Solution Manual for Foundations of Mathematical Economics By Michael Carter A+ verified, ISBN: 978-0262531924

012345160896392
  



!"




,2345689 6 




36





















!"
#




$
%

&'"

36(
)*"

)*"


+


"""
)*"

,"*"*
)*"

)-
$"
./"
#

360
1

)*"

$
"

"%


)*"

2%%
)*"



)*"'

3'

)*"





"


4#
"%


56*"

76*'

,"8-
)*"

9"#"+"*
)*"


"
./"
#


,12345679





!

"#
$
$$#!
!% !

% $& !'(
) 

* +
1234567,
-./
0

1
./
'(!%) 
0(!%) 
12345672
) 
3'!3 
./4
!
'(!# 4
!
5$
0

$% #!
6!0


 

,


9  







!"#$
1%
&$
'("))!
*+,-./0#12
13
45("605"78(9":;./0#12
1:
45("605"78(9"3;<07=03
:
>4/0
?#0='(0519>0(4@0050@079(;
*+A-.B4@?50("CD7'90(09(4#0
EF 9>050990#("C9>045?>4609GHIHJH
HK!
EF 9>0(09"C="7(L@0#('7470="7"@1
EF 9>0(09"C8"")('7470="7"@1
EF 9>0(09"C?5410#('74
84@0;.B4@?50("CMNDNMOP(09(4#0
EF 9>0#0457L@60#(Q
EF 9>0749L#457L@60#(R
EF 9>0(09"C455?"(('650="5"#(
EF 9>0(09"C?"(('650?#'=0("C="??0#"79>0S"#5)@4#T09
EF 9>0(09"C?"(('65090@?0#49L#0("C5'UL')S490#;
VW
X
Y


Z


[!H\
XY
Z
[!;
*+]-^>0?5410#(09'(%
X_0771
J>#'(!;^>0'#4=9'"7(?4=0(4#0
3`
Xa"=T
b='(("#(
c4?0#! `
X_0771
J>#'(
*+d-^>0(09"C?5410#('(%
X3
Y 
$
; 4^>0(9#49081(?4=0"C04=>?5410#'(9>0(09
"CC04('650"L9?L9(
3`
Xe`
1Qf&e`
gh`
!
S>0#0e`

'(9>0"L9?L9"C)4@`;
*+i-^>0?5410#(09'(%
X
Y

!;^>0#04#0Y
Xj="45'9'"7(H74@051
kl%
mXn
!
Y!

!

Y!


!
Y

!

Y

!!
^>0#04#0Y ="45'9'"7('74907?5410#84@0;
o
psG((L@09>s492

1lW
qr
ms;^>s49'(s2

1t

W
qr
;^>'('@?5'0s(2

s1t

W

47)2

1t

r
H"#2

1sW
47)2
1rs
;J"7(0UL07951H2
1W
ur
;J"7/0#(051H4((L@02
1W
ur
;^>'('@?5'0(9>492
1
W

47)2
1r
;J"7(0UL079512
1t

W

s47)2
1t

r

47)9>0#0C"#0
21t
W
qr
;^>'('@?5'0(9>492
1lW
qr
m;^>0"9>0#')079'91'(?#"/0)('@'54#51;
v
7W
X%

W1w
5W
X n
W1w



, 01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
4
564
4 5
4
4
4
4
4
574
4 5 4 5
4
4
4
4
4
4
89:;<=47>7?4@A=4<=5BCD9EF4G4
H5IJKL4?4564M4K64N474O4
4 PQPRS@A=4TCUVB=4TVCW=4EX4C4T9F:B=4WE9F4DETT49T4YI+JZ,J>4@A-=,4T=D4EX4VETT9[B=4E;DWEU=T49F4DA<==4
DETT=T49T4DA=4V<E\;WD4
GYIJZJO]4GYIJZJO]4GYIJZJO4N4HYIJY./IJYLIJHYIJYIJZJLIJHYIJZJIJYLIJ
4


