Microeconomics Problem Set: Consumer Choice, Utility, and Budget Constraints

Binary Relations and Transitivity

Understanding Binary Relations in Economics

Binary relations are fundamental in microeconomics, especially in the study of preferences and choice. A binary relation on a set X is a rule that assigns, for some or all pairs (A, B) in X, a truth value to the statement "A is related to B." Transitivity is a key property for rational choice.

• Transitive Relation: A relation R on set X is transitive if whenever A R B and B R C, then A R C for all A, B, C in X.

• Examples in Economics: Preference relations ("at least as good as") are typically assumed to be transitive for rational decision-making.

Application: Determining which relations (such as "has at least as many pages as" or "is at least as far from the origin as") are transitive helps in modeling consumer preferences and choices.

Consumer Choice and Indifference Curves

Bundles, Trade-offs, and Marginal Rate of Substitution

Consumers choose between bundles of goods to maximize their utility, subject to their budget constraints. Indifference curves represent combinations of goods that provide the same level of satisfaction.

• Bundle: A specific combination of quantities of two or more goods (e.g., apples and bananas).

• Indifference Curve: A curve showing all bundles that provide the consumer with the same utility.

• Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade one good for another while remaining on the same indifference curve. Mathematically, $MRS = -\frac{MU_x}{MU_y}$, where $MU_x$ and $MU_y$ are the marginal utilities of goods x and y, respectively.

Example: If a consumer is indifferent between (6, 2) and (3, 4), the MRS between apples and bananas at these points can be inferred from the trade-off.

Marginal Rate of Substitution and Utility Functions

Calculating MRS from Utility Functions

The marginal rate of substitution can be calculated directly from a utility function $U(A, B)$, where A and B are quantities of two goods.

• Marginal Utility: The additional utility gained from consuming one more unit of a good, holding other goods constant.

• Formula: For $U(A, B)$, $MRS = -\frac{\partial U/\partial A}{\partial U/\partial B}$

Example: For $U(A, B) = A + 2B$, $MU_A = 1$, $MU_B = 2$, so $MRS = -\frac{1}{2}$.

Budget Constraints and Consumer Choice

Budget Lines and Affordability

Consumers face budget constraints that limit the combinations of goods they can afford. The budget line represents all combinations of goods that exhaust the consumer's income.

• Budget Constraint Equation: $P_A A + P_B B = I$, where $P_A$ and $P_B$ are prices of goods A and B, and $I$ is income.

• Budget Line: The set of all bundles (A, B) that satisfy the budget constraint.

• Affordable Bundle: A bundle is affordable if it lies on or below the budget line.

Example: If income $I = 12$, $P_A = 3$, $P_B = 2$, the budget line is $3A + 2B = 12$.

Utility Functions and Convex Preferences

Properties of Utility Functions

Utility functions represent consumer preferences. Convex preferences imply that consumers prefer averages to extremes, leading to convex indifference curves.

• Convex Preferences: If a consumer prefers a mix of two bundles to either bundle alone, preferences are convex.

• Utility Functions: Common forms include Cobb-Douglas ($U(A, B) = A^\alpha B^{1-\alpha}$), perfect substitutes ($U(A, B) = aA + bB$), and perfect complements ($U(A, B) = \min\{aA, bB\}$).

Example: $U(A, B) = A^2 + B^2$ is not convex, while $U(A, B) = \sqrt{A} + \sqrt{B}$ is convex.

Effects of Taxes on Consumer Choice

Budget Line Shifts and Purchasing Power

Taxes on goods change their prices, affecting the consumer's budget constraint and purchasing power.

• Tax Impact: If a tax increases the price of bananas from $P_B$ to $P_B(1 + t)$, the budget line pivots inward, reducing the maximum affordable quantity of bananas.

• Purchasing Power: The consumer's ability to buy goods decreases as the tax reduces the set of affordable bundles.

Example: If income $I = 24$, $P_A = 3$, $P_B = 2$, and a 20% tax is imposed on bananas, the new price of bananas is $2.4$, and the budget line becomes $3A + 2.4B = 24$.

Summary Table: Utility Functions and Convexity

Utility Function	Convex Preferences?	Notes
$U(A, B) = A^2 + B^2$	No	Indifference curves are not convex to the origin
$U(A, B) = A^{1/2} + B^{1/2}$	Yes	Convex indifference curves
$U(A, B) = A^{1/2} B^{1/2}$	Yes	Cobb-Douglas, always convex
$U(A, B) = A + B$	Yes	Perfect substitutes, linear indifference curves
$U(A, B) = A + \sqrt{B}$	Yes	Convex indifference curves

Key Formulas

• Budget Constraint: $P_A A + P_B B = I$

• Marginal Rate of Substitution: $MRS = -\frac{MU_A}{MU_B}$

• Marginal Utility: $MU_A = \frac{\partial U}{\partial A}$, $MU_B = \frac{\partial U}{\partial B}$