DEFAULT PENALTY AS A
DISCIPLINARY AND
SELECTION MECHANISM IN
PRESENCE OF MULTIPLE
EQUILIBRIA
Shyam Sunder
(From the joint work of Juergen Huber,
Martin Shubik and Shyam Sunder)
Workshop on Experimental Social Sciences
Mumbai Vidyapeeth
December 28-29, 2009
A basic question and several
solutions
• In the microeconomic theory of
the price system it is possible that
several equilibria may be present.
• Macro economists tend to ignore
this possibility as essentially
irrelevant
• Are they right? Does this
represent a gap between macro
practice and micro theory?
• Is there a satisfactory solution,
and does it matter?
2
A Summary of
Experimental Evidence
• Default penalties and bankruptcy
laws needed to implement
general equilibrium as a playable
game and to mitigate strategic
defaults can also provide
conditions for uniqueness
• Assignment of default penalty on
fiat money  economy goes to
selected equilibrium
• Role of accounting/bankruptcy
aspects of social mechanisms in
resolving mathematically
intractable multiplicity problem
3
• The general equilibrium proof of
the existence of a competitive
price system was both a triumph
and a disaster
• It provided a deep
mathematization for the existence
of equilibrium conditions for
efficient prices that cut out time
and uncertainty
4
• A whole school of mathematical
economics was born with
considerable sophistication but
little connection with the context
and realities of the ongoing
economy
5
• The work of Arrow, Debreu and
Mckenzie proved the existence of
efficient prices but did nothing
about their evolution or their
fairness.
• This can be seen easily when we
observe that an economy may
have many equilibria
6
•
One of the tasks of extreme
mathematical difficulty is what
necessary and sufficient
conditions are required for a
unique competitive equilibrium
to exist?
•
We believe that the answer that
society gives to both uniqueness
and fairness lies in extending
the model to embed it an a
dynamic model of society that
permits default and requires
default laws.
7
• Our program involves both theory
and experimentation.
• We utilize an example of an
exchange economy with three
equilibrium points constructed by
Shapley and Shubik. This is
illustrated in the next slide
8
Figure 1: An Exchange Economy with
Two Goods and Three Competitive
Equilibria
9
• We observe that it has three
equilibrium points the two
extreme ones favoring either
traders of type a or b.
• The middle is more equitable
(and is not stable under Walrasian
dynamics)
10
• When this exchange economy is
remodeled as a playable strategic
market game, if borrowing is
permitted default rules must be
specified to take care of every
possibility in the system.
• But these rules will entail some
action of negative worth to the
defaulter and are denominated in
money. Thus they link money to
the individual’s utility
11
• Technically if the penalties are set
equal to or above the Lagrangian
multipliers of the related general
equilibrium model this will be
sufficient to prevent strategic
bankruptcy. Thus by selecting the
penalties we can select the
equilibrium point to be chosen.
12
• In selecting the penalties
associated with the middle
equilibrium the society resolves
its “fairness” problem but from
the viewpoint of finance it does
not select the minimal cash flow
equilibrium.
• This is shown in the next slide
13
Comparison of Prices in the
Three Competitive Equilibria
Trade 1
Trade 2
Final 1
Final 2
Price1
Price2
Trade
CE1
32.26
10.74
7.74, 10.74
32.26, 39.26
3.97
20.11
383.66
CE2
13.17
19.82
26.83, 29.82 13.17, 20.18
12.5
9.375
570.13
CE3
3.21
39.77
36.97, 39.77
19.53
5.47
475.66
3.21, 10.23
14
• In higher dimensions it is always
straightforward to select the
minimum cash flow equilibrium,
but this is not true for the
selection of “fair” as well as
efficient equilibrium
• In fact government selects the
penalties more or less blindly and
in the course of the application of
legal and political pressures they
are adjusted
15
• There is a considerable literature
on multiple equilibria as is
summarized by Morris and
shin(2000)
• They deal primarily with Bayesian
equilibria with noise
• Our approach here is different
from, but complementary with,
this literature. We stress the laws
of society as providing direct
strong coordinating and coercive
devices for the economy
16
Experimental Design
Balances Carried
Forward Period to
Period
Endowments
Refreshed each
Period
Exchange of Goods T1a (A is
without Money
numeraire),
T1b (B is
numeraire)
Fail to converge to
any CE
Exchange of Goods T2a,
with Money and
T2b,
Penalty Targeted
T2c
at one of the CEs
Converge to
selected CE
T2a-R,
T2b-R,
T2c-R
Converge to
selected CE
Exchange of Goods T3
with Money and
Converge to other
non-CE Penalty
outcomes
determined by
selected penalty
T3-R
Converge to other
outcomes
determined by
selected penalty
17
Treatment 1: Conjectures
1. In Treatment 1 the process fails
to converge to any of the three
competitive equilibria.
