OOBAMACARE AND A FIX FOR THE IRS ITERATION
SAMUEL J. FERGUSON Introduction
In 2018, I took an Uber ride. Although my driver qualified for help with payingfor health insurance under the Affordable Care Act, he couldn’t determine theamount of his benefit. Worse, tax software and government calculators said heshould receive $0 to help him pay for health insurance, instead of the roughly$3,000 that the law seemed to prescribe. He asked me to look into the matter,and my efforts led to a mathematical odyssey captured by
Time ’s film crew anda senior writer at
Money magazine in an online film clip and article [3]. I ammotivated to publish my findings by a communication [6] that the IRS will includereference to it in its guidance after publication in a peer-reviewed journal. Then,tax software companies will be able to implement procedures proposed here withoutlegal liability, relieving the current computational issues affecting Affordable CareAct beneficiaries. I am also motivated by the opportunity to bear witness to theresolution of a civic concern by means of mathematical modeling and proof.2.
Obamacare’s Premium Tax Credit
A tax device created by the Affordable Care Act plays a key role in my driver’sproblem, so we first review this law. In 2010, the United States Congress passed thePatient Protection and Affordable Care Act [5], also called Obamacare. A coupleof its provisions are relevant. First, it provides for the setup of online exchanges,so American households can directly purchase health insurance meeting certainminimum standards. These standards apparently give rise to the “patient protection”part of the law’s name. Second, the law makes qualified health insurance affordablefor every American household with household income M in the range F ≤ M ≤ F. Here, F is the federal poverty line for the household, a governmentally-prescribednumber depending on household size and state which adjusts annually according toa specified notion of inflation.Now, how does the law make health insurance affordable when F ≤ M ≤ F ? Itdoes so by creating a tax credit to help eligible households pay the premiums of We postpone giving definitions of “affordable” and “household income,” but we note thatthe latter may be referred to as the household’s modified adjusted gross income (MAGI) in theliterature, hence our choice of the letter “M” to denote it. Worksheets for calculating M may befound in the Instructions for Form 8962 [10, p 6]. The value of F used in Obamacare calculations for a given tax year may be found in theInstructions for Form 8962 for that year [10, pp 6–7]. For example, in the continental UnitedStates in 2018, for a household with n people, F is approximately $8,000 + n · $4,000 [9, p 7].Thus, at that time, a household of one person had a federal poverty line of about $12,000, and ahousehold of four people had a federal poverty line of about $24,000. a r X i v : . [ q -f i n . GN ] A ug S. J. FERGUSON
Figure 1.
A graph of the 2018 applicable figure f ( m ) as a function of m . For m <
1, the value of the dashed line is f (1). This value is used if an applicablefigure is required in that case. No applicable figure is needed when m > fig:ApplicableFiguresGraph qualified health insurance. For such households, and for at least some choices ofhealth plan, the credit pays all of the cost of the insurance premiums except for aportion which is considered affordable. Moreover, this credit may be received inadvance, to help pay the cost of premiums right away, and is refundable, so the fullcredit is receivable whether or not the household owes taxes that offset it.Having introduced Obamacare’s tax credit for premiums, how can we find it?Following along with Form 8962 [8], which taxpayers must file to claim the premiumtax credit, we see that the computation requires an applicable figure f . Theapplicable figure represents the percentage of M which is affordable for the householdto pay for health insurance. Working with decimals rather than percentages, wemodel f as a governmentally-determined function of m , where m = M/F.
Here, the division is exact, so m is a real number in [1 ,
4] obtained without roundingwhen F ≤ M ≤ F . If M > F , then household income is too high to receive thepremium tax credit (PTC), so P T C = $0. When
M < F we again have
P T C = $0unless the household qualifies for an exception [10, p 8], in which case the applicablefigure f (1) is used. Thus, applicable figures aren’t needed for m outside [1 , BAMACARE AND A FIX FOR THE IRS ITERATION 3
Example.
Say we are considering the 2018 tax year. Then the applicable figure isappropriately modeled by defining f ( m ) by f ( m ) = j, ≤ m < . ,k + ( (cid:96) − k ) m − . . − . , . ≤ m < . ,(cid:96) + ( a − (cid:96) ) m − . − . , . ≤ m < ,a + ( b − a ) m − . − , ≤ m < . ,b + ( c − b ) m − . − . , . ≤ m < ,c, ≤ m ≤ , where ( j, k, (cid:96), a, b, c ) = (0 . , . , . , . , . , . . As Figure 1 shows, our function has a discontinuity at m = 1 .
33 and values inthe interval (0 , . f ( m ). For all taxyears, however, this function is defined similarly to the above example, is monotoneincreasing, and, by the grace of Congress, possesses right continuity on [1 , expected contri-bution , found by multiplying the applicable figure f ( m ) by household income M ,is what the government expects the household to be able to affordably contributetowards health insurance. The expected contribution f ( m ) · M is compared withthe premium P , the sum of the unsubsidized or “sticker price” costs of benchmarkannual health insurance premiums for the household members. The government iswilling to “pick up the tab,” that is, pay all of the cost of the benchmark premiumswhich is not covered by the expected contribution. Thus, the government canpay the remaining amount, P − f ( m ) · M . More precisely, as the government’scontribution is never negative, it pays up to max (0 , P − f ( m ) · M ). Simplified Example.
