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tÑ�Ü�¿F]�S�hfZpUWV\b�VnhfZ

Qt ≤ W

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qt =∑

i qi,t

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qi,t

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tembt¿F]�Â�Tqb�]N^�TG[GTfV\jXTsÓpj\]�Vn_

W − Qt

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Qt+1 = Qt + qt

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pt(qt, Qt)VX_�bt¿F]2ÁF^�VXS{]2hfk�TKÀFZFV\b�hsk�jnTfZpU×V\Z�Á;]�^tV\hWU

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qt + Qt

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Vi,t(Qt) = maxq

{[at − b(qt + Qt)] q + βVi,t+1(Qt+1)}

_�ÀFÓWåd]NS{b�bthKb�¿F]2S{h�Zp_dbt^tTfV\Z}bqi,t ≥ 0

TsZpUQt+1 ≤ W

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Qt+1 = qt + Qt

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β = 1/(1+r)VX_�T�UWVX_tS{hfÀpZ�b�kÄT�S b�h�^NÑ Ø hf^�_�V\Ã2ÁFjnVXS{V\bdcfesV\b�Vn_�Tf_t_�ÀFÃ2]NU�bt¿pTqb�bt¿F]�Ã2Tf^�Å�V\Z"Tsj

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[aT − b(qT + QT )] q = 0

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gÇ¿F]�Z�b�¿F]©Â�Tqbt]�^²S�hfZp_ab�^`TsVnZ}b¦Vn_²ÓFVnZpUWVnZFÅ5UWÀF^tVnZFÅ�b�¿F]�jnT�_db¦Á;]�^tVnhmU®e�VnZpUWVn[mVnUWÀ"TsjpjXTsZ"UUW]N[f]�jnhfÁ;]�^`_²Tf^�]�ÀFZpTfÓFj\]�b�h2TfSNÞ�ÀpV\^t]�_�ÀWê�S{Vn]�Z}b�Â�Tqbt]�^1klhf^�Tsjnjpbt¿F]$jnTfZpU8bt¿pTqb�b�¿F]Nc8Â�TfZ}bb�h2UW]N[f]�jnhfÁ�Ñ Ø hf^©_�V\Ã2ÁFjnVXS{V\bdcfeWÂ�]$T�_�_�ÀFÃ2]�bt¿pTqb�Â�¿F]NZ�bt¿FVX_�Vn_1bt¿F],S�T�_a]�e�bt¿F]�^�]NÃ8TsVnZFV\ZpÅÂ�Tqb�]N^1VX_�TsjnjnhmSNTqbt]NU�]~Þ}ÀpTsjnj\c8Ó;]{bdÂ�]�]NZ�bt¿F]$Ûp^�Ã8_NѦÜ�¿F]$[GTfj\Àp]�klÀpZpS btV\h�Z�klh�^�h�ZF]$Ûp^tÃ�Vn_b�¿p]�Z

Vi,T (QT ) = (aT − bW )(W − QT )/n � É

Ü�¿F]N^�],VX_�ZFh�hfÁFb�VnÃ�VnÙNTsb�VnhfZ&Áp^�h�ÓFj\]NÃ�Tsb�bt¿FVn_�ÛpZpTfj¯_abtTfÅf]�e"_�V\ZpS�]�Tfj\j°^�]NÃ2TfV\ZpV\ZFÅ�Â�Tsb�]�^VX_�Tfj\jnhWS�Tqbt]NU�]~Þ�À"Tsjnj\c�Ó;]{bdÂ�]�]�Z×Tsjnj�bt¿F]5Û"^�Ã8_Ne;TfZpU]NT�S`¿Ûp^tÃ�Ú _�Ã8Ts^tÅfVnZpTsj°Ó"]NZF]{ÛFb$klhf^TsZ�TfUpUWV�btV\h�ZpTsj�ÀFZpV�b,hfk1Â�Tqb�]N^�Vn_�Á;h�_�V�btV\[�]fÑ�g×]�SNTsZáb�¿F]N^�]�klhf^t]�Ã2hq[f]�b�hKbt¿F]8ZF]{îmb�b�hjXTf_ab�Á;]�^tVnhmU®Ñ

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Vn_

Vi,T−1(QT−1) = maxq

{(aT−1 − bQT )q + βVi,T (QT )} � Î�

Â�¿F]N^�]β

Vn_�T�UWVX_�S�hfÀFZ}b$kÄTfS{b�hf^~Ñ8rmV\Z"S{]QT = QT−1 + qT−1

e®b�¿p]2UFÀF^tTfÓFVnj\V\bdc&hsk1b�¿F]jXTsZpU2^t]N_�hfÀF^`S{]©SNTsZ8Ó;]�]�îWÁF^�]~_�_�]NU�bt¿F^�h�ÀFÅf¿2bt¿F]�UF]�Ã8TsZpU�V\Z8bt¿F]�S�ÀF^t^�]NZ�b¦Á;]�^tVnhmU�Ó"]NV\ZFÅT�klÀFZpS{b�VnhfZ�hsk

QT

ѦÜ�¿F]�Û"^t_ab�hf^`UW]�^©S�hfZpUWV\b�VnhfZ�klhf^�bt¿F],_�]�jn]NS btV\h�Z�hfkqVn_

aT−1 − bQT − bq + β∂Vi,T /∂QT = 0 � ó�

Â�¿F]N^�],À"_a],VX_©Ã8T�UW],hsk�bt¿F],^t]N_�ÀFj\b�b�¿"Tqb∂QT /∂q = 1

Ñ©Y�_�VnZFÅ8b�¿p],kÄTfS{b©b�¿"TqbqT−1 =

qi,T−1emb�¿FVX_©SNTsZ�Ó"]�^t]NTf^�^`TsZFÅ�]NU2bth8Vn_�hfjXTqbt]�bt¿F]�^t]NTfS{b�VnhfZ�klÀFZpS{b�VnhfZ

qi,T−1 =1

2b

[

aT−1 − b(q−i,T−1 + QT−1) −β

n(aT − bW )

]

� û

Â�¿F]N^�]q−i,T−1 =

i6=j qj,T−1eWTsZpU�Àp_�]�VX_1Ã8T�UW]�hfk

∂Vi,T /∂QT = −(aT − bW )/nÑ

gÇV\b�¿2VnUF]�Z}b�VXS�TfjWjnTfZpU2UW]�[�]�jnhfÁFÃ2]�Z}b�Ûp^tÃ8_�eqbt¿F]�b�hfbtTfjWjnTfZpU2UW]�[�]�jnhfÁ;]NU�V\Z2Á;]�^tV\hWUT −1

eTqb�bt¿F],o�T�_a¿K]NÞ}ÀFVnj\VnÓF^tV\ÀpÃKepVn_

qT−1 =n

n + 1

[aT−1

b− QT−1

]

−β

n + 1

[aT

b− W

] � ø�

ã�Z�S{h�Z}b�^`Tf_ab�Â�V\b�¿�b�¿F]$S{hfÃ2Ã2hfZ2Á;hmhfj;hskªÂ�Tsb�]N^¦j\VXS{]NZp_�]N_Ne�TsZ�Tfj�bt]�^tZpTqbtV\[�]©TsÁFÁp^�h}TfS`¿VX_�b�h×Tsjnj\hWS�Tsb�]8Â�Tqbt]�^,jnVnS�]�Zp_�]N_,b�h&]~TfS`¿ÔUW]N[f]�jnhfÁ;]�^5ÁF^�Vnhf^,b�hbt¿F]�jXTsZpUÔUF]�[f]Nj\h�ÁFÃ2]�Z}bNÑö�TfZpUiUW]�[�]�jnhfÁ;]�^`_$Tf^�]�bt¿F]�Zikl^t]�]�b�hUW]N[f]�jnhfÁ×b�¿p]�Vn^�jnTfZpU�Tqb�b�¿F]8Á"TfS{]2]~TfS`¿�S`¿Fhmh�_�]N_NeÂ�V\b�¿Fh�ÀWb8kl]NTf^2b�¿pTsb�bt¿F]Â�Tqb�]N^8Â�V\jnj�Ó;]&S�TsÁFb�ÀF^t]NUBÓmc£^tV\[qTsj�jXTsZpUßUW]N[f]Nj\h�Á"]N^t_NÑÒÜ�¿pVn_klhf^tÃñhfkªÁF^�h�Á"]N^abdc5^�VnÅf¿}bt_²S�TsZ2Ó;]©VnÃ�Ápj\]NÃ�]NZ}b�]NU�Ó}c�¿pTG[mV\ZpÅ,T$b�hsb`TsjFÂ�Tqbt]�^

Wi

klh�^¦]NT�S`¿Ûp^tÃ�eFÂ�V�bt¿ ∑

Wi = WÑ1Ü�¿F],_db`Tqbt]$[qTf^�VXTsÓFjn]

Qt

Vn_�ZphqÂè^t]�ÁFjXTfS�]NU�Ó}cKT8_db`Tqbt]$[�]NS{b�hf^

~Qt = [Q1,t, . . . , Qn,t]ѦÜ�¿F],_abtTqbt]�[GTf^�VXTsÓpj\]~_�Tf^�]�ÀFÁ=UFTsb�]NU�T�_

Qi,t+1 = Qi,t + qi,t

ÑØ h�^

nVXUW]NZ�btVnSNTsj;Ûp^tÃ8_�e}Â1]�Tf_t_�ÀFÃ2]©b�¿"Tqb�]NTfS`¿�Ûp^�à klh�j\jnhq©_¦bt¿F]�_�TfÃ�]�jnTfZpU�UF]�[f]Nj�Ý

ô

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Vi,T ( ~QT ) = (aT − bW )(Wi − Qi,T ) � ÷�

R�_©klhf^�b�¿F]�S{h�Ã�Ã2h�ZÁ"hmhfj¯S�T�_a],h�ÀWb�jnVnZF]NU&TfÓ"hq[�]fepb�¿F]�[GTfj\Àp],hsk�bt¿F],Ûp^tÃ�V\Zbt¿F],Û"ZpTsjÁ;]�^tV\hWUKUWhm]N_�ZFhfb�j\]~TG[f]�^�hmh�à klhf^�S`¿Fh�VnS�]$UFÀF^�VnZFÅ�b�¿pVn_�Á;]�^tVnhmU®Ñ²Ü�¿F]$Ûp^tÃ*_�V\Ã2ÁFjnc�Àp_�]N_Tsjnjmb�¿p]©^�]NÃ8TsVnZFV\ZpÅ$Â�Tqbt]�^�jnVnS�]�Zp_�]N_�V\b¦hqÂ�Zp_NÑ°gi]�b�¿p]�^t]{klhf^t]�Ã2hq[f]�b�h$b�¿p]©Á"]N^�VnhWU5Ó;]{klh�^�]b�¿p]�jnT�_db~Ñ