HYIJZJIJZJLIJHZIJYIJYLIJHZIJYIJZJLIJHZIJZIJYLIJHZIJZIJZJL4 ‘/
44
^4D_V9WCB4E;DWEU=49T4DA=4T=`;=FW=4HYIJYIJZJL4EX4DaE4A=C\T4XEBBEa=\4[_4C4DC9B>4
PQPPS
bJc4d0eJN4GRO4
aA=<=4RSN4HfIJfIgggJIJfL49T4DA=4V<E\;WD9EF4VBCF4;T9F:4FE49FV;DT4CF\4V<E\;W9F:4FE4E;DV;DT>4@E4
T==4DA9Th4i<TD4FED=4DACD4ReJS9T4C4X=CT9[B=4V<E\;WD9EF4VBCF>4@A=<=XE<=h4RS14bJ>4^BTEh4
RS14deJ0CF\4DA=<=XE<=4RS14bJc4d >4 0
@E4TAEa4DACD4DA=<=49T4FE4EDA=<4X=CT9[B=4V<E\;WD9EF4VBCF49F44444eJ2, h4a=4CTT;U=4DA=4WEFD<C<_>4@ACD4
9Th4a=4CTT;U=4DA=<=49T4TEU=4X=CT9[B=4V<E\;WD9EF4VBCF4jSSSSSSSS0e1,2,3, JJJJJJRS+,-,
S>44@A9T49UVB9=T4DA=4
=k9TD=FW=4EX4C4VBCF4V<E\;W9F:4C4VET9D9l=4E;DV;D4a9DA4FE49FV;0 DT>4@A9T4D=WAFEBE:9WCB4
9FX=CT9[B=h4TE4DACD4KJ1mJbJ>4
PQPnS 7>44o=D44pS14qJHKL>44@A9T449UVB9=T44DACD44HKIJrpL414bJ>44o=D44ps4t4p>44@A=F44HKIJrpsL4u4
HKIJrpL4CF\4X<==4\9TVETC[9B9D_49UVB9=T44DACD4HKIJrpsL414bJ>4@A=<=XE<=4ps414qJHKL>4
6>44^:C9F44CTT;U=44pSSs14qJHKL>4444@A9T449UVB9=T44DACD44HKIsJrpL4414 bJ>4444v_44X<==44\9TVETCBh4HKsIJ
rpL414bJJXE<4=l=<_4K4u4Kh4aA9WA49UVB9=T44DACD4pS14qJHKL>44qJHKsL4w4qJHKL>4
xyxz{@A=4\EUC9F4EX4|}~49T4G7IJ6O4N4JCF\4DA=4<CF:=49T4G6IJ€O44bJ>4
xyx‚{89:;<=47>7>4
xyxƒ{@A=4<=BCD9EF4|„4TD<9WDB_4A9:A=<4†‡ˆ‰~49T4D<CFT9D9l=h4CFD9T_UU=D<9W4CF\4CT_UU=D<9W>4ŠD49T4
FED4WEUVB=D=h4‹ŒŒŽ„Œ4E<4T_UU=D<9W>4




