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HW5_SOL - Economics 314 Suggested Solutions to HW 5 TA Yan...

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Economics 314: Suggested Solutions to HW 5 TA: Yan Li March 10, 2009 1 Case 1, Handout 2 From the budget constraint, c 1 = ° 0 ( ° 1 ° c 0 ) : U ( c 0 ; c 1 ) = ±c 0 + (1 ° ± ) c 1 ; 0 ± ± ± 1 Plugging in the c 1 from the budget constraint, we get a function of only c 0 given by: f ( c 0 ) = [ ± ° (1 ° ± ) ° 0 ] c 0 + (1 ° ± ) ° 0 ° 1 1. [(i)] 2. If ± ° (1 ° ± ) ° 0 > , then consuming more c 0 increases utility. So we would want to consume the maximum possible c 0 ; which is obtained when we plug in c 1 = 0 in the budget constraint. This gives c 0 = ° 0 : Solution: ( c 0 ; c 1 ) = ( ° 1 ; 0) 3. If ± ° (1 ° ± ) ° 0 < , consuming more c 0 decreases utility. So c 0 = 0 : This gives c 1 = ° 0 ° 1 : Solution ( c 0 ; c 1 ) = (0 ; ° 0 ° 1 ) : 4. If ± ° (1 ° ± ) ° 0 = , then any ( c 0 ; c 1 ) is optimal subject to the budget constraint. 1
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2 Case 2, Handout 2 f ( c 0 ; c 1 ) = ± ln c 0 + (1 ° ± ) ln( ° 0 : ( ° 1 ° c 0 )) Here because of log functions, we can rule out corner solutions, so c 0 is optimal if f 0 ( c 0 ) = 0 : f 0 ( c 0 ) = 0 = ) ± c 0 ° 1 ° ± ° 1 ° c 0 = 0 Solving for the equation gives: c 0 = ±° 1 = ) c 1 = (1 ° ± ) ° 0 ° 1 3 Case 3, Handout 2 The equation for f 0 ( c 0 ) = 0
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