Possibilities, Preferences, and Choices: Microeconomic Consumer Theory

Consumption Possibilities

Budget Constraints and the Budget Line

Household consumption choices are limited by income and the prices of available goods and services. The budget line represents all combinations of goods a consumer can afford given their income and the prices of those goods.

• Budget Line: Shows the limits to consumption choices. Points on or inside the line are affordable; points outside are not.

• Indivisible Goods: Must be purchased in whole units (e.g., movies).

• Divisible Goods: Can be purchased in any quantity (e.g., cola).

• Example: Lisa has $40, movies cost $8 each, cola costs $4 per case. Seven possible combinations of movies and cola can be plotted on a graph, forming the budget line.

The Budget Equation

The budget equation mathematically describes the budget line:

• General Form: $P_C Q_C + P_M Q_M = Y$

• Where $P_C$ = price of cola, $Q_C$ = quantity of cola, $P_M$ = price of movies, $Q_M$ = quantity of movies, $Y$ = income.

• Rearranged: $Q_C = \frac{Y}{P_C} - \frac{P_M}{P_C} Q_M$

• Real Income: $\frac{Y}{P_C}$ (in terms of cola)

• Relative Price: $\frac{P_M}{P_C}$ (price of a movie in terms of cola)

Effects of Price and Income Changes

• Change in Price: A rise in the price of a good on the x-axis decreases the affordable quantity and increases the slope of the budget line (rotation).

• Change in Income: An increase or decrease in income shifts the budget line parallel to itself; the slope remains unchanged.

• Example Equations:

  • Initial: $8Q_M + 4Q_C = 40$

  • After price change: $4Q_M + 4Q_C = 40$

  • After income change: $8Q_M + 4Q_C = 20$

Possibility	Movies (Quantity)	Cola (Quantity)	Expenditure on Movies	Expenditure on Cola
A	0	10	$0	$40
B	1	8	$8	$32
C	2	6	$16	$24
D	3	4	$24	$16
E	4	2	$32	$8
F	5	0	$40	$0

Preferences and Indifference Curves

Indifference Curves

An indifference curve shows all combinations of two goods that provide the consumer with the same level of satisfaction (utility). The consumer is indifferent between any two points on the same curve.

• Points above an indifference curve are preferred; points below are less preferred.

• Example: Lisa is indifferent between consuming (2 movies, 6 cola) and (3 movies, 4 cola).

Preference Map

A preference map is a series of indifference curves. Curves farther from the origin represent higher levels of utility.

• Notation: $I_0$ (lower utility), $I_1$ (higher utility), $I_2$ (even higher utility).

• Lisa prefers any point on $I_2$ to any point on $I_1$, and any point on $I_1$ to any point on $I_0$.

Utility Functions and Indifference Curves

Utility functions assign a numerical value to each bundle of goods, representing the consumer's satisfaction.

• General Form: $U = U(M, C)$ or $U = U(Z, B)$

• Indifference curve for utility level $\overline{U}$: $\overline{U} = U(M, C)$

• Example: If $U = \sqrt{ZB}$, then $\overline{U} = 4$ includes all $(Z, B)$ such that $ZB = 16$ (e.g., (4,4), (2,8), (8,2), (1,16), (16,1)).

Marginal Utility

Marginal utility is the additional satisfaction gained from consuming one more unit of a good, holding the quantity of the other good constant.

• Formula: $MU_z = \frac{\Delta U}{\Delta Z}$

Marginal Rate of Substitution (MRS)

The marginal rate of substitution (MRS) measures the rate at which a consumer is willing to give up one good (y) to obtain an additional unit of another good (x), while remaining on the same indifference curve.

• Formula: $MRS = \frac{\Delta B}{\Delta Z}$

• Also: $MRS = -\frac{MU_z}{MU_b}$

• Example: If Lisa moves from bundle a to b, she gives up 3 burritos for 1 pizza ($MRS = -3$); from b to c, she gives up 2 burritos for 1 pizza ($MRS = -2$).

Diminishing Marginal Rate of Substitution

The diminishing marginal rate of substitution is a key assumption in consumer theory. As a consumer acquires more of good x, they are willing to give up less of good y for additional units of x.

• Indifference curves are typically convex to the origin.

• MRS approaches zero as we move down and to the right along an indifference curve.

Special Cases of Indifference Curves

• Perfect Substitutes: Indifference curves are straight, parallel lines. Example: Bill is willing to exchange one can of Coke for one can of Pepsi ($MRS = -1$).

• Perfect Complements: Indifference curves are L-shaped. Example: Maureen gets the same utility from one piece of pie and one scoop of ice cream as from one piece of pie and two scoops.

• Imperfect Substitutes: Standard convex indifference curves, showing a consumer views two goods as imperfect substitutes.

Best Affordable Choice and Constrained Consumer Choice

Optimal Consumption Bundle

The best affordable choice is the bundle that:

• Lies on the budget line

• Is on the highest attainable indifference curve

• Has an MRS equal to the relative price of the two goods

Given preferences and budget constraints, the consumer's optimal bundle maximizes utility.

Summary Table: Types of Indifference Curves

Type	Shape	Example	MRS
Perfect Substitutes	Straight lines	Coke & Pepsi	Constant
Perfect Complements	L-shaped	Pie & Ice Cream	Zero or Infinite
Imperfect Substitutes	Convex curves	Pizzas & Burritos	Diminishing

Key Formulas

• Budget Equation: $P_C Q_C + P_M Q_M = Y$

• Real Income: $\frac{Y}{P_C}$

• Relative Price: $\frac{P_M}{P_C}$

• Marginal Utility: $MU_z = \frac{\Delta U}{\Delta Z}$

• Marginal Rate of Substitution: $MRS = \frac{\Delta B}{\Delta Z} = -\frac{MU_z}{MU_b}$

Additional info:

• Indifference curves farther from the origin represent higher utility.

• Convexity of indifference curves reflects diminishing MRS and typical consumer behavior.

• Optimal choice occurs where the budget line is tangent to the highest attainable indifference curve.