All right,

let's go through a real price optimization situation with a very realistic scenario.

And use that demand information and the information from the situation I'm about

to provide you to come up with an optimal price.

So you're going to be the retailer in this situation.

Now and you're setting price for month 56 in the data.

Why month 56?

Because that's the next month in the data.

The data that we estimate the model on goes all the way through month 55.

And now we're going to use all 55 months of that data to try to say,

what should we do in month 56?

What price should we set?

So we're going to be in month 56 and

you're going to price chicken at $1.75 a pound.

You're the retailer, you have the ability to do that.

And the wholesale price of chuck, that means the person selling it to you,

that's the wholesale price, is $1.50.

And we're going to be trying to decide based on that information,

what should the retail price of chuck be?

We also know that the individuals in this sample have a disposable income around

$10,000 and that's stable, and we'll also include that information.

So let's do it.

So here's our demand model again.

These are just the estimated coefficients, and

I've written down sort of the theoretical demand model down below it.

Well, what have I done?

I have substituted in the coefficients, right, so there's my intercept 779.

I'm going to multiply P by its coefficient rounded which is -50.2, and so forth.

But notice, the only thing I don't know here is price.

I know the price of chicken, that's $1.75, so I can also substitute that in.

I also know the income is stable around $10,000.

I can substitute that in.

And I know that I'm in the 56th month of the data, so I can substitute that in.

What that's going to allow us to do is simplify this whole thing a lot, right?

I can just do those multiplications and

those additions, simplify it down to these numbers.

And ultimately, once I do just a little bit of algebra, I can simplify that down.

It's really arithmetic.

I can just simplify it down to a demand function that looks like that.

Now what do we do with that?

Well, we do the optimization like we did before, but now we have this

real data coming from both the retailer and the demand side.

So we've got P- C, there's the margin,

multiplied by how many units I'm going to sell, that's the profit function.

Well, what's P- C?