6.2 Kinked Indifference Curves

The gravitational pull argument holds that if the $MRS > MRT$, Chuck can do better by moving to the right along his PPF. In the cases we’ve looked at thus far, Chuck has stopped when he reached a point of tangency, where $MRS = MRT$, or when he reaches a corner. But what happens if indifference curves or PPFs have kinks — that is, have discontinuous slopes? Then if the optimum occurs at the kink, either the MRS or the MRT (or both!) will be undefined, so the tangency condition will not hold.

In order for this to occur, it must be the case that either the agent’s indifference curves or PPF are defined in a piecewise manner. One example of a utility function with this characteristic is the perfect complements utility function (shown the left diagram below). An example of a kinked PPF is shown in the right diagram below.

In the left graph, Chuck maximizes his utility by consuming at the kink of his utility function, and in the right-hand graph he maximizes his utility at the kink of the production function. Let’s examine these two cases mathematically to see what’s going on; and especially, in the right-hand case, why even when a PPF is kinked it doesn’t mean the optimum will always be there.

In each case, as in our analysis of corner solutions, we’ll assume that Chuck has 100 hours of labor, and plot his MRS and MRT along his PPF as $L_1$ increases from 0 to 100.

Example: Perfect Complements

Let’s look at the production functions we often use: \(x_1 = f_1(L_1) = 10\sqrt{L_1}\) \(x_2 = f_2(L_2) = 6\sqrt{L_2}\) As we’ve shown many times, this results in the PPF with equation \(\frac{x_1^2}{100} + \frac{x_2^2}{36} = 100\) For Chuck’s utility function, let’s use the perfect complements (or Leontief) function \(u(x_1,x_2) = \min\{4x_1,5x_2\} = \begin{cases}4x_1 & \text{ if }4x_1 < 5x_2\\5x_2 & \text{ if }4x_1 > 5x_2\end{cases}\) This has marginal utilities \(MU_1 = \begin{cases}4 & \text{ if }4x_1 < 5x_2\\0 & \text{ if }4x_1 > 5x_2\end{cases}\) \(MU_2 = \begin{cases}0 & \text{ if }4x_1 < 5x_2\\5 & \text{ if }4x_1 > 5x_2\end{cases}\) Let’s recall how we derived the gravitational pull argument: we said that Chuck should spend more time producing fish (good 1) if the utility from an hour of fishing would exceed the utility from an hour producing coconuts: \(MP_{L1} \times MU_1 > MP_{L2} \times MU_2\) Given this utility function, the right-hand side of this is zero if $4x_1 < 5x_2$, so it would always hold; and conversely, the left-hand side of this is zero if $4x_1 > 5x_2$, so it would never hold. In terms of the marginal rate of substitution, the MRS is infinite when $4x_1 < 5x_2$, and therefore always greater than the MRT; and the MRS is zero when $4x_1 > 5x_2$, and consequently less than MRT.

We can see this in the diagram of MRS vs. MRT:

Hence the optimal point must be the bundle along the PPF where $4x_1 = 5x_2$, as shown by the dotted line in the top diagram above. To solve this mathematically, we plug $x_2 = \frac{4}{5}x_1$ into the budget constraint gives us \(\frac{x_1^2}{100} + \frac{(\left[\frac{4}{5}x_1\right]^2}{36} = 100\) Some tedious algebra gives us $x_1 = 60$, and therefore $x_2 = \frac{4}{5} \times 60 = 48$. You can confirm that this corresponds to the labor allocation $(L_1 = 36, L_2 = 64)$.