Aristotle’s Critique of Plato’s Definition of the Good

Book one of Aristotle’s Nicomachean Ethics is devoted to investigation of the nature of the good. As a former student of Plato, Aristotle used his teacher’s view towards the subject as the starting point of inquiry. However, his approach was different than Plato’s because he established the notion of direction of arguments being important for their outcomes. Aristotle’s idea was that arguments should lead up to fundamental principles but not proceed from them. This concept of directionality is also important in establishing a logical conclusion that there should be universal science in examining the good. Since in practice, Aristotle argued, such science does not exist and the good cannot be universal. He further extended his argument against Plato’s view of the universal good by inquiring whether a longer lasting good is more of a good than a temporary one. These three arguments are important in a broader context since they present a foundational structure for his concepts of morality and ethics in human life.

Aristotle’s first critique of Plato’s view of the good is that Plato failed to recognize that forms could be subject to development. This development involves the notions of priority and posteriority, since, according to Aristotle, concepts of quality and relatedness must be derived from substance of objects first. In that sense the good has to be first described for what it constitutes of and only then one can investigate its particular qualities. In general, comparing the qualities of different objects would reveal their “relatedness”, which suggests that good is not a single and universal entity, but rather a relationship encompassing various goods of different qualities. Therefore, Aristotle’s critique contrasts with Plato’s view of the good because it implies opposite direction for the approach of defining the good. In Plato’s Republic, the good was presented as the universal and unchangeable truth illuminating some representations of the forms. In that sense the good was independent and higher in hierarchy to the subjects it is related to. This signifies Plato’s top-down approach, where elements of human life are determined by the good and not the other way around. However, as described before, in Aristotle’s view, “good cannot be universal, common to all cases, and single” because various definitions of the good derive from different categories of substance.

Aristotle further argued against Plato’s view of a single good noting that existence of the universal good would mean that there should be “a single science dealing with all things.” Aristotle’s critique seems valid by an analogy: for if intelligence is good in the soul, as sight is good in the body, then the universal science of the good should be similar in relation to the good in both soul and body. If such a science does not exist then it must be true that such a relationship is not universal. Therefore, if Plato’s conception of the universal good is correct, then such science should necessarily exist by pure logic. In that case it is natural to ask if this science actually exists. Aristotle argues that it does not, since “in actual fact there are many sciences dealing even with the goods that fall into a single category.” This was problematic for Plato’s notion of the good because every science should have only one function, like every sailor, doctor or any human being has his own proper function. Therefore, if the good has many functions it is either divisible or these functions are not distinct. But if every good is derived from the single universal good, as claimed by Plato, then they all would share similar qualities and, therefore, would share the same function and would not be distinct. In that sense, it is mathematically precise and logical for Aristotle to conclude that good is divisible and many sciences exist in relation to different goods.

Aristotle’s final argument took on the endurance of good, since Plato argued that the “good-as-such” is more of the good because it is everlasting. Aristotle tried to object this difference by an analogy to whiteness: “whiteness which lasts for a long time is no whiter than whiteness which lasts only for a day”; therefore, goodness is no more of a good if it lasts longer. However, I think, this analogy is correct only if we assume that whiteness and universal good share the same kind of goodness. But if Aristotle truly believed that there is no universal good, then such an analogy would not be very meaningful because humans would be subject to different kind of good than colors, animals and other things. More precisely, good for humans would be necessarily some sort of mediation between pain and pleasure, since happiness, which is a part of Aristotle’s definition of a human good, could not be achieved without experiencing both of them. Hence, proper analogy should be considered in terms of how much pain good inflicts and how much pleasure it delivers. Having made the distinction the question of permanence becomes easier: if bad lasts longer, then it inflicts more pain, therefore, less of it is preferred to more. Similarly, if good lasts longer, then it delivers more pleasure and more of it is preferred to less. Thus, it logically follows that good which is everlasting will be more of a good, because it will deliver more pleasure and would be preferred by everyone. Hence, it seems that Aristotle’s notions of pleasure and pain, in fact, help to support Plato’s view that the duration of good is important determinant of the quality of goodness.

