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Unformatted text preview: Midterm Exam, Econ 210A, Fall 2008 1) Elmer Kink’s utility function is min { x 1 , 2 x 2 } . Draw a few indifference curves for Elmer. These are Lshaped, with the corners lying on the line x 1 = 2 x 2 . Find each of the following for Elmer: • His Marshallian demand function for each good. Elmer’s chosen bundle always lies on the point where his budget line meets a kink of the L. This means that his Marshallian demand satisfies the two equations x 1 = 2 x 2 and p 1 x 1 + p 2 x 2 = m . Solving these equations, we have (2 p 1 + p 2 ) x 2 = m and hence x 2 ( p 1 ,p 2 ,m ) = m/ (2 p 1 + p 2 ) . (1) Then x 1 ( p 1 ,p 2 ,m ) = 2 x 2 ( p 1 ,p 2 ,m ) = 2 m 2 p 1 + p 2 (2) • His Indirect utility function. His indirect utility function is v ( p,m ) = u ( x 1 ( p,m ) ,x 2 ( p,m )) = 2 m/ (2 p 1 + p 2 ) . (3) • His Hicksian demand function for each good. His Hicksian demand func tion for x i can be found by differentiating the expenditure function, which we solve for in the next question. In particular, this function is x 1 ( p,u ) = ∂e ( p 1 ,p 2 ,u ) ∂p 1 = u (4) and x 2 ( p,u ) = ∂e ( p 1 ,p 2 ,u ) ∂p 2 = u/ 2 (5) • His Expenditure function. His expenditure function is e ( p 1 ,p 2 ,u ) = (2 p 1 + p 2 ) u . We can find this by noting that v ( p,e ( p,u )) = u . Using the solution for v ( p,u ) found in Equation 3 we see that 2 e ( p,u ) 2 p 1 + p 2 = u. Rearranging terms, we have e ( p,u ) = ( p 1 + p 2 2 ) u. (6) 1 • Verify that Roy’s Law applies in Elmer’s case. Roy’s law requires that x i ( p,m ) = ∂v ( p 1 ,p 2 ,m ) ∂p i ÷ ∂v ( p 1 ,p 2 ,m ) ∂m . (7) In this case, ∂v ( p 1 ,p 2 ,m ) ∂p 1 = 2 m (2 p 1 + p 2 ) 2 ) (8) and ∂v ( p 1 ,p 2 ,m ) ∂m = 1 (2 p 1 + p 2 )) . (9) Substituting frow Equations 8 and 9 into Equation 7, we have x 1 ( p,m ) = 2 m 2 p 1 + p 2 (10) This coincides with the solution for x 1 ( p,m ) found in Equation 2. A similar argument shows that Roy’s law applies for Good 2.similar argument shows that Roy’s law applies for Good 2....
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
 Spring '10
 fallahi
 Utility

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