quantity constrained equilibria · ti 2001-116/1 tinbergen institute discussion paper quantity...

31
TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings 1 Gerard van der Laan 2 Dolf Talman 3 1 Department of Economics, Universiteit Maastricht, 2 Department of Econometrics, Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam, and Tinbergen Institute, 3 Department of Econometrics & Operations Research, and CentER, Tilburg University

Upload: others

Post on 01-May-2020

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

TI 2001-116/1Tinbergen Institute Discussion Paper

Quantity Constrained Equilibria

P. Jean-Jacques Herings1

Gerard van der Laan2

Dolf Talman3

1 Department of Economics, Universiteit Maastricht,2 Department of Econometrics, Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam, and Tinbergen Institute,3 Department of Econometrics & Operations Research, and CentER, Tilburg University

Page 2: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

Tinbergen Institute The Tinbergen Institute is the institute for economic research of the Erasmus Universiteit Rotterdam, Universiteit van Amsterdam and Vrije Universiteit Amsterdam. Tinbergen Institute Amsterdam Keizersgracht 482 1017 EG Amsterdam The Netherlands Tel.: +31.(0)20.5513500 Fax: +31.(0)20.5513555 Tinbergen Institute Rotterdam Burg. Oudlaan 50 3062 PA Rotterdam The Netherlands Tel.: +31.(0)10.4088900 Fax: +31.(0)10.4089031 Most TI discussion papers can be downloaded at http://www.tinbergen.nl

Page 3: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

"�@?|�|) �L?t|h@�?i_ ,^��*�Mh�@��� ai@?�a@U^�it Oih�?}t5 Bih@h_ �@? _ih w@@?6 #L*u A@*4@?7

�L�i4Mih 2Hc 2ff�

�Wklv uhvhdufk lv sduw ri wkh uhvhdufk surjudp _Frpshwlwlrq dqg Frrshudwlrq%12S1M1M1 Khulqjv/ Ghsduwphqw ri Hfrqrplfv/ Xqlyhuvlwhlw Pddvwulfkw/ S1R1 Er{ 949/ 9533 PGPddvwulfkw/ Wkh Qhwkhuodqgv/ S1KhulqjvCdojhf1xqlpddv1qo�J1 ydq ghu Oddq/ Ghsduwphqw ri Hfrqrphwulfv dqg Wlqehujhq Lqvwlwxwh/ Yulmh Xqlyhuvlwhlw/Gh Erhohoddq 4438/ 43;4 KY Dpvwhugdp/ Wkh Qhwkhuodqgv/ joddqChfrq1yx1qoeD1M1M1 Wdopdq/ Ghsduwphqw ri Hfrqrphwulfv ) Rshudwlrqv Uhvhdufk dqg FhqwHU/ WloexujXqlyhuvlw|/ S1R1 Er{ <3486/ 8333 OH Wloexuj/ Wkh Qhwkhuodqgv/ wdopdqCnxe1qo

Page 4: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

�Mt|h@U|W? @ t|@?_@h_ }i?ih@* i^��*�Mh��4 4L_i* �| �t @tt�4i_ |�@| |�ihi @hi ?L Th�Ui hit|�U|�L?t @?_|�@| Th�Uit @_��t| �?�?�|i*) u@t| |L |�i�h i^��*�Mh��4 �@*�it� �h�Ui h�}�_�|�it 4@) U@�ti |�@|@ UL4Ti|�|��i i^��*�Mh��4 U@??L| Mi @||@�?i_c @?_ h@|�L?�?} L? ?i| _i4@?_t Lh t�TT*�it�t ?ii_i_ |L U*i@h |�i 4@h!i|t� ��i?@tt)c #hi3i @?_ vL�?it ThL�i_ �? |�i 4�_ �b.ft |�@||�ihi i �t| i^��*�Mh�@ ��|� h@|�L?�?}� �LhiL�ihc h@|�L?�?} �t t�U� |�@| ?L| ML|� _i4@?_ @?_t�TT*) Lu @ }LL_ @hi h@|�L?i_ t�4�*|@?iL�t*)c @| *i@t| L?i UL44L_�|) �t ?L| h@|�L?i_ @| @**c@?_ |�ihi �t h@|�L?�?} L? |�i ?i| t�TT*) Lh ?i| _i4@?_ Lu @ }LL_ L?*) �u |�i Th�Ui Lu |�@|}LL_ �t L? �|t *L�ih Lh �TTih ML�?_c hitTiU|��i*)� W? |�i �bHft _�ti^��*�Mh��4 4L_i*t ��|�L?*) h@|�L?�?} L? ?i| t�TT*�it �ihi �?|hL_�Ui_� W? @** |�iti 4L_i*t Th�Uit @hi hit|h�U|i_ M)TLt�|��i *L�ih @?_ �TTih ML�?_t�W? |��t T@Tih |�i ti| Lu @_4�tt�M*i Th�Uit �t @**L�i_ |L Mi @? @hM�|h@h) UL?�i ti|�6Lh t�U� @? @hM�|h@h) ti| �| U@??L| Mi }�@h@?|ii_ |�@| |�ihi i �t|t @ UL?t|h@�?i_ i^��*�M�h��4 t@|�tu)�?} |�@| @ Th�Ui ��** Mi L? �|t �TTih Lh *L�ih ML�?_ �? U@ti Lu h@|�L?�?}� i�?|hL_�Ui @ 4Lhi }i?ih@* i^��*�Mh��4 UL?UiT|c U@**i_ "�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4E"�,�� �| t�U� @? i^��*�Mh��4 |�i *i�i*t Lu t�TT*) @?_ _i4@?_ h@|�L?�?} @hi UL4T*i|i*)_i|ih4�?i_ M) |�i UL4TL?i?|t Lu @ _�hiU|�L? �? ���U� |�i Th�Ui t)t|i4 U@??L| Mi 4L�i_u�h|�ih ��|�L�| *i@��?} |�i ti| Lu @_4�tt�M*i Th�Uit� `�i? |�i ti| �t UL4T@U|c �i t�L�|�i i �t|i?Ui Lu @ UL??iU|i_ ti| Lu "�,tc UL?|@�?�?} |�L |h���@* ?L�|h@_i i^��*�Mh�@� �Lhi�L�ihc |�i ti| UL?|@�?t uLh i�ih) UL44L_�|) @ }i?ih@*�3i_ #hi3i i^��*�Mh��4c Mi�?} @ "�,@| ���U� |��t UL44L_�|) �t ?L| Mi�?} h@|�L?i_c @?_ @*tL @ }i?ih@*�3i_ t�TT*)�UL?t|h@�?i_i^��*�Mh��4 ��|�L�| _i4@?_ h@|�L?�?}� i @TT*) |��t 4@�? hit�*| |L ti�ih@* tTiU�@* U@titc�?U*�_�?} |�i U@ti Lu @? �?ML�?_i_ ti| Lu @_4�tt�M*i Th�Uit�ki) �Lh_tG i U�@?}i iUL?L4)c Th�Ui h�}�_�|�itc i^��*�Mh��4c h@|�L?�?}�a,w�UL_iG �S2c �S�c �SHc #D��

