# The Left- and Right-Wing Political Power Design: The Dilemma of Welfare Policy with Low-Income Relief

## Abstract

**:**

## 1. Introduction

**Table 1.**Numerical simulation behind the left- and right-wing political power design; LWP—left-wing politicians, RWP—right-wing politicians.

Obtained by Means of Income Density Distribution (Figure 1); Personal Allowance φ = 4.03; θ = 61.9; h = −0.18; m = 2.07; r = ${\scriptscriptstyle \raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}$: a Proportion to (ξ − σ) | Policy of Equal (Politically Symmetric) Power | LWP Proposal Accepted by RWP | Proposal Minimizing Wealth-tax | Poverty Line = ½ of Median Income | RWP Proposal Accepted by LWP | Policy of Disagreement, the Breakdown | |
---|---|---|---|---|---|---|---|

η | λ_{1}, q = 5% | λ, q ≈ 0% | ½μ | λ_{2}, q = 5% | δ | ||

Poverty line; welfare policy | $\mathsf{\xi}=$ | 79.23 | 40.79 | 45.50 | 41.15 | 50.28 | 6.39 |

Poverty rate: percentage of citizens below the poverty line | 47.36% | 15.73% | 19.15% | 15.99% | 22.81% | 0.41% | |

Political power of left-wing politicians | $\mathsf{\alpha}(\mathsf{\xi})$ | 0.50 | 0.18 | 0.21 | 0.18 | 0.24 | Not defined |

LI netto; the after-tax residue of ξ | $\mathrm{u}(\mathsf{\xi})$ | 58.02 | 31.02 | 34.50 | 31.29 | 37.99 | 6.44 |

Account for public goods expenses | $\mathrm{g}(\mathsf{\xi})$ | 19.02 | 27.63 | 26.70 | 27.56 | 25.75 | −2.49 |

Account for LI relief transfers | $\mathrm{B}(\mathsf{\xi})$ | 10.61 | 1.57 | 2.17 | 1.62 | 2.91 | 0.01 |

Account for public spending, the size of the wealth-pie | $\mathrm{z}(\mathsf{\xi})$ | 29.63 | 29.20 | 28.87 | 29.18 | 28.66 | −2.48 |

Average taxable income—the wealth amount | $\mathrm{W}(\mathsf{\xi})$ | 105.04 | 109.95 | 108.86 | 109.87 | 107.88 | 120.46 |

Wealth-tax, marginal tax rate | $\mathsf{\tau}(\mathsf{\xi})$ | 28.21% | 26.56% | 26.52% | 26.56% | 26.56% | −2.06% |

**Figure 1.**At the sample $\mathrm{P}(\mathsf{\sigma},\mathsf{\theta}+\mathrm{h}\cdot {\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathsf{\mu})$ of the income density distribution, $\mathsf{\mu}$ solves the equation ${\int}_{0}^{\mathrm{x}}\mathrm{P}(\mathsf{\sigma},\mathsf{\theta}+\mathrm{h}\cdot \mathrm{x})}\mathrm{d}\mathsf{\sigma}=0.5$ for $\mathrm{x}$; $\mathsf{\mu}=82.30$ . Appendix A1 contains the analytical form for the sample expression in Figure 1.

## 2. Preliminaries

#### Settings

## 3. Relevant Trends and Issues

- Fiscal policy: During the delivery to its final destinations, provided that the books accounting for the relief payments finance have been balanced a priori, the wealth-pie must remain balanced throughout and in spite of volatility in the economy;
- Negotiations: The left- and right-wing political bargaining on how to share the wealth-pie complies with the rules and norms of the alternating-offers bargaining game;
- Pre-equity of breakdown: Political breakdown, or threat, directly affects the bargaining solution. Pre-equity guarantees equal conditions for players before the bargaining game commences;
- Political power design: Bringing a motion to a vote is necessary to address the majority opposition to high taxes and excessive public spending. Whether it is viewed as positive or negative, or whether it ought to be acknowledged or not, rejected or accepted, this motion must be politically designed in advance.

