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8.slutsky-answers - 98 SLUTSKY EQUATION(Ch 8 Chapter 8 NAME...

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Chapter 8 NAME Slutsky Equation Introduction. It is useful to think of a price change as having two dis- tinct effects, a substitution effect and an income effect. The substitution effect of a price change is the change that would have happened if in- come changed at the same time in such a way that the consumer could exactly afford her old consumption bundle. The rest of the change in the consumer’s demand is called the income effect . Why do we bother with breaking a real change into the sum of two hypothetical changes? Because we know things about the pieces that we wouldn’t know about the whole without taking it apart. In particular, we know that the substitution ef- fect of increasing the price of a good must reduce the demand for it. We also know that the income effect of an increase in the price of a good is equivalent to the effect of a loss of income. Therefore if the good whose price has risen is a normal good, then both the income and substitution effect operate to reduce demand. But if the good is an inferior good, income and substitution effects act in opposite directions. Example: A consumer has the utility function U ( x 1 , x 2 ) = x 1 x 2 and an income of $24. Initially the price of good 1 was $1 and the price of good 2 was $2. Then the price of good 2 rose to $3 and the price of good 1 stayed at $1. Using the methods you learned in Chapters 5 and 6, you will find that this consumer’s demand function for good 1 is D 1 ( p 1 , p 2 , m ) = m/ 2 p 1 and her demand function for good 2 is D 2 ( p 1 , p 2 , m ) = m/ 2 p 2 . Therefore initially she will demand 12 units of good 1 and 6 units of good 2. If, when the price of good 2 rose to $3, her income had changed enough so that she could exactly afford her old bundle, her new income would have to be (1 × 12)+(3 × 6) = $30. At an income of $30, at the new prices, she would demand D 2 (1 , 3 , 30) = 5 units of good 2. Before the change she bought 6 units of 2, so the substitution effect of the price change on her demand for good 2 is 5 6 = 1 units. Our consumer’s income didn’t really change. Her income stayed at $24. Her actual demand for good 2 after the price change was D 2 (1 , 3 , 24) = 4. The difference between what she actually demanded after the price change and what she would have demanded if her income had changed to let her just afford the old bundle is the income effect. In this case the income effect is 4 5 = 1 units of good 2. Notice that in this example, both the income effect and the substitution effect of the price increase worked to reduce the demand for good 2. When you have completed this workout, we hope that you will be able to do the following: Find Slutsky income effect and substitution effect of a specific price change if you know the demand function for a good.
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