Forelesning 2

Utility and Efficiency in Exchange

Introduction

This topic formalizes the concept of efficiency in exchange, characterizes efficient equilibria, and explores the role of individual preferences in determining economic outcomes. The analysis is foundational for understanding how markets allocate resources and the trade-offs between equity and efficiency.

2.1 Basic Assumptions about Preferences

Definition of Preferences

• Preferences describe whether and why consumers prefer one set of goods ("basket") to another. They reflect desirability, not actual choices, which also depend on prices and income.

Basket	Units of Food	Units of Clothing
A	20	30
B	16	30
C	40	20
D	18	20
E	40	30

• Completeness: Consumers can compare and rank all possible baskets. For any two baskets A and B, either A ≽ B, B ≽ A, or both (indifference).

• Transitivity: If A ≽ B and B ≽ C, then A ≽ C (consistency in preferences).

• More is better (Local Non-Satiation): In the relevant range, more of a good is always weakly preferred to less.

• Convexity (Plausible Assumption): The more you consume of one good, the more you are willing to substitute that good for the other.

2.2 Indifference Curves and Indifference Maps

Indifference Curves

• Indifference curves represent combinations of goods that provide the consumer with the same level of satisfaction.

• Any two baskets on the same indifference curve are equally desirable to the consumer.

• Higher indifference curves represent higher utility levels.

Example: If basket A is on curve U3, B on U2, and C on U1, then A ≻ B ≻ C.

2.3 Marginal Rate of Substitution (MRS)

Definition and Properties

• MRS: The maximum amount of one good a consumer is willing to give up to obtain one additional unit of another good.

• Graphically, MRS is the slope of the indifference curve at a given point.

• Under convex preferences, MRS falls as we move towards more consumption of a good along the indifference curve.

Formula:

$MRS_{xy} = -\frac{dY}{dX}$ (holding utility constant)

2.4 The Advantages of Trade

Pareto Improvements and Efficient Allocations

• Trade between agents can lead to Pareto improvements if their MRS differ.

• Pareto-improving trades are possible whenever agents have different MRS.

• Such trades are characterized by terms of trade that lie between the agents' MRS.

• When $MRS_{Karen} = MRS_{James}$, no further Pareto improvements are possible—this is a Pareto-efficient equilibrium.

Example: Two agents (James and Karen) with initial endowments of clothing and food can trade to reach a mutually beneficial allocation.

2.5 Reformulation of Pareto Efficiency in Exchange

• An allocation of goods is efficient only if the marginal rate of substitution between any pair of goods is the same for all consumers.

Formula:

$MRS_{Karen} = MRS_{James}$

2.6 Edgeworth Box

Graphical Representation of Allocations

• The Edgeworth box is a graphical tool to represent all possible allocations of two goods between two agents, given total endowments.

• Each point inside the box (including edges) is a possible allocation; points outside are not possible.

	Food	Clothing
Total Endowment	10	6

Additional info: The box's width and height correspond to the total quantities of food and clothing, respectively. The origin for one agent is at the bottom left, and for the other at the top right.

2.7 The Contract Curve

Efficient Allocations and Bargaining Power

• The contract curve is the set of all Pareto-efficient allocations where the agents' indifference curves are tangent.

• The specific allocation reached depends on the allocation mechanism, initial endowment, and bargaining power of the agents.

• In a competitive market, trade leads to a subset of the contract curve; with a central planner, any point on the contract curve may be chosen.

2.8 Utility Possibility Frontier

Trade-offs Between Agents' Utilities

• The utility possibility frontier represents the maximum utility combinations for two agents, given the contract curve.

• Allocations below the frontier are inefficient.

2.9 Equity and Efficiency

Trade-offs and Social Welfare

• If only efficiency is considered, all allocations on the contract curve are equally good.

• Equity concerns whether some allocations are more "fair" than others.

• Economists debate whether Pareto efficiency should be supplemented with criteria like fairness when evaluating outcomes.

Social Welfare Functions (SWFs)

• SWFs aggregate individual preferences into a single criterion and incorporate value judgments.

• Egalitarian: Equalize utilities.

• Rawlsian (maximin): Maximize the utility of the least well-off.

• Utilitarian: Maximize the sum of utilities.

• Weighted utilitarian: Maximize a weighted sum of utilities.

Equity-Efficiency Trade-off

• A social planner can enforce redistribution to achieve a more equitable outcome, but may face constraints (e.g., only able to use taxes).

• Redistribution can reduce incentives and total output, leading to allocations that are more equitable but less efficient (inside the utility possibility frontier).

• There may be a trade-off between equity and efficiency when the planner is constrained.