lOMoARcPSD|8220114
EC202 Exam 2016 Answers
Microeconomic Principles II (The London School of Economics and Political Science)
StuDocu is not sponsored or endorsed by any college or university
Downloaded by Juan Esnal ()
, lOMoARcPSD|8220114
◗✉❡st✐♦♥s
✶✳ ❆ ❝♦♠♣❡t✐t✐✈❡ ✜r♠ ♣r♦❞✉❝❡s ❛ s✐♥❣❧❡ ♦✉t♣✉t ❢r♦♠ t❤r❡❡ ✐♥♣✉ts❀ ♣r♦❞✉❝t✐♦♥
1
✐s ❣✐✈❡♥ ❜② q ≤ [z1 z2 z3 ] 3 ✇❤❡r❡ q ✐s t❤❡ q✉❛♥t✐t② ♦❢ ♦✉t♣✉t ❛♥❞ z1 , z2 , z3
❛r❡ q✉❛♥t✐t✐❡s ♦❢ t❤❡ ✐♥♣✉ts✳
✭❛✮ ■❢ t❤❡ ✐♥♣✉t ♣r✐❝❡s ❛r❡ ❣✐✈❡♥ ❜② w1 , w2 , w3 s❤♦✇ t❤❛t t❤❡ ❝♦st ❢✉♥❝t✐♦♥
1
✐s 3q [w1 w2 w3 ] 3 ✳❬✸ ♠❛r❦s❪
✭❜✮ ❙❦❡t❝❤ t❤❡ ✭❧♦♥❣✲r✉♥✮ ▼❈ ❛♥❞ ❆❈ ❝✉r✈❡s✳ ❬✷ ♠❛r❦s❪
✭❝✮ ■♥♣✉t ✸ ✐s ✜①❡❞ ✐♥ t❤❡ s❤♦rt r✉♥✳ ❯s✐♥❣ t❤❡ ❞✐❛❣r❛♠ ❢r♦♠ ♣❛rt ✭❜✮
❡①♣❧❛✐♥ ✇❤❛t t❤❡ s❤♦rt✲r✉♥ ▼❈ ❛♥❞ ❆❈ ♠✉st ❧♦♦❦ ❧✐❦❡✳ ❬✸ ♠❛r❦s❪
✷✳
✭❛✮ ■♥ ❡❝♦♥♦♠② ❆ ❣♦♦❞s ✶ ❛♥❞ ✷ ❝❛♥ ❜❡ ♣r♦❞✉❝❡❞ ✉s✐♥❣ ❛ s✐♥❣❧❡ ❣♦♦❞ ✸
❛s ❛♥ ✐♥♣✉t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str❛✐♥t
a13 q1 + a23 q2 ≤ R3
✇❤❡r❡ a13 , a23 ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✱ q1 , q2 ❛r❡ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡
t✇♦ ❣♦♦❞s ❛♥❞ R3 ✐s t❤❡ ❣✐✈❡♥ r❡s♦✉r❝❡ st♦❝❦ ♦❢ ❣♦♦❞ ✸✳ ❉r❛✇ t❤❡
♣r♦❞✉❝t✐♦♥ ♣♦ss✐❜✐❧✐t② s❡t ❢♦r ❣♦♦❞s ✶ ❛♥❞ ✷ ✐♥ ❡❝♦♥♦♠② ❆✳ ❬✷ ♠❛r❦s❪
✭❜✮ ■♥ ❡❝♦♥♦♠② ❇ ♣r♦❞✉❝t✐♦♥ ♦❢ ❣♦♦❞s ✶ ❛♥❞ ✷ r❡q✉✐r❡s t❤❡ ✉s❡ ♦❢ ❣♦♦❞s
✸✱✹ ❛♥❞ ✺ ❛s ✐♥♣✉ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝♦♥str❛✐♥ts ✭✇❤✐❝❤
