6.3 Kinked Constraints
Kinked constraints are incredibly important in economics. One simple reason they might arise is due to a change in technology.
For example, suppose Chuck can produce one fish per hour using a fishing pole, and one coconut per hour by climbing trees: \(x_1 = f_1(L_1) = L_1\) \(x_2 = f_2(L_2) = L_2\) Now suppose he crafts a net that will allow him to catch 2 fish per hour rather than 1. However, the net has a capacity of 80 fish; so if he wants to produce more than 80 fish, he needs to go back to his pole. If he wants to go over that amount — that is, if he spends 40 hours fishing with a net plus $(L_1 - 40)$ hours fishing with a pole — he will produce $80 + (L_1 - 40) = 40 + L_1$ fish: \(f_1(L_1) = \begin{cases}2L_1 & \text{ if }L_1 \le 40\\40 + L_1 & \text{ if }L_1 \ge 40\end{cases}\) In this case \(MP_{L1} = \begin{cases}2 & \text{ if }L_1 \le 40\\1 & \text{ if }L_1 \ge 40\end{cases}\) and since his $MP_{L2} = 1$, this means his MRT is \(MRT = \begin{cases}\frac{1}{2} & \text{ if }L_1 \le 40\\1 & \text{ if }L_1 \ge 40\end{cases}\) This results in a kinked budget constraint:
- if he spends all his time collecting coconuts, he can produce $x_2 = 100$ coconuts (and no fish); so his vertical intercept is at (0, 100).
- if he starts fishing, he’ll use the net for his first 40 hours, so each fish he catches will have an opportunity cost of only half a coconut; so his MRT is $\frac{1}{2}$ in the upper-left portion of his PPF.
- if he spends exactly 40 hours filling the net, he can get 80 fish and have 60 coconuts in his remaining time; this is the kink point at (80, 60).
- if he wants more than 80 fish, he can catch 1 per hour with his pole, so his opportunity cost of each fish at that point is one coconut and his MRT is 1 in the lower-right portion of his PPF.
- if he only wants to produce fish, he can catch 140 in total (80 in the net and 60 more using his pole); so his horizontal intercept is at (140, 0).
How does this MRT compare to his MRS? Well, to answer that, we have to consider the relationship between MRT and MRS at the kink point. Suppose his utility function is \(u(x_1, x_2) = x_1x_2\) so his MRS is \(MRS(x_1,x_2) = \frac{x_2}{x_1}\) At the kink point of (80,60), therefore, his MRS would be \(MRS(80,60) = \frac{60}{80} = \frac{3}{4}\) In other words, he would be approximately indifferent between 3 coconuts and 4 fish at the point (80, 60). How does this compare with his MRT?
- The MRT to the left of the kink point is $\frac{1}{2}$, which is less than his MRS; so he wouldn’t want to move to the left. In other words, if he produced 4 less fish with his net, he could only get 2 coconuts; so he wouldn’t want to do that.
- The MRT to the right of the kink point is 1, which is more than his MRS; so he wouldn’t want to move to the right. In other words, if he produced 4 more fish, he’d need to use his pole (since his net is exactly full at the kink point), so producing 4 more fish would mean giving up 4 coconuts; and he’s not willing to do that either!
Since $MRS > MRT$ to the left of the kink, and $MRS < MRT$ to the right of the kink, his optimal point is the kink. If we plot MRS vs MRT along the PPF as usual, we can see how the discontinuity in the MRT results in this behavior:
Now, it’s not always the case that a kink is the optimal point! Suppose Chuck found a second net, so he could catch up to 160 fish in his nets using 80 hours of his labor. In that case his problem would look like the following graph, and the optimal point would involve spending exactly half of his labor producing fish and half producing coconuts:
In this case, Chuck’s MRS at the kink point is $20/160 = 0.125$; since this is less than his MRT of 0.5 to the left of the kink, this means Chuck wants to move to the left from the kink point, and the optimal solution is characterized by the tangency condition at (100, 50).