Question

Following in the spirit of the labor market game described in Example \(22.4,\) suppose the firm's total revenue function is given by \\[ T 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 \(22.4 ?\) 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, the Nash equilibrium wage contract is (0, 0), but an alternative wage contract of (4, 4) is shown to be Pareto superior. For the alternative wage contract to be sustainable as a subgame perfect equilibrium, both players must be better off under the alternative wage contract compared to any other possible contracts, and there should be credible threats for both players to revert to the Nash equilibrium if one deviates from the alternative contract. Factors such as repeated interactions, reputation, and enforcement mechanisms can help facilitate the sustainability of the alternative wage contract as a subgame perfect equilibrium.

Step-by-step solution

Part (a): Finding the Nash equilibrium wage contract

Step 1: Write the total revenue and utility functions. Total revenue function: \\[ TR = 10L - L^2 \\] Union's utility function: \\[ U(w, L) = wL \\] Step 2: Compute the firm's profit function and the wage bill. Firm's profit function: \\[ \pi = TR - wL = 10L - L^2 - wL \\] Wage bill: \\[ WB = wL \\] Step 3: Write down the first-order conditions for Nash equilibrium. Firm's first-order condition (maximizing profit): \\[ \frac{\partial \pi}{\partial w} = -L = 0 \\] Union's first-order condition (maximizing utility): \\[ \frac{\partial U}{\partial L} = w - \frac{\partial w}{\partial L}L = 0 \\] Step 4: Solve the first-order conditions to find the Nash equilibrium wage contract \((w^*, L^*)\). From the firm's first-order condition, \(L = 0\). Substituting this into the union's first-order condition, we get: \\[ w - \frac{\partial w}{\partial L}(0) = 0 \\] As \(\frac{\partial w}{\partial L} = 0\), the Nash equilibrium wage contract is \((w^*, L^*)=(0, 0)\).

Part (b): Pareto superiority of alternative wage contract

Step 1: Calculate the utility and profit levels for the Nash equilibrium and alternative wage contract. Nash equilibrium wage contract \((0, 0)\): \\[ U(0, 0) = 0 \cdot 0 = 0 \\] \\[ \pi(0, 0) = 10 \cdot 0 - 0^2 - 0 \cdot 0 = 0 \\] Alternative wage contract \((4, 4)\): \\[ U(4, 4) = 4 \cdot 4 = 16 \\] \\[ \pi(4, 4) = 10 \cdot 4 - 4^2 - 4 \cdot 4 = 16 \\] Step 2: Compare the utility and profit levels. Since \(U(4, 4) > U(0, 0)\) and \(\pi(4, 4) > \pi(0, 0)\), the alternative wage contract \((4, 4)\) is Pareto superior to the Nash equilibrium wage contract \((0, 0)\).

Part (c): Conditions for the sustainability of the alternative wage contract as a subgame perfect equilibrium

For the alternative wage contract to be sustainable as a subgame perfect equilibrium, it must meet the following conditions: 1. Both players (the firm and the union) must be better off under the alternative wage contract compared to any other possible wage contracts. 2. The alternative wage contract should be enforced by credible threats. If one player deviates from the contract, the other player should have a credible threat to deviate from it as well and revert to the Nash equilibrium contract. In our case, we have shown that the alternative wage contract \((4, 4)\) is Pareto superior to the Nash equilibrium wage contract \((0, 0)\). Hence, it satisfies the first condition. The second condition can be satisfied if both the firm and the union have a credible threat of reverting to the Nash equilibrium contract if the other player deviates from the alternative contract. This requires cooperation and trust between the two players, which can be facilitated by repeated interactions, reputation, or other enforcement mechanisms such as contracts and laws.

Key concepts

Understanding the Total Revenue Function

In an economic context, the total revenue function represents the total income a firm receives from selling its output of goods or services. The function is often defined in terms of the quantity of goods sold. In the exercise we're focusing on, the total revenue function is given by the equation:
\[ TR = 10L - L^2 \]
Here, 'TR' stands for total revenue, and 'L' is the quantity of labor. This quadratic function reflects the revenue generated as labor input varies. As labor increases, revenue firstly increases at a decreasing rate until it reaches a maximum point, then starts to decrease as labor input continues to increase. The shape of this function can be attributed to concepts such as diminishing returns, where each additional unit of labor contributes less to output than the previous unit.

Deciphering the Union's Utility Function

A union's utility function in economics is a way to express the well-being or satisfaction derived from various possible scenarios for its members, often involving wages and employment levels. In our specific labor market exercise, the union's utility function is tied to the total wage bill and is represented as:
\[ U(w, L) = wL \]
The 'U' denotes the utility, 'w' is the wage rate, and 'L' denotes the level of employment or labor. The product of the wage rate and the amount of labor indicates the total amount of wages paid to the workers, which the union looks to maximize. The simplicity of this function suggests that the union derives utility solely based on the sum of the wages its members receive, without considering other factors such as working conditions or job security.

Analyzing Pareto Superiority

Pareto superiority, a core concept in microeconomics, refers to a situation where an allocation of resources makes at least one individual better off without making anyone else worse off. In the context of the wage contract problem, comparing the Nash equilibrium contract to the alternative contract \(w'=L'=4\) raises the question of Pareto superiority.

The alternative contract's utility for the union and profit for the firm are both \(16\), an improvement over the Nash equilibrium utility and profit, which are \(0\). Therefore, the alternative contract is considered Pareto superior because it benefits both parties without causing detriment to either. It's essential to recognize that in real-world negotiations, achieving Pareto improvements can be crucial for securing mutually beneficial agreements.

Exploring Subgame Perfect Equilibrium

The concept of subgame perfect equilibrium pertains to a consistent strategy within dynamic games, where players make sequential decisions. It requires that each player's strategy is optimal, considering the other players' strategies, for any point in the game onward.

In the scenario of the wage contract, for the alternative contract \(w'=L'=4\) to be a subgame perfect equilibrium, it must be sustainable; both the firm and the union must have no incentive to deviate from this agreement in any subgame. Two main conditions are highlighted: the contract must provide mutual benefit overriding any alternative, and there must be credible mechanisms to enforce it. Trust, repetition, reputation, and formal agreements can help ensure that neither side will gain from deviating. A subgame perfect equilibrium aligns incentives to maintain a stable agreement, thus it's a powerful concept when analyzing how rational players might negotiate and sustain contracts over time.