Across-the-Board Spending Cuts Are Very Inefficient: A Proof

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1 Uiversity of Texas at El Paso Departmetal Techical Reports (CS) Departmet of Computer Sciece Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof Vladik Kreiovich Uiversity of Texas at El Paso, vladik@utep.edu Olga Kosheleva Uiversity of Texas at El Paso, olgak@utep.edu Hug T. Nguye New Mexico State Uiversity - Mai Campus, huguye@msu.edu Sogsak Sriboochitta Chiag Mai Uiversity, sogsakeco@gmail.com Follow this ad additioal works at: Commets: Techical Report: UTEP-CS Recommeded Citatio Kreiovich, Vladik; Kosheleva, Olga; Nguye, Hug T.; ad Sriboochitta, Sogsak, "Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof" (2015). Departmetal Techical Reports (CS). Paper This Article is brought to you for free ad ope access by the Departmet of Computer Sciece at DigitalCommos@UTEP. It has bee accepted for iclusio i Departmetal Techical Reports (CS) by a authorized admiistrator of DigitalCommos@UTEP. For more iformatio, please cotact lweber@utep.edu.

2 Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof Vladik Kreiovich, Olga Kosheleva, Hug T. Nguye, ad Sogsak Sriboochitta Abstract I may real-life situatios, whe there is a eed for a spedig cut, this cut is performed i a across-the-board way, so that each budget item is decreased by the same percetage. Such cuts are ubiquitous, they happe o all levels, from the US budget to the uiversity budget cuts o the college ad departmetal levels. The mai reaso for the ubiquity of such cuts is that they are perceived as fair ad, at the same time, ecoomically reasoable. I this paper, we perform a quatitative aalysis of this problem ad show that, cotrary to the widely spread positive opiio about across-the-board cuts, these cuts are, o average, very iefficiet. 1 Formulatio of the Problem: Are Across-the-Board Spedig Cuts Ecoomically Reasoable Across-the-board spedig cuts are ubiquitous. Whe a departmet or eve a coutry faces a uexpected decrease i fudig, it is ecessary to balace the budget by makig some spedig cuts. Vladik Kreiovich Departmet of Computer Sciece, Uiversity of Texas at El Paso, 500 W. Uiversity, El Paso, Texas 79968, USA, vladik@utep.edu Olga Kosheleva Uiversity of Texas at El Paso, 500 W. Uiversity, El Paso, TX 79968, USA, olgak@utep.edu Hug T. Nguye Departmet of Mathematical Scieces, New Mexico State Uiversity, Las Cruces, New Mexico 88003, USA, ad Faculty of Ecoomics, Chiag Mai Uiversity, Chiag Mai, Thailad, huguye@msu.edu Sogsak Sriboochitta Faculty of Ecoomics, Chiag Mai Uiversity, Chiag Mai, Thailad, sogsak@eco.chiagmai.ac.th 1

3 2 V. Kreiovich, O. Kosheleva, H. T. Nguye, ad S. Sriboochitta I may such situatios, what is implemeted is a across-the board cut, whe all the spedig items are decreased by the same percetage. For example, all the salaries are decreased by the same percetage. The ubiquity of such cuts is motivated largely by the fact that sice they apply to everyoe o the same basis, they are fair. Across-the-board cuts may soud fair, but are they ecoomically efficiet? The fact that such cuts are fair do ot ecessarily mea that they are ecoomically efficiet. For example, if we cosistetly take all the wealth of a coutry ad divide it equally betwee all its citizes, this may be a very fair divisio, but, because of its lack of motivatios to work harder, this clearly will ot be a very ecoomically efficiet idea. Curret impressio. The curret impressio that across-the-board cuts may ot be ecoomically optimal, but they are ecoomically reasoable; see, e.g., [1, 3, 4, 5, 6, 8, 9, 10, 11]. What we show i this paper. I this paper, we perform a quatitative aalysis of the effect of across-the-board cuts, ad our coclusio is that their ecoomic effect is much worse tha it is usually perceived. Commet. To make our argumet as covicig as possible, we tried our best to make this paper ad its mathematical argumets as mathematically simple ad easy-to-read as we could. 2 Let Us Formulate The Problem i Precise Terms Formulatio of the problem i precise terms. Let us start by formulatig this problem i precise terms. What is give. First, we eed to describe what we had before the eed appeared for budget cuts. Let us deote the overall spedig amout by x, ad the amout origially allocated to differet spedig categories by x 1,x 2,...,x, so that x i = x. Sometimes, it turs out that the origial estimate x for the spedig amout was too optimistic, ad istead we have a smaller amout y < x. What we eed to decide. Based o the decrease amout y < x, we eed to select ew allocatios, i.e., select the values y 1 x 1,...,y x for which x i = x. What is a across-the-board spedig cut. A across-the-board spedig cut meas that for each i, we take y i = (1 δ) x i, where the commo value δ > 0 is determied by the coditio that (1 δ) x = y. Thus, this value δ is equal to