Page 5: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

� W?|hL_�U|�L?�ihuiU| UL4Ti|�|�L? �t @ M@t�U @tt�4T|�L? �? iUL?L4�U |�iLh)� �** @}i?|t @hi @tt�4i_ |LMi Th�Ui |@!iht @?_ |�ihiuLhi i Thitt |�i�h _i4@?_ @?_ t�TT*) @| |�i Thi�@�*�?} Th�Uit L?|�i 4@h!i|� Ah@_i |@!it T*@Ui L?*) @| @ Th�Ui t)t|i4 uLh ���U� uLh i�ih) UL44L_�|) |L|@*_i4@?_ i^�@*t |L|@* t�TT*)� W| �t @tt�4i_ |�@| |�ihi �t ?L hit|h�U|�L? L? |�i Th�Uit @?_|�@| Th�Uit @_��t| �?�?�|i*) u@t|� @*h@t E�H.e� UL?t�_ihi_ |�i ThLM*i4 Lu |�i i �t|i?Ui Lu@ }i?ih@* UL4Ti|�|��i i^��*�Mh��4c ULhhitTL?_�?} |L @ Th�Ui t)t|i4 @| ���U� @** 4@h!i|tU*i@h t�4�*|@?iL�t*)� N?_ih �ih) }i?ih@* UL?_�|�L?t |�i i �t|i?Ui Lu t�U� @ @*h@t�@?i^��*�Mh��4 �@t Mii? t�L�? M) �hhL� @?_ #iMhi� E�bDe��N?i4T*L)4i?| @?_ i Uitt t�TT*) L? UL44L_�|) @?_ u@U|Lh 4@h!i|t @hi @TT@hi?|*)tih�L�t ThLM*i4t �? 4@?) UL�?|h�it� A�i *�|ih@|�hi �@t �_i?|��i_ @ �@h�i|) Lu hi@tL?t ��)Th�Uit 4�}�| Mi Thi�i?|i_ uhL4 @_��t|�?} |L @*h@t�@? �@*�it� �L*�|�U@**) _it�hi_ Th�Uihit|h�U|�L?t @?_ hi}�*@|�L?tc t�U� @t Th�Ui UL?|hL*t |L hi_�Ui �?�@|�L?c 4�?�4�4 �@}itc LhTh�Ui �?_i @|�L? @hi 4i?|�L?i_ M) �L E�bHf�c �}�)i? @?_ `�@**i) E�bHSc �bbf�c B�?t�M�h}� @?_ V@? _ih Oi)_i? E�bHH�c Oih�?}t E�bb.� @?_ A��?t|h@ E2fff�� ��i?@tt) E�bb��UL?t�_iht ?L?�UL4Ti|�|��i Th�Ui t)t|i4t hit�*|�?} uhL4 4L_i*t ��|� �4TihuiU| UL4Ti|�|�L?�`�i? 4@h!i|t @hi �?UL4T*i|ic Th�Ui hi}�*@|�L?t 4@) Mi �ti_ |L }i?ih@|i �@hi|L �4ThL�i�4i?|t @t t|hitti_ M) #hi3i @?_ BL**�ih E�bb��c Oih�?}t @?_ �L*i4@hU�@!�t E2fff�c @?_#hi3i E2ff���`�i? |h@_i LUU�ht @}@�?t| _�ti^��*�Mh��4 Th�Uitc 4@h!i|t U*i@h |�hL�}� ^�@?|�|)@_��t|4i?|tc i�}� M) �4TLt�?} ^�@?|�|) h@|�L?�?}c ^�i��?} t)t|i4t Lh ThL_�U|�L? ^�L|@�W? |�i 4�_ �b.ft ��i?@tt) E�b.D�c #hi3i E�b.D� @?_ vL�?it E�b.D� �?_iTi?_i?|*) _i�i*�LTi_ 4@h!i| U*i@h�?} 4iU�@?�t4t uLh iUL?L4�it ��|� Th�Ui hit|h�U|�L?t M) �t�?} ^�@?|�|)h@|�L?�?}� W? |�@| @TThL@U�c @? @}i?| U�LLtit @ 4Lt| Thiuihhi_ UL?t�4T|�L? M�?_*ic t�M��iU| |L ML|� @ M�_}i| UL?t|h@�?| @?_ ^�@?|�|) UL?t|h@�?|t L? ?i| _i4@?_t @?_ t�TT*�it�AL i Thitt |�i |h@?tT@hi?U) Lu 4@h!i|tc ^�@?|�|) h@|�L?�?} �t ?L| @**L�i_ |L @giU| i Uittt�TT*) @?_ i Uitt _i4@?_ t�4�*|@?iL�t*)� `�i? Th�Ui h�}�_�|�it @hi uLh4�*@|i_ M) *L�ih@?_ �TTih ML�?_t L? |�i Th�Uit Lu |�i UL44L_�|�itc #hi3i E�b.D� ThL�i_ |�i i �t|i?Ui Lu@? i^��*�Mh��4 @| ���U� @| *i@t| L?i @ Th�Lh� U�Lti? UL44L_�|) �t ?L| Mi�?} h@|�L?i_ @| @**�6�h|�ih t�TT*) E_i4@?_� h@|�L?�?} L? @ UL44L_�|) U@? L?*) Mi M�?_�?} ��i? �|t Th�Ui �tL? �|t *L�ih E�TTih� ML�?_� 5�U� @? i^��*�Mh��4 �t U@**i_ @ #hi3i i^��*�Mh��4�W? |�i �bHft ML|� �@? _ih w@@? @?_ k�h3 @h}�i_ |�@| �? Th@U|�Ui h@|�L?�?} L? ?i|_i4@?_ �t h@hi*) LMtih�i_ @?_ _��U�*| |L �4T*i4i?|� A��t 4L|��@|i_ |�iti @�|�Lht |LUL?t�_ih i^��*�Mh�@ ��|� L?*) h@|�L?�?} L? |�i ?i| t�TT*�it� W? �@? _ih w@@? E�bHfc �bH2�@?_ k�h3 E�bH2� |�i i �t|i?Ui Lu @? i^��*�Mh��4 ��|� L?*) h@|�L?�?} L? |�i ?i| t�TT*�it@?_ @| *i@t| L?i UL44L_�|) ?L| h@|�L?i_ @| @** �@t Mii? ThL�i?� 5�U� @? i^��*�Mh��4 �tU@**i_ @ t�TT*)�UL?t|h@�?i_ Lh �?i4T*L)4i?| i^��*�Mh��4�

Page 6: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

W? Oih�?}t E�bbH� |�i i?|�hi ti| Lu i^��*�Mh�@c U@**i_ UL?t|h@�?i_ i^��*�Mh�@c �@t Mii?U�@h@U|ih�3i_� A�ihi i �t|t @ UL??iU|i_ ti| Lu UL?t|h@�?i_ i^��*�Mh�@c UL?|@�?�?} |�L |h����@* i^��*�Mh�@� W? L?i |h���@* i^��*�Mh��4c |�i |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4c @**UL44L_�|�it @hi u�**) h@|�L?i_ �? |�i�h t�TT*) @?_ @** Th�Uit @hi L? |�i�h *L�ih ML�?_c��ihi@t @| |�i L|�ih |h���@* i^��*�Mh��4c |�i |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4c @**UL44L_�|�it @hi u�**) h@|�L?i_ �? |�i�h _i4@?_ @?_ @** Th�Uit @hi L? |�i�h �TTih ML�?_�A�i UL??iU|i_ ti| @*tL UL?|@�?t uLh i�ih) UL44L_�|) @ UL?t|h@�?i_ i^��*�Mh��4 @| ���U�|�@| UL44L_�|) �t ?L| Mi�?} h@|�L?i_ @| @**� W? T@h|�U�*@hc |��t �4T*�it |�i i �t|i?Ui Lu@ #hi3i i^��*�Mh��4� �LhiL�ihc L?i Lu |�iti i^��*�Mh�@ �t @ t�TT*)�UL?t|h@�?i_ i^��*�M�h��4 @?_ @?L|�ih L?i @ _i4@?_�UL?t|h@�?i_ i^��*�Mh��4� W| �t TLtt�M*i |L UL4T�|i |��tUL??iU|i_ ti| Lu i^��*�Mh�@ M) �t�?} |�i @*}Lh�|�4 Thiti?|i_ �? Oih�?}tc A@*4@? @?_ v@?}E�bbS�� 6Lh @? �?|ihThi|@|�L? Lu |�iti hit�*|t �? |ih4t Lu ti*u�u�*�**�?} ULLh_�?@|�L? u@�*�hitc@t �i** @t uLh @? i |i?t�L? |L iUL?L4�it ��|� ThL_�U|�L?c tii #hi3i E�bb.� @?_ ��|@??@c�hitc #hi3ic Oih�?}t @?_ V�**@?@UU� E2ff���W? #i�i3 @?_ #hi3i E�bHe� @?_ �@? _ih w@@? E�bHe�c iUL?L4�it ��|� Th�Ui �?_i @�|�L?t @hi UL?t�_ihi_� W| �@t Mii? t�L�? |�@| |�ihi i �t|t @ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4��|�L�| h@|�L?�?} L? |�i ?�4ih@�hi UL44L_�|)c ��i? |�i ti| Lu ui@t�M*i Th�Uit @**L�t uLhi?L�}� �i �M�*�|) Lu |�i Th�Ui *i�i* Lu |�i ?L?�?�4ih@�hi UL44L_�|�it� i__iTL�* E�bH.�_�tU�ttit i |i?t��i*) *�?!@}it Mi|�ii? Th�Uit |�hL�}� �?_i u�?U|�L?t @?_ t�L�t |�@| @t�TT*) UL?t|h@�?i_ i^��*�Mh��4 �? ���U� @| *i@t| L?i UL44L_�|) �t ?L| h@|�L?i_ i �t|t �u|�i t)t|i4 Lu �?_i it �t ?L?�U�hU�*@hc ��i� ?L UL44L_�|) �t _�hiU|*) Lh �?_�hiU|*) �?_i i_M) �|ti*u� W? |�iti T@Tiht �| �t @*tL @h}�i_ |�@| |�i i^��*�Mh��4 UL?_�|�L? |�@| h@|�L?�?} Lu@ UL44L_�|) �t ?L| @**L�i_ �?*itt �|t Th�Ui hit|h�U|�L? �t M�?_�?}c U@??L| Mi 4@�?|@�?i_��i? Th�Uit @hi |�i_ |L i@U� L|�ih |�hL�}� �?_i u�?U|�L?t�W? |��t T@Tih �i UL?t�_ih |�i }i?ih@* U@ti ��ihi Th�Ui hit|h�U|�L?t *i@_ |L @? @h�M�|h@h) UL?�i ti| Lu ?L??i}@|��i Th�Uit� A��t ti| UL�*_ Mi @ TL�?| @t �? |�i � �Th�Ui*�|ih@|�hic @ U�Mi ��|� *L�ih @?_ �TTih ML�?_t uLh |�i Th�Uit @t �? |�i 4L_i* Lu #hi3iE�b.D�c @ t*�Ui Lu @ UL?�i UL?i �? U@ti Lu Th�Ui �?_i @|�L? @t �? |�i 4L_i*t Lu #i�i3@?_ #hi3i E�bHe� @?_ i__iTL�* E�bH.�� 6Lh |�i }i?ih@* U@ti �| U@??L| Mi @tt�hi_ |�@||�ihi �t L?*) h@|�L?�?} uLh @ UL44L_�|) �u �|t Th�Ui �t i^�@* |L �|t 4�?�4�4 Lh 4@ �4�4�@*�i� A�ihiuLhi �i }i?ih@*�3i |�i i^��*�Mh��4 UL?UiT| |L |�i L?i Lu "�@?|�|) �L?t|h@�?i_,^��*�Mh��4� �| t�U� @? i^��*�Mh��4c UL44L_�|�it 4@) Mi h@|�L?i_ L?*) �? U@ti |�i Th�Uit)t|i4 *�it L? |�i ML�?_@h) Lu |�i ti| Lu @_4�tt�M*i Th�Uit� +@|�L?�?} �t _i|ih4�?i_ M) @_�hiU|�L? �? ���U� |�i Th�Uit U@??L| @_��t|c ��i� M) @ �iU|Lh �? |�i ?Lh4@* UL?i Lu |�i ti|Lu @_4�tt�M*i Th�Uit� � ?i}@|��i UL4TL?i?| Lu |��t �iU|Lh �?_�Uit t�TT*) h@|�L?�?} L? |�ihi*@|i_ UL44L_�|)c @?_c *�!i��tic @ TLt�|��i UL4TL?i?| �?_�Uit _i4@?_ h@|�L?�?}�i ��** t�L� |�@| �? U@ti |�i ti| Lu Th�Uit �t UL4T@U| @?_ L?*) UL?|@�?t TLt�|��i Th�Ui2

