prelims_Micro Prelim June 2005

prelims_Micro Prelim June 2005 - University of California...

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Unformatted text preview: University of California, Davis Date: June 27, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE- ================================================================================== Please answer four of the following five questions 1. Choice with status-quo bias. A consumer’s consumption set is X . Her choice depends on both her attainable (or budget) set B ⊂ X and on her status-quo point s ∈ B . She is endowed with both a utility function u : X → ℜ and a moving-cost function c : X → ℜ ++ , where c ( s ) is interpreted as the cost of leaving the status quo s . We restrict our attention to attainable sets B with the property that u has a unique maximizer on B , denoted x *( B ). At ( B , s ) (i. e., when facing the attainable set B from the status-quo point s ), she chooses (1) the status-quo point s if u ( s ) > u ( x *( B )) − c ( s ), (2) point x *( B ) if u ( s ) < u ( x *( B )) − c ( s ) , (3) either the status-quo point s or point x *( B ) if u ( s ) = u ( x *( B )) − c ( s ) . (Note that in this case her choice is not unique.) (a) Show that her choices satisfy the following property: if x ∈ B 1 ⊂ B is chosen at ( B , s ), then x is also chosen at ( B 1 , s ). Interpret this property in terms of revealed preference. (b) Show that there exists a function f that has both the consumption point x and the status-quo point s as vector arguments such that solving the problem max x f ( x , s ) subject to x ∈ B is equivalent to applying the choice rules (1) and (2). Interpret f . Is f continuous when X = ℜ N + and both u and c are continuous? (c) Let X = ℜ 2 + , u ( x 1 , x 2 ) = ( x 1 ) 0.5 ( x 2 ) 0.5 , and ( ω 1 , ω 2 ) = (2, 2). Consider budget sets of the form B [ p ] = {( x 1 , x 2 ) ∈ ℜ 2 + : x 1 + p x 2 < ω 1 + p ω 2 }, parameterized by the positive number p (understood as the normalized price of good 2). In this section, graphs can be approximate, but they should clearly indicate the ranges over which the curves are increasing or decreasing. (c.1) Assume that the consumer maximizes u in the usual manner, without status-quo bias, i.e., given p > 0, the consumer chooses ( x 1 , x 2 ) in order to maximize u ( x 1 , x 2 ) subject to ( x 1 , x 2 ) ∈ B [ p ]. Compute the solution function ( x 1 *( p ), x 2 *( p )) and the value function v ( p ) of this maximization problem. What is ( x 1 *(1), x 2 *(1))? What is v (1)? Separately graph x 2 *( p ) and v ( p ), with p on the horizontal axis. (c.2) Assume now that the consumer has a status-quo bias, as defined above, with status-quo point s = ( ω 1 , ω 2 ) = (2, 2), and with moving-cost function c ( x ) = 0.2. Obtain and graph the relationship between p and her choice of x 2 (again, with p on the horizontal axis)....
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This note was uploaded on 02/09/2010 for the course ECON 200D taught by Professor Pontusrendahl during the Winter '06 term at UC Davis.

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prelims_Micro Prelim June 2005 - University of California...

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