4 Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof 3 δ = 1 y x. What we pla to aalyze. We wat to check whether the across-the-board spedig cut y i = (1 δ) x i is ecoomically reasoable, e.g., to aalyze how it compares with the optimal budget cut. We eed to describe, i precise terms, what is better ad what is worse for the ecoomy. To make a meaigful compariso betwee differet alterative versios of budget cuts, we eed to have a clear uderstadig of which ecoomical situatios are preferable. I other words, we eed to be able to cosistetly compare ay two differet situatios. It is kow that such a liear (total) order o the set of all possible alteratives ca be, uder reasoable coditios, described by a real-valued fuctios f (y 1,...,y ) defied o the set of such alteratives: for every two alteratives (y 1,...,y ) ad (y 1,...,y ), the oe with the larger value of this fuctio is preferable (see, e.g., [12]): if f (y 1,...,y ) > f (y 1,...,y ), the the alterative (y 1,...,y ) is preferable; o the other had, if f (y 1,...,y ) > f (y 1,...,y ), the the alterative (y 1,...,y ) is preferable. The objective fuctio should be mootoic. The more moey we allocate to each item i, the better. Thus, the objective fuctio should be icreasig i each of its variables: if y i < y i for some i ad y i y i for all i, the we should have f (y 1,...,y ) < f (y 1,...,y ) We cosider the geeric case. I this paper, we do ot assume ay specific form of the objective fuctio f (y 1,...,y ). Istead, we will show that the same result that across-the-board cuts are ot efficiet holds for all possible objective fuctios (of course, as log as they satisfy the above mootoicity coditio). So, whether our mai objective is: to icrease the overall GDP, or to raise the average icome of all the poor people, or, alteratively, to raise the average icome of all the rich people, o matter what is our goal, across-the-board cuts are a far-from-optimal optimal way to achieve this goal. Resultig formulatio of the problem. We assume that the objective fuctio f (y 1,...,y ) is give. We have the iitial amout x. Based o this amout, we selected the values x 1,...,x for which f (x 1,...,x ) attais the largest possible value uder the costrait that x i = x. Let us deote the value of the objective fuctio correspodig to this origial budget allocatio by f x.

5 4 V. Kreiovich, O. Kosheleva, H. T. Nguye, ad S. Sriboochitta Now, we are give a differet amout y < x. Ideally, we should ow select the values y 1,...,y for which f (y 1,...,y ) attais the largest possible value uder the costrait that y i = y. Due to mootoicity, the resultig best-possible value f y of the objective fuctio f (y 1,...,y ) is smaller tha the origial value f x. I the across-the-board arragemet, istead of selectig the optimal values y i, we select the across-the-board values y i = (1 δ) x i, where δ = 1 y. The resultig x allocatio of fuds is, i geeral, ot as good as the optimal oe. Thus, the resultig value of the objective fuctio f δ is, i geeral, smaller tha f y. To decide how ecoomically reasoable are across-the-board cuts, we eed to compare: the optimal decrease f x f y i the value of the objective fuctio, with the decrease f x f δ caused by usig across-the-board spedig cuts. 3 Aalysis of the Problem Possibility of liearizatio. Usually, the relative size of the overall cut does ot exceed 10%; usually it is much smaller. By ecoomic stadards, a 10% cut is huge, but from the mathematical viewpoit, it is small i the sese that terms which are quadratic i this cut ca be safely igored. Ideed, the square of 0.1 = 10% is 0.01 = 1% 10%. Thus, if we expad the depedece of the objective fuctio f (y 1,...,y ) i Taylor series aroud the poit (x 1,...,x ), i.e., if we cosider the depedece f (y 1,...,y ) = f (x 1,...,x ) c i (x i y i )+... = f x c i (x i y i )+..., (1) where c i def = f y i, the we ca safely igore terms which are quadratic i terms of the differeces ad coclude that f (y 1,...,y ) = f x where we deoted y i def = x i y i 0, ad thus, that: f x f (y 1,...,y ) = c i y i, c i y i. (2) Commet. Sice the objective fuctio f (x 1,...,x ) is mootoic i each of the variables, all the partial derivatives c i are o-egative: c i 0.

