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Flourishing as Productive Tension: Theory and Model

  • Aviad Heifetz EMAIL logo and Enrico Minelli
Published/Copyright: November 20, 2019
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 20 Issue 1

Abstract

Based on insights from psychology, we argue that individual well-being has two incommensurable dimensions, gratification and flourishing. We provide a model in which the individual’s vitality is tunneled to different goals/practices. Gratification is represented by discharge of flow, while flourishing corresponds to generated power. The two dimensions are correlated in a way that depends on the characteristics of the available goals/practices, and their relationship need not be monotonic. We also extend the model to social interaction, prove existence of an equilibrium and show the possibility of ‘flourishing traps’.

Keywords: gratification; flourishing; flow; social network; flourishing trap

Acknowledgements

We would like to thank Steffen Huck, Carmen Marchiori, Herakles Polemarchakis, and two anonymous referees for very useful comments.

Appendix

A

Proof of Proposition 1

In the case of two pipes we have

R=11R1+1R2=11Rp1+Z1+1Rp2+Z2

and

V=Q01Rp1+Z1+1Rp2+Z2

The flow through pipe k is

Qk=VRpk+Zk=Q01Rp1+Z1+1Rp2+Z2Rpk+Zk

The hydroelectric power produced by turbine k is

Pk=ZkQk2=ZkQ01Rp1+Z1+1Rp2+Z2Rpk+Zk2

The overall electric power produced by the turbines – representing the individual’s flourishing – is

P=∑k=12Pk=Z1Q01Rp1+Z1+1Rp2+Z2Rp1+Z12+Z2Q01Rp1+Z1+1Rp2+Z2Rp2+Z22=Z1Q01+Rp1+Z1Rp2+Z22+Z2Q01+Rp2+Z2Rp1+Z12

Recalling (footnote 6) that

Rpk=cAk2

the ‘attractiveness’

αk=1Rpk

of goal/practice k is monotonically increasing in its value/meaningfulness (i. e. in the corresponding pipe’s cross-section area) Ak.

The partial derivative of P2 w.r.t. α2 is

∂P2∂α2=2Q02α1α2Z2α1Z1+12α1+α2+α1α2Z1+α1α2Z23>0

so the more attractive is goal/practice 2 deemed, the higher the power produced by turbine 2. At the same time, the partial derivative of P1 w.r.t. α2 is

∂P1∂α2=−2Q02α12Z1α1Z1+1α2Z2+1α1+α2+α1α2Z1+α1α2Z23<0

so the more attractive is goal/practice 2, the lower the power produced by turbine 1. Together, the partial derivative of overall flourishing, P=P1+P2 w.r.t. α2 is

∂P∂α2=2Q02α1α1Z1+1α2Z2−α1Z1α1+α2+α1α2Z1+α1α2Z23

Since all variables are non-negative,

∂P∂α2=0

only when

α2=α1Z1Z2

and this is a minimum point of flourishing (overall turbines’ power) because

∂2P∂α22=
=2Q02α1α1Z1+13α12Z12+3α12Z1Z2−2α2α1Z1Z2+3α1Z1−2α2α1Z22+α1Z2−2α2Z2α1+α2+α1α2Z1+α1α2Z24
=α2=α1Z1Z22Q02Z24α12α1Z1+12Z1+Z23>0

Reverting back to the original variables A1,A2, minimum flourishing is reached at

A2∗=A1Z1Z2

ᅛᅛᅛᅛᅛᅛᅛᅛ                                                                                                                                                        ■

Proof of Proposition 2

When multiplying Z1,Z2,Rp1,Rp2 (and hence R) by the same factor β > 1, flourishing

P=Z1Q01+Rp1+Z1Rp2+Z22+Z2Q01+Rp2+Z2Rp1+Z12

will also increase by the factor β – as long as

Q0R=Q01Rp1+Z1+1Rp2+Z2≤Vˉ

However, once Q0R=Q01Rp1+Z1+1Rp2+Z2>Vˉ, the flow through pipe k will be VˉRpk+Zk, and overall flourishing will be

P=Z1VˉRp1+Z12+Z2VˉRp2+Z22

thus decreasing by the factor β > 1 when Z1,Z2,Rp1,Rp2 are all multiplied by the factor β.ᅛᅛᅛᅛᅛᅛ                   ■

Proof of Social Equilibrium Existence

For each individual n and goal/practice k,

(6)Akn=Fkn∑m≠nPkmdkmn

where Fkn is a non-negative, increasing continuous function of the other individuals’ flourishing in goal/practice k, Pkm, divided by their corresponding distance dkmn. Recall also that overall flourishing of individual n is

Pn=∑k=1KPkn

and her vitality flow is

(7)Q0n=F0n∑m≠nPmd0mn=F0n∑m≠n∑j=1KPjmd0mn

where F0n is a non-negative, increasing continuous function of the other individuals’ Pm overall flourishing, divided by their corresponding distance d0mn.

From the definitions in the main text we also have:

Rpkn=cAkn2
(8)Rkn=Rpkn+Zkn=c+Akn2ZknAkn2
(9)Rn=1∑j=1K1Rjn=1∑j=1KAjn2c+Ajn2Zjn

so that

RnRkn=1∑j=1KAjn2c+Ajn2Zjnc+Akn2ZknAkn2=Akn2c+Akn2Zkn∑j=1KAjn2c+Ajn2Zjn

and

(10)Qkn=minQ0nRnRkn,gD+ℓRkn=minF0n∑m≠n∑j=1KPjmd0mn⋅Akn2c+Akn2Zkn∑j=1KAjn2c+Ajn2Zjn,Akn2gD+ℓc+Akn2Zkn.