♠✉st ❛❧❧ ❤♦❧❞✮✿
a1j q1 + a2j q2 ≤ Rj , j = 3, 4, 5,
✇❤❡r❡ a1j , a2j ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ❛♥❞ Rj ✐s t❤❡ ❣✐✈❡♥ r❡s♦✉r❝❡
st♦❝❦ ♦❢ ❣♦♦❞ j ✳ ❉r❛✇ t❤❡ ♣r♦❞✉❝t✐♦♥ ♣♦ss✐❜✐❧✐t② s❡t ❢♦r ❣♦♦❞s ✶ ❛♥❞
✷ ✐♥ ❡❝♦♥♦♠② ❇✿ ❡①♣❧❛✐♥ ✇❤② ✐t ♠✉st ❜❡ ❝♦♥✈❡①✳ ❬✸ ♠❛r❦s❪
✭❝✮ ❊①♣❧❛✐♥ ✇❤❛t ✇♦✉❧❞ ❤❛♣♣❡♥ t♦ t❤❡ ♣r♦❞✉❝t✐♦♥ s❡ts ♦❢ ❡❝♦♥♦♠② ❆ ❛♥❞
❡❝♦♥♦♠② ❇ ✐❢ ❛❞❞✐t✐♦♥❛❧ st♦❝❦s ♦❢ ❣♦♦❞ ✸ ❜❡❝❛♠❡ ❛✈❛✐❧❛❜❧❡✳❬✸ ♠❛r❦s❪
√
✸✳ ❆ ♣❡rs♦♥ ❤❛s t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ U (x1 , x2 ) = 2 x1 + x2 ✳ ❲r✐t❡ t❤❡ ♣r✐❝❡
♦❢ ❣♦♦❞ ✶ ❛s p✱ ❛ss✉♠❡ t❤❛t t❤❡ ♣r✐❝❡ ♦❢ ❣♦♦❞ ✷ ✐s ✜①❡❞ ❛t ✶ ❛♥❞ ❛ss✉♠❡
t❤❡ ✐♥❝♦♠❡ y ✐s s✉✣❝✐❡♥t❧② ❤✐❣❤ t♦ ❡♥s✉r❡ t❤❛t ♣♦s✐t✐✈❡ ❛♠♦✉♥ts ♦❢ ❜♦t❤
❣♦♦❞s ❛r❡ ❜♦✉❣❤t✳
✭❛✮ ■❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ p ✐s ❛❧s♦ ✶ s❤♦✇ t❤❛t t❤❡ ❝♦♠♣❡♥s❛t✐♥❣ ✈❛r✐❛t✐♦♥
♦❢ ❛ ♣r✐❝❡ r✐s❡ ∆p ✐s −∆p [1 + ∆p]−1 ✳ ❬✹ ♠❛r❦s❪
✭❜✮ ❆❣❛✐♥ ❛ss✉♠❡ t❤❛t p ✐♥✐t✐❛❧❧② ❤❛s t❤❡ ✈❛❧✉❡ ✶✳ ❚❤❡ ❣♦✈❡r♥♠❡♥t ♥♦✇
✐♠♣♦s❡s ❛ t❛① t ♦♥ ❣♦♦❞ ✶ ✐♥ ♦r❞❡r t♦ r❛✐s❡ r❡✈❡♥✉❡✳ ❙❤♦✇ t❤❛t t❤❡
❧♦ss t♦ ❛ ❝♦♥s✉♠❡r ❡①❝❡❡❞s t❤❡ r❡✈❡♥✉❡ r❛✐s❡❞ ❜② t❤❡ ❣♦✈❡r♥♠❡♥t ❜②
❛♥ ❛♠♦✉♥t t2 [1 + t]−2 ✳ ❬✹ ♠❛r❦s❪
✹✳ ❴❴
✶
Downloaded by Juan Esnal ()
, lOMoARcPSD|8220114
✺✳ ❴❴
✻✳ ❴❴
✼✳ ❆ ♣❡rs♦♥ ❧✐✈❡s ❢♦r T + 1 ②❡❛rs ❛♥❞ ❝♦♥s✉♠❡s ❛ q✉❛♥t✐t② xit ♦❢ ❣♦♦❞ i ✐♥
♣❡r✐♦❞ t✱ ✇❤❡r❡ i = 1, 2, ..., n ❛♥❞ t = 0, 1, ..., T.