Ü�¿F]�[qTsjnÀF]�hsk¯b�¿F]�Û"^�ÃðVnZKb�¿F]�Zp]{îmb�b�h2jXTf_ab�Á;]�^tVnhmUKVX_

Vi,T−1 = maxq

{

(aT−1 − bQT )q + βVi,T ( ~QT )} � Ë�

Â�¿FVXS`¿áVX_�_dbt^�ÀpS{b�ÀF^`TsjnjncKVXUW]NZ�btVnSNTsj¯b�h�b�¿p]2S�hfÃ2Ã2hfZ&Á;hmhfj�UW]�ÛpZFV\b�VnhfZ�TsÓ;hq[f]fÑ�Ü�¿F]�Ûp^`_dbhf^`UW]N^�S�hfZpUFV�btV\h�Z�klh�^

qi,T−1Vn_

aT−1 − bQT − bq + β∂Vi,T ( ~QT )/∂Qi,T = 0 � ô�

gÇV\b�¿�b�¿F]�ÁF^tV\[qTsb�]�jnc�TsjnjnhmSNTqbt]NU8Â�Tqbt]�^�j\VXS{]NZp_�]N_Ne�bt¿F]�UW]�^tVn[GTsb�Vn[f]�hsk�bt¿F]$[GTfj\Àp]�klÀpZpS btV\h�ZVX_−(aT − bW )

ѦY�_�VnZFÅ�b�¿FVX_�kÄTfS{bNemÛp^tÃiÚ _�^�]~TfS{b�VnhfZ�klÀpZpS btV\h�ZKVX_

qi,T−1 =1

2

[aT−1

b− (q−i,T−1 + QT−1) − β(

aT

b− W )

] � �

ù1hfÃ2ÁpTf^�VnZFÅ�]NÞ}ÀpTsb�VnhfZÔÊÂ�V\b�¿Ô]~Þ�À"Tqb�VnhfZÔû"e�bt¿F]�UWV\æ=]�^t]�ZpS�]�Vn_,b�¿pTsb,b�¿p]�Ã5ÀFj\b�VnÁFjncmV\ZFÅkÄTfS{b�h�^�h�Z&b�¿p]�jXTf_ab�bt]�^tÃ/Vn_

β/nVnZá]NÞ}ÀpTsb�VnhfZ&û"eªÂ�¿FVnj\]2V\b$VX_

βVnZá]~Þ�À"Tqb�VnhfZiÊFÑ,Ü�¿FVX_

_�¿Fhq©_1b�¿pTsb�bt¿F],UW^tV\[�]�^�klhf^�bt¿F]�UWV\æ;]N^�]NZpS{]�VnZKjXTsZ"U�UW]N[f]Nj\h�ÁFÃ2]�Z}b�Á"Tqb�¿"_�Ó"]�bdÂ1]N]�Z�b�¿F]bdÂ�h�_�V\b�ÀpTsb�VnhfZp_8VX_2bt¿F]kÄTfS{b8b�¿pTsb�Â�V\b�¿Òb�¿F]áÁF^�Vnhf^�TfÁFÁF^thfÁF^tVXTqb�VnhfZÒ_acW_ab�]NÃKe�bt¿F]jnT�_dbÁ;]�^tV\hWU�S�h�_ab¦b�h5Ûp^�Ã

ihsk®VnZpS{^t]NT�_aVnZFÅ�bt¿F]�TfÃ�h�ÀFZ}b1hfk®jnTfZpU8UW]�[�]�jnhfÁ;]NU8VnZ�Á;]�^tV\hWU

T − 1

VX_2hfZFjnc1/n

Â�¿pTsb�V�b�VX_2Â�¿F]�ZßÂ�Tqbt]�^2^�VnÅf¿}b`_8Ts^t]KÁp^�Vn[qTqb�]�e�S�hfZp_�VX_dbt]�Z}b8Â�V�bt¿Ò^t]N_�ÀFj\bt_

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UW]N[f]�jnhfÁ�Ã2hf^t]�jXTsZpU�VnZ�Á;]�^tV\hWUT − 1

ÀFZ"UW]�^�ÁF^tVnhf^�TsÁFÁp^�h�ÁF^�VXTqbtV\h�Z8b�¿"TsZKÂ�V�bt¿�ÁF^tVn[GTsb�]Â�Tqb�]N^�^�VnÅf¿}bt_NÑ�Y�_�V\ZpÅ2_�cmÃ2Ã�]�b�^tcfeWbt¿F]�b�hfbtTfj®TsÃ2hfÀFZ}b�hsk¯jnTfZpU�UW]�[�]�jnhfÁ;]NU�V\Z

T − 1VX_

qT−1 =n

n + 1

[aT−1

b− QT−1

]

−βn

n + 1

[aT

b− W

] � ÉNÏ�

ù1hfÃ2ÁpTf^�VnZFÅ�b�¿FVX_�^t]N_�ÀFj�b�bthK]~Þ}ÀpTqbtV\h�ZiøK_a¿phq©_�bt¿pTqb�bt¿F]�o�T�_a¿�Þ}ÀpTsZ}b�V\bdcVX_�j\hqÂ�]�^�VnZ

T − 1Â�V\b�¿�ÁF^tVn[GTsb�],Â�Tqbt]�^�^tV\Å�¿}bt_�b�¿"TsZV\b�Vn_©Â�V�bt¿ÁF^�Vnhf^©TfÁFÁF^thfÁF^tVXTqb�VnhfZ�ѲÜ�¿FVX_©TsÅ}TsVnZ

^t]�ÁF^thWUWÀpS{]~_�bt¿F]�_abtTfZpUFTs^`U�^t]N_�ÀFj�b�b�¿pTsb$Â�¿p]�ZáT�^�]~_ah�ÀF^tS�]�Vn_$UW^`TGÂ�Z�kl^�h�Ã/T�S{h�Ã2Ã�h�ZÁ;h}h�j¤eWV\b©Â�Vnj\j®Ó;]�hq[f]N^�]�îWÁFj\h�V�bt]NUªÑ

ù1hfÃ2ÁpTf^�VnZFÅK]NÞ}ÀpTqbtV\h�Z�øKTfZpUá]~Þ}ÀpTqbtV\h�Z£ÉNÏ�Tfjn_�hK_�¿Fhq©_�b�¿p]8V\Ã2ÁpT�S b�hsk�S`¿pTsZpÅfVnZFÅÛp^tà ZmÀFÃ�Ó"]N^t_NÑ�gÇ¿F]�ZÕb�¿F]N^�]�VX_5T&Ã�h�ZFhfÁ;hfjncfe

n = 1e�bt¿F]�ZÕb�¿F]N^�]�VX_,ZpháUWV\æ;]N^�]NZpS{]

Ó;]{bdÂ�]�]�Z�bt¿F]�bdÂ1h�]~Þ}ÀpTqbtV\h�Zp_�Ñ©RñÃ2hfZFh�Á"h�j\VX_db$UWhm]N_�ZFhsb$ZF]�]~U�b�h�Â�hf^t^tc�TsÓ;hfÀFb�hsbt¿F]�^Ûp^tÃ�Ú _�S�TsÁFb�ÀF^tV\ZpÅáÂ�Tqb�]N^5^tV\Å�¿}bt_NÑ&ã�ZpS{^t]NT�_aVnZFÅ

nVnZpS{^t]NT�_a]~_�bt¿F]�bthsbtTfj�TfÃ2hfÀFZ}b5hfk�UW]{Ý

[f]Nj\h�ÁFÃ2]�Z}b�UWÀp^�VnZFÅT − 1

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qT /qT−1Vn_

b

(

n + 1

n

) (

W − QT−1

(aT−1 − QT−1) − β(aT − W )

)

− 1

klhf^�bt¿F],ÁF^tVn[GTsb�]�Â�Tqbt]�^�^tV\Å�¿�b`_©S�T�_a]�ѲR�_n

Å�^�hq©_Nemb�¿FVX_�[qTsjnÀF],UW]~S{jnV\Zp]N_Ne"_�¿FhqÂ�V\ZpÅ�bt¿pTqbVnÃ�Á;]�^�kl]NS{b1S{h�Ã2Á"]�b�V\b�VnhfZ2V�b`_a]Nj�k=bt]�ZpUF_¦b�h,UW]NjnTGc�UW]�[�]�jnhfÁFÃ2]NZ�b~Ñ°Ü�¿FVX_¦VX_¦S�hfZp_�Vn_ab�]NZ}b¦Â�V�bt¿_abtTsZ"UFTs^`U�^�]~_aÀpj�b`_,klh�^�^t]N_�hfÀp^tS�]8]{îmb�^`TfS{b�VnhfZÕÂ�¿F]NZÕbt¿F]�^t]�VX_�V\Ã2Á;]�^�kl]NS b8S{h�Ã2Á"]�b�V\b�VnhfZ®ÑØ h�^�b�¿F]�Áp^�Vnhf^�TsÁpÁF^�h�ÁF^tVnTsb�VnhfZ�SNTf_�]fe}b�¿p]�^tTsb�Vnh�VX_

b(n + 1)(W − QT−1)

n(aT−1 − QT−1) − β(aT − W )− 1

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jnV\Ã2V\b©]�[qTsjnÀpTqbt]N_�bthb(W − QT−1)

aT−1 − QT−1− 1

Â�¿FVXS`¿�Vn_5ZF]�Å}Tqb�Vn[f]2V\kaT−1/b > W

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T − 1eW^`Tqbt¿F]�^

b�¿"TsZ5VnZTÑ°Ü�¿F]�V\Ã2ÁFjnVXS�TqbtV\h�Z,VX_�bt¿pTqb�Â�V\b�¿2T©jXTs^tÅf]²]NZFhfÀpÅf¿

nesTfj\jWUW]N[f]�jnhfÁpÃ�]NZ}b°hWS�S�ÀF^t_

UWÀF^tVnZFÅ2b�¿F]�Û"^t_ab©Á"]N^�VnhWUªÑ�ã�Z�bt¿F]5TfÓp_a]NZpS{]�hfk�]�îmb�]�^tZpTfj\V\b�Vn]N_�T�_�_�hWS{VXTqb�]~U�Â�V\b�¿�bt¿F],^`Tqbt]hsk²UF]�[f]Nj\h�ÁFÃ2]�Z}bNe;bt¿FVn_$VX_©bt¿F]�]{ê�S{Vn]�Z}b�_ah�j\ÀWbtV\h�ZÕÆ ¾ TsjnÀF]�ÅKTsZpU×rmhfjnhqÂ�e°É~ÊfÊ�ÏqÐ�Ñ���]�[f]Nj�ÝhfÁpÃ�]NZ}b5Â�V\jnj²h�ZFjnc×hWS�S�ÀF^5VnZÕjXTqbt]�^,Á;]�^tV\hWUF_5V�k�UF]�Ã8TsZpUÔ_�¿FV\kçbt_�ÀFÁ � V\ZpS�^�]~Tf_�]N_�VnZ

at

eÂ�¿FVXS`¿×Ã8Tfâf]N_�UW]N[f]Nj\h�ÁFÃ2]�Z}b�b�¿pTsb�Â�Tf_$ZFhfb�ÁF^�hfÛFbtTfÓFjn]�VnZi]NTs^tjnV\]N^$Á;]�^tVnhmUp_�ÁF^�hfÛFbtTfÓFjn]VnZKjXTqbt]�^�Á;]�^tV\hWUF_NÑ