Say an unmarried 60-year-old nonsmoker forms a householdof 1 person in Dutchess County, in the state of New York, in 2018. Assumetheir benchmark premium is $500 per month, F = $12,000, M = $48,000, and f (4) = 0 .
09. Then, their benchmark annual health insurance premium is P = 12 · $500 = $6,000 , and their expected contribution is f ( m ) · M = 0 . · $48,000 = $4,320 . The values j = f (1), k = f (1 . (cid:96) = f (1 . a = f (2), b = f (2 . c = f (3) are from Table 2in the Instructions for Form 8962 for 2018 [9, p 9]. They are found by locating applicable figuresfor household income equal to 100%, 133%, 150%, 200%, 250%, and 300% of the federal povertyline, respectively. That our function f ( m ) is an appropriate model for the rest of the applicablefigures in the table may be seen by noting that, given n % on the right column of Table 2, roundingthe value of f ( n ) to the nearest ten-thousandth yields the corresponding decimal on the leftcolumn. To model the applicable figures for 2019, say, we use the same functional form with thevalues j = 0 . k = 0 . (cid:96) = 0 . a = 0 . b = 0 . c = 0 . If enrollees, plans, or premiums change from month to month, then monthly premiums mustbe recorded individually. And, in certain circumstances, costs are split between multiple taxreturns. We assume, for simplicity, that neither of these occur.
S. J. FERGUSON
If they buys the benchmark insurance, then the government pays the rest, which is P − f ( m ) · M = $6,000 − $4,320 = $1,680 . Thus, $1,680 is the amount of Obamacare’s premium tax credit for the household.We recall that the premium tax credit can be taken in advance. After doing this,the remaining balance on the benchmark premiums can be paid with the expectedcontribution. As the government considers the expected contribution affordable,we have seen how, according to government definitions, Obamacare makes healthcoverage affordable when M satisfies F ≤ M ≤ F .A household need not buy the benchmark insurance to receive a premium taxcredit, however, and can purchase other qualified insurance from the exchange. Let Q denote the sum of the unsubsidized annual costs of such qualified insurance, forhealth plans actually purchased for the household members. The government isstill willing to contribute max (0 , P − f ( m ) · M ) or the full cost Q of the chosenqualified insurance, if this is less, since the government cannot pay more than thefull cost. Thus, in general, the premium tax credit is given by P T C = min ( Q, max (0 , P − f ( m ) · M )) . Enrollees receive a form giving the values they should use for Q and P when filingtaxes. To recap, the instructions for claiming the premium tax credit give thevalue of f ( m ) from tables once M is known. Then, using Q, P read off a form, thepremium tax credit
P T C is calculated by the above formula. With such a preciseprocess available, why was my Uber driver unable to find his premium tax credit?3.
The Problem
We have yet to define or calculate household income M . There’s a reason for that.For households with income from self-employment —an independent contractor, aprivate tutor, and a driver associated with a ridesharing app are all likely to beconsidered self-employed—the value of M can be tricky to find, particularly if thehousehold is eligible for a premium tax credit. In 2014, self-employed workers were“almost three times more likely” than other workers to obtain health insurance fromthe government exchanges created by Obamacare, according to the Treasury [1],so self-employed households form a sizable proportion of beneficiary households.Thus, we are motivated to address any computational issues they may face; suchissues could potentially affect a large number of people.Pinning down the household income M for self-employed households like that ofmy Uber driver requires detective work. This is because self-employed householdsare eligible for a tax deduction D involving health insurance costs, which may bedifficult to determine when those costs are being shared with the government. Rather The benchmark insurance for a household member is the “second lowest cost silver plan” onthe government exchange for the household’s county of residence, and depends on the age andsmoking habits of the enrollee, in addition to the county and tax year. In particular, if all available employer-sponsored health insurance plans require an employeecontribution that exceeds the household’s expected contribution f ( m ) · M , so they are “unafford-able,” then the household may generally purchase suitable insurance on the exchange, and receivea premium tax credit, to get affordable qualified insurance. Forms 1095-A list annual values for Q and P on lines 33A and 33B, respectively, for purchasers[8]. They list annual values of advance payments of premium tax credits, AP T C , on line 33C.
