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 \\[ \text { minimize } c-w(24-h) \text { s.t. } U(c, h)=c^{a} 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 \(\partial l^{c} / \partial w>0\) (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

Answer: In a Cobb-Douglas utility function for consumption and leisure, a change in the real wage has no effect on labor supply. This is because the substitution effect, which causes leisure to decrease when the wage increases, is precisely offset by the income effect, resulting in no changes in labor supply.

Step-by-step solution

In order to derive the expenditure function, we'll use the Lagrangian method to solve the expenditure-minimization problem. The Lagrangian function for this problem is given by: \(\mathcal{L}(c,h,\lambda)=c-w(24-h)+\lambda[\bar{U}-c^{\alpha}h^{1-\alpha}]\) #Step 2: Find the first-order conditions (FOCs)#

Next, we will find the FOCs with respect to \(c\), \(h\), and \(\lambda\). This will give us the following equations: (1) \(\frac{\partial\mathcal{L}}{\partial c} = 1 - \alpha\lambda c^{\alpha-1}h^{1-\alpha} = 0\) (2) \(\frac{\partial\mathcal{L}}{\partial h} = w - (1-\alpha)\lambda c^{\alpha}h^{-\alpha} = 0\) (3) \(\frac{\partial\mathcal{L}}{\partial \lambda} = \bar{U}-c^{\alpha}h^{1-\alpha}=0\) #Step 3: Solve the FOCs for the expenditure function#

Now we need to solve the system of FOCs to obtain the expenditure function. First, we will find the Marshallian demands for \(c\) and \(h\), and then plug them into the budget constraint: (1) \(\Rightarrow c^{\alpha-1}=\frac{1}{\alpha\lambda h^{1-\alpha}}\) (2) \(\Rightarrow h^{-\alpha}=\frac{1}{(1-\alpha)\lambda c^{\alpha}}\) Divide (1) by (2): \(c^{-1}h=h^{-1}c\Rightarrow c=h\) Now, substitute this result into (3): \(c^{\alpha}h^{1-\alpha}=\bar{U}\Rightarrow c^{2\alpha-1}\bar{U}^{1-\alpha}\Rightarrow c=\bar{U}^{\frac{1-\alpha}{2\alpha-1}}h\) Now we can find the expenditure function by plugging the Marshallian demands into the budget constraint: \(E(\bar{U},w)=c+w(24-h)=\bar{U}^{\frac{1-\alpha}{2\alpha-1}}h+w(24-\bar{U}^{\frac{1-\alpha}{2\alpha-1}}h)\) #Step 4: Derive the compensated demand functions for consumption and leisure#

To derive the compensated demand functions, we will use the Envelope Theorem. The Envelope Theorem states that the derivative of the expenditure function with respect to a parameter is equal to the partial derivative of the Lagrangian with respect to that parameter, holding the optimum constant: \(\frac{\partial E}{\partial w}=\frac{\partial\mathcal{L}}{\partial w}\big|_{c^*,h^*}=-24+h^*\) Now we can use this relationship along with the expenditure function to find the compensated demand for consumption: \(c^c\equiv\frac{\partial E}{\partial\bar{U}}=h^*\) For the compensated demand for leisure: \(h^c\equiv\frac{\partial E}{\partial w}=-24+h^*\) #Step 5: Derive the compensated labor supply function and its partial derivative with respect to the wage#

The compensated labor supply function is given by: \(l^c=24-h^c\) Now, we need to find the partial derivative of the compensated labor supply function with respect to the wage: \(\frac{\partial l^c}{\partial w}=-\frac{\partial h^c}{\partial w}=-1\) #Step 6: Use the Slutsky equation to explain the income and substitution effects of a change in the real wage#

According to the Slutsky equation, the total effect of a change in the wage on leisure demand can be decomposed into income and substitution effects: \(\frac{\partial h^u}{\partial w}=\frac{\partial h^c}{\partial w}+\frac{\partial h^c}{\partial I}\frac{\partial I}{\partial w}\) In this case, we found that \(\frac{\partial h^c}{\partial w} = -1\), which means the substitution effect is equal to 1 (leisure decreases when the wage increases). Since the Cobb-Douglas utility function has no income effect, the substitution effect is precisely offsetting the income effect. That means the total effect of a wage change on leisure is zero, so there will be no changes in labor supply due to changes in the wage. This is a unique property of the Cobb-Douglas utility function.