R�_5Â�Tsb�]�^�^t]N_�hfÀp^tS�]N_$Ó"]~S{h�Ã�]8Ã2h�^�]8¿F]~TG[mV\jncá_�ÀFÓp_tS{^tV\Ó;]NUªe°Åfhq[�]�^tZFV\ZpÅKTfÀWb�¿Fh�^�V\b�Vn]N_Ts^t]�V\ZpS�j\VnZF]~Uábth�^�]~UWÀpS{]�bt¿F]�TsÃ2hfÀpZ�b�hfk�Â�Tqbt]�^�j\VXS{]NZp_�]NUáklhf^�¿mÀFÃ8TsZ�Àp_�]fÑ8ö�TfZpUiUW]{Ý[f]Nj\h�Á"]N^t_�Ts^t]bdcmÁFVXS�TsjnjncßTGÂ�Ts^t]hsk5b�¿FVX_�Á;h�_t_�V\ÓFVnjnV�bdc�e�TsZ"UÇÃ8Tsâf]&b�¿F]NV\^UW]N[f]�jnhfÁpÃ�]NZ}bUW]~S{VX_aVnhfZp_�T�S�S�hf^`UWV\ZpÅfjncfÑ Ø ^�h�Ã.bt¿F]©Á;]�^`_aÁ;]NS{b�Vn[f]�hskªT�UW]N[f]�jnhfÁ;]�^~eqbt¿F]�TsÃ2h�ÀFZ}b¦hsk=Â�Tsb�]�^TG[qTsVnjnTfÓFjn]©klhf^�jnVXS{]�Z"_aVnZFÅ�V\ZKÁ"]N^�VnhWU

tVn_1bt¿F]�^t]{klh�^�]�T�^tTfZpUWhfà [qTs^tVXTsÓFjn]femÂ�¿Fh}_a]�[qTsjnÀF]$Vn_

UW]�b�]�^tÃ2V\Zp]NU�Ó;]{klhf^t]5jXTsZ"UUW]�[�]�jnhfÁFÃ2]�Z}b�hWSNS{ÀF^`_�VnZ&Á"]N^�VnhWUte=ÓFÀWb$Tskçb�]N^�UW]N[f]�jnhfÁpÃ�]NZ}b

VnZt − 1

Vn_�S�hfÃ2ÁFjn]{bt]fÑ�ã�Zib�¿p]8Ã�hWUW]Nj¤e�b�¿F]�TsÃ2hfÀFZ}b�hsk�Â�Tqb�]N^�TG[GTfV\jXTsÓpj\]2VX_�V\Z"S{jnÀpUW]NUTf_�TsZáT�UFUWV\b�VnhfZpTfj°_abtTsb�]5[qTf^�VXTsÓFjn]fe

Wt

Ñ$Ü�¿F]5[qTfj\ÀF]�hskWT

VX_$S`¿Fh�_�]�ZÓmc�b�¿F]�Åfhq[�]�^tZWÝVnZFÅiTfÀWb�¿phf^tV�bdcÔTqkçbt]�^�b�¿p]�S�ÀF^t^�]NZ�b2Á;]�^tV\hWU£UF]�[f]Nj\h�ÁFÃ2]�Z}b8S`¿FhfVXS{]~_5ÓFÀFb2Áp^�Vnhf^�b�hiZF]�î}bÁ;]�^tV\hWUªÚ _�Ñ8R�_�_�ÀFÃ2V\ZpÅ�bt¿pTqb,jXTsZ"U×UW]N[f]Nj\h�Á"]N^t_�Ts^t]2]{îWÁ;]NS bt]NU×^�]�b�ÀF^tZiÃ2TsîWV\Ã2VnÙ�]�^`_Neªb�¿F][qTsjnÀF]�hsk�T2UW]�[�]�jnhfÁFÃ2]�Z}b�Ûp^tÃðÂ�V\b�¿�hfÁ;]�ZTfSNS{]~_�_�Â�Tqb�]N^�^�VnÅf¿}b`_1Ó;]NS�hfÃ2]N_

Vi,t(Qt, Wt) = maxq

{(at − bQt+1)q + βEt [Vi,t+1(Qt+1, Wt+1)]}

Â�¿F]N^�]Et[•]

Vn_¦b�¿F]©]�îWÁ"]~S btTsb�VnhfZp_¦hfÁ;]�^`Tqb�h�^¦Tqb²Á;]�^tV\hWUte}]�[qTsjnÀpTqbt]NU2hq[�]�^�b�¿p]©^tTfZFÅf]�hfk

Á;h�_t_aVnÓFjn],[qTsjnÀF]N_©klhf^Wt

Ñ�Ü�¿p]5Â�Tqb�]N^�TG[qTsVnjnTfÓFVnj\V\bdcKS�hfZp_ab�^`TsVnZ}b©Vn_�ZFhqÂQt + qt ≤ Wt

ÑÜ�¿F]&_abtTsb�]�b�^`TsZ"_aV\b�VnhfZÒ]NÞ}ÀpTsb�VnhfZÒklhf^

Qt

^�]NÃ8TsVnZp_Qt+1 = Qt + qt

Ñ Ø hf^Wt

e�b�¿F]

ÉfÉ

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eÂ�¿F]N^�]

W t

Vn_�bt¿F]�ÀFÁFÁ;]�^�j\VnÃ2V�bh�ZIb�¿p]i[qTfj\ÀF]ihskWt+1

Ñ�ã�k5bt¿F]ÔUW]N_�Vn^�]~UÇj\]N[f]Nj$hfkjnVnS�]�Zp_�]N_1hfZFjnc8UW]NS�j\VnZF]~_�e}b�¿p]�Z

W t = Wt

emÂ�¿FV\jn]$V�k�V\b�S�TsZ�Åfh5VnZ�]NV�bt¿F]�^�UWVn^�]~S btV\h�Z®e}b�¿F]NZ

W t = WÑIR�Ã2h�^�]_�hfÁF¿FVX_ab�VXS�Tqbt]NUBÃ2hmUF]�j©Â�hfÀFjXUB¿pTG[f]�b�¿F]ZmÀFÃ�Ó"]N^�hsk�TG[qTsVnjnTfÓFj\]

jnVnS�]�Zp_�]N_1UW]{bt]�^tÃ�VnZF]~U2Ó}c2T�Å�hq[f]�^tZFÃ2]�Z}b¦UW]~S{VX_aVnhfZ8ÁF^thWS{]~_�_NÑ°Ü�¿FVX_1Vn_²Ó"]NcfhfZ"U�b�¿F]$_�S�hfÁ;]hsk°b�¿F],S�ÀF^t^�]NZ�b©TfZpTsjncW_aVX_�Ñ

ë�c�VnÃ�Á;h�_�VnZFÅ8_ah�Ã�]�klÀp^abt¿F]�^©_ab�^tÀpS btÀF^t]�hfZ�b�¿F]�]�îmÁ;]NS{btTsb�VnhfZp_NeWb�¿p]�V\Ã2ÁpT�S b©hsk¯b�¿pVn_ÀFZpS�]�^�btTfV\Z}bdc8Ó;]NS{h�Ã2]N_©S�j\]~Ts^~Ñ�ö�]{b

g(w)Ó;]5T2UW]�Z"_aV\bdc8klÀFZpS{b�VnhfZh�Z

[0, W ]emb�¿p]�^tTfZFÅf]

hsk=jnVnS�]�Zp_tTsÓFjn]�Â�Tqb�]N^NѯR�_t_aÀpÃ�VnZFÅ�bt¿pTqb¦^t]N_ab�^tVXS b�VnhfZ"_�Ts^t]©S{ÀFÃ�ÀFjnTsb�Vn[f]�e�TsZ"U�Å�V\[�]�Z2[qTsjnÀF]N_klhf^

Qt+1TsZpU

Wt

eFb�¿F],UW]NZp_�V�bdc8klh�^�b�¿F]�Â�Tqbt]�^©TG[qTsVnjXTsÓFjn]�VnZt + 1

S�TfZ�Ó;]�Â�^�V\babt]�Z�Tf_

g(w|Qt+1, Wt) = g(w)/

∫ Wt

Qt+1

g(z)dz

Â�¿FVXS`¿áklh�^�_aVnÃ2ÁFj\VXS{V\bdc&Â�]8UW]{ÛpZp]2T�_f(w)

Ñ2gi]�Tf_t_aÀpÃ�]�b�¿"Tqb�b�¿F]8jXTf_ab�Á"]N^�VnhWUá[qTsjnÀF]klÀFZpS{b�VnhfZ�VX_�ZFhqÂñVXUW]�Z}btVnSNTsj�bthK]~Þ}ÀpTqbtV\h�ZBÉ�e®Â�V�bt¿

W^t]�ÁFjXTfS�]NU×Ómc

WT

Ñ8gÇV\b�¿ib�¿F]~_a]UW]�ÛpZFV\b�VnhfZp_Nemb�¿F]

T − 1[qTfj\ÀF]$klÀFZ"S b�VnhfZVX_

Vi,T−1 = maxq

{

(aT−1 − bQT )q + β

∫ WT−1

QT−1

Vi,T (w)f(w)dw

}

� ÉfÉ

Â�¿F]N^�]�bt¿F]5[qTsjnÀF]�klÀpZpS btV\h�ZZFhqÂéUW]�Á;]�ZpUp_©hfZ�bt¿F]5Â�Tqbt]�^�TG[qTfV\jXTsÓFVnjnV�bdc�Ñ�R�^�Å�ÀFÃ2]�Z}bt_�hfkb�¿p]1[qTsjnÀF]²klÀFZpS btV\h�Z5Â�¿FVXS`¿5Tf^�]²ÛFîW]NU�Â�V\b�¿�^�]~_aÁ;]NS{b�b�h�b�¿p]²VnZ�bt]�Å�^tTsb�VnhfZ5Tf^�]²_�ÀFÁFÁp^�]~_�_�]NUªÑÜ�¿F]$Ûp^`_db©hWUW]N^©S{h�ZpUWV\b�VnhfZ�S�TsZKZFhqÂíÓ"]�Â�^tV�b�b�]NZ�T�_

aT−1 − bQT − bq = −∂

∂QT

[

β

∫ WT−1

QT

Vi,T (w)f(w)dw

]

