‡ MRS: Application to Broadband Pricing
‡ Insights: Cobb-Douglas Utility Model
‡ ಯ,PSHUIHFW6XEVWLWXWHVರ3UHIHUHQFHV
‡ Insights: Linear Utility Model
‡ ಯPerfect 6XEVWLWXWHVರPreferences
‡ ಯ5HODWLYH,QWHQVLW\ರRIPreferences
‡ Convex Combinations
‡ (Strictly) Convex Preferences
‡ Budget Set with Uniform Prices and Exogenous Income
‡ Utility Maximization Problem (Intro & Types of Utility Models in this course)
‡ (On your own) UMP Math: Watch 2020-2021 Videos and Read Consumer
Theory Chapter
‡ UMP Summary
1
Today:
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 6
Micro-Case: Broadband Service
Household Demand for Broadband Internet in 2010 BE Journal of Public Economics
3
9XMPMX]SJ&VSEHFERH7IVZMGI!0MRIEV*YRGXMSRSJXLIJSPPS[MRKŰ'LEVEGXIVMWXMGWű
ܷ ൌ െͲǤͲʹͳܥ ൅ ͲǤͻͶͷܨܵ ൅ ͳǤͲͲͻܸܨܵ ൅ ͲǤͶͳ͹ܴ ൅ ͲǤͳ͵Ͷܲ ൅ ͲǤͲͻʹܶܪ ൅ ͲǤͲͲͲͲͳܯܮ ൅ ͲǤͳͲ͸ܸܨ ൅ ͲǤͲ͸ͻܯܴ
Ű&VSEHFERH7IVZMGI'LEVEGXIVMWXMGWű Coefficient(Marginal Utility)
ࡹࢁࢉ࢕࢙࢚
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁࢉ࢕࢙࢚
ܥ ൌ Cost to Broadband Users െ0.021 1 1
ܨܵ ൌ Fast speed internet service 0.945 0.022 45
ܸܨܵ ൌ Very fast speed internet service 1.009 0.021 48.05
ܴ ൌ Reliable internet service 0.417 0.05 19.86
ܲ ൌ Priority of download service 0.134 0.157 6.39
ܶܪ ൌ Tele-health connectivity 0.092 0.229 4.38
ܯܮ ൌ Mobile laptop connectivity 0.00001 2,100 0.000476
ܸܨ ൌ Video phone capabilities 0.106 0.198 5.05
ܯܴ ൌ movie rentals 0.069 0.304 3.29
How can a Broadband service provider use these numbers (say, for cost and ܸܨܵሻ to make more money?
Micro-Case: Broadband Service
6XSSRVHDFDEOHFRPSDQ\UDLVHVWKHಫFRVWRIEURDGEDQG
VHUYLFHVಬ
What can/should this company do to prevent customers from
ಯVZLWFKLQJGHIHFWLQJರWRRWKHUEURDGEDQGSURYLGHUV"
Suppose a cable company ಯLPSURYHVರEURDGEDQGVHUYLFH
How much can the company raise cost without risking customers
ಯVZLWFKLQJGHIHFWLQJರWRRWKHUEURDGEDQGSURYLGHUV"
4
How can a Broadband service provider use these numbers (say, for cost and ܸܨܵሻ to make more money?
Cost 以 AttributeTUT 00 0
AttributeTUX lostA lil out
以
Micro-Case: Broadband Service
Use the following table and graphs to answer the questions on the previous slide
5

‘•–
‘•–
Ű&VSEHFERH7IVZMGI'LEVEGXIVMWXMGWű Coefficient(Marginal Utility)
ࡹࢁࢉ࢕࢙࢚
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁࢉ࢕࢙࢚
ܥ ൌ Cost to Broadband Users െ0.021 1 1
ܸܨܵ ൌ Very fast speed internet service 1.009 0.021 48.05
How can a Broadband service provider use these numbers (say, for cost and ܸܨܵሻ to make more money?
dans
acostu⼆
MRS⼆咒器 on
MRS⼆qq.cn mu 不 21muz
Micro-Case: Broadband Service
Use the following table and graphs to answer the questions on the previous slide
6

‘•–
‘•–
Ű&VSEHFERH7IVZMGI'LEVEGXIVMWXMGWű Coefficient(Marginal Utility)
ࡹࢁࢉ࢕࢙࢚
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁ࢜ࢇ࢘࢏ࢇ࢈࢒ࢋ
ࡹࢁࢉ࢕࢙࢚
ܥ ൌ Cost to Broadband Users െ0.021 1 1
ܸܨܵ ൌ Very fast speed internet service 1.009 0.021 48.05
How can a Broadband service provider use these numbers (say, for cost and ܸܨܵሻ to make more money?
