38 Mathematical Appendix

38 Mathematical Appendix - In this Appeaj '11 provide a...

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Unformatted text preview: In this Appeaj '11 provide a brief review of some of the mathematical in the text. This material is meant to serve as a remifidéi‘of? itions of various terms used in the text. It is emphat- ically not 3:3th ' mathematics. The definitions given will generally be the _:___;fi:=-Ehe.most rigorous. A describes relationship’between numbers. For eachliiumberjgz, reaction assigns a unique number y according to some rule; Thus aQ-fnnetion' can be indicated by describing the rule, as "take a " ’3 number and 1 , or “take a. number and multiply it by 2," and so on. We Iparticular functions as y i 332, y 2 23:. Functions are sometimes refefiéd toes. transformations. Often we tonindicate that some variable 3,: depends on some other variable :8, but we don’t know the specific algebraic relationship between the two Variables. this case we write y = f which should be interpreted as saying that the variable 3; depends on 2: according to the rule f. Given a _fimi:t1:on_y_= f (:L'), the number I. is often called the indepen- dent Varia_b1_e,.egd__'t_he number 3; is often called the dependent variable. A2 MAYH EMATICAL APPEN DIX The idea is that :1: varies independently, but the value of y Edepends ou-the value of 3:. " '_ 3' ' . - _ Often some variable 3; depends on several other variablliés. :61, 3:2, and so on, so we write 3; = f(;!:1_.,$2) to indicate that b0th_"vafiables tOgether determine the value of y. A.2 Graphs A graph of a function depicts the behavior cf a functicgr Figure Al skews two graphs of. functions. In mathematics't ; ' " ' ' ' able is usually depicted on the horizontal'axis-, and thé is depicted on the vertical axis”. Thejgfaph tilenindicfifi between the indepehcie'nt and the dependent variables __ __ _ However, in ecohomics it is common'to graph functioesl dent variable on the Vertical axis and the dependent" Sheri. zontal axis. Demand functionsgfor exarnple,- arejusu H @igged withthe price on the vertical axis and the-amount demanded oh ._ 711'“ 5. efisontal-aécis.. Figure A .1 ' A3 Properties of Functions A continuous function is one that can be drawn wit-bout. lifting-a. pencil from the paper: there are no jumps in a continuous function-A emboth EQUATlONS AND IDENTETIES fili- function is one ithat has no “kinks” or corners. A monotonic function is one that always increases or always decreases; a positive monotonic function always increases as 3: increases. while a. negative monotonic function always- decreases as :1: increases. AA inverse Functions Recall that a function has the property that for each value of a: there is a unique value 3}; associated with it and that a monotonic function is one-that is alwaygl'ifiereasing or always decreasing. This implies that for a just jiby Solving .ias a function of y. If y = 29:, then the inverse function is 2. {fay then there is no inverse function; given any y, both .1“ Li's-fl and z'fr— have the property that their square is equal to 31. Thus there is-not ajtmiqne value of :1: associated with each value of y. as is required by the definition of a function. Equations Identities An eqHation- askswhen a function is equal to some particular number. Examples of equations are 2.13:8 1:229 fix):0- Ther,SDIi}-ti9ni.t§":ailft- equation is a value of m that satisfies the equation. The first a solution of c z 4. The second equation has two solutions, 1:]: == —3. The third equation is just a general equation. Weldon’t ktié’iifitssolution until we know the actuai-rule that f stands for, but'fwie can denote its solution by it“. This simply means that :r“ is a number such that f(2:*)£= D. We that 33* satisfies the equation fix) 2 0. An identity a relationship between variables that holds for all values of the variables: Here are some examples of identities: (:1: + y)2 3172 + 21:31 + '1}? 2(Jr + 1) E 211: + 2. The special symbol E means that. the left-hand side and the right-hand side are equal for all Values of the variables. An equation only holds for some values of the variables, whereas an identity is true for all values of the variables. Often an identity is true by the definition of the terms hurrah-red. Ad MATHEMATiCAL APPENDIX AI: Linear Functions A linear function is a function of the form y 2 an: + b, where a and b are constants. Examples of linear functions are y=2$+3‘ 9:3'499. Strictly speaking,‘ a- function ofthe for-.121 'y. = a2: affine function,- and Only Elections ofthe'form y '_ -_ linear functions. However, we will not insist on this . 