BackLecture 3: Optimization, Game Theory, and Equilibrium in Microeconomics
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Optimization, Games, and Equilibrium
Introduction to Optimization
In microeconomics, optimization refers to the process by which economic agents (such as individuals, firms, or governments) select the best feasible option from a set of alternatives. The goal is to maximize their benefit or utility given constraints such as limited resources or information.
Optimization using total value: This technique involves calculating the total value (benefit minus cost) of each feasible option and selecting the one with the highest total value.
Optimization using marginal analysis: This method focuses on the change in total value when switching from one option to another, using these marginal comparisons to choose the best option.
Key result: Both total value and marginal analysis approaches yield identical answers when applied correctly.
Challenges in Optimization
Making optimal choices can be difficult due to several factors:
Limited information: Agents may not have access to all relevant data.
Complexity: Sorting through information and alternatives can be complicated.
Inexperience: Lack of experience with a situation can hinder decision-making.
Uncertainty: Even the best choice may lead to a bad outcome due to unpredictable factors.
Optimization Techniques
Total Value Approach
This approach involves the following steps:
Translate all costs and benefits into common units (e.g., dollars per month).
Calculate the total net benefit for each alternative:
Select the alternative with the highest net benefit.
Example: Tablet or Phone Usage
Suppose you are deciding how many hours per day to spend watching videos or reading news on a tablet or smartphone. The following table summarizes the benefits and costs:
Hours (watching/reading) | Total Benefits (amusement or knowledge) | Total Costs (mental & physical fatigue; missing out on lecture, homework, and social activities) |
|---|---|---|
1 | $35 | $20 |
2 | $70 | $40 |
3 | $90 | $65 |
4 | $100 | $100 |
Net benefit for each option:
Hours | Benefit | Cost | Net Benefit |
|---|---|---|---|
0 | 0 | 0 | 0 |
1 | 35 | 20 | 15 |
2 | 70 | 40 | 30 |
3 | 90 | 65 | 25 |
4 | 100 | 100 | 0 |
Optimal choice: 2 hours, with a net benefit of $30.
Marginal Analysis Approach
Marginal analysis focuses on the incremental changes in benefit and cost when moving between alternatives.
Translate all costs and benefits into common units.
Calculate the marginal benefit (change in benefit) and marginal cost (change in cost) for each additional unit.
Choose the alternative where Marginal Benefit = Marginal Cost.
Formulas:
Hours | Benefit | Marginal Benefit | Cost | Marginal Cost | MB-MC | Net Benefit |
|---|---|---|---|---|---|---|
0 | 0 | - | 0 | - | - | 0 |
1 | 35 | 35 | 20 | 20 | 15 | 15 |
2 | 70 | 35 | 40 | 20 | 15 | 30 |
3 | 90 | 20 | 65 | 25 | -5 | 25 |
4 | 100 | 10 | 100 | 35 | -25 | 0 |
Optimal choice: 2 hours, where MB-MC is maximized and net benefit is highest.
Example: New Tablet with Lower Costs
Hours | Benefit | Cost | Net Benefit |
|---|---|---|---|
0 | 0 | 0 | 0 |
1 | 35 | 20 | 15 |
2 | 70 | 35 | 35 |
3 | 90 | 50 | 40 |
4 | 100 | 75 | 25 |
Optimal choice: 3 hours, with a net benefit of $40.
Game Theory and Strategic Play
Introduction to Game Theory
Game theory is the study of strategic interactions among multiple decision-makers (players). Each player's payoff depends not only on their own choices but also on the choices of others.
Game: A multi-person decision problem.
Nash equilibrium: A situation in which each agent chooses a strategy that maximizes their benefit, given the strategies of every other agent.
Applications of Game Theory
International trade agreements
Firm strategy and competition
Manager-employee relations
Tax compliance and regulation
Political negotiations and voting
Biology and medicine (competition, pandemics, etc.)
Formalizing a Game
Players: Who are the decision-makers?
Strategy set: What actions can each player take?
Timing: Who moves when?
Information structure: What does each player know when making decisions?
Payoffs: What does each player receive for each possible outcome?
Example: Rock, Paper, Scissors
Players: Alex and Betty
Strategy set: Rock (R), Paper (P), Scissors (S)
Timing: Simultaneous moves
Information: Both know the rules
Payoffs: Win (1), Lose (-1), Draw (0)
Zero-Sum Games and Mutual Gains
Zero-sum games are situations where one player's gain is exactly another player's loss (e.g., chess, poker). However, many real-world interactions allow for mutual gains through voluntary exchange, as illustrated by the tariff game.
Zero-sum thinking: Ignores the possibility of mutual gains from trade.
Nash Equilibrium
Definition: Each player chooses a strategy that maximizes their payoff, given the strategies of others.
Best response: For each player, their strategy is the best response to the equilibrium strategies of the other players.
Summary Table: Optimization Techniques
Technique | Definition | Key Formula | Optimality Condition |
|---|---|---|---|
Total Value | Choose option with highest net benefit | Maximum net benefit | |
Marginal Analysis | Compare marginal benefit and marginal cost |
|
Key Terms
Optimization: Selecting the best feasible option to maximize benefit.
Marginal analysis: Evaluating the impact of small changes in choice.
Net benefit: Total benefit minus total cost.
Game theory: Study of strategic interactions among agents.
Nash equilibrium: Each agent's strategy is optimal given others' strategies.
Zero-sum game: One player's gain is another's loss.
Example Applications
Choosing study hours to maximize exam performance (balancing benefit of practice vs. cost of fatigue).
Deciding production levels in a firm (where marginal revenue equals marginal cost).
Strategic interactions in international trade or business competition.
Additional info: The notes provide foundational concepts for microeconomic decision-making, including optimization, marginal analysis, and game theory, with practical examples and tables for clarity.
