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ECON30020: Mathematical Economics

创建日期：2024-06-15 16:33:30

所属分类：数学mathematics

标签： ECON30020

Mathematical Economics

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ECON30020: Mathematical Economics

topology
the real line R and Euclidean space Rn
closed and open sets, compactness
dot (scalar) product
functions
level sets
examples in economics
S. R. Williams (UniMelb) Lecture 2 01/03/2024 2 / 1
The Real Line
a depiction of the set R of real numbers
open interval (a, b) = {x ∈ R |a < x < b}
closed interval [c , d ] = {x ∈ R |c ≤ x ≤ b}
S. R. Williams (UniMelb) Lecture 2 01/03/2024 3 / 1
Euclidean Space
Rn = {x = (x1, x2, . . . , xn) |xi ∈ R for 1 ≤ i ≤ n}
x denotes a vector, a point, or an n-tuple
“bold” used for vectors
Euclidean distance: for x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn),
the distance ∥x− y∥ between y and x is
∥x− y∥ =
√
(x1 − y1)2 + (x2 − y2)2 + . . .+ (xn − yn)2
length of a vector x (or distance from 0 = (0, 0, . . . , 0)) is
∥x∥ =
√
x21 + x
2
2 + . . .+ x2n
on the real line R,
∥x − y∥ = |x − y | (absolute value)
∥x∥ = |x |
S. R. Williams (UniMelb) Lecture 2 01/03/2024 4 / 1
Open Balls
topos: “place” (Greek)
generalizing open and closed intervals to Euclidean space Rn
open ball of radius ε at the point x0 ∈ Rn : Bε(x0)
all points whose distance from x0 is less than ε
expresses “nearness” to the center x0
S. R. Williams (UniMelb) Lecture 2 01/03/2024 5 / 1
Closed Balls
closed ball of radius ε at the point x0 ∈ Rn: all points whose
distance from x0 is less than or equal to ε
S. R. Williams (UniMelb) Lecture 2 01/03/2024 6 / 1
Open and Closed Sets
X ⊂ Rn is open if, for every x0 ∈ X , there exists an ε (x0) > 0 such
that Bε(x0)
(
x0
) ⊂ X
expresses the ability to move around any point x0 without leaving the
set X
S. R. Williams (UniMelb) Lecture 2 01/03/2024 7 / 1
Open and Closed Sets (Cont.)
Y ⊂ Rn is closed if its complement Rn\Y is an open set
S. R. Williams (UniMelb) Lecture 2 01/03/2024 8 / 1
Open and Closed Sets in the Real Line
every open interval is an open set, every closed interval is a closed set
an open set need not be an open interval, and a closed set need not
be a closed interval
can be disjoint, separated sets
S. R. Williams (UniMelb) Lecture 2 01/03/2024 9 / 1
Bounded Set in Euclidean Space
A set X ⊂ Rn is bounded if there exists an open ball Bε (x0) that
contains it
ε is not required here to be small
a set is bounded if and only if there is an upper bound on how far apart
any two of its elements can be
S. R. Williams (UniMelb) Lecture 2 01/03/2024 10 / 1
Boundary Point of a Set
x ∈ Rn is a boundary point of the set X ⊂ Rn if for every δ > 0,
each of the sets
Bδ(x) ∩ X ,Bδ(x) ∩Rn\X
is non-empty.
every ball Bδ(x) centered at x includes points in X and points outside
of X (in Rn\X )
example: the boundary points of the ball Bε(x0) consists of all points
that are exactly ε in distance from x0
S. R. Williams (UniMelb) Lecture 2 01/03/2024 11 / 1
A Set is Closed
It Contains All of its Boundary Points
Suppose X ⊂ Rn is closed
show that every boundary point of X lies in X
proof by contradiction: suppose x∗ is a boundary point of X and
x∗ /∈ X
because X is closed, its complement Rn\X is open
x∗ ∈ Rn\X
there exists Bε(x∗) ⊂ Rn\X
this contradicts the assumption that x∗ is a boundary point of X
conclude: every boundary point of X lies in X
S. R. Williams (UniMelb) Lecture 2 01/03/2024 12 / 1
A Set Contains All of its Boundary Points
It is Closed
Suppose every boundary point of X ⊂ Rn is in X
prove that X must be closed
proof by contradiction: suppose x∗ ∈ Rn\X has the property that
every Bε(x∗) ∩ X is nonempty
we also have:Bε(x∗)∩Rn\X is nonempty because x∗ ∈ Bε(x∗),Rn\X
x∗ is therefore a boundary point of X
this contradicts the fact that every boundary point of X lies in X
conclude: X is closed
S. R. Williams (UniMelb) Lecture 2 01/03/2024 13 / 1
