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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. - PowerPoint PPT Presentation
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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 SciencesMumbai Vidyapeeth
December 28-29, 2009
2
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?
3
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
4
• 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
5
• A whole school of mathematical economics was born with considerable sophistication but little connection with the context and realities of the ongoing economy
6
• 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
7
• 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.
8
• 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
9
Figure 1: An Exchange Economy with Two Goods and Three Competitive
Equilibria
10
• 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)
11
• 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
12
• 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.
13
• 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
14
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 3.21, 10.23 19.53 5.47 475.66
15
• 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
16
• 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
17
Experimental DesignBalances Carried Forward Period to Period
Endowments Refreshed each Period
Exchange of Goods without Money
T1a (A is numeraire), T1b (B is numeraire)Fail to converge to any CE
Exchange of Goods with Money and Penalty Targeted at one of the CEs
T2a, T2b, T2cConverge to selected CE
T2a-R, T2b-R,T2c-RConverge to selected CE
Exchange of Goods with Money and non-CE Penalty
T3Converge to other outcomes determined by selected penalty
T3-RConverge to other outcomes determined by selected penalty
18
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).
19
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 1-1 of T1a
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 1-2 of T1a
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 2-1 of T1a
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 2-2 of T1a
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
20
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 1-1 of T1b
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 1-2 of T1b
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 2-1 of T1b
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
Holdings of goods A and B in Run 2-2 of T1b
0
10
20
30
40
50
0 10 20 30 40
holdings of good A
ho
ldin
gs
of
go
od
B
21
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.
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tra
din
g V
olu
me
Period
Cumulative Trading Volume of Good A in T1a
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tra
din
g V
olu
me
Period
Cumulative Trading Volume of Good A in T1b
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tra
din
g V
olu
me
Period
Cumulative Trading Volume of Good B in T1a
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tra
din
g V
olu
me
Period
Cumulative Trading Volume of Good B in T1b
Efficiency in T1a
50%
60%
70%
80%
90%
100%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Period
Eff
iicie
ncy
.
Efficiency in T1b
50%
60%
70%
80%
90%
100%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Period
Eff
icie
ncy
.
22
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.
23
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.
24
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)
0
10
20
30
40
50
0 10 20 30 40
ho
ldin
gs
of
go
od
B
holdings of good A
Holdings of goods A and B in Run 1 of T2a
0
10
20
30
40
50
0 10 20 30 40
ho
ldin
gs
of
go
od
B
holdings of good A
Holdings of goods A and B in Run 1 of T2b
0
10
20
30
40
50
0 10 20 30 40
ho
ldin
gs
of
go
od
B
holdings of good A
Holdings of goods A and B in Run 1 of T2c
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 T2a
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12
Period
Vol
ume
i
Cumulative Trading Volume of Goods A and B in T2b
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12
Period
Vol
ume
i
Cumulative Trading Volume of Goods A and B in T2c
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 11 12
Period
Vol
ume
i
25
Figure 7: Path of Average Holdings of Goods A and B in Treatment 2-R (with holdings reinitialized)
Run 1
0
10
20
30
40
50
0 10 20 30 40
Holdings of good A
Ho
ldin
gs
of
go
od
B
Run 2
0
10
20
30
40
50
0 10 20 30 40
Holdings of good A
Ho
ldin
gs
of
go
od
B
Figure 8: Time Series of Efficiency in Treatment 2-R (with holdings reinitialized) (Period 0 = autarky)
0%
25%
50%
75%
100%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Eff
icie
ncy
Period
Efficiency in Run 1 of T2-R
T2a-R T2b-R T2c-R
0%
25%
50%
75%
100%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Eff
icie
ncy
Period
Efficiency in Run 2 of T2-R
T2a-R T2b-R T2c-R
26
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)
-75
-50
-25
0
25
50
75
1 2 3 4 5 6 7 8 9 101112131415
Ne
t Mo
ne
y H
old
ing
s
Period
Treatment 2a and 2a-R (μ2 = 0.28)
-75
-50
-25
0
25
50
75
1 2 3 4 5 6 7 8 9 101112131415
Ne
t Mo
ne
y H
old
ing
s
Period
Treatment 2b and 2b-R (μ2 = 0.75)
-75
-50
-25
0
25
50
75
1 2 3 4 5 6 7 8 9 101112131415
Ne
t Mo
ne
y H
old
ing
s
Period
Treatment 2c and 2c-R (μ2 = 5.07)
27
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.
28
Treatment 3: Conjectures
• 6. In Treatment 3 the unique equilibrium defined by the default penalties μ1 and μ2 is approached.
29
Figure 10: Holdings of Goods A and B in the two Runs of Treatment 3
0
10
20
30
40
50
0 10 20 30 40
Ho
ldin
gs
go
od
B
Holdings good A
Holdings of goods A and B in Run 1 of T3
0
10
20
30
40
50
0 10 20 30 40
Ho
ldin
gs
go
od
B
Holdings good A
Holdings of goods A and B in Run 2 of T3
30
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
0
10
20
30
40
50
0 10 20 30 40
Holdings of good A
Ho
ldin
gs
of
go
od
B
...
60%
70%
80%
90%
100%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Eff
icie
ncy
Period
Efficiency over Time in T3-R
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.
0
25
50
75
100
125
150
1 2 3 4 5 6 7 8 9 10 11
Cu
mu
lati
ve
Vo
lum
e
Period
Cumulative Trading Volume over time in T3
Run 1 good A Run 2 good A Equilibrium A
Run 1 good B Run 2 good B Equilibrium B
Efficiency over time in Treatment 3
60%
70%
80%
90%
100%
1 2 3 4 5 6 7 8 9 10 11
Period
Eff
icie
ncy
.