BAMACARE AND A FIX FOR THE IRS ITERATION 5 than discussing household income in general, we just discuss it for householdswhose sole income source is self-employment in a single business. Let us denotethe earned income generated from this activity by I . If the household’s healthinsurance is all purchased on the exchange by this business, then some nonnegativeamount D of that cost can be deducted from I , so taxes are only paid on theamount I − D . If there are no other sources of “above the line ” deductions besides D and the ones used to compute I , then the household income M is given by M = I − D. We say that the household has a “simple” tax return in this case, since it only hasone income source and one above the line deduction besides those used to find I .For simple tax returns, the premium tax credit can be determined from M = I − D ,and hence from D , but what range of values can D have? We introduce two legalconstraints on D . The first is that the government doesn’t permit more to bededucted than the household was billed for during the year, so D ≤ Q . If advancepayments of premium tax credits were sent, and we denote the total amount sentby AP T C , then the balance billed for was Q − AP T C . So, D ≤ Q − AP T C inthis case, but for simplicity we take
AP T C = $0 for now. The second constraint,which we call the “no double-dipping rule,” is that the government doesn’t permitmore in deductions and credits than was possible to pay. Without this restriction,an enterprising person might buy health insurance at a negative effective cost,presumably contrary to the taxpayers’ wishes. In our case, we write the rule as D + P T C ( D ) ≤ Q, where P T C ( D ) is the amount of the premium tax credit for a household withincome M = I − D . We can make this function explicit by replacing m with M/F and M with I − D in the equation for P T C from the previous section, giving
P T C ( D ) = min (cid:18) Q, max (cid:18) , P − f (cid:18) I − DF (cid:19) · ( I − D ) (cid:19)(cid:19) . Since the second constraint implies the first for simple tax returns with
AP T C = $0,we ignore the first for now, and may refer to the second simply as “the constraint.”We now come to my Uber driver’s dilemma. To find his premium tax credit, hemust know D . But D ≤ Q − P T C ( D ), so he must know his premium tax credit tofind out how large D can be. But the premium tax credit is what he wanted to findin the first place! So, there is a “circular relationship” in the United States tax code The Instructions for Form 8962 [10, p 6] say that household income in general is the sum ofthe modified adjusted gross incomes of the household members. Each modified adjusted grossincome is the sum of the adjusted gross income (AGI) on the corresponding tax return and certaintax-exempt income. The AGI itself is obtained from total or gross income by subtracting certainadjustments, called “above the line” deductions. Earned income is defined to be the net profit from a business minus the self-employment taxdeduction—corresponding to the half of Social Security and Medicare taxes normally paid byemployers—and tax-deductible self-employed retirement plan contributions. We assume, for simplicity, that earned income I is at least as large as Q . Otherwise, we mustalso require that the self-employed health insurance deduction D satisfy D ≤ I . The origin of the name is that on Form 1040 [7] for 2017 and many prior years, AGI appearsat the bottom of the front page, with a line underneath. The deductions needed to computeAGI—the “above the line” deductions—appear above that line, while all other deductions do not.
S. J. FERGUSON between the premium tax credit and the self-employed health insurance deduction[13]. This means that the Internal Revenue Service (IRS) has the following problem:
Problem 3.1.
What is a procedure, computable by hand in a reasonable time, thatfinds the appropriate health insurance deduction D for any self-employed householdeligible for Obamacare’s premium tax credit?What does “appropriate” mean? The appropriate choice of D is the nonnegativevalue which maximizes the tax benefit for the household. The tax benefit is thesum of the tax credit P T C ( D ) and the amount of taxes saved by reducing incomeby the deduction D . If the tax function T ( · ) assigns, to a given income in dollars,the federal income tax on that value for a household, ignoring the tax credit,then the amount of taxes saved by reducing taxable income from I to I − D is T ( I ) − T ( I − D ). With these definitions, the tax benefit is P T C ( D ) + ( T ( I ) − T ( I − D )) . As T ( I ) is independent of D , the optimal solution is unaffected by dropping it fromthe problem formulation. Thus, the appropriate deduction is the value of D solvingmax D + P T C ( D ) ≤ Q P T C ( D ) − T ( I − D ) . Essentially, we want to maximize the tax credit while minimizing the tax, subject tothe constraint. As an increase in D causes a decrease in I − D , whence an increasein P T C ( D ) − T ( I − D ), the largest value of the latter occurs for the largest D satisfying the constraint. That is, the appropriate nonnegative value of D ismax ( { D : D + P T C ( D ) ≤ Q } ) , provided M = I − D ≥ F for this value, so the household is eligible for the credit.What does “computable by hand in a reasonable time” mean, above? Practically,it means the IRS can put it into its tax guidance. Informally, this means the IRSdoes not consider it overly onerous to require of a typical taxpayer with accessto its instructions, even if removed from modern computing. For example, if wetry all possible whole dollar values for D that satisfy the constraint, then, tothe nearest dollar, some value will yield the maximum tax benefit, and thus willgive the appropriate D . But the IRS would likely consider having to try everypossible constrained value D to be an overly burdensome computational task for anAmerican unable to access a computer or smartphone. Thus, although guaranteedto succeed, this is not a procedure that any household can “compute by handin a reasonable time.” On the other hand, the maximization problem for D caneventually be converted to an algebraic equation in D for each taxpayer. This isbecause f ( m ) · M is a piecewise-quadratic function of D and the constraint is ananalyzable inequality. However, the IRS would probably find it unreasonable torequire an American removed from the internet to discover the necessary algebraand numerical computation of square roots. The task is to create an algorithm orprocedure which can be implemented in a reasonable number of steps that justinvolve addition, subtraction, multiplication, division, and rounding. While this In case m = I − DF falls outside of [1 ,