V1我 ⼼ lost不以 ⼼
想占1MRSkmi⼆4805
costni
_SEiiiii.n.ci
148.05
independent
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 7
I
lkfcq.iq
Interior VsBoundary
goodbadlneutral
intensityofpreference
ಯ&REE-'RXJODVರInsights from Utility Model & Indifference Curves
ܷ ൌ ݍଵ
ఈ ݍଶ
ఉ ֜ – ‹–‡”‹‘” „—†Ž‡•ǡ „‘–Š ‰‘‘†• ƒ”‡ ̶‰‘‘†̶ ‰‘‘†• •‹…‡ ܯ ଵܷǡܯ ଶܷ ൐ Ͳ
ݍଶ ൌ
ܷ
ݍଵ
ఈ
ଵ
ఉ
୍୬ୢ୧୤୤ୣ୰ୣ୬ୡୣ େ୳୰୴ୣ ୍େ ୉୯୳ୟ୲୧୭୬
֜
ܣݏ ݍଵ ՜ Ͳǡ ݍଶ ՜ λ
ܣݏ ݍଶ ՜ Ͳǡ ݍଵ ՜ λ
֜
‹Ǥ ‡Ǥ †‘‡•ᇱ– –‘—…Š ݕ െ ƒš‹•
‹Ǥ ‡Ǥ †‘‡•ᇱ– –‘—…Š ݔ െ ƒš‹•
ܯܴܵ ൌ
݀ݍଶ
݀ݍଵ
ቚ
௎
ൌ െ
ߙ
ߚ
ݍଶ
ݍଵ
֜
ۼ܍܏܉ܜܑܞ܍ ܁ܔܗܘ܍ ֜
‘‘†• ͳ ƒ† ʹ ƒ”‡ ܘ܍ܚ܋܍ܑܞ܍܌ ܉ܛ ܛܝ܊ܛܜܑܜܝܜ܍ܛ
܁ܔܗܘ܍ ܞ܉ܚܑ܍ܛ ܊ܡ ܊ܝܖ܌ܔ܍ ֜ ۾܍ܚ܋܍ܑܞ܍܌ ܉ܛ ۷ܕܘ܍ܚ܎܍܋ܜ ܁ܝ܊ܛܜܑܜܝܜ܍ܛ
ሺŠ‡ ™‘”† •—„•–‹–—–‡• ‹• ۼ۽܂ ‹ –Š‡ •‡•‡ ‘ˆ ͳͲͳሻ
8
A
B
C
Notice
ܯ ஺ܷǡܯܷ஻ǡܯܷ஼ ൐ Ͳ
ܷ ܣ ൌ ܷ ܤ ൌ ܷ ܥ ൐ Ͳ
ܯܴ ஺ܵ ൐ ܯܴܵ஻ ൐ ܯܴܵ஼
Suppose @ this High ௤మ
௤భ
ratio ֜ ൐ ͳ
'SRWYQIVLEWKVIEXIVMRXIRWMX]SJTVIJIVIRGIJSVŷ#
Suppose @ this Low ௤మ
௤భ
ratio ֜ ൏ ͳ
'SRWYQIVLEWKVIEXIVMRXIRWMX]SJTVIJIVIRGIJSVŷ#
⼩Bso
Lax.y
intercept
PositiveinteriorBoundk pos.tw NegativeTradeoff
mnvur
notaconstant
出品
tnanqgungupiotq.tnin
togeta
gmnguplessqtogetquwq.ouk.az
more
thana
ಯ/LQHDUರ Insights from Utility Model & Indifference Curves
ܷ ൌ ߙݍଵ ൅ ߚݍଶ ֜ – ‹–‡”‹‘” „—†Ž‡•ǡ „‘–Š ‰‘‘†• ƒ”‡ ‰‘‘† ‰‘‘†• •‹…‡ ܯ ଵܷǡܯ ଶܷ ൐ Ͳ
ݍଶ ൌ
ܷ
ߚ
െ
ߙ
ߚ
ݍଵ
୍୬ୢ୧୤୤ୣ୰ୣ୬ୡୣ େ୳୰୴ୣ ୍େ ୉୯୳ୟ୲୧୭୬
֜
ܣݏ ݍଵ ൌ Ͳǡ ݍଶ ൌ ܷȀߚ
ܣݏ ݍଶ ൌ Ͳ ݍଵ ൌ ܷȀߙ
ܯܴܵ ൌ
݀ݍଶ
݀ݍଵ
ቚ
௎
ൌ െ
ߙ
ߚ
֜
ۼ܍܏܉ܜܑܞ܍ ܁ܔܗܘ܍ ֜
‘‘†• ͳ ƒ† ʹ ƒ”‡ ܘ܍ܚ܋܍ܑܞ܍܌ ܉ܛ ܛܝ܊ܛܜܑܜܝܜ܍ܛ
۱ܗܖܛܜ܉ܖܜ ܁ܔܗܘ܍ ֜ ۾܍ܚ܋܍ܑܞ܍܌ ܉ܛ ۾܍ܚ܎܍܋ܜ ܁ܝ܊ܛܜܑܜܝܜ܍ܛ
ሺŠ‡ ™‘”† •—„•–‹–—–‡• ‹• ۼ۽܂ ‹ –Š‡ •‡•‡ ‘ˆ ͳͲͳሻ
9
A
B
C
ܷȀߚ
ܷȀߙ Notice
ܯ ஺ܷǡܯܷ஻ǡܯܷ஼ ൐ Ͳ
ܷ ܣ ൌ ܷ ܤ ൌ ܷ ܥ ൐ Ͳ
ܯܴ ஺ܵ ൌ ܯܴܵ஻ ൌ ܯܴܵ஼ ൌ
ߙ
ߚ
@ This High ௤మ
௤భ
ratio ֜ ൌ ఈ
ఉ
'SRWYQIVLEWKVIEXIVMRXIRWMX]SJTVIJIVIRGIJSVŷ#
@ This Low ௤మ
௤భ
ratio ֜ ൌ ఈ
ఉ
'SRWYQIVLEWKVIEXIVMRXIRWMX]SJTVIJIVIRGIJSVŷ#
⼩p 70
䶲gunmore
givealess
MN䤐缃景 台 i likegoodmorethangood2卡4 Likegood2morethangoodI
Cobb-Douglas Utility Model: Characterizing Preferences
ܷ ൌ ݍଵ
ఈ ݍଶ
ఉ ™Š‡”‡ ߙǡ ߚ ൐ Ͳ
ܯ ଵܷ ൌ ߙݍଵ
ఈିଵ ݍଶ
ఉ
ܯܷଶ ൌ ߚݍଵ
ఈ ݍଶ
ఉିଵ
Indifference Curves
ݍଶ
ఉ ൌ
ܷ
ݍଵ
ఈ
ଵ
ఉ
ݔ ƒš‹• ‹–‡”…‡’– ՜ λ
ݕ ƒš‹• ‹–‡”…‡’– ՜ λ
݀ݍଶ
݀ݍଵ
ቚ
௎ ௖௢௡௦௧௔௡௧
ൌ െ
ߙ
ߚ
ݍଶ
ݍଵ
Indifference curves are
ಯFXUYHGtowards originರ
10
At interior bundles, consumer
’‡”…‡‹˜‡•‰‘‘†•ͳƒ†ʹƒ•Dz‰‘‘†dzgoods
At interior bundles, consumer perceives goods as Dz‹’‡”ˆ‡…–substitutesdzሺNOT in the ECO 101 sense)
At interior bundles, consumer has
