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

In summary, for the three given utility functions, the indifference curves for utility functions \(U(x, y) = \min(x, y)\) and \(U(x, y) = x + y\) are convex, while the indifference curves for the utility function \(U(x, y) = \max(x, y)\) are not convex.

Step-by-step solution

Utility Function: \(U(x, y)=\min(x, y)\)

For this utility function, the utility level is equal to the minimum of the two values of \(x\) and \(y\). Now let's consider two points \((x_1, y_1)\) and \((x_2, y_2)\) on the same indifference curve, which means their utility values are equal and \(k =\min(x_1, y_1) = \min(x_2, y_2)\). The midpoint between these two points is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). By applying the utility function to the midpoint, we get: $$U\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \min\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ If either \((x_1, y_1)\) or \((x_2, y_2)\) are bad bundles and lie on the diagonal (where \(x = y\)), the utility is greater than or equal to \(k\). When both points are strictly good bundles, the midpoint's utility is also equal to \(k\). Thus, in all cases, the midpoint's utility is greater or equal to \(k\). Therefore, the indifference curves for this utility function are convex.

Utility Function: \(U(x, y)=\max(x, y)\)

Now, let's consider the case where the utility function is defined as \(U(x, y) = \max(x, y)\). In this case, the utility from consuming a bundle is equal to the maximum of the two values \(x\) and \(y\). Let's consider two points \((x_1, y_1)\) and \((x_2, y_2)\) on the same indifference curve, meaning \(k = \max(x_1, y_1) = \max(x_2, y_2)\). The midpoint between these two points is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Now, apply the utility function to the midpoint: $$U\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \max\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ If both points \((x_1, y_1)\) and \((x_2, y_2)\) lie on the diagonal (where \(x = y\)), the utility from the midpoint is equal to \(k\). However, if either of the points are bad bundles, the utility from the midpoint is less than \(k\). Therefore, the indifference curves for this utility function are not convex.

Utility Function: \(U(x, y)=x + y\)

For the last case, the utility function is given by \(U(x, y) = x + y\). In this case, the utility is the sum of the values of \(x\) and \(y\). Consider again two points \((x_1, y_1)\) and \((x_2, y_2)\) on the same indifference curve, implying \(k = x_1 + y_1 = x_2 + y_2\). The midpoint between these two points is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Now, apply the utility function to the midpoint: $$U\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \frac{x_1 + x_2}{2} + \frac{y_1 + y_2}{2} = \frac{1}{2}(x_1 + y_1) + \frac{1}{2}(x_2 + y_2)$$ Since \(k = x_1 + y_1 = x_2 + y_2\), the utility from the midpoint simplifies to: $$\frac{1}{2}k + \frac{1}{2}k = k$$ Therefore, the utility from the midpoint is equal to \(k\), and the indifference curves for this utility function are convex.