Question

What is the formula for the wage elasticity of labor supply?

Short answer

The formula for the wage elasticity of labor supply is: \(\epsilon = \frac{dL/dw}{L/w}\), where \(\epsilon\) represents the wage elasticity of labor supply, \(L\) represents the quantity of labor supplied, and \(w\) represents the wage rate.

Step-by-step solution

Define the labor supply function

Let's assume that the labor supply is a function of the wage rate. The labor supply function can be expressed as: \(L = L(w)\), where \(L\) represents the quantity of labor supplied and \(w\) represents the wage rate.

Introduce the concept of elasticity

Elasticity measures the responsiveness of one variable to changes in another variable. In the context of labor supply, the wage elasticity of labor supply measures the proportional change in labor supply in response to a proportional change in the wage rate. Mathematically, it can be expressed as: \(\epsilon = \frac{\text{% change in labor supply}}{\text{% change in wage rate}}\)

Write the formula for the wage elasticity of labor supply

Using the definition of elasticity and the labor supply function, we can write the formula for the wage elasticity of labor supply as: \(\epsilon = \frac{\Delta L / L}{\Delta w / w}\)

Rearrange the formula for the wage elasticity of labor supply

Rearrange the formula to make it a single ratio: \(\epsilon = \frac{\Delta L}{\Delta w} \cdot \frac{w}{L}\)

Derive the formula using partial derivatives

Now, replace the change in labor supply and change in the wage rate with partial derivatives. In other words, the derivative of the labor supply function with respect to the wage rate can be used to represent the change in labor supply with respect to the change in the wage rate. Therefore, the formula for the wage elasticity of labor supply becomes: \(\epsilon = \frac{dL/dw}{L/w}\) So, the wage elasticity of labor supply is the ratio of the partial derivative of the labor supply function with respect to the wage rate to the ratio of labor supply to the wage rate.