To sum up Aristotle provided a strong critique of Plato’s view of the good. His arguments are largely based on mathematical and logical reasoning, which Plato acknowledged as means of understanding true forms of things. However, it is also important to use this reasoning properly so that substances and qualities would correspond to proper analogies. On the other hand, in the case of everlasting good it might have been beneficial for Aristotle to acknowledge that longer lasting good is more of a good, since life of a human being is a continuous process and, on moral grounds, humans should be rewarded with becoming “more good” if they act morally and justly throughout their lives, equally as wrongdoers should be punished for their misdeed and faults.

Hermann Hesse "Jaunajam Menininkui"

Dėkoju už naujametinį laišką. Jis liūdnas ir slegiantis, ir man tai suprantama. Bet jame yra sakinys apie tai, kad neduoda ramybės mintis, jog Tau ir Tavo gyvenimui lemta prasmė, užduotis, kuriuos neįvykdęs kankinsies. Kad ir kaip ten būtų, tai teikia vilties, nes yra šventa teisybė, ir norėčiau, kad retkarčiais prisimintum ir apmąstytum ir vieną kitą mano pastabą dėl to. Šios mintys nėra mano, jos senos kaip pasaulis, ir tai bene vertingiausia iš viso, ką žmonės kada nors sugalvojo mąstydami apie save ir savo paskirtį.
Ko tu pasiektumei gyvenime, - ir ne tik kaip menininkas, bet ir kaip žmogus, kaip vyras ir tėvas, draugas ir kaimynas ir t.t., - amžinoji pasaulio „prasmė“, amžinasis teisingumas visa tai matuos ne kokiu nors bendru, bet Tavo paties vienkartiniu ir asmenišku matu. Dievas savo teisme neklaus tavęs: „Ar tapai Hodleriu ar Pikasu, Pestalociu ar Gothelfu?“ Priešingai, jis paklaus: „Ar tu iš tikrųju tapai ir buvai tuo J.K., kurio galimybes ir paveldą esi gavęs?“ ir niekada žmogus negalės, be gėdos ir siaubo, prisiminti savo gyvenimo ir savo klystkelių – jis tegalės pasakyti: „Ne, aš juo netapau, tačiau bent jau stengiausi iš visų jėgų“. Jeigu tai galės pasakyti garbingai, jis bus išteisintas ir išlaikęs bandymą.
Jei tave trikdo tokie įvardžiai kiaip „Dievas“ ar „amžinasis teisėjas“, gali juos paprasčiausiai praleisti, ne jie čia svarbu. Svarbu vien tai, kad kiekvienas mūsų yra gavęs palikimą ir užduotį, kiekvienas yra paveldėjęs iš tėvo ir motinos, iš daugybės protėvių, iš savo tautos, savo kalbos tam tikras savybes – geras ir blogas, malonias ir sunkias, talentus ir trūkumus, - ir visa tai kartu yra Jis, ir tą Nepakartojamą, kas šiuo atveju vadinasi J.K., jis turi įvaldyti ir iki galo išgyventi, subrandinti ir galiausiai grązinti daugiau ar mažiau ištobulintą.
Yra tokių įspūdingų pavyzdžių, pasaulio ir meno istorijose jų kiek nori: sakysim, šeimoje vienas, kaip dažnai pasakoje, yra kvailys ir nevykėlis, bet kaip tik jam tenka pagrindinis vaidmuo, ir kaip tik dėl to, kad jis toks ištikimas savo esmei, visi gabieji ir sėkmingieji greta jo lieka menki.
Arba kitas pavyzdys: praėjusio amžiaus pradžioje Frankfurte gyveno didžiais gabumais apdovanota Brentanų šeima; iš jos beveik dvidešinties vaikų du dar ir šiandien žymūs – tai poetai Klemensas ir Betina. Taigi visi iš to didelio būrio brolių ir seserų buvo labai gabūs, įdomūs neiliniai žmonės, ieškančios dvasios, spindidntys talentai; tik vyriausiasis buvo ir liko ribotas, visą gyvenimą praleido tėvų namuose kaip tyli namų dvasia, niekam nenaudingas; jis, ramus katalikas, kantrus ir geraširdis brolis ir sūnus, sąmojingame, linksmame brolių ir seserų būryje, kur nevengta elgtis tikrai ekscentriškai, pamažu tapo tyliu vidurio ir ramybės tašku, nuostabia namų brangenybe, spinduliuojančią taiką ir gerumą. Apie šį ribotą, vaiku likusį vyrą broliai ir seserys kalbėjo su tokia pagarba ir meile, kaip apie nieką kitą. Tad ir šiam nesubrendėliui, šiam kvaileliui buvo lemta sava prasmė, sava užduotis, kurią jis atliko tobuliau negu visi jo nepaprasti broliai ir seserys.
Trumpai tariant, jei žmogus jaučia poreikį pateisinti savo gyvenimą, svarbu ne objektyvus, bendras pasiekimų lygis, bet kaip tik tai, kad jis savo esmę ir visa tai kas jam duota, kiek įmanoma visapusiškiau ir gryniau išreikštų savo gyvenime ir veikloje.
