Deriving_Demand_Functions_Examples

Deriving_Demand_Functions_Examples - Deriving Demand...

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Unformatted text preview: Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x 1 and x 2 , with respective prices p 1 and p 2 , and income m . 1 Perfect Substitutes For perfect substitutes, we have to look at respective prices. After all, if goods are per- fect substitutes, then the consumer is indifferent between them, and will have no problem adjusting consumption to get the good with the lowest price. 1.1 The basic case (1:1) For 1:1 perfect substitutes, the situation is about as plain as can be. Say p 1 > p 2 . The consumer will spend all their income on good 2. How do we know without doing any of that fancy math stuff? If the consumer is just as happy with a unit of good 1 as they are with a unit of good 2, and good 2 is less expensive, then they might as well use all their income on good 2 (they get more stuff that way). Similarly, if p 1 < p 2 , the consumer will choose only good 1. What if p 1 = p 2 ? Then any combination of good 1 and good 2 that uses all their budget is fine with them. So for each good, we have three possible demand functions depending on the prices. For example, demand for good 1 can be expressed as x 1 ( p 1 .p 2 ,m ) = 0 if p 1 > p 2 m p 1 if p 1 < p 2 Any ( x 1 , x 2 ) that satisfies p 1 x 1 + p 2 x 2 = m if p 1 = p 2 and similarly for good 2 (with the inequalities reversed, of course). 1.2 A more complicated example (2:3) Problem : Let the individual have a utility function u ( x 1 ,x 2 ) = 2 x 1 + 3 x 2 and an income of 120. They face prices p 1 = 2 and p 2 = 6. What is their demand for x 1 ? For x 2 ? 1 Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancy between this handout and lecture material, the lecture material should be considered the correct source. Despite all efforts, typos may find their way in - please read with a wary eye. Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2007 Solution : The easiest way to do this is to look at how much x 1 they can buy with all their income and how much x 2 they can buy with all their income, then see which gives the higher utility. If they spend all their money on x 1 , their utility is u ( x 1 , 0) = 2 * m p 1 = 2 * 120 2 = 120. If they spend all their income on x 2 , their utility is u (0 ,x 2 ) = 3 * m p 2 = 3 * 120 6 = 60. Since they get a higher utility from consuming only x 1 , their demand functions will be x 1 ( p 1 ,p 2 ,m ) = m p 1 and x 2 ( p 1 ,p 2 ,m ) = 0 We could also have solved this by figuring out the slope of the budget line relative to the slope of the indifference curves (i.e. the MRS). The slope of the budget line is- p 1 /p 2 =- 1 / 3, and the MRS is- 2 / 3. Since the absolute value of the price ratio is lower than the absolute value of the MRS at all point, we know the individual is never going to trade...
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This note was uploaded on 01/29/2012 for the course ECONOMICS 101 taught by Professor Tikk during the Spring '11 term at University of Toronto- Toronto.

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Deriving_Demand_Functions_Examples - Deriving Demand...

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