#### 3.1. Fiscally Safe Welfare Policies, to Be Continued in Section 4

#### 3.2. Bargaining the Welfare State Rules and Norms, to be Continued in Section 5

- • X
- is the first actor’s share of $1, with $\mathsf{\alpha}$ as the first actor’s asymmetric power indicator, $0\le \mathrm{x}\le 1$, $0\le \mathsf{\alpha}\le 1$;
- • u(x)
- denotes the first actor’s payoffs of the first actor’s $1 share $\mathrm{x}$;
- • y
- is the second actor’s share of $1, where $1-\mathsf{\alpha}$ is the second actor’s asymmetric power indicator, $0\le \mathrm{y}\le 1$;
- • g(y)
- denotes the second actor’s payoffs of the second actor’s $1 share $\mathrm{y}$.

Leibenstein argued that there are two components to the productivity problem: one relates to the determination of the size of the pie, while the second relates to the division of the pie. Looked upon independently, all agents can jointly gain by increasing the pie size…the situation need not be a zero-sum game. Tactics that determine pie division can affect the size of the pie. It is this latter possibility that is especially significant.

#### 3.3. Pre-Equity of Political Breakdown

We can interpret a breakdown as the result of the intervention of a third party, who exploits the mutual gains. A breakdown can be interpreted also as the event that a threat made by one of the parties to halt the negotiations is actually realized. This possibility is especially relevant when a bargainer is a team (e.g., government), the leaders of which may find themselves unavoidably trapped by their own threats.

**Figure 4.**The graph depicts two different motions for a vote. For the higher tax τ = 29.01%, marked by the horizontal line, and the lowest tax τ = 26.52%, marked by the vertical dash. Indicated by $\to $ , at cross-points of the contract curve with the horizontal line, we observe controversial expectations of voters. The shares of lower basic but higher public goods are shown on the left, while this payoff reverses towards the right side of the graph, as the shares of basic goods increase while those of public goods decrease. Thus, the higher tax τ = 29.01% cannot lead to a political consent, in line with Observation 5.

#### 3.4. Voting and Political Power Design, to Be Continued in Section 6

## 4. Analysis of Fiscally Safe Welfare Policies, Continued from Section 3.1

#### Idempotent Rules and Norms of Wealth Redistribution

- ϕ
- the personal allowance establishing the tax bracket $\left[\mathsf{\phi},\infty \right)$; it is an ex-ante control (tuning) variable, $0<\mathsf{\phi}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}<\mathsf{\xi}$;
- ξ
- the income frame, the poverty line; a policy determining who is living in poverty, as well as the choice or the control parameter;
**z**- the size $\mathbf{z}=\mathsf{\tau}\cdot \mathrm{W}(\mathsf{\xi})$ of the wealth-pie; the amount of the wealth-pie that is equal to public spending per capita when taxes are proportional;
- X
- the share of the wealth-pie of size
**z**; a portion $\mathrm{X}$ of**z**to be deposited in favor of the left-wing politicians for funding the relief payments, $0\le \mathrm{x}\le 1$; - α
- the political power of the left-wing politicians, $0<\mathsf{\alpha}<1$;
- τ
- the marginal tax rate, the rate $\mathsf{\tau}(\mathsf{\xi},\mathrm{x})$ of the wealth amount $\mathrm{W}(\mathsf{\xi})$ determined by (1);
- u
- the after-tax residue of the income frame equal to the poverty line $\mathsf{\xi}$, the wants function $\mathrm{u}(\mathsf{\xi},\mathrm{x})$ of the left-wing politicians, as determined by (2) and (3);
- g
- the objective function $\mathrm{g}(\mathsf{\xi},\mathrm{x})$ of the right-wing politicians, determined by (1) and (2); the account for the refund of public goods expenses per capita.

## 5. Analysis of the Welfare State Bargaining and Rules and Norms, Continued from Section 3.2

#### 5.1. Left- and Right-Wing Politicians’ Bargaining Procedure

**Figure 5.**The aspirations of left-wing politicians expressed when opposing the right-wing political objectives are depicted on the vertical and horizontal axes, respectively. The graph shows the contract curve sloping from ${\mathsf{\xi}}_{2}$ toward ${\mathsf{\xi}}_{1}$ , projected on the surface of basic goods vs. vital goods—the projection of efficient poverty lines $\mathsf{\xi}\in \left[{\mathsf{\xi}}_{1},{\mathsf{\xi}}_{2}\right]$ resolving the contract constraint (5).

