Pay What You Want - A Self Image Perspective
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Transcript of Pay What You Want - A Self Image Perspective
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Working Paper 2014:6
Department of EconomicsSchool of Economics and Management
Pay-What-You-Want PricingSchemes: A Self-Image Perspective
Goytom Abraha KahsayMargaret Samahita
February 2014
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†
‡
∗
†
‡
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http://www.wired.com/listening_post/2008/10/radioheads-in-r/
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i
U i(c) = ui(x) − c
+ f i(c, r)
c, r ≥ 0.
x ui(x)
c
f
c
r
c f c > 0, f cc < 0
r
r
r
c
f r 0
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r = v c
r = p = c.
ui(x) ∈R+
j i
∂f j
∂c (c, r) >
∂f i
∂c (c, r) ∀c.
j i
r = v
maxc
U i(c) = ui(x) − c + f i(c, v).
∂f i
∂c (c, v) = 1.
c
c∗
j
i c∗i < c∗ j
v
v
i j v ∂f i∂c (v, v) < 1 ≤∂f j∂c (v, v) c
∗i < v ≤ c
∗ j
j
i
∂f j∂c
(c, v) > ∂f i∂c
(c, v) ∀ c
i
∂f j∂c
(c∗i , v) > ∂f i∂c
(c∗i , v) = 1 = ∂f j∂c
(c∗j , v) f c∗i < c
∗j
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i c∗i < v j
c∗ j ≥ v
U i(c∗i ) = ui(x)−c
∗i +f i(c
∗i , v)
ui(x)
c∗i v
ui(x) − c∗i
c∗i v
c∗i ui(x)
p ≥ v
p ≤ c∗i − f i(c∗i , v) (≤ v) ∀i
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v
p
c∗i p
p
v
v
f r < 0
∂c∗
∂v > 0
U i(c) = ui(x) − c + µi
(c − v) −
1
2(c − v)2
µi > 0; c < v + 1.
µ
µi
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c
c∗i = 1 + v − 1
µi.
ui(x) ∼ U (0, 2v)
µi ∼ U
1
1 + v, 1 +
v
1 + v
c∗i > 0 ∀i
1 ≤ v ≤ 10 p v + 1 v − 1
v
p = v + 1
v
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v
v
p
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c∗
v
p
v
v
v
v
v
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e e e e
v e e
http://www.nytimes.com/2010/05/21/us/21free.html?_r=0
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http://www.budgettravel.com/feature/0804_PayWhatYouCan,1342/http://www.budgettravel.com/feature/0804_PayWhatYouCan,1342/
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i ∂f i∂c (c∗i , v) = 1
∂f i
∂c (v, v) < 1 =
∂f i
∂c (c∗i , v).
c c∗i < v
j ∂f j∂c (c∗ j , v) = 1.
∂f j
∂c (c∗ j , v) = 1 ≤
∂f j
∂c (v, v).
c
v ≤ c∗ j
i p
r = p
U i( p) = ui(x) − p.
i
c∗i
U i(c∗i ) = ui(x) − c
∗i + f i(c
∗i , v)
≥ ui(x) − v + f i(v, v)
= ui(x) − v
≥ ui(x) − p ∀ p ≥ v
= U i( p)
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c∗ U i(c∗i ) ≥ U i( p)
p
v
U i( p) = ui(x) − p.
U i(c∗i ) = ui(x) − c
∗i + f i(c
∗i , v).
p ≤ c∗i − f i(c
∗i , v) U i(c
∗i ) ≤ U i( p)
c∗i − f i(c∗i , v) ≤ v ∀i c
∗
c∗ v
U i(c∗(v)) = ui(x) − c
∗(v) + f i(c∗(v), v).
v
∂U (c∗(v))
∂v = −
dc∗(v)
dv +
∂f
∂c∗ ·
dc∗(v)
dv +
∂ f
∂v
=
∂f
∂v v.
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ui(x) ∼ U (0, 2v)
µi ∼ U 1
1 + v, 1 + v
1 + v .
v
g(ui(x), µi) = v + 1
4v2
Pr(U i(c∗i ) ≥ 0)
Pr(U i(c∗i ) ≥ 0) = 1 − Prui(x) < v − (1 − µi)
2
2µi = 1 −
1+
v1+v
11+v
v− (1−µi)22µi0
g(ui(x), µi) dui(x)dµi
= 1 −
v + 1
4v2
2v2 + v
v + 1 −
1
2 ln(2v + 1
= 2v − 1
4v +
v + 1
4v2 · ln(2v + 1)1/2
0.5
v = 1.7383
U i( p) = ui(x) − p
Pr(U i( p) ≥ 0) = 1 − Pr(ui(x) < p)
= 1 − p
2v.
0.5 p ≥ v
p < v
0.5
p
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v
p = v − 1v
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p = v + 1
v
