Derivation of the Elasticity of Substitution:

To derive the elasticity of substitution of the production function, we first need to determine the relationship between the inputs of labor and capital. The elasticity of substitution (σ) is defined as the percentage change in the ratio of inputs to bring about a 1% change in output. Mathematically, it can be expressed as:

σ = (d(ln(K/L)) / d(ln(MP_L/MP_K))

where MP_L and MP_K are the marginal products of labor and capital, respectively.

Explanation of Economic Usefulness of Elasticity of Substitution:

The elasticity of substitution provides valuable insights into the production process of a firm. Here's why it is economically useful:

1. **Efficiency**: A high elasticity of substitution indicates that labor and capital are easily substitutable for each other. This implies that the firm can adjust its input mix to achieve the same level of output at a lower cost. On the other hand, a low elasticity of substitution suggests that inputs are complements, and the firm may face challenges in optimizing its production process.

2. **Technology Adoption**: Understanding the elasticity of substitution can help firms make decisions about adopting new technologies. For instance, if a new technology increases the substitutability between labor and capital, the firm can increase its efficiency by adjusting its input mix accordingly.

3. **Wage and Capital Allocation**: The elasticity of substitution can also influence wage and capital allocation decisions. In a production function with high substitutability, firms may choose to substitute labor with capital, which can impact wages and employment levels.

In conclusion, the elasticity of substitution plays a crucial role in determining the efficiency, technology adoption, and input allocation decisions of firms, making it a key concept in economic analysis.