#### 5.2. Alternating-Offers Bargaining Game Analysis

- Politician No. 1, u
- the left-wing political aspirations, the marginal citizens’ $\mathsf{\sigma}=\mathsf{\xi}$ after-tax residue, basic necessities of the needy, cost of living;
- Politician No. 2, g
- the right-wing political objective, expenses that benefit all citizens—expenses upon vital goods alone, without relief payments;
- Third Partaker, q, τ
- voters’ electoral maneuvering facing higher taxes $\mathsf{\tau}$ expressing an implicit risk $0<\mathrm{q}<<1$ of the negotiations collapsing prematurely.

$\begin{array}{c}\mathrm{Q}(\mathsf{\xi},\mathsf{\tau},\mathrm{g})=0\\ \mathrm{L}(\mathsf{\xi},\mathrm{x},\mathrm{u})=0\\ \mathrm{D}(\mathsf{\xi},\mathrm{x},\mathrm{u})=0\end{array}$ | $\begin{array}{c}\to \mathbf{\text{Delivery (1);}}\hfill \\ \to \mathbf{\text{Volatility (4);}}\hfill \\ \to \mathbf{\text{Contract (5);}}\hfill \end{array}\}$ | enforcing constraints on rules and norms of the wealth redistribution, |

$\mathrm{u}(\mathsf{\xi})=\mathsf{\xi}-\frac{(\mathsf{\xi}-\mathsf{\phi})}{\mathrm{v}(\mathsf{\xi})}$, where $\nu (\mathsf{\xi})=1+\left(\mathsf{\xi}-\mathsf{\phi}\right)\cdot \left(\frac{\dot{\mathrm{B}}(\mathsf{\xi})}{\mathrm{B}(\mathsf{\xi})}-\frac{\dot{\mathrm{W}}(\mathsf{\xi})}{\mathrm{W}(\mathsf{\xi})}\right)$4; $\mathsf{\tau}(\mathsf{\xi})=\frac{1}{\nu (\mathsf{\xi})}$. |

$\mathrm{g}(\mathsf{\xi})=\frac{\mathrm{W}(\mathsf{\xi})}{\nu (\mathsf{\xi})}-\mathrm{B}(\mathsf{\xi})$; the size of wealth-pie $\mathbf{z}(\mathsf{\xi})=\mathrm{B}(\mathsf{\xi})+\mathrm{g}(\mathsf{\xi})=\frac{\mathrm{W}(\mathsf{\xi})}{\nu (\mathsf{\xi})}$. |

## 6. Analysis of Voting and Political Power Design, Continued from Section 3.4

To count on ${\mathrm{x}}^{\circ}$ share of $1 is a realistic attitude toward the first actor’s position of negotiations. Indeed, even if the second actor might have a stronger negotiating power than the first actor, ${\mathsf{\alpha}}^{\circ}<1-{\mathsf{\alpha}}^{\circ}$, the first actor, sooner rather than later, might predict the second actor’s preferences and thus force a concession.

## 7. Discussion

## 8. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## Appendix A1. Example and Results

## Appendix A2. Simulation Foundation and Illustration

- η
- the policy on poverty with equal left- and right-wing political power; the left- and right-wing political organizations are in symmetrical positions or in equal roles;
- λ
_{1} - the outcome of the alternating-offers game—representing what the right-wing politicians accept;
- λ
- the policy on poverty minimizing wealth-tax;
- ½μ
- ½ of the median income, indicating that half of the population earns income above $\mathsf{\mu}$, while the income of the remaining half is below $\mathsf{\mu}$;
- λ
_{2} - the outcome of the alternating-offers game—representing what the left-wing politicians accept;
- δ
- the least desirable outcome, resulting in the policy breakdown or disagreement, which naturalizes the risk of negotiations’ premature collapse, caused, for instance, by mutual traps.

