Marginal analysis 

A baseball team plays in a stadium that holds 80,000 spectators. With ticket prices at $12, the average attendance had been 42,000. When ticket prices were lowered to $9, the average attendance rose to 54,000.
(i) Find the demand function, assuming that it is linear.
(ii) How should ticket prices be set to maximize revenue? 
Sol:
(i) Let p(x) be the demand function. Then, 12 = p(42000) and 9 = p(54000). Since 'p' is assumed to be linear, this gives a slope of:
Thus, using the pointslope form of the equation of a line, we have:
p(x) 
= 
(x – 42000) + 12 

= 
x + 22.5 
(ii) The revenue function is:
R(x) 
= 
x.p(x) 

= 
(22.5)x – 
Since R'(x) = 22.5 – , we see that R'(x) = 0 when x = 45000. This value of 'x' gives an absolute maximum by the first derivative test. The corresponding price is p(45000) = $11.25. Therefore, to maximize revenue, ticket prices should be set at $11.25. 
Applications to Economics
Cost function: The total cost 'C' of producing and marketing 'x' units of a product depends upon the number of units (x). So the function relating 'C' and 'x' is called "Cost function" and is written as "C = C(x)". The total cost of producing 'x' units of the product consists of two parts: (i) fixed cost and (ii) variable cost.
 Fixed cost: The fixed cost consists of all types of costs which do not change with the level of production. Ex: salaries, rent, insurance, utilities, etc.
 Variable cost: The variable cost is the sum of all costs that are dependent on the level of production. Ex: the cost of material, cost of packaging, etc.
Demand function: An equation that relates price per unit and quantity demanded at that price is called a demand function or price function. If 'p' is the price per unit of a certain product and 'x' is the number of units demanded, then 'p(x)' is called the demand function or price function.
Revenue function: If 'x' is the number of units of certain product sold at a rate of Rs. 'p' per unit, then the amount derived from the sale of 'x' units of a product is the total revenue. Thus, if 'R' represents the total revenue from 'x' units of the product at the rate of Rs. 'p' per unit, then R = [x.p] is the total revenue. Thus, if R(x) is the revenue function, then R(x) = x . p(x).
Profit function: The profit is calculated by subtracting the total cost from the total revenue obtained by selling 'x' units of a product. Thus, if P(x) is the profit function, then P(x) = R(x) – C(x).
Average cost function: Let C = C(x) be the total cost of producing and selling 'x' units of a certain commodity, then the average cost (AC) is the cost per unit, that is, AC = . Thus, if c(x) is the average cost function, then c(x) = .
Marginal cost function: Let C = C(x) be the total cost of producing 'x' units of a certain product, then the marginal cost (MC) is the rate of change of C with respect to 'x'. In other words, the marginal cost function is the derivative of the cost function C(x). Thus, MC = .
Average revenue function: If R is the total revenue obtained by selling 'x' units of a certain commodity at a price 'p' per unit, then the average revenue (AR) is the revenue per unit, that is, AR = .
Marginal revenue function: If R is the total revenue obtained by selling 'x' units of a certain commodity at a price 'p' per unit, then the marginal revenue (MR) is the rate of change of total revenue with respect to the quantity demanded (x). In other words, the marginal revenue function is the derivative, R'(x), of the revenue function R(x). Thus, MR = .
Points to remember 
• If the average cost is minimum, then marginal cost = average cost. 
• If the profit is a maximum, then marginal revenue = marginal cost. 