Question

(a) Does the assumption of diminishing marginal utility to goods \(x\) and \(y\) imply strictly convex indifference curves? (b) Does the assumption of strict convexity in the indifference curves imply diminishing marginal utility to goods \(x\) and \(y ?\)

Short answer

(a) Yes, it implies strictly convex indifference curves. (b) No, strict convexity does not imply diminishing marginal utility.

Step-by-step solution

Understanding Diminishing Marginal Utility

Diminishing marginal utility refers to the decrease in additional satisfaction a consumer gains from consuming one more unit of a good, given that they are already consuming some quantity of that good. If each additional unit of goods \( x \) and \( y \) provides less additional utility than the previous one, then the utility function exhibits diminishing marginal utility.

Analyzing Convexity of Indifference Curves

Strictly convex indifference curves occur when the combination of two different bundles of goods provides a higher utility than sticking to one single bundle. This means that a mixture of goods \( x \) and \( y \) is preferred to having only one of the goods. Thus, if utility decreases less rapidly for combinations than for extremes, it suggests convexity.

Implication of Diminishing Marginal Utility on Indifference Curves

If marginal utility diminishes for both \( x \) and \( y \), consuming combinations of these goods tends to be preferred. This leads to strictly convex indifference curves, because combinations provide more consistent utility than extremes. Hence, diminishing marginal utility typically implies strict convexity of indifference curves.

Implication of Strict Convexity on Diminishing Marginal Utility

Strict convexity of indifference curves implies that average combinations of \( x \) and \( y \) are preferred to extremes, suggesting that marginal rates of substitution change as quantities are substituted. However, this does not necessarily mean that each additional unit of \( x \) or \( y \) provides less satisfaction. Thus, strict convexity does not directly imply diminishing marginal utility.

Key concepts

Diminishing Marginal Utility

The concept of diminishing marginal utility is fundamental in consumer theory. It describes a pattern where each additional unit of a consumed good adds less to overall satisfaction or utility than the previous one. Imagine enjoying slices of pizza. The first slice might be delightful, the second still tasty, but by the time you get to the fourth or fifth slice, the extra satisfaction from each new slice gets smaller. This pattern demonstrates the idea of diminishing marginal utility. In terms of mathematics, if utility is denoted by a function for good \( x \), diminishing marginal utility means that the derivative of this utility function decreases as \( x \) increases. This principle is crucial in understanding consumer choice behavior, as it explains why consumers tend to diversify their consumption rather than consuming only one type of good.

Indifference Curves

Indifference curves are a graphical representation of combinations of goods that provide the same level of satisfaction to the consumer. Imagine you have two goods, apples and bananas. An indifference curve shows all the different mixtures of apples and bananas that make you equally happy. You might have lots of bananas and fewer apples, or vice versa, and remain equally satisfied. These curves are typically downward sloping because if you give up one good, you'll need more of the other to maintain the same level of utility. They never intersect because that would imply inconsistent preferences, which defy the basic principles of consumer theory. Strictly convex indifference curves indicate that a balanced combination of goods provides higher satisfaction, suggesting preferences for variety and diversity in consumption.

Convexity

Convexity in the context of indifference curves signifies that the consumer prefers a mixture of goods over extremes. Consider a curved line on a graph, bowing inward towards the origin. This curve implies that the satisfaction (or utility) derived from a combination of goods is higher than consuming an extreme amount of one over the other. In more technical terms, convexity implies that the slope of the indifference curve becomes flatter as you move along it. This concept ties closely to the diminishing marginal rate of substitution—meaning, as you consume more of one good, you'll be willing to give up fewer units of the other good to maintain the same utility. Convexity illustrates the preference for balanced consumption, which aligns with everyday behaviors where people generally like having a variety of goods rather than just one.