Page 7: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

�iU|Lhtc |�ihi i �t|t @ UL??iU|i_ ti| Lu "�@?|�|) �L?t|h@�?i_ ,^��*�Mh�@ E"�,� UL?|@�?�?}|�L |h���@* i^��*�Mh�@c @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 ��|� u�** h@|�L?�?} L? |�it�TT*) Lu @** UL44L_�|�it @?_ Th�Uit t�U� |�@| |�i �@*�i Lu |�i |L|@* �?�|�@* i?_L�4i?|t �t4�?�4�3i_c @?_ @ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4 ��ihi @** UL44L_�|�it @hi u�**)h@|�L?i_ �? |�i�h _i4@?_ @?_ |�i �@*�i Lu |�i |L|@* �?�|�@* i?_L�4i?|t �t 4@ �4�3i_�A�i UL??iU|i_ ti| @*tL UL?|@�?t uLh i�ih) UL44L_�|) @ }i?ih@*�3i_ #hi3i i^��*�Mh��4cMi�?} @ "�, @| ���U� |�@| UL44L_�|) �t ?L| Mi�?} h@|�L?i_ @| @**c @ }i?ih@*�3i_ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4c @| ���U� |�ihi �t ?L _i4@?_ h@|�L?�?} @?_ @| *i@t| L?i UL44L_�|)�t ?L| h@|�L?i_ @| @**c @?_ @ }i?ih@*�3i_ _i4@?_�UL?t|h@�?i_ i^��*�Mh��4c @| ���U� |�ihi�t ?L t�TT*) h@|�L?�?} @?_ @| *i@t| L?i UL44L_�|) �t ?L| h@|�L?i_ @| @**� A��t i �t|i?Uihit�*| �t ThL�i_ M) �t�?} |�i � i_ TL�?| |�iLhi4 Lu �hL�_ih E�bSf�� A�i 4@�? hit�*| �t@TT*�i_ |L ti�ih@* tTiU�@* U@tit�i @*tL UL?t�_ih U@tit �? ���U� |�i ti| Lu @_4�tt�M*i Th�Uit �t ?L| ML�?_i_ LhUL?|@�?t 3ihL Th�Uitc @**L��?} uLh 4L_i*t �? ���U� tL4i Th�Uit U@? �@h) uhii*) @?_ L|�iht@hi |�i_ M) �?_i t)t|i4t� W? U@ti |�i ti| Lu Th�Uit �t �?ML�?_i_c |�i UL??iU|i_ ti| Lu"�,t �t @*tL �?ML�?_i_ ��|� tL4i Lu |�i Th�Uit }L�?} |L �?�?�|)c ���*i t�4�*|@?iL�t*)uLh @?) UL44L_�|) ��|� Th�Ui ML�?_i_ uhL4 @ML�ic �|t hi*@|��i Th�Ui |i?_t |L 3ihL @?_i�i?|�@**) |�i UL44L_�|) MiUL4it u�**) UL?t|h@�?i_ �? �|t _i4@?_c �4T*)�?} ?L |h@_i uLh|��t UL44L_�|)� W? |�i *�4�| |�i iUL?L4) hi_�Uit |L @? iUL?L4) ��|� |h@_i L?*) �? |�iUL44L_�|�it ��|� Th�Uit �?ML�?_i_ uhL4 @ML�i� `�i? |�i Th�Uit Lu |�iti UL44L_�|�it @hi?L| |�i_ |L Th�Uit Lu L|�ih UL44L_�|�itc |�iti Th�Uit |i?_ |L @*h@t�@? i^��*�Mh��4 �@*�ituLh |��t hi_�Ui_ iUL?L4)� W? U@ti |��t �L*_t uLh @** UL44L_�|�itc |�ihi �t @ UL??iU|i_ti| Lu "�,t *i@_�?} uhL4 |�i |h���@* u�** t�TT*) UL?t|h@�?i_ i^��*�Mh��4 |L @ @*h@t�@?i^��*�Mh��4 ��|�L�| h@|�L?�?}�A�i T@Tih �@t Mii? Lh}@?�3i_ @t uL**L�t� 5iU|�L? 2 _itUh�Mit |�i 4L_i* @?_ �?|hL�_�Uit |�i }i?ih@* UL?UiT| Lu "�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4� 6Lh |�i UL4T@U|c UL?�i U@ti ��|� TLt�|��i Th�Uit |�i i �t|i?Ui hit�*|t @hi }��i? �? 5iU|�L? �� 5TiU��U U@tit @hi_�tU�tti_ �? 5iU|�L? ec @t �i** @t i |i?t�L?tc �?U*�_�?} |�i �?ML�?_i_ U@ti�2 A�i 4L_i*i UL?t�_ih @? i U�@?}i iUL?L4) H ' Ei[ �c��c ��j6�'�c � �� W? |��t iUL?L4) |�ihi @hi6 UL?t�4ihtc �?_i i_ � ' �c � � � c6c @?_ ? UL44L_�|�itc �?_i i_ � ' �c � � � c ?� 6Lh & @TLt�|��i �?|i}ihc �i _i?L|i U& ' i�c � � � c &j� ,@U� UL?t�4ih � 5 U6 �t U�@h@U|ih�3i_ M) @UL?t�4T|�L? ti| [ �c @ Thiuihi?Ui ThiLh_ih�?} �� L? [ �c @?_ @ �iU|Lh Lu �?�|�@* i?_L�4i?|t��� A�i |L|@* i?_L�4i?| � �t _i�?i_ M) � ' S�MU6 ��� i @tt�4i |�@| |�i @_4�tt�M*iTh�Ui t)t|i4t �? |�i iUL?L4) H @hi _itUh�Mi_ M) |�i ti| � � W+?n� A�i uL**L��?} t|@?_@h_

Page 8: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

@tt�4T|�L?t jc N @?_ ` ��|� hitTiU| |L |�i iUL?L4) H @hi 4@_i��tt�4T|�L? j6Lh i�ih) UL?t�4ih � 5 U6c |�i UL?t�4T|�L? ti| [ � �t @ U*Lti_ @?_ UL?�i t�Mti| Lu W+?n@?_ [ � n W+?n � [ ���tt�4T|�L? N6Lh i�ih) UL?t�4ih � 5 U6c |�i Thiuihi?Ui ThiLh_ih�?} �� L? [ � �t UL4T*i|ic UL?|�?�L�tct|hL?}*) 4L?L|L?�Uc @?_ t|h�U|*) UL?�i ���tt�4T|�L? `6Lh i�ih) UL?t�4ih � 5 U6c |�i �iU|Lh Lu �?�|�@* i?_L�4i?|t �� Mi*L?}t |L |�i �?|ih�Lh Lu [ ��A�i @tt�4T|�L? Lu t|h�U| UL?�i �|) @**L�t �t |L �Lh! ��|� _i4@?_ u�?U|�L?t �?t|i@_ Lu_i4@?_ ULhhitTL?_i?Uit @?_ |�ihiM) t�4T*��it L�h ?L|@|�L?� �** L�h hit�*|t U@hh) L�ih |L|�i U@ti Lu UL?�i Thiuihi?Uit� �*tL |�i @tt�4T|�L? Lu t|hL?}*) 4L?L|L?�U Thiuihi?Uit �t4@_i uLh |�i t@!i Lu t�4T*�U�|) @?_ U@? Mi hi*@ i_ UL?t�_ih@M*)�A�i ti| � Lu @_4�tt�M*i Th�Uit 4@) Lh 4@) ?L| UL?|@�? @ @*h@t�@? Th�Ui t)t|i4uLh |�i iUL?L4) H � W? }i?ih@* |��t �t ?L| |�i U@tic @?_ L?i ?ii_t @? i^��*�Mh��4 UL?UiT|�?�L*��?} �iU|Lht Lu ^�@?|�|) UL?t|h@�?|t L? |�i ?i| _i4@?_t @?_ |�i ?i| t�TT*�it Lu |�iUL44L_�|�it� W? |��t T@Tihc �i @?@*)3i |�i U@ti Lu �?�uLh4 h@|�L?�?} t)t|i4tc 4i@?�?}|�@| h@|�L?�?} UL?t|h@�?|t @hi |�i t@4i uLh @** �L�ti�L*_t� ��h hit�*|t U@hh) L�ih |L @�@h�i|) Lu L|�ih h@|�L?�?} t)t|i4tc tii Oih�?}t E�bbS@� uLh @ }i?ih@* |hi@|4i?| Lu h@|�L?�?}t)t|i4t�B��i? @ Th�Ui t)t|i4 R 5 � c @ h@|�L?�?} tU�i4i L? t�TT*) � 5 �W+?nc @?_ @ h@|�L?�?}tU�i4i L? _i4@?_ � 5 W+?nc |�i UL?t|h@�?i_ M�_}i| ti| Lu UL?t�4ih � 5 U6 �t }��i? M)��ERc �c �� ' i%� 5 [ � m R � %� � R � �� @?_ �& � %�& � ��& � �&c ;& 5 U?j�A�i ?�4Mih �& E�&� �t |�i ?i| @4L�?| Lu UL44L_�|) & |�@| @ UL?t�4ih U@? ti** EM�)�@| 4Lt|� A�iti UL?t|h@�?|t @hi _i|ih4�?i_ i?_L}i?L�t*) @?_ tih�i |L i^��*�Mh@|i 4@h!i|t��i?c MiU@�ti Lu Th�Ui hit|h�U|�L?tc |�i Th�Ui 4iU�@?�t4 �t ?L| U@T@M*i Lu _L�?} tL�A�i ULhhitTL?_�?} UL?t|h@�?i_ _i4@?_ _�ERc �c �� Lu UL?t�4ih � �t _i�?i_ @t |�iMit| i*i4i?| uLh �� �? ��ERc �c ��� �iU@�ti Lu |�i t|h�U| UL?�i �|) @?_ t|hL?} 4L?L|L?�U�|)@tt�4T|�L?tc |��t i*i4i?| �t �?�^�i @?_ *�it L? |�i M�_}i| �)TihT*@?ic ��i� R � _�ERc �c �� 'R � ���

4D suhihuhqfh suhrughulqj �l lv vdlg wr eh vwurqjo| prqrwrqlf li {l> e{l 5 [ l> {l � e{l> dqg {l 9@ e{llpsolhv e{l "l {l= D suhihuhqfh suhrughulqj �l lv vdlg wr eh vwulfwo| frqyh{ zkhq iru dq| sdlu {l> e{l 5 [ l>vxfk wkdw {l 9@ e{l> {l �l e{l> lw krogv wkdw �{l . +4� �,e{l "l {l iru � 5 +3> 4,1e