4] during our analysis, we take
P T C ( · ), as a functionof D , to have the value P T C ( D ) given by the above formula if m lies in [1 , P T C ( D ) = 0 if m > m <
1, for now. However, there can be exceptions to this when m < BAMACARE AND A FIX FOR THE IRS ITERATION 7 may be possible for the above-mentioned algebraic equations, it would likely usemany specific details about the function f ( m ), so the guidance would have to berewritten each year using the new year’s function. It would be preferable for theIRS to derive a dependable procedure that is independent of the tax year.If we can solve the above problem for simple tax returns with AP T C = $0, thenwe can check whether our solution, appropriately generalized, handles simple taxreturns with positive values of
AP T C , and general tax returns. We turn now tocurrent IRS guidance for taxpayers who qualify for both a self-employed healthinsurance deduction D and a premium tax credit P T C ( D ). This guidance can beviewed as an attempted solution of the above problem.4. The IRS Fixed Point Iteration
Current IRS guidance offers self-employed Obamacare beneficiaries two methodsfor determining allowable values of their self-employed health insurance deduction D [12, pp 62–65]. The second, “simplified calculation method” is a truncation ofthe first, “iterative calculation method,” so we focus primarily on motivating andanalyzing the IRS iterative method here.To motivate the IRS iteration for finding the appropriate self-employed healthinsurance deduction D , we ask what equations D might satisfy, in plausible scenarios.Certainly, if we write the premium tax credit as a variable, C , we have the equation C = P T C ( D )by definition. In addition, we might hope that the appropriate D , the largestnonnegative value satisfying D + P T C ( D ) ≤ Q , attains the equality D + P T C ( D ) = Q . This means each dollar earmarked for health insurance leads to a dollar of taxcredit or a dollar of insurance deductions, a plausible property for the D giving thegreatest tax benefit to provide. Subtracting P T C ( D ) in this equation, we get D = Q − P T C ( D ) , so we arrive at a system of two equations in two unknowns given by( C, D ) = (
P T C ( D ) , Q − P T C ( D )) . Rather than try to solve the above system of two equations in two unknownsalgebraically, the IRS uses the two earlier equations to define a fixed point iteration.Given ( C n , D n ) for some integer n ≥
1, we can define ( C n +1 , D n +1 ) by( C n +1 , D n +1 ) = ( P T C ( D n ) , Q − P T C ( D n )) . This indeed defines a fixed point iteration, for if we define G on R by G ( x, y ) = ( P T C ( y ) , Q − P T C ( y )) , then the above equation for ( C n +1 , D n +1 ) becomes( C n +1 , D n +1 ) = G ( C n , D n ) . As we shall see, this iteration seeks a limiting point X = ( C, D ) such that X = G ( X );such a point is said to be fixed by G .For an eligible household with a simple tax return and no advance premium taxcredit, the IRS iterative method generally begins with the point ( C , D ) given by C = $0, D = Q and defines points ( C n , D n ) sequentially by the above equation( C n +1 , D n +1 ) = G ( C n , D n ). This is the case, for example, if I − Q ≥ F . For our S. J. FERGUSON model, if we define the sequence { ( C n , D n ) } ∞ n =1 in this way, then we can ask aboutconvergence. If ( C n , D n ) → ( C, D ) as n → ∞ , then, taking limits on both sides ofthe equation ( C n +1 , D n +1 ) = G ( C n , D n ), we can prove that( C, D ) = (
P T C ( D ) , Q − P T C ( D ))holds. From this, we see that ( C, D ) = G ( C, D ), so (
C, D ) is a fixed point of G ,whence D + P T C ( D ) = Q . This shows D satisfies the constraint. Moreover, as D + P T C ( D ) is a strictly increasing function of D when m = I − DF ≥
1, no largervalue of D satisfies the constraint. Thus, D = max ( { D : D + P T C ( D ) ≤ Q } ), thatis, lim n →∞ D n is the appropriate value of D .Having motivated the IRS fixed point iteration with the usual notion of conver-gence, we point out that the IRS uses its own test to determine convergence. First,let us say that a sequence { ( C n , D n ) } ∞ n =1 converges in the IRS sense if and onlyif, when rounding to the nearest penny after each intermediate calculation, thereexists a positive integer N such that (cid:107) ( C k , D k ) − ( C n , D n ) (cid:107) ∞ < ε for all integers k, n ≥ N , with ε = $1. The above norm is defined by (cid:107) ( x, y ) (cid:107) ∞ =max( | x | , | y | ). It is straightforward to show, for the sequences { ( C n , D n ) } ∞ n =1 definedin the preceding paragraph, that the “iterative calculation method” amounts totaking D to be the y -coordinate of ( C n , D n ) after appropriate rounding, where n is the smallest value of N that satisfies our definition of IRS convergence [12, p 63].The actual text of IRS guidance asks taxpayers to “not use the iterative calculationmethod” if (cid:107) ( C n +1 , D n +1 ) − ( C n , D n ) (cid:107) ∞ ≥ ε for all n ≥
1. The oscillatory nature ofthe sequences makes it possible to prove that this holds if and only if { ( C n , D n ) } ∞ n =1 fails to converge in the IRS sense [12, p 64]. Thus, for the sequences coming fromits iteration, the IRS provides a simple, accurate “do not use” test for divergence.IRS guidance also offers its simplified calculation method, which amounts toasking beneficiaries to take D as their health insurance deduction and, hence, C as their premium tax credit. When { ( C n , D n ) } ∞ n =1 fails to converge in the IRS sense,so that we cannot use the IRS fixed point iteration, these are the values that IRSguidance currently arrives at for D and C . The best of the tax software may extendthe simplified procedure, so taxpayers take at most D := lim inf n →∞ D n as their deduction, and hence P T C ( D ) as their premium tax credit. When wedon’t have convergence in the IRS sense, however, D is generally smaller than theappropriate value, and in many cases P T C ( D ) yields a premium tax credit of $0.This is apparently the cause of my Uber driver’s difficulty; we emphasize that theseinappropriate values are what tax software and government calculators give now. Proof.
