Consumer Theory: Axioms, Utility, MRS, and Demand Curves

Axioms Determining Indifference Curve Shape

Six fundamental axioms determine the shape and properties of indifference curves:

1. Completeness: Individuals can compare and rank any two market baskets (A or B, or indifferent between A and B).
2. Transitivity: Guarantees the logical consistency of the ordering among any set of market baskets (If A is preferred to B, and B is preferred to C, then A is preferred to C).
3. Non-Satiation (More is Better): Market baskets with higher quantities (q) are always preferred.
4. Continuity: Market baskets in the same indifference set are distributed continuously.
5. Strict Convexity: Indifference curves are always strictly convex with respect to the origin of coordinates.
6. Smoothness: Indifference curves are smooth, lacking any angular points.

Implications of Non-Satiation

The Non-Satiation axiom has several key implications for indifference sets:

• Indifference sets are decreasing lines (downward sloping).
• Indifference sets further from the origin represent higher utility.
• Indifference sets are represented by single lines, not broad areas.
• A satiation point (a basket preferred to all others) cannot exist in the choice set.
• It allows for standard goods (good, bad, neutral good, perfect complements).
• Indifference curves cannot intersect each other.

Utility Function and Marginal Rate of Substitution (MRS)

The Utility Function (UT FUNCT) represents a numerical assignment criterion. It assigns a real number to each market basket such that this number accurately reflects the basket's place in the consumer preference ordering.

The Marginal Rate of Substitution (MRS) answers the question: How much are we willing to give up from the consumption of one good, in exchange for an additional unit of another good, while keeping total Utility constant?

Special Cases of Indifference Curves

• Perfect Substitutes: Goods that can be substituted at a constant rate (e.g., coffee/tea). Indifference curves are straight lines.
• Perfect Complements: Goods consumed in fixed proportions (e.g., left and right shoes). Indifference curves are L-shaped.
• Neutral Goods: Goods that do not provide higher or lower utility. They do not strictly satisfy the "more is better" aspect of the Non-Satiation axiom.
• Bads: Quantity $q_1$ is a bad when increasing its consumption reduces satisfaction.
• Satiable Goods: Consuming quantities further from a given satiation point diminishes consumer's satisfaction. These goods violate the Non-Satiation axiom beyond the optimal point.

Consumer Equilibrium and Demand Curves (Unit 4)

Consumer Equilibrium

Equilibrium (Eq) is the point where an indifference curve is tangent to the budget line. This point lies on the upper frontier of the budget set and yields the highest possible utility, satisfying the condition: $MRS = P_1/P_2$.

Key Demand Curves

These curves illustrate how the quantity demanded changes in response to variations in price or income:

1. Ordinary Demand Curve (ODC)

• Definition: Shows the quantity demanded of a good ($q_1$) as a function of its own price ($p_1$).
• Function: $q_1 = q_1(p_1, \bar{p_2}, \bar{Y})$.
• Related Curve: Price Consumption Curve (CPC): Shows how the quantities demanded ($q_1, q_2$) change when $p_1$ changes, holding $p_2$ and Income ($Y$) constant.
• Condition: $MRS = p_1/p_2$ (where $p_2$ and $Y$ are constant, $p_1$ is free).

2. Cross Demand Curve (CDC)

• Definition: Shows the quantity demanded of a good ($q_1$) as a function of the price of the other good ($p_2$).
• Function: $q_1 = q_1(\bar{p_1}, p_2, \bar{Y})$.
• Related Curve: Price Consumption Curve of Good 1 (CPC2): Shows how the quantities demanded ($q_1, q_2$) change when $p_2$ changes, holding $p_1$ and $Y$ constant.
• Condition: $MRS = p_1/p_2$ (where $p_1$ and $Y$ are constant, $p_2$ is free).

3. Engel Curve (EC)

• Definition: Shows the quantity demanded of a good ($q_1$) as a function of the consumer's income ($Y$).
• Function: $q_1 = q_1(\bar{p_1}, \bar{p_2}, Y)$.
• Related Curve: Income Consumption Curve (ICC): Shows how the quantities demanded ($q_1, q_2$) change when income ($Y$) changes, holding $p_1$ and $p_2$ constant.
• Condition: $MRS = p_1/p_2$ (where $p_1$ and $p_2$ are constant, $Y$ is free).