Page 9: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

� �i**�!?L�? U@ti _�tU�tti_ �? |�i *�|ih@|�hi �t ��i? |�i ti| � �t }��i? M)�? ' iR 5 W+?nmR& � R& � R&c & 5 U?jc��ihi R& @?_ R& @hi @ Th�Lh� }��i? *L�ih @?_ �TTih ML�?_t uLh |�i Th�Ui R& Lu }LL_ & 5 U?ct@|�tu)�?} f R& � R&� W? #hi3i d�foc @ h@|�L?i_ i^��*�Mh��4 ��|� hitTiU| |L t�U� @ U�M�Uti| Lu @_4�tt�M*i Th�Ui t)t|i4t @?_ @ �?�uLh4 h@|�L?�?} t)t|i4 �t _i�?i_ @t uL**L�t�

#i�?�|�L? 2�� E�L?t|h@�?i_ ,^��*�Mh��4�� �L?t|h@�?i_ ,^��*�Mh��4 uLh |�i iUL?L4) H ' Ei[ �c��c ��j6�'�c �?� �t @ Th�Ui t)t|i4RW 5 �?c @ h@|�L?�?} tU�i4i E�Wc �W� 5 �W+?n � W+?nc @?_c uLh i�ih) UL?t�4ih � 5 U6c @UL?t�4T|�L? M�?_*i %W� 5 [ � t�U� |�@|E�� uLh @** � 5 U6c %W� ' _�ERWc �Wc �W�(E��� S6�'� %W� ' �(E���� uLh @** & 5 U?G %W�& ���& ' �W& uLh tL4i � 5 U6 �4T*�it %W�& � ��& �W&c ;� 5 U6c @?_c@?@*L}L�t*)c %W�& � ��& ' �W& uLh tL4i � 5 U6 �4T*�it %W�& � ��& : �W&c ;� 5 U6(E��� uLh @** & 5 U?c �u |�ihi �t � 5 U6 t�U� |�@| �W& ' %W�& � ��&c |�i? RW& ' R& @?_ �u |�ihi�t � 5 U6 t�U� |�@| �W& ' %W�& � ��&c |�i? RW& ' R&��L?_�|�L? E�� hi^��hit |�@| |�i UL?t�4T|�L? Lu i@U� UL?t�4ih i^�@*t ��t UL?t|h@�?i_ _i�4@?_c ���*i �L?_�|�L? E��� �t |�i 4@h!i| U*i@h�?} UL?_�|�L?� �L?_�|�L? E���� �4T*�it |�@||�ihi �t ?L| t�4�*|@?iL�t*) h@|�L?�?} L? ML|� t�_it Lu @ 4@h!i|� �L?_�|�L? E��� ThiU*�_itt�TT*) h@|�L?�?} �? |�i 4@h!i| Lu tL4i UL44L_�|) @t *L?} @t �|t Th�Ui �t ?L| L? �|t *L�ihML�?_c ��ihi@t _i4@?_ h@|�L?�?} �? |�i 4@h!i| Lu @ UL44L_�|) _Lit ?L| |@!i T*@Ui �u �|tTh�Ui �t ?L| L? �|t �TTih ML�?_� �Mtih�i |�@| |�ihi i �t| |�L |h���@* i^��*�Mh�@� �?i �t |�i|h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4c ��|� u�** h@|�L?�?} L? |�i t�TT*) Lu @** UL44L_�|�itctL �W ' f?c Th�Ui t)t|i4 RW ' R @?_ @**LU@|�L? %W� ' ��c � 5 U6� A�i L|�ih �t |�i |h���@*_i4@?_�UL?t|h@�?i_ i^��*�Mh��4c ��|� u�** h@|�L?�?} L? |�i _i4@?_ Lu @** UL44L_�|�itc tL�W ' f?c Th�Ui t)t|i4 RW ' R @?_ @**LU@|�L? %W� ' ��c � 5 U6�W| �@t Mii? t�L�? �? Oih�?}t E�bbH� |�@| |�ihi i �t|t @ UL??iU|i_ ti| Lu UL?t|h@�?i_i^��*�Mh�@ |�@| UL?|@�?t ML|� |h���@* i^��*�Mh�@� A��t hit�*| @_4�|t �ih) t�4T*i ThLLut Lu@ ?�4Mih Lu tTiU�@* U@tit |�@| �ihi |hi@|i_ �? |�i *�|ih@|�hi MiuLhic *�!i |�i i �t|i?UiLu @ #hi3i i^��*�Mh��4 ��|� hitTiU| |L UL44L_�|) & E#hi3i E�b.D�� @?_ |�i i �t|i?UiLu @ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 E�@? _ih w@@? E�bHfc �bH2��� � #hi3i i^��*�Mh��4��|� hitTiU| |L UL44L_�|) & �t @ UL?t|h@�?i_ i^��*�Mh��4 ��|�L�| h@|�L?�?} �? |�i 4@h!i|uLh UL44L_�|) &� � t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 �t @ UL?t|h@�?i_ i^��*�Mh��4 ��|�L�|h@|�L?�?} L? |�i _i4@?_c @?_ ��|�L�| h@|�L?�?} L? |�i 4@h!i| Lu @| *i@t| L?i UL44L_�|)�

D

Page 10: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

�? @*}Lh�|�4 |L UL4T�|i |�i UL??iU|i_ ti| Lu UL?t|h@�?i_ i^��*�Mh�@ �@t Mii? ThLTLti_ �?Oih�?}tc A@*4@? @?_ v@?} E�bbS���L?t|h@�?i_ i^��*�Mh�@ ��|� _�gihi?| h@|�L?�?} tU�i4it 4@) �i** )�i*_ |�i t@4iUL?t|h@�?i_ i^��*�Mh��4 @**LU@|�L?� `�i? |�ihi �t ?L h@|�L?�?} L? |�i t�TT*) t�_i Lu @4@h!i|c |�i ThiU�ti tTiU��U@|�L? Lu �W& �t �44@|ih�@*c @t *L?} @t �W& %W�& � ��& uLh @**� 5 U6� �?@*L}L�t*)c ��i? |�ihi �t ?L h@|�L?�?} L? |�i _i4@?_ t�_i Lu @ 4@h!i|c @?)h@|�L?�?} tU�i4i �W& t@|�tu)�?} �W& : %W�& ���& uLh @** � 5 U6 �t UL4T@|�M*i ��|� @ UL?t|h@�?i_i^��*�Mh��4� A�i uhii_L4 �? |�i tTiU��U@|�L? Lu @ ?L?�M�?_�?} h@|�L?�?} tU�i4i U@? Mi�ti_ |L t�4T*�u) ?L|@|�L? @?_ ThLLut M) 4@!�?} @ T@h|�U�*@h U�L�Ui uLh @ ?L?�M�?_�?}h@|�L?�?} tU�i4i� i uLU�t L? h@|�L?�?} tU�i4it E,c �� |�@| @hi hiThiti?|i_ M) @ t�?}*i�iU|Lh o 5 d��c�oc U@**i_ |�i h@|�L?�?} �iU|Lhc ��ihi d��c�o ' i% 5 W+?m � � � % � �j�� h@|�L?�?} �iU|Lh o 5 d��c�o �?_�Uit @ h@|�L?�?} tU�i4i E,c �� }��i? M), ' �� � o � f? @?_ � ' � � o � f?�

i 4L_�u) |�i _i�?�|�L? Lu @ UL?t|h@�?i_ i^��*�Mh��4 @t uL**L�t�#i�?�|�L? 2�2 E�L?t|h@�?i_ ,^��*�Mh��4�� �L?t|h@�?i_ ,^��*�Mh��4 uLh |�i iUL?L4) H ' Ei[ �c��c ��j6�'�c �?� �t @ Th�Ui t)t|i4RW 5 �?c @ h@|�L?�?} �iU|Lh oW 5 d��c�oc @?_c uLh i�ih) UL?t�4ih � 5 U6c @ UL?t�4T|�L?M�?_*i %W� 5 [ � t�U� |�@|E�� uLh @** � 5 U6c %W� ' _�ERWc�� � oWc � � oW�(E��� S6�'� %W� ' �(E���� uLh @** & 5 U?c �u |�ihi �t � 5 U6 t�U� |�@| ��& � oW& ' %W�& ���&c |�i? RW& ' R& @?_ �u|�ihi �t � 5 U6 t�U� |�@| �& � oW& ' %W�& � ��&c |�i? RW& ' R&�5�?Ui f? � %W� � � @?_ f? �� � @?_ |�ihiuLhi �� %W� � uLh @?) �c @ ?i}@|��i�@*�i Lu oW& �4T*�it |�@| |�ihi U@? Mi L?*) h@|�L?�?} L? |�i t�TT*) Lu UL44L_�|) &c ���*i |�iLTTLt�|i �L*_t �? U@ti Lu @ TLt�|��i �@*�i Lu oW&�`�i? oW& ' fc |�ihi U@? Mi ?i�|�ih h@|�L?�?}L? |�i _i4@?_ t�_i ?Lh L? |�i t�TT*) t�_i Lu |�i 4@h!i| uLh UL44L_�|) &� A�ihiuLhi�L?_�|�L? E���� hiT*@Uit ML|� �L?_�|�L? E���� @?_ �L?_�|�L? E��� Lu #i�?�|�L? 2���W? |��t T@Tih �i @**L� uLh @ }i?ih@* ti| � Lu @_4�tt�M*i Th�Uitc uLh �?t|@?Ui ��i?tL4i Lu |�i Th�Uit @hi |�i_ |L Th�Uit Lu L|�ih UL44L_�|�it |�hL�}� �?_i @|�L?� �iU@�ti�? t�U� U@tit � 4@) ?L| Mi @ U�Mic @?_ |��t |�ihi 4@) Mi ?L �?_iTi?_i?|*) }��i? *L�ih@?_ �TTih ML�?_t uLh |�i Th�Uitc �i �@�i |L 4L_�u) |�i UL4T*i4i?|@h�|) �L?_�|�L? E���� Lu#i�?�|�L? 2�2� 6Lh tL4i tTiU�@* U@tit |��t �tt�i �@t h@�ti_ @*hi@_) M) ��i||) @?_ �@)@!

