1.17 Solutions at Kinks

The gravitational pull argument holds that if the $MRS > p_1/p_2$, a consumer can do better by moving to the right along her budget constraint, and vice versa. In the cases we’ve looked at thus far, the consumer has stopped when she reached a point of tangency, where $MRS = p_1/p_2$, or when she reaches a corner. But what happens if indifference curves or budget constraints have kinks — that is, have discontinuous slopes? Then if the optimum occurs at the kink, either the MRS or the price ratio (or both!) will be undefined, so the tangency condition will not hold. For the purposes of this course, we’ll be focused on the situation of a kinked budget constraint, which will some important applications in Econ 51.

Kinked Budget Constraints

Up to now, we’ve been looking at the optimal choice along a budget line, so the price ratio was a constant. However, some budget constraints have price ratios that vary along the constraints. With such kinked budget constraints, we need to examine the relationship between the MRS and the price ratio along each of the segments of the budget constraint to solve the consumer’s optimization problem.

For example, consider the case of nonlinear electricity rates: you pay a low rate for the first kilowatt-hours (kWh) of electricity you use, up to a threshold $\hat x$; then you pay a higher rate. Such mechanisms are used to ensure that people have at least some affordable electricity, but to deter excessive use; the budget constraint is characterized by an initial relatively flat portion reflecting the low initial price, then a steeper portion reflecting the higher price for additional use. Where the slope of the budget constraint changes, there is a “kink:”

It is entirely possible to have an MRS at the kink point that is simultaneously higher than the price ratio to the left of the kink, and lower than the price ratio to the right of the kink. In such a case the gravitational pull will always draw you toward the kink: your utility will always be increasing along the left-hand segment (since the MRS is greater than the price ratio everywhere in that region), while your utility will always be decreasing along the right-hand segment (since the MRS is less than the price ratio everywhere in that region):

Note, however, that the optimum doesn’t have to be at the kink. You can drag the threshold $\hat x$ in the diagram above to set it higher or lower than 300 kWh. Notice what happens if you do:

• If you drag the threshold $\hat x$ all the way to the left, at 120 kWh, the MRS at the kink is 0.45, which is greater than even the higher price of 0.20. This threshold is so low that when the threshold is met, you’re willing to pay the higher price to continue using electricity; you can see that the optimum occurs at a point of tangency to the right of the kink, with $x_1 = 220$.

• On the other hand, if you drag the threshold $\hat x$ all the way to the right, at 600 kWh, the MRS at the kink is 0.05, which is less than even the lower price of 0.10. In this case the threshold is set so high that the higher price above that level is irrelevant: the optimum is a point of tangency to the left of the kink, with $x_1 = 400$.

The bottom line is that even when there is a kinked constraint or a kinked indifference curve, it doesn’t follow that the optimum is automatically at the kink itself. You should compare the MRS and the price ratios at each kink (and each corner), to get a sense as to where the optimal choice will be.