Let f(x1, x2) be a production function with two factors and...

Let $f\left(x_{1}, x_{2}\right)$ be a production function with two factors and let $w_{1}$ and $w_{2}$ be their respective prices. Show that the elasticity of the factor share $\left(w_{2} x_{2} / w_{1} x_{1}\right)$ with respect to $\left(x_{1} / x_{2}\right)$ is given by $1 / \sigma-1$.

Key Concepts

Production Function

A production function represents the relationship between input factors and the resulting output in an economy. It encapsulates how various inputs, such as labor and capital, are combined to produce goods or services. The structure of a production function highlights the trade-offs and complementarities between different inputs, and it serves as the foundational concept for analyzing productivity and efficiency in production analysis.

Factor Shares

Factor shares refer to the proportion of total production expenditure allocated to each input. They measure how the cost of production is distributed across different factors, such as labor and capital. This concept is crucial in understanding the pricing and employment decisions in an economy, as well as in analyzing income distribution and the impact of technological changes on input utilization.

Elasticity

Elasticity is a measure of the responsiveness of one variable to a percentage change in another variable. In the context of production and factor allocation, elasticity quantifies how sensitive the factor shares or factor ratios are to changes in the relative prices or marginal rates of substitution. It is an essential tool in economic analysis, helping to understand how changes in economic conditions or policies impact production decisions.

Elasticity of Substitution

The elasticity of substitution is a specific type of elasticity that measures the ease with which one input can be substituted for another in the production process without altering the level of output. In the context of the problem, it is used to relate changes in the ratio of factor inputs to changes in the ratio of their cost shares, providing insights into the degree of substitutability between the factors. A higher elasticity of substitution implies that a firm can more easily switch between inputs in response to changes in their prices.

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