We first observe, inductively, that D ≤ D ≤ D ≤ · · · when ( C , D ) = ($0 , Q ).Then, we apply the left continuity of the function P T C ( · ), inherited from the right continuity of f ( · ), to the sequence { D n } ∞ n =1 . Our hypothesis that ( C n , D n ) → ( C, D ) as n → ∞ then yields P T C ( D ) = lim n →∞ P T C ( D n ) = C . We note that this test for divergence is equivalent to using the well-known Cauchy criterionfor this with ε = $1. BAMACARE AND A FIX FOR THE IRS ITERATION 9
Unsurprisingly, the IRS says that self-employed taxpayers “may have difficulty”computing their premium tax credit, according to the IRS document which in-troduced the fixed point iteration [13]. However, IRS guidance says that “any”computation method may be used to find the appropriate deduction, provided itrespects the constraint and the separate rules for the deduction and credit [12, p64]. The below example proves that neither method given by the IRS always worksto compute appropriate values, so the fact that any valid method may be used givesus a fresh opportunity to solve the IRS problem—and my Uber driver’s dilemma.
Example.
Say we are considering the 2018 tax year. Then the example of anapplicable figure function f ( m ) given previously is the appropriate one to use. Inparticular, we can use it to calculate values of the extended tax credit function P T C ( D ) = (cid:40) min (cid:0) Q, max (cid:0) , P − f ( I − DF ) · ( I − D ) (cid:1)(cid:1) , ≤ I − DF ≤ , , I − DF > I − DF < . Suppose we have a household in Brooklyn, New York, consisting of one individualand one dependent child who is less than 26 years old. The household’s relevantfederal poverty line is F = $16,240 [9, p 7]. Looking up benchmark prices for thecounty, Kings County, we find that the unsubsidized cost of benchmark healthinsurance premiums for the household is $865.81 per month or, rounding to thenearest dollar, P = $10,390 annually [2]. Suppose that the household, altogether,has earned self-employment income from a single business which amounts to I =$71,150, and take Q = P . Following the IRS fixed point iteration, and rounding tothe nearest dollar in intermediate steps for simplicity, we obtain( C , D ) = ($0 , $10,390)and C = $10,390 − . · $60,760 , as $71,150 − $10,390 = $60,760. Hence, after rounding, C is $4,581. Thus, D = $10,390 − $4,581 = $5,809 . In turn, this makes I − D = $65,341 > F = $64,960, so by our above formula for P T C ( D ), we get C = $0 . Unfortunately, this yields D = $10,390 , putting us back where we started. Hence, the sequence doesn’t converge in the IRSsense. On the other hand, if we follow the simplified calculation method from theIRS, we arrive at a deduction of D =$5,809 and a premium tax credit of C = $0.This is even worse than not claiming the premium tax credit at all, and letting D =$10,390. It turns out that the $0 value for the premium tax credit is notappropriate, as we shall see. If we progressively narrow our search for the deduction D by performing repeated bisections, for example, then we can do better. The Bisection Method
We now propose a bisection procedure, and prove that it always gives theappropriate self-employed health insurance deduction D for simple tax returns.The proof works because, although there may, in general, be discontinuities in theunderlying structures that affect potential computations, we have monotonicityand left continuity in the function P T C ( D ), the latter inherited from the rightcontinuity of f ( m ) through m = I − DF . We first motivate the use of bisection byadapting the well-known proof of the Intermediate Value Theorem by bisection toleft continuous, monotone increasing functions. Theorem 5.1.