 3

, 01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
3
4546789:3;<==<>?@A3BCD=:3=?EBE3B9:?F3F:EG:HB?I:3GF<G:FB?:EJ3
KL N3 M33 M
OBFPCQ@PER?BS?TIP:33 U3 3
M M33M3
EVWW:BF?H3 U3 M 3M33M3
CCE@VB?WW:
XEVWW: BF?H3BF?H3 M33U3 M3UM3
H<WG=:B:3 M3M3U3
Y<B:3B9CB3B9:3GF<G:FB?:E3<;3EVWW:BFV3C@Z3C@B?XEVWW:BFV3CF:3@<B3W[B[C==V3:\H=[E?I:J3
454]7^:B3,- D:3C@3:_[?IC=:@H:3F:=CB?<@3<;3C3E:B3`/L-a0- 3J389CB3?Eb3B9:3F:=CB?<@,- 3ScdOPQPRSTPe3
cfggPhOSidjkldhOjkcShSTPmdnPdoOchdcpqrdhpjhdPTPOfdsL`LD.- :=<@AE3B<3E<W:3:_[?IC=:@H:3H=CEEJ3
^:B3tLD:3C@V3:=:W:@B3?@3`LC@Z3=:B3utv3D:3B9,- :3H=CEE3<;3:=:W:@BE3:_[?IC=:@B3B<3
tb3B9CB3?E3
,utv3w3x3sL.3̀Ly3sL,3tLz3
{?@H:3,-?E3OPQPRSTPe3tL,-tLC@Z3E<3tL.-,-utvJ3|I:FV3tL.-`LD:=<@AE3B<3E<W:3:_[?IC=:@H:3
H=CEE3C@Z3B9:F:;<F:3
`La3t1 .`L
2,utv3
Y:\Bb3>:3E9<>3B9CB3B9:3:_[?IC=:@H:3H=CEE:E3CF:3:?B9:F3Z?E}<?@B3<F3?Z:@B?HC=b33B9CB3?E3
,utv3~a3,uv3?;3C@Z3<@=V3?;3;,utv3€3 ,uv3a3J3
‚?FEBb3CEE[W:3,utv3€3,uv3a3J389:@3tL.3,utv3D[B33t.,uƒL vJ389:F:;<F:3,utv3~a3,uvJ3
„<@I:FE:=Vb33CEE[W:33,utv3€3,uv33~a333C@Z33=:B33sLL.3,utv3€3,uvJ33389:@33sLL,3tLLC@Z33DV3EVWW:BFV3
tL,3sJ333=E<3sL,3LC@Z3E<3DV3BFC@E?B?I?BV3tL,3J333^:B3†LD:3C@V3:=:W:@B3?@33,utv33E<33B9CB33†LL
,3tJ333AC?@33DV33BFC@E?B?I?BV33†LL,3LLC@Z33B9:F:;<F:33†LL.3,uvJ333‡:@H:3
,utv3ˆ3,uvJ3{?W?=CF33F:CE<@?@A3?WG=?:E33B9CB3,uv3ˆ3,utvJ389:F:;<F:3,utv3a3,uvJ3
‰:3H<@H=[Z:3B9CB3B9:3:_[?IC=:@H:3H=CEE:E3GCFB?B?<@3`J3
454Š789:3E:B3<;3GF<G:F3H<C=?B?<@E3?E3@<B3C3GCFB?B?<@3<;3B9:3E:B3<;3G=CV:FEb3E?@H:3C@V3G=CV:F3
HC@3D:=<@A3B<3W<F:3B9C@3<@:3H<C=?B?<@J3‚<F3:\CWG=:b3G=CV:F3‹3D:=<@AE3B<3B9:3H<C=?B?<@E3
x‹zb3x‹ŒLz3C@Z3E<3<@J3
454Ž7 3
sL3†La3sL‘3†LC@Z3†L~‘3sL
†L,3’La3†L‘3’LC@Z3’L‘3†L
8CFECE[@W:
E?B?I3?’BLV‘3
3<;s3L‘3 ?E3?W=?:GE=3?s:EL‘3
89?WG 3’’L,3
J3‰: B?I?BB9VC3B†3’LL,3
sLC3@@Z:3:DZV33B<BF3CE9@<E?>3 ~‘s3s
J3J“3[BE3EB[9W:
?E3?3W<GB9=?::FE>?
3B9E:CbB33B9CB3?E3
†L‘3sL>9?H93H<@BFCZ?HBE3B9:3CEE[WGB?<@3B9CB3sL3†J389:F:;<F:3>:3H<@H=[Z:3B9CB3’L~‘3sL
C@Z3B9:F:;<F:3sL3’J389:3<B9:F3F:E[=B3?E3GF<I:Z3?@3E?W?=CF3;CE9?<@J3
45”•7–—˜™™š›œžŸEE[W:3sL3†J3
3 sL3†La3†L~‘3sL
3
3
>9?=:3 L †L3sLa3†L‘3sL
3
89:F:;<F:3 L sL3†La3†L~3sL