ƒDz•–”‘‰–ƒ•–‡ˆ‘”˜ƒ”‹‡–›dz(strictly convex preferences)
Consumer would rather consume both goods together than each good separately („‡…ƒ—•‡ǥǫ)
n 9 1 go
Cobb-Douglas Utility Model: Characterizing Preferences
ܷ ൌ ݍଵ
ఈ ݍଶ
ఉ ™Š‡”‡ ߙǡ ߚ ൐ Ͳ
ܯ ଵܷ ൌ ߙݍଵ
ఈିଵ ݍଶ
ఉ
ܯܷଶ ൌ ߚݍଵ
ఈ ݍଶ
ఉିଵ
Indifference Curves
ݍଶ
ఉ ൌ
ܷ
ݍଵ
ఈ
ଵ
ఉ
ݔ ƒš‹• ‹–‡”…‡’– ՜ λ
ݕ ƒš‹• ‹–‡”…‡’– ՜ λ
݀ݍଶ
݀ݍଵ
ቚ
௎ ௖௢௡௦௧௔௡௧
ൌ െ
ߙ
ߚ
ݍଶ
ݍଵ
Indifference curves are
ಯFXUYHGtowards originರ
11
At interior bundles, consumer
’‡”…‡‹˜‡•‰‘‘†•ͳƒ†ʹƒ•Dz‰‘‘†dzgoods
At interior bundles, consumer perceives goods as Dz‹’‡”ˆ‡…–substitutesdzሺNOT in the ECO 101 sense)
At interior bundles, consumer has
ƒDz•–”‘‰–ƒ•–‡ˆ‘”˜ƒ”‹‡–›dz(strictly convex preferences)
Consumer would rather consume both goods together than each good separately („‡…ƒ—•‡ǥǫ)
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 8
Suppose ܷ ൌ ݂ ݍଵǡ ݍଶ and ܯ ଵܷǡܯܷଶ ൐ Ͳ
At an interior bundle in the consumption set:
ˆ ܯܴܵ ൌ
ௗ௤మ
ௗ௤భ ௎
ቐ
൐ ͳ
ൌ ͳ
൏ ͳ
then ൞
‡Žƒ–‹˜‡Ž› ‰”‡ƒ–‡” ‹–‡•‹–› ‘ˆ ’”‡ˆ‡”‡…‡• ˆ‘”ǣ ̴̴̴̴̴̴̴̴̴̴̴̴̴̴
“—ƒŽ ‹–‡•‹–› ‘ˆ ’”‡ˆ‡”‡…‡• ˆ‘” ݍଵ ƒ† ݍଶ
‡Žƒ–‹˜‡Ž› ‰”‡ƒ–‡” ‹–‡•‹–› ‘ˆ ’”‡ˆ‡”‡…‡• ˆ‘”ǣ ̴̴̴̴̴̴̴̴̴̴̴̴̴̴
7KLVLVDSRZHUIXOPHFKDQLVPIRULGHQWLI\LQJWHVWDEOHಯPDUNHWVHJPHQWVರ
13
Intensity of Preferences IRUಯ*RRGರ*RRGV
n NE
Likegoodmove
goodI
likegoodmove
good2
A denttradeoff B
ifanswersarethesame Bothgoodsinthesamesegment
交换的数量相同
Example: ܯܴܵ and the Intensity of Preferences ಯ6HJPHQWDWLRQರ
14
Horsepower
Interior
Space
Extra
50 hp
Source: A. PetrinŰQuantifying the Benefits of New Products: The Case of the MinivanűJournal of Political Economy (2002)
How much interior space would parents vs. teenagers ŰKMZIYTűsubstitute/tradeoff) in exchange for 50
more horsepower engine and remain indifferent to the initial bundle?
Ű4YVTPIű7IKQIRX
Ű&PYIű7IKQIRX
Parents ǔteenagers
teenagerswnnngtoqueupmorensm.eu
inexchangeforpower
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 9
Tasteforvariety
16
:HLJKWHG$YHUDJHಯ&RQYH[&RPELQDWLRQVರ mmslope
icon7525
A ò
Avg ixinot
wavgwixigaom.ua
cini
a㠒收 innit偕 灬偕时以加
un B㴞 1㵘 1点悲品 华然邀 WeightedAvg
qxnxx.snAunxB
Examples: Towards (Strict) Taste for Variety [(Strictly) Convex Preferences]
17
Suppose that ܣ ̱ ܤ (where ܣ ് ܤ in the interior of the consumption set)
Now consider ܥ ൌ ߙܣ ൅ ͳ െ ߙ ܤ where ߙ א Ͳǡͳ
Is ܥ ظ ܣ ̱ ܤ?
