Optimal Linear Income Taxation PDF

Reality  Texrate depends on your income level linear Tex  Linear Tex the tax rate is independent of income but the total taxpa...

Reality  Texrate depends on your income level linear Tex  Linear Tex the tax rate is independent of income but the total taxpayment increases withyour income Mary's salary is 10 she 10 x 10 1 Total taxpayment pays he Peter's salary is 100 pays 10 x.ISoo co Usually this is not optimal in improving welfare Because the redistribution a week Linear tax system is good for efficiency since it power minimizes the disincentive to work the distortion of labour tax system is progressive The tax rate increases with income In reality our T Problem flexible the notation Tiz where z let's define the tax in is a way using the income reported by the agent Ritt where Ret Egf taf measures how much the agent can get out of total income 2 After toxincome  if TCA is differentiable ddTtz represents the mongrel tox rate It measures how much the agent gets taxed out one additional dollar of income Taxation in a model with no behavioral responses no labor supply We start with a simple version of an optimal income taxation problem that ignores the labor supply response to taxation or tax rates Suppose the agent has utility uic smh thet vice o and u c so diminishing marginel utility or concave function Labor does not enter the function and it is supplied inelastically constant supply of utility labor force regardless of the income The agent consumes everything that is left after taxes so that C Z there is no saving After tax income The is populated economy several agents and their income is by distributed to hit according with 2 e o.o pdf distribution we study the problem of a government whose goal is to maximize the total utility of the Imax social welfare economy Evey agent in the economy is equally weighted smh that Ucc hczydz sum of everyone's utility utilitarian  we call this type of social welfare function utilitarian The government targets a level of tax revenue E and its budget constraint is Tax payment foreachperson Targeted government T CZ hez dz E expenditure education e.g medical collected tax from everyone Total Tax revenue The Lagrangian for the problem L htt dz ucc If 1oz hee dz E objective function Tax budget constant social welfarefanton for the government where X is constant across individuals and measures the value of government revenues in equilibrium The optimal choice of T't delivers the following first order condition L I ucc htt dz X TCH ha DZ E Z Tiz ITCH hit de XEN constant 2L Ty z f U IZ TIZ x X hot O JU LUIZ TIZ 2CZ TCH U Z Tiz L l 8TH J CZ Ta  Rearrange the terms u IZ Tiz X IZ Equilibrium Condition Notice that since 1 is constant and all agents have the same preferences or function the condition implies that utility equilibrium consumption is equalized across all indirdnels 14 Assume Za s z from the done result w U Zz Taza X i U IZ Tct X i Z Tiz c U Zz T Zz U Z Taz U so I n is a strictly decreasing function  ng Zz T Zu Z T Z z C Equal consumption acroes agents too tax rate Suppou TIZ t z 2z t Zz Zi Tz Za Z t Zz Z t 1 or too optimal linear tax rate This is direct the utilitarian social welfare function a consequences of and the concavity of the utility  Intuition of why too tax rate is optimal who would otherwise suppose that we faxed a rich indolent have a high level of consumption to redistribute to a poor who would otherwise have low The marginal utility gain of consumption the poor would be higher than the Margaret utility toes of the rich if the utility has decreasing marginal returns 1 a co implied by the conoarity function of the utility levels This implies that until all consumption are equalized across the economy the government can increase social welfare through redistribution from the rich to the poor Redistribute conspton MU gan of the MU loss of the stops when poor nah  the government every agent has Since the same weight in social welfare function the optimal policy will treat gain for individual equelly Thos B no the government all from giving a higher level of consumption to a particular goop of all of indirdnd Tax rate 1007 to collect the is income and redistribute Godly to all yuh Redistribute Consptin 100 tax rate Tax Payment 4110 55 A to Z co B too 2 Revenue 110  In reality this is not practiced Person B is Behavior Response unhappy by reducing his work effort J Tiz hit dz E head't E J IZ c c Inez dz J Z head't E total population average income assume it is I c E E  Prograsive Taxation A progressive tax is a tax that imposes a lower marginal tax rate on low income earners compared to those with a higher income making it based on the taxpayer's ability Taiwan has implemented a tax system progressive for indwhel income taxes Brackets Annual Income Magnet tax rates 1st 0 5 540 ooo 2nd 540001 1210000 12 3rd 1210001 2420000 2070 4th 2420001 4530000 30 5th 4530001 1031 oooo 407 6th 10310001 and over 45 600,000 540.000 t 60,000 Tex 540.000 570 t 60000 x 12  After tax income ya 45 cslope zero tax i or ya y rate 1 n I Income Brackets ly 1st Bracket After tax income ye Cl 5 y 0 95 y 2nd Bracket ya I 12 y o 88 y 3rd Bracket ya I l 207 y 0.8 y we use the following labor income tax function to approximate the progressive tax system Let denote pre tax income and income y 1 O ya after tax ya Oo y The parameter Oo captures the level of taxes  ya i't The 0 the parameter captures the progress.nity f taxes or curvature Higher Or more progressive a y o 5 0 y if 9 so o ga o.gl O Tax function T O y y y pre tax After tax