0:00

The last topic that I'm going to talk about

Â in this module is that of optimization.

Â Now earlier on, I talked a little bit about

Â optimization when we have linear functions and linear constraints.

Â That was known as linear programming.

Â Now I'm going to show you some optimization in what I would term,

Â a more classical sense.

Â So I'm going to use the mathematical technique calculus now

Â to help solve an optimization problem.

Â And remember, I had said previously that one of the key

Â uses of quantitative models is as inputs to optimization decisions.

Â So businesses are always trying to optimize their performance in some way.

Â So optimization problems can sometimes be solved using a calculus approach,

Â and that's what I'm going to show you.

Â 0:53

So, let's go back to the model that we had used to understand

Â the relationship between the price of a product and the quantity demanded.

Â And ask ourselves the question,

Â can we find the optimal price in order to maximize our profits?

Â Now, to do that, I've gotta put a little bit of a notation in place.

Â So that's going to happen on this slide.

Â So let's consider the model we had looked at before.

Â We had called it demand model, where the quantity demanded of a product,

Â and here's the model, is equal to 60,000 times price to the power -2.5.

Â So I'm presenting you with that model.

Â Now you're sitting there thinking, where does he get a model like that from?

Â Well, that question actually isn't what I'm trying

Â to do right now in this particular module.

Â I'm going to talk about where did these models come from

Â when I talk about regression in one of the other modules.

Â So even though this looks like it's being pulled out of thin air,

Â there is a basis for creating these models that we will discuss.

Â But for right now, I just want to show you this deterministic model.

Â So let's say our model for

Â demand is quantity equal to the 60,000 times price to the power -2.5.

Â That shows me how quantity is related to price.

Â Further, I'm going to assume that the price of production is a constant

Â at $2 for each unit.

Â So every unit that I produce costs me $2.

Â Now here's the question, what price is profit maximized at?

Â So you can choose price.

Â It's your product, get whatever you want.

Â You could set a very low price, and you'd probably sell a lot.

Â But if the price was lower than your cost of production,

Â you wouldn't be making any money.

Â You could set a really, really high price for

Â one of these objects that you're selling.

Â 3:07

And revenue can be written as the price that you sell the object,

Â at times the quantity that you sell.

Â So if you're selling candy bars, they cost $2 for someone to purchase.

Â I mean the price is $2 and you sell ten of them, then your revenue is $20.

Â That's right, 2 times 10.

Â So that's all that's going on there.

Â We write that more generally as p times q, price times quantity.

Â Now, the profit is the revenue minus the cost.

Â So the profit equals pq, which is our revenue.

Â Now what is the cost of producing q units?

Â Each unit costs c dollars to produce, and I'm going to produce q of them.

Â So the total cost according to this model is c times q.

Â 3:54

So I can simplify that equation into q(p- c).

Â So that's the profit that we're going to make.

Â But we've got a model for q in terms of price, and

Â that model says quantity equals 60,000 times price to the power of -2.5.

Â So putting it together, we can see that our profit is equal to 60,000 times

Â price to the -2.5, that's the quantity, times (p- c).

Â And in this particular example, I'm taking the cost of production at $2 per unit, so

Â that's (p- 2).

Â So now now you're looking at an equation that has come out of

Â the quantitative model for quantity.

Â And I am now going to ask,

Â at what value of p is the profit maximized?

Â So choose profit p to maximize this equation.

Â So this is what we mean by an optimization,as I said before.

Â Optimization is one of the things that we tend to do with our quantitative models.

Â So how are we going to do it?

Â Well, there is a brute force approach to this.

Â We've got a function for profit.

Â 5:07

Let's just choose different values of price, which I'm writing as little p,

Â and plug them into the function and see what the profit looks like.

Â And so in the table on this slide,

Â you can see I've plugged in different values for price.

Â That's in the price column.

Â And I've used the equation, the model, to figure out what the profit is.

Â So if I charge $1.75 for

Â this product, I actually don't make any profit at all.

Â There's a negative profit, otherwise known as a loss.

Â And of course that makes perfect sense because 1.75 is less than the cost of

Â production, which is $2.

Â If I were to price at $2, then I don't make any profit whatsoever because my

Â price is exactly equal to my cost.

Â So you get zero for the second one.

Â And then the subsequent numbers in there

Â are just coming out of the profit equation.

Â Now, if I look down through that table,

Â the optimization just corresponds to finding the biggest number in there.

Â And I've drawn a graph that shows you the profit as a function of price.

Â And you're really trying to figure out at which value of p, the x-axis,

Â is the profit the highest?

Â Where is the top of that graph, in other words?

Â So this is a brute force approach because I haven't actually tried every value of p.

Â If I'm implementing this in a spreadsheet, spreadsheets have cells.

Â And in each cell, you can only put in one number.

Â And so, it's a discrete approach to solving this problem.

Â And it looks to me that the best value

Â of price is somewhere sitting between three and four.