 +

, 01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
0
12345616789:;;<=>?@0A?B0CDE?B0A?FG?
@0A?B0HI?@0J?B0CDE?B0 @0
B0A?F00HI?B0J?F00CDE?F0JK??B0
LMDN>?J?M;?OPCD;MOMQ>R?S>?NTDNU<E>?OVCO?@0J?FG?
WOOV?CP>O?=C
@0J? PCDOV;MCOMOQ?MFOZ0?J?
MD;B?0OTC?D;EV?TOS? M=[@GU?M>:;;?;O<V=>
CO??FT0OJ?
V>BPSM C;?;<F=[
R?N;T>DR?OOPVCCPZO??MO;T??CO;V;><?=> OM@TDG??X>?YDTS?
0J?
OVCO?B0A?FG?X>?NTDNU<E>?OVCO?F0 @0CDE?
@0J?F0CDE?F0KJ?@0HI?@0A?F0
\VM;?;VTS;?OVCO?A?M;?OPCD;MOMQ>G?
]^_]`abcbdefb9LMDN>?J?M;?ghihjklhm?@0J?@0SVMNV?M=[UM>;?@0n?@G?
12345616789:;;<=>?@0n?B0CDE?B0n?FG?oTS?
@0n?B0pI?@0J?B0CDE?B0J?@0
B0n?F0pI?B0J?F0CDE?F0J?B0
?
\PCD;MOMQMOZ?Tq?J?M=[UM>;? 9 @0J?B0CDE?B0J?F0HI?@0J?F0
?
? rT=sMDMDt? 0 F0J?B0CDE?B0J?@0HI?F0J?@0
?
??
5uvv8126w9 0 @0J?F0CDE?F0J?@0HI?@0n?F0

@0n?B0pI?@0J?B0CDE?B0J?@0
pI?B0J?@0CDE?@0J?B0
pI?B0n?@0
]^__`abcbdefb9xQ>PZ?MDO>t>P?M;?C?=<UOM[U>?Tq?MO;>UqR?OVCO?M;?y0H?zyG?
123456167899:;;<=>?y0H?{|0CDE?|0H?}~0SV>P>?{0}0 ,-€0G?\V>D?y0H?{}~0;T?OVCO?y0M;?C?
=<UOM[U>?Tq?~G?
41995uvv8126w99Wq??y00H??{|R??{00,?€0R??OV>D??|00H??z./?y00CDE??{00,‚00€0G???ƒTP??>„C=[U>R????M;??C?
? ]^_†`??=<UOM[U>?Tq?ˆ?s<O?ˆ?M;?DTO?C?=<UOM[U>?Tq?G?
9 ? ‰Š0‹Œ?H?? Š0B0‹0F0Ž?
9
9 ]^_‡`?  Š ‹
0 ?H ?
 B?0Ž?
? J?B ?H?‹0B0F0Ž?
‘?B?H?‹0F0Ž?
’?B ?H?Š0@0B0Ž?
“?B ?H?Š0@0Ž?