7 Linear functions can also be expressed implicitly in c.- In such a case, we often like to Solve for y as a to the “standard” form: c a: y=—e—x. bb_ A.7 Changes and Rates of Change The notation Ax is'read as ‘thechangeiin'jcf‘ :It m. If m Changes from :17". to w**_,.'.the3_1 the: Change in 'x I egg. We canalsowrite . _ _ _ to indicate that -a:’.*"" is 27* 3g 25-. Typically Ac will refer to a small changein m. 7 this by saying that .Azc represents a marginal change A rate of change is the ratio of two changes. If: _ , _ given by y 2 fix), then the rate of change of y with _ bv _A_$I u f($+A$) "Nil .533: w A2: I The rate of change measures how y changes as :5 changes, : _ \ A linear function has the property that the rate of-ch'ange of y With respect to x is constant. To prove this, note that if y -_.- a '+ 62, then ' fi—aezw———-—-———--—: [5.2: A3: E'_ .- Bl Ol’FS AM.) IN TFRFl'P'iS A3 For nonlinearfilnctimjh. flu: rate of rlmnge (if 1110 fimvticm will {lisp-Tull {H} the value of Consider. {or {ixanu'ilsiu The lilnrt‘inn t —- 333‘ For l'lllh function _ Afr; “ (3: {r A333 —- .153 1’2 2313.1' T‘ {3.2:}: 1'2 r 23.;- IL". AH.” Arr. Here the rat-e of: 01131ng from J.‘ to .‘r —l— A3“ depends on tlu- mlme of ,f' 21ml 0!] the Size. Of the Chang"; £1.17. Bill. if we. consider wry HUN-[ll Changes in .I'. A1? will be nearly zero. 5.0 the“). rate of change of i; with respect to :i' will be approximately A.8 Slopes. arid intercepts The rate of__ciza)nge of a function can be interpreted graphicafly as the slope of the fufic'fi'on. In Figure A29. we have depicted a. linear funmzimi y :7 W225 + 4, -The"fvertica1 intercept of this function is the value of y when .1: I G, wfiicli if; y = 4. The horizontal intercept is the value 0f .‘IT Whén y = 0, which is :2", : 2. The slope of the function tho mm ail" (fll'dl'lgi‘ of y as :1: changes. In this case, the 510139 of the function is —2. A I. I B? rtgrceptSQ -'Pa;n¢lA'depic-i_s the function y =: 93991 B" _d_epi_¢t5fflfizeffihfiion' sy x32 In gonemL if a: linear {lineman has: Elli-f figui'm y :2 my + b. tiu- \'::1":%r:1l intercept. will be y“ : E1 and the liorimmnl ini'm‘t'x‘pt. will he .r-" “hf-=91. ll" 21 linear function is expi'etthl in llw form {£1.31 + (131‘2 2:. a”. ' Figure A“? A6 MATHEMATICAL APPENDIX then the horizontal intercept will be the. value of 11 when 3:; = D,- which is 2:? : a1, and the vertical intercept will occur when 311 = 0, which means 13:; : dog. The slope of this function is mn.1/a2. _ A nonlinear function has the property that its slope changes as 9: changes. A tangent to a function at some point :r is e linear function that has the same slope. In Figure AQB we have depicted the function 3:2 and the ' tangent line at. I -—~ 1. 1 ' If y increases whenever I increases, then Ag; wili always'have the same sign as Ami so that the slope of the function will be positive. If on the other hand y decreases when 3: increases, or 3; increases when a: decreaSes, ._ Ag; and As; will have opposite signs, so that the slope of the function will be negative. I A.9 Absolute Values and Logarithms The absolute value of a number is a function f defined the following rule: “f > U ' a? l 2: “10(1)— {mat ifar < 0. Thus the absolute value of a number can be found by dropping the signof the number. The absolute value function is usually Wfittenz'as- ' The (natural) logarithm or log of :1: describes a particrflar functioniof it, which we write as y : inx or y = 111(3). The logarithmifinlctionsis the ' _ unique function that has the properties my) =1n(w) + me) for all positive numbers 2: and and .. ln(e) = 1. (In this last equation, 6 is the base of natural logarithms is equalto . 2.7183 . . .J In words, the log of the preduct of two nugrrbeiis'fisithe sumi'of _ the individual logs. This property implies another importenit'property of logarithms: " infiri’) : yln(:r), which says that the log of I raised to the power y isiieqnal' to 3; times the log of :r. ' I AJ 0 Derivatives The derivative of a. function y = f (.r) is defined to be (We) _ 1. f(m+n:i:) — fer) -— lIfl W. (is: nan—+0 Ax SECOND DERIVATIVES AT In the defiiaative is the limit of the rate of change of y with respect to as the :2: goes to zero. The derivative gives precise meaning . to the irate of change of y with respect to a: for small changes in The-gain fix) with respect to x is also denoted by f’ (35:). We shove that the rate of change of a linear function y = e331+ b is for this linear function $3011 the. eerie 0?. 1*! With-res?“ ‘0 x will ofjgflx). 