401-feqa - E 401 1 Final Examination questions: Answers 1....

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E frqrplfv 401 1 Final Examination questions: Answers 1. The “Gorman form” means that the indirect utility function can be written as a + bw where a and b are functions of prices alone. Quasi-linear preferences do not necessarily f t the Gorman form. For example, if u ( x 1 ,x 2 )=2 x 1 / 2 1 + x 2 then x 1 x 2 V ( p 1 ,p 2 ,w ) Range ( p 2 /p 1 ) 2 w/p 2 p 2 /p 1 p 2 /p 1 + w/p 2 w p 2 2 /p 1 w/p 1 02 ( w/p 1 ) 1 / 2 w<p 2 2 /p 1 The f rst row conforms to the Gorman form but the second row doesn’t. When wealth is su ciently low quasilinear preferences may not f t the Gorman form. If preferences do f ttheGormanformandeveryonehasthesamepreferencesthen redistributions of wealth will not a f ect aggregate demands. To see this in the form of a two-good example write v j ( p 1 2 j )= a ( p 1 2 )+ b ( p 1 2 ) w j ,j = A,B. Using Roy’s identity for good 1 x A 1 ( p 1 2 A x B 1 ( p 1 2 B 2 a ( p 1 ,p 2 ) p 1 b ( p 1 2 ) b ( p 1 ,p 2 ) p 1 b ( p 1 2 ) ( w A + w B ) , from which the result is clear. 2. Using the de f nition of the pro f t function π ( p,w 1 2 Max z 1 0 ,z 2 0 p ( z 1 + z 2 ) 1 / 2 w 1 z 1 w 2 z 2 This yields y ( 1 2 ) ( z 1 + z 2 ) 1 / 2 z 1 ( 1 2 ) z 2 ( 1 2 ) π ( 1 2 ) Range p/ (2 w 1 )( p/ (2 w 1 )) 2 0 p 2 / (4 w 1 ) w 1 w 2 p/ (2 w 2 )0 ( p/ (2 w 2 )) 2 p 2 / (4 w 2 ) w 1 >w 2 Note that the standard properties hold here.
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E frqrplfv 401 2 Output supply and input demnds are homogeneous of degree 0 in prices The pro f t function is homogeneous of degree 1 in prices And y p 0 , z 1 w 1 0 , z 2 w 2 0 3. Write short-run total cost as C S ( q ) and long-run total cost as C L ( q ) . In C S ( q ) z 2 is f xed at z 2 and so C S ( q ) C L ( q ) and we know the two are equal at q . So q is a minimizer of g ( q ) C S ( q ) C L ( q ) . Thus g ± ( q )= C ± S ( q ) C ± L ( q )=0 g ±± ( q C ±± S ( q ) C ±± L ( q ) > 0 . So at q the short-run and long-run marginal costs are equal to each other and the short-run MC is steeper than the long-run MC. Given p ± >p , then, the q at p ± = C ± L ( q ) will exceed the q at p ± = C ± S ( q ) . Se the picture on the last page. 4. (a) A pro f t function is the value function for a price-taking competitive f rm, that is, π ( p,w 1 ,w 2 Max z 1 ,z 2 pf ( z 1 2 ) w 1 z 1 w 2 z 2 , where w i are input prices, p is output price and f ( z 1 2 ) is the production function.
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This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.

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401-feqa - E 401 1 Final Examination questions: Answers 1....

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