income income dT Tdy we denote by Tig's the magnet tax rate and by the tax rate a household with Tig average income y pays  Average tax rate Topayment Income y Q Y Eog T O l Dog Marginal Tax rate did d Additional tax payment for 0 an extra 1 you earn ly Oof I 1 I I O I I Oo l a 5 12 207 3 407 marginal tax rate y 0.95 0.88 0.8 of o b After tax income Total tax 0.05 t 0.12 t 0.2 t 0.3 t 0.4 Total income 1 07 l t l t t l l t l 5 Average tox rate 10 2 21.4 The impact of tax on labor supply and tax progressivity revenue Consider a representative household Assume all people are identical This household has preference over streams of consumption and hours c hi UIC h logo xh dButility of working The household faces the budget constraints O C Oof wwi.hn After tax income HI total labor supply for all individuals hi w wage rate WH Total labor income Ziv h wh indicidnel labor income  o L loge y xfo.ci j c Uce h Budget Constrant Foc 2L Gc X O O Mah i th t too Il o Tha Fn O ph t o j E o O Tty I Ooh 8 y O Doll a y I Tig Tn l T city yh if one increases tax Oi this increases progressiity marginal tax rates T.lu which leads to a decrease in hours worked h LHS O T T ly t d T't't RHS ht Higher progressivity discourage hours of work Labor supply distortion over efficiency  Household are different in wages the laborsupply of households at the the distribution will respond differently low and the high and of income Assume each family is composed of an equal number of low productivity a with we e and member permanent hourly product.it wge a member with Wu e high productivity wage ex ear X a e i a 0 x a The family maximizes a.meL.h.llogc Tt ago yh.IT collective decision low income member high incomemember max the total Ca consumption c consumption Instead of hi laborsupply utility h labor apply mere their own utilil subject to a family budget constraint o o at Cu o gingham o feaI Aug Income After tax income a e uh lowwage Total consumption Total after tox income expenditure  We can show that hours worked of high earnings members respond more strongly to an increase in tax progressivity than those of low members Marginal utility of consumption versus rate of earnings return of labor Poor people have relatively higher marginal than the marginal dis of The utility of consumption utility working then the marginal utility of consumption is higher decrease of increase of magnet drutility of working relative hours worked hYh Think about 4 L lega yh.IT ileogcri yhiII tXfooleIghinc.I hTI oolae G Cu Order Condition logx Ix First Yaa Ya d 2L Igc ca Gcn Yc I O Thi noon o Ohio a h g noon a again hi ez h i Th O 11 lO f hit ten 2a e O 7 I We want to study the effect of progressivity 0 on the labor supply ratio hYh fat adf.io Remark date e fixe e dazax e a ot i Fae e foal o zac1 O use product rate eat idk 2am O Oi iz Zac1 O f 2e O 01 2 t Oi12 tdd 2aci oil e y 2cLl O 2 2 act O l 1 0 12 to 125 c 2e e 2aci ewf.FI f O 21 o o co c 0 Q T hath had i odd 1 c o the degree of hours worked in are strictly decreasing The relative the the bigger one the product.ity tax progressing more so More income differences wit WL ftp.lhyhe i3 Lager differences The parameter a increases o.ch dd h at income differences T e2c i i e at i i ii e ai Za a a 2h Since an increase in the progressivity increases the marginal tax rate household member respond more of high productivity and the high income whose marginal tax rate strongly than that of low income earners high income member might actually decline hours worked of the family decline relative to those of low income member of Retirement Saving “How much can I withdraw from my pension portfolio, to avoid the risk of running out?” Nobel Prize winner William F. Sharpe called this the single “nastiest, hardest problem in finance”. How much to save for retirement Imagine, for a moment, your ideal retirement. Perhaps you're picturing yourself on a beach, book in hand, with the waves gently lapping at your feet. Or maybe you're visualizing a cozy cabin in the mountains, surrounded by pristine snow. Now, keep that image in your mind and ask yourself, 'Am I financially prepared to turn that dream into a reality? There is widespread concern that many families in the U.S. and other countries are not saving enough in general, and particularly for retirement. Only 36% of Americans feel confident about their retirement savings? That leaves a vast majority unsure about their financial future. This statistic isn't just a number—it's a wake-up call. It's time to take control of our retirement planning, and I'm here to guide you through that journey. Potential Risks facing in retirement 1. Longevity Risk Living longer than expected is a potential risk in retirement. While it may seem like a blessing to have more years to enjoy life, it also means needing to stretch savings and investments further. The fear of outliving one's retirement funds is a valid concern, which highlights the importance of planning and saving appropriately for the future. 1. Inflation Risk Another risk to be aware of in retirement is inflation. Over time, the cost of living tends to rise, and if retirement savings do not keep pace with inflation, their purchasing power can significantly decrease. It is essential to consider this risk when planning for retirement and to have strategies in place to combat inflation's impact on finances. 