[MUSIC PLAYING] This time we're going to talk about profit maximization when we're considering the level of inputs to use. So this is a really very powerful use of the production function concept. Because the production function helps us to determine the best level of a production input, such as a fertilizer or a herbicide or water. So we saw in the last segment about the relationship between inputs and outputs being captured by a production function and production functions having a reasonably characteristic shape. In this graph you can see two different production functions for the same crop-- it's wheat in both cases-- but growing on two different soil types. So it illustrates-- and again, it's nitrogen fertilizer-- so again, it's a relationship between the inputs and outputs. But you can see that the relationship is a little bit different depending on the circumstances. And in this case, it's different between two different soil types, a sandy soil or a limey soil. Production's a little bit more responsive, a bit steeper on a sandy soil because it has less nutrients to start with. So that previous graph was the relationship between fertilizer rates and yield. This graph is the relationship between fertilizer rate and revenue. So what I've done to produce this graph is multiply the numbers in the previous graph by the price, the sale price, of the wheat so that you can calculate the amount of revenue that would be received for each level of fertilizer. So it has the same shape. It's just at a different level. And the axis on the left-hand side now is revenue. But broadly it's a very similar looking graph. Next, we need to consider, if we're trying to determine the optimal level of this input-- nitrogen fertilizer, in this case-- we need to worry about how much that's going to cost. So we're going to plot on this same graph a variable cost curve. So the variable cost function is simply the quantity of fertilizer multiplied by the fertilizer price. And naturally enough, that's a straight line. The more fertilizer we put on, the more it's going to cost us. And it increases in a linear way like that. So the left-hand axis now is fertilizer cost. The bottom axis is, again, nitrogen fertilizer rate. And now I want to bring those two things together. The profit is simply the revenue function minus the cost function. And we're going to determine the optimal level of the input by looking for the fertilizer level which has the greatest difference between revenue and cost. Where does the revenue exceed the cost by the greatest amount? Or where's the biggest gap between the two curves? So there's the two curves, the two revenue curves, for the two different soil types. And the cost function will be the same in each case. The fertilizer costs the same no matter which soil type you apply it to. But you can see the two dotted lines indicate that the optimal fertilizer rate is different on these two different soil types. On the limey soil, which has the higher and less steep response function, production function, at the top, you can see that the optimal level of fertilizer is a bit lower. Whereas for the sandy soil, which is a bit more responsive to fertilizer, the optimal fertilizer rate is understandably a bit higher. So this is a really helpful concept. Depending on the relationship between inputs and outputs and the cost of the fertilizer, you can calculate which is the optimal fertilizer rate. And if you eyeball those two different cases, you can see that the optimal fertilizer rate occurs where the slope of the cost function is exactly the same as the slope of the revenue function. So think about that. The two slopes are identical. At those points where the dashed lines are, if you run your eyes up and down, you can see that the slope of the revenue function is the same as the slope of the cost function. And because the slopes of the curves are different for the two production functions, the point where the slopes are equal with the cost function is also different. Now in this case, I've just looked at one of these production functions or revenue functions, the one for sandy soil. And I've calculated the difference between the revenue and the cost. And that's the new curve in the middle, the purple curve, which is labeled profit. So this is just the difference. So you can find out the optimal level of fertilizer just by finding the point of this curve, which is as high as possible, which is the peak of the hill. And it's the same as we saw in the previous graph. You can see it's the point where the dashed line is there. And if we look at the profit curves for both of our soil types, you can see, again, we've got the-- the peak of the curve is at two different places. It's the same as we saw when we looked at the revenue curve and the cost curve. Just this time we've only shown the curve that is the difference between those two curves in each case. So there's various ways you can identify the optimal herbicide rate or the optimal fertilizer rate or the optimal water rate, application rate. One way is just to judge it from looking at the graph in the way that we were doing just then. Another is to use calculus to calculate a function for the maximum point. And a third way is to calculate profit numerically for a range of different input levels. So just do the maths. Calculate it. So for example, if we assume that the wheat cost is $250 a ton. And the fertilizer price is $1.90 a kilogram. And we have a function for the yield and how it changes in response to fertilize, f, which is that function down on the bottom there. Then I can calculate the profit for any particular level of fertilizer that I'm interested in. So that's what I've done here. If you look across the columns, I've got different fertilizer rates. I've got the wheat yield. I've got the revenue, then the cost, and then the profit. Profit is last column. So you can see that the-- looking down the column of the profit figures, I can identify which of those profit figures is the highest, the one that's red. And then look across to the left to see the level of fertilizer that corresponds to that highest level of profit. So this is a fairly simple calculation to do. You could do it in a spreadsheet. The hardest part is knowing what the function is for the relationship between fertilizer and yield. So in summary, the revenue function-- or the production function multiplied by the output price gives you the revenue function. And profit is simply the revenue minus the cost. And we can find the level of an input that maximizes profit by seeing where the difference between the revenue and the cost is the greatest.