2. In Treatment 1 the middle CE is
favored.
3. In Treatment 1 the choice of the
medium of exchange or
numeraire does not influence
the outcomes (prices and
distribution of goods).
18
Figure 2: Holdings of Goods A and B in the Four Runs of Treatment 1a
(with Good A as the Numeraire)
Holdings of goods A and B in Run 1-2 of T1a
50
50
40
40
holdings of good B
holdings of good B
Holdings of goods A and B in Run 1-1 of T1a
30
20
10
30
20
10
0
0
0
10
20
30
40
0
10
holdings of good A
Holdings of goods A and B in Run 2-1 of T1a
30
40
Holdings of goods A and B in Run 2-2 of T1a
50
50
40
40
holdings of good B
holdings of good B
20
holdings of good A
30
20
10
30
20
10
0
0
10
20
holdings of good A
30
40
0
0
10
20
30
40
holdings of good A
19
Figure 3: Holdings of Goods A and B in the Four Runs of Treatment 1b
(with Good B as the Numeraire)
Holdings of goods A and B in Run 1-2 of T1b
50
40
40
holdings of good B
50
30
20
30
20
10
10
0
0
0
10
20
30
0
40
10
20
30
40
holdings of good A
holdings of good A
Holdings of goods A and B in Run 2-1 of T1b
Holdings of goods A and B in Run 2-2 of T1b
50
50
40
40
holdings of good B
holdings of good B
holdings of good B
Holdings of goods A and B in Run 1-1 of T1b
30
20
10
30
20
10
0
0
10
20
holdings of good A
30
40
0
0
10
20
30
40
holdings of good A
20
Figure 4: Time Series of Cumulative Trading Volume (Top Panels) and Efficiency
(Bottom Panels) per Period in Treatment 1. T1a is presented on the left side, while T1b
is on the right side. The Second Run of each Student Cohort is shaded in grey.
Cumulative Trading Volume of Good A in T1b
200
200
150
150
Trading Volume
Trading Volume
Cumulative Trading Volume of Good A in T1a
100
50
100
50
0
0
1
2
3
4
5
6
7
1
8 9 10 11 12 13 14 15 16
Period
200
150
150
100
50
3
5
6
7
8 9 10 11 12 13 14 15 16
Period
100
50
0
0
1
2
3
4
5
6
7
1
8 9 10 11 12 13 14 15 16
Period
2
3
100%
90%
90%
Efficiency .
100%
80%
70%
4
5
6
7
8 9 10 11 12 13 14 15 16
Period
Efficiency in T1b
Efficiency in T1a
Effiiciency .
4
Cumulative Trading Volume of Good B in T1b
200
Trading Volume
Trading Volume
Cumulative Trading Volume of Good B in T1a
2
80%
70%
60%
60%
50%
50%
1
2 3
4
5
6 7
8
9 10 11 12 13 14 15 16
Period
1 2 3
4 5
6 7 8
9 10 11 12 13 14 15 16
Period
21
Treatment 1
• No clear convergence to any of
the three CEs.
• It does not matter which of the
two goods in this exchange
economy is chosen as the
numeraire.
• Efficiency in all markets is high,
demonstrating that such simple
markets serve well as
coordination mechanisms.
22
Treatment 2: Conjectures
•
•
4. In Treatment 2 the system
converges and can be made to
converge to any of the three
equilibria guided by the
selection of parameter μ
(default penalty).
5. In Treatment 2 net money
holdings will be equal to the
equilibrium level of zero.
23
Figure 5: Individual and Average Holdings of Goods A and B in Treatment 2
(holdings of goods and money carried over from one period to the next)
Holdings of goods A and B in Run 1 of T2b
Holdings of goods A and B in Run 1 of T2c
50
50
40
40
40
30
20
10
holdings of good B
50
holdings of good B
holdings of good B
Holdings of goods A and B in Run 1 of T2a
30
20
10
0
10
20
30
40
0
0
holdings of good A
20
10
0
0
30
10
20
30
40
0
holdings of good A
10
20
30
40
holdings of good A
Figure 6: Time Series of Cumulative Trading Volume in Treatment 2. T2a is presented
on the left side, T2b in the center, and T2c on the right.
