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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
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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.

In order for this to occur, it must be the case that either the agent’s indifference curves or budget constraint 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 budget constraint is shown in the right diagram below.

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

In each case, we’ll plot three graphs: the consumers budget constraint/indifference curve diagram; her total utility along the budget constraint; and the MRS and price ratio along the budget constraint.

Example: Perfect Complements

Let’s choose a very simple budget constraint: \(x_1 + x_2 = 100\) This has a price ratio of 1.

Consider the perfect complements (or Leontief) function \(u(x_1,x_2) = \min\{3x_1,2x_2\} = \begin{cases}3x_1 & \text{ if }3x_1 < 2x_2\\2x_2 & \text{ if }3x_1 > 2x_2\end{cases}\) This has marginal utilities \(MU_1 = \begin{cases}3 & \text{ if }3x_1 < 2x_2\\0 & \text{ if }3x_1 > 2x_2\end{cases}\) \(MU_2 = \begin{cases}0 & \text{ if }3x_1 < 2x_2\\2 & \text{ if }3x_1 > 2x_2\end{cases}\) Let’s recall how we derived the gravitational pull argument: we said that a consumer should buy more good 1 if her “bang for her buck” from good 1 is greater than tha from good 2: \({MU_1 \over p_1} > {MU_2 \over p_2}\) Given this utility function, the right-hand side of this is zero if $3x_1 < 2x_2$, so it would always hold; and conversely, the left-hand side of this is zero if $3x_1 > 2x_2$, so it would never hold. In terms of the marginal rate of substitution, the MRS is infinite when $3x_1 < 2x_2$, and therefore always greater than the price ratio of 1; and the MRS is zero when $3x_1 > 2x_2$, and consequently less than the price ratio.

Visually, we can see that when $3x_1 < 2x_2$ (i.e., everywhere above and to to the left of the “ridge condition” $3x_1 = 2x_2$), there is an area of overlap between the preferred set and the budget set; utility is increasing; and the MRS is infinite, and therefore greater than the price ratio. Likewise, when $3x_1 > 2x_2$, there’s also an overal; utility is decreasing; and the MRS is zero, and therefore less than the price ratio:

Hence the optimal point must be the bundle along the budget line $x_1 + x_2 = 100$ where $3x_1 = 2x_2$, which occurs at the point $(40, 60)$. At this point, the utility function along the budget line (top-right) reaches a pointy “peak,” but is not flat; and the MRS (bottom-right) is undefined. So this optimum is not characterized by the tangency condition $MRS = p_1/p_2$. However, the gravitational pull argument still holds, and draws us to this optimal point.

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:

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.

Next: The Lagrange Method of Finding an Optimal Bundle
Copyright (c) Christopher Makler /