601-fin-10fa

# 601-fin-10fa - E 601 1 Final Examination Answer seven(7 of...

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E frqrplfv 601 1 Final Examination Answer seven (7) of the following nine (9) questions in the exam booklet provided. Number each question carefully so that the instructor can f nd your answer to any question easily. All questions are of equal weight. 1. You are given the following information about a consumer’s purchases. Goods 1and2aretheon lygoodsconsumed . Year 1 Year 2 Quantity Price Quantity Price Good 1 100 100 120 p 2 1 Good 2 100 100 x 2 2 80 For what values of p 2 1 will there exist values of x 2 2 that contradict the weak axiom of revealed preference? Justify your answer carefully. ANSWER In order that the data contradict WARP it must be that (a) the year 2 bundle could have been bought with year 1 wealth and prices and (b) the year 1 bundle could have been bought with year 2 wealth and prices. So (a) (100) (100) + (100) (100) (100) (120) + (100) x 2 2 or x 2 2 80 and (b) p 2 1 (120) + (80) x 2 2 p 2 1 (100) + (80) (100) or x 2 2 100 1 4 p 2 1 . The two ranges in (a) and (b) intersect only if 100 1 4 p 2 1 80 or 80 p 2

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E frqrplfv 601 2 2. Derive the pro f t function for the following one output, two input technology: y = s min ( z 1 , 2 z 2 ) . Do the associated output supply and factor demand functions have the expected properties? Explain. ANSWER When the f rm is maximizing pro f ts with this technology y 2 = z 1 =2 z 2 so Pro f ts py w 1 z 1 w 2 z 2 = py w 1 y 2 w 2 y 2 / 2 Setting dPro f ts /dy =0 obtain p 2 w 1 y w 2 y =0 or y ( p, w 1 .w 2 )= p 2 w 1 + w 2 And then z 1 ( p, w 1 .w 2 )= y ( p, w 1 .w 2 ) 2 = ± p 2 w 1 + w 2 ² 2 z 2 ( p, w 1 .w 2 )= 1 2 y ( p, w 1 .w 2 ) 2 = 1 2 ± p 2 w 1 + w 2 ² 2 π ( p, w 1 .w 2 )= p 2 2 w 1 + w 2 w 1 ± p 2 w 1 + w 2 ² 2 w 2 2 ± p 2 w 1 + w 2 ² 2 = p 2 2 w 1 + w 2 1 2 ± p 2 w 1 + w 2 ² 2 (2 w 1 + w 2 ) = 1 2 p 2 2 w 1 + w 2 Note that output supply and input demands are homogeneous of degree zero in ( p, w 1 .w 2 ) ;th ep r o f t function is homogeneous of degree one in ( p, w 1 .w 2 ) ;and dy/dp > 0 ,dz 1 /dw 1 < 0 , and dz 2 /dw 2
E frqrplfv 601 3 3. Answer both parts. (a) What does it mean to say that a utility function, u ( · ) , represents a preference relation on some choice set X ?P r o v et h a ti f u ( · ) represents preference relation ± , this preference relation must be complete and transitive. (b) Now suppose X = R 2 + and ( x 1 1 ,x 1 2 ) " ( x 2 1 ,x 2 2 ) when x 1 1 >x 2 1 , or x 1 1 = x 2 1 and x 1 2 >x 2 2 . Is this preference relation complete, transitive and continuous? Defend your answers carefully. ANSWER (a) A utility function u : X R represents a preference relation ± on the choice set X if x, y X, x ± y u ( x

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601-fin-10fa - E 601 1 Final Examination Answer seven(7 of...

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