S

Page 11: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

E�b.H�c k�h3 E�bH2�c #i�i3 @?_ #hi3i E�bHe�c �@? _ih w@@? E�bHe� @?_ i__iTL�* E�bH.���t @ t�4T*i i @4T*ic UL?t�_ih |�i ti| � }��i? M)� ' iR 5 W+?nmR� � fc [

�MU R� ' �c R� � R� � R�c � 5 ajc E����ihi U � U? qi�j �t @ ti| Lu �?_i UL44L_�|�it @?_ a ' U? q EU^i�j� �t |�i ti| Lu �?_i i_UL44L_�|�it� �L44L_�|) � �t |�i ?�4ih@�hi UL44L_�|)� W? �@? _ih w@@? E�bHe� @?_i__iTL�* E�bH.�c �| �@t Mii? t�L�? |�@| |�ihi i �t|t @ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4��|�L�| h@|�L?�?} L? |�i ?�4ih@�hi UL44L_�|)c ���*i t�4�*|@?iL�t*) @** UL44L_�|�it �?U @hi �?h@|�L?i_ Lh @| *i@t| L?i Lu |�i UL44L_�|�it �? a �t �?h@|�L?i_� W? |�i *@||ihU@tic �L?_�|�L? E���� Lu #i�?�|�L? 2�2 �t ?L| t@|�t�i_� W? }i?ih@* |�ihi �t h@|�L?�?} L? @**UL44L_�|�it �? Uc @*|�L�}� MiU@�ti Lu |�i u@U| |�@| S�MU R� ' �c @| *i@t| L?i Lu |�i Th�UitR�c � 5 Uc �t TLt�|��i @?_ �t ?L| _L�?�@h_t h�}�_� W| �@t Mii? ThL�i_ |�@| �? |��t U@ti|�ihi i �t|t @ UL?t|h@�?i_ i^��*�Mh��4 t�U� |�@| |�i *i�i* Lu h@|�L?�?} L? i@U� Lu |�itiUL44L_�|�it �t }��i? M) |�i t@4i uh@U|�L? Lu |�i �?�|�@* i?_L�4i?|��? i^��*�Mh��4 UL?UiT| uLh }i?ih@* Th�Ui hit|h�U|�L?t t�L�*_ UL?|@�? |�i UL?UiT| Lu#i�?�|�L? 2�2 @t @ tTiU�@* U@ti @?_ t�L�*_ @*tL _i@* ��|� @ ti| Lu Th�Uit @t �? E�� �? @t@|�tu@U|Lh) �@)� W? L�h i^��*�Mh��4 UL?UiT| �i hi^��hi |�@| h@|�L?�?} �t ?L| @**L�i_ ��i?|�i i^��*�Mh��4 Th�Ui �iU|Lh RW *�it �? |�i �?|ih�Lh Lu |�i ti| � c MiU@�ti |�i? |�i Th�Uit)t|i4 U@? 4L�i uhii*) �?|L @?) _�hiU|�L? @?_ RW �t @ @*h@t�@? i^��*�Mh��4 Th�Ui �iU|Lh�W? U@ti �? i^��*�Mh��4 |�i Th�Ui �iU|Lh RW _Lit ?L| *�i �? |�i �?|ih�Lh Lu |�i ti| � c ���U� �t@*�@)t |�i U@ti ��i? � �t ?L| u�**�_�4i?t�L?@*c |�ihi @hi �iU|Lht |�@| TL�?| L�|�@h_t |L� @| RW @?_ ���U� @hi _�hiU|�L?t �? ���U� |�i Th�Ui t)t|i4 U@??L| 4L�i ��|�L�| *i@��?}|�i ti| Lu @_4�tt�M*i Th�Uit � � A�i UL**iU|�L? Lu @** |�iti _�hiU|�L?t uLh4t |�i ?Lh4@* UL?i@| RW |L �� 6Lh @?) RW 5 �c |�i ?Lh4@* UL?i CERW� |L � @| RW �t }��i? M)

CERW� ' io 5 W+?mR � o � RW � o uLh @?) R 5 �j�A�i ?Lh4@* UL?i }��it �?uLh4@|�L? L? |�i _�hiU|�L?t �? ���U� |�i Th�Ui t)t|i4 �t hit|h�U|i_|L 4L�ic @?_ |�ihiuLhi L? |�i !�?_ Lu h@|�L?�?} |�@| LUU�ht �? i^��*�Mh��4� W? L�h i^���*�Mh��4 UL?UiT| �i hi^��hi |�@| �? i^��*�Mh��4 |�i ?Lh4@* UL?i CERW� UL?|@�?t @? i*i4i?|oW 5 d��c�o ���U� UL4T*i|i*) _i|ih4�?it |�i *i�i*t Lu t�TT*) @?_ _i4@?_ h@|�L?�?}�`�i? oW� �t TLt�|��i E?i}@|��i�c �? ���U� U@ti |�i Th�Ui Lu UL44L_�|) � U@??L| Mi �?Uhi@ti_E_iUhi@ti_� ��|�L�| *i@��?} � ��i? 4L��?} �? |�i _�hiU|�L? oWc �i @**L� L?*) uLh _i4@?_Et�TT*)� h@|�L?�?} L? UL44L_�|) �� A�i *i�i* Lu h@|�L?�?} �t _i|ih4�?i_ M) |�i �@*�i LuoW� � A�i ��}�ih |�i @MtL*�|i �@*�i Lu oW� c |�i |�}�|ih |�i h@|�L?�?} ��** Mi� `�i? oW� ' fc�? ���U� U@ti |�i Th�Ui Lu UL44L_�|) � �t ?L| U�@?}�?} ��i? 4L��?} �? |�i _�hiU|�L? LuoWc ?L h@|�L?�?} �t @**L�i_ L? UL44L_�|) �� A��t }��it �t |�i uL**L��?} _i�?�|�L? Lu |�i}i?ih@* UL?UiT| Lu @ "�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4�

.

Page 12: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

#i�?�|�L? 2�� E"�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4�� "�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4 E"�,� uLh |�i iUL?L4) H ' Ei[ �c��c ��j6�'�c � � �t@ Th�Ui t)t|i4 RW 5 � c @ h@|�L?�?} �iU|Lh oW 5 d��c�oc @?_c uLh i�ih) UL?t�4ih � 5 U6c @UL?t�4T|�L? M�?_*i %W� 5 [ � t@|�tu)�?}E�� uLh @** � 5 U6c %W� ' _�ERWc�� � oWc � � oW�(E��� S6�'� %W� ' �(E���� oW 5 CERW���t 4L|��@|i_ @ML�ic �L?_�|�L? E���� *�?!t |�i h@|�L?�?} �iU|Lh |L |�i Th�Ui hit|h�U|�L?t� A�ih@|�L?�?} tU�i4i �t UL4T*i|i*) _i|ih4�?i_ M) @ �iU|Lh oW �? |�i ?Lh4@* UL?i CERW� @| |�ii^��*�Mh��4 Th�Ui RW�5�?Ui f � %W� � � @?_ f �� �c �| �L*_t |�@| �? @ "�, ��& %W�& � ��& �&uLh @** & 5 U? @?_ � 5 U6� Oi?Uic �? @ "�,c @ UL?t�4ih � 5 U6 U@? L?*) Mi h@|�L?i_ �? ��t_i4@?_ uLh }LL_ & �u oW& ' �& � E%W�& � ��&� : f� �?@*L}L�t*)c tL4i UL?t�4ih � 5 U6 U@?L?*) Mi h@|�L?i_ �? ��t t�TT*) Lu }LL_ & �u oW& ' ��& � E%W�& ���&� f� Oi?Ui |�ihi U@??L|Mi h@|�L?�?} t�4�*|@?iLt*) L? ML|� t�_it Lu @?) 4@h!i|� �L|�Ui |�@| |�i h@|�L?�?} L? |�it�TT*) E_i4@?_� Lu UL44L_�|) & �t |�}�|ihc |�i U*Ltih oW& �t |L ��& E�&c hitTiU|��i*)�� W?T@h|�U�*@hc |�ihi �t u�** t�TT*) h@|�L?�?} �u oW& ' ��& @?_ |�ihi �t u�** _i4@?_ h@|�L?�?} �uoW& ' �&� `�i? RW �t �? |�i �?|ih�Lh Lu � c |�i? oW ' f?c �? ���U� U@ti |�i h@|�L?�?} tU�i4iL? _i4@?_ i^�@*t � @?_ |�i h@|�L?�?} tU�i4i L? t�TT*) ��c �4T*)�?} |�@| ?L?i Lu |�ih@|�L?�?} UL?t|h@�?|t U@? Mi M�?_�?}�5�44@h�3�?}c �? @ "�, UL?t�4iht 4@ �4�3i |�i�h �|�*�|) �? |�i�h UL?t|h@�?i_ M�_�}i| ti|tc |L|@* _i4@?_ i^�@*t |L|@* t�TT*)c |�ihi �t ?L t�4�*|@?iL�t h@|�L?�?} L? ML|�t�TT*) @?_ _i4@?_c @?_ @ �iU|Lh �? |�i ?Lh4@* UL?i |L � _i|ih4�?it |�i |�}�|?itt Luh@|�L?�?}� W? U@ti � ' �?c oW& f �4T*�it RW& ' R& @?_ oW& : f �4T*�it RW& ' R&c )�i*_�?}ThiU�ti*) �L?_�|�L? E���� Lu #i�?�|�L? 2�2�6Lh |�i ti| � }��i? �? E��c �| �L*_t uLh @?) TLt�|��i Th�Ui �iU|Lh R @?_ @?) o 5 CER�|�@| o& ' o� uLh i�ih) |�L �?_�Uit & @?_ � �? |�i �?_i ti| U� Oi?Uic �u |�ihi �t h@|�L?�?}L? |�iti UL44L_�|�itc |�i? @UULh_�?} |L �L?_�|�L? E���� Lu #i�?�|�L? 2��c |�i @4L�?| Luh@|�L?�?} �t |�i t@4i uLh @** UL44L_�|�it �? U�6hL4 @? iUL?L4�U TL�?| Lu ��i�c �| �t Lu Uh�U�@* �4TLh|@?Ui |�@| L�h i^��*�Mh��4UL?UiT| �t �?_iTi?_i?| Lu |�i �?�|t Lu 4i@t�hi4i?| |�@| @hi �ti_ �? |�i _i�?�|�L? Lu @UL44L_�|)� 5�TTLti |�@| |�i �?�| Lu 4i@t�hi4i?| �ti_ �? |�i _i�?�|�L? Lu UL44L_�|)& �t 4�*|�T*�i_ M) @ TLt�|��i UL?t|@?| k&c & ' �c � � � c ?� �? iUL?L4) HEk� ��|� �?�|�@*i?_L�4i?|t ��Ek� }��i? M) ��&Ek� ' �&*k&c & ' �c � � � c ?c ti| Lu @_4�tt�M*i Th�Uit