1. Sequence Risk (Market Volatility Risk) Retirement savings are often invested in financial markets, meaning they are exposed to market volatility. Not knowing future investment returns and the order in which they come. If investments experience significant losses early on, it can have a lasting impact on the overall portfolio and its ability to generate income throughout retirement. It's important to diversify investments and have a plan in place that considers potential market fluctuations to mitigate sequence risk. Importance of Saving for Retirement As we navigate our careers, the concept of retirement can often seem distant, but understanding its importance is critical for long-term financial health. Without a substantial retirement fund, one may face financial challenges during what should be carefree golden years. The importance of retirement savings cannot be overstated—it's not merely saving for an extended holiday but preparing for a period of life that may last as long as or longer than an entire career. Properly saving for retirement secures financial independence, allowing individuals to maintain their standard of living, manage healthcare costs, and pursue post-retirement dreams without the constraints of employment. It's a testament to a life's work, a safeguard against uncertainty, and a legacy that can support future generations. 1 Financial 2 Maintained 3 Legacy Building Independence Lifestyle Apart from covering own Retirement savings are Having a sufficient needs, retirement savings pivotal in ensuring that retirement fund means can serve as an one can enjoy their later preserving the lifestyle one inheritance, impacting the years without the need for is accustomed to without financial well-being of continued employment or making significant heirs. financial support from sacrifices. others. Overview of the 4% Rule The 4% Rule is a widely recognized strategy for retirement withdrawals, hailed for its simplicity and practicality. The premise of the rule is to withdraw 4% of one's retirement savings in the first year, with subsequent withdrawals adjusted for inflation annually. Pioneered by financial advisor William Bengen in 1994, the rule was designed to help retirees manage their savings over the span of a 30-year retirement period. By following the 4% Rule, retirees aim to strike a balance between preserving their wealth and having enough income to lead a comfortable life. This guideline serves as a starting point for retirement income planning, thwarting the risk of depletion too quickly, as well as the fear of overly conservative spending that could lead to an overly restrictive lifestyle. Retirement Saving * 4% = Retirement Spending Assumption: No other retirement incomes 1 Simple Approach 2 Proven Longevity 3 Flexibility Factor The 4% rule offers an Rooted in historical data, While it provides a base uncomplicated method for the 4% rule is based on the framework for retirees to determine how worst-case scenarios of withdrawals, the rule can much they can safely market performance, be adjusted depending on withdraw each year seeking to ensure that individual circumstances without exhausting their savings persist through and market conditions. savings prematurely. economic downturns. Explanation of the 4% Rule and Its Origins The 4% Rule emerged from an exhaustive study of historical market returns and aimed to address a fundamental question faced by retirees: How much can one safely withdraw from retirement savings without fear of outliving one's assets? William Bengen, examining market behavior over half a century, observed that a 4% withdrawal rate, adjusted each subsequent year for inflation, would have left the nest egg intact over a 30-year retirement, regardless of the market conditions at the start. This revolutionary idea shifted the retirement landscape, highlighting the potential for a rule-based approach to maintain a steady income. The 4% Rule considers typical asset allocation between stocks and bonds, acting as a guiding principle rather than an immutable law and accommodating modifications based on investment performance and personal needs. Rule Concept Study Insights Adjustable Strategy Grounded in historical analysis, Bengen's studies offered a Despite its strict percentage, the the rule was designed to tangible framework that rule is adaptable, allowing for withstand volatile markets and retirees could apply, fostering informed adjustments to provide a sustainable confidence in the sustainability address individual risk withdrawal rate over the long of their retirement funds. tolerance and spending needs. term. How does the 4% rule work? Living off your portfolio carries the risk of inflation. Withdrawals may lose purchasing power over time. For instance, if you withdraw $70,000 annually from a $1 million portfolio with a 7% withdrawal rate, and inflation averages 3% per year, the purchasing power of $70,000 will halve in 25 years. One way to plan for the impact of inflation is to adopt a withdrawal rate smaller than the rate of return on the portfolio; that allows the portfolio value to grow annually. 