14127lec6_bndrat

14127lec6_bndrat - 14.127 Lecture 6 Xavier Gabaix March 11,...

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Unformatted text preview: 14.127 Lecture 6 Xavier Gabaix March 11, 2004 0.0.1 Shrouded attributes. A continuation • Rational guys U i = q − p + max ( V − p,V − e ) + σε i ¯ = q − p + V − min ( p,e ) + σε i = U i + σε i • Rational demand for good 1 D 1 = P U 1 > max U i = P ¯ i =2 ,...,n U 1 + σε 1 > max U i i =2 ,...,n ∗ • We look for symmetric equilibrium with ( p i ,p i ) = ( p ,p ∗ ) for i = 2 , ..., n . • Profits π 1 = D 1 p + p 1 p ≤ e • Assume c = 0 and c = 0 • Call α — fraction of rational guys. • Then ∗ π = α p + p 1 p ≤ e D ( A ) + (1 − α ) p + p 1 p ≤ V D ( − p + p ) where A = − p − min ( p,e ) + p ∗ + min ( p ∗ , e ) • The slope of profit ∂π = αD ( A ) − α p + p 1 p ≤ e D ′ ( A ) + (1 − α ) D ( A ) − (1 − α ) p + p 1 p ≤ ∂p ′ = αD (0) − α p + p 1 p ≤ e D (0) + (1 − α ) D (0) − (1 − α ) p + p 1 p ≤ V = D (0) − p + pQ D ′ (0) where Q denotes the fraction of consumers that buy the add-on. Q = α 1 p ≤ e + (1 − α ) 1 p ≤ V • Hence D (0) p + pQ = D ′ (0) = µ thus the total industry profits do not depend by the add-on structure ( α,V,e ). • Proposition. An optimum p ∈ { e,V } . • Proof. — If e < p < V , then it is better to set p = V . — If ≤ p < e , then it is better to increase p to e and lower p by e − p . Then rational demand stay the same, and irrational demand increases. QED • So, p ≥ e and min ( e,p ) = e , and π 1 = D ( − p + p ∗ ) α p + p 1 p ≤ e + (1 − α ) ( p + p ) = D ( − p + p ∗ ) p + p α 1 p ≤ e + 1 − α = D ( − p + p ∗ ) [ p + F ] • If p = V then F = V (1 − α ) . if p = e then F = e . Consequently: — p = V iff V (1 − α ) > e , or in other words, when e α < α ∗ = 1 − V — p = e iff e α > α ∗ = 1 − V • Take α < α ∗ . Then p = µ − pQ + c = µ − V α (1 − α ) + c So if µ = 0 then p < c and the base good is a loss leader. 1 Advertising. Does it solve the problem?...
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This note was uploaded on 12/26/2011 for the course ECON 14.127 taught by Professor Staff during the Fall '10 term at MIT.

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14127lec6_bndrat - 14.127 Lecture 6 Xavier Gabaix March 11,...

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