� Ek� ' ieR 5 W+? m eR& ' k&R&c & ' �c � � � c ?c uLh tL4i R 5 �jH

Page 13: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

@?_ @TThLTh�@|i*) hi_i�?i_ UL?t�4T|�L? ti|t [ �Ek� @?_ Thiuihi?Ui hi*@|�L?t �� Ek� t�L�*_�@�i @? i^���@*i?| ti| Lu "�,t @t |�i iUL?L4) H � �Lhi ThiU�ti*)c uLh i@U� "�, ERWc oWc %W�Lu H |�ihi t�L�*_ Mi @ "�, ERWEk�c oWEk�c %WEk�� Lu HEk� @?_ ��Ui �iht@c ��ihi RWEk� �tLM|@�?i_ uhL4 RW M) UL4TL?i?|��ti 4�*|�T*�U@|�L? M) kc @?_ oWEk� @?_ %WEk� @hi LM|@�?i_uhL4 oW @?_ %W M) UL4TL?i?|��ti _���t�L? M) k� A�i uL**L��?} hit�*| U*@�4t |�@| i^���@*i?Ui�L*_t�A�iLhi4 2�eA�i ti| Lu i^��*�Mh�@ Lu H ' Ei[ �c��c ��j6�'�c � � �t i^���@*i?| |L |�i ti| Lu i^��*�Mh�@ LuHEk� ' Ei[ �Ek�c�� Ek�c ��Ek�j6�'�c � Ek�� uLh @?) U�L�Ui Lu k f��hLLu�W| �t LM��L�t |�@| ERWc oWc %W� t@|�t�it �L?_�|�L?t E�� @?_ E��� Lu #i�?�|�L? 2�� Lu @ "�, LuH �u @?_ L?*) �u ERWEk�c oWEk�c %WEk�� t@|�t�it �L?_�|�L?t E�� @?_ E��� Lu #i�?�|�L? 2�� Lu @"�, Lu HEk�� W| �t i@t�*) �ih��i_ |�@| oW 5 CERW� �u @?_ L?*) �u oWEk� 5 CkERWEk��c ��ihiCk _i?L|it |�i ?Lh4@* UL?i @| � Ek�� A��t t�L�t |�@| ERWc oWc %W� t@|�t�it �L?_�|�L? E����Lu |�i _i�?�|�L? Lu @ "�, Lu H �u @?_ L?*) �u ERWEk�c oWEk�c %WEk�� t@|�t�it �L?_�|�L? E����Lu |�i _i�?�|�L? Lu @ "�, Lu HEk�� "�,�#�� , �t|i?Ui hit�*|tAL UL?t�_ih |�i i �t|i?Ui Lu "�,tc �? |��t tiU|�L? �i hit|h�U| L�hti*�it |L |�i U@ti |�@| �t@|�t�it |�i uL**L��?} �tt�4T|�L? �� W? |�i ?i | tiU|�L? �i ��** @*tL @**L� uLh 3ihL Th�Uit@?_ @? �?ML�?_i_ ti| Lu @_4�tt�M*i Th�Uit��tt�4T|�L? �A�i ti| � Lu @_4�tt�M*i Th�Uit �t @ ?L?�i4T|)c UL?�i @?_ UL4T@U| t�Mti| �? |�i �?|ih�LhLu W+?n�

i �ht| t�L� |�@| |�ihi @hi |�L |h���@* ?L�|h@_i "�,t ��|� u�** h@|�L?�?} L? |�it�TT*)c hitTiU|��i*) |�i _i4@?_� wi| � f @?_ � � Mi }��i? M)� f ' iR 5 � m� � R � � � hR uLh @** hR 5 �j

@?_� � ' iR 5 � m� � R � � � hR uLh @** hR 5 �j�

�L|�Ui |�@| |�i �?|ihtiU|�L? Lu � f @?_ � � �t i�|�ih i4T|) Lh i^�@* |L � � �h�Uit �? � f @hit�U� |�@| |�i �@*�i Lu |L|@* �?UL4i R � � �t 4�?�4�3i_c @?_ Th�Uit �? � � t�U� |�@| |��tb

Page 14: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

�@*�i �t 4@ �4�3i_� A�i ?i | hit�*| U*@�4t |�@| |�ihi �t @ |h���@* ?L�|h@_i "�, ��|� u�**h@|�L?�?} L? |�i t�TT*) @| @?) Th�Ui �? � f @?_ @ |h���@* ?L�|h@_i "�, ��|� u�** h@|�L?�?}L? |�i _i4@?_ @| @?) Th�Ui �? � ��A�iLhi4 ���6Lh @?) R 5 � f |�ihi �t @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 ��|� h@|�L?�?} �iU|LhoW ' �� @?_ @**LU@|�L? %W� ' ��c � 5 U6� 6Lh @?) R 5 � � |�ihi �t @ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4 ��|� h@|�L?�?} �iU|Lh oW ' � @?_ @**LU@|�L? %W� ' ��c � 5 U6��hLLu�A@!i @?) R 5 � f� �*i@h*)c R 5 M?_E� �� �) _i�?�|�L? Lu � f �i �@�i |�@| �� 5 CER��A@!�?} o ' �� }��it uLh @?) � 5 U6 |�@| ��ERc fc 2�� ' i��j @?_ �i?Ui _�ERc fc 2�� ' ��c�4T*)�?} |�@| 4@h!i|t U*i@h @?_ |��t @ ?L�|h@_i i^��*�Mh��4 ��|� u�** h@|�L?�?} L? t�TT*)�t LM|@�?i_��?@*L}L�t*)c |@!i @?) R 5 � �� W| �L*_t |�@| � 5 CER�� i �@�i uLh @?) � 5 U6 |�@|��ERc�2�c f� ' i%� 5 [ �m%� � ��j @?_ �i?Ui M) �tt�4T|�L? N |�@| _�ERc�2�c f� ' ��c�4T*)�?} |�@| 4@h!i|t U*i@h @?_ |��t @ ?L�|h@_i i^��*�Mh��4 ��|� u�** h@|�L?�?} L? _i4@?_�t LM|@�?i_� "�,�#�A�i i �t|i?Ui Lu ?L?�|h���@* i^��*�Mh�@ uL**L�t uhL4 |�i ?i | hit�*| t@)�?} |�@| |�ihi �t @UL??iU|i_ ti| � Lu "�,t UL?|@�?�?} ML|� |�i |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_|�i |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4�A�iLhi4 ��2wi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc Nc ` @?_ �� A�i?|�ihi i �t|t @ UL??iU|i_ ti| Lu "�@?|�|) �L?t|h@�?i_ ,^��*�Mh�@ Lu |�i iUL?L4) H c UL?|@�?�?}@ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 ERc��c��c � � � c �6� uLh @?) R 5 � fc @?_ @ |h���@*_i4@?_�UL?t|h@�?i_ i^��*�Mh��4 ERc�c��c � � � c �6� uLh @?) R 5 � ��i UL?|�?�i |��t tiU|�L? ��|� |�i ThLLu Lu A�iLhi4 ��2� i �ht| uLU�t L? |�i i^��*�Mh@|�?}4iU�@?�t4 |L �?_ @ "�,� AL _L tLc �i �?|hL_�Ui @ ti| ' � W+? UL?|@�?�?} � @?_ _i�?iuLh i�ih) ^ 5 ' @ Th�Ui RE^� 5 � @?_ @ h@|�L?�?} �iU|Lh oE^� 5 d��c�o� A�i ti| ' �t |@!i?|L Mi ' ' i^ 5 W+? m n^ � Rn2 � � uLh tL4i R 5 �jctL ' �t |�i ti| Lu i*i4i?|t �? W+? *)�?} @| 4Lt| @| @ _�t|@?Ui � uhL4 � c �t�?} |�i ,�U*�_i@??Lh4c @?_ |��t �?U*�_it |�i ti| � � W? |�i ti^�i* M?_E'� _i?L|it |�i ML�?_@h) Lu ' @?_�?|E'� �|t �?|ih�Lh� 6�h|�ihc uLh ^ 5 'c _i�?i |�i ThL�iU|�L? RE^� Lu ^ L? � M)RE^� ' @h} 4�?RM� nR� ^n2�

