The Market Power Function
Theoretical results
Experimental results
Role of Network and Production Capacity in Allocating Market Power
Jiangzhuo Chen
Matthew Macauley
Achla Marathe
Network Dynamics and Simulation Science Laboratory
Virginia Bioinformatics Institute
Virginia Polytechnic Institute and State University
Trans-Atlantic INFRADAY
University of Maryland, College Park
November 2, 2007
The Market Power Function
Theoretical results
Experimental results
Outline
1 The Market Power Function
Definitions
Networks
Market power spectrum
2 Theoretical results
Local capacity changes
Threshold capacities
Global capacity changes
3 Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
The Market Power Function
Theoretical results
Experimental results
Definitions
Networks
Market power spectrum
Definitions: Market power function
◮ Setting: ς distinct sellers, distributed across a network, all selling a uniform commodity.
◮ Production capacity vector: c = (c1 , . . . , cς ).
Definition (Market power)
Seller i has Pi (c) units of market power if the total amount of the demand that can be
fulfilled ∆(c), decreases by MPi (c) in the absence of supplier i .
◮ The market power function is defined by
mp : Rς+ −→ [0, 1]ς ,
mpi : Rς+ −→ [0, 1],
mpi (c) =
Pi (c)
.
∆(c)
◮ The global market power function is the sum of the individual market functions:
MP : Rς+ −→ [0, 1],
MP(c) =
ς
X
i =1
mpi (c) = mp(c) · (1, 1, . . . , 1).
The Market Power Function
Theoretical results
Experimental results
Definitions
Networks
Market power spectrum
Modeling a market over a network
◮ Let G = (V , E ) be an undirected graph with edge capacities C : E −→ R+ .
◮ There is a set of suppliers (or sellers, generators, etc) S ⊂ V , with capacity vector
|S|
c : S −→ R+ .
|T |
◮ There is a P
set of consumers T ⊂ V , with demand vector d : T −→ R+ , and total
demand D = i ∈T di .
◮ The goods must be delivered from the sellers to the consumers across the network G .
The Market Power Function
Theoretical results
Experimental results
Definitions
Networks
Market power spectrum
Market graphs
◮ Without loss of generality, we may assume that the graph G contains:
a super-source s, and edges {s, si } for all si ∈ S, each with capacity ci .
a super-sink t, and edges {ti , t} for all ti ∈ T , each with capacity di .
◮ Such a graph is called a market graph.
◮ Sending the goods from the suppliers to the consumers is modeled as an s-t network
flow. The maximum that can be delivered is the max-flow, or min-cut of the network.
Definition
For a market graph G , the submarket graph G (S ′ ) of a subset of suppliers S ′ ⊂ S is
constructed from G by setting the capacity of the edge {s, sj } = 0 for all sj 6∈ S ′ .
◮ Thus, G (S ′ ) is the market graph when all suppliers not in S ′ are off-line.
◮ Observe that the market power of seller i is:
mpi (c) =
mincut(G (S \ {si }), s, t)
mincut(G , s, t) − mincut(G (S \ {si }), s, t)
=1−
.
mincut(G , s, t)
mincut(G , s, t)
The Market Power Function
Theoretical results
Experimental results
Definitions
Networks
Market power spectrum
Resource constraints
Example: Consider 3 sellers with capacity vector c = (9, 5, 3) and one consumer with
demand 10 (and no network). This can be modeled by the following market graph:
s1
9
s
5
3
9
s2
s3
5
t1
10
t
3
◮ The market powers of the individual sellers are
mp1 (c) =
2
,
10
mp2 (c) = 0,
mp3 (c) = 0 .
◮ Observe that if sellers 1 and 2 merge into one entity, then together, they will have 7
units (or 70%) of market power. The behavior of market power under merging was the
topic of an earlier paper.
The Market Power Function
Theoretical results
Experimental results
Definitions
Networks
Market power spectrum
The market power spectrum
◮ Given a market graph G = (V , E ), with capacity vector c, consider two extreme cases:
Scale the capacity vector kc, for k → 0.