6 Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof 5 Liearizatio simplifies the problem: geeral idea. Let us describe how the use of liearizatio simplifies the computatio of the two differeces f x f y ad f x f δ. Liearizatio simplifies the problem: case of optimal spedig cuts. Let us start with the computatio of the differece f x f y correspodig to the optimal spedig cuts. The optimal arragemet (y 1,...,y ) is the oe that maximizes the value of the objective fuctio f (y 1,...,y ) uder the costrait y i = y. Maximizig the value of the objective fuctio f (y 1,...,y ) is equivalet to miimizig the differece f x f (y 1,...,y ), which, accordig to the formula (2), is equivalet to miimizig the sum c i y i. To make the problem easier to solver, let us also describe the costrait y i = y i terms of the ew variables y i. This ca be achieved if we subtract this costrait from the formula x i = x. As a result, we get a equality y i = y, where we deoted y def = x y. Thus, due to the possibility of liearizatio, the correspodig optimizatio problem takes the followig form: miimize the sum c i y i uder the costrait y i = y. Let us prove that this miimum is attaied whe y i0 = y for the idex i 0 correspodig to the smallest possible value of the derivative c i, ad y i = 0 for all other idices i i 0. Ideed, for the arragemet whe y i0 = y ad y i = 0 for all i i 0, the miimized sum attais the value ( y i = c i0 y = mic i i ) y. Let us prove that for every other arragemet, we have a larger (or equal) value of the differece f x f (y 1,...,y ). Ideed, by our choice of i 0, we have c i c i0 for all i. Thus, due to y i 0, we have c i y i c i0 y i, ad therefore, c i y i c i0 y i = c i0 ( y i ) = c i0 y. Thus, the differece f x f y correspodig to the optimal spedig cuts is equal to

7 6 V. Kreiovich, O. Kosheleva, H. T. Nguye, ad S. Sriboochitta ( ) f x f y = mic i y. (3) i Liearizatio simplifies the problem: case of across-the-board spedig cuts. For across-the-board spedig cuts, we have y i = (1 δ) x i ad hece, y i = x i y i = δ x i. The coefficiet δ ca be obtaied from the coditio that (1 δ) x = y, i.e., that y = x y = δ x, thus δ = y x. Substitutig the correspodig values y i ito the liearized expressio for the objective fuctio, we coclude that c i y i = where we deoted δx i def = x i c i δ x i = δ c i x i = y x c i x i = y c i δx i, x. From the costrait x i = x, oe ca coclude that δx i = 1. Thus, the resultig decrease f x f δ is equal to: f x f δ = y c i δx i. (4) What we eed to compare. To compare the decreases i the value of the objective fuctio correspodig to the optimal cuts ad to the across-the-board cuts, we therefore eed to compare the expressios (3) ad (4). Let us treat the values c i ad δx i as radom variables. The values of c i ad x i deped o may factors which we do ot kow beforehad, so it makes sese to treat them as radom variables. I this case, both expressios (3) ad (4) become radom variables. How we compare the radom variables. Because of the related ucertaity, sometimes, the differece f x f δ may be almost optimal, ad sometimes, it may be much larger tha the optimal differece f x f y. A reasoable way to compare two radom variables is to compare their mea values. This is what we mea, e.g., whe we say that Swedes are, o average taller tha Americas: that the average height of a Swede is larger tha the average height of a America. It is reasoable to assume that the variables c i ad δx i are all idepedet. Sice we have o reaso to believe that the variables c i correspodig to differet budget items ad/or the variables δx j are correlated, it makes sese to assume that these variables are idepedet. This coclusio is i lie with the geeral Maximum Etropy approach to dealig with probabilistic kowledge: if there are several

