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6. Demand - Solutions

6. Demand - Solutions - Chapter 6 NAME Demand Introduction...

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Chapter 6 NAME Demand Introduction. In the previous chapter, you found the commodity bundle that a consumer with a given utility function would choose in a specific price-income situation. In this chapter, we take this idea a step further. We find demand functions , which tell us for any prices and income you might want to name, how much of each good a consumer would want. In general, the amount of each good demanded may depend not only on its own price, but also on the price of other goods and on income. Where there are two goods, we write demand functions for Goods 1 and 2 as x 1 ( p 1 , p 2 , m ) and x 2 ( p 1 , p 2 , m ). When the consumer is choosing positive amounts of all commodities and indifference curves have no kinks, the consumer chooses a point of tangency between her budget line and the highest indifference curve that it touches. Example: Consider a consumer with utility function U ( x 1 , x 2 ) = ( x 1 + 2)( x 2 + 10). To find x 1 ( p 1 , p 2 , m ) and x 2 ( p 1 , p 2 , m ), we need to find a commodity bundle ( x 1 , x 2 ) on her budget line at which her indifference curve is tangent to her budget line. The budget line will be tangent to the indifference curve at ( x 1 , x 2 ) if the price ratio equals the marginal rate of substitution. For this utility function, MU 1 ( x 1 , x 2 ) = x 2 + 10 and MU 2 ( x 1 , x 2 ) = x 1 + 2. Therefore the “tangency equation” is p 1 /p 2 = ( x 2 + 10) / ( x 1 + 2). Cross-multiplying the tangency equation, one finds p 1 x 1 + 2 p 1 = p 2 x 2 + 10 p 2 . The bundle chosen must also satisfy the budget equation, p 1 x 1 + p 2 x 2 = m . This gives us two linear equations in the two unknowns, x 1 and x 2 . You can solve these equations yourself, using high school algebra. You will find that the solution for the two “demand functions” is x 1 = m 2 p 1 + 10 p 2 2 p 1 x 2 = m + 2 p 1 10 p 2 2 p 2 . There is one thing left to worry about with the “demand functions” we just found. Notice that these expressions will be positive only if m 2 p 1 + 10 p 2 > 0 and m +2 p 1 10 p 2 > 0. If either of these expressions is negative, then it doesn’t make sense as a demand function. What happens in this For some utility functions, demand for a good may not be affected by all of these variables. For example, with Cobb-Douglas utility, demand for a good depends on the good’s own price and on income but not on the other good’s price. Still, there is no harm in writing demand for Good 1 as a function of p 1 , p 2 , and m . It just happens that the derivative of x 1 ( p 1 , p 2 , m ) with respect to p 2 is zero.
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68 DEMAND (Ch. 6) case is that the consumer will choose a “boundary solution” where she consumes only one good. At this point, her indifference curve will not be tangent to her budget line. When a consumer has kinks in her indifference curves, she may choose a bundle that is located at a kink. In the problems with kinks, you will be able to solve for the demand functions quite easily by looking at diagrams and doing a little algebra. Typically, instead of finding a tangency equation, you will find an equation that tells you “where the kinks are.”
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