[MUSIC] Let's look at some further aspects of processing, and bring in your knowledge that you gain from your course module. In fact, what we we're doing is looking at using differential calculus to ascertain the optimum setting price, and the optimum quantity. To start, you need to develop your research into the expected demand, for your particular product or service. In other words, you need to figure out the demand function, if you wish. But most importantly, it's how a change in price will impact on the demand for your product or service. This actual change in price and change in demand, or what we often refer to or economist refer to is price elasticity of demand, is something that's really key, for you as owners, managers, to know about your particular business. It's difficult information to find out one way could be going through, and doing a customer survey for instance, looking at how they would react to a change in price. And you could also bring in your further quantitative knowledge here, by using probability to estimate what a change in price would have as an effect, on the demand for your product course service. Let's have a look at an example using the following data. So, we have a product that costs 40 pounds to make. I'm going to call that our variable costs. We currently sell 500 units of this product, at a price of 100 pounds. So, we've got a variable cost of 40 pounds, and a selling price of 100 where we currently sell 500 units. It has been determined that sales will drop by ten units, for every one pound increase in that selling price. Fixed costs in our example are 5,000 pounds. So, these are independent of that change in demand, in terms of the variable cost. The total costs we can therefore say is, the fixed costs plus the variable cost per unit multiplied by the number of units. So, there are three elements in our total cost. In other words, it's the 5,000 pounds plus the 40 pounds multiplies by, at the moment, the current number of units which is 500 units. In this case, we can therefore see that the total cost equals 25,000 pounds. We're interested in finding the point where the marginal cost, alternative name, if you wish, for the variable cost is the same or is equal to marginal revenue. In other words, where MC=MR. The first step, is to work out at what price would we not sell, any of our product, or service, for that matter. Remember, currently, we're selling 500 at a selling price. Each one of those has a selling price of 100 pounds. And we will use 10 units for every one pound increase in the selling price. Therefore, I would sell 490 units if I had a selling price of £101 or 480 units at a selling price of £102, etc., etc. At what price do we sell no units? Well, how many increments, how many changes, if you wish, are there between our current number of units that we're selling. And zero, if a one pound change causes a 10 unit change, to simpler out of course is to divide 500 units by 10 ,in other words 50. So, a 50 pound change in on top of in addition to the current selling price. It's that point, we wouldn't sell any unit subtle. So in another words, we can therefore establish that the selling price for demand falls to zero is a 150,000 pounds. 100 pounds, the current selling price plus 50 increments of one pound each. Once we know this, then at that price of 150, that can help us actually establish, what we can build here in fact, is a formula. We need to ask the question, "If at 150 pounds I don't sell any units, at what price would I sell one unit?" Well, we know that a one pound change is a ten unit change. What I would like to know, is how much will cause a one unit change? And simply dividing one pound or 100 pence by 10 units means that a 10P per unit change. In other words, if I was to increase or decrease the selling price by 10P, that would cause a one unit change. An increase, one unit would be less, a decrease in the selling price, I'd sell on extra unit. We can actually build a formula into that, so therefore, we could say that the maximum selling price, that point where we don't sell any units at all, Minus 10P would equate to one unit sold. In other words, 150 pounds minus 10P, or 149.90 would equal 1 unit sold. We can use this knowledge to determine, what the selling price is for the optimum number of units and cells, and develop a formula as follows. The selling price equals the price where we don't sell any, 150 pounds, minus the rate that adds one sale multiplied by the number of sales. The number of units if you wish, therefore our selling price formula is 150 pounds, the maximum where the amount has fall to zero, minus 10P that change for one unit multiplied by x. In other words, is our selling price formula 150- 0.1x, 0.1 representitive of one-tenth pence, one pound? Clearly here, x represents the number of units made and sold. It's the same number in both equations, we just don't know what that number is yet. That's what we're going to find out. So we know that the total cost, our fist equation, is 5,000 + 40x. And our selling price formula is 150- 0.10x. What we don't know is, what is x? But. also the point we are interested in, is finding out when marginal costs, in other words the 40 pounds, is equal to the marginal revenue. In other words, the revenue gained from selling that one extra unit. So, we know that the marginal cost, is the same as the variable cost per unit, in other words 40 pounds. So, let's work on the marginal revenue aspect of this. How do we calculate marginal revenue? Well, the first thing we need to calculate, is a total revenue. And total revenue of the optimum level. Now, the word is this is total number that we sold, at the price that we sold them at. As we don't know how many we sold yet, we just let x represent that particular value. What we do have is a selling price formula, so what we know here is 150- 10x represents the selling price, at any given level of output where x will just represent the number of units. If that's the selling price, we can turn that into our total revenue by simply multiplying that formula by x, in other words, the total revenue will be 150X- 0.10X squared. The marginal revenue is related to the total revenue, by the number of units effectively. If we divide by X, we can then equalize the two sides of that equation. In other words, if we differentiate the total revenue. So, whereas we had 150x- 0.10x squared. If we were to differentiate that equation that becomes 150- 0.2 x. We can therefore say that the marginal revenue is 150 pounds minus 20 pence times x the number of units. Now, I know that what marginal revenue is, and the point that I'm looking for is when marginal cost equals marginal revenue. Then we should be able to solve and find out the optimum number of units, and the optimum selling price based upon that. Okay, so let's look at what we know, we know the marginal cost, the variable cost is 40 pounds. We now know that the marginal revenue is 150 pounds minus 20 pence times x. We are interested in that point, where the marginal cost equals the marginal revenue. In other words, where 40 = 150- 0.2x, we can restate this if you wish, simply manipulating the equation. At no point 2X- 150- 40 or alternatively 0.20X = 110, and then the final step is simply divide 110 by 0.20, and we found that x the optimum number of units Is therefore 550. In other words, when we make and sell 550 units, that will optimize our profit. We can now recalculate the selling price. We know the formula for our selling price, and we know that relationship between the number of units and the price. So, in this instance go to substitute inter original selling price formula if you can remember that, 150- 10x, where x represents 550. In other words, the selling price is 150- 55, or 95 pounds. The optimum selling price is 95 pounds, and at that selling price, we will sell 550 units. [MUSIC]