8 Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof 7 possible probability distributios cosistet with our kowledge, it makes sese to select the oe which has the largest ucertaity (etropy; see, e.g., [2, 7]), i.e., to select a distributio for which the etropy S = ρ(x) l(ρ(x))dx attais the largest possible value, where ρ(x) is the probability desity fuctio (pdf). I particular, for the case whe for two radom variables, we oly kow their margial distributios, with probability desities ρ 1 (x 1 ) ad ρ 2 (x 2 ), the Maximum Etropy approach selects the joit probability distributio with the probability desity ρ(x 1,x 2 ) = ρ 1 (x 1 ) ρ 2 (x 2 ) that correspods exactly to the case whe these two radom variables are idepedet. Cosequece of idepedece. I geeral, the mea E[X +Y ] of the sum is equal to the sum E[X] + E[Y ] of the meas E[X] ad E[Y ]. So, from the formula (4), we coclude that E[ f x f δ ] = y E[c i δx i ]. Sice we assume that for each i, the variables c i ad δx i are idepedet, we coclude that E[ f x f δ ] = y E[c i ] E[δx i ]. (5) Here, we have o reaso to believe that some values δx i are larger, so it makes sese to assume that they have the same value of E[δx i ]. From the fact that δx i = 1, we coclude that formula (5) takes the form E[δx i ] = 1, i.e., that E[δx i ] = 1. Thus, E[δx i ] = 1, ad the E[ f x f δ ] = y 1 E[c i ]. (6) Let us select distributios for c i. Now, we eed to compare: the value (6) correspodig to across-the-board cuts with the expected value of the optimal differece (3): [ ] E[ f x f y ] = y E mic i. (7) i I both cases, the oly remaiig radom variables are c i, so to estimate these expressios, we eed to select appropriate probability distributios for these variables. We do ot have much iformatio about the values c i. We kow that c i 0. We also kow that these values caot be too large. Thus, we usually kow a upper

9 8 V. Kreiovich, O. Kosheleva, H. T. Nguye, ad S. Sriboochitta boud c o these values. Thus, for each i, the oly iformatio that we have about the correspodig radom variable c i is that it is located o the iterval [0,c]. Uder this iformatio, the Maximum Etropy approach recommeds that we select the uiform distributio o this iterval. This recommedatio is i perfect accordace with commo sese: if we have o reaso to believe that some values from this iterval are more probable or less probable the others, the it is reasoable to assume that all these values have the exact same probability, i.e., that the distributio is ideed uform. Let us use the selected distributios to estimate the desired mea decreases (6) ad (7). For the uiform distributio o the iterval [0,c], the mea value is kow to be equal to the midpoit c 2 of this iterval. Substitutig E[c i] = c ito the formula 2 (6), we coclude that E[ f x f δ ] = 1 y c. (8) 2 To compute the estimate (7), let us first fid the probability distributio for the miimum m def = mic i. This distributio ca be deduced from the fact that for each i value v, the miimum m is greater tha v if ad oly if each of the coefficiets c i is greater tha v: m v (c 1 > v)&... &(c > v). Thus, Prob(m > v) = Prob((c 1 > v)&... &(c > v)). Sice the variables c 1,...,c are all idepedet, we have Prob(m > v) = Prob(c 1 > v)... Prob(c > v). For each i, the radom variable c i is uiformly distributed o the iterval [0,c], so Prob(c i > v) = c v, ad thus, c ( ) c v Prob(m > v) =. c So, the cumulative distributio fuctio (cdf) is equal to: F m (v) = Prob(m v) = 1 Prob(m > v) ( ) c v F m (v) = 1. c By differetiatig the cdf, we ca get the formula for the correspodig probability desity fuctio (pdf) ρ m (v) = df m(v) dv = c (c v) 1.

