ProblemSet1-2009

ProblemSet1-2009 - MRS and the Marshallian demand functions...

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Econ 6202, Fall 2009 Dmitry Shapiro Problem Set 1 Due Tuesday September 8 1. In the class it was stated that utility functions u and v represent the same preference relation if and only if there exists a strictly increasing function g such that v = g ( u ). (a) Prove this statement. Make sure to prove both if and only if parts. (b) Using the statement above prove that indifference curves do not depend on a particular functional form of utility function. Explicitly show where in your reasoning you use the fact that g is strictly increasing function. (c) Using the statement above prove that the MRS does not depend on a particular functional form of utility function. Explicitly show where in your reasoning you use the fact that g is strictly increasing function. 2. Consider the following utility functions: u ( x 1 ,x 2 ) = x 1 x 2 and v ( x 1 ,x 2 ) = ln( x 1 ) + ln( x 2 ) . Verify that u and v have the same indifference curves and the same MRS . Explain why. 3. Graph an indifference curve, and compute the
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Unformatted text preview: MRS and the Marshallian demand functions for the following utility functions: (a) Perfect substitutes u ( x 1 ,x 2 ) = αx 1 + βx 2 , where α > 0 and β > 0; (b) Perfect complements: u ( x 1 ,x 2 ) = min { αx 1 ,βx 2 } , where α > 0 and β > 0. 4. (JR 1.21) We have noted that u ( x ) is invariant to positive monotonic transformation. One com-mon transformation is the logarithmic transform, ln( u ( x )). Take the logarithmic transform of the Cobb-Douglas utility function; then using that as the utility function, derive the Marshallian demand functions and verify that they are identical to those derived in class. 5. (JR 1.27). A consumer of two goods faces positive prices and has a positive income. Her utility function is u ( x 1 ,x 2 ) = max { ax 1 ,ax 2 } + min { x 1 ,x 2 } , where 0 < a < 1 . Derive the Marshallian demand functions. 1...
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This note was uploaded on 10/20/2010 for the course ECON 6202 taught by Professor Shapiro during the Fall '09 term at UNC Charlotte.

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