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Unformatted text preview: Lecture 11 - Business and Economics Optimization Problems and Asymptotes 11.1 More Economics Applications Price Elasticity of Demand One way economists measure the responsiveness of consumers to a change in the price of a product is with what is called price elasticity of demand . For example, changing the price on vegetables usually strongly affects the demand while changing the price of milk or water doesnt affect that demand that much. Definition 11.1 If p ( x ) is a differentiable demand function, then the price elasticity of demand is given by = p ( x ) /x dp/dx where the lowercase Greek letter eta. For a given price, the demand is said to be elastic if | | > 1 and the demand is said to be inelastic if | | < 1 . The demand is unit elasticity if | | = 1 . The Price Elasticity of Demand measures the ratio of the percentage change of de- mand of a product to the percentage change of price of the product. If demand is elastic ( | | > 1), the percentage increase in demand is greater than the percentage increase in price, and hence the demand is sensitive to changes in price. Likewise if demand is inelastic( | | < 1), the percentage increase in demand is less than the percentage increase in price, and hence the demand is insensitive to changes in price. When elasticity is equal to 1, the percentage changes are roughly equal. For example, rough values of elasticity for some common commodities are: Tomatoes = 4 . 60 Automobiles = 1 . 35 Housing = 1 . 00 Mail = 0 . 05 Examples: 1. Let the demand function for a product be modelled by p ( x ) = 21- 3 2 x . Find the price elasticity of demand when x = 36 and x = 400. solution: Here dp dx =- 3 4 x . So for x = 36: p (36) = 21- 3 2 36 = 12 p (36) =- 3 4 36 =- 1 8 Thus = 12 / 36- 1 / 8 =- 8 3 Here | | > 1 so the demand is elastic. For x = 400: p (400) = 21- 3 2 400 =- 9 p (400) =- 3 4 400 =- 3 80 Thus =- 9 / 400- 3 / 80 = 3 5 Here | | < 1 so the demand is inelastic. 2. Suppose the demand function for ice cream bars is given by p ( x ) = 8- 2 x . For what values of x do ice cream bars have unit elasticity?...
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- Fall '09