Run 1 Run 2
31
Treatment 3
• The unique equilibrium defined by the chosen penalty is approached in Treatment 3.
32
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.
33
Trading Screen without Fiat Money
34
Results Screen without Fiat Money
35
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 0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 5 39.3 44.3 49.3 54.3 59.3 64.3 69.3 74.3 79.3 84.3 89.3
10 63.2 68.2 73.2 78.2 83.2 88.2 93.2 98.2 103.2 108.2 113.2 15 77.7 82.7 87.7 92.7 97.7 102.7 107.7 112.7 117.7 122.7 127.7 20 86.5 91.5 96.5 101.5 106.5 111.5 116.5 121.5 126.5 131.5 136.5 25 91.8 96.8 101.8 106.8 111.8 116.8 121.8 126.8 131.8 136.8 141.8 30 95.0 100.0 105.0 110.0 115.0 120.0 125.0 130.0 135.0 140.0 145.0 35 97.0 102.0 107.0 112.0 117.0 122.0 127.0 132.0 137.0 142.0 147.0 40 98.2 103.2 108.2 113.2 118.2 123.2 128.2 133.2 138.2 143.2 148.2 45 98.9 103.9 108.9 113.9 118.9 123.9 128.9 133.9 138.9 143.9 148.9
Units
of B
you
hold
50 99.3 104.3 109.3 114.3 119.3 124.3 129.3 134.3 139.3 144.3 149.3
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
0 5 10 15 20 25 30 35 40 45 50 0 0.0 43.3 69.5 85.5 95.1 101.0 104.5 106.7 108.0 108.8 109.3 5 5.0 48.3 74.5 90.5 100.1 106.0 109.5 111.7 113.0 113.8 114.3
10 10.0 53.3 79.5 95.5 105.1 111.0 114.5 116.7 118.0 118.8 119.3 15 15.0 58.3 84.5 100.5 110.1 116.0 119.5 121.7 123.0 123.8 124.3 20 20.0 63.3 89.5 105.5 115.1 121.0 124.5 126.7 128.0 128.8 129.3 25 25.0 68.3 94.5 110.5 120.1 126.0 129.5 131.7 133.0 133.8 134.3 30 30.0 73.3 99.5 115.5 125.1 131.0 134.5 136.7 138.0 138.8 139.3 35 35.0 78.3 104.5 120.5 130.1 136.0 139.5 141.7 143.0 143.8 144.3 40 40.0 83.3 109.5 125.5 135.1 141.0 144.5 146.7 148.0 148.8 149.3 45 45.0 88.3 114.5 130.5 140.1 146.0 149.5 151.7 153.0 153.8 154.3
Units
of B
you
hold
50 50.0 93.3 119.5 135.5 145.1 151.0 154.5 156.7 158.0 158.8 159.3
36
Trading Screen with Fiat Money
37
Results Screen with Fiat Money
38
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
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 39.3 44.3 49.3 54.3 59.3 64.3 69.3 74.3 79.3 84.3 89.3
10 63.2 68.2 73.2 78.2 83.2 88.2 93.2 98.2 103.2 108.2 113.2 15 77.7 82.7 87.7 92.7 97.7 102.7 107.7 112.7 117.7 122.7 127.7 20 86.5 91.5 96.5 101.5 106.5 111.5 116.5 121.5 126.5 131.5 136.5 25 91.8 96.8 101.8 106.8 111.8 116.8 121.8 126.8 131.8 136.8 141.8 30 95.0 100.0 105.0 110.0 115.0 120.0 125.0 130.0 135.0 140.0 145.0 35 97.0 102.0 107.0 112.0 117.0 122.0 127.0 132.0 137.0 142.0 147.0 40 98.2 103.2 108.2 113.2 118.2 123.2 128.2 133.2 138.2 143.2 148.2 45 98.9 103.9 108.9 113.9 118.9 123.9 128.9 133.9 138.9 143.9 148.9
Units
of B
you
hold
50 99.3 104.3 109.3 114.3 119.3 124.3 129.3 134.3 139.3 144.3 149.3
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
0 5 10 15 20 25 30 35 40 45 50 0 0.0 8.5 13.7 16.9 18.8 19.9 20.6 21.0 21.3 21.5 21.6 5 1.0 9.5 14.7 17.8 19.7 20.9 21.6 22.0 22.3 22.4 22.5
10 2.0 10.5 15.7 18.8 20.7 21.9 22.6 23.0 23.3 23.4 23.5 15 3.0 11.5 16.7 19.8 21.7 22.9 23.6 24.0 24.3 24.4 24.5 20 3.9 12.5 17.7 20.8 22.7 23.9 24.6 25.0 25.2 25.4 25.5 25 4.9 13.5 18.6 21.8 23.7 24.8 25.5 26.0 26.2 26.4 26.5 30 5.9 14.5 19.6 22.8 24.7 25.8 26.5 27.0 27.2 27.4 27.5 35 6.9 15.4 20.6 23.8 25.7 26.8 27.5 27.9 28.2 28.4 28.5 40 7.9 16.4 21.6 24.7 26.6 27.8 28.5 28.9 29.2 29.3 29.4 45 8.9 17.4 22.6 25.7 27.6 28.8 29.5 29.9 30.2 30.3 30.4
Units
of B
you
hold
50 9.9 18.4 23.6 26.7 28.6 29.8 30.5 30.9 31.2 31.3 31.4
Thank You.
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