� É~Î�

Ü°hi]�îWÁFj\h�^�]�b�¿p]V\Ã2ÁpT�S b8hfZBb�¿F]&S`¿phfVXS{]Khskqi,T−1

e1b�¿F]^tV\Å�¿}b8_�VXUW]�hsk�]~Þ}ÀpTqbtV\h�ZíÉGÎVX_�S{hfÃ2ÁpTf^�]~U&bth

βV ′i,t(QT )

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βVi,T (QT )f(QT ) − β

∫ WT−1

QT

[

∂Vi,T (w)

∂QT

f(w) + Vi,T (w)∂f(w)

∂QT

]

dw

Y�_�VnZFÅ�bt¿F]�UF]{ÛpZFV\b�VnhfZihskf(w)

e®V\b�S�TsZ×Ó"]2_�¿FhqÂ�Zb�¿"Tqb∂f(w)/∂QT = f(QT )f(w)

ÑØ Àp^abt¿F]�^~eWV�b�Ã�Àp_db�Ó;]�b�^tÀF]$b�¿pTsb

Vi,T (QT ) = 0eF_�V\Z"S{]�Â�¿F]�ZKZFh2Â�Tqb�]N^�^�]NÃ2TfV\Z"_²bth2Ó"]

jnVnS�]�Zp_�]NUªe"ZFh8jXTsZpUSNTsZÓ;]5UW]�[�]�jnhfÁ;]NUªÑ�Ü�¿F]5TfÓ"hq[�]�]NÞ}ÀpTqbtV\h�ZS�TsZ�bt¿F]�ZÓ;]5^t]�Â�^tV\bab�]NZTf_

−β

∫ WT−1

QT

(

∂Vi,T (w)

∂QT

+ f(QT )Vi,T (w)

)

f(w)dw � É~ó�

Ü�¿F]¦TfÃ�h�ÀFZ}b®hskmjnTfZpU�UW]N[f]�jnhfÁ;]NU$UWÀF^tVnZFÅT−1

Â�V\jnjfVnZpS{^t]NT�_a]�V\kW]NÞ}ÀpTqbtV\h�Z�É~ó�Vn_®j\]~_�_=bt¿pTsZ

−β∂Vi,T (WT−1)/∂QT

ѯã�k"]~Þ}ÀpTqbtV\h�Z�ÉNó�VX_�_aÀFÓFb�^`TfS bt]NU,kl^�h�Ã−β∂Vi,T (WT−1)/∂QT

e�]$�]{b

−β

∫ WT−1

QT

(

∂Vi,T (WT−1)

∂QT

−∂Vi,T (w)

∂QT

− f(QT )Vi,T (w)

)

f(w)dw � ÉNû

R�_aÀWê�S�V\]NZ�b8S{h�ZpUWV\b�VnhfZ£klhf^�b�¿FVX_2]{îWÁF^t]N_t_aVnhfZ�bth×Ó;]KÁ"h}_aV\b�Vn[f]KVn_�bt¿pTqb�b�¿F]�UWV\æ;]N^�]NZpS{]Ó;]{bdÂ�]�]�Z�bt¿F]5Á"Ts^�b�VXTsj°UF]�^tV\[qTqbtV\[�]�b�]N^�Ã8_�Vn_�ZF]�Å}TqbtV\[�]fÑ�ã�k

∂V 2i,T (QT , w)/∂QT ∂w < 0

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b(WT−1 − w)/neªÂ�¿pVnS`¿×Vn_�Á"h}_aV\b�Vn[f]5klhf^�Tsjnj

w < WT−1e;TfZpU

∂V 2i,T (QT , w)/∂QT ∂w = b/n > 0

Ñ©Ü�¿mÀp_�epklhf^�b�¿F],jnVnZF]NTf^�S�T�_a]�epV\bVX_�ÀFZpS{jn]NTf^�Â�¿F]{bt¿F]�^�VnZ}b�^thWUWÀpS{VnZFÅ�ÀFZpS�]�^�btTsVnZ}bdc�Â�Vnj\jªVnZpS{^t]NT�_a]$hf^�^t]NUFÀpS{]�UW]N[f]�jnhfÁpÃ�]NZ}bVnZ

T − 1ѦÜ�¿F]�V\Ã2ÁpT�S b©hfk�b�¿p]�jnT�_db�bt]�^tÃðV\Z�]~Þ�À"Tqb�VnhfZiÉ�û2UW]�b�]�^tÃ2V\Zp]N_�bt¿F]�^�]~_aÀpj�b~Ñ

Ö�îWÁpTsZ"UWV\ZpÅÇb�¿F]£j\VnZF]NTf^&S�T�_a]Ô^t]�[�]NTsjX__�hfÃ2]Ôhsk�b�¿F]BUW]{b`TsVnjn_NÑ gÇV\b�¿E(wn) ≡

∫ WT−1

QT

wnf(w)dwep]~Þ�À"Tqb�VnhfZiÉ�û2Ó;]NS�hfÃ2]N_

−β

n

{

bWT−1 + f(QT )[

aT QT + bE(w2)]

− [(aT + bQT )f(QT ) + b]E(w)}

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f(QT )eE(w2)

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WT−1e=V\b�klhfjnj\hq©_©b�¿pTsb

f(QT ) → 0Ñ,rmVnZpS{]

E(w)TfÁFÁF^th�TfS`¿p]N_

WT

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WT

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VX_¦VnZpS{^t]NT�_a]~UªÑ¯R�_1UF]�Ã2hfZp_ab�^`Tqbt]NU2Ó;]�jnhqÂßV\Z�T,Z}ÀpÃ�]N^�VXS�Tfjp]�îWTfÃ2ÁFj\]�efbt¿F]�UFV\jnÀWb�VnZFÅ�]{æ=]NS bhsk

nTsjX_ah�ÁFjXTGcW_�TsZ�V\Ã2Á;hf^�btTsZ}b¦ÁpTf^ab¦VnZ8UW]{bt]�^tÃ2V\ZFVnZFÅ�bt¿F]�ÓpTsjXTsZ"S{]�klhf^�bt¿F]�hfÁ;]�Z8T�S�S{]~_�_

S�T�_a]�Ñë�^�VnZFÅfVnZFÅ8ÀFZ"S{]�^�btTfV\Z}bdc�VnZ}b�h8bt¿F]5SNTf_�]�Â�V�bt¿ÁF^�Vn[qTqbt]�Â�Tsb�]�^�^�VnÅf¿}bt_�^t]NÞ}ÀFVn^t]N_©TfZ�T�_dÝ

_�ÀFÃ2ÁWb�VnhfZTsÓ;hfÀFb©¿FhqÂIbt¿F]5TsÀFb�¿Fh�^�V\bdc�Â�Vnj\j�T�S`¿FVn]�[f]�T2UW]~_aVn^�]~UK^�]~UWÀpS btV\h�Z�V\ZKb�¿p]�b�hsb`TsjZmÀFÃ5Ó;]�^²hsk=j\VXS{]NZp_a]~_�Ñ�ï$Vn[f]�Z2bt¿pTqb�bt¿F]©^tV\Å�¿}bt_¦Tf^�]�ÁF^tV\[qTsb�]�jnc5¿p]�jXUªe�Â�]©Tf_t_�ÀFÃ2]�b�¿pTsb¦b�¿F]TsÀFb�¿Fh�^�V\bdc&ÁFÀF^`S`¿pT�_a]~_�bt¿F]N_�]8^tV\Å�¿�b`_�Ñ8ã�Z�ÁpTf^abtVnS�ÀFjXTs^~e®Â1]8T�_�_�ÀFÃ2]2b�¿pTsb�b�¿F]�TfÀWb�¿phf^tV�bdcÁpTGcW_�hfZF]�ÁF^tVXS{]Kklhf^8Tsjnj�b�¿p]j\VXS{]NZp_a]~_2ÁFÀF^`S`¿pTf_�]NU®e�b�¿pTsb�ÁF^tVnS�]�Â�¿FVXS`¿ß^�]~_aÀpj�b`_�VnZÒb�¿F]UW]~_aVn^�]~U�^t]NUWÀ"S b�VnhfZKVnZ�bt¿F]�b�hfbtTfj®TsÃ2hfÀFZ}b�hsk�j\VXS{]NZp_�]NU�Â�Tsb�]�^~ѯg×],SNTsZ�b�¿F]NZKÂ�^tV\b�]$b�¿F][qTsjnÀF]$klÀFZpS{b�VnhfZTf_

Vi,t(~zt) = maxq,s

{(at − bQt+1)q + tts + βEt [Vi,t+1(~zt+1)]}

Â�¿F]N^�]~zt = [ ~Q′

t, ωt, ~W ′t ]′Vn_�T_db`Tqb�]�[f]NS{b�h�^�klhf^5b�¿F]�_�cW_dbt]�Ã�ÑÜ�¿F]�_abtTsb�]N_�Ts^t]8ZFhqÂ

~Qt

e;b�¿F]2Þ}ÀpTsZ}btV�bdc�hsk1jnTfZpU&UW]N[f]�jnhfÁ;]NU&Ó}c�]~TfS`¿áUW]N[f]Nj\h�Á"]N^Neωt

e=b�¿p]5bthsbtTfj¯TfÃ2hfÀFZ}b�hfkÂ�Tqb�]N^�b�¿pTsb©b�¿p]5TsÀWbt¿Fhf^tV\bdc�Tsjnj\hq©_�bth�Ó;]�j\VXS{]NZp_a]~UªepÀpÁ�b�h�TsZ"U�VnZpS�j\ÀpUFV\ZFÅ�UFTqbt]

te"TsZ"U

~Wt

e=bt¿F]2[fh�j\ÀFÃ2]�hsk²Â�Tqbt]�^$j\VXS{]NZp_a]~_$¿F]�jXU&Ómc]NT�S`¿áUW]N[f]�jnhfÁ;]�^~Ñ�Ö²TfS`¿iUW]�[�]�jnhfÁ;]�^$ZFhqÂ

É�û

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S`¿Fhmh�_�]N_�bdÂ�h[qTs^tVnTfÓFj\]~_�e®b�¿F]�TfÃ2hfÀFZ}b,hsk�jXTsZpU×b�h&UF]�[f]Nj\h�Á®eqi,t

e�TsZ"Uábt¿F]�[�hfjnÀFÃ2]�hfkjnVnS�]�Zp_�]N_�b�hK_a]Nj\j¯ÓpTfS`âKbth�b�¿p]�Å�hq[f]�^tZFVnZFÅ�TsÀFb�¿Fh�^�V\bdcfe

si,t

Ñ�Ü�¿F]�Â�Tsb�]�^�j\VXS{]NZp_�]5Ã8Ts^tâf]�bS{jn]NTf^t_�Tsb�Áp^�VXS{]

ttÑ�Ö�îWÁ;]NS b`TqbtV\h�Zp_�Ts^t]�]�[qTsjnÀpTqbt]NU�hq[f]N^�T2UWVX_ab�^tV\ÓFÀFb�VnhfZKklhf^