✭❛✮ ❍✐s ✉t✐❧✐t② ✐s ❣✐✈❡♥ ❜②
T X
X n
αi βt log (xit − γit )
t=0 i=1
✇❤❡r❡ t❤❡ αi , βt , γit ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♣❛r❛♠❡t❡rs s✉❝❤ t❤❛t ni=1 αi =
P
t=0 βt = 1 ❛♥❞ βt /βt−1 = δ, t = 1, ..., T ✱ ✇❤❡r❡ 0 < δ < 1✳
PT
✐✳ ❙❦❡t❝❤ t❤❡ ✐♥❞✐✛❡r❡♥❝❡ ❝✉r✈❡s ❢♦r ❣♦♦❞s i ❛♥❞ j ❛t ❛ ❣✐✈❡♥ t✐♠❡
t❀ ✐♥t❡r♣r❡t t❤❡ s❧♦♣❡✳ ❬✶ ♠❛r❦❪
✐✐✳ ■♥t❡r♣r❡t t❤❡ ♣❛r❛♠❡t❡rs γit ✳ ❬✶ ♠❛r❦❪
✐✐✐✳ ❙❦❡t❝❤ t❤❡ ✐♥❞✐✛❡r❡♥❝❡ ❝✉r✈❡s ❢♦r ❣♦♦❞ i ❝♦♥s✉♠❡❞ ✐♥ ♣❡r✐♦❞ t
❛♥❞ t❤❡ s❛♠❡ ❣♦♦❞ i ❝♦♥s✉♠❡❞ ✐♥ ♣❡r✐♦❞ t − 1❀ ✐♥t❡r♣r❡t t❤❡
s❧♦♣❡✳ ❬✶ ♠❛r❦❪
✭❜✮ ❚❤❡ ♣❡rs♦♥ r❡❝❡✐✈❡s ❛♥ ❡①♦❣❡♥♦✉s ✐♥❝♦♠❡ yt ✐♥ ♣❡r✐♦❞ t❀ t❤❡ ♣r✐❝❡
pit ♦❢ ❣♦♦❞ i ✐♥ ♣❡r✐♦❞ t ✐s ❦♥♦✇♥ ✐♥ ❛❞✈❛♥❝❡❀ ✐t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t
t❤❡r❡ ✐s ❛ ♣❡r✲♣❡r✐♦❞ ✐♥t❡r❡st r❛t❡ r t❤❛t ✐s ❝♦♥st❛♥t t❤r♦✉❣❤ t✐♠❡✳ ■❢
t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r r❡str✐❝t✐♦♥s ♦♥ ❜♦rr♦✇✐♥❣ ❛♥❞ ❧❡♥❞✐♥❣ ✇❤❛t ✐s t❤❡
♣❡rs♦♥✬s ❧✐❢❡t✐♠❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t❄ ❬✹ ♠❛r❦s❪
✭❝✮ ❋✐♥❞ t❤❡ q✉❛♥t✐t✐❡s x∗it t❤❛t ♠❛①✐♠✐s❡ t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ✐♥ ♣❛rt ✭❛✮
s✉❜❥❡❝t t♦ t❤❡ ❧✐❢❡t✐♠❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ✐♥ ♣❛rt ✭❜✮✳ ❬✺ ♠❛r❦s❪
✭❞✮ ❋✐♥❞ t❤❡ ❡✛❡❝t ♦❢ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♣r✐❝❡ pit ♦♥ t❤❡ ❞❡♠❛♥❞ ❢♦r ❣♦♦❞
