Approximating the gains from trade in large markets

Approximating the gains from trade in large
markets
Ellen Muir
Supervisors: Peter Taylor & Simon Loertscher
Joint work with Kostya Borovkov
The University of Melbourne
7th April 2015
Outline
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Setup market model
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Define the Walrasian quantity traded and level of welfare
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Approximating welfare
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Approximating the scaled quantity traded
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Example computation
Setup
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Consider a market for a homogeneous, indivisible good
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There are N buyers who demand one unit of the good
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There are M sellers with the capacity to produce one unit
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Buyer valuations: V1 , . . . , VN are IID RVs with AC DF F
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Seller costs: C1 , . . . , CM are IID RVs with AC DF G
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This is a canonical mechanism design model
Ordering of buyers and sellers
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Sellers are ordered from lowest cost to highest cost
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For j = 1, . . . , M let C(j) denote the jth lowest seller cost
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This forms a supply curve
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Buyers are ordered from highest valuation to lowest valuation
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For i = 1, . . . , N let V[i] denote the ith highest buyer valuation
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This forms a demand curve
Ordering of buyers and sellers
C(M )
V[1]
C(1)
V[N ]
1
N
M
Figure 1 : The ordered costs and valuations form demand and supply curves
Economic welfare
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The level of welfare generated by a market is essentially the gains
from trade
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The welfare generated by a market is equal to the sum of trading
buyer valuations less the sum of trading seller costs
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If N T and MT are the sets of trading buyers and sellers
respectively, the level of welfare is given by
X
i∈N T
Vi −
X
j∈MT
Cj
How can we maximise welfare?
C(M )
V[1]
C(1)
V[N ]
1
K
N
M
Figure 2 : The shaded region shows the maximum level of welfare
The Walrasian quantity and level of welfare
Definition
The Walrasian quantity, K, is the quantity traded that maximises
welfare. That is,
K=
arg max
k
X
V[i] − C(i) .
k=0,1,...,N ∧M i=1
The associated level of welfare, W , is then
W =
K
X
i=1
V[i] − C(i) .
The problem at hand
We want to approximate
W =
K
X
i=1
for large N and M .
V[i] − C(i)
Background on empirical quantile functions
Definition
Let X1 , . . . , Xn be an IID sample of RVs. Then
Qn (t) =
n
X
i=1
X(i) 1
i−1
i
<t≤
n
n
is known as the empirical quantile function of the sample.
Background on empirical quantile functions
Qn (t)
1
X(n)
X(1)
1
n
n−1
n
1
t
Figure 3 : An example of an empirical quantile function
Background on empirical quantile functions
Proposition (Cs¨org˝o and R´ev´esz, 1978)
Let U1 , . . . , Un be an IID sample of U (0, 1) RVs and Qn (t) be the
associated uniform quantile function. As n → ∞, for t ∈ (0, 1),
1
Qn (t) = t + n− 2 B(t) + θn (t)n−1 log n,
where B(t) is a Brownian bridge process and kθn k∞ = O(1) a.s.
Here we use the standard notation k·k∞ := supt∈(0,1) |·|.
Approximating welfare
Returning to the problem at hand,
W =
K
X
i=1
K
K
X
X
V[i] − C(i) =
V[i] −
C(i) ≡ W V − W C .
i=1
Approximate W V and W C separately.
i=1
Approximating W C
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Let G(−1) be the quantile function of G
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C
Let U1C , . . . , UM
be an IID sample of U (0, 1) RVs
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Let QC
M denote the associated uniform quantile function
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We have
WC =
K
X
i=1
C[i] =
K
X
K
X
i
1 (−1)
C
G
QC
G(−1) U(i)
=M
M
M
M
i=1
i=1
Approximating W C
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Rewriting the sum on the previous slide as an integral,
W
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C
K
Z M
K
X
i
1 (−1)
C
G
QM
=M
G(−1) QC
=M
M (u) du
M
M
0
i=1
Then letting BC denote a Brownian bridge process,
W
C
Z
=M
K
M
1
C
(u)M −1 log M du,
G(−1) u + M − 2 BC (u) + θM
0
C
= O(1) a.s.
where θM
∞
Approximating W V
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Repeating this procedure for W V we obtain
WV = N
Z
K
N
1
V
F (−1) 1 − u + N − 2 BV (1 − u) + θN
(u)N −1 log N du,
0
V
= O(1) a.s.
where θN
∞
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Here, F (−1) is the quantile function of F
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BV is a Brownian bridge process independent of BC
To proceed, we must approximate K/N and K/M .
