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 di¤er 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
1
2
h
while the bad Zwarte Pieten utility function is
Ub = w h
While the time spent in the cold by potential Zwarte Pieten does not
a¤ect their productivity it does a¤ect their well-being.
(a) Draw a diagram in (w; h) space and establish that the single-
crossing condition is satis…ed. In the same diagram indicate the
zero pro…t conditions for the Santa Claus committee. [20 marks]
Also, depict the zero pro…t condition for a …rm that hires an g-
type ZP and for a …rm that hires an b-type ZP, respectively. [Note:
These zero-pro…t conditions are represented by lines along which
the di¤erence between wage and productivity is equal to zero, i.e.
for which wg = 4000 and wb = 400.]
Your utility function is unreadable. Also, what is the point of substituting ... for letters in certain words? Are you cut-and-pasting from some assignment with special Dutch language characters?
Good luck
To illustrate the single-crossing condition and the zero-profit conditions in the given scenario, we will create a diagram in the (w; h) space.
In this diagram, we will plot the wages (w) on the horizontal axis and the time spent in the cold (h) on the vertical axis.
Let's start by plotting the utility functions of the good Zwarte Pieten (Ug) and the bad Zwarte Pieten (Ub).
For the good Zwarte Pieten:
Ug = w - (1/2)h
For the bad Zwarte Pieten:
Ub = w - h
To represent these utility functions graphically, we can draw two lines. The slope of the lines will be 1 (corresponding to the coefficients of w) for both functions. However, the slope of the line for the good Zwarte Pieten will be -1/2 (corresponding to the coefficient of h).
So, we have:
Line 1: Ug = w - (1/2)h
Line 2: Ub = w - h
Now, let's plot these lines on the diagram.
Next, we need to draw the zero-profit conditions for the Santa Claus committee and the firms hiring g-type and b-type Zwarte Pieten, respectively.
The zero-profit condition for the Santa Claus committee is when the total wages paid equal the total productivity. In this case, the committee needs to distribute 4000 gifts.
So, we can represent this zero-profit condition with a horizontal line labeled "wg = 4000" on the diagram.
The zero-profit condition for a firm hiring a g-type Zwarte Piet is when the wages paid equal the productivity of 4000 gifts.
Similarly, the zero-profit condition for a firm hiring a b-type Zwarte Piet is when the wages paid equal the productivity of 400 gifts.
Both of these conditions can be represented with horizontal lines labeled "wg = 4000" and "wb = 400" respectively on the diagram.
Once all these lines are plotted on the diagram, we can analyze their intersections and observe whether the single-crossing condition is satisfied.
The single-crossing condition requires that the lines representing utility functions of different types of workers do not intersect. Instead, they should have a region of separation, where one type of worker always has a higher utility (in this case wages minus time spent in the cold) than the other, for any given wage level.
If we observe that the lines representing Ug and Ub do not intersect, and the zero-profit conditions for the Santa Claus committee, a g-type Zwarte Piet, and a b-type Zwarte Piet are all satisfied, then we can conclude that the single-crossing condition is indeed met in this scenario.
Note: Without access to a specific diagram or visual representation, it is challenging to provide a detailed step-by-step explanation. However, the general procedure outlined above should allow you to construct the required diagram and showcase the single-crossing and zero-profit conditions for the Santa Claus committee and the different types of Zwarte Pieten.