Is ܥ غ ܣ ̱ ܤ
,S[[MPP]SYYWIXLMWWPMHIXSŰTVITűJSVXIWXWERHUYM^^IW#
as strongtasteforvariety
AKAistnctgconuxpnetenenea.sn
Goodgood
Q Qiyesa.iay ehauayntty_ymhidg.uaNoQiNo Q_QyesA A 的
A
Ba B is 13 13
(Strict) Taste for Variety; (Strictly) Convex Preferences
18
Convex Preferences have nothing to do with convex functions but do involve convex combinations
Assume ܷሺݍଵǡ ݍଶ) is differentiable
Suppose that ܣ ̱ ܤ (where ܣ ് ܤ in the interior of the consumption set)
Now consider ܥ ൌ ߙܣ ൅ ͳ െ ߙ ܤ where ߙ א ͲǡͳIf ܥ ൌ ߙܣ ൅ ͳ െ ߙ ܤ ظ ܣ ̱ ܤ
Or equivalently
ܷ ܥ ൐ ܷ ܣ ൌ ܷሺܤሻ
for ߙ א Ͳǡͳ and all bundles in the consumption
set ܣ ് ܤ ǡ then the consumer has Strictly
Convex Preferences
(aka Strong Taste for Variety)
;MXLŰKSSHűKSSHWWXVMGXP]GSRZI\TVIJIVIRGIW
implies that ௗ ெோௌ
ௗ௤భ
൏ Ͳ
If ܥ ൌ ߙܣ ൅ ͳ െ ߙ ܤ ذ ܣ ̱ ܤ
Or equivalently
ܷ ܥ ൒ ܷ ܣ ൌ ܷሺܤሻ
for ߙ א Ͳǡͳ and all bundles in the consumption
set ܣ ് ܤ ǡ then the consumer has Convex
Preferences
(aka Taste for Variety)
;MXLŰKSSHűKSSHWGSRZI\TVIJIVIRGIWMQTPMIW
that ௗ ெோௌ
ௗ௤భ
൑ Ͳ
Onlyfolaigoodmnnnnnn
Examples: (Strict) Taste for Variety [(Strictly) Convex Preferences]
19
Strictly Convex
Preferences
At Interior Bundles:
ܯ ଵܷǡܯܷଶ ൐ Ͳ
݀ȁܯܴܵȁ
݀ݍଵ
൏ Ͳ
Strictly Convex
Preferences
At Interior Bundles:
ܯ ଵܷǡܯܷଶ ൐ Ͳ
݀ȁܯܴܵȁ
݀ݍଵ
൏ Ͳ
Convex
Preferences
At Interior Bundles:
ܯ ଵܷǡܯܷଶ ൐ Ͳ
݀ȁܯܴܵȁ
݀ݍଵ
ൌ Ͳ
Strictly Non-Convex
Preferences
At Interior Bundles:
ܯ ଵܷǡܯܷଶ ൐ Ͳ
݀ȁܯܴܵȁ
݀ݍଵ
൐ Ͳ
Strictly Convex
Preferences
At Interior Bundles:
ܯ ଵܷǡܯܷଶ ൏ Ͳ
݀ȁܯܴܵȁ
݀ݍଵ
൐ Ͳ
A A A A A
B B B B B
C
ߙܣ ൅ ͳ െ ߙ ܤ
C
ߙܣ ൅ ͳ െ ߙ ܤ
C C C
ߙܣ ൅ ͳ െ ߙ ܤ ߙܣ ൅ ͳ െ ߙ ܤ ߙܣ ൅ ͳ െ ߙ ܤ
Assume ܷሺݍଵǡ ݍଶ) is differentiable
Suppose that ܣ ̱ ܤ (where ܣ ് ܤ in the interior of the consumption set)
Now consider ܥ ൌ ߙܣ ൅ ͳ െ ߙ ܤ where ߙ א Ͳǡͳ
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 10
$&RQVXPHUಬV%XGJHW6HW
Suppose all ܰ ಯJRRGVರLQWKHFRQVXPHUಬVFKRLFHVHWDUHsold at uniform prices ଵܲǡ ଶܲǡ ǥ ǡ ேܲ ൐
Ͳ
Let ܻ̈́ ൐ Ͳ GHQRWHWKHFRQVXPHUಬVincome DOORFDWHGIRUSXUFKDVLQJJRRGVಯOXPS-VXPರ
exogenous income)
The set of affordable bundles ಯEXGJHWVHWರLV
ܤ ൌ ݍଵǡ ݍଶǡ ǥ ǡ ݍே ǣ ෍
௜ୀଵ
ே
௜ܲݍ௜ ൑ ܻ
The 2-good budget set consists of all bundles ݍଵǡ ݍଶ א ܥ s.t:
ଵܲݍଵ ൅ ଶܲݍଶ ൑ ܻ
or
š’‡†‹–—”‡ ൑ ܻ
ܧ ൑ ܻ
21
$&RQVXPHUಬV%XGJHW6HW
22
ݍͳ
ݍʹ
ܤǣ the set of
affordable bundles in
the choice set
The equation of the budget line is:
ݍଶ ൌ
ܻ
ଶܲ
െ ଵܲ
ଶܲ
ݍଵܻ
ଶܲ
݀ݍଶ
݀ݍଵܤǣ ሼ ଵܲǡ ଶܲǡ ܻሽ
A
B
The 2-good budget set consists of all bundles ݍଵǡ ݍଶ א ܥ s.t: ଵܲݍଵ ൅ ଶܲݍଶ ൑ ܻ
ܻ
ଵܲ
The Slope of the budget line:
݀ݍଶ
݀ݍଵ
ቚ
ୡ୭୬ୱ୲ୟ୬୲ ୉୶୮ୣ୬ୢ୧୲୳୰ୣ
ൌ െ ଵܲ
ଶܲ
What happens to the set of affordable choices if all pecuniary parameters Ĺ by the same % amount?