Cumulative Trading Volume of Goods A
and B in T2b
Cumulative Trading Volume of Goods A
and B in T2c
200
200
150
150
150
100
Volume i
200
Volume i
Volume i
Cumulative Trading Volume of Goods A
and B in T2a
100
50
50
0
100
50
0
0
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12
Period
Period
Period
24
Figure 7: Path of Average Holdings of Goods A and B in Treatment 2-R (with holdings
reinitialized)
Run 2
50
50
40
40
Holdings of good B
Holdings of good B
Run 1
30
20
10
30
20
10
0
0
0
10
20
30
40
0
10
Holdings of good A
20
30
40
Holdings of good A
Figure 8: Time Series of Efficiency in Treatment 2-R (with holdings reinitialized)
(Period 0 = autarky)
Efficiency in Run 2 of T2-R
Efficiency in Run 1 of T2-R
T2b-R
T2a-R
T2c-R
100%
100%
75%
75%
Efficiency
Efficiency
T2a-R
50%
T2b-R
T2c-R
50%
25%
25%
0%
0%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Period
Period
25
Figure 9: Development of Average Net Money Holdings for Subjects endowed with
Good A in Treatment 2 (holdings carried over) and T2-R (holdings re-initialized)
Treatment 2b and 2b-R
(μ2 = 0.75)
Treatment 2c and 2c-R
(μ2 = 5.07)
75
75
50
50
50
25
25
25
0
-25
-50
Net Money Holdings
75
Net Money Holdings
Net Money Holdings
Treatment 2a and 2a-R
(μ2 = 0.28)
0
-25
1 2 3 4 5 6 7 8 9 101112131415
Period
-25
-50
-50
-75
0
-75
-75
1 2 3 4 5 6 7 8 9 101112131415
1 2 3 4 5 6 7 8 9 101112131415
Period
Period
26
Treatment 2
• These results of T2 and T2-R
broadly confirm the results from
T1—the introduction of a money
allows convergence to the unique
equilibrium that is defined by the
value/default penalty associated
with the money.
27
Treatment 3: Conjectures
• 6. In Treatment 3 the unique
equilibrium defined by the default
penalties μ1 and μ2 is
approached.
28
Figure 10: Holdings of Goods A and B in the two Runs of Treatment 3
Holdings of goods A and B in Run 2 of T3
50
50
40
40
30
30
Holdings good B
Holdings good B
Holdings of goods A and B in Run 1 of T3
20
20
10
10
0
0
0
10
20
Holdings good A
30
40
0
10
20
30
40
Holdings good A
29
Figure 11: Holdings of goods A and B (Left Panel) and Efficiency (Right Panel) per
Period in Treatment 3-R.
Run 1 of T3-R
Efficiency over Time in T3-R
50
100%
40
30
Efficiency
Holdings of good B ...
90%
20
80%
70%
10
60%
0
0
10
20
30
1
40
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Period
Holdings of good A
Figure 12: Time Series of Trading Volume (Left Panel) and Efficiency (Right Panel) per
Period in Treatment 3. The Second Run is shaded in grey.
Cumulative Trading Volume over time in T3
Efficiency over time in Treatment 3
Run 1 good A
Run 2 good A
Equilibrium A
Run 1 good B
Run 2 good B
Equilibrium B
Run 1
Run 2
100%
150
90%
Efficiency .
Cumulative Volume
125
100
75
80%
50
70%
25
60%
0
1
2
3
4
5
6
7
Period
8
9
10
11
1
2
3
4
5
6
7
8
9
10
11
Period
30
Treatment 3
• The unique equilibrium defined
by the chosen penalty is
approached in Treatment 3.
31
Conclusions
• Treatment 1: Empirical support for
theoretical indeterminacy.
• Treatment 2: Salvage value/default penalty of
a fiat money can be chosen to achieve any of
the competitive equilibria of the economy.
• Other penalties generate specific equilibrium
outcomes (not necessarily economize on use
of money)
• Institutional arrangements in a society
provide the means to resolve the possibility
of multiple equilibria in an economy.
• Empirical support for the attitudes of
macroeconomists who do not regard the
non-uniqueness of competitive equilibria as a
major applied problem.
