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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 aQfnnetion' 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 outhe
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 theamount demanded oh ._ 711'“ 5. eﬁsontalaécis.. Figure
A .1 ' A3 Properties of Functions A continuous function is one that can be drawn witbout. liftinga. pencil
from the paper: there are no jumps in a continuous functionA 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
onethat 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'sﬂ and z'fr— have the property that their square is equal to 31.
Thus there isnot 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 actuairule 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 lefthand side and the righthand
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 hurrahred. 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 ofch'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 {lispTull {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 rate 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'depici_s the function y =: 93991 B" _d_epi_¢t5fﬂﬁzefﬁhﬁion' sy x32 In gonemL if a: linear {lineman has: Ellif 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 Thegain 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*! Withres?“ ‘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:
Thesecond_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; itsslope 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 functionsy :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 PartialDerivatives _ _: "if Suppose that y depends on both at; Sandi:22, so that ::
_.____ the partial derivative of f(e1_,eg; wiggram 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. Similzirlygthe 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
wedeﬁne 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 begshown that if f (.T) is a smooth function that aci'iievos its
maximum valuesat :3“, then one“) _
(LI; ’0 2 at L “'5' l < 0.
dzir~ "' These expressions are referred to as the ﬁrstorder condition and the
secondorder Condition for a maximum. The first—order condition says
that the function is ﬂat: at 1*; while the secondorder condition says that
thefunction 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 secondorder condition now says that the. function is convex near 31*. If Iy 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 ﬁrstorder 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? Mien7&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 theabjéCtive 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.
 Fall '09
 Franchsica

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