10 Across-the-Board Spedig Cuts Are Very Iefficiet: A Proof 9 Based o this pdf, we ca compute the desired mea value: c c E[m] = v ρ m (v)dv = v 0 0 c (c v) 1 dv. By movig the costat factor outside the itegral ad by itroducig a ew auxiliary variable w = c v for which v = c w ad dv = dw, we ca reduce this itegral expressio to a simpler-to-itegrate form E[m] = c c (c w) w 1 dw = ( c c ) 0 c c w 1 dw w dw = 0 0 ) (c c c c+1 = ( c c+1 1 ) = c + 1 Substitutig the resultig expressio [ E ito the formula (7), we coclude that mic i i ] = c + 1 which is ideed much smaller tha the expressio (8). 1 ( + 1) = c + 1. E[ f x f δ ] = 1 y c, (9) + 1 Coclusio: across-the-board spedig cuts are ideed very iefficiet. I this paper, we compared the decreases i the value of the objective fuctio for two possible ways of distributig the spedig cuts: the optimal spedig cuts, ad the across-the-board spedig cuts. The resultig mea decreases are provided by the expressios (8) ad (9). By comparig these expressios, we ca coclude that the average decrease caused by the across-the-board cuts is + 1 larger tha what is optimally possible, where is the 2 overall umber of differet budget items. This result shows that o average, across-the-board cuts are ideed very iefficiet. Ackowledgmets We ackowledge the partial support of the Ceter of Excellece i Ecoometrics, Faculty of Ecoomics, Chiag Mai Uiversity, Thailad.

11 10 V. Kreiovich, O. Kosheleva, H. T. Nguye, ad S. Sriboochitta This work was also supported i part by the Natioal Sciece Foudatio grats HRD ad HRD (Cyber-ShARE Ceter of Excellece) ad DUE Refereces 1. J. Berger, The Case for Across-the-Board Spedig Cuts, Mercatus Ceter, George Maso Uiversity, March B. Chokr ad V. Kreiovich, How far are we from the complete kowledge: complexity of kowledge acquisitio i Dempster-Shafer Approach, I: R. R. Yager, J. Kacprzyk, ad M. Pedrizzi (Eds.), Advaces i the Dempster-Shafer Theory of Evidece, Wiley, New York, 1994, pp Cogressioal Budget Office, CBO Estimated Impact of Automatic Budget Eforcemet Procedures Specified i the Budget Cotrol Act, Washigto, DC, September Cogressioal Budget Office, Ecoomic Effects of Policies Cotributig to Fiscal Tighteig i 2013, Washigto, DC, November Cogressioal Budget Office, The Budget ad Ecoomic Outlook: Fiscal Years 2013 to 2023, Washigto, DC, February C. Edwards, We Ca Cut Govermet: Caada Did, Cato Policy Report, 2012, Vol. 34, No. 3 (May/Jue), pp E. T. Jayes ad G. L. Bretthorst, Probability Theory: The Logic of Sciece, Cambridge Uiversity Press, Cambridge, UK, R. Koga, How the Potetial 2013 Across-The-Board Cuts i the Debt-Limit Deal Would Occur, Ceter o Budget ad Policy Priorities, Washigto, DC, USA, November R. Koga, How the Across-The-Board Cuts i the Budget Cotrol Act Will Work, Ceter o Budget ad Policy Priorities, Washigto, DC, USA, April R. Koga, Timig of the 2014 Sequestratio, Ceter o Budget ad Policy Priorities, Washigto, DC, USA, August A. Schotter, Microecoomics: A Moder Approach, Cegage Learig, Bosto, Massachusetts, P. Suppes, D. M. Kratz, R. D. Luce, ad A. Tversky, Foudatios of Measuremet, Vols. I III, Academic Press, Sa Diego, Califoria, 1989.

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