, for a point that yields a positive profit. 0 < a < 1 and verify that it is homogeneous and convex in (p, w).

Discover how to calculate the profit function for the technology y = x^a, where 0 < a < 1. The solution covers key concepts such as convexity, homogeneity, and optimization of the profit function in terms of input price (w) and output price (p). Suitable for advanced economics or mathematics students.

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