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answersmid11 - Midterm Examination: Economics 210A October...

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Midterm Examination: Economics 210A October 2011 The exam has 6 questions. Answer as many as you can. Good luck. 1) A) Must every quasi-concave function must be concave? If so, prove it. If not, provide a counterexample. (In all answers where you provide a counterexample, you must show that your example is really a counterexample.) Answer: Not every quasi-concave function is concave. Here is ac oun t e r ex amp l e . D eFn eth efun c t i on F with domain ° + (the positive real numbers.) such that F ( x )= x 2 .W eshowtw oth ing s : 1) This function is quasi-concave. 1 To see this, note that F is a strictly increasing function on ° + .The re fo rei f F ( y ) F ( x ), it must be that y x and hence for any t [0 , 1], ty +(1 t ) x x .S in c e F is an increasing function, it follows that F ( ty +(1 t ) x ) F ( x ) . Therefore F is quasi-concave. 2) The function F is not concave. To see this, note that F (2) = 4 and F (0) = 0, but F ( 1 2 2+ 1 2 0) = F (1) = 1 1 2 F (2) + 1 2 F (0) = 2 . This cannot be the case if F is a concave function. B) Must every concave function be quasi-concave? If so, prove it. If not, provide a counterexample. Answer: Every concave function is quasi-concave. Proof. If f is concave, its domain is a convex set A o ra l l x and y in A ,and t between 0 and 1, if f ( tx +(1 t ) y ) tf ( x )+(1 t ) f ( y ) . (1) ±rom the Expression ?? it follows that f ( tx +(1 t ) y ) f ( y )+ t ( f ( x ) f ( y )) . (2) If f ( x ) f ( y ), then it follows from Expression
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2) Let F and G be real-valued concave functions with the same domain, A . DeFne the function H so that for all x A , H ( x )= F ( x )+ G ( x ). Is H a concave function? If so, prove it. If not, provide a counterexample. Answer: Since F and G are concave functions with domain A ,it must be that if x A and y A ,thenfora l l t between 0 and 1, F ( tx +(1 t ) y ) tF ( x )+(1 t ) F ( y ) and G ( tx +(1 t ) y ) tG ( x )+(1 t ) G ( y ) Given these two inequalities, we see that H ( tx +(1 t ) y )= F ( tx +(1 t ) y )+ G ( tx +(1 t ) y ) tF ( x )+(1 t ) F ( y )+ tG ( x )+(1 t ) G ( y ) = t ( F ( x )+ G ( x )) + (1 t )( F ( y )+ G ( y )) = tH ( x )+(1 t ) H ( y ) By deFnition, the left side of Expression ?? equals H ( tx +(1 t ) y ). The right side of Expression ?? equals tF ( x )+ tG ( x )+(1 t ) F ( y )+ (1 t ) G ( y )= tH ( x )+(1 t ) HY ). Therefore H ( tx +(1 t ) y tH ( x )+(1 t ) H ( y ) for all t between 0 and 1, which means that
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answersmid11 - Midterm Examination: Economics 210A October...

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