Scale the capacity vector kc, for k → ∞.
◮ As k → 0, then for small enough k, the edge capacities in the network will be
irrelevant, and the market will be a purely resource-constrained market.
◮ As k → ∞, then for large enough k, the relative production capacities will be
irrelevant, and the market will be a purely network-constrained market.
◮ The range of possible markets between these two extremes is called the market power
spectrum.
Question: How is the market power function vary across the market power spectrum?
Where on the spectrum do certain real-world markets lie?
The Market Power Function
Theoretical results
Experimental results
Local capacity changes
Threshold capacities
Global capacity changes
Local capacity changes
◮ Consider how the market power function changes as we change the capacity of a
single generator.
◮ Let f ∗ (G , ci ) denote max-flow in the market graph G when capacity of generator i is ci .
∗ := f ∗ (G , c = 0), the max-flow when s drops out.
◮ Let f−i
i
i
◮ Using the theory of residual flows, we can derive:
(
if ci ≤ ci∗ (G ),
f ∗ + ci
f ∗ (G , ci ) = −i
∗
∗
f−i + ci (G ) otherwise.
where ci∗ (G ) is the maximum possible flow that generator i can provide.
The Market Power Function
Theoretical results
Experimental results
Local capacity changes
Threshold capacities
Global capacity changes
Theorem
When ci increases, market power of si is non-decreasing; market power of sj , j 6= i , is
non-increasing.
Remark. The total market power MP(ci ) is non-decreasing in ci when ci ≤ ci∗ (G ),
non-increasing when ci ≥ ci∗ (G ).
◮ In our experimental results with the Portland electrical network, the total market
power is always non-increasing in ci . However, this need not always be true.
Example.
s1
1
s
3
3
s2
t1
3
10
t
3
3
s3
3
◮ If c1 = 1, then mp1 = 0.25, mp2 = 0, mp3 = 0, and MP = 0.25.
◮ If c1 = 2, then mp1 = 0.40, mp2 = 0, mp3 = 0, and MP = 0.40.
The Market Power Function
Theoretical results
Experimental results
Local capacity changes
Threshold capacities
Global capacity changes
Threshold capacities
◮ The local threshold capacity of generator i , denoted Tiℓ , is the minimum non-negative
value such that if ci > Tiℓ , then ∇ei mpi = 0
◮ The local threshold capacity of generator i , denoted Tiℓ , is the minimum non-negative
value such that if ci > Tiℓ , then ∇ei MPi = 0
◮ Mathematically,
Til = sup{Supp (∇ei mpi ) ∪ {0} },
Tig = sup{Supp (∇ei MP)},
where sup is supremum of a set, and Supp is the support of a function.
◮ Note. ∇ei MP has a non-empty support, but ∇ei mpi may not.
´
`
Prop. If we define Til = sup{Supp ∇ei mpj ∪ {0}}, then for any i 6= j,
Tiℓ ≤ Tij ≤ Tig .
The Market Power Function
Theoretical results
Experimental results
Local capacity changes
Threshold capacities
Global capacity changes
Global capacity changes
◮ Consider the scenario when the capacity vector is scaled by k > 0.
◮ It is intuitive to conjecture that as k increases, the total market power should decrease,
i.e., ∇c MP(c) ≤ 0. However, this is not always the case, as the following example
shows.
Example.
s1
1
s
3
3
s2
t1
3
10
t
3
3
s3
3
◮ If k = 1, then mp1 = 0.25, mp2 = 0, mp3 = 0, and MP = 0.25.
◮ If k = 2, then mp1 = 0.40, mp2 = 0, mp3 = 0, and MP = 0.40.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Experimental set-up
◮ Electrical grid of Portland, Oregon.
◮ Over 600 nodes and 700 transmission lines.
◮ 41 generators locations, but 11 have zero production capacity.
◮ Peak demand vector is taken from FERC, and rescaled by a factor of 0.337 so that
total supply is 105% of the total demand.
