Source: Practice MCQs for Exam 1, Texas A&M University
Tags: consumer preferences, completeness, transitivity, utility function, indifference curve, marginal rate of substitution, MRS, diminishing MRS, convexity, marginal utility, perfect substitutes, perfect complements, Cobb-Douglas utility
Consumer theory starts with preferences that are complete and transitive, then represents them with utility functions and indifference curves. The slope of an indifference curve is the marginal rate of substitution (MRS). Standard "well-behaved" indifference curves are convex to the origin because of diminishing MRS. Perfect substitutes and perfect complements are special cases with straight-line and L-shaped indifference curves respectively.
Complete preferences
The consumer can rank any pair of alternatives. For any two bundles A and B, the consumer either prefers A, prefers B, or is indifferent.
Transitive preferences
If the consumer prefers A to B and B to C, then the consumer prefers A to C. Transitivity ensures consistent ranking.
Utility function
A mathematical representation of preferences that assigns a number to each bundle. Higher numbers mean more preferred bundles. The specific numbers do not matter, only their ordering.
Indifference curve
A curve showing all combinations of goods that give the consumer the same level of utility. Movement along the curve keeps utility constant.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while staying on the same indifference curve. It is measured by the slope of the indifference curve (in absolute value). MRS typically diminishes as you move along the curve.
Diminishing MRS
As you consume more of good X and less of good Y, you are willing to give up less Y per additional unit of X. This is the reason indifference curves are convex to the origin.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding consumption of all other goods constant. MRS_xy = MU_x / MU_y.
Diminishing marginal utility
The more of a good you consume, the less additional satisfaction each extra unit provides. Example: the more pizza Alex eats, the less he enjoys another slice.
Perfect substitutes
Goods that can be exchanged at a constant rate. Indifference curves are straight downward-sloping lines. The MRS is constant (does not diminish).
Perfect complements
Goods that are consumed in fixed proportions. Indifference curves are L-shaped. The utility function takes the form u(x, y) = min{x, y} or similar.
Completeness means the consumer can rank any pair of bundles, not just bundles they "like."
Transitivity means rankings are internally consistent.
Together, these two properties allow the consumer to rank all bundles in a coherent order.
A complete and transitive preference relation does not require a linear utility function or any specific shape of utility.
Each curve represents a single utility level.
The slope of the indifference curve at any point is the MRS between the two goods.
The slope does not show the ratio of market prices (that is the budget line) or transitivity (that is an axiom of preferences, not a slope).
Indifference curves are convex to the origin because of the assumption of diminishing MRS. As you get more of one good, each additional unit is worth less to you in terms of the other good. This is distinct from transitivity, completeness, or the "more is better" assumption.
For the utility function u(x, y) = 4x²y³:
MU_x = ∂u/∂x = 8xy³
MU_y = ∂u/∂y = 12x²y²
Take the partial derivative with respect to the variable in question, treating the other variable as a constant.
Linear utility (e.g., u(x, y) = 200x + 199y):
Indifference curves are straight downward-sloping lines.
The goods are perfect substitutes.
MRS is constant at 200/199.
Min utility (e.g., u(x, y) = min{x, y}):
Indifference curves are L-shaped.
The goods are perfect complements.
The consumer always buys x and y in equal amounts (or whatever ratio the min function specifies).
Cobb-Douglas utility (e.g., u(x, y) = x^a × y^b):
Indifference curves are smooth, convex curves.
MRS diminishes along the curve.
If X and Y are perfect substitutes, the MRS is constant. The diminishing MRS assumption is not satisfied. All other standard assumptions (completeness, transitivity, more is preferred to less) are still satisfied.
The MRS of one good for another is measured by moving along a single indifference curve, not between different curves, not along a budget line, and not along a demand curve.
MRS formula:
MRS_xy = MU_x / MU_y = -(dy/dx) along an indifference curve
Partial derivative example:
u(x, y) = 4x²y³
MU_x = ∂(4x²y³)/∂x = 8xy³
MU_y = ∂(4x²y³)/∂y = 12x²y²
⚠️ Complete preferences mean the consumer can rank any pair, not only pairs they enjoy. "He can rank only alternatives that he likes" is wrong.
⚠️ Convexity of indifference curves comes from diminishing MRS, not from transitivity, completeness, or "more is better."
⚠️ Perfect substitutes violate the diminishing MRS assumption. All other standard preference axioms hold.
⚠️ The slope of the indifference curve is the MRS, not the price ratio. The price ratio is the slope of the budget line.
⚠️ For MU_x of u = 4x²y³, the answer is 8xy³. Watch for answers that incorrectly multiply all exponents or forget the chain rule.
Q: Steven has complete and transitive preferences. What can he do?
A: He can rank any pair of alternatives. Completeness covers all bundles, not just ones he likes. Transitivity ensures the ranking is consistent, but it does not require a linear utility function.
Q: Jane's utility function is u(x, y) = 200x + 199y. What shape are her indifference curves?
A: Straight downward-sloping lines. This is a linear utility function, making the goods perfect substitutes.
Q: What situation is consistent with the law of diminishing marginal utility?
A: The more pizza Alex eats, the less he enjoys another slice. Each additional unit adds less satisfaction.
Q: Indifference curves are convex to the origin because of which assumption?
A: The assumption of diminishing marginal rate of substitution.
Q: What is MU_x for u(x, y) = 4x²y³?
A: 8xy³. Take the partial derivative with respect to x: bring down the exponent 2, multiply by the coefficient 4, reduce the exponent by 1.
Q: If X and Y are perfect substitutes, which assumption about indifference curves is not satisfied?
A: Diminishing MRS. The MRS is constant for perfect substitutes.
Q: The MRS is measured by moving along what?
A: Along a single indifference curve.
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