Course:MATH110/Archive/2010-2011/003/Teams/Solothurn/Homework 11

Write a linear model to predict the cost of producing flags of your team's Canton under the assumptions that the marginal cost is $7 per unit and that at the current production level of 20 items, the cost is $100.

C= mx+b

where C = total cost

    mx = variable cost 
    b  = fixed cost 
    m  = marginal cost 
    x  = quantity produced

$100= $7(20units) + b

b = $100 - $140

b = -$40

Describe your model. What does your model predict for a production of 150 items? According to your model, what happens to the average cost per item as production levels increase?

Our model predicts the total cost for producing x number of flags given the marginal cost ($7) and a fixed cost (-$40). In the model above, when producing 20 units, the cost of producing a single unit is $5. The reason that this number is lower than the marginal cost is because of the negative fixed cost. One reasoning for the negative fixed cost could be government subsidies for the production of flags.

For the production of 150 items our model predicts a total cost of $1010.

C = $7(150units) + (-$40)

C = $1010

The average cost per unit in this example is $1010/300 = $6.73

According to our model, as the production level increases, the average cost per item increases

examples:

C = $7(300units) + (-$40)

C = $2060

The average cost per unit in this example is $2060/300 = $6.87

C = $7(1000units) + (-$40)

C = $6960

The average cost per unit in this example is $2060/1000 = $6.96

Finally, find some other models (not necessarily linear) for which you get other behaviours such as:

The average cost remains constant as production increases.

An example would b: TC=3Q

Q would be quantity and 3 is the marginal cost

Because in economics, AVC = TC/Q ( average cost = total cost / quantity)

If you make the cost function 3Q, then that means the average cost will be 3, which is constant no matter what your Q value is.

* there is no fixed cost or 'b' in the y=mx+b formula *

The average cost diminishes as production increases.

An example would be: TC=100+3Q

The average cost remains constant but now there is a fixed cost added or 'b' because as the quantity increase, the fixed cost is shared by an increasing number, therefore the average cost diminishes.

The average cost increases as production increases.

An example would be: TC=Q^2

Because AVC=TC/Q, and TC is Q^2, then AVC would then equal Q^2/Q which would make AVC=Q. Therefore as quantity increases average cost increases.

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.

Economies of scale occur when firms get really big. The idea is that by reaching a certain size more opportunities to cut costs or to allocate fixed costs over a greater number of units occurs. For example, if you are buying so much of a certain input that the producers need you to survive than it may be possible to negotiate a better long-term price. It is a general Microeconomic principal that as a firm reaches the upper limits of it's capacity for production it's marginal costs rise. This idea can be easily understood if you just think about how much harder it would be for you to perform any challenging exercise, like a squat for example, after you had just done a dozen or more. For an economy of scale, the idea would be for you to be a giant army of Olympic athletes so that you could perform what would seem like an almost limitless number of repetitions of any exercise between you. In this case you would also have high fixed costs.

So, for example: C=100000+1x

with high fixed costs and low variable costs, no matter where you start your marginal costs (just 1 dollar) will always be less than your average costs which will be falling constantly as they reflect an ever lesser portion of the high fixed costs. They will always be above your average costs, though.

Any other interesting properties that you can think of and create a model for.

What if a firm had no variable costs whatsoever. All it had was a certain amount of fixed costs that it had to pay regardless of production and then it had it's quantity produced. Then it would have a very interesting equation that would look like:

C=1000/x

This could occur if, for example, a company had set up a facility to filter air and once the operational costs were paid it could produce limitless amounts of air filtration at no extra cost. Or perhaps it could be a license fee to a massive waterway that a firm offers people a drink from out of a special filtered fountain.