Throughout the class, youve learned many ways in which the study of derivatives can be applied to business operations. The

last unit covers some additional topics that will come up again in later marketing and economics classes, including marginals

and elasticity of demand. Your goal for this

2-3 page writing project

is to demonstrate your ability to think critically

about business operations, model them with mathematics, and make astute recommendations to improve performance. These

skills are crucial in the world of business management!

Pick a consumer product that is reasonably mainstream and a specific market. For example, you could choose to study

exercise mat sales in a particular state, or sales of a particular kind of bicycle in Florida. Its best to choose a product with

one (or only a few)

discrete price points

. An example of a bad choice would be personal computers, which have a wide

range of internal components that affect pricing and desirability. Once youve picked a product and market, spend some time

researching recent sales data (units moved in your market, and an average price in the time window youre investigating).

Its ok to place some limitations on your data set at this juncture, and round or estimate when necessary. Your goal here is

to find:

a specific market

a decent estimate for how many products were sold in a time window (say, 2017, or March 2016) in that market, and

a decent estimate for the price of the product (say, $250)

This part of the project should take you no more than a few hours. Please track the sources (websites) that you use, and

include them in a reasonably-formatted

Works Cited page

. If you absolutely cant find sales numbers, its OK to make

them up, but please indicate that, and be aware that it might make your calculations look weird.

You now have an estimate for how many units of your product tend to sell at one specific price point in your market.

Construct a

consumer demand function D(x)

that would predict approximately the correct number of sales at your price

point (see point 3, below). This functions will only be a good approximation near the price point that theyre based on;

its ok if they would yield unreliable data when you extrapolate far from the price point.

A method for constructing a

demand function D(x) is provided at the end of this rubric.

Using your function D(x), find the associated

elasticity function E(x)

. Is demand for the product at your price point elastic

or inelastic? According to your model, what would happen to

net revenue

if the products price were slightly increased or

decreased? What is the optimal sales price for your product (when E(x) = 1)? How many units should you expect to sell at

that price (from your demand function)?

All of the questions in the previous paragraph will have very specific answers that pop out of the model function that you

defined. After discussing them, consider what other, more intangible factors might affect demand and profitability. Is your

product seasonal? Is it facing competition from a changing market? Is it controversial in some way, or subject to government

regulation? What can a business do to hedge against these potential concerns? The answers to these questions will vary

widely depending on the topic you chose.

A substantial part of your grade for this assignment (see attached rubric) is based on your ability to clearly communicate the

answers of these questions to someone with a weaker mathematical background. Its important to use precise and appropriate

mathematical terminology, but also to explain what youre saying in a comprehensible way.

Your grade will also be determined by the extent to which you demonstrate your ability to think critically about the market

and about the material covered in this course. Throughout the semester, youve been asked to make market predictions based

on limited data sets; here, youre making predictions based on your own data. Its important that the mock recommendations

that you make to improve performance be expressed in a clear and mathematically rigorous way. Explain to your reader

why

these business changes are worthwhile

, and why they are

the best of several options

. Clearly identify the money

lost by continuing to run the business without these optimizations. Make it clear why its important to listen to your advice!

1

Heres a suggested flow for your paper; feel free to deviate from this blueprint, as long as you hit all of the required components

of the assignment listed in the grading rubric.

1. Clearly state your product, market, and time-frame. State whether youre using actual sales data, or made up numbers.

Some companies dont make sales figures public.

2. List and explain some factors influencing price that are specific to your product. Is it affected by climate, health trends,

or other major factors? Does it have a major competitor (eg, Pepsi vs Coke) with which its competing for sales? Does

it make sense to have discounts (sales) on the product, or is it a strictly regulated commodity like gasoline? This part

of the paper should be very unique to your product and market, and should be

a few paragraphs long

.

3. List the number of sales (Y) at the stated price point (x). Find a demand function D(x) which outputs Y when you

input x. Just like the demand curves from our class, this function should be decreasing (negative derivative). For

example, suppose you tracked 1000 sales at a price of $2 per product. One function that works is D(x) = 2000 - 500x.

You can see that when you plug in your price, x=2, the function outputs 1000 sales. There are infinitely many functions

that work; pick one that you can easily take a derivative of!

4. Find D(x), and use it to find the elasticity function E(x). Just like in the elasticity unit, set E(x) = 1 and solve for x

to find the optimal price point. Is it higher or lower than the price that you found through research? What does the

elasticity function output at the model price (here, at $2)?

5. You now have a new suggested price for your product. Explain (in words) what happens to demand when you increase

or decrease price to that point. What happens if you increase or decrease

past

that point?

6. Lastly, does the new price point make sense for your product? You started the project by completely making up a

demand function; a bad function might suggest that you change your price to something ridiculous (like $10 per can

of soda). Explain to your readers whether the price that the mathematical model suggests is a reasonable one, or if its

more likely that your demand function was unrealistic.

Your paper should be a few pages long to meet all of the requirements of the assignment (see the rubric). Im also hoping

that it wont be more than a few pages long. You have the final week of classes to work on this project (and can certainly

start early!), so Id expect it to take about the same time as studying for and taking an exam, or doing a week of assignments.

Time may vary a bit depending on your luck Googling project ideas.

As long as you follow the rubric and answer the questions fully, the project should be a low-stress A or B to round out

your course grade. Let me know if you have any questions or run into problems, especially when coming up with a function

D(x).

Here is a quick way to make a demand function for your product:

1. Record your number of current sales,

S

(units sold, not revenue), and current price,

P

2. Let

B

be the number of sales (units sold) you expect to lose for every dollar you raise your price

3. Let

A

=

S

+

B

P

4. Your demand function is

D

(

x

) =

A

Bx

You dont need to show these steps, or your values of A or B, in the paper - just the final demand function. This is just

scratch work that gets you there.