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 11
2-Good Utility Maximization Problem (UMP)
Assume 2-JRRGಯ&KRLFH6HWರ ݍଵǡ ݍଶ ൒ Ͳ
$VVXPHFRQVXPHUSHUFHLYHVLQWHULRUEXQGOHVDVಯJRRGರJRRGVLH
interior bundles have ܯ ଵܷǡܯܷଶ ൐ Ͳ)
Suppose all pecuniary parameters: ̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
Consumer chooses optimal bundle by solving UMP (Maximizing her
utility subject to budget and other constraints)
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ǡ ݍଶ ൒ ݍଶ
௠௜௡
Typically: ݍଵ௠௜௡ ൌ ݍଶ௠௜௡ ൌ Ͳ
24
2-Good UMP: Intuition
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ǡ ݍଶ ൒ ݍଶ
௠௜௡
25
ܻ
ଶܲ
ܻ
ଵܲ
ܤ ൌ ሼ ଵܲǡ ଶܲǡ ܻሽ
Infeasible bundle
*
Optimal choice
Sub-optimal bundle
Sub-optimal bundle
ݍଵ
כ
ݍଶ
כ
ݍଵ
ݍଶ
ܷכ
ݍଵ
௠௜௡
ݍଶ
௠௜௡
An illustrative example
26
ಯ&OHDQ,QWHULRU%RXQGDU\8036ROXWLRQV"ರ
27
ECO204 2022-2023 UMP
We will deal with ŰKSSHűKSSHutility functions in two
categories:
Ű(MJJIVIRXMEFPIű#
Only those utility models
with (strictly) convex
preferences
ŰUYEWM-GSRGEZIJYRGXMSRWű
Ű2SR-(MJJIVIRXMEFPIű#
Ű0ISRXMIJ9XMPMX]1SHIPű
(Does it have (strictly)
convex preferences?)
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 12
2-Good UMP
29
914Ű-RTYXWű
ଵܲ ൌ ”‹…‡ ‘ˆ
‘‘† ͳ ൐ Ͳ
ଶܲ ൌ ”‹…‡ ‘ˆ
‘‘† ʹ ൐ Ͳ
ܻ ൌ …‘‡ ‘” —†‰‡– ൐ Ͳ
Utility Model and its Parameters ሺߙǡ ߚǡ Ǥ Ǥ ሻ
914Ű3YXTYXWű
ݍଵ
כ in terms of ଵܲǡ ଶܲǡ ܻ and Utility Parameters
(Good 1 Demand Function)
ݍଶ
כ in terms of ଵܲǡ ଶܲǡ ܻ and Utility Parameters
(Good 2 Demand Function)
ܷכ
;IGERTIVJSVQŰGSQTEVEXMZIűWXEXMGW*SVexample:
݀ݍଵ
כ
݀ ଵܲ
ǡ
݀ݍଵ
כ
݀ ଶܲ
ǡ
݀ݍଵ
כ
ܻ݀
ǡ
݀ݍଵ
כ
݀ݍଵ
௠௜௡
݀ݍଵ
כ
݀ –‹Ž‹–› ƒ”ƒ‡–‡”
ƒ†
ܷ݀כ
݀ ଵܲ
ǡ
ܷ݀כ
݀ ଶܲ
ǡ
ܷ݀כ
ܻ݀
ǡ
ܷ݀כ
݀ݍଵ
௠௜௡
ܷ݀כ
݀ –‹Ž‹–› ƒ”ƒ‡–‡”
(On your own) UMP Math: Watch 2020-2021 Videos and Read Consumer Theory Chapter
7KHಯ*HQHUDOರ-Good UMP
ܷ ݍଵǡ ݍଶ is a differentiable utility function with ܯ ଵܷǡܯܷଶ ൐ Ͳ at interior bundles and
”‡’”‡•‡–‹‰Dz–ƒ•–‡ˆ‘”˜ƒ”‹‡–›dz’”‡ˆ‡”‡…‡•
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ǡ ݍଶ ൒ ݍଶ
௠௜௡
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ െ ܻ ൌ Ͳǡെݍଵ൑ െݍଵ
௠௜௡ǡ െݍଶ൑ െݍଶ
௠௜௡
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ െ ܻ ൌ Ͳǡെݍଵ ൅ ݍଵ
௠௜௡ ൑ Ͳǡെݍଶ ൅ ݍଶ
௠௜௡ ൑ Ͳ
ƒš
௤భǡ௤మǡఒభǡఒమǡఒయ
/ ൌ ܷ ݍଵǡ ݍଶ െ ߣଵ ଵܲݍଵ ൅ ଶܲݍଶ െ ܻ െ ߣଶ െݍଵ ൅ ݍଵ௠௜௡ െ ߣଷ െݍଶ ൅ ݍଶ௠௜௡
30
̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
7KHಯ*HQHUDOರ-Good UMP
ƒš
௤భǡ௤మ
ܷ ݍଵǡ ݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ǡ ݍଶ ൒ ݍଶ
௠௜௡
ƒš
௤భǡ௤మǡఒభǡఒమǡఒయ
/ ൌ ܷ ݍଵǡ ݍଶ െ ߣଵ ଵܲݍଵ ൅ ଶܲݍଶ െ ܻ െ ߣଶ െݍଵ ൅ ݍଵ௠௜௡ െ ߣଷ െݍଶ ൅ ݍଶ௠௜௡
߲/
߲ݍଵ
ൌ
ต
߲ܷ
߲ݍଵ
ெ௎భ
െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
߲/
߲ݍଶ
ൌ
ต
߲ܷ
߲ݍଶ
ெ௎మ
െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
߲/
߲ߣଵ
