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Unformatted text preview: © 2011 - Steven Tschantz Profit maximization Steven Tschantz 2/1/11 Problem A firm has a unique product it will sell to consumers at a uniform price. How should it set the price so as to maximize its profit? Model In our supply and demand model, we had many producers selling a product to many consumers. Producers sell to the con- sumers that offer them the highest price. Consumers buy from producers that will take the lowest price. No seller will accept significantly less than any other seller, and no buyer will pay much more than what other buyers are paying. Thus there will be a (nearly) uniform price for the product, which much be high enough so that sufficiently many producers are willing to sell, and low enough so sufficiently many consumers are willing to buy, such that the supply will just meet the demand. With only one producer, consumers can no longer shop for a lower price. Each consumer must strike a deal with the pro- ducer, and each could in principle get a different price, with the producer demanding higher prices from consumers with higher values for the product. In practice, individual price discrimination is likely to be difficult and costly. More often, the producer sets a single price. Consumers who value the product more than this price will buy, and those who value the product less will not, defining the consumer demand for the product as a function of the price as before. There is no supply function; the producer simply decides on the price they will ask. Suppose the firm sets the price at p . We describe the demand as a function of this variable, q H p L , the amount that can be sold at price p , say in each month. The firm should know how much raw material and labor it will take to produce a given quantity q , and so will have some idea of the cost C H q L of producing this much product. The profit of the firm is then P H p L = pq H p L- C H q H p LL each month. We assume that the firm will seek to maximize profit, concluding that the price charged will be determined by the demand and cost functions, q H p L and C H q L , and so predicting how price will be affected by changes in demand and cost. We have a simple model of how firms set prices, expressed as a simple profit maximization problem. Mathematically, we solve for the profit maximizing price using simple calculus. At a critical point, = ¶P H p L ¶ p = q H p L + p ¶ q H p L ¶ p- C ' H q H p LL ¶ q H p L ¶ p We can identify important factors in determining the best price. For example, C ' H q L is the change in cost per unit change in quantity, what economists call the marginal cost. We might rearrange the terms in this condition to conclude p- C ' H q H p LL p =- 1 Ε where the quantity on the left is the profit margin and Ε = ¶ q H p L ¶ p p q H p L is the own price elasticity of demand, the fractional change in demand as a multiple of the fractional change in price....
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This document was uploaded on 10/28/2011 for the course MATH 256 at Vanderbilt.
- Spring '11