Background
In a previous module you learned some basic functions that occur in economics:
- , the cost function, gives the total cost a company needs to pay for producing x units of a good.
- , the price function, gives the price a company can charge per good at a level of output of
x units. It is the inverse of the demand function, which gives the number of units of a good
customers will demand at a certain price.
- , the revenue function, is the amount of money a company takes in from selling x units of a
good. We have .
- , the profit function, gives the amount of money a company actually earns from selling x units
of a good after taking into account costs, so that .
In the previous module, we worked with a linear cost function. Although it is convenient to model costs with a
linear function, in real life costs are not usually increasing linearly (can you think about why?). In this module,
we will introduce a more realistic cost function and also an average total cost function. Then we will
consider another goal that a company might have: minimizing the average total cost per unit produced. In
order to do this, we want to keep the following theorem from calculus in mind:
If has a maximum or
minimum value at c, then either or does not exist.
Section 1
The following video will introduce the functions we will be working with.
_
The derivative of the total cost curve is
always positivealways negativealways zeroeither
. The derivative of the total cost curve is
always increasingalways decreasingalways constanteither
.
.
The derivative of marginal cost is the same as
total costderivative of total costsecond derivative of
total cost
. This function is
always positivealways negativealways zeroeither
and
always increasingalways decreasingalways constanteither
The average total cost function is
concaveconvexneither
.
Suppose an ice cream truck has the following cost function, where is the number of cones sold: .
Referring to the cost function given above, what is the ice cream truck’s marginal cost function?
What is the ice cream truck’s average total cost function? .
How much does it cost the ice cream truck to sell 15 cones? Round all answers to the nearest penny. $. What
is the cost of selling the 16th cone? $.
Remember, C(16) will give the total cost of making 16 ice cream cones.
Instead, we are looking for the cost of the 16th cone alone. This will be the total cost of making 16 cones
minus the cost of all the cones that we already made.
What is ? . Compare the cost of the 16th cone to the marginal cost of the 16th cone.
Section 2
Now we will think about the real goals of a company. Watch the following video to learn how a company finds
the level of production that will minimize it’s average total costs.
_
Select all that apply: When average total cost is minimized...
profit is maximized. total cost is also
minimized. marginal cost is also minimized. the company is making money. MC=ATC. MC=MR.
Scarlet and Joe are trying to open a Durfee’s swipe delivery service on Yale’s campus. Suppose they have the
cost function: representing the total cost in dollars for deliveries made during the Durfee’s swipe period.
For Scarlet and Joe...
dollarsdeliveriesdollars per delivery
dollarsdeliveriesdollars per delivery
.
If Scarlet and Joe’s only goal is to minimize their cost per delivery, how many deliveries should they aim to
do each day? Round to the closest delivery.
dollarsdeliveriesdollars per delivery
If you solve for the level of production that would minimize ATC, you will not get an integer. However,
Scarlet and Joe cannot perform fractional deliveries. Try the integers around the minimum ATC to see which
one yields a lower cost per delivery.
What is the lowest average cost per delivery that Scarlet and Joe can achieve? Round to the nearest penny. $
dollarsdeliveriesdollars per delivery
.
Scarlet and Joe are thinking of charging a fixed price $13.00 for each swipe (regardless of how
many deliveries are requested). Then Scarlet and Joe’s revenue function would be...
Recall that at profit maximizing level of output, MR=MC. How many deliveries should Scarlet and Joe
perform per day to maximize their profits? Round to the nearest whole number.
dollarsdeliveriesdollars
per delivery