Question

As we saw in Figure \(3.5,\) one way to show convexity of indifference curves is to show that, for any two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) on an indifference curve that promises \(U=k\), the utility associated with the point \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) is at least as great as \(k\). Use this approach to discuss the convexity of the indifference curves for the following three functions. Be sure to graph your results. a. \(U(x, y)=\min (x, y)\) b. \(U(x, y)=\max (x, y)\) c. \(U(x, y)=x+y\)

Short answer

Based on the analysis of the given utility functions, determine if the associated indifference curves are convex. Utility function: U(x, y) = min(x, y) Answer: The indifference curve for U(x, y) = min(x, y) is not strictly convex, but it shows a non-linear relationship between x and y. Utility function: U(x, y) = max(x, y) Answer: The indifference curve for U(x, y) = max(x, y) is not strictly convex, but it shows a non-linear relationship between x and y. Utility function: U(x, y) = x + y Answer: The indifference curve for U(x, y) = x + y is convex, as the consumer always prefers more of both goods.

Step-by-step solution

Utility function U(x, y) = min(x, y)

To examine the convexity of the indifference curve under this given utility function, we first find the utility associated with the midpoint between points \((x_1, y_1)\) and \((x_2, y_2)\): $$U\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) = \min\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ Now, we need to compare this utility value to \(k\). If the value is greater than or equal to \(k\), the indifference curve is convex. The utility function has \(U(x, y) = k\), which implies that \(k = \min(x, y)\) for the two points \((x_1, y_1)\) and \((x_2, y_2)\). We can write these conditions as: $$k = \min(x_1, y_1) \quad \text{and} \quad k = \min(x_2, y_2)$$ Under these conditions, it can be deduced that the midpoint utility is greater than or equal to \(k\), implying a convex indifference curve for \(U(x, y) = \min(x, y)\).

Utility function U(x, y) = max(x, y)

We follow the same process as in Step 1 to evaluate the convexity of the indifference curve under this utility function: $$U\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) = \max\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ Given that the utility for participants in \((x_1, y_1)\) and \((x_2, y_2)\) is \(k\), we have \(k = \max(x_1, y_1)\) and \(k = \max(x_2, y_2)\). Under these conditions, it can be deduced that the midpoint utility is greater than or equal to \(k\), implying a convex indifference curve for \(U(x, y) = \max(x, y)\).

Utility function U(x, y) = x + y

As before, we first find the utility associated with the midpoint between points \((x_1, y_1)\) and \((x_2, y_2)\): $$U\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) = \frac{x_1+x_2}{2} + \frac{y_1+y_2}{2}$$ The utility of participants in \((x_1, y_1)\) and \((x_2, y_2)\) is \(k\), which implies that \(k = x_1 + y_1\) and \(k = x_2 + y_2\). Under these conditions, we can deduce that the midpoint utility is exactly \(k\), implying a convex indifference curve for \(U(x, y) = x + y\).

Graph the results

To provide a visual representation of the convex indifference curves for each utility function, create graphs to showcase the curves: For \(U(x, y)=\min(x, y)\), draw 45-degree diagonal lines as the indifference curves. These curves are not strictly convex but show a non-linear relationship between \(x\) and \(y\). For \(U(x, y)=\max(x, y)\), similar to the previous example, the indifference curves will be 45-degree diagonal lines. These curves are also not strictly convex, but show a non-linear relationship between \(x\) and \(y\). For \(U(x, y)=x + y\), the indifference curves will be downward-sloping straight lines. Since they run parallel to each other, they imply that the consumer always prefers more of both goods and the resulting indifference curves are convex. In conclusion, by analyzing the convexity of indifference curves associated with different utility functions, we can better understand the consumer preferences and the properties of these functions.

Key concepts

Microeconomic Theory

Microeconomic theory is a branch of economics that deals with the behavior of individual economic agents, such as consumers, firms, and workers, and the markets where they interact. It tries to understand how these agents make decisions given their preferences, the resources they have, and the rules of the institutions they operate in.

Within microeconomics, the concept of indifference curves is crucial. These curves represent different combinations of two goods that provide the same level of utility—or happiness—to a consumer. The convexity of these curves is related to the idea of the diminishing marginal rate of substitution, meaning, as a consumer substitutes one good for another, the amount of the second good they are willing to give up decreases. This reflects a general preference for balanced consumption rather than extremes.

Utility Function

A utility function in microeconomics quantifies the satisfaction or happiness a consumer derives from consuming goods and services. These functions help economists model consumer preferences and predict consumer behavior. Utility can be a tricky concept because it doesn't have a standard unit of measurement like meters for distance or kilograms for weight. Still, it allows for the comparison of different levels of satisfaction.

The exercise presents different utility functions, such as the minimum or maximum of two goods and their sum. By examining these functions, economists can determine an individual's preferred consumption bundle. It's important to note that while utility functions can be complex, the idea behind them is to simplify consumer choices into a format that can be easily analyzed and compared.

Consumer Preferences

Consumer preferences refer to the subjective tastes, as measured by the utility function, that govern the choice of one good or combination of goods over another. These preferences are assumed to be complete (the consumer can always say which of two alternatives they prefer or if they're indifferent), transitive (if option A is preferred to B, and B is preferred to C, then A is preferred to C), and more is preferred to less (more of a good is better, holding everything else constant).

The convexity of indifference curves is directly linked to the assumption that consumers prefer a diverse bundle of goods, as they prefer variety and balance in their consumption patterns. These concepts of consumer preferences play a huge role in shaping market dynamics and influencing how companies offer products and services. By understanding the preferences of consumers, businesses can optimize their offerings to match what their customers value most.