Question

Following in the spirit of the labor market game described in Example \(16.6,\) suppose the firm's total revenue function is given by \\[ R=10 l-l^{2} \\] and the union's utility is simply a function of the total wage bill: \\[ U(w, l)=w l \\] a. What is the Nash equilibrium wage contract in the two-stage game described in Example \(16.6 ?\) b. Show that the alternative wage contract \(w^{\prime}=l^{\prime}=4\) is Pareto superior to the contract identified in part (a). c. Under what conditions would the contract described in part (b) be sustainable as a subgame-perfect equilibrium?

Short answer

In summary, we analyzed a two-stage game between a firm and a union. The Nash equilibrium wage contract was determined to be (w, l) = (\\(\frac{10}{3}\\), \\(\frac{10}{3}\\)). We then showed that an alternative wage contract (w', l') = (4, 4) is Pareto superior to the Nash equilibrium contract. Finally, we determined that the alternative wage contract can be sustained as a subgame-perfect equilibrium, as both parties have no incentives to deviate from the agreed contract.

Step-by-step solution

a. Finding the Nash equilibrium wage contract

To find the Nash equilibrium wage contract, we will first determine the best response functions for both the union and the firm. Given the total revenue function R and the union's utility function U, we set up the following optimization problems: 1. Union's maximization problem: \\[ Max_{l} U(w, l) = wl \\] 2. Firm's maximization problem: \\[ Max_{w} \pi(w, l) = R(l) - wl \\] First, we need to find the union's best response function by solving its maximization problem. The utility function is linear in l, so the best response function will be simply the highest wage possible given w:

Find the best response for the union

Since the utility function is linear, the union's best response function will be: \\[ l^{u}(w) = w \\] Now let's find the firm's best response function.

Find the best response for the firm

The firm's maximization problem is given by: \\[ \pi(w, l) = R(l) - wl = 10l - l^2 - wl \\] Taking the first-order derivative with respect to l and setting it equal to 0: \\[ \frac{\partial \pi}{\partial l} = 10 - 2l - w = 0 \Rightarrow l^{f}(w) = \frac{10 - w}{2} \\] Having obtained both best response functions, we can find the Nash equilibrium by setting them equal to each other.

Find the Nash equilibrium wage contract

To find the Nash equilibrium, equate the best response functions: \\[ l^{u}(w) = l^{f}(w) \\] \\[ w = \frac{10 - w}{2} \Rightarrow w = \frac{10}{3} \\] Plugging this value back into one of the best response functions, back into the Nash equilibrium l: \\[ l = \frac{10 - \frac{10}{3}}{2} = \frac{10}{3} \\] So, the Nash equilibrium wage contract in the two-stage game is (w, l) = (\\(\frac{10}{3}\\), \\(\frac{10}{3}\\)).

b. Showing the alternative wage contract is Pareto superior

Now to show that the alternative wage contract (w', l') = (4,4) is Pareto superior to the contract (w, l) = (\\(\frac{10}{3}\\), \\(\frac{10}{3}\\)), we need to verify if both the union and the firm are better off with the alternative contract. 1. Union's utility with the alternative contract: \\[ U(w', l') = w'l' = 4 \times 4 = 16 \\] 2. Union's utility with the Nash equilibrium contract: \\[ U(w, l) = wl = \frac{10}{3} \times \frac{10}{3} = \frac{100}{9} \\] Since \\( U(w', l') > U(w, l)\\), the union is better off with the alternative contract. 3. Firm's profit with the alternative contract: \\[ \pi(w', l') = R(l') - w'l' = 10 \times 4 - 4^2 - 4 \times 4 = 8 \\] 4. Firm's profit with the Nash equilibrium contract: \\[ \pi(w, l) = R(l) - wl = 10 \times \frac{10}{3} - \frac{10^2}{9} - \frac{100}{3} \Rightarrow \pi(w, l) = \frac{20}{9} \\] Since \\( \pi(w', l') > \pi(w, l)\\), the firm is also better off with the alternative contract. Thus, the alternative wage contract (w', l') = (4, 4) is Pareto superior to the contract (w, l) = (\\(\frac{10}{3}\\), \\(\frac{10}{3}\\)).

c. Conditions for alternative contract to be a subgame-perfect equilibrium

For the alternative contract (w', l') = (4, 4) to be sustainable as a subgame-perfect equilibrium, both the union and the firm must have no incentives to deviate from the agreed contract. In other words, each party's payoff from the alternative contract should be greater than or equal to their best response payoffs given the other party's choice. Mathematically, this means: 1. For the union, \\( U(w', l') \ge U(w, l^{u}(w'))\\) 2. For the firm, \\( \pi(w', l') \ge \pi(w^{f}(l'), l')\\) But since we established in part (b) that both parties are better off with the alternative contract, these conditions are satisfied, and thus the alternative wage contract (w', l') = (4, 4) can be sustained as a subgame-perfect equilibrium.

Key concepts

Labor Market Game

In the context of microeconomics, the term 'labor market game' refers to a strategic interaction between two key players in the labor market: a firm and a union. This game is structured to reflect the negotiation process for wages and employment levels. To simplify, imagine the labor market as a kind of tug-of-war, where the firm aims to maximize profits and the union strives to secure the best possible wage for its members.

The 'game' starts with each player possessing a set of strategies. In our textbook example, the firm can choose a level of labor, which indirectly affects the wage rate since the union’s utility depends on the total wage bill (the product of wages and employment level). The outcome of this strategic interaction is typically a wage contract that both parties agree upon.

However, what's intriguing about such a game are the concepts of Nash equilibrium and Pareto superiority. In Nash equilibrium, neither party can improve their situation without making the other party worse off, given the choice of the other player. Meanwhile, a Pareto superior outcome implies a situation where at least one party is better off, and no one is worse off—a concept that is closely related to the idea of efficiency in economics.

Total Revenue Function

The total revenue function is a mathematical representation of how a firm’s revenue changes with varying levels of output or, in the case of our labor market game, the level of labor hired. In simple terms, the total revenue function illustrates how the money coming into a firm increases or decreases as it hires more or fewer workers.

Within our provided textbook problem, the total revenue function is expressed as \( R = 10l - l^2 \), where \( R \) stands for total revenue and \( l \) represents the level of labor. This quadratic function suggests that initially, hiring more labor increases total revenue at a decreasing rate due to the negative squared term, reflecting the law of diminishing returns. However, beyond a certain point, hiring additional labor actually decreases total revenue.

This total revenue function serves as a critical foundation for the firm's decision-making process. By comparing the cost of labor against the revenue labor produces, the firm can determine the optimal level of labor \textemdash the exact point where the difference between the revenue generated and the cost of labor is at its maximum, or in other words, where the firm's profit is maximized.

Union's Utility Function

The union's utility function is key to understanding the union’s objectives and how it responds to different wage contracts. Think of 'utility' as a measure of satisfaction or preference—it encapsulates what the union values. For the union in our labor market game, utility is derived from the total wage bill, as represented by the function \( U(w, l) = wl \).

This linear utility function signifies that the union’s utility increases directly with the wage rate, \( w \), and the level of labor, \( l \). The higher the total wage bill, the happier the union members are because it represents the total amount of remuneration paid to them for their work.

When analyzing scenarios of wage contracts, the union will aim for the outcome that maximizes its utility. However, it must also consider the firm's response, as well as any equilibria that can arise from the strategic interplay between the union's desire for higher wages and the firm's pursuit of profit maximization. A deeper grasp of the union’s utility function assists us in predicting and explaining the nature of the wage contracts that are agreed upon, including how variations in these contracts could lead to more efficient or equitable outcomes for both parties.