ωt

Ñã�Z�]~TfS`¿�Á;]�^tV\hWUªem]~TfS`¿�UW]N[f]Nj\h�Á"]N^�SNTsZFZFhfb�UW]N[f]Nj\h�Á�Ã2hf^t]�jXTsZpU�b�¿pTfZ�jnVnS�]�Zp_�]N_1¿F]�jXUªÑ

Ü�¿FVX_,Â�Tqbt]�^5TG[qTsVnjXTsÓFVnj\V\bdc×S�hfZp_ab�^`TsVnZ}b,Vn_Qi,t + qi,t + si,t ≤ Wi,t

Ñ&Ü�¿F]�TfÀWb�¿Fh�^�V\bdc=Ú _UW]NÃ2TfZpUKklhf^�Â�Tsb�]N^©V\ZáÁ"]N^�VnhWU

tVn_ ∑

Wi,t − ωt

e"b�¿F]�UWV�æ=]�^t]�Z"S{]5Ó;]{bdÂ�]�]NZ�b�¿F]�ÀFÁFÁ;]�^Ó;hfÀFZpU�bt¿F]©TsÀWbt¿Fhf^tV\bdc5UW]N_�Vn^�]~_�V\Z

tTfZpU�b�¿F]�bthsb`TsjFZmÀFÃ�Ó"]N^�hsk=j\VXS{]NZp_a]~_�Ñ

ωt+1VX_¦S`¿Fh}_a]NZ

Tqb�bt¿F]©]NZpU2hsk=Á"]N^�VnhWUte�TfZpU�j\Vn]N_²V\Z�b�¿p]©V\Z}bt]�^t[GTfj

[Qt+1, Wt+1]efÂ�¿F]N^�]�klh�^¦ZFhfbtTqbtV\h�ZpTsj

_�V\Ã2ÁFjnVnS�V�bdc�eQt =

Qi,t

TfZpUWt =

Wi,t

ÑÇÜ�¿F]Kb�hfbtTsj©Þ}ÀpTfZ}b�V\bdc£UW]�[�]�jnhfÁ;]NU£Vn_ÀFÁ=UFTqbt]NU×Tf_

Qi,t+1 = Qi,t + qi,t

Â�¿FVnj\]�b�¿F]2jnVnS�]�Zp_�]N_$¿F]NjnUáVn_$ÀFÁ=UFTsb�]NUiTf_Wi,t+1 =

Wi,t−si,t

Ñ°gÇV\b�¿FVnZ8Á;]�^tV\hWU�Ã2Tf^�â�]{b¦S{jn]NTf^tTfZpS{]1klhf^�Â�Tqbt]�^�jnVnS�]�Zp_�]N_�VnÃ2ÁFj\Vn]N_�b�¿pTsb�Tsb�b�¿F]]NÞ}ÀFVnjnV\ÓF^tVnÀFÃ�e ∑

si,t = Wt −ωt

e}Â�¿F]�^t]�Â�]�UW]�ÛpZF]©bt¿F]�_aÀFÃ�T�_st

Ñ�gÇV�bt¿n

VnUF]�Z}b�VXS�Tfj¤e_�c}Ã2Ã2]{bt^�VXS�Ûp^�Ã8_NeWTqb�b�¿F]�o�T�_a¿�]NÞ}ÀFVnj\VnÓF^tV\ÀFÃ

nsi,t = Wt −ωt

TfZpUnqi,t = qt

eWÂ�¿p]�^t]

qt =∑

qi,t

ÑR�_�klhf^5b�¿F]K]NTf^�jnVn]�^,]�îWTfÃ2ÁFj\]~_�e�Â�]�Ó;]�Å�V\ZBÂ�V�bt¿ÕÁ"]N^�VnhWU

Te�bt¿F]�Á;]�^tV\hWUÔÂ�¿p]�Z£Tfj\j

TG[qTsVnjnTfÓFjn]©Â�Tsb�]N^1VX_�S{hfZ"_aÀFÃ2]~UªÑ�Ü�¿FVn_�VnÃ2ÁFj\Vn]N_�bt¿pTqb�b�¿p]$Â�Tqbt]�^�TG[qTsVnjnTfÓFV\jnV\bdc�S�hfZp_ab�^`TsVnZ}bÓFVnZpUF_�Â�V\b�¿�]~Þ�À"TsjnV�bdc�ep_aÀ"S`¿�bt¿pTqb

Qi,t + qi,t + si,t = Wi,t

ѲÜ�¿F],UW]~S{VX_aVnhfZ�ÁF^thfÓFjn]�à Vn_b�¿p]�^t]{klhf^t]�^�]~UWÀpS{]~U,Ómc,h�ZF]1[qTf^�VXTsÓFjn]fѯR�_t_aÀpÃ�VnZFÅ$b�¿p]�jnVnZF]NTf^�_�Á;]NS{V\Û"SNTqb�VnhfZ�Àp_�]NU�TsÓ;hq[f]feb�¿p]�jnT�_db�Á;]�^tV\hWU�[GTfj\Àp]�klÀFZpS btV\h�Z�Vn_

Vi,T (~zT ) = maxs

{[aT − b(WT − sT )] (Wi,T − Qi,T − s) + tT s}

rmh�j\[mVnZFÅ�klhf^�b�¿p]�hfÁWbtV\Ã2VnÙNTqbtV\h�Z®emb�¿p]�Ûp^t_ab�hf^`UW]�^�S�hfZpUWV\b�VnhfZKVX_

− [aT − b(WT − sT )] + b(Wi,T − Qi,T − s) + tT = 0

Â�¿FVXS`¿éS�TsZõÓ;]ÕVnZ�bt]�^tÁF^t]{b�]~UõTf_&_�¿FhqÂ�VnZFÅÒb�¿pTsbáÂ�Tsb�]N^j\VXS{]NZp_a]~_&Ts^t]Õ_�hfjXUíÀpZ�btV\j5b�¿F]Ã8Ts^tÅfVnZpTsj¯jnh�_t_©kl^thfÃ�klh�^�Å�hfVnZFÅ8b�¿p]�^tVnÅf¿}b�bthKUF]�[f]Nj\h�ÁáT�ÀFZFV\b�hfk²jXTsZpU×Vn_$]~Þ�À"Tsj¯b�h�b�¿F]

É~ø

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ÁF^tVnS�]2b�¿pTsb5bt¿F]�c�S�TfZiÓ;]�_�hfjXU×klhf^~Ñ�ã�Ã�Á;h�_�VnZFÅ&_acmÃ2Ã2]{bt^�c�TsZpUÔ_�VnÃ�Ápj\V\klcmV\ZFÅác}Vn]�jXUF_,TÂ�Tqb�]N^�j\VXS{]NZp_a]�Áp^�VXS{]�VnZ�Á;]�^tV\hWU

t]NÞ}ÀpTsj=bth

tT = (aT − bωT ) −b

n(ωT − QT )

gÇV\b�¿8T�_ah�j\ÀWbtV\h�Z,klh�^�b�¿F]�jnVnS�]�Zp_�]�ÁF^�VXS{]�eqb�¿p]�[GTfj\Àp]1klÀpZpS btV\h�Z�klh�^¯b�¿F]�jXTf_ab�Á"]N^�VnhWUªesÅfVn[f]NZÁF^thsÛFb�Ã8TqîWVnÃ2V\Ù~Tqb�VnhfZ�eWÃ2Tf^�â�]{b�S{jn]NTf^tTfZpS{]�emTfZpU�_acmÃ2Ã2]{bt^�c�eWVn_

Vi,T (~zT ) = (aT − bωT ) (Wi,T − Qi,T ) −b

n(ωT − QT )

(

Wi,T −ωT

n

)

� ÉN÷�

Ü�¿F]�[qTsjnÀF]©hfk=b�¿F]�Û"^�ÃüV\Z2bt¿F]©jXTf_ab²Á"]N^�VnhWU2S{hfÃ2]~_�kl^�h�Ã.bdÂ�h,_ah�ÀF^`S{]N_Neqbt¿F]©ZF]�b²^�]N[f]NZ}Àp]kl^thfà _a]Nj\jnV\ZpÅ5jnhsbt_1Â�V\b�¿�Â�Tqbt]�^~e}TfZpU�b�¿F]$^t]�[f]NZmÀF]�kl^thfà _a]Nj\jnV\ZpÅ5Â�Tqb�]N^²jnVnS�]�Zp_�]N_²ÓpTfS`â�b�hb�¿p]�Åfhq[f]N^�ZpV\ZFÅ�TsÀWbt¿Fhf^tV\bdcfÑ�R�_�_�ÀpS`¿®eWbt¿F]�[qTsjnÀF]$klÀFZpS{b�VnhfZUW]�Á;]�Z"UF_�hfZTfj\j=bt¿F^�]N]�_db`Tqb�][qTs^tVnTfÓFjn]N_NÑ

ë�]{klhf^t]5Áp^�hWS{]N]NUWVnZFÅ�b�hKÁ"]N^�VnhWUT − 1

eªS{h�Zp_�VnUW]N^�bt¿F]�V\Ã2ÁpT�S b�hsk1T�S`¿pTsZFÅ�]5VnZωT

ÑÜ�¿F]$Ûp^`_db�UF]�^tV\[qTqbtV\[�]�hfk�É~÷�Â�V�bt¿K^t]N_�Á;]NS b�bth

ωT

Vn_

∂Vi,T (~zT )

∂ωT

= −b

n2[WT (n + 1) − QT (n − 1)] � ÉqË�

Â�¿FVXS`¿®e�Ómc×[mVn^abtÀF]�hfk�b�¿F]�kÄTfS{b5bt¿pTqbWT ≥ QT

e�Ã�Àp_ab�Ó;]�ZF]NÅ�Tsb�Vn[f]fÑ×Ü�¿FVn_�VnÃ�Ápj\Vn]N_b�¿"Tqb,bt¿F]�jXTs^tÅf]�^$bt¿F]�[fh�j\ÀpÃ�]8hfk�Â�Tsb�]�^�jnVXS{]�Z"_a]~_,TG[qTsVnjnTfÓFj\]�UFÀF^�VnZFÅ�bt¿F]8ÛpZpTfj¦Á;]�^tV\hWUªeb�¿p],j\hqÂ�]�^�bt¿F]�[qTsjnÀF]�klÀFZ"S b�VnhfZ�Ñ1̦^thq[mVnUF]NU

nVX_©Zphsb�b�¿"Tqb�jXTs^tÅf]�epUW]�[�]�jnhfÁ;]�^`_�S�TfZ�b`Tsâ�]

