Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the \(M R S\) declines as \(x\) increases). a. \(U(x, y)=3 x+y\) b. \(U(x, y)-\sqrt{x \cdot y}\) c. \(U(x, y)=\sqrt{x}+y\) \(\mathrm{d} U(x, y)=\sqrt{x^{2}-y^{2}}\) e. \(U(x, y)=\frac{x y}{x+y}\)

Utility functions are fundamental concepts within microeconomics that are used to model preferences of consumers. Simply put, a utility function quantifies the satisfaction or happiness that a consumer derives from consuming various bundles of goods and services. Each function generates a numerical value representing utility, with higher numbers indicating preferred consumption bundles.

For example, consider the utility function U(x, y)=3x+y. Here, x and y could represent quantities of two different goods. This function suggests that for each additional unit of good x, the consumer receives three times more utility than from each additional unit of good y. This function helps us visualize and analyze how choices and trade-offs are being made between two goods.

The Marginal Rate of Substitution (MRS) is an important concept that is intricately linked to indifference curves. It measures the rate at which a consumer is willing to give up one good in exchange for another good while keeping the same level of utility. In mathematical terms, the MRS is the negative slope of an indifference curve and is calculated as the negative ratio of the marginal utilities of two goods.

For example, if the MRS between two goods x and y is 2, it means that a consumer would give up 2 units of y for 1 additional unit of x without undergoing any change in their overall utility. This ratio changes along the indifference curve, capturing the trade-offs that a consumer makes as they substitute one good for another.

The shape of an indifference curve conveys powerful information about consumer preferences. Typically, indifference curves are convex to the origin. Convexity indicates that as the consumer substitutes one good for another, the willingness to exchange (MRS) decreases. In other words, as you consume more of good x and give up units of good y, you value the additional units of x less compared to the units of y you are losing.

However, not all indifference curves are convex. For instance, perfect substitutes and perfect complements produce straight-line indifference curves and L-shaped curves, respectively. Convexity can be checked by examining how MRS changes with the increase of x; in a convex curve, MRS decreases as x increases.

Marginal Utility (MU) represents the additional satisfaction a consumer gains from consuming an additional unit of a good or service. It is derived from the utility function, determined by taking the partial derivative with respect to one of the goods.

In our previous examples, the marginal utility of x from the utility function U(x, y)=3x+y is 3. This means that for every extra unit of x consumed, the utility increases by three units. Put differently, the marginal utility is a measure of the incremental utility that helps explain how consumers make decisions and achieve the equilibrium of utility maximization.