 +

, 01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
2
3456789:2;<=92>?@9?9@2=A2BC2D<,2;<EF2G2HEIEHGJ29J9H9I:2:K9?92EF2I>29J9H9I:2LKEMK2F:?EM:JA2
N?9M9@9F2E:O2:KG:2EF2:K9?92EF2I>29J9H9I:2P<,2;<FQMK2:KG:2P<R2DC2D<,2;<EF2:K92S?F:29J9H9I:2ET2E:2
N?9M9@9F29U9?A2>:K9?29J9H9I:O2:KG:2EF2D<V2P<T>?2GJJ2P<,2;C2
345W7XK92HGYEHGJ29J9H9I:F2>T2;<G?92Z<GI@2[C2XK92HEIEHGJ29J9H9I:2>T2;<EF2DC2XK9F92G?92
GJF>2=9F:2GI@2L>?F:29J9H9I:F2?9FN9M:EU9JAC2
345\7]FFQH92:KG:2D<EF2G2=9F:29J9H9I:2EI2;<>?@9?9@2=A2BC2XKG:2EFO2D<B2P<T>?2GJJ2P<<<<<;,- C2XKEF2
EHNJE9F2:KG:2:K9?92EF2I>2P<<<<;<,- LKEMK2F:?EM:JA2@>HEIG:9F2DC2XK9?9T>?9O2D<EF2HGYEHGJ2EI2;C2^I2
_=Y9GF:HN29JJ99H9 2`CIa:bC2O2:K92IQH=9?F2cd<ed<fd<gd<h2G?92GJJ2HGYEHGJ29J9H9I:FO2=Q:2I>I92>T2:K9H2EF2G2
345i7]FFQH92:KG:2:K929J9H9I:F2G?92@9I>:9@2D`d<Dadjjjd<DkC2l92MGI2E@9I:ETA2:K92HGYEHGJ2
2
9J9H9I:2=A2M>IF:?QM:EIm2GI>:K9?2JEF:2QFEIm2:K92T>JJ>LEIm2?9MQ?FEU92GJm>?E:KH2
n`2o2D. `2
/p< ET2Dp<q2npr`2
2 D
np<o2npr`2 >:K9?LEF92
2
sA2M>IF:?QM:E>IO2:K9?92EF2I>2Dp<LKEMK2F:?EM:JA2FQMM9@9F2nkC2nk<EF2G2HGYEHGJ29J9H9I:C2
345t7 2
u
u :KG:2EF2 2
Dv2EF2HGYEHGJ2wx2:K9?92@>9F2I>:29YEF:2D<q2 Dv2
qyDvz2o2{D2<|2D<q2Dv2}2o2~2
Dv2EF2=9F:22wx2Dv2B2D<T>?29U9?A2D<,2;<
wx2D<V2Dv2T>?29U9?A2D<,2;<
XKG:2EFO29U9?A2D<,2;<=9J>ImF2:>2VyDvz2>?2VyDvz2o2;C2
34€789:2<=92G2I>I9HN:A2F9:2>T2G2F9:2;<>?@9?9@2=A2BC2D<,-;<EF2G2J>L9?2=>QI@2T>?2
=><EQTI2@E:22NET?29E:M29@@>9HE
F2I9GU:99?FA2929UJ99?H9 2EI?22=O>2:QKIG@:O22E:FK2GD:<2V2
A2JI>:L9 EF2nD<<TB2
>?P2G<TJJ>2?n2<9U9,-
?A2J>C2L9
^:2?E2F2=G>2QmI@?92GP:<9>FT:22J>C2L9?2
3437]IA2HQJ:ENJ922>T2e‚2EF22GI2QNN9?2=>QI@2T>?2C22XKQFO2:K922F9:2>T2QNN9?22=>QI@F2>T2<EF2
e‚0d<-`a‚d<aƒ‚djjj<C2X1K-92J9GF:2QNN9?2=>QI@2>T2<EF2e‚C2XK92>IJA2J>L9?2=>QI@2EF2`O2K9IM92
E:2EF2:K92m?9G:9F:2J>L9?2=>QI@C2
3457XK92J9GF:2QNN9?2=>QI@F2>T2EI:9?UGJ2„nd<Z2G?92Z<GI@2[C2XK92J9GF:2QNN9?2=>QI@2>T2ynd<
Zz2EF2PC2
347 2
u
u
u
D<EF2GI2QNN9?2=>QI@2>T2<wx2D<B2n<T>?29U9?A2n<,2<
u wx2n<V2D<T>?29U9?A2n<,2<
wx2<‡2VyDz2
u
†EHEJG?JA2
2 D<EF2G2J>L9?2=>QI@2>T2<wx2D<V2n<T>?29U9?A2n<,2<
wx2n<B2D<T>?29U9?A2n<,2<
wx2<‡2ByDz2



