Course:MATH110/Archive/2010-2011/003/Teams/Thurgau/Homework 11
Linear model
Since the rate of producing a 20 items is 100$, and the cost to produce additional units is 7$, it is fair to conclude that the equation can be written as: Y = 7x +100
For the production of 150 items, it is necessary that we subtract 20 from 150, since the numerical value of 100 is already been accredited into the equation for the first 20 flags, hence:
f(130) = 7(130)+100
f(130) = 1010$
Therefore, according to this linear model, in order to produce 150 units, it would cost about 1010$
Average Cost
Lets compare the average cost when for 100 units and 150 units:
f(80) = 7(80) + 100
f(80) = 660$, this is the total cost. Divide this number by 80 to find the average cost.
(660/80) = $8.25 to create each unit on average.
f(130) = 1010$
(1010/150) = $6.733
As shown in the comparison above, according to this linear equation, the more units produce, the lower will the average cost be for each unit.
Finally, find some other models (not necessarily linear) for which you get other behaviours such as:
* The average cost remains constant as production increases.
For such a function, we first look at the equation we use to get the average cost provided some sort of cost function. Since average cost will be total cost over number of goods produced, then it will pretty much be .
Now that we know this, we can say that we want that function of x to become constant even as x changes, meaning that the ratio between the two must remain the same. And what better way to do that then to have a linear function without any constant.
For any value x, this function will always yield 5x/x, or as you cancel the two x's, the value 5.
* The average cost diminishes as production increases.The average cost increases as production increases.
For this function we can simply use the cost function we used in the first modelling question in this homework which is .
Although its average cost approaches the value 7 by infinitely smaller increments, the value of its average cost nevertheless is still increasing and thus fulfills this part's conditions.
* You obtain an economy of scale. This means that starting at some specific production level, the marginal cost is always less than the average cost.
For this equation, we can simply have a linear function that has a start-up or initial cost which would be represented as any plain added constant. This way, the marginal cost becomes some constant and the average cost will never go below the marginal cost no matter how insignificant the difference between the two will become.
In practical terms we can use the following function:
At C(1), this function would have an average cost of:
and a marginal cost of 5.
At C(1000), this function would have an average cost of:
and still a marginal cost of 5.
This means because of that constant number + 1, or the initial start-up cost, marginal cost will always be lower than average cost.