Homework 11

Yuri

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.

A linear equation is knows as y=mx+f In this case, we will use familiar letters to make it clearer to see the relationship:
C=mx+b
where,

C = Total Cost
m = marginal cost
x = number of flags produced
f = fixed cost

Because we are given the information that the marginal cost is $7, and it costs $100 to produce 20 items, we are able to get:
100=7(20)+f
100=140+f
f=-40

Hence, the linear model is: C = 7x - 40

Describe your model

The model shows what the total cost will be if x number of items are produced at $7 marginal cost with the fixed cost to be $-40.

What does your model predict for a production of 150 items?

C = 7(150)-40
C = 1050-40
C = $1010
Therefore the total cost will be $1010.
The cost per item will be:
$1010/150 = $6.73 each

According to your model, what happens to the average cost per item as production levels increase?

As production level increases, our model tells us that the average cost per item would increase.
An example of this would be to use x = 200 which is greater than 150 as used in the previous question.

C = 7(200) - 40
C = 1400 - 40
C = $1360

Therefore the average cost per item would be:
$1360/200 = $6.80 which is greater than the cost per item of when 150 items are produced.

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

The average cost remains constant as production increases

y = 5x (yis the total cost, 5 is the marginal cost, x is the quantity produced)

So, if there were 40 units produced,
y = 5(40) = $200
Now to find the average cost per unit we:
200/40 = $5

If we increase the units produced to 50,
y = 5(50) = $250
To find the average cost we:
250/50 = $5 and this shows how the average cost stays at a constant of $5 as production increases.

The average cost diminishes as production increases

y = 5x + 100

So, if there were 100 units produced,
y = 5(100) + 100 = $600
The average cost:
600/100 = $6.00

Now increasing the production to 200 units,
y = 5(200) + 100 = $1100
The average cost:
1100/200 = $5.50
Therefore this model agrees with the statement as the average cost diminishes as production increases.

The average cost increases as production increases

y = 5x - 100
So, if there were 100 units produced,
y = 5(100) - 100 = $400
The average cost:
400/100 = $4.00

Now increasing the production to 200 units,
y = 5(200) - 100 = $900
The average cost:
900/100 = $9.00
Therefore this model agrees with the statement as the average cost increases as production increases.