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final09answers - Final Exam Economic 210A Fall 2009 Answer...

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Final Exam Economic 210A, Fall 2009 Answer any 7 questions. For a person with income m , let us define the compensating variation of a price change from price vector p to price vector p 0 to be the amount of additional income (positive or negative) that the person would have to be given to make him exactly as well off after the price change as before. Thus compensating variation is the solution CV to the equation v ( p 0 , m + CV ) = v ( p, m ) where v is the indirect utility function. Let us define equivalent variation to be the amount that someone who currently has income m and prices p would be willing to pay in order to avoid a price change such that the new price vector is p 0 and her income is m . Thus equivalent variation is the solution to the equation v ( p, m - EV ) = v ( p 0 , m ). In general, compensating variation and equivalent variation are not the same, but when preferences are quasi-linear, they are the same. Question 1. Consider a person who consumes two commodities x and y and has utility function u ( x, y ) = x + y - 1 2 y 2 . Let good x be the numeraire and consider price vectors of the form p = (1 , p y ) where p y is the price of good y . a. For what price-income combinations does this consumer choose to consume positive amounts of both goods. For price income combinations such that he consumes positive amounts of each good, write an equation for this person’s Marshallian demand function for each good. ANSWER He consumes a positive amount of good 1 if p y 1 and a positive amount of good 2 if m > p y (1 - p y ). If he consumes positive amounts of both goods, demand for y is 1 - p y and demand for x is m - p y (1 - p y ). b. Let v (1 , p y , m ) be this person’s indirect utility function at price vector (1 , p y ) and income m . Write an equation for v (1 , p y , m ) that applies at all price-income situations such that he chooses some of each good. Write your equation in as simple a form as possible. ANSWER v (1 , p y , m ) = m + 1 2 (1 - p y ) 2 . c. Suppose that this consumer initially consumes positive amounts of both goods at income m and prices (1 , p y ). The prices change to (1 , p 0 y ). Solve for the compensating variation of this price change. Solve for the equivalent variation of this price change. Show that compensating and equivalent variation are equal. ANSWER CV = 1 2 ( (1 - p y ) 2 - (1 - p 0 y ) 2 ) . EV = 1 2 ( (1 - p y ) 2 - (1 - p 0 y ) 2 ) . 1
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d. If p y = 1 / 2, p 0 y = 1 / 4, and m = 2, solve for the compensating variation of a change in prices from (1 , p y ) to (1 , p 0 y ). ANSWER -5/32 Question 2. Consider a person who consumes two commodities x and y and has utility function u ( x, y ) = x 1 / 2 y 1 / 2 . Let good x be the numeraire and consider price vectors of the form p = (1 , p y ) where p y is the price of good y .
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