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The optimum rate of an input doesn't only

depend on specific issues like plant nutrition.

It also depends on more general factors like the output price,

the input cost, and crop yield.

And these things, of course, are not fixed from year to year.

So as prices or costs or yields vary over time,

that will cause the optimal input rate to also vary.

So let's look at an example, and we'll

look at the wheat and nitrogen fertilizer example

that we've been using.

So here is the original revenue function and cost function

for different fertilizer rates showing

that the point where revenue exceeds

the cost by the greatest amount is at 80 kilograms per hectare

for this particular example.

And here's the profit function for the same example,

again, showing 80 kilograms per hectare

as being the point where the profit is at its maximum level.

Now let's look at an increased price

of the output, wheat in this case.

So in the original example, the price of wheat

was $250 per ton.

We're going to increase it now to $300 per ton.

And the result of that, the consequence,

is that the optimal fertilizer rate increases

from 80 to 90 kilograms per hectare.

You can see that in this figure there

are two revenue curves, the original one

for a price of $250 per ton and the new one

for a price of $300 per ton.

And because the price is multiplied

by the production, or the yield, the revenue

goes up by more at high yields.

And that means that the revenue curve tends to be lifted more

on the right-hand side, and that causes

the optimal level of fertilizer to increase, in this case

from 80 to 90 kilograms per hectare.

And here are the two profit functions for the same example.

And again, you can see that the peak of the profit function

was at 80, and now it's at 90 kilograms per hectare.

So there's one example where a change in the output price

causes a change in the input level, that

is, optimum or profit maximizing.

Now let's look at a different example.

This is one where the yield is much lower than we originally

expected.

So suppose that we're able to predict that a drought will

occur and that the yield of wheat,

instead of being three tons, which we normally expect

in this environment, is only 1.2 tons per hectare.

We'll stick with the original price of $250 per ton.

And the result of this lower yield, 1.2 instead of 3,

is that the optimal fertilizer rate

falls from 80 kilograms per hectare

down to 30 kilograms per hectare.

It's quite a large fall in the optimal fertilizer rate

because it's a really large fall in the yield.

You can see in the graph that because of this big reduction

in the potential yield, the revenue curve

has fallen quite dramatically.

The top curve there is for a yield of 3 tons per hectare.

The bottom curve is for a yield of only 1.2 tons per hectare.

They're both for the same wheat price of $250 per ton.

And now the two dashed lines, which

show the optimal fertilizer rates, are quite far apart.

It was originally 80 kilograms per hectare, as you know.

And now it's only 30 kilograms per hectare

where the revenue curve exceeds the cost curve by the greatest

amount.

And that's even easier to see on the profit curves.

The original profit curve, as you can see,

the top one has an optimal fertilizer rate

of 80 kilograms per hectare, and the bottom one

has an optimal rate of only 30.