Let g be a real-valued, monotone increasing, left continuous functionon an interval [ a, b ] , and let a real number k be given. If g ( a ) ≤ k , then there existsa real number c in [ a, b ] such that g ( c ) ≤ k and g ( d ) > k for all d > c in [ a, b ] .Proof. If g ( b ) ≤ k then, as there is no d > b in [ a, b ], we can set c = b . Otherwise, if g ( b ) > k , we denote the midpoint of [ a, b ] by c = a + b . If g ( c ) > k , we set a = a , b = c , and reduce our search to [ a , b ] as, by monotonicity, g ( d ) > k for all d ≥ c in [ a, b ]. If g ( c ) ≤ k , we set a = c , b = b , and again reduce our search to[ a , b ]. Having defined [ a n , b n ] for some n ≥ g ( a n ) ≤ k < g ( b n ), we let c n +1 = a n + b n . If g ( c n +1 ) > k , we set a n +1 = a n , b n +1 = c n +1 , whereas if g ( c n +1 ) ≤ k we set a n +1 = c n +1 , b n +1 = b n . In either case, we have g ( a n +1 ) ≤ k < g ( b n +1 ).Having defined { [ a n , b n ] } ∞ n =1 recursively as above, it follows that a unique c liesin all of the intervals [ a n , b n ], as [ a n +1 , b n +1 ] is a subset of [ a n , b n ] and [ a n , b n ] haslength b − a n for all n ≥
1. Note that c is the limit of the increasing sequence { a n } ∞ n =1 .Since g ( a n ) ≤ k for all n ≥
1, and a n → c as n → ∞ , it follows that g ( c ) ≤ k byleft continuity of g , as desired. Given d > c in [ a, b ], for n sufficiently large we have b − a n < d − c . For such n , we have c ≤ b n < d . As g ( b n ) > k and g is increasing, itfollows that g ( d ) > k . Thus, as d > c in [ a, b ] was arbitrary, c is as desired. (cid:3) We can apply the proof of this theorem to justify the appropriateness of usingbisection to calculate D as in the following corollary. This corollary is for simpletax returns with no advance premium tax credit. For such returns, if I < F , then
P T C = $0 automatically, so the premium tax credit cannot be taken and theoptimal deduction is D = Q . The following corollary handles the remaining case,where I ≥ F . We recall that D must lie in [0 , Q ] and that we must have D ≤ I − F ,so that M = I − D ≥ F , to ensure eligibility for the premium tax credit in ourcurrent setup. The tax benefit when D > I − F , so that P T C = $0 and D = Q ,can be considered separately. However, this poses no computational issue and thebenefit is smaller than taking the premium tax credit in practical situations. Thatis why we only consider D ≤ I − F here. Corollary 5.2.
Suppose
F, P, Q > , I ≥ F are given real numbers, and f is apositive, monotone increasing, right continuous function on [1 , . Define P T C ( · ) If I < Q is possible, then the IRS requires us to further constrain D by D ≤ I . So, thehighest possible deduction is actually D = min( I, Q ) when
I < F , but we have assumed I ≥ Q above for simplicity and will continue to do so below. BAMACARE AND A FIX FOR THE IRS ITERATION 11 on ( −∞ , I − F ] by letting P T C ( d ) = (cid:40) min (cid:0) Q, max (cid:0) , P − f ( I − dF ) · ( I − d ) (cid:1)(cid:1) , ≤ I − dF ≤ , , I − dF > . Then, if
P T C (0) ≤ Q , there is D in [0 , min ( Q, I − F )] such that D + P T C ( D ) ≤ Q and d + P T C ( d ) > Q for all d > D in [0 , min ( Q, I − F )] . Such D may becomputed by bisection. Letting a = 0 and b = min( Q, I − F ) , this means that if b + P T C ( b ) ≤ Q then D = b and if b + P T C ( b ) > Q then we can proceed asfollows. For each integer n ≥ , having obtained a n and b n with a n + P T C ( a n ) ≤ Q < b n + P T C ( b n ) , let c n +1 = a n + b n . If c n +1 + P T C ( c n +1 ) ≤ Q , then let a n +1 = c n +1 , b n +1 = b n . Otherwise, if c n +1 + P T C ( c n +1 ) > Q , let a n +1 = a n , b n +1 = c n +1 . Theincreasing sequence { a n } ∞ n =0 defined by this procedure converges to the number D = max ( { d in [0 , min( Q, I − F )] : d + P T C ( d ) ≤ Q } ) with the property above.Proof. Let a = 0, b = min ( Q, I − F ), and define g on [ a, b ] by g ( d ) = d + P T C ( d ).By the preceding theorem, if P T C ( a ) ≤ Q , then D in [ a, b ] with the desired propertyexists. The proof of that theorem justifies the rest of the assertion. (cid:3) If the function g in the above proof is such that g (0) > Q , then no D with thedesired property exists, but this is not a problem in our application as P T C = $0in this case and we can take D = Q . If g (0) ≤ Q and g ( b ) ≤ Q too, where b = min( Q, I − F ), then the appropriate D must be D = b . In all remaining cases,we can perform repeated bisections of [0 , b ] as described above. Example.
We can perform the bisection procedure on the example from thepreceding section. After more than a dozen bisections, rounding to the nearestdollar after each step, we find that the appropriate deduction is D = $6,208.From this, we find that the appropriate premium tax credit for the household is P T C ( D ) = $4,182, substantially more than the $0 it would receive by the simplifiedcalculation method. Given D , and without performing the bisection, it is readilychecked that this value cannot be improved because D + $4,182 = Q . By means ofsuch checking, the IRS apparently verifies the correctness of tax returns preparedusing values of D found by methods outside of its guidance, such as bisection.Similarly to the above example, when P T C ( D ) = P − f ( m ) · M and Q ≥ f (4) · F ,there is generally an interval of incomes I ≤ F + f (4) · F for which the IRS fixedpoint iteration breaks down. In this case, the simplified calculation method gives P T C = $0, yet an appropriate deduction D can be found by bisection yieldingsubstantial premium tax credits, often worth thousands of dollars.The bisection method also offers improvement over IRS guidance near m = 1 . f ( m ) at 1 .