ൌ െ ଵܲݍଵ ൅ ଶܲݍଶ െ ܻ ൌ Ͳ
ߣଶ ൒ Ͳǡ ݍଵ ൒ ݍଵ
௠௜௡ǡ ߣଶ െݍଵ ൅ ݍଵ
௠௜௡ ൌ Ͳ
ߣଷ ൒ Ͳǡ ݍଶ ൒ ݍଶ
௠௜௡ǡ ߣଷ െݍଶ ൅ ݍଶ
௠௜௡ ൌ Ͳ
31
First, split this into 2 cases
Next, split this into 2 cases
FOCs:
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
7KHಯ*HQHUDOರ-Good UMP
32
Case A Case B Case C Case D
Suppose ݍଵ ൌ ݍଵ௠௜௡ If so, then: ݍଵ ൒ ݍଵ௠௜௡ Suppose ݍଵ ൌ ݍଵ௠௜௡ If so, then: ݍଵ ൒ ݍଵ௠௜௡
Suppose ݍଶ ൌ ݍଶ௠௜௡ Suppose ݍଶ ൌ ݍଶ௠௜௡ If so, then: ݍଶ ൒ ݍଶ௠௜௡ If so, then: ݍଶ ൒ ݍଶ௠௜௡
If so, then: ߣଶ ൒ Ͳ Suppose ߣଶ ൌ Ͳ If so, then: ߣଶ ൒ Ͳ Suppose ߣଶ ൌ Ͳ
If so, then: ߣଷ ൒ Ͳ If so, then: ߣଷ ൒ Ͳ Suppose ߣଷ ൌ Ͳ Suppose ߣଷ ൌ Ͳ
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
ߣଷ ൒ Ͳǡ ݍଶ ൒ ݍଶ
௠௜௡ ǡ ߣଷ െݍଶ ൅ ݍଶ
௠௜௡ ൌ Ͳ
Suppose ݍଶ ൌ ݍଶ௠௜௡
If so, then: ߣଷ ൒ Ͳ
ߣଶ ൒ Ͳǡ ݍଵ ൒ ݍଵ
௠௜௡ ǡ ߣଶ െݍଵ ൅ ݍଵ
௠௜௡ ൌ Ͳ
Suppose ݍଵ ൌ ݍଵ௠௜௡
If so, then: ߣଶ ൒ Ͳ
Suppose ߣଶ ൌ Ͳ
If so, then: ݍଵ ൒ ݍଵ௠௜௡
Suppose ߣଷ ൌ Ͳ
If so, then: ݍଶ ൒ ݍଶ௠௜௡
ߣଶ ൒ Ͳǡ ݍଵ ൒ ݍଵ
௠௜௡ ǡ ߣଶ െݍଵ ൅ ݍଵ
௠௜௡ ൌ Ͳ
Suppose ݍଵ ൌ ݍଵ௠௜௡
If so, then: ߣଶ ൒ Ͳ
Suppose ߣଶ ൌ Ͳ
If so, then: ݍଵ ൒ ݍଵ௠௜௡
FOCs:
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
ಯ*HQHUDOರ-Good UMP Cases: Intuition
33
8SWIIXLIMRXYMXMSRPIXŭWWYTTSWIݍଵ௠௜௡ ൌ ݍଶ௠௜௡ ൌ Ͳ
A B
C
D
Case A
Consume Nothing /
Case B
Purchase good 1 only
Case C
Purchase good 2 only
Case A Case B Case C Case D
Suppose ݍଵ ൌ ݍଵ௠௜௡ If so, then: ݍଵ ൒ ݍଵ௠௜௡ Suppose ݍଵ ൌ ݍଵ௠௜௡ If so, then: ݍଵ ൒ ݍଵ௠௜௡
Suppose ݍଶ ൌ ݍଶ௠௜௡ Suppose ݍଶ ൌ ݍଶ௠௜௡ If so, then: ݍଶ ൒ ݍଶ௠௜௡ If so, then: ݍଶ ൒ ݍଶ௠௜௡
If so, then: ߣଶ ൒ Ͳ Suppose ߣଶ ൌ Ͳ If so, then: ߣଶ ൒ Ͳ Suppose ߣଶ ൌ Ͳ
If so, then: ߣଷ ൒ Ͳ If so, then: ߣଷ ൒ Ͳ Suppose ߣଷ ൌ Ͳ Suppose ߣଷ ൌ Ͳ
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
Case D
Purchase both goods
Violates
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ constraint
We will show that
Indifference curve slope here
is ൐ Budget line slope
We will show that
Indifference curve slope here
is ൏ Budget line slope
We will show that
Indifference curve slope here
is ൌ Budget line slope
̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
ಯ*HQHUDOರ-Good UMP Cases B, C and D
34
Case B Case C Case D
If so, then: ݍଵ ൒ ݍଵ௠௜௡ ൌ Ͳ Suppose ݍଵ ൌ ݍଵ௠௜௡ ൌ Ͳ If so, then: ݍଵ ൒ ݍଵ௠௜௡ ൌ Ͳ
Suppose ݍଶ ൌ ݍଶ௠௜௡ ൌ Ͳ If so, then: ݍଶ ൒ ݍଶ௠௜௡ ൌ Ͳ If so, then: ݍଶ ൒ ݍଶ௠௜௡ ൌ Ͳ
Suppose ߣଶ ൌ Ͳ If so, then: ߣଶ ൒ Ͳ Suppose ߣଶ ൌ Ͳ
If so, then: ߣଷ ൒ Ͳ Suppose ߣଷ ൌ Ͳ Suppose ߣଷ ൌ Ͳ
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
Sub ݍଶ ൌ Ͳ in ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
֜ ݍଵ ൌ
ܻ
ଵܲ
൐ Ͳ ሺ•ƒ–‹•ˆ‹‡• ݍଵ ൒ Ͳ …‘†‹–‹‘ሻ
If Case B is solution, then: ߣଷ ൒ Ͳ
ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ֜ ߣଷ ൌ െܯܷଶ ൅ ߣଵ ଶܲ
To see when ߣଷ ൒ Ͳ we need ߣଵ.
