Question

The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure. Hence, the entire body of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well. Here are three examples. a. Roy's identity. In the Extensions to Chapter 5 we showed how demand functions can be derived from indirect utility functions by using Roy's identity. Use a similar approach to show that the labor supply function associated with the utility. maximization problem described in Equation 16.20 can be derived from the indirect utility function by \\[ l(w, n)=\frac{\partial V(w, n) / \partial w}{\partial V(w, n) / \partial n} \\] Illustrate this result for the Cobb-Douglas case described in Example 16.1 b. Substitutes and complements. A change in the real wage will affect not only labor supply, but also the demand for specific items in the preferred consumption bundle. Develop a Slutsky-type equation for the cross-price effect of a change in \(w\) on a particular consumption item and then use it to discuss whether leisure and the item are (net or gross) substitutes or complements. Provide an example of each type of relationship. c. Labor supply and marginal expense. Use a derivation similar to that used to calculate marginal revenue for a given demand curve to show that \(M E_{l}=w\left(1+1 / e_{l} w\right)\).

Short answer

In this exercise, we connected labor supply theory with demand theory through the analysis of three examples. First, we applied Roy's identity to derive the labor supply function using the Cobb-Douglas utility function. Next, we developed a Slutsky-type equation for the cross-price effect of a real wage change on specific consumption items and discussed the relationships between leisure and consumption items. Finally, we derived the marginal expense for labor supply using a similar process as the calculation for the marginal revenue on a demand curve. Overall, this exercise demonstrated the interconnectivity of various economic concepts in understanding labor markets and consumer behavior.

Step-by-step solution

Understand Roy's Identity

Roy's identity is a method to derive demand functions from indirect utility functions. It is an important result in consumer theory that states that the demand for a good depends on both its price and the total income. In the context of labor supply and leisure, we will use Roy's identity to derive the labor supply function from the indirect utility function.

Apply Roy's Identity to labor supply function

Given the indirect utility function \(V(w, n)\), we can derive the labor supply function \(l(w, n)\) using Roy's Identity as follows: \\[ l(w, n)=\frac{\partial V(w, n) / \partial w}{\partial V(w, n) / \partial n} \\]

Illustrate for the Cobb-Douglas case

Refer to Example 16.1, we have the Cobb-Douglas utility function \(u(c, l)=c^{\alpha}l^{\beta}\), where \(c\) is consumption and \(l\) is leisure. Then, we derive the indirect utility function \(V(w, n)\), the budget constraint, and the labor supply function \(l(w, n)\). Substitute the budget constraint \(c=wn\) into the utility function: \\[ V(w, n)= \max_l \{(wn)^{\alpha}l^{\beta}\} \\] Solve the maximization problem, and substitute the solution back into the indirect utility function to obtain: \\[ V(w, n) = (wn)^{\alpha}\left(\frac{\alpha}{\alpha + \beta}\right)^{\beta} \\] Now applying Roy's identity to derive labor supply function \(l(w, n)\): \\[ l(w, n)=\frac{\partial V(w, n) / \partial w}{\partial V(w, n) / \partial n} = \left(\frac{\alpha}{\alpha + \beta}\right)n \\] #b. Substitutes and complements#

Develop a Slutsky-type equation for the cross-price effect

We will derive the cross-price effect of a change in \(w\) on a particular consumption item (\(x_i\)) using the Slutsky equation. The Slutsky equation can be written in terms of cross-price effects: \\[ \frac{\partial x_i}{\partial w}=\frac{\partial h_i(w, n)}{\partial w} + x_i\frac{\partial l}{\partial w} \\] Where \(h_i(w, n)\) is the Hicksian (compensated) demand for good \(x_i\) regarding wage \(w\) and income \(n\).

Discuss the relationship between leisure and consumption items

The sign of the cross-price effect \(\frac{\partial x_i}{\partial w}\) determines whether leisure and the particular consumption item are net substitutes or complements. If \(\frac{\partial x_i}{\partial w}>0\), then they are net substitutes, because a rise in \(w\) will lead to an increase in the quantity demanded of \(x_i\). On the other hand, if \(\frac{\partial x_i}{\partial w}<0\), then they are net complements, because a rise in \(w\) will lead to a decrease in the quantity demanded of \(x_i\). For gross substitutes or complements, we would instead analyze the relationship between \(x_i\) and leisure, not labor supply. #c. Labor supply and marginal expense#

Understand the relationship between labor supply and marginal expense

Marginal expense is the additional cost incurred by a firm when it hires one more unit of labor. We will derive the marginal expense for labor supply using a similar process as the calculation for the marginal revenue on a demand curve.

