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Stationary and Non-Stationary Revenue Models [Business Models
Posted on June 26, 2013 @ 12:31:00 PM by Paul Meagher

In my last blog I proposed a simple revenue model for lobster fishing. In today's blog I want to refine that model to make it more realistic. The particular refinement that I want to make to the model today involves taking into account the fact that lobster catches tend to decline over the course of a lobster fishing season. The number of lobsters available to catch declines after each catch because you are extracting lobsters from the grounds and are they are not being replaced with new lobsters at a rate that equals the extraction rate. That means you have less lobsters to catch after each running of the lobster traps.

Our simple revenue model for lobster fishing assumed that the size of the catch stayed the same from the beginning of the season to the end of the season. I selected a catch size that was somewhere between the largest and smallest expected catch sizes in order to wash out these differences, however, it would be better if I acknowledged that the catch size distribution I was sampling from was not "stationary" over the course of the fishing season, but rather is "non-stationary", with the mean catch size decreasing over the course of the lobster fishing season. By acknowledging the non-stationary nature of the lobster biomass over the course of a fishing season, we can better estimate what lobster fishing revenue looks like over the course of a season instead of assuming that it is roughly constant around a single mean with some variation due to chance factors.

In general, when you are modelling revenue you need to think deeply about whether the revenue generating process is stationary over time or is non-stationary. When selling gardening supplies, for example, we might expect there to be minor sales outside of the growing season with sales picking up at the beginning of the growing season and then tapering off towards the end. Your revenue model for garden supply sales might be best captured by factors in your revenue modelling equations that takes into account the seasonal and bursty nature of such sales.

In the case of lobster fishing we will attempt to capture the non-stationary nature of the catch size distribution (and therefore the revenue distribution) by assuming that the availability of lobsters to catch decays in an exponential manner. Exponential decay of lobster catch size means that there is a percentage decrease in available lobsters after each catch which will lead to lower mean catch sizes as the season progresses.

Exponential Decay Formula

The exponential decay/growth formula looks like this:

N = N0 * e k*t

The symbols in the equation have the following meaning:

N is the amount of some quantity at time t.

N0 is the initial amount of some quantity at time 0.

e is the natural exponent which is roughly equal to 2.71.

t is the amount of time elapsed.

k is the decay factor or percentage decrease per unit of time.

To use this formula in the context of my lobster fishing revenue model I need to figure out what values to plug into this equation. What I have to work with is some estimates of what the lobster catch will be at the beginning of the season (1000 lbs), what it might be at the end of the season (300 lbs), and how many trips in the boat they will make to the lobster fishing grounds during the lobster fishing season (40 trips - Note: the number of trips will be the t value in our exponential formula). Given these values, we can figure out what the decay rate should be so that we begin the season with a catch of 1000 lbs and end the season with a catch of 300 lbs and we do this over 40 fishing trips.

So here is the exponential growth/decay formula (depending on the sign of the k term):

N = N0 * e k*t

Now substitute in our values:

300 = 1000 * e k*(39)

I use 39 for the value of t rather than 40 because there is no decay in lobster catch size for the first trip. The decay only kicks in on the subsequent 39 trips. The math works out correctly this way as you will see later.

To solve for k, we need to rearrange some terms:

300/1000 = ek*(39)

Applying the natural logarithm function, ln(), to both sides allows us to get rid of the natural exponent e as follows:

ln(300/1000) = ln(ek*(39))

Which evaluates to:

-1.20397 = k * 39

The solution looks like this:

-1.20397/39 = k

Doing the division, we are left with:

k = -0.03087

So the exponential decay formula for the mean catch size looks like this:

N = 1000 * e -0.03087 * t

Where t is the trip number which varies between 1 (for the second trip of the season) and 39 (for the 40th trip of the season).

Testing The Formula

To test his formula we can plug in the relevant values and verify that in our last catch of the season our expected catch would be 300 lbs. The PHP program below verifies that this is the case.

The output of the script tells us that on the final trip we expect to catch 300 lbs of lobster which is how we want our decay function to work.

The mean catch size is not the only parameter that we might expect to vary through the lobster fishing season; we might also expect that the standard deviation in catch sizes would also decrease along with the smaller catch sizes. A simple and reasonable approach to decreasing the expected standard deviation in catch sizes would involve decreasing the standard deviation over catches/trips using the same exponential decay formula but using the initial standard deviation (250) as the N0 initial value in a similar exponential decay equation (i.e., N = 250 * e -0.03087 * t).

The script below is used to verify that our expected catch sizes and expected standard devation in catch sizes start and end at appropriate values.

Conclusion

So what we have done in today's blog is to come up with an exponential decay formula that we will be using to define the mean and standard deviation values that we will plug into our catch size distribution function which we assume to be normally distributed. For our lobster fishing revenue model to become more realistic we have to acknowledge that the revenue obtained from lobster fishing is non-stationary through the season, in particular, that we generate less income each time we go out fishing because the available stock of lobsters is reduced after each catch. We can model this decreased revenue by sampling from a catch size distribution that has a smaller mean and standard deviation after each trip. In my next blog I will show you how the PHP routine above can be incorporated into our lobster fishing revenue model to provide us with a more realistic revenue model, one that might provide us with more realistic expectations regarding cash flow through the season.

As a final note, the lobster fishing season this year is quite unusual as catches are larger than normal and they are still getting good catches towards the end of the season (800 lbs on their last trip). The revenue model ignores certain unusual aspects of this season which might make it a better revenue model for predicting lobster fishing revenue next season. The model does not attempt to overfit the data from this season because the numbers are quite unusual (they might, however, reflect the effect of better conservation measures which might persist in their effects). Predictive revenue modelling can be more of an art than a science as it involves judgement calls regarding what is signal and what is noise.

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