j 6= i ❛t t✐♠❡ t′ 6= t✳ ■♥t❡r♣r❡t ②♦✉r ❛♥s✇❡r✳ ❬✹ ♠❛r❦s❪
✭❡✮ ❙✉♣♣♦s❡ t❤❛t ❢♦r ❝♦♠♠♦❞✐t② i t❤❡ ♣❛r❛♠❡t❡r γit ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ t✳
■❢ pit ✐s ❦♥♦✇♥ t♦ ❜❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ t ✇✐❧❧ t❤❡ ♦♣t✐♠❛❧ ❝♦♥s✉♠♣t✐♦♥
♦❢ ❣♦♦❞ i ❜❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ t❄ ❊①♣❧❛✐♥ ②♦✉r ❛♥s✇❡r✳ ❬✹ ♠❛r❦s❪
✽✳ ■♥ ❛ t✇♦✲❝♦♠♠♦❞✐t② ❡①❝❤❛♥❣❡ ❡❝♦♥♦♠② t❤❡r❡ ❛r❡ t✇♦ t②♣❡s ♦❢ ♣❡♦♣❧❡✳
P❡♦♣❧❡ ♦❢ t②♣❡ a ♦✇♥ ❛❧❧ ♦❢ ❝♦♠♠♦❞✐t② ✶ ✭✶✵✵ ✉♥✐ts ♣❡r ♣❡rs♦♥✮ ❛♥❞ ♣❡♦♣❧❡
♦❢ t②♣❡ b ♦✇♥ ❛❧❧ ♦❢ ❝♦♠♠♦❞✐t② ✷ ✭✷✵✵ ✉♥✐ts ♣❡r ♣❡rs♦♥✮✳ ▲❡t xhi ❞❡♥♦t❡
t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❣♦♦❞ i ❜② ❛ ♣❡rs♦♥ ✐♥ ❣r♦✉♣ h✱ i = 1, 2✱ h = a, b✳
✭❛✮ ❆ss✉♠❡ t❤❛t ♣r❡❢❡r❡♥❝❡s ♦❢ t②♣❡ a ❛r❡ ❣✐✈❡♥ ❜② log xa1 + log xa2 ✳ ❙❤♦✇
t❤❛t ✐♥ ❛ ❝♦♠♣❡t✐t✐✈❡ ♠❛r❦❡t ❡❛❝❤ ♣❡rs♦♥✬s ❞❡♠❛♥❞ ❢♦r t❤❡ t✇♦ ❝♦♠✲
♠♦❞✐t✐❡s ✇✐❧❧ ❜❡ xa = (50, 50p1 /p2 )✳ ❋✐♥❞ t❤❡ ♦✛❡r ❝✉r✈❡ ❢♦r ❛ ♣❡rs♦♥
♦❢ t②♣❡ a✳ ❬✺ ♠❛r❦s❪
✭❜✮ ❆ss✉♠❡ t❤❛t ♣r❡❢❡r❡♥❝❡s ♦❢ ❛ t②♣❡✲b ♣❡rs♦♥ ❛r❡ ❣✐✈❡♥ ❜② 3 log xb1 +
log xb2 ✳ ❋✐♥❞ ❛ t②♣❡✲b ♣❡rs♦♥✬s ❞❡♠❛♥❞ ❢♦r t❤❡ t✇♦ ❣♦♦❞s ❛♥❞ t❤❡
♦✛❡r ❝✉r✈❡✳ ■❢ t❤❡r❡ ❛r❡ ❡q✉❛❧ ♥✉♠❜❡rs ♦❢ ♣❡♦♣❧❡ ♦❢ ❡❛❝❤ t②♣❡ ✉s❡ t❤❡
r❡s✉❧t ❢r♦♠ ♣❛rt ✭❛✮ t♦ ✜♥❞ t❤❡ ❝♦♠♣❡t✐t✐✈❡✲❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❛♥❞
❛❧❧♦❝❛t✐♦♥✳ ❬✺ ♠❛r❦s❪
✷
Downloaded by Juan Esnal ()
EC202 Exam 2016 Answers
