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May 21, 2014

Set the Right Selling Price for Christmas Cookies

In my last blog post, I found the best tasting recipe for the Christmas cookies I want to sell. In this post, I will use JMP's Profiler and Simulator to decide the selling price of the cookies and to investigate profitability. If the price is set too high, then fewer cookies will be sold. If the price is set too low, then lots of cookies will be sold, but the revenue will be lower than it could be. (The example I present here is adapted from one I encountered in Making Hard Decisions: An Introduction to Decision Analysis, by Robert T. Clemen.)

Let’s consider a simple model for Profit:

Profit = Market*Proportion*(Price - Variable) - Fixed.

Market represents the daily size of the market for cookies. Proportion represents the proportion of that market I will capture. Price is the selling price of a cookie. Variable is the variable cost associated with each cookie. Fixed is the fixed costs. Consider the following example.

Assume that the daily market size for cookies is 100,000, the proportion I will capture is 20%, the price of a cookie is $0.50, the variable cost per cookie is $0.10, and the fixed costs are $8000. The Profit formula evaluates to

Profit = 100,000*0.20*(0.75 - 0.10) - 8000 = $2000.

Those numbers are based on my best guesses of the variables in the formula. What if the actual values are different? How will that affect Profit? Using JMP’s Profiler, I can visualize the affect of the variables on Profit. Furthermore, using the Simulator, I can estimate profitability under the uncertain business conditions.

Before I do that, let’s consider one thing. It’s obvious that, all else being equal, charging a higher price is better. But, economics dictates that all else will not be equal. I can anticipate less sales as the price increases. This means the proportion will decrease. I choose to model that in the following way: For every $0.10 increase in price above $0.50, the proportion decreases by 3.5%. And likewise, if the price goes below $0.50, the proportion increases.

I’ve incorporated this model into a JMP data table, called, and you can download it from the JMP File Exchange. The table has a column called Profit, which contains the formula for profit. Run the attached script to get the Profiler and Simulator shown below.

All of the factors in the Profit model are shown. The current value (shown in red under each plot) of the factors is listed, as well as the resulting Profit (shown in red on the left). Each factor has a vertical dashed red line that you can grab and move with the mouse to change the value of the factor to see its effect on Profit, as well as the interaction with other factors. As you can see from the Profiles, when either Market or Prop goes up, the predicted Profit goes up. And likewise, when either of the costs go up, the Profit goes down. The Price profile is most interesting: It is curved, and there is an optimal value for Price. Profit goes down if Price is above or below the optimal value.

I can’t precisely predict market conditions (the model factor values), which means I can't know with certainty whether my business will be profitable. Therefore, instead of guessing at the values of the factors in the model, I will assume a distribution (a range of values). For example, instead of just assuming Market size will be 100,000 cookies, I will assume that Market will be somewhere in the range of 70,000 to 130,000.

More precisely, I model Market size as a Normal variable with mean of 100,000 and standard deviation of 10,000. Proportion is modeled as a triangular variable between 18% and 26%, with the most likely value at 22%. Variable costs are uniform between $0.08 and $0.12. And finally, Fixed costs are triangular between $6,500 and $9,000, with the most likely value at $8,000. I'm using distributions to represent the uncertain market conditions I'm faced with.

The Simulator draws random values from these distributions, calculates Profit, and repeats many thousand times. This results in a distribution for Profit. In other words, I've assessed Profit under thousands of different market conditions.

The important question is: Given the uncertain conditions, how likely is my cookie business to be profitable? The Simulator calculates this as well. Of the thousands of simulated conditions, JMP tracks how many result in a negative profit.

With Price at $0.50, I click the Simulate button to estimate the probability that Profit is greater than or equal to 0. JMP produces a histogram of simulated Profit values and gives the probability as shown below.

As shown, if I set price at $0.50, my average profit will be $971. But, because I'm uncertain about the model factors, that uncertainty gets transferred to uncertainty in Profit. Notice that part of the distribution of Profit is below 0 (the red line on the histogram). In this case, the probability of being profitable is 77.5%.

But Price is not at its optimal value. Let’s move Price to the top of its profile, say 0.61. After making the change, I click Simulate again.

Notice the average Profit increased to $1,448, and the probability of being profitable increased to 86%. I think I’ll charge $0.61 per cookie for now at least. If in the future the market changes, or my costs change, I can rerun the simulation to investigate what Price to charge.

Using JMP’s Profiler I was able to see that Price does affect Profit and also that there is an optimal Price. Using the Simulator, I was able to decide on a reasonable cost and quantify the risks associated with that cost.

The Profiler and Simulator are much richer than what I have demonstrated. This example is the simplest of models, with only one response, Profit. You can model multiple responses and have JMP solve for optimal solutions across all responses.

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