32
Trading Screen without
Fiat Money
33
Results Screen without
Fiat Money
34
Payoff Tables
Table for those initially endowed with A: A + 100 * (1-e(-B/10))
Units of good A you hold at the end of a period
0
5
10
15
20
25
30
35
40
45
50
Units
of B
you
hold
0
0.0
39.3
63.2
77.7
86.5
91.8
95.0
97.0
98.2
98.9
99.3
5
5.0
44.3
68.2
82.7
91.5
96.8
100.0
102.0
103.2
103.9
104.3
10
10.0
49.3
73.2
87.7
96.5
101.8
105.0
107.0
108.2
108.9
109.3
15
15.0
54.3
78.2
92.7
101.5
106.8
110.0
112.0
113.2
113.9
114.3
20
20.0
59.3
83.2
97.7
106.5
111.8
115.0
117.0
118.2
118.9
119.3
25
25.0
64.3
88.2
102.7
111.5
116.8
120.0
122.0
123.2
123.9
124.3
30
30.0
69.3
93.2
107.7
116.5
121.8
125.0
127.0
128.2
128.9
129.3
35
35.0
74.3
98.2
112.7
121.5
126.8
130.0
132.0
133.2
133.9
134.3
40
40.0
79.3
103.2
117.7
126.5
131.8
135.0
137.0
138.2
138.9
139.3
45
45.0
84.3
108.2
122.7
131.5
136.8
140.0
142.0
143.2
143.9
144.3
50
50.0
89.3
113.2
127.7
136.5
141.8
145.0
147.0
148.2
148.9
149.3
Table for those initially endowed with B: B + 110 * (1-e(-A/10))
Units of good A you hold at the end of a period
Units
of B
you
hold
0
5
10
15
20
25
30
35
40
45
50
0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
5
43.3
48.3
53.3
58.3
63.3
68.3
73.3
78.3
83.3
88.3
93.3
10
69.5
74.5
79.5
84.5
89.5
94.5
99.5
104.5
109.5
114.5
119.5
15
85.5
90.5
95.5
100.5
105.5
110.5
115.5
120.5
125.5
130.5
135.5
20
95.1
100.1
105.1
110.1
115.1
120.1
125.1
130.1
135.1
140.1
145.1
25
101.0
106.0
111.0
116.0
121.0
126.0
131.0
136.0
141.0
146.0
151.0
30
104.5
109.5
114.5
119.5
124.5
129.5
134.5
139.5
144.5
149.5
154.5
35
106.7
111.7
116.7
121.7
126.7
131.7
136.7
141.7
146.7
151.7
156.7
40
108.0
113.0
118.0
123.0
128.0
133.0
138.0
143.0
148.0
153.0
158.0
45
108.8
113.8
118.8
123.8
128.8
133.8
138.8
143.8
148.8
153.8
158.8
50
109.3
114.3
119.3
124.3
129.3
134.3
139.3
144.3
149.3
154.3
159.3
35
Trading Screen with Fiat
Money
36
Results Screen with Fiat
Money
37
Payoff Table with Fiat
Money (T2c)
Table for those initially endowed with A: (A + 100 * (1-e^(-B/10))) + Net Money
Units of good A you hold at the end of a period
Units
of B
you
hold
0
5
10
15
20
25
30
35
40
45
50
0
0.0
39.3
63.2
77.7
86.5
91.8
95.0
97.0
98.2
98.9
99.3
5
5.0
44.3
68.2
82.7
91.5
96.8
100.0
102.0
103.2
103.9
104.3
10
10.0
49.3
73.2
87.7
96.5
101.8
105.0
107.0
108.2
108.9
109.3
15
15.0
54.3
78.2
92.7
101.5
106.8
110.0
112.0
113.2
113.9
114.3
20
20.0
59.3
83.2
97.7
106.5
111.8
115.0
117.0
118.2
118.9
119.3
25
25.0
64.3
88.2
102.7
111.5
116.8
120.0
122.0
123.2
123.9
124.3
30
30.0
69.3
93.2
107.7
116.5
121.8
125.0
127.0
128.2
128.9
129.3
35
35.0
74.3
98.2
112.7
121.5
126.8
130.0
132.0
133.2
133.9
134.3
40
40.0
79.3
103.2
117.7
126.5
131.8
135.0
137.0
138.2
138.9
139.3
45
45.0
84.3
108.2
122.7
131.5
136.8
140.0
142.0
143.2
143.9
144.3
50
50.0
89.3
113.2
127.7
136.5
141.8
145.0
147.0
148.2
148.9
149.3
Table for those initially endowed with B: 1/5.07 * (B + 110 * (1-e^(-A/10)) + Net Money
Units of good A you hold at the end of a period
Units
of B
you
hold
0
5
10
15
20
25
30
35
40
45
50
0
0.0
1.0
2.0
3.0
3.9
4.9
5.9
6.9
7.9
8.9
9.9
5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.4
16.4
17.4
18.4
10
13.7
14.7
15.7
16.7
17.7
18.6
19.6
20.6
21.6
22.6
23.6
15
16.9
17.8
18.8
19.8
20.8
21.8
22.8
23.8
24.7
25.7
26.7
20
18.8
19.7
20.7
21.7
22.7
23.7
24.7
25.7
26.6
27.6
28.6
25
19.9
20.9
21.9
22.9
23.9
24.8
25.8
26.8
27.8
28.8
29.8
30
20.6
21.6
22.6
23.6
24.6
25.5
26.5
27.5
28.5
29.5
30.5
35
21.0
22.0
23.0
24.0
25.0
26.0
27.0
27.9
28.9
29.9
30.9
40
21.3
22.3
23.3
24.3
25.2
26.2
27.2
28.2
29.2
30.2
31.2
45
21.5
22.4
23.4
24.4
25.4
26.4
27.4
28.4
29.3
30.3
31.3
50
21.6
22.5
23.5
24.5
25.5
26.5
27.5
28.5
29.4
30.4
31.4
38
Thank You.