Microeconomic Principles II (The London School of Economics and Political Science)
StuDocu is not sponsored or endorsed by any college or university
Downloaded by Juan Esnal ()
, lOMoARcPSD|8220114
◗✉❡st✐♦♥s
✶✳ ❆ ❝♦♠♣❡t✐t✐✈❡ ✜r♠ ♣r♦❞✉❝❡s ❛ s✐♥❣❧❡ ♦✉t♣✉t ❢r♦♠ t❤r❡❡ ✐♥♣✉ts❀ ♣r♦❞✉❝t✐♦♥
1
✐s ❣✐✈❡♥ ❜② q ≤ [z1 z2 z3 ] 3 ✇❤❡r❡ q ✐s t❤❡ q✉❛♥t✐t② ♦❢ ♦✉t♣✉t ❛♥❞ z1 , z2 , z3
❛r❡ q✉❛♥t✐t✐❡s ♦❢ t❤❡ ✐♥♣✉ts✳
✭❛✮ ■❢ t❤❡ ✐♥♣✉t ♣r✐❝❡s ❛r❡ ❣✐✈❡♥ ❜② w1 , w2 , w3 s❤♦✇ t❤❛t t❤❡ ❝♦st ❢✉♥❝t✐♦♥
1
✐s 3q [w1 w2 w3 ] 3 ✳❬✸ ♠❛r❦s❪
✭❜✮ ❙❦❡t❝❤ t❤❡ ✭❧♦♥❣✲r✉♥✮ ▼❈ ❛♥❞ ❆❈ ❝✉r✈❡s✳ ❬✷ ♠❛r❦s❪
✭❝✮ ■♥♣✉t ✸ ✐s ✜①❡❞ ✐♥ t❤❡ s❤♦rt r✉♥✳ ❯s✐♥❣ t❤❡ ❞✐❛❣r❛♠ ❢r♦♠ ♣❛rt ✭❜✮
❡①♣❧❛✐♥ ✇❤❛t t❤❡ s❤♦rt✲r✉♥ ▼❈ ❛♥❞ ❆❈ ♠✉st ❧♦♦❦ ❧✐❦❡✳ ❬✸ ♠❛r❦s❪
✷✳
✭❛✮ ■♥ ❡❝♦♥♦♠② ❆ ❣♦♦❞s ✶ ❛♥❞ ✷ ❝❛♥ ❜❡ ♣r♦❞✉❝❡❞ ✉s✐♥❣ ❛ s✐♥❣❧❡ ❣♦♦❞ ✸
❛s ❛♥ ✐♥♣✉t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str❛✐♥t
a13 q1 + a23 q2 ≤ R3
✇❤❡r❡ a13 , a23 ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✱ q1 , q2 ❛r❡ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡
t✇♦ ❣♦♦❞s ❛♥❞ R3 ✐s t❤❡ ❣✐✈❡♥ r❡s♦✉r❝❡ st♦❝❦ ♦❢ ❣♦♦❞ ✸✳ ❉r❛✇ t❤❡
♣r♦❞✉❝t✐♦♥ ♣♦ss✐❜✐❧✐t② s❡t ❢♦r ❣♦♦❞s ✶ ❛♥❞ ✷ ✐♥ ❡❝♦♥♦♠② ❆✳ ❬✷ ♠❛r❦s❪
✭❜✮ ■♥ ❡❝♦♥♦♠② ❇ ♣r♦❞✉❝t✐♦♥ ♦❢ ❣♦♦❞s ✶ ❛♥❞ ✷ r❡q✉✐r❡s t❤❡ ✉s❡ ♦❢ ❣♦♦❞s
✸✱✹ ❛♥❞ ✺ ❛s ✐♥♣✉ts ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝♦♥str❛✐♥ts ✭✇❤✐❝❤