TfUF[GTfZ}btTsÅ�]2hsk�b�¿F]�j\VnÃ2V�bt]NUÕS�hfÃ2Á"]�b�V\b�VnhfZ®ÑÜ�¿F]�]{æ=]NS btV\[�]�VnZpS{^t]NT�_a]2VnZ�b�hsb`Tsj�UW]�Ã8TsZ"UVnZ�bt^�hWUWÀ"S{]NU�kl^thfÃñb�¿F]�Å�hq[f]�^tZFVnZFÅ�TsÀWbt¿Fhf^tV�bdc5V\ZpS�^�]~Tf_�]N_�ÁF^thsÛpbt_�bth�b�¿F]©¿phfjXUW]�^`_�hsk;b�¿F]~_a]^tV\Å�¿�b`_�e®TfZpUbt¿F]8jXTs^tÅf]�^�b�¿F]8VnZpS�^�]~Tf_�]5VnZiUW]NÃ8TsZpU � bt¿F]8_aÃ8Tsjnjn]�^

ωT

e®bt¿F]2jnTf^�Å�]�^�b�¿F]Ó;]�ZF]�ÛFb�b�h2bt¿F]�_a]Nj\jn]�^`_�Ñ

ã�ZIS{h�Z�bt^tT�_db~e¦kl^thfà b�¿F]&Á;hmhfjn]NUßS�T�_a] � ]NÞ}ÀpTqbtV\h�ZÇ÷� e�bt¿F]&TsZpTfj\h�Åfh�Àp_2UW]�^tV\[qTsb�Vn[f]fe

ÉN÷

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btTfâf]NZ�Â�V\b�¿�^t]N_�Á"]~S b�b�hW

eFVX_

∂Vi,T (QT , ωT )

∂ωT

= −b

n(ωT − QT ) +

1

n(aT − bωT ) � ÉNô�

Ü�¿FVX_�]{îWÁF^t]N_t_aVnhfZKS�TfZ�Ó;]�]�V\b�¿F]N^�Á"h}_aV\b�Vn[f]�hf^�ZF]NÅ�Tsb�Vn[f]feWUW]NÁ"]NZpUWVnZFÅ8hfZKÂ�¿F]�b�¿F]N^ωT −

QT

VX_�jnTf^�Å�]�]NZFhfÀFÅ�¿ábth�Ã2Tfâf]�b�¿F]2Û"^t_ab�b�]�^tà UWh�Ã2V\ZpTfZ}bNÑ2gÇ¿F]�Zibt¿F]2]{îWÁF^t]N_t_aVnhfZ×Vn_ZF]NÅ�TqbtV\[�]feWT�^�]~UWÀpS{b�VnhfZ�VnZ

ωT

VX_�_�TG[mVnZFÅ,bt¿F]�Ûp^tÃ2_�kl^thfà bt¿F]�Ã8_�]�jn[f]~_�Ñ�Ü�¿F]�Å�hq[f]�^tZFVnZFÅTsÀFb�¿Fh�^�V\b�Vn]N_°TfS btV\h�Zp_°Ã2TGc�esTf_¯TsZ,VnZpT�UW[f]N^abt]�Z}b¯S�hfZp_�]NÞ}ÀF]NZpS{]�e~Ó"TsVnjfbt¿F]�S�hfÃ2Á;]{b�VnZFÅ�Ûp^�Ã8_hfÀFb©hsk¯b�¿F]NV\^©Áp^�VX_ah�ZF]�^~Ú _�UWV\jn]�Ã2Ã8TFÑ�ò©hqÂ�]�[�]�^~e}Â�¿p]�Z

ωT

kÄTfj\jX_�bth�T2j\hqÂí]�ZFh�ÀFÅf¿�jn]�[f]Nj¤eb�¿p]�ZKb�¿F],TfÓp_�]�ZpS�]$VnZ�S{hfÃ2Á;]�Zp_tTqbtV\h�Z�^t]N_�ÀFj\b�V\Z�jXTsZpUKhqÂ�ZF]N^�^t]�[�]�ZmÀF]~_²kÄTfj\jnVnZFÅpÑ

ã�Z�Á"]N^�VnhWUT − 1

emb�¿F]�[qTfj\ÀF]�hfk�bt¿F]�jnTfZpU�UW]�[�]�jnhfÁFÃ2]NZ�b�Ûp^tà VX_

Vi,T−1 = maxq,s

{

(aT−1 − bQT )q + tT−1s + β

∫ WT−1

QT

Vi,T (~zT )f(w)dw

}

� ÉNÊ�

Â�¿F]N^�]�b�¿F]8_�ÀFÓp_tS{^tV\ÁFbt_$¿pTG[f]�Ó;]�]�Z�UW^thfÁFÁ;]NU&kl^�h�Ã/b�¿p]�Â�V\b�¿pV\ZiÁ"]N^�VnhWU×UF]NS{VX_�V\h�Zá[qTs^tV�ÝTsÓpj\]~_�Ñ�Ü�¿F]�Ûp^`_db©h�^tUW]N^©S{h�ZpUWV\b�VnhfZKklhf^�bt¿F]5S`¿Fh�VnS�]�hsk

seFb�¿p],ZmÀFÃ5Ó;]�^©hfk¯Á;]�^tÃ2V�b`_�_�hfjXU

b�h�bt¿F]�Åfhq[�]�^tZFV\ZpÅ�TsÀFb�¿Fh�^�V\bdcfeWVX_

tT−1 + βVi,T (WT )f(WT−1) + β

∫ WT−1

QT

[

∂Vi,T (w)

∂ωT

− Vi,T (w)f(WT−1)

]

f(w)dw

Â�¿F]N^�]�b�¿p]©hfZFjnc5Tf^�Å�ÀFÃ2]�Z}b�hskVi,T

_�¿FhqÂ�Z�VX_�bt¿pTqb¦klh�^�b�¿F]�_db`Tqb�]�[qTs^tVXTsÓFjn]ωT

Ѧrm]�babtV\ZFÅb�¿pVn_1]NÞ}ÀpTfjpbth5Ù�]N^�h�TsZpU8^t]NTf^�^`TsZpÅfVnZFÅ�ÅfVn[f]N_¦b�¿F]�ÁF^tVnS�]�ÁpTfVnU2klh�^1T5ÀFZFV\b�Â�hf^�b�¿8hfk®Â�Tsb�]�^jnVnS�]�Zp_�]N_�VnZ

T − 1VX_

tT−1 = β

∫ WT

QT

[

(Vi,T (w) − Vi,T (WT−1)) f(WT−1) −∂Vi,T (w)

∂ωT

]

f(w)dw � ÎfÏ�

rmVnZpS{]w ≤ WT−1

TsZpU∂Vi,T (w)/∂ωT < 0

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Ñ°Ü�¿p]1Û"^�ÃéVX_�_a]Nj\jnVnZFÅ�jnVXS{]�Z"_a]~_¯_�ÀF^tÁFjnÀp_°b�h�b�¿F]�hfZp]N_¯ZF]�]~UW]NU5b�h�S{hq[�]�^°b�¿FVX_Á;]�^tV\hWUªÚ _5UW]N[f]Nj\h�ÁFÃ2]�Z}bNe�_ah×S{ÀF^t^�]NZ}b,Á"]N^�VnhWUÕUW]�[�]�jnhfÁFÃ2]NZ�b,VX_�ÀFZpTqæ=]NS{b�]~UªÑ&ò©hqÂ�]�[�]�^~eklÀWbtÀF^�]UW]N[f]Nj\h�ÁFÃ2]�Z}b�TfZpUB_tTsjn]KhfÁFÁ;hf^�b�ÀFZpV�btV\]~_�Â�Vnjnj�Ó;]Tqæ=]NS{b�]NUBÓmc�b�¿F]�ZmÀFÃ�Ó"]N^�hfkjnVnS�]�Zp_�]N_�bt¿F]�Ûp^tà S{h�Z�btV\ZmÀF]~_�b�h2¿Fh�jnUªÑ

gÇV�bt¿&^t]N_�Á"]~S b�b�h2bt¿F]5TfÃ�h�ÀFZ}b©hsk�jXTsZpU�UW]N[f]�jnhfÁ;]NUKVnZT − 1

epb�¿F],Ûp^t_ab�UW]N^�Vn[qTqb�Vn[f]VX_

aT−1−bQT−bq−βVi,T (QT )f(QT )+β

∫ WT−1

QT

[

∂Vi,T (w)

∂Qi,T

+ Vi,T (w)f(QT )

]

f(w)dw

rm]�bab�VnZFÅ�bt¿FVX_²]~Þ�À"TsjFbth,Ù�]N^�h,TfZpU2^t]NTs^t^`TsZFÅ�V\ZFÅ"eGb�¿F]�hfÁFÁ;hf^�b�ÀpZFV�bdc�S{h}_db~e�VnZ2b�]N^�Ã8_¦hfkªb�¿F]S`¿pTfZFÅf]$V\ZKbt¿F]�]{îWÁ;]NS bt]NUK[qTsjnÀF]�hsk¯b�¿F]�Û"^�ÃðVnZ

TeWVX_

β

∫ WT−1

QT

[

(Vi,T (QT ) − Vi,T (w)) f(QT ) −∂Vi,T (w)

∂Qi,T

]

f(w)dw � ÎFÉ

rmVnZpS{]QT ≤ w

TsZpU∂Vi,T /∂ωT < 0

e°b�¿p]�Ûp^`_db�bt]�^tà VnZp_�VnUF]8b�¿F]K_�Þ}ÀpTf^�]2ÓF^`TfS�]N_�VX_Á;h�_�V�btV\[�]fÑ�o©hsbtVnS�]�b�¿"Tqb1Â�V\b�¿�Â�V�bt¿�S{h�Ã�Á;]�Z"_�Tsb�]NU2]�îWÁF^�h�ÁF^tVnTsb�VnhfZ®e

Vi,T (QT ) 6= 0Â�¿F]�Z

Wi,T > Qi,T

Ñ�ã�k∂Vi,T /∂Qi,T

Vn_�ZF]�Å}Tqb�Vn[f]�eFhf^�ZFhsb�b�hmh�jXTs^tÅf]feFb�¿F]NZ�b�¿FVX_�]�îWÁF^�]~_�_�VnhfZVX_�Á"h}_aV\b�Vn[f]�ѯRßZp]�Å�Tsb�Vn[f]�[qTsjnÀF]�Â�hfÀFjXU�Ó"]�VnZpS{h�Zp_�Vn_ab�]NZ�b¦Â�V\b�¿

TÓ;]�VnZFÅ�b�¿F]�jXTf_ab¦Á;]�^tV\hWUªÑ

ã�Z2b�¿p]�TsÓp_�]�Z"S{]�hsk=bt¿F]©^tVn_�â,bt¿pTqb1b�¿F]©Å�hq[f]N^�ZFVnZFÅ�TsÀWbt¿Fhf^tV\bdc5Â�Vnj\j"]{îWÁF^thfÁF^tVnTsb�]�Â�Tqb�]N^¦jnV�ÝS{]NZp_�]N_Neqb�¿F]�]~Þ}ÀFV\[qTfj\]NZ�b�h�ÁFÁ"h�^abtÀFZFV\bdc5S{h}_db�VX_

