Allow for upward kinks

[trilnick]

I've heard this several times. While upward kinks exist in real life (e.g. overtime pay), I don't know about much literature on them. In fact, Kleven (2016) writes that it is possible to calculate an elasticity with them, but researchers are not really trying to because the positive kink should create a hole in the distribution, but it is not observed in real life data. Nevertheless, let's think of how this would work:

1. Think of a upward, concave kink, i.e. t2<t1.
2. Think of nice, well behaved, indifference curves between earnings and consumption. They are convex, increasing, and with increasing slope. For a specific type n, indifference curves of different budgets may not intersect.
3. For an upward, concave kink, there would be a marginal unaffected type, n*, whose curve has two tangents with the kinked budget line, $n*$ and $n*+\Delta n$, where $n* \le z$ and $n*+\Delta n \ge z*$. Note that Then, since curved are nicely behaved (smooth), there will be such an agent (an agent with tangent at z* will intersect with the kinked budget line). Since agents with higher ability have curves with lower slope at any given n, we know that:
   a. All agents with types n s.t. $n* \leq n\leq z*$ will have a new tangent point in the post-kink area, and since their slopes are lower than $n*$'s slope at any given earning, this point will be greater than $n*+\Delta n$. That means, there should be a hole in the actual distribution of earnings between $n*$ and $z*$.
   b. All Agents with types n s.t. $n* \ge z*$ will change their earnings. At first, it seems like some might decrease their total earnings (income effect?), but if we are dealing with the quasi-linear, iso-elastic function we see that: $$\delta z / \delta (1-t) =n * e * (1-t)^{e-1} \ge 0$$ and therefore all their respective earnings increase. In fact, since their slope is lower than n*'s at any earning point, their new tangent point would be at a higher z than $n*+\Delta n$, so the hole extends at least until then.
4. With the location of the hole, representing the two tangent point of the indifferent type, one could figure out the elasticity of that marginal type. (note: Kleven writes that " the width of the hole can be linked to the compensated earnings elasticity", I don't immediately see how you can do this with just the width, as I don't see any guarantee of symmetry around z*).