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From Wikipedia
In economics, profit maximization is the (short run) 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 costs and variable costs. 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
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 profitmaximizing 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. First, the profit curve is at its maximum at this point (A). Secondly, at the point (B) the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), meaning that the surplus of revenue net of costs (B,C) is at its 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 profitmaximizing output is the optimum price to sell the product.
Marginal revenuemarginal cost method
An alternative argument says that for each unit sold, marginal profit (MÏ€) equals marginal revenue (MR) minus marginal cost (MC). 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. If there are two points where this occurs, maximum profit is achieved where the producer has collected positive profit up until the intersection of MR and MC (where zero profit is collected), but would not continue to after, as opposed to vice versa, which represents a profit minimum. In calculus terms, the correct intersection of MC and MR will occur when:
 \frac{dMR}{dQ} < \frac{dMC}{dQ}
The intersection of MR and 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 profit 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 noncompetitive 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 noncompetitive environment, more complicated profit maximization solutions involve the use of game theory.
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 MÏ€ = 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. 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 \Pi_max conditions by maximizing revenues.
Changes in total 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
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Answers:This is an optimization problem using linear programming where your objective function is to Maximize profits, P = 4A + 6B subject to the following constraints A + 3B 240 3A + 4B 360 2A + B 180 I will leave it up to you to iterate the above. Simply follow the steps in the linear programming iteration (I am sure that any algebra textbook has this). Hope this helps.
Answers:P'(x) =3x^2 +27x 60 =0 x^2 9x +20=0 (x4)(x5) =0 x=4,5 P(5) = answer
Answers:let the no. of gadgets be x and the no. of widgets be y the equations will be 2x + 3y = 1200 2x + y = 800 Profit function is 3x + 2y = Z
Answers:Cons  The cost, which reduces profit margins Pros  Ecological responsibility
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