Tūkstančiai pagundų mus nuolat stumia iš šio kelio, bet visų didžiausioji yra toji, kuri skatina troškimą būti visai kitam negu esi, skatina sekti pavyzdžiais ir idealais, kurių pasiekti neįmanoma ir net nereikia siekti. Ši pagunda, ypač stipri gabesniems žmonėms, yra pavojingesnė už vulgarų paprastą egoizmą, nes atrodo kilni ir dora.
Kiekvienas vaikėzas, pasiekęs tam tikrą amžių, iš pradžių nori tapti karvedžiu ar garvežio mašinistu, paskui medžiokliu ar generolu, paskui Gėte ar Don Žuanu – tai natūralu, toks yra normalus brendimas bei saviaukla: fantazija tam tikru mastu apčiuopia ateities galimybes. Bet gyvenimas neliedžia įvykdyti šių norų, ir vaikiški bei jaunuoliški idealiai miršta savaime. O vis dėl to nuolat trokšti kažko, kas tau nepriklauso, ir žmogus kankina savo prigimtį prievartaujančiais reikalavimais. Taip būna mums visiems. Tačiau vidinio budrumo valandėlėmis mes pajuntame, kad nėra kelio pabėgti nuo savęs pačių kur nors kitur, kad privalome eiti per gyvenimą nešini grynai asmeniškais gabumais ir trūkumais, ir tik tada kartais atsitinka taip, kad pasistumiame truputėlį pirmyn, mums pavyksta kas nors padaryti, ko ansčiau neįstengėme, ir vieną akimirką neabejodami sveikiname save ir galime būti savimi patenkinti. Ilgam, to aišku, neužtenka, bet juk pati mūsų dvasia nesiekia nieko kita, kaip tik natūraliai augti ir bręsti. Tiktai tada išgyvename harmoniją su pasauliu, - o mums tai retai pavyksta, - bet juo stipresnis toks išgyvenimas.
Aš privalau pabrėžti, kad, kalbėdamas apie kiekvienam lemtą užduotį, turiu omeny anaiptol ne tai, ką jauni ir seni meno diletantai vadina savo individualybės ir originalumo gynimu bei įtvirtinimu. Juk savaime suprantama, kad menininkas, kai jis meną trapatina su savo profesija ir gyvenimo turiniu, iš pradžių turi išmokti visko, ko įmanoma išmokti kaip amato, ir neturi manyti, kad privalo šio mokymosi vengti, bijodamas prarasti savo brangią asmenybę ir originlumą. Vebgiantis mokytis ir varginti save darbu menininkas bus toks ir žmogus, jis netaps teisingas nei draugams, nei moterims, nei savo vaikams, nei visuomenės bendrijai, tik su savo originalumu nenaudingai tūnos šalia ir smuks. Žinome šiek tiek panašių pavyzdžių. Rūpestis išmokti to, kas įmanoma, mene yra tokia pat natūrali užduotis , kaip ir gyvenime. Kiekvienam vaikui reikia įdiegti valgymo ir valyvumo, skaitymo bei rašymo įgūdžius; mokytis to, ką galima išmokti, - ne kliūtis individualybės augimui, priešingai, tai ją skatina ir turtina. Man truputį gėda rašyti apie šiuos savaime suprantamus dalykus, bet taip jau yra, kad niekas, atrodo, nebeturi instinkto savaime suprantamiems dalykams, o vietoj to kyla primityvus ko nors negirdėto ir išsiskiriančio kultas. Aš, kaip žinai, neniekinu naujovių mene,priešingai; tačiau moralės sferoje atsiskleidžia tai, kaip žmonės atlieka jiems iškeltas užduotis; čia man įtartinos visos mados bei naujenybės, ir aš su dideliu nepasitikėjimu klausausi protingus žmones šnekant apie naujas morales ir etikas kaip apie meno madas ar stilius.
Šių dienų pasaulyje žmogui iškyla dar vienas reikalavimas, propaguojamas partijų, tėvynių ar tarptautinės moralės mokytojų. Jie reikalauja iš žmogaus visiškai atsisakyti savęs ir idėjos, galinčios reikšti šį tą asmenišką, kas vienintelis, tam, kad prisitaikytų prie ar idealios ateities žmonijos, taptų mašinos sraigteliu, statybine plyta tarp milijonų tokių pat plytų. Nenorėčiau spręsti apie moralinę šio reikalavimo vertę – jis turi savoherojiškų ir didngų sąvybių. Bet jis manęs neįtikina. Unifikacija, kad ir kokia geranoriška ji būtų, prieštarauja prigimčiai, veda ne į taiką ir džiaugsmą, bet į fanatizmą ir karą. Tai iš principo vienuoliškas reikalvimas, leistinas tiktai tada, kai turime reikalą su vienuoliais, su savanoriškai įstojusiais ordino nariais. Bet aš nemanau, kad šis madingas reikalavimasgalėtų būti Tau labai pavojingas.
Matau, kad iš mano laiško išėjo kone straipsnis. Todėl aš jį nusirašysiu ir, jei neprieštarauji, progai pasitaikius, pasiūlysiu perskaityti kitiems.