2 3:2, we had A9p1yigg _. the__ definition of the derivative with to sis 23¢. _ __ were:me methods that. if y = him, then we ..; l 'dzc. m" 7. _ fie _-of e'function is the derivative of the derivative of f-(w), the second derivative of f (:12) with respect to a: " {ding or f”[3:). We know that flog); _ Thiis (£1132 dr 6%.?) *. d(2:::) (£332 - _ do: The-second-_dei'ivfitive measures the curvature of a function. A function with a. negative'seeond derivetiVe at some point is concave near that point; ' its'siope is A function with a positive second derivative at e " point is convex'neer. that point; its-slope is increasing. A function with a zero. deitiyifiive at a point is fist near that point. =2. iAe MATHEMATICAL APPENDIX ' Afléz The Product Rule and the Chain Rule Suppose that g(a::) and 11(3) are both functions of r. define tile function f (2:) that represents their product by f (1;) x -Then the derivative of f (:13) is given by ' - (3(3) dg(:c) d1: d1: '- dh(a:) dx + Mm) = 90"?) Given two functions-y :5 9(3) z' them 3 '-rf<m)--=_1i(e(e)s;i-- .- - I' _ ' For exampie, if 9(13) 2 3:2 end My) =' 23; + 3, then thef-’eet§fiesite fifectien '._. is ' - ' f = 2:132 + 3. I The chain rule says. that the derivative of e with respect to .12 is given by w wedge. dz: ' dy"da:‘ In our example, dh(y)/dy r: 2, anddg(9:)/ds: = 2x. I- -. that df = 2 x 2:1:- :._.4_:I:.-Direct caleulation var—‘1. fihétithiefi-‘isuthe . derivative of the function fix} = 2x2 5E3. ' ' " A.1 3 Partial-Derivatives _- _: "if Suppose that y depends on both at; Sandi-:22, so that :: _.____ the partial derivative of -f(e1_,eg; wig-gram tote aflxlixfl __ ' Hm. +A$1,$2) T.’ ‘I I. The partial derivative of 3" (3:1, 2:2) with respect to 3:1 Cierivetive " of the function with respect to 3:1, balding :62 fixed. Similzir-lygthe partial ' derivative with respect to 3:2 is ' ' . ' LW“) = Kim 6172 ASUQHO A372 2 I . Partial derivatives have exactly the same properties as erdina’ry' derivatives; " only the name has been changed to. protect the innocent'tfthat is, peeple who haven’t seen the a symbol). ' '- OPTIMiZATION in particular, partial (lorivaiive'zs obey the. chain rule, but with an extra twist. Suppose that $1 and x3: both depend on some variatfle t and that we-define the function eff.) b}: 5(1) : f{_1’1(t).333(t)). Then the derivative of git} with respect. to f. is given by dgfit) __ 8f(_:r:1._;i!2} afxrfit} ‘ 5f{r::;,.rz_} digit) alt 51‘ 1 (it 811:2 (if When It changes, it affects both xiii.) and 9:26.). Therefore, we neeé to calculate the derivative of f {'_ 3:11 3:2) with respect to each of those. cl‘ianges. A."1':_4 Optimiiafion If y = aclneves a maximum at cc“ if f{;r*) 2 f(:E) for all m. It can beg-shown that if f (.T) is a smooth function that aci'iievos its maximum values-at :3“, then one“) _ (LI; ’0 2 at L “'5' l < 0. dzir~ "' These expressions are referred to as the first-order condition and the second-order Condition for a maximum. The first—order condition says that the function is flat: at 1*; while the second-order condition says that the-function is concave near Clearly both of these properties have to hold if 1:“ is indeed a maximum. we say that achieves its minimum value at :r’“ if f (3") g f for ail If ffx) isle. smooth function that achieves its minimum at if“, then dféx ) :0 TC 2 ‘r :9: l 20. =r The fil‘shorder condition again says that the function is flat at .15 , the second-order condition now says that the. function is convex near 31*. If I-y 2 f(;E1.,_'.'EQ) is :1. smooth function that achieves its maximum or minimum at some point {31.163}. then we must satisfy new} while ‘ _ I U (313’ i : U' 511: 2 ‘ These are referred to as the first-order conditions. There are also second- order conditions for this probiern: but they are more difficnit to describe. A10 MATHEMATiCAL APPENDIX AJS Constrained Optimization Often we want to consider the maximUm or minimum (Jimefiinctidn dvgr some restricted values of (2:1, 9:2). "The notation max fish, 3:2) mg: . .' " finch" that? Mien-7&1). I: Ci . means find 931' and so; thai fry-€39 f0? that satisfy the equation g-(‘331',$2)'=1c:_ . ' The function f (:51, 2:2) is the-abjéCtive- tion g($1,$2] : c is called-‘tiie_'c0rifitédifit._ 'Metfiodsg of constrained muirflzation" prablerh fire" desér‘ibed'fs Chapter 5. . ' I ' ...
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This note was uploaded on 09/06/2010 for the course FBE ECON2113 taught by Professor Franchsica during the Fall '09 term at HKU.

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38 Mathematical Appendix - In this Appeaj '11 provide a...

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