−β[∂Vi,T (WT−1)/∂Qi,T ]ѦrmÀFÓFb�^`TfS btV\ZpÅ

b�¿pVn_�kl^thfà ]NÞ}ÀpTqbtV\h�Z�ÎFÉ�cmVn]�jXUF_

β

∫ WT−1

QT

[

(Vi,T (QT ) − Vi,T (w)) f(QT ) +

(

∂Vi,T (WT−1)

∂Qi,T

−∂Vi,T (w)

∂Qi,T

)]

f(w)dw

Â�¿FVXS`¿�Â�V\jnj®Ó;]�Á"h}_aV\b�Vn[f]�V\k�b�¿F],_�]NS�hfZpUKUWV\æ;]N^�]NZpS{]�VnZp_�VnUW]�bt¿F]5_tÞ}ÀpTs^t]�ÓF^tT�S{]~_1VX_�]NV�bt¿F]�^Á;h�_�V�btV\[�]�h�^�V�k�ZF]�Å}TqbtV\[�]feªZFhfb$bthmhKjXTs^tÅf]�Ñ Ø hf^�b�¿p]�jnVnZF]NTf^Ne�_�cmÃ2Ã�]�b�^tVnS�]�îWTfÃ2ÁFj\]�eªb�¿pVn__�]NS{h�ZpU�UWV�æ=]�^t]�Z"S{]fe�]�[qTsjnÀpTsb�]NU2Tsb�b�¿F]©]~Þ}ÀFV\jnVnÓF^�VnÀFÃ�efVX_

b(n2−1)(WT−1−w)/n2ѯÜ�¿F]

Ûp^`_db�UWV\æ;]N^�]NZpS{]�]�[qTsjnÀpTsb�]N_�b�hb(w−QT )(WT−1−w)/n2+b(w−QT )(WT−1−QT )/n

ÉNô

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WT−1hq[f]�^�bt¿F]�^tTfZFÅf]$hsk¯b�¿p]$VnZ}b�]NÅf^`Tsj eFS{hfZFÛp^�Ã2VnZFÅ�b�¿F]�^t]N_�ÀFj\bNÑ

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n

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T

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T − 1eFÁF^tV\h�^�b�h8bt¿F]5TfÀWb�¿Fh�^�V\bdc�S`¿Fhmh}_aVnZFÅ2b�¿p]�ÛpZpTsj°Á"]N^�VnhWUKj\]N[f]�j e

WT

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W − WT

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pW

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Ü�¿FVX_�^t]N_�ÀFj\b$Â�Tf_�UW]�[�]�jnhfÁ;]NUáÓ}c�]�îWTfÃ2V\ZFVnZFÅ�b�¿F]2jXTf_ab�bdÂ�h�Á;]�^tV\hWUF_$VnZ×T�UFc}Z"TsÃ2VnSÁF^thfÅ�^tTfÃ�Ã2VnZFÅ2^�]NÁF^t]N_�]�Z}btTsb�VnhfZ®Ñ Ø ÀF^�b�¿p]�^�V\b�]N^tTsb�VnhfZ�VnZpS{^t]NT�_a]~_�b�¿F]�S{h�Ã�Ápj\]�îmV\bdc�hfk�b�¿F]]{îWÁF^t]N_t_�V\h�Zp_�eqÓpÀWb¦ZFhsb�bt¿F]�âf]Nc,^t]N_�ÀFj�b`_�Ñ Ø h�^¦T�Ûpîm]~U2_aÀFÁpÁFj\c�hsk;Â�Tsb�]�^�bt¿pTqb¦VX_�TG[qTsVnjnTfÓFjn]b�h�Ó"]1j\VXS{]NZp_a]~U�Ómc$UF]�[f]Nj\h�Á"]N^t_Ne~Â1]¦]�îWÁ"]~S b�bt¿pTqb�UW]�[�]�jnhfÁFÃ2]NZ�b°Â�V\jnj�hWSNS{ÀF^°Ã�h�^�]²^tTfÁFVnUFj\cÂ�¿F]N^�]�b�¿F]�Â�Tsb�]�^�VX_�Tfj\jnhWS�Tqbt]NU�kl^thfà T×S{hfÃ2Ã2hfZÔÁ;hmhfj e¦VnZ£S{h�Z�bt^tT�_db5b�h×b�¿F]�^tTsb�]�hfkUW]N[f]�jnhfÁpÃ�]NZ}b�Â�¿F]NZ×Â�Tqbt]�^$^�VnÅf¿}bt_�Tf^�]�ÁF^�Vn[qTqbt]�TfZpUá¿p]�jXU&VnZpUW]�Á;]�Z"UW]�Z}b�hsk1b�¿F]NV\^�Àp_�]fÑY©Z"S{]�^�btTfV\Z}bdcÒ]�ZF¿"TsZpS�]N_2bt¿FVn_�]{æ=]NS{bNÑügÇ¿F]�ZÇbt¿F]�^t]&VX_�^tVX_aâBb�¿pTsb�bt¿F]áÂ�Tqbt]�^�_�ÀFÁFÁFjncÂ�Vnj\j$Ó"]ijn]�ÅfVX_�jnTsb�Vn[f]�jncÒ^�]~UWÀpS{]~UßV\Zèb�¿F]×klÀWbtÀF^�]�e�TsZ"UIÂ�Tqb�]N^�VX_�Zphsb�hqÂ�Zp]NUßbtV\jnj�V�bKVn_Àp_�]NUªeFUF]�[f]Nj\h�ÁFÃ2]�Z}b�Ã2TGc2Ó;]$klÀF^abt¿F]�^©T�S�S�]�jn]�^`Tqb�]~UªÑ¯ã�ZKS�hfZ}b�^`Tf_abNe}Â�¿F]�Z�b�¿p]�TfÀWb�¿phf^tV�bdc^t]NUWÀpS�]N_�Â�Tqbt]�^�TG[qTsVnjnTfÓFV\jnV\bdcÓ}cáÓFÀFcmV\ZpÅÂ�Tsb�]�^�^tV\Å�¿}bt_�VnZiT�Ã8Ts^tâf]�bNe®UW]N[f]Nj\h�ÁFÃ2]�Z}b�Vn__�j\hqÂ�]NU�]�[f]NZ�Ã2h�^�]�Ñ

$ %'&#(!)+*��,�

Ü�¿F]©^t]N_�ÀFj\bt_¦UF]�[f]Nj\h�Á"]~U2TsÓ;hq[f]©Ts^t]�Vnj\jnÀp_ab�^`Tqb�]~U�Àp_aVnZFÅ�bt¿F]�klh�j\jnhqÂ�V\ZpÅ�Z}ÀpÃ�]N^�VXS�TfjF]{îFTfÃ�ÝÁFjn]fѦÜ�¿F]�jnVnZF]NTf^©UW]�Ã8TsZ"U�klÀFZ"S b�VnhfZVX_�ÁpTf^tTfÃ�]�b�]N^�VnÙ�]~U�Â�V\b�¿

aT = aT−1 = 30eb = 1

e

β = 1eFTsZpU

n = 2hf^

n = 6Ñ Ø h�^�b�¿F]�bdÂ�h2S�Tf_�]N_�VnZm[fhfjn[mV\ZpÅ�^tVn_�â=emb�¿F],UFVn_ab�^tV\ÓpÀWb�VnhfZKhsk

ÛpZpTfjªÁ"]N^�VnhWU�Tsjnj\hqÂ�TsÓFjn]©bthsb`Tsj®Â�Tqbt]�^�j\VXS{]NZp_�]N_�VX_

f(w) =αeαw

eαWT−1 − eαQT

Â�¿FVXS`¿�¿"Tf_�T�Ã2hfÃ2]�Z}b©Åf]NZF]�^`TqbtV\ZpÅ5klÀFZ"S b�VnhfZ�Å�V\[�]�ZKÓ}c

Mw(t) = E(etw) =

(

α

t + α

) (

e(t+α)WT−1 − e(t+α)QT

eαWT−1 − eαQT

)

Ü�¿F]Ã2hfÃ2]NZ�b�Å�]�ZF]N^tTsb�VnZFÅiklÀFZpS{b�VnhfZI_�V\Ã2ÁFjnV�Û"]N_8]N[GTfj\À"Tqb�VnhfZÒhsk�bt¿F]]{îWÁ;]NS b`TqbtV\h�Zp_8VnZ]NÞ}ÀpTsb�VnhfZ8ÉGø�TsZpU$bt¿F]²Áp^�Vn[qTqb�]1Â�Tsb�]N^®^�VnÅf¿}b�]~Þ}ÀFV\[qTfj\]NZ�b~Ñ°Ü�¿FVX_¯UW]�Zp_�V\bdc�klÀFZ"S b�VnhfZ�¿pTf_®b�¿F]ÁF^thfÁ;]�^�bdc�bt¿pTqb�Tf_

αVn_°V\Z"S{^t]NTf_�]NU®eNbt¿F]�ÁF^�h�ÓpTsÓpV\jnV�bdc$Ã8Tf_t_®VX_¯V\ZpS�^�]~Tf_�V\ZpÅfjnc�S{h�ZpS{]NZ}b�^`Tqb�]~U

ÎWÉ

Page 22: ZBb ¿F] ¿FVX ab hf^tcÔhsk ¿}ÀpÃ2TfZÒ ]{b b jn]Ã2]Z}b8VnZneumann.hec.ca/cree2005/janmaat.pdf · V\jnjn]^~e¯ÎsÏ ÏFÉÐ Ñ gÇV bt¿Ôb ¿F]KTs^t^ Vn[qTsj¦hfkÖ¦ÀF^thfÁ;]NTsZÕ_

hfZWT−1

Ñ�ã�Z&b�¿F]�j\VnÃ2V�b~e®Tf_α → ∞

e"bt¿F]2UWVn_ab�^tVnÓFÀWb�VnhfZ×UW]�Å�]�ZF]N^tTsb�]N_©b�h�b�¿F]2S{]N^ab`TsVnZS�T�_a]�Â�V\b�¿

WT = WT−1e�Â�¿pV\jn]&Tf_

α → 0e�b�¿FVX_�UWVX_ab�^tV\ÓFÀFb�VnhfZÇS{h�Z}[�]�^tÅf]~_�h�ZÒb�¿F]

ÀFZFV\klhf^tÃ/UWVX_dbt^�VnÓFÀWbtV\h�Z®Ñ ¾ TfZFVnÁFÀFjXTqb�VnhfZhfkα

bt¿F]�^t]{klh�^�]�Tsjnj\hq©_�TsZ&]N[GTfj\À"Tqb�VnhfZhfk¦¿FhqÂÓ;]�¿pTG[mVnhf^�Vn_�Tsæ;]~S bt]NUKÓmc�S`¿pTsZpÅf]N_�VnZKb�¿F],UWVX_ab�^tV\ÓFÀFb�VnhfZ®Ñ