Some simulations
Figure 4 : Uniformly distributed costs and valuations with N = 100 and
M = 110
Some simulations
Figure 5 : Uniformly distributed costs and valuations with N = 200 and
M = 220
Some simulations
Figure 6 : Uniformly distributed costs and valuations with N = 500 and
M = 550
Some simulations
Figure 7 : Uniformly distributed costs and valuations with N = 2000 and
M = 2200
Approximating K/N
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Take M = λN and approximate K/N only
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Consider the functional
H(k) = sup{t : k(t) < 0}
(−1)
(−1)
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Let GM (t) and FN (t) denote the empirical quantile functions
associated with the seller costs and buyer valuations respectively
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Then
K
(−1)
(−1)
= H GM (λ−1 t) − FN (1 − t)
N
Approximating K/N
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For t ∈ (0, 1 ∧ λ) define
k0 (t) = G(−1) (λ−1 t) − F (−1) (1 − t)
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Let t0 = H(k0 ) so that, as n → ∞,
K d
−
→ t0
N
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We wish to determine the distribution of the leading order error term
A visual illustration
Figure 8 : A simulation of K/N with uniform types, N = 50 and M = 50
Asymptotic normality of sample quantiles
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It is well known that, as n → ∞,
√
BG (λ−1 t)
,
λg(G(−1) (λ−1 t))
√
−BF (t)
d
(−1)
N (FN (1 − t) − F (−1) (1 − t)) −
→
f (F (−1) (1 − t))
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√
d
(−1)
N (GM (λ−1 t) − G(−1) (λ−1 t)) −
→√
Combining these gives, as n → ∞,
(−1)
d
−
→√
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√
(−1)
N [(GM (λ−1 t) − FN
(1 − t)) − (G(−1) (λ−1 t) − F (−1) (1 − t))]
BG (λ−1 t)
BF (t)
+
λg(G(−1) (λ−1 t)) f (F (−1) (1 − t))
But we want the convergence of
(−1)
(−1)
N [H(GM (λ−1 t) − FN
(1 − t)) − H(G(−1) (λ−1 t) − F (−1) (1 − t))]
The delta method
We need a functional version of the delta method.
Proposition
Let θ, σ 2 be finite valued constants and suppose {Xn }n∈N satisfies
√
d
n [Xn − θ] −
→ N (0, σ 2 ).
Then for any function r with r0 (θ) 6= 0,
√
d
n [r(Xn ) − r(θ)] −
→ N (0, σ[r0 (θ)]2 ).
The derivative of H
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Compute the functional derivative of H
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Do this by modifying an example in Section 8, Chapter I of
Borovkov (1998)
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For any v ∈ C(−∞, ∞),
H 0 (k0 , v) =
v(t0 )λg(G(−1) (λ−1 t0 ))f (F (−1) (1 − t0 ))
λg(G(−1) (λ−1 t0 )) + f (F (−1) (1 − t0 ))
The Functional Delta Method
By Proposition 1 in Borovkov (1985),
√
(−1)
(−1)
N [H(GN (t) − FN (1 − t)) − H(G(−1) (t) − F (−1) (1 − t))]
d
λg(G(−1) (λ−1 t0 ))f (F (−1) (1−t0 ))
t0 (1−λ−1 t0 )
t0 (1−t0 )
−
→ λg(G
0,
(−1) (λ−1 t ))+f (F (−1) (1−t )) N
2 g 2 (G(−1) (λ−1 t )) + f 2 (F (−1) (1−t ))
λ
0
0
0
0
≡ Z.
Thus, we have
1
K d
= t0 + N − 2 Z + O N −1 log N ,
N
where Z is normally distributed.
Uniform costs and valuations with N = M
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In this case, our expression for K/N simplifies to
1
K d 1
= + N−2 N
N 2
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1
0,
8
1
Y
+ O N −1 log N ≡ + √ + O N −1 log N
2
N
Our expression for W C simplifies to
C d
Z
−1
1
2
2 +N
W =N
√
Y
u du +
N
0
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W
1
2
Z
BC (u) du + O (log N )
0
Our expression for W V simplifies to
V d
Z
−1
1
2
2 +N
Y
(1 − u) du +
=N
0
√
Z
N
0
1
2
BV (1 − u) du + O (log N )
Uniform costs and valuations with N = M
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Direct computation then gives
W =W
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V
N √
+ NN
−W =
4
C d
10
0,
+ O(log N )
192
The dependence of K on BV and BC affects higher order terms in
the expansion
Example error simulation
√
Figure 9 : The distribution of (W − N/4)/ N vs. the density of
d
Y = N (0, 10/192) for N = M = 1000 with 5000000 simulations
References
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Borovkov, A. A. (1998), Mathematical Statistics, number 26 – 27,
Gordon & Breach, chapter 1.
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Borovkov, K. A. (1985), ‘A note on a limit theorem for differentiable
mappings’, The Annals of Probability 13(3), 1018 – 1021.
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Cs¨
org˝
o, M. and R´ev´esz, P. (1978), ‘Strong approximations of the
quantile process’, The Annals of Statistics 6(4), 882 – 894.