Now let's move onto the revenue & profit functions.
Demand function or the price function
Firstly, let's assume that the price which some item can be sold at if there is a demand for x units is specified by p (x ) . This function is typically called either the demand function or the price function
Revenue function
Then the revenue function is how much money is made through selling x items and is,
R ( x ) = x p ( x )
The profit function is then,
P ( x )= R ( x ) - C ( x ) = x p ( x ) - C ( x )
Be careful to not confuse the demand function, p ( x ) - lower case p, & the profit function, P ( x ) - upper case P. Bad notation possibly, but there it is.
marginal revenue function
the marginal revenue function is R′ ( x ) and
Profit function
The marginal profit function is P′ ( x)
and these revel the revenue & profit respectively if one more unit is sold.
Let's take a quick look at an example of using these.
Example The weekly cost to generate x widgets is specified by
C ( x ) = 75, 000 + 100 x - 0.03x^{2} + 0.000004 x^{3 } 0 ≤ x ≤ 10000
and the demand function for the widgets is specified by,
p ( x ) = 200 - 0.005x 0 ≤ x ≤ 10000
Find out the marginal cost, marginal revenue & marginal profit while 2500 widgets are sold and while 7500 widgets are sold. Suppose that the company sells accurately what they produce.
Solution
The first thing we have to do is get all the several functions which we'll require. Following are the revenue & profit functions.
R ( x ) = x ( 200 - 0.005x ) =200 x - 0.005x^{2}
P ( x ) = 200x - 0.005x^{2} - (75, 000 + 100x - 0.03x^{2}+ 0.000004x^{3} )
= -75, 000 + 100 x + 0.025x^{2} - 0.000004 x^{3}
Now, all the marginal functions are following,
C′ ( x ) = 100 - 0.06 x + 0.000012 x^{2}
R′ ( x ) =200 - 0.01x
P′ ( x ) = 100 + 0.05x - 0.000012x^{2}
The marginal functions while 2500 widgets are sold are following,
C′ ( 2500) = 25 R′ ( 2500) = 175 P′ ( 2500) = 150
The marginal functions while 7500 are sold are following
C′ (7500) = 325 R′ (7500) = 125 P′ (7500) = -200
Therefore, upon producing & selling the 2501st widget it will cost the company approximately $25 to generate the widget and they will illustrates an added $175 in revenue and $150 in profit.
Alternatively while they generate and sell the 7501st widget it will cost an additional $325 and they will attain an extra $125 in revenue, however lose $200 in profit.