Compact
a set X is compact if it is both closed and bounded
along with continuity of a function, this condition ensures existence of
extreme values (maxima and minima) in X
this will come next week, after we discuss continuity
Weierstrass Theorem
S. R. Williams (UniMelb) Lecture 2 01/03/2024 14 / 1
Dot (Scalar, or Inner) Product of Two Vectors
Let x = (x1, x2, . . . , xn) and y = (y1, y2, , . . . , yn). Then
x · y = x1y1 + x2y2 + . . .+ xnyn
start with two vectors x, y, end up with a real number x · y
∥x∥ = √x · x
S. R. Williams (UniMelb) Lecture 2 01/03/2024 15 / 1
Use: Budget Constraint
let:
x = (x1, x2, . . . , xn) denote a bundle of goods, where each xi ≥ 0
p = (p1, p2, , . . . , pn) a vector of prices, where each pi > 0
I > 0 the income or wealth of a consumer.
the consumer’s budget constraint is
p1x1 + p2x2 + . . .+ pnxn = p · x ≤ I
use dot product when > 2 goods
his choice set is
C (p, I ) = {x ∈ Rn |each xi ≥ 0,p · x ≤ I }
= {x ∈ Rn+ |p · x ≤ I } .
S. R. Williams (UniMelb) Lecture 2 01/03/2024 16 / 1
Interpretation
x · y = ∥x∥ · ∥y∥ · cos θ
where θ is the angle between the vectors x and y
x · y = 0 implies that the vectors x and y are perpendicular
used in interpreting gradients and directional derivatives in
multivariable calculus
S. R. Williams (UniMelb) Lecture 2 01/03/2024 17 / 1
Functions
Let X and Y be sets. A function f from X to Y is a rule that assigns an
element of Y to each element of X
denoted:
y = f (x), f : X → Y
starting point: X , Y ⊂ R
a function f in this case may be given by a formula, e.g.,
f (x) = x2 − 1,
or, it may be defined in some other way e.g., f (x) = the largest integer
that is less than or equal to x
more generally: X and Y can be any sets
essential point is that a unique element of Y is specified for each
element of X
not every element of Y need be assigned
S. R. Williams (UniMelb) Lecture 2 01/03/2024 18 / 1
Terminology of Functions
f : X → Y
X : the domain of the function f
Y : the range of the function f
sometimes called the codomain
f (X ): the image of the function f
f (X ) ⊂ Y is the set consisting of all elements of the range Y that are
assigned by f to some element of X
f (X ) = {y ∈ Y |y = f (x) for some x ∈ X }
for x ∈ X , f (x) is an element of Y , and f (X ) is a subset of Y
S. R. Williams (UniMelb) Lecture 2 01/03/2024 19 / 1
Terminology of Functions (Cont.)
for y ∈ Y , f −1(y) ⊂ X is the inverse image (or preimage)
determined by y
the set of all elements in X such that are assigned y by f :
f −1(y) = {x ∈ X |f (x) = y }
when X ⊂ Rn and y ∈ R, an inverse image is sometimes called a
level set of f
more generally: for any subset Y ′ ⊂ Y of the range,
f −1(Y ′) =
{
x ∈ X ∣∣f (x) ∈ Y ′ }
S. R. Williams (UniMelb) Lecture 2 01/03/2024 20 / 1
Examples of Functions in Economics
the consumption set for a consumer is
Rn+ = {(x1, x2, . . . , xn) ∈ Rn |each xi ≥ 0}
xi denotes a nonnegative quantity of good i
a utility function is a function
u : Rn+ → R
that represents the consumer’s preferences over bundles of goods
x ∈ Rn+
a level set u−1(k) of u determined by k ∈ R is an indifference curve
for the consumer, i.e., all bundles x ∈ Rn+ that provide the consumer
with utlity k :
u−1(k) = {x ∈ Rn+ |u(x) = k }
S. R. Williams (UniMelb) Lecture 2 01/03/2024 21 / 1
Examples of Functions in Economics (Cont.)
the input set for a firm is
Rn+ = {(x1, x2, . . . , xn) ∈ Rn |each xi ≥ 0}
here, xi denotes a nonnegative quantity of input i
a production function is a function
q : Rn+ → R
that represents the maximal amount of an output that the firm can
produce using the inputs x ∈ Rn+
a level set q−1(k) of q determined by k ∈ R is an isoquant for the
firm, i.e., all bundles x ∈ Rn+ of inputs that the firm could use to
produce the output k
q−1(k) = {x ∈ Rn+ |q(x) = k }

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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