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Berkeley MBA 201A - Lecture 1: Introduction

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Additional Lecture NotesLecture 1: IntroductionOverviewThe purposes of this lecture are (i) to take care of introductions; (ii) to transactcourse business; (iii) to consider the value of models; (iv) to conduct the world’sfastest course in calculus; and (v) to introduce decision trees. The last of theseis the main focus of the lecture.Notes1. Introduction(a) Introduce course(b) Introduce selfi. Want you to succeed — see me or gsi.ii. Office is F675, phone 2-7575, email: [email protected](c) Introduce Simon Wakeman2. Course business(a) Class reps.(b) Seating chart(c) Cold calling(d) Minimal use of PowerPoint3. Value of Models(a) From Sylvie and Bruno Concluded by Lewis Carroll (1893):“That’s another thing we’ve learned from your Nation,” saidMein Herr, “map-making. But we’ve carried it much furtherthan you. What do you consider the largest map that wouldbe really useful?”“About six inches to the mile.”“Only six inches!” exclaimed Mein Herr. “We very soongot to six yards to the mile. Then we tried a hundred yardsto the mile. And then came the grandest idea of all! Weactually made a map of the country, on the scale of a mileto the mile!”Copyrightc2004 Benjamin E. Hermalin. All rights reserved.mba 201a Lecture 1 —Fall2004“Have you used it much?” I enquired.“It has never been spread out, yet,” said Mein Herr: “thefarmers objected: they said it would cover the whole country,and shut out the sunlight! So we now use the country itself,as its own map, and I assure you it does nearly as well.”(b) What does economics offer:• A framework from which to better understand the information,objectives, and circumstances faced in business.• A framework from which to make better decisions.• A systematic approach to solving business problems more effi-ciently and successfully.(c) Two types of analysis:• Normative: What you should do to accomplish your objectives.• Positive: What others are likely to do.(d) Models are the tools used• A model is an abstraction from reality.• Its goal is, as Sherlock Holmes put it, to “recognize out of anumber of facts which are incidental and which vital.”• Those that are incidental are ignored, those that are vital arekept.• Good models and bad models—one goal of this course is to helpyou tell them apart.(e) “The ideas of economists ..., both when they are right and whenthey are wrong, are more powerful than is commonly understood.Indeed, the world is ruled by little else. Practical men [and women],who believe themselves to be quite exempt from any intellectual in-fluences, are usually the slaves of some defunct economist.” – JohnMaynard Keynes4. Mathematics — World’s Fastest Calculus Course(a) A wee bit of calculus—fortunately calculus is really easy2mba 201a Lecture 1 —Fall2004value/unitunits (x)vMF(x)x1F(x1)x1+hFigure 1: TherateofchangeinF (·) at x1is MF(x1).CalculusCalculus can be seen as means of determining areas under curves and determin-ing the rate at which these areas are increasing (or decreasing).Consider Figure 1. It shows a function, called MF(·), which is a flat (hori-zontal) line at height v. Thatis,forallx, MF(x)=v. Let the area under theMF(·) curve from 0 to x be denoted F (x). Using the formula for the area of arectangle, height × width, we see that F (x)=vx.One question we might ask is the rate at which F (·) increases as we increaseits argument. For instance, if we thought of x as time (e.g., hours) and v asspeed (e.g., km/hour), then vx would be distance traveled (e.g., kilometers).Recall that vx = F (x). Recall too that speed is the rate at which distanceincreases. Hence, the rate at which F (·) increases is v, which is to say the rateof change of F (·)atanyx is MF(x).The “M” in MF(·) stands for marginal. Marginal means rate of change. Inmathematics, MF(·) would be referred to as the derivative of F (·).∗Observe that we could have arrived at this conclusion another way. One wayto calculate the rate is to consider the difference in F (·) at two points normalizedby the difference between the two points. That is, calculateF (x1+ h) − F (x1)(x1+ h) − x1.The numerator is the difference in F (·) evaluated at two points x1and x1+ h(see Figure 1). The denominator is the difference between the points. We dividethe numerator by the denominator because the rate needs to be expressed ona per-whole-unit basis (e.g., we talk about km./hr., not km. per half hour).∗In mathematics, we might, therefore, write dF (x)/dx = MF(x)orF(x)=MF(x).3mba 201a Lecture 1 —Fall2004value/unitunits (x)MG(x)x1G(x1)x1+hFigure 2: TherateofchangeinG(·) at x1is MG(x1).Notice we can rewrite the above fraction asF (x1+ h) − F (x1)(x1+ h) − x1=v × (x1+ h) − v × (x1)h=vhh= v.Figure 2 shows a somewhat more complex situation. Here we’ve graphedthe function MG(·). We’ve defined the function G(·) so that G(x) is the areaunder MG(·)from0tox. We would like to know the rate of change of G(·)atx1.To calculate that rate of change, observe that we would know how to cal-culate that rate of change were MG(·) a flat function like MF(·) in Figure 1.Were it flat, then the rate of change would just be the height of the rectangle(i.e., MG(x1)). But observe, from Figure 2, that for a small change, from x1to x1+ h, where h is small, the change in G(·) is approximately equal to thecross-hatched rectangle whose height is MG(x1) and whose width is h.Andwe know the rate of change of area in this rectangle—it’s MG(x1). As h getssmaller, the rectangle becomes a better and better approximation of the changein G(·). Hence, we can conclude that MG(x1) is the rate of change in G(·)atx1.Note that the argument just given approximates the same algebra we de-4mba 201a Lecture 1 —Fall2004ployed with respect to F (·)andMF(·):Rate of change in G(·)atx1≈G(x1+ h) − G(x1)(x1+ h) − x1=G(x1+ h) − G(x1)h.This approximation gets better the smaller is h. Indeed, if we let h shrinkall the way to zero, then this approximation will be exact. For example, ifG(x)=ax2+ bx + c, where a–c are constants, then we haveRate of change in G(·)atx1≈a(x1+ h)2+ b(x1+ h)+c−ax21+ bx1+ ch=ax21+2ax1h + ah2+ bx1+ bh + c − ax21− bx1− ch=2ax1h + ah2+ bhh=2ax1+ ah + b.If we shrink h all the way to zero, this becomes 2ax1+ b. In other words, ifG(x)=ax2+ bx + c,thenMG(x)=2ax1+ b.Finally, observe that we’ve also shown that the area under a function, sayt(·),


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