♠✉st ❛❧❧ ❤♦❧❞✮✿
a1j q1 + a2j q2 ≤ Rj , j = 3, 4, 5,
✇❤❡r❡ a1j , a2j ❛r❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts ❛♥❞ Rj ✐s t❤❡ ❣✐✈❡♥ r❡s♦✉r❝❡
st♦❝❦ ♦❢ ❣♦♦❞ j ✳ ❉r❛✇ t❤❡ ♣r♦❞✉❝t✐♦♥ ♣♦ss✐❜✐❧✐t② s❡t ❢♦r ❣♦♦❞s ✶ ❛♥❞
✷ ✐♥ ❡❝♦♥♦♠② ❇✿ ❡①♣❧❛✐♥ ✇❤② ✐t ♠✉st ❜❡ ❝♦♥✈❡①✳ ❬✸ ♠❛r❦s❪
✭❝✮ ❊①♣❧❛✐♥ ✇❤❛t ✇♦✉❧❞ ❤❛♣♣❡♥ t♦ t❤❡ ♣r♦❞✉❝t✐♦♥ s❡ts ♦❢ ❡❝♦♥♦♠② ❆ ❛♥❞
❡❝♦♥♦♠② ❇ ✐❢ ❛❞❞✐t✐♦♥❛❧ st♦❝❦s ♦❢ ❣♦♦❞ ✸ ❜❡❝❛♠❡ ❛✈❛✐❧❛❜❧❡✳❬✸ ♠❛r❦s❪
√
✸✳ ❆ ♣❡rs♦♥ ❤❛s t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ U (x1 , x2 ) = 2 x1 + x2 ✳ ❲r✐t❡ t❤❡ ♣r✐❝❡
♦❢ ❣♦♦❞ ✶ ❛s p✱ ❛ss✉♠❡ t❤❛t t❤❡ ♣r✐❝❡ ♦❢ ❣♦♦❞ ✷ ✐s ✜①❡❞ ❛t ✶ ❛♥❞ ❛ss✉♠❡
t❤❡ ✐♥❝♦♠❡ y ✐s s✉✣❝✐❡♥t❧② ❤✐❣❤ t♦ ❡♥s✉r❡ t❤❛t ♣♦s✐t✐✈❡ ❛♠♦✉♥ts ♦❢ ❜♦t❤
❣♦♦❞s ❛r❡ ❜♦✉❣❤t✳
✭❛✮ ■❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ p ✐s ❛❧s♦ ✶ s❤♦✇ t❤❛t t❤❡ ❝♦♠♣❡♥s❛t✐♥❣ ✈❛r✐❛t✐♦♥
♦❢ ❛ ♣r✐❝❡ r✐s❡ ∆p ✐s −∆p [1 + ∆p]−1 ✳ ❬✹ ♠❛r❦s❪
✭❜✮ ❆❣❛✐♥ ❛ss✉♠❡ t❤❛t p ✐♥✐t✐❛❧❧② ❤❛s t❤❡ ✈❛❧✉❡ ✶✳ ❚❤❡ ❣♦✈❡r♥♠❡♥t ♥♦✇
✐♠♣♦s❡s ❛ t❛① t ♦♥ ❣♦♦❞ ✶ ✐♥ ♦r❞❡r t♦ r❛✐s❡ r❡✈❡♥✉❡✳ ❙❤♦✇ t❤❛t t❤❡
❧♦ss t♦ ❛ ❝♦♥s✉♠❡r ❡①❝❡❡❞s t❤❡ r❡✈❡♥✉❡ r❛✐s❡❞ ❜② t❤❡ ❣♦✈❡r♥♠❡♥t ❜②
❛♥ ❛♠♦✉♥t t2 [1 + t]−2 ✳ ❬✹ ♠❛r❦s❪
✹✳ ❴❴
✶
Downloaded by Juan Esnal ()
, lOMoARcPSD|8220114
✺✳ ❴❴
✻✳ ❴❴
✼✳ ❆ ♣❡rs♦♥ ❧✐✈❡s ❢♦r T + 1 ②❡❛rs ❛♥❞ ❝♦♥s✉♠❡s ❛ q✉❛♥t✐t② xit ♦❢ ❣♦♦❞ i ✐♥
♣❡r✐♦❞ t✱ ✇❤❡r❡ i = 1, 2, ..., n ❛♥❞ t = 0, 1, ..., T.
