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. \(U(x, y)=x+y\)

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 \(U(x, y) = \min(x, y)\).

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 \(U(x, y) = \max(x, y)\).

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 \(U(x, y) = x + y\).

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.