◮ Only six generators have any market power, and the total market power is
MP(c) = 0.398.
◮ Two sets of experiments:
Scale the individual production capacity ci independently (local change).
Scale the production capacity vector c (global change).
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Summary of results
Gen. ID
Prod-cap (ci )
Deg-cap.
mpi
Max mpi
Max MP
Min MP
Degree
Til
g
Ti
17
608
688
0.145
0.145
1
0.270
1
451
688
Highest Market Power (mpi ) Generators
19
152
485
601
18
390
372
333
260
241
429
348
456
300
276
0.075
0.061
0.057
0.033
0.027
0.075
0.061
0.057
0.033
0.027
1
1
1
1
1
0.335
0.398
0.287
0.334
0.342
1
1
1
1
1
233
191
176
103
84
429
348
402
300
276
Highest Deg-cap Generators
83
55
356
100
89
32
15
2
4542
2075
2265
1162
0
0
0
0
0
0
0
0
0.669
0.486
0.434
0.402
0
0
0
0
11
3
5
2
0
0
0
0
540
483
466
453
Table: Characteristics of Select Generators (in Kw)
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Remarks about the results
For each seller, ci > Tiℓ , i.e., production capacity is greater than its local threshold
capacity.
5 of the 6 sellers with market power have Tig equal to their “degree-capacity” (sum
of capacities of all incident edges).
For 9 of 10 sellers, ci < Tig . This suggests that the market is more
resource-constrained than network constrained.
For the 4 sellers with the highest degree-capacity, Til = 0. They have no market
power, but can eliminate global market power completely by increasing their
capacity.
Tig > Til . A generator may be able to affect global market power without being
able to affect its individual market power, but not vice-versa.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Individual market power function plots
Figure: Individual market power (mpi ), as a function of ci .
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Global market power with respect to local changes
Figure: Total market power (MP), as a function of ci .
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Global market power
Figure: Total market power as a function of
the supply scaling factor.
Figure: Individual market power of the top 7
sellers, as a function of the supply scaling
factor.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Remarks about the results
For k > 1, MP monotonically decreases with k (this is in general not guaranteed!)
ci
For k < 1, the values of mpi quickly converge to P
. Thus even the slightest
cj
excess demand in the market will make it predominately resource-constrained.
j
∇c MP(kc) increases with k, i.e., MP is convex. A small increase in supply can
wipe out a large amount of the market power.
The graphs of mpi for sellers 152 and 485 intersect, an explicit example of how
network topology can play a role in market power.
Overall, the values of ∇ei mpi (kc) for the top seven generators are not significantly
different, suggesting that the market is more resource-constrained rather than
network-constrained.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Experiment
◮ Base case: scale demand so that supply = demand.
◮ Scale both supply and demand vector independently. (Recall: the base case in the
previous experiments were when supply was 105% of demand, and global market power
was 0.398.)
Observations.
Changing the ratio of supply to demand by 2% makes the value of k required to
eliminate market power decrease by 10%.
The market power function converges very quickly. In particular, the market power
functions are virtually identical for the cases when supply is 130%, 150%, 200%,
and 300% of demand.
All functions take on a value of 1 for k < 1 but for k = 1.2, the total market power
is less than 10% in every case. This implies that demand elasticity can play a
significant role in mitigating market power.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Figure: Market power as a function of the supply scaling factor, for different initial ratios of supply
and demand.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
3-D plot of the previous figure
Figure: Market power as a function of supply and demand.
The Market Power Function
Theoretical results
Experimental results
Set-up
Local capacity changes
Global capacity changes
Elasticity
Acknowledgments
Thank you for your attention!
Special thanks:
Network Dynamics and Simulation Science Laboratory (NDSSL), at the Virginia
Bioinformatics Institute, at Virginia Tech.
Los Alamos National Laboratory.
Contact info: {chenj, mmacaul, amarathe}@vbi.vt.edu.
NDSSL:
Web: http://ndssl.vbi.vt.edu