�f

Page 15: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

5�?Ui M) �tt�4T|�L? �c |�i ti| � �t UL?�i @?_ UL4T@U|c uLh i�ih) ^ 5 ' �| �L*_t |�@| RE^��t �?�^�i*) _i�?i_ @?_ UL?|�?�L�t �? ^c @?_ ^�RE^� 5 CERE^��� �LhiL�ihc n^�RE^�n2 � �c��|� i^�@*�|) �u @?_ L?*) �u ^ 5 M?_E'�� W| �L*_t |�@| RE^� ' ^ ��i? ^ 5 �� A�i ti| ' �@t|�i uL**L��?} ThLTih|�it�wi44@ ���E�� A�i ti| ' �t @ UL?�i c UL4T@U|c u�**�_�4i?t�L?@* t�Mti| Lu W+? @?_ UL?|@�?t � �? �|t�?|ih�Lh(E��� A�i ML�?_@h) Lu ' �t t4LL|���hLLu� A�i UL4T@U|?itt uL**L�t uhL4 |�i UL4T@U|?itt Lu �� A�@|' �t u�**�_�4i?t�L?@* @?_UL?|@�?t � �? �|t �?|ih�Lh uL**L�t �44i_�@|i*) uhL4 �|t _i�?�|�L?� AL ThL�i UL?�i �|)c |@!i@?) ^�c ^2 5 ' @?_ f � b � �c @?_ *i| ^Eb� ' b^�nE��b�^2 @?_ REb� ' bRE^��nE��b�RE^2��5�?Ui � �t UL?�i c �i !?L� |�@| REb� 5 �� �LhiL�ihc n^Eb� � REb�n2 � bn^� � RE^��n2 nE�� b�n^2 � RE^2�n2 � �� A�ihiuLhic ^Eb� 5 '��hLTih|) E��� uL**L�t uhL4 |�i �ti Lu |�i 2�?Lh4 �? |�i _i�?�|�L? Lu '� A@!i @?)^W 5 M?_E'�� A�i? n^W�RE^W�n2 ' � @?_ @?) ^ ��|� n^�RE^W�n2 � � Mi*L?}t |L '� Oi?Uic|�i ?Lh4@* UL?i @| ^W UL?|@�?t @| 4Lt| L?i �iU|Lh ��|� *i?}|� L?i� 5�?Ui ' �t UL?�i c�i !?L� |�@| |�i ?Lh4@* UL?i @| ^W �t ?L?�i4T|) @?_ �TTih�ti4�UL?|�?�L�t� Oi?Uic |�i?Lh4@* UL?i @| @?) ML�?_@h) TL�?| ^ Lu ' UL?|@�?t @ �?�^�i �iU|Lh ��|� *i?}|� L?i @?_|��t �iU|Lh �t UL?|�?�L�t �? ^c ��i� |�i ML�?_@h) Lu ' �t t4LL|�� "�,�#�i _i�?i |�i u�?U|�L? o G '$ W+? M)

oE^� ' f?c ^ 5 �@?_c uLh & 5 U?co&E^� ' E4�?�MU? ��m^� � R�E^�m�E^& � R&E^��n^ � RE^�n2c ^ 5 ' q ��5�?Ui n^� RE^�n2 }Lit |L 3ihL ��i? ^� RE^� }Lit |L f? @?_ ��& � E4�?�MU? ���^�3R�E^���E^& �R&E^�� � �&c �| uL**L�t |�@| oE^� }Lit |L f? ��i? ^ UL?�ih}it |L @ TL�?| �? �� A��t �4T*�it|�@| oE^� �t UL?|�?�L�t �? ^� A�i L|�ih ThLTih|�it Mi*L� �44i_�@|i*) uL**L� uhL4 wi44@�����LhL**@h) ��eA�i u�?U|�L? o G '$ W+? t@|�t�it |�i uL**L��?} ThLTih|�itGE�� o �t UL?|�?�L�t(E��� ��& � o&E^� � �& uLh @** ^ 5 '(E���� o&E^� ' �& �u @?_ L?*) �u �&^&3R&E^� ' 4�?�MU? ���^�3R�E^�� @?_ ^ 5 M?_E'�(

��

Page 16: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

E��� o&E^� ' ��& �u @?_ L?*) �u � �&^&3R&E^� ' 4�?�MU? ���^�3R�E^�� @?_ ^ 5 M?_E'�(E�� oE^� 5 CERE^�� uLh @** ^ 5 '�i ?L� _i�?i uLh @?) UL?t�4ih � 5 U6 |�i hi_�Ui_ M�_}i| ULhhitTL?_i?Ui �� G '$ W+?M) ��E^� ' ��ERE^�c�� � oE^�c � � oE^��c ^ 5 '�6�?@**)c uLh @?) UL?t�4ih � 5 U6 �i _i�?i |�i hi_�Ui_ _i4@?_ ULhhitTL?_i?Ui _� G ' $W+? M)

_�E^� ' i%� 5 ��E^�m%� �� +�c uLh @** +� 5 ��E^�j��iU@�ti |�i ti| � �t �? |�i �?|ih�Lh Lu W+?nc �i �@�i |�@| uLh @?) ^ 5 ' |�i Th�Ui �iU|Lh RE^� 5� �t t|h�U|*) TLt�|��i� �L|�Ui |�@| t�?Ui �� � oE^� U@? Mi i^�@* |L 3ihLc |�i U�i@Tih TL�?|@tt�4T|�L? |�@| �t �t�@**) hi^��hi_ |L t�L� UL?|�?��|) Lu |�i M�_}i| ULhhitTL?_i?Uic �t��L*@|i_� �i�ih|�i*ittc �| uL**L�t uhL4 A�iLhi4 2�2 �? Oih�?}t E�bbSM� |�@| �� �t UL?|�?�L�t@| @?) ^ 5 '� `�|� �tt�4T|�L? N �| |�i? uL**L�t |�@| _� �t @ UL?|�?�L�t u�?U|�L? @?_ tL�t |�i hi_�Ui_ i Uitt _i4@?_ u�?U|�L? 5 G '$ W+? _i�?i_ M)5E^� ' [�MU6 _�E^�� ���) �tt�4T|�L? N |�i M�_}i| UL?t|h@�?| RE^� � _�E^� � RE^� � �� �t @*�@)t t@|�t�i_ ��|�i^�@*�|) @?_ �i?Ui @*h@t< *@� �L*_tc ��i� RE^� � 5E^� ' f uLh @** ^ 5 '� A�i u�?U|�L? 5t@|�t�it |�i uL**L��?} ThLTih|�it�

wi44@ ��D N?_ih �tt�4T|�L?t jc Nc ` @?_ �c |�i hi_�Ui_ i Uitt _i4@?_ u�?U|�L?5 G '$ W+? t@|�t�it |�i uL**L��?}GE�� 5 �t UL?|�?�L�t(E��� `@*h@t< *@� �L*_tG RE^� � 5E^� ' f uLh @** ^ 5 '(E���� o&E^� ' ��& �4T*�it 5&E^� � f @?_ o&E^� ' �& �4T*�it 5&E^� � f�A�i ?i | |�iLhi4 t�L�t |�@| @ 3ihL TL�?| Lu 5 L? ' �?_�Uit @ "�,�A�iLhi4 ��Swi| ^W Mi @ 3ihL TL�?| Lu 5 L? 'c ��i� 5E^W� ' f?� A�i? ERE^W�c oE^W�c _�E^W�c � � � c _6E^W�� �t@ "�@?|�|) �L?t|h@�?i_ ,^��*�Mh��4��hLLu�i �@�i |L t�L� |�@| �L?_�|�L?t E��c E��� @?_ E���� Lu #i�?�|�L? 2�� �L*_� �*i@h*)c �L?_��|�L?t E�� @?_ E��� �L*_ M) UL?t|h�U|�L? Lu |�i hi_�Ui_ i Uitt _i4@?_ u�?U|�L?� �) ThLTih|)

�2

Page 17: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

E�� Lu �LhL**@h) ��e �i �@�i |�@| oE^W� �t @? i*i4i?| Lu CERE^W��c ���U� t�L�t |�@| �L?�_�|�L? E���� Lu #i�?�|�L? 2�� �L*_t� "�,�#�6hL4 A�iLhi4 ��S �| uL**L�t |�@| |�i ^�it|�L? Lu i �t|i?Ui Lu @ "�, hi_�Uit |L |�i i �t|i?UiLu @ 3ihL TL�?| Lu 5� �iuLhi t|@|�?} L�h }i?ih@* hit�*| L? |�i UL??iU|i_ ti| Lu UL?t|h@�?i_i^��*�Mh�@ �? |ih4t Lu |�i T@h@4i|ih ^c �i �ht| t�L� |�@| ' UL?|@�?t |�L 3ihL TL�?|t |�@|)�i*_ |h���@* t�TT*)�UL?t|h@�?i_ @?_ _i4@?_�UL?t|h@�?i_ i^��*�Mh�@� wi| 'f @?_ '� Mi}��i? M)

'f ' i^ 5 'm� � ^ � � � h uLh @** h5 'j@?_

'� ' i^ 5 'm� � ^ � � � h uLh @** h5 'j��*i@h*)c t�?Ui ' �t UL4T@U|c |�i ti|t 'f @?_ '� @hi ML|� ?L?�i4T|)� �LhiL�ihc |�i�?|ihtiU|�L? Lu 'f @?_ '� �t i4T|)c t�?Ui ' �t u�**�_�4i?t�L?@*�A�iLhi4 ��.,@U� i*i4i?| �? 'f Lh '� �t @ 3ihL TL�?| Lu 5 @?_ )�i*_t @ |h���@* i^��*�Mh��4GE�� �?) ^ 5 'f �?_�Uit |�i |h���@* t�TT*)�UL?t|h@�?i_ "�, ERE^�c��c��c � � � c �6� ��|�RE^� 5 � f�E��� �?) ^ 5 '� �?_�Uit |�i |h���@* _i4@?_�UL?t|h@�?i_ "�, ERE^�c �c��c � � � c �6� ��|�RE^� 5 � ���hLLu�A@!i @?) ^ 5 'f� �*i@h*)c ^ 5 M?_E'�� �) _i�?�|�L? Lu 'f �i �@�i |�@| oE^� ' �� @?_RE^� 5 � f� W| uL**L�t |�@|c uLh @?) � 5 U6c ��E^� ' i��j @?_ �i?Ui _�E^� ' ��c �4T*)�?}|�@| 5E^� ' f? @?_ |��t ^ �?_�Uit @ ?L�|h@_i i^��*�Mh��4 ��|� u�** h@|�L?�?} L? t�TT*)��?@*L}L�t*)c |@!i @?) ^ 5 '�� A�i? oE^� ' � @?_ RE^� 5 � �� W| uL**L�t |�@|c uLh @?)� 5 U6c ��E^� ' i%� 5 [ �m%� � ��j @?_ �i?Ui M) �tt�4T|�L? N |�@| _�E^� ' ��c �4T*)�?}|�@| 5E^� ' f? @?_ |��t ^ �?_�Uit @ ?L�|h@_i i^��*�Mh��4 ��|� u�** h@|�L?�?} L? _i4@?_�"�,�#�A�i i �t|i?Ui Lu ?L?�|h���@* i^��*�Mh�@ uL**L�t uhL4 |�i ?i | hit�*| t@)�?} |�@| ' UL?|@�?t @UL??iU|i_ ti| � Lu 3ihL TL�?|t Lu 5 �@��?} @ ?L?�i4T|) �?|ihtiU|�L? ��|� ML|� 'f @?_ '�c��i� |�ihi �t @ ti| � � ' Lu 3ihLit Lu 5 Mi�?} @ UL??iU|i_ t�Mti| Lu ' t@|�tu)�?} � _'f 9' >@?_ � _'� 9' >� A��t �4T*�it |�@| |�ihi i �t| @ |h���@* t�TT*) UL?t|h@�?i_ i^��*�Mh��4 @?_@ |h���@* _i4@?_ UL?t|h@�?i_ i^��*�Mh��4c ���U� @hi UL??iU|i_ |L i@U� L|�ih |�hL�}� @ ti|Lu "�,t�

