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Unformatted text preview: Econ 110 TA sections solutions January 14, 2009 1 Problem 1 Suppose the household in the twoperiod model has the utility function u ( c 1 ;c 2 ;l 1 ;l 2 ) = v ( c 1 ) + 1 1 + & v ( c 2 ) + ( l 1 ;l 2 ) where v ( c ) = c 1 & & & 1 1 & and ( l 1 ;l 2 ) is some unspeci&ed function of l 1 and l 2 . The constants & and are both assumed to be positive. (a) What is the households marginal rate of substitution between c 1 and c 2 ? MUC 1 = dv dc 1 = c & & 1 and MUC 2 = dv dc 2 = (1 + & ) = c & & 2 1+ . Hence the marginal rate of substitution is MUC 1 MUC 2 = c & & 1 c & & 2 = (1+ ) = (1 + & ) & c 2 c 1 & . (b) Assuming that the interest rate, R , is equal to & , how will c 1 compare to c 2 ? What is the growth rate of consumption between periods? We know that MUC 1 MUC 2 = 1+ R , so (1 + & ) & c 2 c 1 & = 1+ R . If T = & this means & c 2 c 1 & = 1 or c 2 = c 1 . In other words the growth rate of consumption is 0. (c) If the interest rate rises to some R > R what happens to the growth rate of consumption? How does your answer depend on the value of ? If R rises to R > R then (1 + & ) & c 2 c 1 & = 1 + R implying that c 2 c 1 & = 1 + R 1 + & > 1 + R 1 + & = 1 Hence c 2 c 1 = 1 + R 1 + & 1 & > 1 So consumption will grow at a positive rate because of the rise in interest rates. This makes sense. A rise in interest rates causes an intertemporal substitution e/ect which gives the household an incentive to reduce c 1 and raise c 2 . The 1 change in consumption is smaller the bigger...
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 Spring '08
 SchmittGrohe
 Macroeconomics, Utility

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