 +

, 01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
0
1234567809:98;0<=,0>?@0
0
<=A0B=CD0<E0F=BE0780<E0G0BE0HIJ0<?0F=B?0
KCIL90HMM09M9N9IOP0<=,0>?0H890L7NQH8HRM9@0A0CP0L7NQM9O9S0CO0CP0H0O7OHM078J98T0
1G
23U 0E5\V=?P\P^W^N90
^=\=_0@X09Y=C=OC[P9008L07<NQ M9O900D78009:98;00YT0000Z[
Y=XY=BY=780BY=XY=<Y=780R7O [9 I0O0[D79880009:98;00<\=B= ,-]==HIJ00D7800HMM00Y==
T0̀C
<Y=aY=BY=bcdefggeY=Z[9I0hijki0<=a0BT0
<Y=laY=BYl=ab0BctdTe0msu7
cnoeY=p9O0q=R90O[90j8PO0CIJC:CJWHM0rCO[0H0PO8CLO0Q89D989IL9@0O[HO0CP0q=G0
NQM9O9I9PP007D0XY==9IPW89P0O[HO0q=CP00hijkihvT0Z[9I0hijki0
NCIYs<Y=
Y
<=A0B=CD0<q=AY=Bq=
B=A0 <=7O[98rCP90
123w5p9O0x@0y=HIJ0z=R90PWRP9OP07D0H0jICO90P9O0]T0K9O0CILMWPC7I0{0CP0
|}~}€}ePCIL90x={0xT0
‚dfƒm„‚„oePCIL90x={0y=HIJ0y={0z=CNQMC9P0x={0z=T0
fƒ‚„†m‡nno‚d„ˆePCIL90x={0y=HIJ0y={0x=CNQMC9P0x=G0y=
Z[989D7890{0CP0H0QH8OCHM078J98T0
1X0
23‰‹5 =DV7P8P0WHN9
MM0 ‹ <=HŠITJ006BW8=OH[9898@00RCD70O<[=0CPM90HH0PM9O0HWPQO0QW9Q8Q09R870WRI7JWPI0J7@D00BŠ=X0
0== T0Z<[HT0OŒ0DC0P0B<==CPX0
0H‹0M=9DH7P8O00HWQMM0Q‹9=80ŠR=H7I,-
WJI0JB@0=
<=X0BT0;0HIOCŽP;NN9O8;@0<=G0BT0 ,-
1235 0
<=a0B=G0<=X0B=HIJ0B=X0<=
0
r[CL[0CNQMC9P0O[HO0<=G0B=R;0HIOCP;NN9O8;T0̀HL[09‘WC:HM9IL90LMHPP0
a0s<t0G0’B0=,0]=“0B=a0<=”0
L7NQ8CP9P0•WPO0H0PCI–M909M9N9IO0<T0
—˜™šeeNH›0œs]t0G0]==HIJ0NCI0œs]t 0G0T0
124ž5Z[90PWRP9O0’?\?=Ÿ\=¢”0D78NP0H0L[HCIT0¡7890–9I98HMM;@0O[90P9O07D0CIO9–980Q7r98P07D0H0
–C:9I0IWNR980’_0\=_\=_ \^^^=”0D78NP0H0L[HCIT0
£¤¥£¦VPPWN90<=HIJ0B=H890NH›CNHM09M9N9IOP07D0O[90L[HCI0ŠT0Z[9I0<=X0‹=D780HMM0‹= ,-Š=
HIJ0CI0QH8OCLWMH80<=X0BT0KCNCMH8M;@0B=X0‹=D780HMM0‹= ,-Š=HIJ0CI0QH8OCLWMH80B=X0<T0KCIL90
X0CP0HIOCŽP;NN9O8CL@0<=G0BT0
124§50ET00;0HPPWNQOC7I@0D7809:98;0̈===,-.- ==y====©=@00/- 000s̈t0CP0H0I7I9NQO;0jICO90L[HCIT0ª9IL9@0CO0
[HP0H0WIC‘W90NH›CNHM09M9N9IO@0«s̈tT0
?T00p9O0̈=R90HI;0I7J9T00`CO[980̈=CP00HI0CICOCHM0I7J90780̈=[HP0H00WIC‘W90Q89J9L9PP780«s̈tT0
`C O[9800«s̈t00CP00HI00CICOCHM0I7J9@007800CO00[HP00H00WIC‘W90Q89J9L9PP780«s«s̈ttT00u7IOCIWCI–0CI0
O[CP0rH;@0r90O8HL907WO0H0WIC‘W90QHO[0D87N0¨=RHL¬0O70HI0CICOCHM0I7J9T0­90LHI0R90
PW8907D09:9IOWHMM;089HL[CI–0HI0CICOCHM0I7J90PCIL90y=CP0jICO9T0
e
12435 0
sE\=?t0®0s¢\=Et0G0s¢\=?t0HIJ00sE\=?t0̄0s¢\=?t0G0sE\=?t0



