Trust in the Firm
Some Remarks on the Mathematical Economics of Trust
Horst Albach
Email: [email protected]; Address: Waldstr. 49, 53177 Bonn
Honorary Professor at HHL Leipzig Graduate School of Management
Professor Emeritus at the Humboldt-University Berlin
Talk given at the CASiM Conference 2012
‘The Role of Trust in Business Economics’ on June 28, 2012
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
1/9
Examples of Management Methodology
of the 21st Century
dynamic production function
neo-classical model of the firm
control theory
chaos theory
game theory
principal-agent-theory
prisoner’s dilemma
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
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Prisoner’s Dilemma with Opportunistic Behavior:
Non-Cooperative Game in Static Form
Assumption: Each player wants to maximize his utility (i.e. his individual
profit), even though this may lead to the death of the other player.
Table: Pay-Off Matrix of the Two Players A and B
Actions (A,B)
B1
B2
A1
(-30,-30)
(-3,-60)
A2
(-50,-2)
(-5,-4)
The result of the game is (A1,B1): each player has a loss of 30 units.
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
3/9
Prisoner’s Dilemma with Fellow-Feeling
If player A applies the weight of α1 = 0.8 to his own pay-off and the
weight of α2 = 0.2 to player B’s pay-off, then we get the pay-off matrix for
player A shown below.
Table: Pay-Off Matrix of Player A for α1 = 0.8 and α2 = 0.2
Actions (A,B)
B1
B2
A1
-30
-14.4
A2
-40.5
-4.8
Now A2BA becomes the optimal solution.
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
4/9
Maximizing Aristotle’s Utility Function with Costs
U(x) = α1
p(x) x − K (x)
|
{z
}
+ α2 |{z}
cx
= P(x,p(x),K (x)) = profits
(1)
= F (x)
= fellow-feeling
with
α1 , α2 = weights of profits P and fellow-feeling F
x = production volume
p = p(x) = A − B x = price of the product with production volume x,
K (x) = C x + D x 2 + FK = cost function
FK = fix costs,
A, B, C , D = positive constants.
Maximizing Aristotle’s utility function (1) with respect to x yields:
x∗ =
A−C
α2
c
A−C
+
2 (B + D) α1 2 (B + D) 2 (B + D)
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
with
α2 = 1 − α1
CASiM Conference 2012
5/9
Weighted Profit, Weighted Fellow-Feeling and Total Utility
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Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
6/9
Two-Stage-Two-Persons-Principal-Agent Model
The Lagrange objective function of the principal P is given by:
L s1 (x1 , `1 ), s2 (x1 , `1 , x2 , `2 ), a1 , a2 (x1 , `1 ), λ, µ1 , µ
e2 (x1 , `1 )
Z
Z G1 x1 − s1 (x1 , `1 ) + G2 x2 − s2 (x1 , `1 , x2 , `2 ) f2 x2 , `2 |a2 (x1 , `1 ) f1 (x1 , `1 |a1 )
=
(Z Z
+λ
U1 s1 (x1 , `1 ) + U2 s2 (x1 , `1 , x2 , `2 ) f2 x2 , `2 |a2 (x1 , `1 ) − V2 a2 (x1 , `1 )
)
b
·f1 (x1 , `1 |a1 ) − V1 (a1 ) − H
(Z +µ1
U1 s1 (x1 , `1 ) +
Z
U2 s2 (x1 , `1 , x2 , `2 ) f2 x2 , `2 |a2 (x1 , `1 ) − V2 a2 (x1 , `1 )
)
·f10 (x1 , `1 |a1 )
Z
Z
+
−
µ2 (x1 , `1 )
V10 (a1 )
U2 s2 (x1 , `1 , x2 , `2 ) f20 x2 , `2 |a2 (x1 , `1 ) − V20 a2 (x1 , `1 ) f1 (x1 , `1 |a1 ).
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
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Two-Stage-Two-Persons-Principal-Agent Model
Optimizing the Lagrange function gives the optimal intensity of work for
the agent A
Z G1 x1 − s1 (x1 , `1 ) +
(Z +µ1
Z
G2 x2 − s2 (x1 , `1 , x2 , `2 ) f2 x2 , `2 |a2 (x1 , `1 ) f10 (x1 , `1 |a1 )
U1 s1 (x1 , `1 ) +
Z
U2 s2 (x1 , `1 , x2 , `2 ) f2 x2 , `2 |a2 (x1 , `1 ) − V2 a2 (x1 , `1 )
)
·f100 (x1 , `1 |a1 )
−
V100 (a1 )
=0
and the optimal wage strategy for the principal P
Z Z
Z
G2 x2 − s2 (x1 , `1 , x2 , `2 ) f20 x2 , `2 |a2 (x1 , `1 ) f1 (x1 , `1 |a1 ) + µ2 (x1 , `1 )
Z
·
U2 s2 (x1 , `1 , x2 , `2 ) f200 x2 , `2 |a2 (x1 , `1 ) − V200 a2 (x1 , `1 ) f1 (x1 , `1 |a1 ) = 0.
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
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Conclusion
The paper has shown:
1
Trust is an economic variable.
2
There are many mathematical methods that all come to the same
conclusion: Trust matters.
3
Mathematical methods prove that trust is essential for the firm’s
competitiveness.
4
Mathematical formulations of Aristotle’s ethics and Adam Smith’s
‘fellow-feeling’ prove the importance of Ethics of the Firm as a basis
for responsible management.
5
Long-term relationships reduce the moral hazard problem and thereby
reduce transaction costs of the firm.
6
Good management means management of trust within the firm and
of the long-term relationships with the firms shareholders.
7
Management of resilience and sustainability is based on management
of trust.
Prof. (em.) Dr. Dr. h.c. mult. Horst Albach
() Mathematical Economics of Trust
CASiM Conference 2012
9/9