33 again prevents IRS convergence nearby. Infact, it is possible for the equation D + P T C ( D ) = Q to have no solution for someinterval of incomes I , due to the discontinuity at m = I − DF = 1 .
33. For such I ,there is a value of D such that D + P T C ( D ) < Q yet d + P T C ( d ) > Q for d > D due to a discontinuous jump in P T C ( d ) as d approaches D from the right. Thus,this D is the appropriate value for the deduction, yet D + P T C ( D ) < Q . In ourmodel of the IRS fixed point iteration, if { ( C n , D n ) } ∞ n =1 converges to ( C, D ), then D + P T C ( D ) = Q necessarily follows. For this reason, in this interval of incomes I , it is impossible for the IRS fixed point iteration to converge; it can be shownthat this is true no matter how we select the initial point ( C , D ).6. The Advance Premium Tax Credit
Having developed the bisection procedure for simple tax returns with no advancepremium tax credit, we consider the situation where advance payments of thepremium tax credit are sent. Let
AP T C denote the amount of advanced paymentsin this case, and let us treat $0 as 0. Then, D must lie in [0 , Q − AP T C ],but otherwise the computation of the premium tax credit may begin as before,starting from the interval [0 , Q − AP T C ]. It is usually the case that, if a taxpayerwith
AP T C > m = I − DF ≥ m < P T C ( D ) that is given when 1 ≤ m ≤
4, usingthe actual values of I and D but using f (1) as the applicable figure, as mentionedpreviously. We obtain the following corollary after taking f ( m ) = f (1) for m in[0 , Corollary 6.1.
Suppose
F, P, Q > , I ≥ Q , and < AP T C ≤ Q are given realnumbers, and f is a positive, monotone increasing, right continuous function on [0 , . Define P T C ( · ) on ( −∞ , I ] by letting P T C ( d ) = (cid:40) min (cid:0) Q, max (cid:0) , P − f ( I − dF ) · ( I − d ) (cid:1)(cid:1) , ≤ I − dF ≤ , , I − dF > . Then, if
P T C (0) ≤ Q , there is D in [0 , Q − AP T C ] such that D + P T C ( D ) ≤ Q and d + P T C ( d ) > Q for all d > D in [0 , Q − AP T C ] . Moreover, the number D = max ( { d in [0 , Q − AP T C ] : d + P T C ( d ) ≤ Q } ) with the property above can be computed by bisection, starting from a = 0 and b = Q − AP T C , as in the previous corollary.
Lastly, when
AP T C >
0, the value
AP T C of the advance payments must bereconciled with the appropriate premium tax credit
P T C ( D ) when taxes are filed.If P T C ( D ) ≥ AP T C , the taxpayer receives an additional
P T C ( D ) − AP T C whenfiling taxes, so the total amount of
P T C ( D ) is ultimately received. However, if AP T C > P T C ( D ), then the taxpayer must repay some or all of the advancepayments which were received in excess of the appropriate amount. Precisely,according to IRS guidance, the taxpayer repaysmin (cid:18) AP T C − P T C ( D ) , R (cid:18) I − DF (cid:19)(cid:19) , where, for nonnegative m , the repayment limitation R ( m ) is of the form R ( m ) = r, ≤ m < ,s, ≤ m < ,t, ≤ m < , ∞ , ≤ m BAMACARE AND A FIX FOR THE IRS ITERATION 13 with r ≤ s ≤ t [10, p 16]. For example,( r, s, t ) = (300 , , r, s, t are doubled for any other filing status. For comparison, the valuesof r, s, t for single taxpayers in 2018 were $300, $775, $1,300, respectively [9, p 16].Independent of the tax year, R ( m ) is increasing, positive, and right continuous.The total tax benefit from Obamcare when AP T C > P T C ( D ), which is theamount received minus the amount repaid, is B ( D ) = AP T C − min (cid:18) AP T C − P T C ( D ) , R (cid:18) I − DF (cid:19)(cid:19) , and this need not equal P T C ( D ). However, there is no need to modify the bisectionin the previous corollary to take this new formula into account. To see this, firstnote that the total Obamacare tax benefit B ( D ), for the D found from the bisection,is at least P T C ( D ) and less than AP T C in the case we are considering, where
AP T C > P T C ( D ). As B ( D ) < AP T C , and D ≤ Q − AP T C , the “no double-dipping rule” is automatically satisfied in this case, as summing gives D + B ( D ) < Q .Moreover, as B ( D ) ≥ P T C ( D ), all values d > D in [0 , Q − AP T C ] persist in beinginappropriate when we use B ( D ) as the effective credit. Indeed, d + P T C ( d ) > Q implies that d + B ( d ) > Q in this case. Thus, the appropriate value of D in[0 , Q − AP T C ] found in the previous corollary is still optimal when taking intoaccount the total Obamacare tax benefit, even when
AP T C >
Further Questions
We have seen that we can perform bisection for models of simple tax returns,whether or not there are advance payments of the premium tax credit. What aboutmodels of general tax returns? We briefly sketch the main ideas. Additional sourcesof income, whether tax-exempt or not, do cause translations in M , but as longas earned income from a single self-employed business is at least Q , the previousconsiderations still apply to M = I − D , where now I represents the sum of allincome sources relevant to computing household income. Thus, this increase ingenerality causes no difficulty. If there are further above the line deductions whichare not from a short list of exceptions [12, p 65], or insurance deductions comingfrom dental or other plans that are unrelated to the premium tax credit, then theyagain cause a translation in M . Effectively, the previous considerations apply to M = I − d − D , where we let d represent the sum of these additional deductions.From this, we see generalization again causes no difficulty. Finally, for the taxcode as it stands, there are above the line deductions that can be altered