What do Andy Warhol and Fractals have in common?

This past Friday, January 25, was the culmination of my winter study class "Math 14: Creating Fractal Art". Basically, for three weeks we were coding Matlab to produce some interesting fractal pictures. For the final project we had to make our own fractal and "upgrade" it using various artistic techniques of the Photoshop. I decided to take a rather different approach and use a somewaht simple fractal pictures but present them in the light of modern art ideas of Andy Warhol.

About the Art

Mathematics and fine arts are similar or maybe even the same on many dimensions. Combining a beautiful mathematical concept of a fractal and style of a famous pop artist Andy Warhol my intention was to reveal a delicate relationship between the two.

In the universe of mathematics a local hero, the mathematician, tries to understand the world by formalizing the surrounding into the symbols and numbers of mathematics. Ultimately he explores the world in a purely formalized dimension. He uses this formalism for his advantage to reveal the hidden mysteries of the world and to leave irrelevant details aside.

An artist, alike the mathematician, does not leave the details aside. In the artist’s world, emotions, impressions and feelings are depicted as an inseparable whole. Irrelevant details are accepted and even welcomed in this world. Otherwise the art was boring.

But even though the two worlds where mathematicians and artist live might look very different, they are governed by the same universal laws. The essence does not change, only the form does. In the same way, fractal pictures cycle through the rainbow of colors, gain obscure looks and forms charming us with their thrilling looks yet leaving their mathematical nature intact.

