Rubinstein2005-page94 - ( x 1 , x 2 ) has the slope − √...

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October 21, 2005 12:18 master Sheet number 92 Page number 76 Problem Set 6 Problem 1. ( Easy ) In a world with two commodities, consider a consumer’s preferences that are represented by the utility function u ( x 1 , x 2 ) = min { x 1 , x 2 } . a. Calculate the consumer’s demand function. b. Verify that the preferences satisfy convexity. c. Calculate the indirect utility function v ( p , w ) . d. Verify Roy’s Equality. e. Calculate the expenditure function e ( p , u ) and verify Dual Roy’s Equality. Problem 2. ( Moderate ) Imagine that you are reading a paper in which the author uses the indirect utility function v ( p 1 , p 2 , w ) = w / p 1 + w / p 2 . You suspect that the author’s conclusions in the paper are the outcome of the “fact” that the function v is inconsistent with the model of the rational consumer. Take the following steps to make sure that this is not the case: a. Use Roy’s Equality to derive the demand function. b. Show that if demand is derived from a smooth utility function, then the indifference curve at the point
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Unformatted text preview: ( x 1 , x 2 ) has the slope − √ x 2 / √ x 1 . c. Construct a utility function with the property that the ratio of the partial derivatives at the bundle ( x 1 , x 2 ) is √ x 2 / √ x 1 . d. Calculate the indirect utility function derived from this utility func-tion. Do you arrive at the original v ( p 1 , p 2 , w ) ? If not, can the original indirect utility function still be derived from another utility function satisfying the property in (c). Problem 3. ( Moderate ) A consumer with wealth w is interested in purchasing only one unit of one of the items included in a (Fnite) set A . All items are indivisible . The consumer does not derive any “utility” from leftover wealth. The consumer evaluates commodity x ∈ A by the number V x (where the value of not purchasing any of the goods is 0). The price of commodity x ∈ A is p x > 0. a. ±ormulate the consumer’s problem....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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