The North Sealand Raspberry Plantation
North Sealand Raspberry Plantation
North Sealand Raspberry Plantation is a minor fruit plantation, that exclusively cultivates organic raspberries with organic methods. The plantation rents one hectare of land at an annual cost of 3,000 DKK Furthermore, it costs the firm 15,000 DKK to stretch out a net for protection against birds, to replant, to inspect, and cultivate the raspberries.
Harvest season is coming, which lasts about three weeks. All the raspberries are ready to be picked. The owner has to determine how many kilos are to be picked - and consequently how many pickers to hire. It is out of the question to harvest all of the raspberries as the costs of picking the last ones (the small ones or those that hang low to the ground) exceeds the value of the raspberries. The owner knows from experience that, due to internal competition the first pickers hired work faster, as succeeding pickers are hired. This positive relationship comes quickly to an end, as the pickers are in the way of each other, if too many are hired. The pickers are paid 100 DKK an hour, and with a variation each picks 8-13 kg. per hour. Naturally the best pick more and the worst pick less. Effort is put into hiring the best first.
In this short-term scenario, the costs of renting and cultivating the land are fixed. Only the costs of the raspberry pickers vary with the activity level.
The costs in a table
The costs incurred by North Sealand Raspberry Plantation for the picking of raspberries are listed in table 2.1, showing the costs as a function of the number of berries picked.
Raspberries (100 kg) |
TC |
TVC |
TFC |
ATC |
AVC |
MC* |
AFC |
0 |
18.000 |
- |
18.000 |
- |
- |
- |
- |
10 |
27.800 |
9.800 |
18.000 |
2.780 |
980 |
810 |
1.800 |
20 |
35.200 |
17.200 |
18.000 |
1.760 |
860 |
720 |
900 |
30 |
43.200 |
25.200 |
18.000 |
1.440 |
840 |
930 |
600 |
40 |
54.800 |
36.800 |
18.000 |
1.370 |
920 |
1.440 |
450 |
* MC has been calculated by differentiating the function below at point Q (give it a try).
Table 2.1: The costs as a function of the number of raspberries picked
Costs shown mathematically.
The costs incurred by North Sealand Raspberry Plantation in conjunction with picking raspberries, can be listed as a mathematical function, with cost in terms of the amount of raspberries picked (100 kg.):
The TC function can be divided into fixed and variable costs. The fixed costs do not vary in terms of production. Therefore, the TFC function and the TVC functions are found based on the TC-function:
TFC = 18.000
Based on the total functions, the average functions can be found by dividing by Q:
The MC function is found by differentiating the TVC function (the TC function, as the constant unit is neutralized anyway). Below the TVC function is differentiated:
It is now possible to go "the other way" in the triangle model and see if the original functions result, which is a good way to check the validity of one's calculations.
The TC function is found by multiplying the ATC function with Q:
The TVC function is found by integrating the MC function :
As seen from the calculations above, the original functions appear by "going the other way" in the triangle model, which indicates that the mathematical part of the cost functions are correct.
Costs shown graphically
The cost of picking the raspberries can be shown graphically, as is done in figure 2.1.1 and 2.1.2:
In figure 2.1.1 and 2.1.2 some important cost-related correlations are demonstrated:
o When MC increases and crosses AVC it always happens at a local minimum for AVC. The logic implied here is that when MC is below AVC then AVC decreases, and when MC is above AVC then AVC increase. The MC's crossing of the AVC demarcates the change between the descending and ascending sectors of AVC. This pattern corresponds to that MC, which is respectively below and above the AVC on each side of the crossing. Similar lines of reasoning are involved when MC is descending and crosses AVC.
o The reasoning explained above also applies to the relationship between MC and ATC. Notice that MC crosses ATC at a higher Q-level than when AVC was crossed. This is due to ATC being AFC+AVC, and thus ATC is at a higher level.
o In a continuous relationship, MC expresses the gradient of VC, and thus TC. When MC is low it is synonymous with TC ascending slowly.