 +

,  01344567189 9 
7



!"#
$%
&$#
$

' 7(8 )*
1
23445 6718/9:;<1=>1?@@AB1CD?>E1FDB1G18H:9:IJ1KL=K1;<1M/NOP18:=>E1M/NOP1971Q;R;S=BSTJ1
8:/19:;<1=1SDUAB1CD?>E1FDB1G81H:9:I71
3VWX<<?RA18:P1971YLA>18:;<1=>1?@@AB1CD?>E1FDB1G18H:9:IJ1KL=K1;<18:P18:/1971ZF1[:;<1=>T1?@@AB1
CD?>E1FDB1G18H:9:IJ1KLA>1[:P1871YLABAFDBAJ18:;<1KLA1SA=<K1?@@AB1CD?>E1FDB1G18H:9:I71
Q];DR;>^S=ABBS<TAJ1ST9J:1;=<1<=<?1SRA
DUA
18B:1/1C9D:?_>1E81F7D1YB1LAG18>1H:89::;I<J11==>>1E?1@;<@1A\BB1AC=DK?A>BE11KLFD=B>11=>T1DKLAB1SDUAB1CD?>E71
G81H:9:IJ1KL=K1;<18:P1971
`VWa<;>\1KLA1@BAbAE;>\1Ac?;^=SA>bA1
8:P18:-19::_d18:/1e8:-19f1_18:
8:/19:P18::_d1e8:/19f1-18:_18:
234g5X1bL=;>1h:;<1=1bDR@SAKA1@=BK;=SST1DBEABAE1<AK71iDB1A^ABT18H:9:01h:U;KL18:j_19J1
A;KLAB18:k19:DB19:k1871YLABAFDBAJ1lmnom1KLA1RAAK1=>E1pD;>1CT1
8:, -
.9:_19:;F18:k1
8:;F19:k1
9:
8:
, . 8:;F18:k1
8:/ 9:_19:;F19:k198::
:
h:;<1=1S=KK;bA1U;KL1KLA<A1D@AB=K;D><71
234q5X<<?RA1h61=>E1hr1=BA1S=KK;bA<J1=>E1SAK1h:_1h61s1hr71]D><;EAB1=>T1KUD1ASARA>K<1MO
_11e86H:8rf11=>E11NOO_1e96H:9rf11;>11h711Q;>bA11h611=>E11hr11=BA11S=KK;bA<J11[611_1861/1961101h61
=>E11[r11_118/. r1 11110. 19r1111111hrJ11<D11KL=K11tOO_11e[6H:[r/. f1_11e86/. 111111960.
H:8r1111119rf111111h7111
i
C ?
D B
?K>L EAB
1 FRD
D B B
1 AK
L1At 1O P1 M O = >
E1 t O P1 N O ;> 1KL A 1 > = K ? B= S1 @ BD E ? bK1 D BE A BJ1 <D1 K
L =K1tO;<1=>1?@@AB1