✭❛✮ ❍✐s ✉t✐❧✐t② ✐s ❣✐✈❡♥ ❜②
T X
X n
αi βt log (xit − γit )
t=0 i=1
✇❤❡r❡ t❤❡ αi , βt , γit ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ♣❛r❛♠❡t❡rs s✉❝❤ t❤❛t ni=1 αi =
P
t=0 βt = 1 ❛♥❞ βt /βt−1 = δ, t = 1, ..., T ✱ ✇❤❡r❡ 0 < δ < 1✳
PT
✐✳ ❙❦❡t❝❤ t❤❡ ✐♥❞✐✛❡r❡♥❝❡ ❝✉r✈❡s ❢♦r ❣♦♦❞s i ❛♥❞ j ❛t ❛ ❣✐✈❡♥ t✐♠❡
t❀ ✐♥t❡r♣r❡t t❤❡ s❧♦♣❡✳ ❬✶ ♠❛r❦❪
✐✐✳ ■♥t❡r♣r❡t t❤❡ ♣❛r❛♠❡t❡rs γit ✳ ❬✶ ♠❛r❦❪
✐✐✐✳ ❙❦❡t❝❤ t❤❡ ✐♥❞✐✛❡r❡♥❝❡ ❝✉r✈❡s ❢♦r ❣♦♦❞ i ❝♦♥s✉♠❡❞ ✐♥ ♣❡r✐♦❞ t
❛♥❞ t❤❡ s❛♠❡ ❣♦♦❞ i ❝♦♥s✉♠❡❞ ✐♥ ♣❡r✐♦❞ t − 1❀ ✐♥t❡r♣r❡t t❤❡
s❧♦♣❡✳ ❬✶ ♠❛r❦❪
✭❜✮ ❚❤❡ ♣❡rs♦♥ r❡❝❡✐✈❡s ❛♥ ❡①♦❣❡♥♦✉s ✐♥❝♦♠❡ yt ✐♥ ♣❡r✐♦❞ t❀ t❤❡ ♣r✐❝❡
pit ♦❢ ❣♦♦❞ i ✐♥ ♣❡r✐♦❞ t ✐s ❦♥♦✇♥ ✐♥ ❛❞✈❛♥❝❡❀ ✐t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t
t❤❡r❡ ✐s ❛ ♣❡r✲♣❡r✐♦❞ ✐♥t❡r❡st r❛t❡ r t❤❛t ✐s ❝♦♥st❛♥t t❤r♦✉❣❤ t✐♠❡✳ ■❢
t❤❡r❡ ❛r❡ ♥♦ ♦t❤❡r r❡str✐❝t✐♦♥s ♦♥ ❜♦rr♦✇✐♥❣ ❛♥❞ ❧❡♥❞✐♥❣ ✇❤❛t ✐s t❤❡
♣❡rs♦♥✬s ❧✐❢❡t✐♠❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t❄ ❬✹ ♠❛r❦s❪
✭❝✮ ❋✐♥❞ t❤❡ q✉❛♥t✐t✐❡s x∗it t❤❛t ♠❛①✐♠✐s❡ t❤❡ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ✐♥ ♣❛rt ✭❛✮
s✉❜❥❡❝t t♦ t❤❡ ❧✐❢❡t✐♠❡ ❜✉❞❣❡t ❝♦♥str❛✐♥t ✐♥ ♣❛rt ✭❜✮✳ ❬✺ ♠❛r❦s❪
✭❞✮ ❋✐♥❞ t❤❡ ❡✛❡❝t ♦❢ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♣r✐❝❡ pit ♦♥ t❤❡ ❞❡♠❛♥❞ ❢♦r ❣♦♦❞
j 6= i ❛t t✐♠❡ t′ 6= t✳ ■♥t❡r♣r❡t ②♦✉r ❛♥s✇❡r✳ ❬✹ ♠❛r❦s❪