when thevalue of D is adjusted. For example, the student loan interest deduction for 2019is normally $2,500 or the total interest paid on student loans during the tax year,whichever is less [11, p 37]. Let us denote the smaller of these two amounts by k .When the taxpayer’s modified adjusted gross income (MAGI) is between $70,000and $85,000, the deduction becomes the amount SL = k · $85 , − M AGI $15,000 . Roughly speaking, the above value for MAGI is found using an adjusted grossincome (AGI) computed with a value of $0 for the student loan interest deduction.The above formula shows, for example, that the student loan interest deduction“phases out” from k to $0, with a slope of − k/ $15,000, as income ranges from$70,000 to $85,000. If the self-employed health insurance deduction D is increased,that could “undo” some portion of the phasing out in certain circumstances. Thiscan cause the student loan interest deduction, and other above the line deductionslike it, to get caught up in the circular relationship between D and the premium taxcredit. However, because all of the current exceptional above the line deductions ofthis type are deductions that phase out like this one, increasing D causes each ofthem, when successively calculated, to be monotone increasing as functions of D , ascan be verified by examining them individually [12, p 65]. For this reason, P T C ( D )persists in being monotone in D for general tax returns, and for this reason bisectioncan currently be appropriately adapted for models of general returns.What is perhaps more mathematically interesting than our ability to adapt awell-known algorithm is the task of explaining in detail when and how the IRSprocedures break down. As we have seen, this is equivalent to the question of whenthe IRS fixed point iteration fails to converge in the IRS sense. The explanation,as we have emphasized above, seems to lie in the discontinuities that the function P T C ( D ) possesses in general. However, because the precise way that the pointsof the iteration are jostled about is also influenced by the slopes involved, near m = 1 .
33, we have not computed the precise intervals of IRS divergence in general.In particular, this problem is not fully resolved, and may be of interest, when
AP T C >
P T C ( D )as a function of D and fool the bisection method into finding a point which doesnot maximize tax benefits globally. There are currently no such above the linedeductions of this type, but from 2005 to 2017 the domestic production activitiesdeduction (DPAD) that might affect, for instance, someone who strikes oil in Texas,involved a continuous phase-in. The slope is sufficiently shallow, however, that when D is decreased by ε , causing the other relevant above the line deductions to decreaseby some ε i , DPAD can phase in by at most 0 . ε + (cid:80) i ε i ). So, reversing this, anincrease in D by ε still decreases AGI by at least ε + (cid:80) i ε i − . ε + (cid:80) i ε i ), or0 . ε + (cid:80) i ε i ), leading to an increase in P T C ( D ). Thus, monotonicity is preservedby this deduction.What if a new deduction arises that phases in with a discontinuous jump upwards?In that case, the bisection method can indeed fail, and we should seek an alternative.A naive binned Newton method might succeed in many cases, but when m is near1.33, for example, and a discontinuous phase-in is suitably chosen, this could fail too.If the household’s state has expanded Medicaid past the (1 . F threshold—and BAMACARE AND A FIX FOR THE IRS ITERATION 15 the Affordable Care Act prescribes Medicaid expansion over and beyond this—wewould expect the household to be eligible for Medicaid, so the household wouldlikely be on Medicaid and hence unaffected by this discontinuity at m = 1 . M ≥ F to get started. Further discontinuous numerical analysis might thusbe needed to clarify the situation in states that have not expanded Medicaid, if adeduction with a discontinuous phase-in were to arise.A policy-related question we might ask is how to modify the premium tax credit,as a function of household income, so that the IRS fixed point iteration alwaysconverges in the IRS sense. If P T C ( · ) is changed so that it is given by a monotone,continuous, piecewise-differentiable function, for example, with both of its one-sidedderivatives of suitably small magnitude at each point, then we expect the IRSiteration to always converge. Such a change could be obtained by a modification thattakes f (1 .
33) = f (1), say, to make f ( m ) continuous on [1 , M > F . While Congress mayprefer to achieve objectives which are incompatible with this particular proposal,the fact that a circular relationship in the US tax code exists at all suggests thatthere are appropriate ways to involve mathematicians in realistic future policychoices. We hope this mathematical excursion, resolving the concern raised by myUber driver, inspires some mathematicians to look for these and other ways theycan use mathematics to help address citizens’ concerns and, more generally, issuesthat affect the people they meet in daily life. Acknowledgements.
This paper has been improved as a result of discussions withXiaona Zhou at Brooklyn’s City Tech about how to adapt the results presented hereinto an online self-employed premium tax credit calculator [14]. Any feedback aboutthe calculator should be addressed to
[email protected] inorder to be incorporated. Finally, I wish to thank the anonymous reviewers andmy colleagues for their many helpful suggestions.
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