Sub ߣଶ ൌ Ͳ in ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ֜ ߣଵ ൌ
ெ௎భ
௉భ
Sub ߣଵ ൌ
ெ௎భ
௉భ
in ߣଷ ൌ െܯܷଶ ൅ ߣଵ ଶܲ ൌ െܯܷଶ ൅
ெ௎భ
௉భ
ଶܲ
ƒ•‡ ‘……—”• ™Š‡ ߣଷ ൒ Ͳ
ܯܴܵ ൒ ଵܲ
ଶܲ
ƒ– „—†Ž‡
Sub ݍଵൌ Ͳ in ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
֜ ݍଶ ൌ
ܻ
ଶܲ
൐ Ͳ ሺ•ƒ–‹•ˆ‹‡• ݍଶ ൒ Ͳ …‘†‹–‹‘ሻ
If Case C is solution, then: ߣଶ ൒ Ͳ
ܯ ଵܷ െ ߣଵ ଵܲ ൅ ߣଶ ൌ Ͳ ֜ ߣଶ ൌ െܯ ଵܷ ൅ ߣଵ ଵܲ
To see when ߣଶ ൒ Ͳ we need ߣଵ.
Sub ߣଷ ൌ Ͳ in ܯܷଶ െ ߣଵ ଶܲ ൅ ߣଷ ൌ Ͳ ֜ ߣଵ ൌ
ெ௎మ
௉మ
Sub ߣଵൌ
ெ௎మ
௉మ
in ߣଶ ൌ െܯ ଵܷ ൅ ߣଵ ଵܲ ൌ െܯ ଵܷ ൅
ெ௎మ
௉మ
ଵܲ
ƒ•‡ ‘……—”• ™Š‡ ߣଶ ൒ Ͳ
ܯܴܵ ൑ ଵܲ
ଶܲ
ƒ– „—†Ž‡
Suppose neither Case B nor Case C arise; i.e.
neither of the following expressions arise:
ܯܴܵ ൐ ଵܲ
ଶܲ
ǡ ܯܴܵ ൏ ଵܲ
ଶܲ
If neither Case B nor Case C arise, the solution
must be Case D:
ܯܴܵ ൌ ଵܲ
ଶܲ
Optimal choice found by solving:
ܯܴܵ ൌ ଵܲ
ଶܲ
and
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
̈́ ଵܲǡ ̈́ ଶܲǡ ܻ̈́ ൐ Ͳ
ಯ*HQHUDOರ-Good UMP Cases B, C, D: Summary
35
Case B (Only) Case C (Only) Case D (Only)
ˆ ܯܴܵ ൐ ଵܲ
ଶܲ
ƒ– —†Ž‡ ݍଵ ൌ
ܻ
ଵܲ
ǡ ݍଶ ൌ Ͳ ˆ ܯܴܵ ൏
ଵܲ
ଶܲ
ƒ– —†Ž‡ ݍଵ ൌ Ͳǡ ݍଶ ൌ
ܻ
ଶܲ
If neither Case B nor C arise: ܯܴܵ ൌ ௉భ
௉మ
ƒ– —†Ž‡
ݍଵ ൌ
ܻ
ଵܲ
ݍଵ ൌ Ͳ
Bundle D ݍଵ and ݍଶ found by solving:
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
ܯ ଵܷ
ଵܲ
ൌ
ܯܷଶ
ଶܲ
ݍଶ ൌ Ͳ ݍଶ ൌ
ܻ
ଶܲ
ߣଵ ൌ
ܯ ଵܷ
ଵܲ
ߣଵ ൌ
ܯܷଶ
ଶܲ
ߣଵ ൌ
ܯ ଵܷ
ଵܲ
ൌ
ܯܷଶ
ଶܲ
൐ Ͳ
ߣଶ ൌ Ͳ ߣଶ ൌ ܯܷଶ
ଵܲ
ଶܲ
െ ܯ ଵܷ ൐ Ͳ ߣଶ ൌ Ͳ
ߣଷ ൌ ܯ ଵܷ
ଶܲ
ଵܲ
െ ܯܷଶ ൐ Ͳ ߣଷ ൌ Ͳ ߣଷ ൌ Ͳ
B
C
D
Here: Indifference curve
steeper than budget line
Here: Indifference curve
flatter than budget line
Here: Indifference curve
tangent to budget line
Ajaz Hussain, Economics, University of Toronto (STG)
Ű'SRWYQIV8LISV]ű
Part 13
ಯ*HQHUDOರ-Good UMP Cases B, C, D: Summary
37
Case B (Only) Case C (Only) Case D (Only)
ˆ ܯܴܵ ൐ ଵܲ
ଶܲ
ƒ– —†Ž‡ ݍଵ ൌ
ܻ
ଵܲ
ǡ ݍଶ ൌ Ͳ ˆ ܯܴܵ ൏
ଵܲ
ଶܲ
ƒ– —†Ž‡ ݍଵ ൌ Ͳǡ ݍଶ ൌ
ܻ
ଶܲ
If neither Case B nor C arise: ܯܴܵ ൌ ௉భ
௉మ
ƒ– —†Ž‡
ݍଵ ൌ
ܻ
ଵܲ
ݍଵ ൌ Ͳ
Bundle D ݍଵ and ݍଶ found by solving:
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
ܯ ଵܷ
ଵܲ
ൌ
ܯܷଶ
ଶܲ
ݍଶ ൌ Ͳ ݍଶ ൌ
ܻ
ଶܲ
ߣଵ ൌ
ܯ ଵܷ
ଵܲ
ߣଵ ൌ
ܯܷଶ
ଶܲ
ߣଵ ൌ
ܯ ଵܷ
ଵܲ
ൌ
ܯܷଶ
ଶܲ
൐ Ͳ
