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Â This time we're going to talk about profit maximization

Â when we're considering the level of inputs to use.

Â So this is a really very powerful use

Â of the production function concept.

Â Because the production function helps

Â us to determine the best level of a production input,

Â such as a fertilizer or a herbicide or water.

Â So we saw in the last segment about the relationship

Â between inputs and outputs being captured by a production

Â function and production functions having

Â a reasonably characteristic shape.

Â In this graph you can see two different production functions

Â for the same crop-- it's wheat in both cases--

Â but growing on two different soil types.

Â So it illustrates-- and again, it's nitrogen fertilizer--

Â so again, it's a relationship between the inputs and outputs.

Â But you can see that the relationship is a little bit

Â different depending on the circumstances.

Â And in this case, it's different between two different soil

Â types, a sandy soil or a limey soil.

Â Production's a little bit more responsive, a bit steeper

Â on a sandy soil because it has less nutrients to start with.

Â So that previous graph was the relationship

Â between fertilizer rates and yield.

Â This graph is the relationship between fertilizer rate

Â and revenue.

Â So what I've done to produce this graph

Â is multiply the numbers in the previous graph

Â by the price, the sale price, of the wheat

Â so that you can calculate the amount of revenue that

Â would be received for each level of fertilizer.

Â So it has the same shape.

Â It's just at a different level.

Â And the axis on the left-hand side now is revenue.

Â But broadly it's a very similar looking graph.

Â Next, we need to consider, if we're

Â trying to determine the optimal level of this input-- nitrogen

Â fertilizer, in this case-- we need

Â to worry about how much that's going to cost.

Â So we're going to plot on this same graph a variable cost

Â curve.

Â So the variable cost function is simply

Â the quantity of fertilizer multiplied by the fertilizer

Â price.

Â And naturally enough, that's a straight line.

Â The more fertilizer we put on, the more it's going to cost us.

Â And it increases in a linear way like that.

Â So the left-hand axis now is fertilizer cost.

Â The bottom axis is, again, nitrogen fertilizer rate.

Â And now I want to bring those two things together.

Â The profit is simply the revenue function

Â minus the cost function.

Â And we're going to determine the optimal level of the input

Â by looking for the fertilizer level which has the greatest

Â difference between revenue and cost.

Â Where does the revenue exceed the cost

Â by the greatest amount?

Â Or where's the biggest gap between the two curves?

Â So there's the two curves, the two revenue curves,

Â for the two different soil types.

Â And the cost function will be the same in each case.

Â The fertilizer costs the same no matter

Â which soil type you apply it to.

Â But you can see the two dotted lines indicate

Â that the optimal fertilizer rate is different on these two

Â different soil types.

Â On the limey soil, which has the higher and less steep response

Â function, production function, at the top,

Â you can see that the optimal level of fertilizer

Â is a bit lower.

Â Whereas for the sandy soil, which

Â is a bit more responsive to fertilizer,

Â the optimal fertilizer rate is understandably a bit higher.

Â So this is a really helpful concept.

Â Depending on the relationship between inputs and outputs

Â and the cost of the fertilizer, you

Â can calculate which is the optimal fertilizer rate.

Â And if you eyeball those two different cases,

Â you can see that the optimal fertilizer rate occurs

Â where the slope of the cost function

Â is exactly the same as the slope of the revenue function.

Â So think about that.

Â The two slopes are identical.

Â At those points where the dashed lines are,

Â if you run your eyes up and down,

Â you can see that the slope of the revenue function

Â is the same as the slope of the cost function.

Â And because the slopes of the curves

Â are different for the two production functions,

Â the point where the slopes are equal with the cost function

Â is also different.

Â Now in this case, I've just looked

Â at one of these production functions or revenue

Â functions, the one for sandy soil.

Â And I've calculated the difference

Â between the revenue and the cost.

Â And that's the new curve in the middle, the purple curve,

Â which is labeled profit.

Â So this is just the difference.

Â So you can find out the optimal level of fertilizer

Â just by finding the point of this curve, which

Â is as high as possible, which is the peak of the hill.

Â And it's the same as we saw in the previous graph.

Â You can see it's the point where the dashed line is there.

Â And if we look at the profit curves for both of our soil

Â types, you can see, again, we've got the-- the peak of the curve

Â is at two different places.

Â It's the same as we saw when we looked

Â at the revenue curve and the cost curve.

Â Just this time we've only shown the curve

Â that is the difference between those two curves in each case.

Â So there's various ways you can identify the optimal herbicide

Â rate or the optimal fertilizer rate

Â or the optimal water rate, application rate.

Â One way is just to judge it from looking at the graph in the way

Â that we were doing just then.

Â Another is to use calculus to calculate

Â a function for the maximum point.

Â And a third way is to calculate profit numerically

Â for a range of different input levels.

Â So just do the maths.

Â Calculate it.

Â So for example, if we assume that the wheat cost

Â is $250 a ton.

Â And the fertilizer price is $1.90 a kilogram.

Â And we have a function for the yield

Â and how it changes in response to fertilize,

Â f, which is that function down on the bottom there.

Â Then I can calculate the profit for any particular level

Â of fertilizer that I'm interested in.

Â So that's what I've done here.

Â If you look across the columns, I've

Â got different fertilizer rates.

Â I've got the wheat yield.

Â I've got the revenue, then the cost, and then the profit.

Â Profit is last column.

Â So you can see that the-- looking down

Â the column of the profit figures,

Â I can identify which of those profit figures

Â is the highest, the one that's red.

Â And then look across to the left to see the level of fertilizer

Â that corresponds to that highest level of profit.

Â So this is a fairly simple calculation to do.

Â You could do it in a spreadsheet.

Â The hardest part is knowing what the function

Â is for the relationship between fertilizer and yield.

Â So in summary, the revenue function--

Â or the production function multiplied by the output price

Â gives you the revenue function.

Â And profit is simply the revenue minus the cost.

Â And we can find the level of an input that maximizes profit

Â by seeing where the difference between the revenue

Â and the cost is the greatest.

Â