Derive the marginal expense for labor supply

First, let's suppose the labor supply function is given by \(L(w)\). The elasticity of labor supply with respect to the wage is denoted by \(e_{Lw}\). Then, the relationship between the wage and the labor supply is: \\[ e_{Lw}=\frac{\partial L}{\partial w}\frac{w}{L} \\] Now, we derive the marginal expense for labor supply \(ME_L\): \\[ ME_{L}=\frac{\partial (wL)}{\partial L}=\frac{\partial \left[w\left(\frac{e_{Lw}}{w}L\right)\right]}{\partial L} = w\left(1+\frac{1}{e_{Lw}}\right) \\] Therefore, the marginal expense for labor supply is: \(ME_{L}=w\left(1+1 / e_{Lw}\right)\).

Key concepts

Roy's identity

Roy's identity is a crucial concept in consumer and labor economics. It helps explain how demand functions can be derived from indirect utility functions. Essentially, it highlights the connection between consumer preferences expressed through a utility function and their observable behavior in the form of demand for goods. In the context of labor supply, Roy's identity is used to derive the labor supply function from an indirect utility function.

For example, if you're trying to understand how much labor (or work) an individual would supply given their levels of wages and leisure, Roy's identity provides a formula to do just that. By differentiating the indirect utility function concerning wage and dividing it by the differentiation with respect to income, one can calculate the labor supply function. In simple terms, it provides a way of obtaining the labor supply curve by evaluating how changes in wages affect the choice between work and leisure, wrapped into the indirect utility expression.

Cross-price effects

Cross-price effects are insightful in understanding consumer choice changes when the price of one good changes. This relates to the Slutsky equation, which breaks down the total effect of a price change into two parts: the substitution effect and the income effect.

When we talk about cross-price effects in the context of labor supply, we're interested in how a change in wages (price of leisure) affects the demand for other goods. This can provide insights into whether different goods are substitutes or complements. For instance, if an increase in wages leads to an increased consumption of a particular good, then leisure and that specific good are substitutes. Conversely, if the consumption decreases, then the goods are complements.

• Substitutes example: If a higher wage leads more individuals to buy luxury cars, because as leisure (time off work) becomes more expensive, people work more and earn to afford car.
• Complements example: Higher wages might lead to more gym memberships because people value their health alongside their increased income from working more.

Slutsky equation

The Slutsky equation is a detailed way to understand how changes in a good's price influence its demand. It divides the effects into substitution and income effects — both significant components in economic analysis.

This equation becomes particularly useful when analyzing how wage changes affect the demand for leisure and consumption. When applying the Slutsky equation to labor economics, you see how an increase in wages can influence the demand for non-leisure goods — are individuals buying more or less of other products as their working hours change?

The substitution effect, due to a wage change, would lead people to alter their work-leisure balance and possibly modify consumption patterns to maintain a similar level of utility. Meanwhile, the income effect, resulting from increased buying power, might lead them in different directions, such as affording better quality goods or more leisure.

Marginal expense

Marginal expense represents the additional total cost to a firm from hiring one more unit of labor. It's essential to grasp this concept to understand how firms make labor hiring decisions and the interplay between labor cost and productivity.

Using a parallel from revenue analysis, similar to how marginal revenue is derived, marginal expense encapsulates how responsive the labor demand is to wage changes. An exciting elemen it captures is the elasticity of labor supply, which tells us how many more workers are willing to work when wages go up. For instance, if the labor supply is inelastic, wages will need a significant increase to coax more labor.

The calculation shows that the marginal expense grows with a rise in wage, adjusted by the elasticity of supply. Mathematically, it can be expressed as \( ME_{L} = w (1 + 1 / e_{Lw}) \), revealing that adjusting for elasticity is crucial when considering labor costs.

Cobb-Douglas utility function

The Cobb-Douglas utility function is a prominent tool in economic theory to represent consumer preferences. It assumes that goods are consumed in fixed proportions, giving a straightforward way of examining consumer choices and substitution between goods.

For labor economics, the Cobb-Douglas utility function can model how individuals balance labor and leisure. Individuals derive utility from consumption and leisure, represented by a function such as \( u(c, l) = c^{\alpha}l^{\beta} \), where \(c\) is consumption and \(l\) is leisure. This function implies each additional unit consumed gives diminishing marginal utility, recognizing a balance that mirrors realistic human behavior.

• Individuals adjust their work (labor) and leisure based on changes in wages, reflecting the trade-off between earning more income and enjoying more free time.
• The parameters \(\alpha\) and \(\beta\) represent the preference strength towards consumption and leisure, respectively.

This framework helps economists quantify and predict how shifts in wages or preferences might influence labor supply.