Ø V\Å�ÀF^�]Îi_a¿phq©_,bt¿F]�]{îFTsÃ2ÁFjn]�^�]~_aÀFj\bt_NÑÕã�ZÒÁpTsZp]�jX_ � T��TsZ"U � Ó- e1b�¿F]�]�îmÁ;]NS{b�]~U[qTsjnÀF]N_5hsk

∂Vi,T−1/∂qi,t

Tf^�]8Ápj\hfbab�]~UªÑgÇ¿F]NZn = 2

e°bt¿F]�S�ÀF^t[f]N_5Ts^t]�_dbt]�]�Á;]�^�TsZpUVnZ�bt]�^`_a]~S b$b�¿F]2¿Fh�^�VnÙ�h�Z}btTsj¯Ù�]N^�hKjnV\ZF]8Tsb�jnhqÂ�]�^$j\]N[f]�jX_$hsk

qT−1b�¿"TsZiÂ�¿F]�Z

n = 6Ñ5ã�Z

kÄTfS{bNe�V�kn = 7

efb�¿F]�h�Á"]NZ2T�S�S�]N_t_�S{ÀF^t[f]~_�Â1h�ÀFjnU�ZFhfb²V\Z}bt]�^`_a]~S b�b�¿F]�¿Fhf^tV\ÙNhfZ}btTfjmÙ�]�^th�j\VnZF]Â�V\b�¿FVnZKb�¿F],^`TsZpÅf]$hsk�TG[GTfV\jXTsÓpj\]�Â�Tqb�]N^NѦÜ�¿pVn_�UW]NÃ2hfZp_ab�^`Tqbt]N_�b�¿"Tqb©VnZpS{^t]NT�_aVnZFÅ

n_�¿FV\kçbt_

UW]N[f]�jnhfÁpÃ�]NZ}b5b�háb�¿F]K]NTf^�jnVn]�^5Á;]�^tVnhmU®e²TsZpUÕÂ�V�bt¿BjXTs^tÅf]�]NZFhfÀpÅf¿ne¦Tfj\j�UW]N[f]�jnhfÁpÃ�]NZ}b

hWS�S�ÀF^t_�]NTf^�jnV\]N^NÑ�Ü�¿F]�TfÃ5ÓFVnÅfÀFV\bdcKhsk�b�¿p]�VnÃ2ÁpTfS{b�hsk¦ÀpZpS{]N^ab`TsVnZ�bdcKhfZ&b�¿F]�hfÁ;]�ZáT�S�S{]~_�_S�T�_a]�VX_�TsjX_ahKUW]�Ã2hfZ"_dbt^tTsb�]NU×VnZábt¿F]N_�]2ÁpTsZF]Njn_NÑ5ã�ZiÁpTfZF]�j � T� e®b�¿F] ∂Vi,T−1/∂qi,t

jnV\ZF]klhf^�b�¿p]�h�Á"]NZ×T�S�S{]~_�_�Â�V�bt¿×^tVX_aâS�T�_a]�Vn_�Tfj\Ã2h�_ab�]N[f]N^�cmÂ�¿F]N^�],Ó;]�jnhqÂébt¿F]�h�Á"]NZ×T�S�S{]~_�_Â�V\b�¿Fh�ÀWb°^tVn_�â�S�Tf_�]fÑ�ã�Z,ÁpTs^�b�VXS{ÀpjnTf^NeNb�¿FVX_¯Vn_®b�^tÀF]1Tsb�bt¿F]²¿Fh�^�VnÙ�h�Z}btTsjfÙ�]�^th©jnV\Zp]feGV\Z"UWVnSNTqbtV\ZFÅb�¿"Tqb�Â�V\b�¿�bdÂ�h2Ûp^tÃ2_Nep^tVX_aâ � Â�V\b�¿

α = 1�_�jnhq©_�S{ÀF^t^�]NZ}b�UW]�[�]�jnhfÁFÃ2]�Z}b~Ñ1ã�Z&S�hfZ}b�^`Tf_abNe

ÁpTfZF]�j � Ó-�_a¿Fhq©_�b�¿pTsb©klhf^n = 6

eFbt¿F]�^t],VX_�T�_�Ã8Tsjnj\]N^©^tTfZFÅf]�Â�¿p]�^t]�b�¿F]5hfÁ;]�Z&TfSNS{]~_�_Â�V\b�¿�^�VX_aâ�jnVnZF]�Vn_�Ó;]�jnhqÂÔb�¿p]�h�Á"]NZ�T�S�S�]N_t_°Â�V\b�¿phfÀWb�^tVn_�â�j\VnZF]�ѯR�b�b�¿p]�V\Z}b�]N^t_�]NS{b�VnhfZ�Â�V�bt¿b�¿p]�¿Fhf^tV\ÙNhfZ}btTfj=Ù�]�^th�j\VnZF]fep^�VX_aâ � Â�V\b�¿

α = 1�ZFhqÂIVnZpS{^t]NT�_a]~_�S{ÀF^t^�]NZ}b�UF]�[f]Nj\h�ÁFÃ2]�Z}bNÑ

Ì�TsZF]Njn_ � S®TsZ"U � U.°_�¿FhqÂib�¿F]1V\Ã2ÁpT�S b°hfkFV\Z"S{^t]NTf_�VnZFÅαѯã�Z5Tfj\jmS�T�_a]~_�eGTf_

αÓ;]NS�hfÃ2]N_

jXTs^tÅf]fembt¿F],[qTsjnÀF]~_�hskqT−1

S�hfZm[f]N^�Å�]�h�ZKb�¿F]5SNTf_�]N_�Â�V\b�¿Fh�ÀWb�^tVn_�â=Ñ1Ü�¿FVX_�VX_�S�hfZp_�Vn_ab�]NZ}bÂ�V\b�¿Kb�¿F],kÄTfS b�bt¿pTqb�T�_

αVX_�VnZpS{^t]NT�_a]~Uªemb�¿F]�UWVn_ab�^tVnÓFÀWb�VnhfZTfÁFÁF^th�TfS`¿p]N_�S{]N^ab`TsVnZ�bdc�Â�V�bt¿

WT = WT−1Ñ�ã�Z×ÁpTsZF]Nj � S/{eªÂ�¿F]�^t],bt¿F]�^t]�Tf^�]

n = 2Ûp^tÃ8_�e

qT−1VX_�Ó;]�jnhqÂõb�¿p]�ZFh

^tVn_�â,S�T�_a]�Ó;hsb�¿2Â�V\b�¿2hfÁ;]�Z2TfSNS{]~_�_�TsZ"U5ÁF^tVn[GTsb�]�ÁF^thfÁ;]�^�bdc,Â�Tqb�]N^�j\VXS{]NZp_a]~_�Ñ�Ü�¿F]�VnÃ�Á"TfS bVX_�ÁpTs^�b�VXS{ÀFjXTs^tjnc8ÁF^�h�ZFhfÀpZpS{]~U�klh�^�b�¿F]�ÁF^tVn[GTsb�]�ÁF^thfÁ;]�^�bdc�S�T�_a]�eWÂ�¿F]�^t]$b�¿F]�]~Þ�ÀpV\jnV\Óp^�VnÀFÃ

qT−1[qTfj\ÀF]8hfk$É~ÏKVn_�h�ZFjncá÷fÏ10�hsk�b�¿p]8]NÞ}ÀFVnj\VnÓF^tV\ÀFà [qTsjnÀF]8Â�V�bt¿FhfÀFb,^�VX_aâ=e1ÉN÷FÑ ÷}ËWÑ2ã�Z

b�¿p]�TsÓ"_a]NZpS{]�hsk¦^tVn_�â=e"h�Á"]NZáTfSNS{]N_t_©VnZpS�^�]~Tf_�]N_T − 1

UW]�[�]�jnhfÁFÃ2]�Z}b�kl^�h�à É~÷FÑ ÷}Ë�btháÉNôÀFZFV\bt_NѦgÇ¿F]NZ�b�¿F]N^�]�Tf^�]

n = 6Ûp^tÃ8_�eFb�¿F],UWVn^t]NS btV\h�Z�hfk°bt¿F],^tVn_�â�]�æ;]~S b�S`¿pTfZFÅf]~_�hq[f]�^

ÎfÎ

Page 23: ZBb ¿F] ¿FVX ab hf^tcÔhsk ¿}ÀpÃ2TfZÒ ]{b b jn]Ã2]Z}b8VnZneumann.hec.ca/cree2005/janmaat.pdf · V\jnjn]^~e¯ÎsÏ ÏFÉÐ Ñ gÇV bt¿Ôb ¿F]KTs^t^ Vn[qTsj¦hfkÖ¦ÀF^thfÁ;]NTsZÕ_

16 18 20 22 24

−10

−50

510

a) Marginal Return, n = 2

Development qT−1

∂Vi,

T−1

∂qi,

T−1

open, riskopen, no riskprivate, no riskprivate, risk

1015

2025

10−4 10−3 0.01 0.1 1 10

c) Eq. Development, n = 2

Distribution α

qT

−1

16 18 20 22 24

b) Marginal Return, n = 6

Development qT−1

10−4 10−3 0.01 0.1 1 10

d) Eq. Development, n = 6

Distribution α

Ø V\Å�ÀF^t]�Î2� ¾ Tf^�Å�V\Z"Tsj�^t]{btÀF^tZ&TsZpU]~Þ}ÀFV\jnVnÓF^�VnÀFÃT − 1

UF]�[f]Nj\h�ÁFÃ2]�Z}bNÑ Ø h�^�ÁpTfZF]�jX_ � T�TsZ"U � Ó- e α = 1Ñ Ø hf^©Tfj\j®SNTf_�]N_Ne

QT−1 = 1TfZpU

WT−1 = 26Ñ

Îsó

Page 24: ZBb ¿F] ¿FVX ab hf^tcÔhsk ¿}ÀpÃ2TfZÒ ]{b b jn]Ã2]Z}b8VnZneumann.hec.ca/cree2005/janmaat.pdf · V\jnjn]^~e¯ÎsÏ ÏFÉÐ Ñ gÇV bt¿Ôb ¿F]KTs^t^ Vn[qTsj¦hfkÖ¦ÀF^thfÁ;]NTsZÕ_

Ã2h�_ab�hfk�bt¿F]�^`TsZpÅf]$hskα

[qTsjnÀF]~_�Ñ�o©hqÂè^�VX_aâ8VnZpS�^�]~Tf_�]N_�UW]�[�]�jnhfÁFÃ2]�Z}b�V\ZKÁ;]�^tV\hWUT − 1

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