1 Year 1 $20,000,000 - $800,000 = $19,200,000 With market return rate 7%: $19,200,000×1.07 = $20,544,000 2 Year 2 With inflation-adjustment: $800,000×1.03 = $824,000 $20,544,000 - $824,000 = $19,720,000 With market return rate 7%: $19,720,000×1.07 = $21,100,400 3 Year After Repeat the above calculation 4 Year 30 Retirement Account Remaining: $41,286,259 Withdrawal rates of 3% to 4% continue to produce high portfolio success rates for stock-dominated portfolios. Even the 5% withdrawal rate produces reasonably high portfolio suc- cess rates for all payout periods, but the 6% and 7% rates perform reasonably well only for short payout periods. All withdrawal rates above 7% perform poorly for all payout periods. Pros and Cons of the 4% Rule The 4% Rule is a celebrated guideline in retirement planning, but like any strategy, it is not without its drawbacks. On the plus side, the rule demystifies the process of fund withdrawal, reducing the complexity of financial decisions for retirees. By setting a clear benchmark, retirees can approach their golden years with less anxiety about financial solvency. Additionally, the rule's inherent flexibility allows for personalization based on individual circumstances and market performance. On the other hand, critics argue that the rule may be overly simplistic in today's economic climate. They point to lower expected future returns and increased life expectancies as factors that might challenge the rule's validity. Moreover, life's unpredictability can render a fixed withdrawal rate inappropriate for some, necessitating a more nuanced approach to retirement spending. Clarity & Confidence Historical Resilience Offers retirees a clear guideline for fund Based on historical market data, it holds up withdrawal, fostering confidence in financial against various market conditions and provides decision-making. long-term fund sustainability. Versatility Current Relevance The flexibility to adjust withdrawals based on With evolving economic conditions and individual lifestyles and needs is embedded longevity, the 4% rule may need re-evaluation within the rule. to remain an effective strategy. Factors to Consider When Applying the 4% Rule When applying the 4% Rule to one's retirement strategy, a multitude of factors must be considered. Market volatility can affect investment returns and the value of the retirement portfolio, which in turn impacts the sustainability of withdrawals. Personal factors, such as life expectancy, health, and lifestyle desires also significantly influence the application of the rule. Inflation, an often underestimated factor, can erode the purchasing power of withdrawn funds over time and must be factored into annual adjustments. It's crucial to also consider sources of income outside of retirement savings, such as Social Security benefits, pensions, and any potential part-time work. A comprehensive understanding of these factors will aid in customizing the 4% rule to one's unique retirement plan, resulting in a more resilient and personalized approach to retirement income. 1 Market Conditions Understanding market trends and economic conditions can help anticipate and mitigate the impact on retirement funds. 2 Personal Factors Personal life expectancy, health, and retirement lifestyle choices play a significant role in the practical application of the 4% rule. 3 Income Sources Additional income sources should be integrated into the financial planning process to optimize the withdrawal strategy. 4 Inflation Concerns Annual adjustments for inflation are essential to maintain the purchasing power of retirement funds throughout the years. Alternatives to the 4% Rule While the 4% Rule remains a valid starting point, several alternatives have emerged to accommodate the changing financial landscape and personal preferences. Dynamic spending strategies, such as guardrails or bucket strategies, offer more flexibility in adjusting withdrawal rates in response to market performance. Similarly, the use of annuities can provide a guaranteed income stream, reducing reliance on market- dependent withdrawals. Other considerations, like the incorporation of part-time work during the initial phase of retirement or delaying Social Security benefits to increase payments, can also help in extending the longevity of retirement files. Each of these alternatives presents distinct advantages and considerations, illustrating the importance of a tailored approach to retirement planning. 1 2 Dynamic Spending Annuities Strategies that adjust withdrawals based on These financial products can secure a consistent market performance and personal wealth offer a income stream, ensuring stability regardless of modern, responsive approach to retirement market fluctuations. spending. 3 4 Work in Retirement Social Security Strategies Part-time employment can supplement income Optimizing the timing of Social Security benefits and offset the need for higher withdrawals from can serve as a significant bolster to retirement retirement savings. income planning. Conclusion and Key Takeaways Retirement saving, underpinned by strategies like the 4% Rule, is an intricate art that blends discipline, foresight, and adaptability. As individuals approach their post-career years, understanding the principles behind retirement saving, and the nuances of withdrawal strategies, becomes imperative. The evolving economic environment and personal circumstances dictate that no single rule is a panacea; rather, a toolkit of strategies, including the 4% Rule and its alternatives, can be the foundation of a secure and enjoyable retirement. Key takeaways include the importance of starting early, the necessity of ongoing education, and the value of personalization in retirement planning. Ultimately, the goal of retirement saving is to ensure that one's retirement years are marked by comfort, choice, and financial security to relish the culmination of a life's work. 30 4% 1994 Years Rate Origin The timespan for which the 4% The percentage of retirement The year when the 4% Rule was rule was originally tested to last savings suggested to withdraw introduced by financial advisor without depleting retirement yearly following the 4% Rule William Bengen. savings.