About the Math

The following images are a close-up view of the Θ-Schottky Group circles generated using the Breadth–First Algorithm. The initial four circles were specified in matrix C, where column 1 is x-coordinates, column 2 is y-coordinates and column 3 is radii of each circle:

I then used Mobius transformations a, b and their inverses to generate the fractal image of my initial circles iterated nine times under the following conditions:

This is how the initial fractal looks like (before zooming into the corner and cycling through colours):

Or a 3-D version:

Five Demons

I am a multiple personality. My subconscious demons are always crossing their swords in a fight for dominance and might to control me. I am the fighting arena, where the main prize is to wield immense power upon my thoughts and actions. The victor gains limitless access to the deepest corners of my mind and the darkest chambers of my wishes. I am fated to be controlled by the demon���s desires, notions and values���

���I ��� Thinker��� is the cleverest demon of all. The name of Faust was given him by Lucifer - the Lord of Night and Hell. He comes out only in the darkness of the night and asks the questions which may not be answered by the man. He forces me to look into the celestial sphere of magical stars, think about the beauty and essence of Life. He takes me to travel through the sky of immortals ��� Socrates, Aristotle, Descartes, Nietzsche and numbers of others. He is the priest and the dreamer, he is the question and the answer, he is the beginning and the ending. He knows everything and nothing.

���I ��� Leader��� is a demon tenable and strict. He calls himself The Mighty Caesar and often proclaims the words of success - ���Veni, Vedi, Vici.��� He sees no limits for his quests but is careful. He is like a chameleon changing his color to disguise the enemies of his. He can be either a Diplomat or a Despot, a Democrat or a Socialist, a Liberal or a Conservative ��� anyone who he wants, or has to be. This curious demon feels pleasure to enter a discussion with every thoughtful person he meets. He never gives up. He fights until he commits the opponent to defeat. He is proud about his victories but never about himself. He dares to face the challenges but never the stupid ones.

���I ��� Athlete��� is a venturesome gambler, the demon of adrenaline and adventure. His credo is ���Citius, Altius, Fortius��� and he dreams about being the first in the Olympic Games. He gulps down every glass of a victory champagne; he bathes in a blaze of glory and sends air kisses to the girls when the crowd gets wild. He hates defeats, however, he considers them as challenges of everyday. He doesn���t know what fear is. Authorities do not exist to him. He lives only for the sweet feeling of victory, ten seconds of running and being free as a wind.

���I ��� Creator��� is quite a controversial demon. Once being taken by the enchanting scent of love perfume, he can feel like Shakespeare and write heartwarming sonnets to his Juliet in the dark of the Moon. Just a moment after that, challenged by someone���s dreadfull act, he can manifest a work for Human Rights and then declare the Self-independence Act. This demon is emotional in any case. He can sing the song which he has heard in the Moulen Rouge, even though he knows nothing about music. He creates because he lives, he lives because he creates. He writes what he feels. He writes nothing as well as everything.

���I ��� Friend��� is the softest demon of all the five. Some noble values you may find, which make him special and nice. Maybe he is na��ve as Don Quixote fighting with windmills and sheep, yet sometimes chivalrous as one of the musketeers, believing in saying ��� ���One for all and all for one!��� He knows the meaning of word friendship and does not share it with every person in the street. He feels the difference between friends and acquaintances. He keeps his word. He trusts his friends.

And so I live with five demons of mine. I am just a marionette of my inner inmates, a prisoner of a cell, where no thoughts can see the daylight of the sky. Nothing real is left. Dreams and visions prevail but my eyes can still notice something in the dark. It seems like a new Alexandria, a place where demons are curbed, a place where they are controlled. Not I. Meanwhile my demons fight again and nobody knows who will win this time���

1+1=1?

Let me start this blog by saying that I am not stupid or crazy as you might have thought after reading the title of this blog. The purpose of this article is not to question the basics of the arithmetic, but to show some new ways of understanding mathematics. As I have learned through many conversations with friends, people often perceive mathematics as sort of a scary beast locked in a castle full of ignorant, boring and incomprehensible mathematicians. But it's really not. Let me illustrate my point by an excerpt from an ancient legend of the Ulysses.

"When Ulysses had left the land of the Cyclops, after blinding Polyphemus, the poor old giant used to sit every morning near the entrance to the cave with a heap of pebbles and pick up one for every ewe that he let pass. In the evening when the ewes returned, he would drop one pebble for every ewe that he admitted to the cave. In this way, by exhausting the stock of pebbles that he had picked up in the morning he ensured that all his flock had returned."

