Next example that I want to show you is called a Monte Carlo Simulation.

Now Monte Carlo Simulations are very useful for modeling complicated scenarios.

The example that I have here I wouldn't claim is particularly complicated.

But it will certainly give you a sense of what a Monte Carlo Simulation

can do for you.

So I'm going to go back to the demand model.

The demand model being that the quantity of a product demanded

is equal to 60,000 times its price to the power -2.5.

In the module where we had looked at this model,

we had worked out via calculus that the optimal price given this set

up was equal to $3.33, three and a third dollars.

And that was based off of the equation

that the optimal price was equal to c times b over 1 + b,

where c was the cost, and in the example we had the cost equal to 2.

And b was the elasticity, and in the example we have b equals -2.5.

If you plug those numbers into the optimality equation solution,

you will get three and a third.

So that's where the three and a third came from, but

we were treating this in a deterministic fashion.

In other words, we were saying we know what b is.

b is equal to -2.5, but many situations,

you're not actually going to know exactly what b is.

b is going to have some uncertainty associated with it, and

it would be good if we could propagate the uncertainty that's associated with b

all the way through to some uncertainty associated with the optimal price.

So what if b is not known exactly?

Well, one natural thing to do is to try and

put in different potential values of b.

Maybe you could put b in at -2.4 and see what happens.

Maybe you could put it in at -2.3 and see what happens, to start generating a range.

And what Monte Carlo simulation does is take that idea, try different values of b.

But, it draws those values of b from what we call a probability distribution.

And each time it draws a new value from b,

it calculates the optimal price and stores that, and we will replicate that process.

We will take hundreds or thousands, or

even millions of draws, from that probability distribution and

end up with an entire distribution for the optimal price.

And as I said, in this particular example, there's only one unknown, which is b.

But in many examples, and I've worked on examples where we might have a million

different unknowns with each one having its own probability distribution.

And the same idea follows, you can draw each of those unknowns

from a probability distribution propagated through the formula.

The formula here being c times b over 1 + b, and

get a range of uncertainty on the outcome.

So let's see that working in this particular example.

On this slide I'm showing you the input to a Monte Carlo simulation and

the output from the simulation.

So I'm going to generate the elasticity b from what's termed a uniform distribution.

And my knowledge suggests that b lies somewhere between -2.9 and

-2.1 and essentially each number between -2.9 and

-2.1 is equally likely, so that's what we call a uniform distribution.

And I've drawn a picture of that uniform probability distribution for b.

And it's a straight line going along the top because every

outcome is equally likely.

So let's say we take a b from this particular distribution, and

now drop it in to our optimality equation, which is c times b over 1 + b.

Remember that c is equal to 2 in this particular instance, so we take a b,

a random b, and drop it into the formula, and we save the answer.

And then we keep doing that.

And in this particular example, I have replicated that 100,000 times.

I've drawn 100,000 bs from the uniform probability distribution,

and each time I have calculated what the optimal price is.

And that's what I'm showing you in the histogram

at the bottom right-hand graphic on this slide.

You can see it has an interesting distribution associated with it,

it's not flat.

And the reason it's not flat, even though the input came from a uniform

distribution, is that our formula, c times b over 1 + b, is not a linear equation.

It's non-linear, and

that means that we're not going to have a uniform distribution coming out.

And we can see that some outcomes are more likely than others,

that those are the places where the histogram is higher.

Once we've got this entire probability distribution for

the output, remember that's the optimal price,

we can do some useful things with that output probability distribution.

On the distribution, I have drawn in where our single best guess,

that was the three and a third.

But I've also placed what one might term range of feasible values around there.

I've created intervals that capture 80% of the draws

from this distribution.

And that 80% interval there ranges from 3.1 to 3.7.

And so I could use that interval as a more realistic basis for

understanding the uncertainty and the optimal price,

given that I acknowledge that I don't know exactly what b is.

So that's the basic idea of a Monte Carlo simulation.

It's like a scenario analysis, but you're looking at potentially thousands or

millions of scenarios.

And those scenarios are being generated from inputs that are drawn

from probabilistic models, from probability distributions.

So a very, very common technique in many business situations.