✭❡✮ ❙✉♣♣♦s❡ t❤❛t ❢♦r ❝♦♠♠♦❞✐t② i t❤❡ ♣❛r❛♠❡t❡r γit ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ t✳
■❢ pit ✐s ❦♥♦✇♥ t♦ ❜❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ t ✇✐❧❧ t❤❡ ♦♣t✐♠❛❧ ❝♦♥s✉♠♣t✐♦♥
♦❢ ❣♦♦❞ i ❜❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ t❄ ❊①♣❧❛✐♥ ②♦✉r ❛♥s✇❡r✳ ❬✹ ♠❛r❦s❪
✽✳ ■♥ ❛ t✇♦✲❝♦♠♠♦❞✐t② ❡①❝❤❛♥❣❡ ❡❝♦♥♦♠② t❤❡r❡ ❛r❡ t✇♦ t②♣❡s ♦❢ ♣❡♦♣❧❡✳
P❡♦♣❧❡ ♦❢ t②♣❡ a ♦✇♥ ❛❧❧ ♦❢ ❝♦♠♠♦❞✐t② ✶ ✭✶✵✵ ✉♥✐ts ♣❡r ♣❡rs♦♥✮ ❛♥❞ ♣❡♦♣❧❡
♦❢ t②♣❡ b ♦✇♥ ❛❧❧ ♦❢ ❝♦♠♠♦❞✐t② ✷ ✭✷✵✵ ✉♥✐ts ♣❡r ♣❡rs♦♥✮✳ ▲❡t xhi ❞❡♥♦t❡
t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❣♦♦❞ i ❜② ❛ ♣❡rs♦♥ ✐♥ ❣r♦✉♣ h✱ i = 1, 2✱ h = a, b✳
✭❛✮ ❆ss✉♠❡ t❤❛t ♣r❡❢❡r❡♥❝❡s ♦❢ t②♣❡ a ❛r❡ ❣✐✈❡♥ ❜② log xa1 + log xa2 ✳ ❙❤♦✇
t❤❛t ✐♥ ❛ ❝♦♠♣❡t✐t✐✈❡ ♠❛r❦❡t ❡❛❝❤ ♣❡rs♦♥✬s ❞❡♠❛♥❞ ❢♦r t❤❡ t✇♦ ❝♦♠✲
♠♦❞✐t✐❡s ✇✐❧❧ ❜❡ xa = (50, 50p1 /p2 )✳ ❋✐♥❞ t❤❡ ♦✛❡r ❝✉r✈❡ ❢♦r ❛ ♣❡rs♦♥
♦❢ t②♣❡ a✳ ❬✺ ♠❛r❦s❪
✭❜✮ ❆ss✉♠❡ t❤❛t ♣r❡❢❡r❡♥❝❡s ♦❢ ❛ t②♣❡✲b ♣❡rs♦♥ ❛r❡ ❣✐✈❡♥ ❜② 3 log xb1 +
log xb2 ✳ ❋✐♥❞ ❛ t②♣❡✲b ♣❡rs♦♥✬s ❞❡♠❛♥❞ ❢♦r t❤❡ t✇♦ ❣♦♦❞s ❛♥❞ t❤❡
♦✛❡r ❝✉r✈❡✳ ■❢ t❤❡r❡ ❛r❡ ❡q✉❛❧ ♥✉♠❜❡rs ♦❢ ♣❡♦♣❧❡ ♦❢ ❡❛❝❤ t②♣❡ ✉s❡ t❤❡
r❡s✉❧t ❢r♦♠ ♣❛rt ✭❛✮ t♦ ✜♥❞ t❤❡ ❝♦♠♣❡t✐t✐✈❡✲❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❛♥❞
❛❧❧♦❝❛t✐♦♥✳ ❬✺ ♠❛r❦s❪
✷
Downloaded by Juan Esnal ()
