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

The Flag of Ticino

To produce flags of Ticino, we begin with the information that it costs $100 to make 20 flags.

$\$100/20=5$

shows that to make one flag out of this group of 20, it costs $5.

To make the twenty-first flag, and onwards, it will now cost $7 per flag.

If it costs $7 to produce x units, we can write this as $7x$ making the model

$P(x)=7x$

But we need to consider that the first 20 flags produced are purchased at the cheaper price of $100, or $5 per flag. We will have to subtract something in our model to accommodate for this. To find out what this is, we calculate how much it would cost if there was no cheaper price of $5 per flag and the price was set at $7 per unit. To produce those same 20 flags we would have

$(\$7)(20)=\$140.$

Compared to:

$(\$5)(20)=\$100$

$\$140-\$100=\$40$, we have a difference of $40 saved.

The model now looks like:

$C=7x-40$

Where:

• x = number of items

• 7 = $7 marginal cost

• C = Total production cost

• 40 = the difference between buying 20 flags for $5 each or 20 flags for $7 each.

To produce 150 flags, our model predicts the following:

$\$7(150)-\$40=\$1010$

So it will cost us $1010 to produce 150 flags.

The average cost of each flag if we bought 150 would be given as:

$\$1010/150=\$6.73$

So each flag costs on average $6.73.

Our model shows that as we produce more and more flags, the average cost will approach $7. We should see a horizontal asymptote at $7 as $x->\infty$.

Some Other Interesting Functions=

In order to have a model where the average cost stays the same as production increases, you could have a very simple model. If, say, we are producing toy banks (banking being a huge part of Ticino's economy) we could have a model where each bank produced cost $10, with no scaling. Our model would be

$C=10b$

where C is our cost and b is the number of banks created. Here our average cost would always stay the same.

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In order to have the average cost decrease, we could have a factory producing our toy banks. At such a situation, the cost is quite a bit higher to start production. Let's say that we have a flat rate per unit, $3, but a cost of $10,000 to set up the factory. In this case, our model would be as follows:

$C=10000+3b$

Our average cost/unit would decrease quite quickly, to the point where it would greatly encourage mass production. This model is an example of an economy of scale, as our average cost always decreases after we start production.

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For a more interesting model, let's look at a hypothetical model where we're buying land. Say we are looking at the costs in millions of dollars per $km^{2}$.

We may see the average cost diminish as we start buying larger sections of land. However, once we reach a certain point, the cost of each square kilometer will start increasing. Purely hypothetically, this could be described as follows:

When $L<1000$ then $C={\frac {2L(L-1)}{L+1}}$

When $L>1000$ then $C=L^{2}$

where L is the amount of land in km^2, and C is the cost in millions of dollars. We may see behaviour like this because Ticino is only $2812km^{2}$. If you attempt to buy too much of the canton, the cost may well start increasing very quickly. Obviously this is not an exact representation of real estate in Ticino, but it's an interesting model regardless.