Show that the elasticity of the factor share with respect to...

Key Concepts

Relative Price (Wage Ratio)

The relative price, in this case represented as the ratio w2/w1, is key to understanding how changes in factor prices affect input choices. It serves as the independent variable in measuring the elasticity of the factor share, highlighting the adjustment in the share of income allocated to a factor when its wage changes relative to another factor. This ratio is central to models of production involving multiple inputs.

Elasticity of Substitution

The elasticity of substitution, often denoted by sigma, is a measure of how easily one factor can be substituted for another in the production process as their relative prices change. It influences how the factor shares adjust to changes in the wage ratio because it determines the rate at which firms substitute between inputs in response to cost changes, thus linking relative prices to the distribution of income across factors.

Elasticity

Elasticity measures the responsiveness of one variable to changes in another variable. In economic contexts, it often refers to the percentage change in a dependent variable resulting from a one percent change in an independent variable. In this problem, the elasticity quantifies the responsiveness of the factor share to changes in the relative wage ratio.

Factor Share

The factor share is the proportion of total output or income that is attributed to a specific production factor, such as labor or capital. It is derived from the cost minimization or production function, and reflects how income is distributed among the factors of production. Understanding factor shares is essential for analyzing income distribution and the impact of factor price changes on production.

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