## Appendix A3. Verification

Replace | to implement an improved | by | to make a decline in |

– | better off | – | worse off |

– | improve improvement | – | decline deterioration |

– | to claim for relief payments | – | that relief payments have been revoked |

– | defici | – | surplus |

– | ≥, > | – | ≤, < |

Transpose | an increase | with | a decrease |

- (
**i**) - ${\frac{\partial}{\partial \mathsf{\xi}}\mathrm{L}(\mathsf{\xi},{\mathrm{x}}^{\mathrm{o}},{\mathrm{u}}^{\mathrm{o}})|}_{\mathsf{\xi}=\mathsf{\xi}\xb0}=0$, where ${\mathrm{u}}^{\mathrm{o}}=\mathrm{u}({\mathsf{\xi}}^{\mathrm{o}},{\mathrm{x}}^{\mathrm{o}})$ provided that
- (
**ii**) - ${\frac{\partial}{\partial \mathrm{u}}\mathrm{L}({\mathsf{\xi}}^{\mathrm{o}},{\mathrm{x}}^{\mathrm{o}},\mathrm{u})|}_{\mathrm{u}=\mathrm{u}\xb0}\ne 0$.

**Proof**

**ii)**holds. Let (

**i)**solve for ${\mathsf{\xi}}^{\mathrm{o}}$ at the fiscally idempotent outcome $\mathsf{\phi},{\mathsf{\xi}}^{\mathrm{o}}\Rightarrow {\mathbf{z}}^{\mathrm{o}},{\mathrm{x}}^{\mathrm{o}},\mathsf{\alpha},{\mathsf{\tau}}^{\mathrm{o}},\langle {\mathrm{u}}^{\mathrm{o}},{\mathrm{g}}^{\mathrm{o}}\rangle $. Combining (

**i**) and A(2), we conclude that

## Appendix A4. Mathematical Derivation

$\mathsf{\tau}\cdot \mathrm{W}(\mathsf{\xi})=\mathrm{B}(\mathsf{\xi})+\mathrm{g}$ | Delivery constraint: the size of the welfare pie, i.e., the average amount of tax returns, is equal to the sum of the average monetary value per capita of primary goods and the average of non-primary goods g. |

$\mathrm{B}(\mathsf{\xi})=\mathrm{x}\cdot \mathsf{\tau}\cdot \mathrm{W}(\mathsf{\xi})$ | Budget constraint imposed on the relief payments finance in accordance with the share x of the wealth-pie—the tax-revenue. |

$\mathrm{u}=(1-\mathsf{\tau})\cdot (\mathsf{\xi}-\mathsf{\phi})+\mathsf{\phi}$ | Stability constraint that determines fiscally idempotent policy ξ. |

$\mathrm{u}=\mathsf{\xi}-\mathsf{\tau}\cdot (\mathsf{\xi}-\mathsf{\phi})$ | After-tax residue constraint: an alternative form of stability constraint, where u is the after-tax position of a marginal citizen with income σ = ξ, which concedes with the left-wing political aspirations. |

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^{1}Below, we continue to refer to the average taxable income as “wealth”.^{2}Status and control variables are the prerogatives of control theory.^{3}We already highlighted the worsening quality of welfare services for all citizens when the LI level is “climbing” high.^{4}$\mathrm{\pm}$ Rates $\dot{\mathrm{W}}(\mathsf{\xi})\le 0$, $\dot{\mathrm{W}}(\mathsf{\xi})\ge 0$ of the changes in the wealth amounts $\mathrm{W}(\mathsf{\xi})$ are essential for the analysis, whereas the function $\mathrm{B}(\mathsf{\xi})$ is valid only when $\dot{\mathrm{B}}(\mathsf{\xi})>0$, and $0<\mathsf{\phi}<\mathrm{u}<\mathsf{\xi}$.^{5}Table 1 was created by numerical simulation carried out upon imaginary distribution of citizens’ incomes.^{6}Poverty rate determines the percent of anyone who lives with income below the official poverty line. The poverty line separates the rich (those with an income above the poverty line), from the less fortunate (having income below the line).

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Mullat, J.E. The Left- and Right-Wing Political Power Design: The Dilemma of Welfare Policy with Low-Income Relief. *Soc. Sci.* **2016**, *5*, 7.
https://doi.org/10.3390/socsci5010007

**AMA Style**

Mullat JE. The Left- and Right-Wing Political Power Design: The Dilemma of Welfare Policy with Low-Income Relief. *Social Sciences*. 2016; 5(1):7.
https://doi.org/10.3390/socsci5010007

**Chicago/Turabian Style**

Mullat, Joseph E. 2016. "The Left- and Right-Wing Political Power Design: The Dilemma of Welfare Policy with Low-Income Relief" *Social Sciences* 5, no. 1: 7.
https://doi.org/10.3390/socsci5010007