ߣଶ ൌ Ͳ ߣଶ ൌ ܯܷଶ
ଵܲ
ଶܲ
െ ܯ ଵܷ ൐ Ͳ ߣଶ ൌ Ͳ
ߣଷ ൌ ܯ ଵܷ
ଶܲ
ଵܲ
െ ܯܷଶ ൐ Ͳ ߣଷ ൌ Ͳ ߣଷ ൌ Ͳ
B
C
D
Here: Indifference curve
steeper than budget line
Here: Indifference curve
flatter than budget line
Here: Indifference curve
tangent to budget line
Quick Example: Cobb-Douglas UMP
ƒš
௤భǡ௤మ
ܷ ൌ ݍଵ
ఈݍଶ
ఉݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ ൌ Ͳǡ ݍଶ ൒ ݍଶ
௠௜௡ ൌ Ͳ
Recall: ܷ ൌ ݍଵఈݍଶ
ఉ ֜ ܯܴܵ ൌ
ఈ
ఉ
௤మ
௤భ
38
Case B (Only) Case C (Only) Case D (Only)
Š‡ ܯܴܵ ൐ ଵܲ
ଶܲ
ƒ– ݍଵ ൌ
ܻ
ଵܲ
ǡ ݍଶ ൌ Ͳ Š‡ ܯܴܵ ൏
ଵܲ
ଶܲ
ƒ– ݍଵ ൌ Ͳǡ ݍଶ ൌ
ܻ
ଶܲ
Since neither Cases B or C arise, the
solution is Case D where ܯܴܵ ൌ ௉భ
௉మ
ܯܴܵ ƒ– —†Ž‡ ൌ
ߙ
ߚ
ݍଶ
ݍଵ
ൌ
ߙ
ߚ
Ͳ
ܻȀ ଵܲ
ج ଵܲ
ଶܲ
ܯܴܵ ƒ– —†Ž‡ ܥ ൌ
ߙ
ߚ
ݍଶ
ݍଵ
ൌ
ߙ
ߚ
ܻȀ ଶܲ
Ͳ
ث ଵܲ
ଶܲ
For Bundle D: Solve
ܯܴܵ ƒ– —†Ž‡ ܦ ൌ
ߙ
ߚ
ݍଶ
ݍଵ
ൌ ଵܲ
ଶܲ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
Case B cannot be the solution Case C cannot be the solution
ݍଵ
כ ൌ
ఈ
ఈାఉ
௒
௉భ
and ݍଶכ ൌ
ఉ
ఈାఉ
௒
௉మ
Quick Example: Linear UMP
ƒš
௤భǡ௤మ
ܷ ൌ ߙݍଵ ൅ ߚݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ ൌ Ͳǡ ݍଶ ൒ ݍଶ
௠௜௡ ൌ Ͳ
Recall: ܷ ൌ ߙݍଵ ൅ ߚݍଶ ֜ ܯܴܵ ൌ
ఈ
ఉ
39
Case B (Only) Case C (Only) Case D (Only)
Š‡ ܯܴܵ ൐ ଵܲ
ଶܲ
ƒ– ݍଵ ൌ
ܻ
ଵܲ
ǡ ݍଶ ൌ Ͳ Š‡ ܯܴܵ ൏
ଵܲ
ଶܲ
ƒ– ݍଵ ൌ Ͳǡ ݍଶ ൌ
ܻ
ଶܲ
'EWI ( MW XLI WSPYXMSR [LIRIZIV
ߙ
ߚ
ൌ ଵܲ
ଶܲ
ܯܴܵ ƒ– —†Ž‡ ൌ
ߙ
ߚ
൐ ଵܲ
ଶܲ
ܯܴܵ ƒ– —†Ž‡ ܥ ൌ
ߙ
ߚ
൏ ଵܲ
ଶܲ
Solve
ܯܴܵ ƒ– —†Ž‡ ܦ ൌ
ߙ
ߚ
ൌ ଵܲ
ଶܲ
ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
Case B is the solution whenever ఈ
ఉ
൐
௉భ
௉మ
Case C is the solution whenever ఈ
ఉ
൏
௉భ
௉మ
ݍଵ
כ ൌ
ܻ
ଵܲ
ǡ ݍଶ
כ ൌ Ͳ ݍଵ
כ ൌ Ͳǡ ݍଶ
כ ൌ
ܻ
ଶܲ
ݍଵ
כ ƒ† ݍଶ
כ any bundle on ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻ
See graphs on next slide
Quick Example: Linear UMP
40
B
C
D
Whenever ܯܴܵ ƒ– —†Ž‡ ܤ ൐ ௉భ
௉మ
Whenever ఈ
ఉ
൐
௉భ
௉మ
ݍଵ
כ ൌ
ܻ
ଵܲ
ǡ ݍଶ
כ ൌ Ͳ
Case B Case C Case D
Whenever
ଵܲ
ଶܲ
൏
ߙ
ߚ
Whenever ܯܴܵ ƒ– —†Ž‡ ܥ ൏ ௉భ
௉మ
Whenever ఈ
ఉ
൏
௉భ
௉మ
ݍଵ
כ ൌ Ͳǡ ݍଶ
כ ൌ
ܻ
ଶܲ
Whenever ܯܴܵ ƒ– —†Ž‡ ܦ ൌ ௉భ
௉మ
Whenever ఈ
ఉ
ൌ
௉భ
௉మ
ଵܲݍଵ
כ ൅ ଶܲݍଶ
כ ൌ ܻ
Whenever
ଵܲ
ଶܲ
൐
ߙ
ߚ
Here
ଵܲ
ଶܲ
ൌ
ߙ
ߚ
ƒš
௤భǡ௤మ
ܷ ൌ ߙݍଵ ൅ ߚݍଶ ݏǤ ݐǤ ଵܲݍଵ ൅ ଶܲݍଶ ൌ ܻǡ ݍଵ ൒ ݍଵ
௠௜௡ ൌ Ͳǡ ݍଶ ൒ ݍଶ
௠௜௡ ൌ Ͳ