. As tradition has it, Santa Claus is coming on the 5th of December from

Spain to the Netherlands to bring gifts to Dutch children. Help is
needed to distribute the gifts for which the Santa Claus committee is
hiring helpers, so-called Zwarte Pieten.
There are two types of Zwarte Pieten; good ones who can deliver 4000
gifts in this night, while the bad ones can only manage 400 gifts. Zwarte
Pieten will be paid by their productivity.

While the Dutch Santa Claus committee is not able to observe the
productivity of Zwarte Pieten before the 5th of December, they have to
settle pay beforehand.

The good Zwarte Pieten differ from the bad ones in their cold weather
endurance, which they can use to signal their type by standing in the
cold. The good Zwarte Pieten have a utility function over wages, w,
and hours spend in the cold h
Ug = w - .5h
while the bad Zwarte Pieten utility function is
Ub = w - h

While the time spent in the cold by potential Zwarte Pieten does not
affect their productivity it does affect their well-being.
(a) Draw a diagram in (w; h) space and establish that the single-
crossing condition is satisfi…ed. In the same diagram indicate the
zero pro…fit conditions for the Santa Claus committee. [20 marks]

Also, depict the zero pro…fit conditions for a fi…rm that hires a g-
type ZP and for a …rm that hires an b-type ZP, respectively. [Note:
These zero-profi…t conditions are represented by lines along which
the difference between wage and productivity is equal to zero, i.e.
for which wg = 4000 and wb = 400.]

(b) If the committee could identify the good from the bad Zwarte
Pieten, would any candidate spend any time in the cold?

That is spelled Zwarte Piet.

Sra

Good Luck with the exam on 13th Jan!

To answer these questions, we need to analyze the utility functions of the good and bad Zwarte Pieten and understand their implications.

(a) Drawing the diagram:
In the (w, h) space, we can plot the utility functions for the good and bad Zwarte Pieten. The key is to find the wage at which both utility functions are equal and draw the zero profit conditions for the Santa Claus committee.

1. Good Zwarte Pieten:
The utility function for good Zwarte Pieten is given by Ug = w - 0.5h. We can plot this utility function on the diagram. The wage is represented on the horizontal axis, and the time spent in the cold is represented on the vertical axis.

2. Bad Zwarte Pieten:
The utility function for bad Zwarte Pieten is given by Ub = w - h. We can plot this utility function on the same diagram.

The single-crossing condition is satisfied if the two utility functions cross only once and never intersect again. This means there is a unique wage where the utility functions are equal.

3. Zero-profit conditions:
The Santa Claus committee wants to set the wage in such a way that they pay Zwarte Pieten just enough to cover their productivity. We need to find the wage at which both good and bad Zwarte Pieten deliver the exact number of gifts they are capable of.

For the good Zwarte Pieten, the wage (wg) is determined such that Ug = 4000 - 0.5h.
For the bad Zwarte Pieten, the wage (wb) is determined such that Ub = 400 - h.

Plotting these zero-profit conditions means drawing lines where the difference between the wage and productivity is zero. So, wg = 4000 and wb = 400.

(b) If the committee could identify the good from the bad Zwarte Pieten, no candidate would spend any time in the cold. This is because the utility function for the good Zwarte Pieten (Ug) has a negative coefficient (-0.5) in front of the time spent in the cold (h). As a result, good Zwarte Pieten derive less utility for each hour spent in the cold compared to bad Zwarte Pieten. Therefore, if the committee can differentiate between the two types, the good Zwarte Pieten would not voluntarily spend any time in the cold.