So from the very beginning mathematics was based on association: individuals of the flock could be matched, one by one, with those of a heap of pebbles so that both were exhausted together, then the two groups were equal. The same principle was applied when inventing numbers like 1, 2, 3...etc which eventually became just symbolic representations of quantitative meanings attached to them. In fact, if the Mesopotamian man had decided that 2 has to be "one", then today we would be using 2 instead of 1 to represent the unit quantity. And in fact there is no reason why we shouldn't do that. After all, numbers are just symbols, placeholders we created in order to simplify the world surrounding us. Indeed mathematics is just another language we are operating in order to systemize, rationalize and conceptualize the complex world we are living in. Of course, mathematics can get complicated sometimes, but it's only because of the complexities underlying it. Yet, there is also a direct link of mathematics to a conventional understanding of the language - mathematical logic. This branch of mathematics serves like a translation program converting the language we use every day into its mathematical representation.

Since in logic we deal with statements or propositions which have some meaning, every such proposition is either true or false. We can assign the truth value T=1 when the proposition is true, and T=0 when it is false. Let's consider proposition A, which is true either if statements X or Y are correct. Let's also consider proposition B, which is true if statements W and Z are correct. Following this logic, our arithmetic additions would be as following.

Statement A: X+Y=A, 1+0=1, 1+1=1, 0+1=1
Statement B: W+Z=B, 1+0=0, 1+1=1, 0+1=0

As weird as it looks from the "common sense" it is still true, according to the logical rules we set, that one plus one is one! And in fact, we are not doing anything very complicated here, Polyphemus used exactly the same association technique when he was putting a pebble for each ewe, and we are "putting" "1" for every statement which is true.

So here we are probably at the point where you are asking yourself, if you have just wasted 3minutes of your life reading some impractical, nonsensical mathematical daydreaming. Well, not really. The fact that you are reading this blog means that you are reaping the fruits of the direct application of the mathematical logic described above. How? Just think about computers. In their very early days, they were used just by mathematicians as computation systems spitting out some preprogrammed answers (outcomes of the logical propositions) in numbers, which could later be translated into everyday language. When those systems grew more complex, computers "were taught" to do more complicated task like monitoring patient's heartbeat, connecting us to the internet and many more. But at the end, it's still the same 1's and 0's, pebbles and ewes, making those entire monster Pentiums exist.

Using Calculus of Variations to optimize the cost of production: Ericsson case

Today I will try to combine calculus of variations problem solving methods with strategic decision making in corporate finance. But before I begin the analysis, I will briefly describe all the elements and concepts I am going to use.

Firstly, for those who are now familiar with it, calculus of variations is a branch of mathematics which is devoted to problem solving in infinite-dimensional spaces of functions (as opposed to R1 or Rn). To explain it simply, regular calculus (single or multivariable) seeks to find values of xn for which some function f(x1, x2, x3���xn) is maximized. However, in calculus of variation (CV) we are trying to find the function itself rather than its elements. The most famous and widely used concept from the CV is the Euler���s equation, which we will also use in the Ericsson case analysis.

Secondly, I will also use such concepts as breakeven point, inventory and production cost which I learned at the Corporate Finance course this year. The main goal of this short interdisciplinary-synthesis exercise is to show how mathematics can be used to make well-though management decisions. More precisely, I will try to find and prove mathematically which production strategy should the company (Ericsson in this case) choose in order to minimize its production cost.

