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Economics 202A Final Exam Answers Fall Semester 2007 1.(a) The Hamiltonian for this problem is H = u ( c ) + ° ( y + ra ° c ) : The °rst-order conditions are @H @c = u 0 ( c ) ° ° = 0 ; _ ° = °± ° @H @a = ° ( ± ° r ) : 0 = lim t !1 e ° °t ° ( t ) a ( t ) : (b) Since now u 0 ( c ) = c ° 1 ; we can write the last equation as u 00 ( c )_ c = u 0 ( c ) ( ± ° r ) ; or as _ c c = ° u 0 ( c ) cu 00 ( c ) ( ± ° r ) = ² ( r ° ± ) : (c) We need to solve the equation a (0) = Z 1 0 h c (0)e ± ( r ° ° ) t ° y i e ° rt d t for the initial (optimal) consumption level, c (0) : The solution is c (0) = a (0) + ( y=r ) R 1 0 [e ± ( r ° ° ) t ] e ° rt d t = a (0) + ( y=r ) R 1 0 e ± ( r ° ° ) t ° rt d t = a (0) + ( y=r ) [ ² ( r ° ± ) ° r ] ° 1 n e [ ± ( r ° ° ) ° r ] t j 1 0 o = [ ²± ° ( ² ° 1) r ] [ a (0) + ( y=r )] : The assumption that ( ² ° 1) r ° ²± = ² ( r ° ± ) ° r < 0 ensures that above, lim t !1 e [ ± ( r ° ° ) ° r ] t = 0. (d) Looking at the preceding consumption function, we see the three ways a rise in the interest rate r will a±ect saving: 1

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1. The marginal propensity to consume out of total wealth is ²± ° ( ² ° 1) r: When r rises, that coe²cient falls with an e±ect proportional to ² . This is the substitution e±ect. 2. The substitution e±ect is counteracted by an e±ect proportional to unity that tends to make ²± ° ( ² ° 1) r rise when r rises. This is the income e±ect. The coe²cient ² °
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