# Social Media Star

## The Scenario

You are the marketing executive of a high profile international fashion brand. Your latest handbag line is out and your job is to work out which celebrities to work with in promoting your brand and the new product line. You’ve narrowed your choices down to two celebrities: Kendall Jinker and Sally Burns.

To help in making your decision, you’ve pulled some modelling data on how each superstar’s social media campaigns have run in the past. You find that products Kendall Jinker promotes follow this model:

where *x* is the number of days since the start of the campaign and *y* are the number of people reached by the campaign

Sally Burn’s social media model is this:

You have inside information that your competitors have a campaign scheduled to start in 10 days, so you need to reach as many people as possible before then. Plot the reach of each star’s social media over a 10 day campaign and choose your ambassador based on the results.

If 1% of the people reached by your campaign buys a $1000 handbag, with a profit margin of $500, calculate the profit for the best celebrity. Account for a $200,000 celebrity fee.

## Approach

First, fill out a table with the 10 days of the campaign for each celebrity:

Using the two models given above, you can fill out the table values for each celebrity (or do it using a spreadsheet software on a computer, tablet or graphics calculator). Remember *x* is the campaign day number.

With the values filled out, you can now plot the results as a graph. Typically time (campaign day number) would go along the horizontal, x-axis of the graph, and the number of people reached would therefore go along the vertical, y-axis:

The results are interesting. Sally Burns initially reaches more people than Kendall Jinker over the first few days of the campaign, but then Kendall shoots ahead and is more than **five times** Sally Burns’ total by the end of the 10th day.

The reason for the difference is due to the different types of growth for the two celebrities.

Sally Burns experiences *quadratic* *growth* – the number of people she reaches is proportional to the square of the number of days passed.

Kendall Jinker experiences *exponential growth* – the number of people she reaches grows by a constant multiple each day. For example, from day 5 to day 6 her reach grows by a factor of 2 from 3200 to 6400. From day 6 to day 7 her reach **again** grows by a factor of 2, to go from 6400 to 12800.

To calculate the profits for the campaign:

*Profit = percentage conversion x number of people x profit / converted person – celebrity fee*

*Profit = 0.01 x 102400 x $500 – $200,000*

*Profit = $312,000*

A third of a million dollars profit for a 10 day campaign – not bad. The celebrity fee still leaves the company with a healthy profit so this marketing move is probably a good one. You may be in line for a promotion!

## Real Life Example – Exponential Growth in Nature

One of the best examples of exponential growth in real life is the growth of populations of animals, insects or microorganisms over short time periods. In times of abundance these populations can experience exponential growth until they run out of food or predator numbers rise in response to reduce the rate of growth.

Another example is the initial spread of some types of virus, which will have exponential growth in the number of people affected if there is no immunization present in the population.