Key concepts

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is frequently employed in consumer theory to represent the relationship between two goods and how they contribute to the user's satisfaction or utility. Typically, it is written as
\(U(X,Y) = X^{\alpha}Y^{1-\alpha}\)
where \(0 < \alpha < 1\) and X and Y are quantities of two goods being consumed. In the context of labor supply theory, the function
\(U(c, h) = c^{\alpha}h^{1-\alpha}\)
is used to model the trade-off between consumption (\(c\)) and leisure (\(h\)), with the parameters reflecting the preferences of the individual.

The Cobb-Douglas utility function is revered for its simplicity and the fact that it leads to consistent behavior (i.e., non-increasing marginal utility for each good). It also implies a specific form of demand functions, which are linear in a log-log scale, and can be used not only to analyze consumer choice but also to derive labor supply functions, as outlined in the textbook solution.

Compensated Demand Functions

Compensated demand functions come into play when we consider changes in consumption due to price changes while keeping utility constant. These are also known as Hicksian demand functions. When a price changes, two things happen: the consumer is poorer or richer in real terms because the purchasing power changes (income effect), and the relative price has changed, making goods relatively cheaper or more expensive (substitution effect).

The idea of compensated demands is to isolate the substitution effect by adjusting income to keep the consumer at the same level of utility – hence, they show what the demand would be if the consumer were 'compensated' for the change in wealth. In the provided exercise, the envelope theorem is used to derive these compensated demand functions for consumption and leisure from a Cobb-Douglas utility function.

Envelope Theorem

The Envelope Theorem is a fundamental concept in microeconomics, particularly in optimization problems. It states that if you have an optimization problem with a parameter, the rate at which the value of the optimized objective function changes as you change the parameter is given just by the partial derivative of the object function with respect to the parameter, holding the initial choices constant.

In the expenditure-minimization approach, this theorem helps to find the change in expenditure with respect to wages or the price of a good without fully solving the utility-maximization problem each time that change occurs. Essentially, it simplifies the process of assessing how external changes will affect a consumer's expenditures.

Slutsky Equation

The Slutsky equation is a key result in consumer theory that relates changes in Marshallian (uncompensated) demand to Hicksian (compensated) demand. It shows that the total change in demand for a good due to a change in its price can be broken down into a substitution effect (holding utility constant) and an income effect (change in purchasing power). The mathematical representation of this relationship is
\(\frac{\partial x_i}{\partial p_j} = \frac{\partial h_i}{\partial p_j} + \frac{\partial x_i}{\partial I} \frac{\partial I}{\partial p_j}\)
where \(x_i\) is the Marshallian demand for good \(i\), \(h_i\) is the Hicksian demand for good \(i\), \(p_j\) is the price of good \(j\), and \(I\) is income.

In the labor supply problem given in the exercise, the Slutsky equation illustrates that the income and substitution effects of a real wage change cancel out precisely in the case of a Cobb-Douglas utility function, leading to a labor supply function that is invariant to wage changes.

Labor Supply Theory

Labor supply theory explores how individuals decide between allocating their time to labor (working) or leisure. This trade-off reflects the opportunity cost of leisure, which is the wage an individual could earn by working. According to labor supply theory, the quantity of labor supplied is influenced by both the substitution effect, where an increase in wages makes leisure more costly and thus leads individuals to work more, and the income effect, where an increase in wages allows individuals to afford more leisure.

In the exercise, the individual's utility from consumption and leisure is modeled with a Cobb-Douglas utility function. The labor supply curve derived from this tends to be backward-bending, meaning that initially, as wages increase, labor supply increases due to the substitution effect, but beyond a certain point, the income effect dominates, and increases in wages lead to a decrease in labor supply as individuals can afford to 'buy' more leisure time. However, in the special case of a Cobb-Douglas utility function, the exercise shows that these two effects offset each other, yielding a labor supply curve that is not affected by wage changes.