Profit maximization

In economics, profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem. The total revenue–total cost method relies on the fact that profit equals revenue minus cost, and the marginal revenue–marginal cost method is based on the fact that total profit in a perfectly competitive market reaches its maximum point where marginal revenue equals marginal cost.

Basic definitions

Any costs incurred by a firm may be classed into two groups: fixed cost and variable cost. Fixed costs are incurred by the business at any level of output, including zero output. These may include equipment maintenance, rent, wages, and general upkeep. Variable costs change with the level of output, increasing as more product is generated. Materials consumed during production often have the largest impact on this category. Fixed cost and variable cost, combined, equal total cost.

Revenue is the amount of money that a company receives from its normal business activities, usually from the sale of goods and services (as opposed to monies from security sales such as equity shares or debt issuances).

Marginal cost and revenue, depending on whether the calculus approach is taken or not, are defined as either the change in cost or revenue as each additional unit is produced, or the derivative of cost or revenue with respect to quantity output. It may also be defined as the addition to total cost or revenue as output increase by a single unit. For instance, taking the first definition, if it costs a firm 400 USD to produce 5 units and 480 USD to produce 6, the marginal cost of the sixth unit is approximately 80 dollars, although this is more accurately stated as the marginal cost of the 5.5th unit due to linear interpolation. Calculus is capable of providing more accurate answers if regression equations can be provided.

Total revenue - total cost method

Profit Maximization - The Totals Approach

To obtain the profit maximising output quantity, we start by recognizing that profit is equal to total revenue (TR) minus total cost (TC). Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph. Finding the profit-maximizing output is as simple as finding the output at which profit reaches its maximum. That is represented by output Q in the diagram.

There are two graphical ways of determining that Q is optimal. Firstly, we see that the profit curve is at its maximum at this point (A). Secondly, we see that at the point (B) that the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), the surplus of revenue net of costs (B,C) is the greatest. Because total revenue minus total costs is equal to profit, the line segment C,B is equal in length to the line segment A,Q.

Computing the price at which to sell the product requires knowledge of the firm's demand curve. The price at which quantity demanded equals profit-maximizing output is the optimum price to sell the product.

Mathematical example:

TR = 120Q - 0.5Q²

TC = 420 +60Q + Q2

∏ = TR - TC

∏ = 120Q - 0.5Q² - (420 +60Q + Q2)

∏ = 120Q - 0.5Q² - 420 - 60Q - Q2)

∏ = - 420 + 120Q - 60Q- 0.5Q² - Q2

∏ = - 420 + 60Q - 1.5Q²

Profit is maximized when marginal profit equals zero. Marginal profit is the first derivative of the total profit function,

M ∏ = 0

M ∏ = 60 - 3Q

Q = 20

Marginal cost-marginal revenue method

Profit Maximization - The Marginal Approach

If total revenue and total cost figures are difficult to procure, this method may also be used. For each unit sold, marginal profit equals marginal revenue minus marginal cost. Then, if marginal revenue is greater than marginal cost, marginal profit is positive, and if marginal revenue is less than marginal cost, marginal profit is negative. When marginal revenue equals marginal cost, marginal profit is zero. Since total profit increases when marginal profit is positive and total profit decreases when marginal profit is negative, it must reach a maximum where marginal profit is zero - or where marginal cost equals marginal revenue. This is because the producer has collected positive profit up until the intersection of MR and MC (where zero profit is collected and any further production will result in negative marginal profit, because MC will be larger than MR). The intersection of marginal revenue (MR) with marginal cost (MC) is shown in the next diagram as point A. If the industry is perfectly competitive (as is assumed in the diagram), the firm faces a demand curve (D) that is identical to its Marginal revenue curve (MR), and this is a horizontal line at a price determined by industry supply and demand. Average total costs are represented by curve ATC. Total economic profits are represented by area P,A,B,C. The optimum quantity (Q) is the same as the optimum quantity (Q) in the first diagram.

If the firm is operating in a non-competitive market, minor changes would have to be made to the diagrams. For example, the Marginal Revenue would have a negative gradient, due to the overall market demand curve. In a non-competitive environment, more complicated profit maximization solutions involve the use of game theory.

Example to substantiate MR=MC maximizes Profit

MR = $25, MC = $10 and Total Profit TP = $1000.

On producing one more unit, MR now decreases to $21 and MC rises to $11. It made $10 more profit than it was making before, which is $1010. Production of another unit will make $8 more profit again, which is $1018. Subsequent unit will fetch $5, so total profit increases to $1023.

This continues until MR is equal to MC, at which point profit is maximizing, and hence marginal profit is zero. Say on producing the next unit, MR = $15 and MC = $15. So TP will remain $1023, which is the maximum profit.

If MR > MC, it means that the business could increase total profit by MR-MC on production of an extra unit. Thus the business should increase production until MR=MC, where the total profit is maximized.

Maximizing revenue method

In some cases a firm's demand and cost conditions are such that marginal profits are greater than zero for all levels of production. In this case the Mπ = 0 rule has to be modified and the firm should maximize revenue. In other words the profit maximizing quantity and price can be determined by setting marginal revenue equal to zero. Marginal revenue equals zero when the marginal revenue curve has reached its maximum value = topped out. An example would be a scheduled airline flight. The marginal costs of flying the route are negligible. The airline would maximize profits by filling all the seats. The airline would determine the p-max conditions by maximizing revenues.