Â But I don't know exactly where between three and four it is.

Â So this gives me a sense of where the answer is.

Â And it might be fit for use.

Â It might be enough for you to say,

Â I just want to set the price between three and four.

Â But optimization does give us the potential to be a bit

Â more precise about it, so that's what I'm going to do now.

Â So the calculus approach to these problems involves

Â the mathematical technique of differentiation.

Â And what we need to be able to do is to find the derivative, which

Â means the rate of change of a function, of the profit with respect to price.

Â And we need to see where that derivative equals to zero.

Â So optimization,

Â the actual mathematics of optimization is not the goal of this course.

Â The goal of this course is to talk about modeling, and

Â this is one of the places that models are used.

Â And so I'm not actually going to do the, I'm going to present you with the results.

Â If you're interested in calculus, well, and it's use in business,

Â you can certainly find other courses that will address that.

Â So I'm just going to skip to the answer here.

Â It turns out that by applying calculus to this problem,

Â you can obtain the optimal price.

Â And the optimal price, which I'll write as P-opt,

Â opt for optimal, is equal to c times b over 1+b.

Â Where c is the production cost, and b is the exponent in the power function.

Â So with this neat little mathematical model that we had for quantity demanded as

Â a function of price, I'm able to leverage that equation, leverage that model.

Â And come up with an answer to the question,

Â what's the best price to set in order to maximize my profits?

Â Now going back to this example, c was equal 2, that was the cost of production.

Â And b was equal to -2.5.

Â If I plug in those numbers to those equation,

Â you can convince yourself that the optimal value for

Â p, for the profit, is about equal to 3.33.

Â It's really 3 and a third, is the best value for price.

Â So that is the solution to the problem.

Â And by creating or

Â using a simple model for the quantity, for

Â the demand, I'm able to end up with a simple model.

Â You can even call it a rule of thumb if you want, a simple formula for pricing.

Â 9:17

Now in terms of interpretation again, well, we know what c is.

Â That was the cost.

Â That coefficient, the -2.5 in the power function model that we're looking at.

Â Remember, this model for demand is a power function model.

Â It was 60,000 times price to the power of -2.5.

Â That's a special quantity, the exponent b in this situation,

Â and it gets called the price elasticity of demand.

Â And so oftentimes economists will put a negative in front of that,

Â because the coefficient is -2.5.

Â And one might say the price of elasticity of demand is 2.5 for

Â this particular product.

Â What that -2.5 means in terms of the business

Â process is that as you increase price by 1%,

Â you can anticipate in fall in quantity demanded of 2.5%.

Â So the coefficient relates percent change in x to percent change in y.

Â And the -2.5 means that as x is going up, y is going down.

Â So a 1% increase in price is associated with a 2.5% fall in quantity demanded.

Â And that proportional relationship,

Â proportional change in x to proportional change in y, is true for any value of x.

Â That's what's very special about the power functions,

Â that proportionate change between x and y is a constant.

Â And in this case, it's -2.5.

Â And as I say, people might call the price elasticity of demand here 2.5.

Â So that's the calculus approach.

Â And I'll finish this off with a slide that shows you

Â what the calculus approach is doing.

Â The blue curve is the demand equation,

Â that is the curve 60,000 times price to the -2.5.

Â So that shows how price and quantity demanded are related.

Â Now, for any value of the price, so fix the price.

Â Stop the box moving means fix the price.

Â 11:25

For any value of the price,

Â you can go up to the curve and record what that point is.

Â That will define a box.

Â And that box, that's the light gray shaded area

Â in the graph here actually is the profit that's associated with that price.

Â And you know it's the profit, because the width of the box is p minus c,

Â that was price minus cost, and the height of the box is q, the quantity demanded.

Â Remember, that was how we were able to write our profit here,

Â as q times p minus c.

Â So q is the height is the box, p minus c is the width of the box.

Â The product of those two numbers is the profit, and

Â the product of those two numbers is the area of the box.

Â So what the calculus approach does is take your quantitative model,

Â which is the blue curve here, that's what we contribute with the modeling.

Â And then we do the optimization, which is simply to find the value at p

Â at which the area of the gray shaded box is largest.

Â If I can find that value of p, I've solved the optimization problem.

Â And so again, one of the nice things about these quantitative models is they can help

Â you visualize the solution to a problem.

Â And without this visual here,

Â it's hard to kind of see in the same way as to what we're really trying to achieve.

Â And as I say, what we're trying to achieve when we maximize the profit is essentially

Â to find the value of price at which this gray shaded box is maximized.

Â And we know from having gone through the calculus approach,

Â that it's at p equal to 3 and a third, 3.33.

Â Now that completes the topic that I wanted to talk about in this

Â module on deterministic models.

Â Remember, deterministic was the antithesis of probabilistic or stochastic.

Â Deterministic model says no uncertainty anywhere.

Â