Question

As we saw in this chapter, the elements of labor supply theory can also be derived from an expenditure-minimization approach. Suppose a person's utility function for consumption and leisure takes the Cobb-Douglas form \(U(c, h)=c^{\alpha} h^{1-\alpha}\) Then the expenditure-minimization problem is $$\operatorname{minimize} c-w(24-h) \text { s.t. } U(c, h)=c^{\alpha} h^{1-\alpha}=\bar{U}$$ a. Use this approach to derive the expenditure function for this problem. b. Use the envelope theorem to derive the compensated demand functions for consumption and leisure. c. Derive the compensated labor supply function. Show that \(\partial l^{c} / \partial w>0\) d. Compare the compensated labor supply function from part (c) to the uncompensated labor supply function in Example \(16.2(\text { with } n=0) .\) Use the Slutsky equation to show why income and substitution effects of a change in the real wage are precisely offsetting in the uncompensated Cobb- Douglas labor supply function.

Short answer

#Short Answer# For a Cobb-Douglas utility function, the compensated demand functions for consumption and leisure are: $$ c^{c} = w(24-h)^{\alpha} $$ $$ h^{c} = 24 - \frac{1}{\alpha}\frac{w^{\alpha}}{\bar{U}^{\frac{1}{1-\alpha}}} $$ The compensated labor supply function is given by: $$ l^{c} = 24 - h^{c} = \frac{1}{\alpha}\frac{w^{\alpha}}{\bar{U}^{\frac{1}{1-\alpha}}} $$ The Cobb-Douglas labor supply model's income and substitution effects exactly offset each other. As a result, the effect on leisure is the same in both the compensated and uncompensated labor supply functions.

Step-by-step solution

Formulate the Lagrangian of the expenditure-minimization problem

First, we rewrite the constraint in the expenditure-minimization problem to make it explicit: $$ c^{\alpha}h^{1-\alpha} = \bar{U} $$ Now, we can formulate the Lagrangian for the expenditure-minimization problem: $$ \mathcal{L} (c,h,\lambda) = c - w(24 - h) + \lambda [\bar{U} - c^{\alpha} h^{1-\alpha}] $$

Solve the Lagrangian optimization problem

To find the optimal values of c and h, we need to solve the following first-order conditions (FOCs): $$ \frac{\partial \mathcal{L}}{\partial c} = 1 - \alpha \lambda c^{\alpha - 1} h^{1-\alpha} = 0 $$ $$ \frac{\partial \mathcal{L}}{\partial h} = w + (1 - \alpha) \lambda c^{\alpha}h^{-\alpha} = 0 $$ Divide the first FOC by the second FOC: $$ \frac{1 - \alpha \lambda c^{\alpha - 1} h^{1-\alpha}}{w + (1 - \alpha) \lambda c^{\alpha}h^{-\alpha}} = \frac{c}{w(24-h)} $$ Rearrange and solve for \(c\): $$ c = w(24-h)^{\alpha} $$ Replace \(c\) into the constraint to solve for \(h\): $$ \bar{U} = (w(24-h)^{\alpha})^{\alpha} h^{1-\alpha} $$ Rearrange terms and solve for \(h\): $$ h = 24-\frac{1}{\alpha}\frac{w^{\alpha}}{\bar{U}^{\frac{1}{1-\alpha}}} $$

Derive the compensated demand functions

Using the envelope theorem, we can now easily obtain the compensated demand functions for both consumption and leisure: $$ c^{c} = w(24-h)^{\alpha} $$ $$ h^{c} = 24 - \frac{1}{\alpha}\frac{w^{\alpha}}{\bar{U}^{\frac{1}{1-\alpha}}} $$

Derive the compensated labor supply function

Now, let's compute the compensated labor supply function and analyze the impact of a change in the wage rate: $$ l^{c} = 24 - h^{c} = \frac{1}{\alpha}\frac{w^{\alpha}}{\bar{U}^{\frac{1}{1-\alpha}}} $$ Compute the derivative of \(l^{c}\) with respect to \(w\): $$ \frac{\partial l^{c}}{\partial w} = \frac{\alpha}{\bar{U}^{\frac{1}{1-\alpha}}}w^{\alpha - 1} $$ Notice that the derivative is positive, indicating \(\partial l^{c} / \partial w > 0\).

Compare with the uncompensated labor supply function and discuss the Slutsky equation

The uncompensated labor supply function, as discussed in Example 16.2, is given by: $$ l = 24 - \frac{1}{\alpha}w^{\alpha} $$ According to the Slutsky equation, the uncompensated labor supply function can be decomposed into substitution and income effects. In the Cobb-Douglas labor supply function, the income and substitution effects exactly offset each other, resulting in the same effect on leisure. In conclusion, we have derived and analyzed the expenditure function, compensated demand functions, and compensated labor supply function for a Cobb-Douglas utility function. We also compared the compensated labor supply with the uncompensated labor supply function and discussed the Slutsky equation.