Numerical Example

A promoter decides to rent an arena for concert. The arena seats 20,000. The rental fee is 10,000. (This is a fixed cost.) The arena owner gets concessions and parking and pays all other expenses related to the concert. The promoter has properly estimated the demand for concert seats to be Q = 40,000 - 2000P, where Q is the quantity of seats and P is the price per seat. What is the profit maximizing ticket price?

Because the promoter’s marginal costs are zero the promoter maximizes profits by charging a ticket price that will maximize revenue. Total revenue equals price, P, times quantity. Total revenue is expressed as a function of quantity, so we need to work with the inverse demand curve:

$P(Q)=20-Q/2000$

This gives total revenue as a function of quantity, TR(Q) = P(Q) x Q, or

$TR(Q)=20Q-Q^{2}/2000$

Total revenue reaches its maximum value when marginal revenue is zero. Marginal revenue is the first derivative of the total revenue function: MR(Q)=TR'(Q). So

$MR(Q)=20-Q/1000$

Setting MR(Q) = 0 we get

$0=20-Q/1000$

$Q=20,000$

Recall that price is a function of quantity sold (the inverse demand curve. So to sell this quantity, the ticket price must be

$P(20000)=20-20,000/2,000=10$

It may seem more natural to view the decision as price setting rather than quantity setting. Generally this is not a more natural mathematical formulation of profit maximization, however, because costs are usually a function of quantity (not of price). In this particular example, however, the promoter’s marginal costs are zero. This means the promoter maximizes profits simply by charging a ticket price that will maximize revenue. In this particular case, we characterize total revenue as a function of price:

$TR2(P)=(40,000-2000P)P=40,000P-2000(P)2$

Total revenue reaches its maximum value when marginal revenue is zero. Marginal revenue is the first derivative of the total revenue function so

$MR2(P)=40,000-4000P$

Setting MR2 = 0 we get

$0=40,000-4000P$

$P=10$

Profit = TR2(P) -TC

Profit = [40,000P - 2000(P)2] - 10,000

Profit = [40,000(10) - 2000(10)2] - 10,000

Profit = 400,000 - 200,000 - 10,000

Profit = 190,000

What if the promoter had charged 12 per ticket?

Q = 40,000 - 2000P.

Q = 40,000 - 2000(12)

Q = 40,000 - 24,000 = 16,000 (tickets sold)

Profits at 12:

Q = 16,000(12) = 192,000 - 10,000 = 182,000

Changes in fixed costs and profit maximization

A firm maximizes profit by operating where marginal revenue equal marginal costs. A change in fixed costs has no effect on the profit maximizing output or price. The firm merely treats short term fixed costs as sunk costs and continues to operate as before. This can be confirmed graphically. Using the diagram illustrating the total cost total revenue method the firm maximizes profits at the point where the slope of the total cost line and total revenue line are equal. A change in total cost would cause the total cost curve to shift up by the amount of the change. There would be no effect on the total revenue curve or the shape of the total cost curve. Consequently, the profit maximizing point would remain the same. This point can also be illustrated using the diagram for the marginal revenue marginal cost method. A change is fixed cost would have no effect on the position or shape of these curves.

• What if the arena owner in the example above triples the fee for the next concert but all other factors are the same. What price should the promoter now charge for tickets in light of the fee increase?

The same price of $10. The fee is a fixed cost which the promoter should consider as a sunk cost and simply ignore it in calculating his profit maximizing price. The only effect is that the promoter’s profit will be reduced by $20,000.

Markup pricing

In addition to using methods to determine a firm’s optimal level of output, a firm can also set price to maximize profit. The optimal markup rules is:

(P - MC)/P = 1/ -Ep

or

P = (Ep/(1 + Ep)) MC

Where MC equals marginal costs and Ep equals price elasticity of demand. Ep is a negative number. Therefore, -Ep is a positive number.

In English the rule is that the size of the markup is inversely related to the price elasticity of demand for a good.

MPL, MRPL and profit maximization

The general rule is that firm maximizes profit by producing that quantity of output where marginal revenue equals marginal costs. The profit maximization issue can also be approached from the input side. That is, what is the profit maximizing usage of the variable input? To maximize profits the firm should increase usage "up to the point where the input’s marginal revenue product equals its marginal costs". So mathematically the profit maximizing rule is MRP_L = MC_L The marginal revenue product is the change in total revenue per unit change in the variable input assume labor. That is MRP_L = ∆TR/∆L. MRP_L is the product of marginal revenue and the marginal product of labor or MRP_L = MR x MP_L.

• Derivation:

MR = ∆TR/∆Q

MP_L = ∆Q/∆L

MRP_L = MR x MP_L = (∆TR/∆Q) x (∆Q/∆L) = ∆TR/∆L

See also

• Business organization
• Corporation
• Market forms
• Microeconomics
• Pricing
• Production, costs, and pricing
• Rational choice theory
• Supply and demand
• Marginal revenue
• Total revenue

External links

• Profit Maximization in Perfect Competition by Fiona Maclachlan, Wolfram Demonstrations Project.