Notice, by (10) that Qkn is therefore bounded from above by

Qˉkn=VˉZkn

(since Rkn≥Zkn) and below by 0.

The flourishing of individual n in goal/practice k is in turn defined by:

(11)Pkn=ZknQkn2

which is therefore bounded from above by

Pˉkn=ZknQˉkn2=Vˉ2Zkn

and below by 0.

From (6) it hence follows that Akn is bounded from above by

Aˉkn=Fkn∑m≠nPˉkmdkmn

and below by 0.

Then for the compact convex domain

D=∏n,k[0,Pˉkn]×[0,Qˉkn]×[0,Aˉkn]

the continuous map

Φ:D→D

defined according to (11), (10) and (6) by

ΦknPjm,Qjm,Ajmm,j=ZknQkn2,minF0n∑m≠n∑j=1KPjmd0mn⋅Akn2c+Akn2Zkn∑j=1KAjn2c+Ajn2Zjn,Akn2Vˉc+Akn2Zkn,Fkn∑m≠nPkmdkmn

has a fixed point by Brouwer’s fixed-point theorem, which constitutes a social equilibrium.ᅛᅛᅛ                     ■

Proof of Proposition 3 (Existence of Flourishing Traps)

Assume a social network with n = 1, 2 symmetric individuals, where

F01=F02≡Q0F11=F12≡A1

are constant, i. e. the individuals do not influence each other’s vital flow Q0=Q01=Q02 (do not serve for one another as vitality-enhancing parent figure role model), and do not serve either as role models for goal/practice 1 whose value/meaningfulness is fixed at A1 for both; but do serve as role models for one another regarding goal/practice 2, where

d212=d221≡d2

and where the functions

F21⋅=F22⋅=⋅14

are the fourth root. Assume, for simplicity, that Vˉ is large enough so as not to bind Q, i. e. that Q = Q0.

Given the symmetry between the two individuals in the network, when looking for a symmetric social equilibrium we have to solve

A2=P2d214=Z2d2Q01+cA22+Z2cA12+Z1214

or

A22=Q0Z2d21+cA22+Z2cA12+Z1

which has two solutions:

(12)A˙2=0A¨2=1A12cZ1+Z2+1Q0Z2d2A12cZ1+1−A12c

with the corresponding flourishing levels

P˙=Q02Z1P¨=Q02Z1Z1+cA12Z2+A12cZ1+Z2+1Q0A12cZ1+1Z2d2−A12c+12+Q02Z2Z2+A12cZ1+Z2+1Q0A12cZ1+1Z2d2−A12cZ1+cA12+12

There exist values of Z1, A1, Z2 and d2 for which [10] we have

A¨2>A˙2=0,P¨>P˙

In such a social network the first social equilibrium with values A˙2,P˙ is a ‘flourishing trap’, in which goal/practice 2 is deemed as irrelevant and practically non-existent, and consequently flourishing is lower than in the second social equilibrium with values A¨2,P¨, in which practice 2 is respected and followed.

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Published Online: 2019-11-20

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/bejte-2018-0162
Keywords for this article
gratification; flourishing; flow; social network; flourishing trap

Articles in the same Issue

  1. Research Articles
  2. Optimal Forestry Contract with Interdependent Costs
  3. Bi and Branching Strict Nash Networks in Two-way Flow Models: A Generalized Sufficient Condition
  4. Pay-What-You-Want in Competition
  5. Two Rationales for Insufficient Entry
  6. Students’ Social Origins and Targeted Grading
  7. Pricing, Signalling, and Sorting with Frictions
  8. On the Economic Value of Signals
  9. The Core in Bertrand Oligopoly TU-Games with Transferable Technologies
  10. Reasoning About ‘When’ Instead of ‘What’: Collusive Equilibria with Stochastic Timing in Repeated Oligopoly
  11. Timing Games with Irrational Types: Leverage-Driven Bubbles and Crash-Contingent Claims
  12. Costly Rewards and Punishments
  13. Blocking Coalitions and Fairness in Asset Markets and Asymmetric Information Economies
  14. Strategic Activism in an Uncertain World
  15. On Equilibrium Existence in a Finite-Agent, Multi-Asset Noisy Rational Expectations Economy
  16. Optimal Incentives Under Gift Exchange
  17. Public Good Indices for Games with Several Levels of Approval
  18. Vagueness of Language: Indeterminacy under Two-Dimensional State-Uncertainty
  19. Winners and Losers of Universal Health Insurance: A Macroeconomic Analysis
  20. Behavioral Theory of Repeated Prisoner’s Dilemma: Generous Tit-For-Tat Strategy
  21. Flourishing as Productive Tension: Theory and Model
  22. Notes
  23. A Note on Reference-Dependent Choice with Threshold Representation
  24. Regular Equilibria and Negative Welfare Implications in Delegation Games
  25. Unbundling Production with Decreasing Average Costs
  26. A Simple and Procedurally Fair Game Form for Nash Implementation of the No-Envy Solution
  27. Decision Making and Games with Vector Outcomes
  28. Capital Concentration and Wage Inequality
  29. Annuity Markets and Capital Accumulation
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