Maximum Profit

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In an economic problem, it is important to distinguish between total quantities and unit quantities. Examples are the total cost of a production run, and the cost of one unit. Let's use variables like Tcost and Ucost to make these distinctions. At times we'll use an equation relating these types of variables.
For example, equation for total cost = number produced times the unit cost

equation for total cost = number produced times the unit cost

, where n is the number of units in the production run.
Consider a manufacturing operation in which there is a total setup cost,
TsetupCost, of $150 for a production run and a unit cost, Ucost, of $40. n units are produced in such a run.
The company has found that for this item, the number which can be sold (and therefore the size of a production run n) depends on the unit price, Uprice as follows:
relationship of number sold to unit price

relationship of number sold to unit price

Equation 1>
The company wishes to maximize the total profit, Tprofit, on this product. We are to help them.

First, relate the total profit to the total revenue, Trevenue, and total cost, TCost, for a production run.

total profit in terms of the total revenue and total cost

Equation 2

Now express how total revenue depends on the number sold, n, and the unit price, Uprice.

Total revenue = number of units * unit price:
total revenue in terms of the number of units and the unit price

Equation 3

Find how the total cost depends on the total setup cost and the unit cost of production.

total cost in terms of setup cost and unit cost
Total cost = total setup cost + number of units * unit cost

Equation 4

Combine equations 3 and 4 with equation 2.

We get

Equation 5

Solve equation 1 for Uprice and use that to simplify equation 5.

Using

(from equation 1)
We can rewrite equation 5 as
total profit in terms of only the total number produced

Equation 6

Now we have the total profit expressed in terms of one variable. How do we find the maximum?

We differentiate set the result to 0, and solve for the value which maximizes the profit.

Do that.

Equation 7

What do we do next?

We determine the maximum total profit by substituting n = 20 into equation 6.

We get

Let's try the second derivative test to check that this is a maximum. Find the second derivative by using equation 7.

What can we conclude?

Since the second derivative is negative, we do have a maximum.

As a further confirmation, plot equation 6 versus n. Then check your plot against the figure below.

graph of total profit vs number of units, showing the maximum for n = 20

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