GMH:NI7111v u^ABT11?@@AB11CD?>E11t v11_11e[v6H:[vrf11DF11GMH:NI1R?<K11L=^A11[w::Pw::8w::=>E11[w::Pw::9wJ1
<QD;R;
11KLS==BKS 1tJ111MP1
1T O-N t
1O_1e861-196H:8r1-O;<
7 11 1Y L A B AFDBA J1 1t O 191r1KfL71A11SA=<K11?@@AB11CD?>E11DF11GMH:NIJ11KL=K11;<11tOO_11MO/1N71
234x5yAK1z:CA1=1<?C<AK1DF1h:=>E1SAK1
z{1_1G18:01h:|18:P1}:FDB1A^ABT1}:01z:I1
CA1KLA1<AK1DF1?@@AB1CD?>E<1DF1z71YLA>18{101z{1 ~7 1T1=<<?R@K;D>J1z{1L=<1=1\BA=KA<K1
SDUAB1CD?>E1[71Q;>bA1A^ABT1}:z:;<0. 1=1SDUAB1CD?>E1DF1z{J1[:P1}:FDB1A^ABT1}:z71YLABAFDBA0. 1[:;<1
=>1?@{@AB1CD?>E1DF1z71i?BKLABRDBAJ1[:;<1KLA1SA=<K1?@@AB1CD?>E1DF1zJ1<;>bA1[:€18:FDB1A^ABT111
8:z71Y=L;B;<11DAF<1K=ASCASRA ;<LA<1KL=0. K1A^ABT1<?C<AK1DF1h:=S<D1L=<1=1SA=<K1?@@AB1CD?>E71Z>1@=BK;b?S=BJ1
A;<^1A=B1Tb1D@R@ SAKA1S=KK;bA71 <1=1SA=<K1?@@AB1=>E1=1\BA=KA<K1SDUAB1CD?>E71]D><Ac?A>KST1h:
> K<1L=
2345‚;KLD?K1SD<<1DF1\A>AB=S;KTJ1UA1U;SS1@BD^A1KLA1bSD<AE1;>KAB^=S1b=<A71yAK1ƒ„H:[1CA1=>1
;>KAB^=S1;>1=1S=KK;bA1†71‡Ab=SS1KL=K1„:_1;>Fƒ„H:[1=>E1[:_1<?@ƒ„H:[71]LDD<A1=>T18H:9:;>1ƒ„H:[1
ˆ1†71Q;>bA1†:;<1=1S=KK;bAJ18:/19:01†:=>E1
8:/19:_1<?@G18H:9:I1€1[:
YLABAFDBA18:/19:01ƒ„H:[71Q;R;S=BSTJ18:-19:01ƒ„H:[71ƒ„H:[1;<1=1S=KK;bA71Q;R;S=BSTJ1FDB1=>T1<?C<AK1z:ˆ1ƒ„H:[1
01ƒ„†HJ1:[<?1@1<Dz1:KL01
=K†1:ƒ;„F1H†:[:;<11;b<D1bR@S 71<DJ1<?@1z:€1[:_1<?@ƒ„H:[71YLABAFDBA1<?@1z:01ƒ„H:[71Q;R;S=BST1;>F1z:
ˆ1 AKAA7K1AXS
DR@S




 +