Consider Ericsson Company in 2003. According to Vernimmen Corporate Finance textbook, their total breakeven was 23,156 that year. Let���s try to find out what production model should the company has adopted so that their costs were minimized:

Let x(t)=inventory, with x(0)=0 and x(T)=B where B=breakeven point and T=365 days. Then, x'(t)=���production rate per day���. We assume that production cost of one unit is proportional to production rate, since higher PR requires more expensive machines, more working hours etc. Therefore, c1x'(t)=���production cost��� and c1x'2(t)=���production cost per day���. We also assume that inventory cost per day is proportional to amount of inventories in storage: c2x(t)=���cost of inventories���. Our goal is to minimize total cost, which can be written as:

We assume that the first and the second partial derivatives are continuous; therefore we can use the Euler���s condition:

Here we have, F(t, x, x')= c1x'^2+c2x, therefore solving the system gives us:

This second degree derivative can be easily solved by integrating it twice, and therefore, giving us a result:
We use our initial conditions to find constants:

Therefore, we can find our function x(t):
Also, we know that T=365 and B=23,156. We get:

Although, this formula could be practically applied only if we knew constants c1 and c2 (which would vary from one industry to another due to different nature of the production processes and companies' sales models), we can still draw some important conclusions. One of them is that the cost function is quadratic, meaning that the firm should reduce its rate of production in the early phases of production, but increase it at the latter ones. This finding is also supported by the statement of Prof. Salvi (i.e. my Finance professor) that ���companies should not produce rapidly in the early-life stages of the product, but rather postpone it until it gains the position in the market.��� Could you name another implication of this finding?

The Leverage Equation Insights

Last time at the corporate finance course we were learning about the leverage effect. It is a very nice concept showing that the return on shareholders��� equity can be maximized by financing investments partially through banks, rather than solely by shareholders. The equation looks like this: Return on Equity (ROE) = Return on Capital Employed (ROCE) + (ROCE - Cost of Debt)*(Net Debt/Shareholders Equity). In a very simplified scenario, a shareholder would seek to maximize the Net Debt in order to maximize ROE. Obviously, that is not a realistic result since cost of debt increases as net debt rises. In order, to reflect this relationship I decided to find the optimal level of shareholder���s equity in respect to debt. The solution would be as following:

Let ROE=y(x) where x=Net Debt. Also let ROCE=b and Sh.Equity=c, where b is a constant and c (c>=0) is a variable of interest. Also let the r=interest rate (on debt). Assuming that the cost of debt consists only of interest payments, we then have the leverage equation: y(x,c)=b-(b+xr)*(x/c). This is just a simple quadratic equation y(x,c)=b+bx-(x^2r/c) with two unknowns. Our goal is to maximize y(x,c) in respect to x, therefore, we impose y'(x,c)=0 condition (partial derivative in respect to x). We get: x-2xr/c=0 which can be expressed as cx=2xr, therefore, c=2r. It is interesting to notice that even though x's simplify in our equation, there is an interesting condition when x=0 in the original leverage equation. When that happens (x=0) we get that ROE=ROCE, which means that shareholders receive full benefits of their investments. However, that is usually a corner solution, since very few firms would have net debt=0. But conventional solution is rather interesting as well, since it says that ROE is maximized when shareholders��� equity does not exceed (or is no less) twice the interest rate on debt. Since at this level the ROE is stable (i.e. no further change can cause ROE to increase) the objective of shareholders is to stay at this level of equity financing.

There are some practical applications of this result as well. This finding could partially explain why, for example inflation, is so crucial for economic growth. To explain this statement better, imagine that inflation soars and the central bank decides to increase the interest rate. Then private banks will also increase their lending rates (r), and shareholders will have to provide more of their own funds in order to stay at the optimal ROE level. Moreover, each increase in the interest rate increase will a double effect on businesses, since 1% increase in ���r��� will require shareholders to raise their equity by 2%. This effect would be further amplified by the fact that during the inflationary periods, banks become more risk averse and therefore, might respond by increasing required debt repayment rates more than proportionately to the central bank���s rate. That said, his short analysis shows why today���s credit crunch might have even stronger negative effect on businesses. As the credit provided by banks became more expensive, companies��� shareholders now have to provide bigger ���chunk��� of the investments in their own businesses. But that���s obviously not that easy to do, since shareholders face the full risk of investment, and therefore, require higher returns. Then obviously more risky projects will be abandoned and overall investment levels will fall, leaving people with fewer jobs, lower benefits and overall more depressed future perspective���