Question

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

Short answer

The formula for the wage elasticity of labor supply depends on the functional form of the labor supply function. For linear functions, it is given by Elasticity = \( \frac{bW}{L} \). For log-linear functions, it is Elasticity = \( b \). For log-log functions, it is Elasticity = \( \frac{b}{L} \).

Step-by-step solution

1. Define the Elasticity of Labor Supply

The elasticity of labor supply is a measure of how sensitive the quantity of labor supplied is to changes in the wage rate. It is the percentage change in the quantity of labor supplied for a given percentage change in the wage rate.

2. Identify the Labor Supply Function

The labor supply function is the relationship between the quantity of labor supplied (L) and the wage rate (W). It usually takes one of the following functional forms: - Linear: \(L = a + bW\) - Log-linear: \( L = aW^b\) - Log-log: \( \log{L} = a + b \log{W} \) where a and b are constants.

3. Differentiate the Labor Supply Function with Respect to Wage Rate

To find the wage elasticity of labor supply, we need to differentiate the labor supply function with respect to the wage rate. This will give us the slope of the labor supply curve, which is the change in the quantity of labor supplied for a given change in the wage rate. The differentiation of the labor supply function depends on its functional form: - For linear functions: \( \frac{dL}{dW} = b \) - For log-linear functions: \( \frac{dL}{dW} = abW^{b-1} \) - For log-log functions: \( \frac{dL}{dW} = \frac{b}{W} \)

4. Calculate the Wage Elasticity of Labor Supply

The wage elasticity of labor supply is the ratio of the percentage change in the quantity of labor supplied to the percentage change in the wage rate. It is calculated as follows: Elasticity = \( \frac{(\% \, \Delta L)}{(\% \, \Delta W)} \) which can be rearranged as: Elasticity = \( \frac{dL}{dW} \times \frac{W}{L} \) The elasticity will vary depending on the functional form of the labor supply function. Substitute the appropriate expression from step 3 into the formula and simplify: - For linear functions: Elasticity = \( \frac{bW}{L} \) - For log-linear functions: Elasticity = \( \frac{abW^{b-1}W}{aW^b} = b \) - For log-log functions: Elasticity = \( \frac{\frac{b}{W} W}{L} = \frac{b}{L} \) The formula for the wage elasticity of labor supply depends on the functional form of the labor supply function. If the labor supply function is given, we can use this information to find the wage elasticity using the appropriate formula from the list above.