��

Page 18: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

A�iLhi4 ��Hwi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc Nc ` @?_ �� A�i?|�ihi i �t|t @ UL??iU|i_ ti| � � ' Lu 3ihL TL�?|t Lu 5 t�U� |�@| �_'f 9' > @?_ �_'� 9' >��hLLu�wi| �f @?_ �� Mi t�U� |�@| S?&'��&^& ' �f ��i? ^ 5 'f @?_ S?&'��&^& ' �� ��i? ^ 5 '���L|�Ui |�@| �f @?_ �� @hi �i**�_i�?i_c �� : �f @?_ �f � S?&'��&^& � �� uLh @** ^ 5 '� 6LhtL4i � : fc _i�?i ff � W+? M)

ff ' i% 5 W+?m ?[&'��&%& ' �fc 4@ &MU? %& ��j@?_c uLh f k � �c fk � W+? M)

fk ' i% 5 W+?m% ' %f n kS?&'��2& E�� � �f��c %f 5 ffj��*i@h*)c uLh % 5 fkc f � k � �c �i �@�i |�@| S?&'��&%& ' E� � k��f n k��c @?_ |��tS?&'��&%& ' �f ��i? % 5 ff @?_ S?&'��&%& ' �� ��i? % 5 f�� 6�h|�ihc _i�?i

f ' ^kMdfc�ofk@?_ |@!i� t��U�i?|*) *@h}i |�@| ' � f @?_ |�@| @?) ^ 5 'qE'f^'�� *�it �? |�i �?|ih�LhLu f� �L|�Ui |�@| 'f ' ' _ff @?_ '� ' ' _f��i _i�?i |�i ti| ff ' i% 5 W+?mS?&'��&%& ' �fj� wi| �E%� Mi |�i Lh|�L}L?@*ThL�iU|�L? Lu % 5 W+? L? ffc ��i� �E%� ' %� b� uLh tL4i b 5 W+� �i |c *i| |�i ti| iff Mi_i�?i_ M)

iff ' i% 5 ffm% ' �E^ n 5E^��c ^ 5 'j ^ i% 5 ffm% ' �ERE^��c ^ 5 M?_E'�j�5�?Ui ' �t UL4T@U| @?_ 5 @?_ � @hi UL?|�?�L�t u�?U|�L?tc �| uL**L�t |�@| iff �t @ ML�?_i_t�Mti| Lu ff @?_ |��t � U@? Mi |@!i? tL *@h}i |�@| ff UL?|@�?t iff �? �|t hi*@|��i �?|ih�Lh�6Lh % 5 ff @?_ k 5 dfc �oc _i?L|i %k ' %n kS?&'� �2& E�� � �f�� 5 fk @?_ _i�?i |�iTL�?|�|L�ti| 4@TT�?} ) G ff � dfc �o$ ff M)

)E%c k� ';AA?AA=i�E%k n 5E%k��j �u %k 5 �?|E'�c�L?�Ei�E%k n 5E%k��j ^ i�ERE%k��j� �u %k 5 M?_E'�ci�ERE^E%k���j �u %k 5 f q'cuLh @** E%c k� 5 ff � dfc �oc ��ihi �L?�E�� _i?L|it |�i UL?�i ��** Lu @ ti| @?_ ^E%� �t |�iLh|�L}L?@* ThL�iU|�L? Lu % 5 W+? L? '� �*i@h*)c ff �t UL4T@U| @?_ UL?�i c |�i 4@TT�?} )�t �TTih ti4��UL?|�?�L�tc @?_ uLh i�ih) E%c k� 5 ff� dfc �o �| �L*_t |�@| )E%c k� �t UL4T@U|cUL?�i @?_ ?L?�i4T|)� �UULh_�?} |L �hL�_ih<t |�iLhi4c tii �hL�_ih E�bSf�c |�ihi i �t|t

�e

Page 19: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

@ UL??iU|i_ ti| � Lu � i_ TL�?|t Lu ) �? ff � dfc �o t�U� |�@| � _ Eff � ifj� 9' > @?_� _ Eff � i�j� 9' >c ��i� |�ihi i �t|t @ UL??iU|i_ ti| � �? ff � dfc �o t@|�tu)�?}% 5 )E%c k�c uLh @** E%ck� 5 �c

UL?|@�?�?} @ TL�?| �? ff�ifjc @?_ UL?|@�?�?} @ TL�?| �? ff�i�j� Oi?Uic |�i ti| � � f_i�?i_ M)� ' i% 5 fm% ' %kc E%ck� 5 �j

�t @ UL??iU|i_ ti| �? f t�U� |�@| � _ ff 9' > @?_ � _ f� 9' >� W| hi4@�?t |L Mi t�L�?|�@| i�ih) i*i4i?| Lu � *�it �? ' @?_ �t @ 3ihL TL�?| Lu 5�A@!i @?) % 5 �c |�i? % ' %k uLh tL4i E%c k� 5 �� 5�TTLti �ht| |�@| %k 5 f q'�6hL4 |�i _i�?�|�L? Lu ) �| |�i? uL**L�t |�@|% ' �ERE^E%k����

5�?Ui �E%� �t |�i ThL�iU|�L? Lu % L? ffc |�ihi i �t|t b 5 W+ t�U� |�@| % ' �ERE^E%k��� 'RE^E%k��� b�� 5�?Ui %k � ^E%k� ' qE^E%k�� RE^E%k��� uLh tL4i q : fc �| uL**L�t |�@|Eq n ��E^E%k�� RE^E%k��� ' %k � RE^E%k��' %k � Eb� n %�' E kS?&'��2& E�� � �f�� b���A��t �4T*�it |�@| ^E%k�� RE^E%k�� ' B� uLh tL4i B 5 i� �8�82 c �8�82j� Oi?Uic i�|�ih k ' f@?_ %k 5 'fc Lh k ' � @?_ %k 5 '�� A��t UL?|h@_�U|t |�@| %k 5 f q '� �L?ti^�i?|*)c%k 5 ' uLh i�ih) %k 5 ���i |c t�TTLti |�@| %k 5 �?|E'�� A�i? %k� kS?&'� �2& E����f�� ' % ' �E%kn5E%k���5�?Ui �E%k n 5E%k�� ' %k n 5E%k�� b� uLh tL4i b 5 W+c �| uL**L�t |�@|5E%k� ' Eb� kS?&'��2& E�� � �f�����iU@�ti Lu @*h@t< *@�c �i LM|@�?RE%k� � 5E%k� ' Eb� kS?&'��2& E�� � �f��RE%k� � � ' f�

Oi?Uic t�?Ui RE%k� �� : fc �i �@�i |�@| b ' kS?&'� �2& E����f� @?_ |��t 5E%k� ' f?c t�L��?}|�@| %k �t @ 3ihL TL�?| Lu 5�6�?@**)c t�TTLti %k 5 M?_E'�� A�i? |�ihi i �t|t q� � f @?_ q2 � f ��|� q�nq2 ' �t�U� |�@|q��E%k n 5E%k�� n q2�ERE%k�� ' % ' %k � kS?&'��2& E�� � �f����D

Page 20: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

Nt�?} |�@| �E%� �t |�i ThL�iU|�L? Lu % L? ffc �| uL**L�t |�@| |��t U@? Mi hi�h�||i? @tq�%k n q�5E%k� n q2RE%k�� >� ' %k � kS?&'��2& E�� � �f��uLh tL4i > 5 W+� `�|� q� n q2 ' � @?_ *i||�?} b ' >� kS?&'� �2& E�� � �f� �i LM|@�?q�5E%k� ' q2E%k � RE%k�� n b��6�ht|c t�TTLti q� ' f� A�i?c @t MiuLhic �| uL**L�t |�@| %k � RE%k� ' B� uLh tL4i B 5i� �8�82 c �8�82jc �4T*)�?} |�@| %k 5 'f ^'�� Oi?Uic @UULh_�?} |L A�iLhi4 ��.c 5E%k� ' f?@?_ |��t %k �t @ 3ihL TL�?| Lu 5� 5iUL?_c t�TTLti q� : f� A�i?5E%k� ' q2q� E%k � RE%k�� n bq�� ' Bq�oE%k� n bq��cuLh tL4i B � f� 5�?Ui %k 5 M?_E'�c �i �@�i |�@| |�ihi �t & t�U� |�@| mo&E%k�m ' �&�`�i?o&E%k� ' ��& �| uL**L�t |�@|5&E%k� ' � Bq��& n bq��& � fc

tL b � B� Oi?Uic �B n b : f @?_ |��t5�E%k� ' Bq�o�E%k� n bq��� � � Bq��� n bq��� � f

uLh @** � 5 U? @?_ |�ihiuLhi 5E%k� ' f? _�i |L @*h@t< *@�� 5�4�*@h*)c ��i? o&E%k� ' �& �|uL**L�t |�@|5&E%k� ' Bq��& n bq��& � fcuLh tL4i B � f @?_ |��t B n b � f� A�i?

5�E%k� ' Bq�o�E%k� n bq��� � Bq��� n bq��� � fuLh @** � 5 U? @?_ |�ihiuLhi 5E%k� ' f? _�i |L @*h@t< *@�� "�,�#�A�i |�iLhi4 }�@h@?|iit |�@| |�ihi �t @ UL??iU|i_ ti| Lu "�,t UL??iU|�?} |�L |h���@* i^���*�Mh�@� ,@U� "�, �? |��t ti| �t �?_�Ui_ M) tL4i ^W 5 � @?_ �t i^�@* |L ERWc oWc %W� 'ERE^W�c oE^W�c %E^W��c ��ihi %E^W� �t |�i _i4@?_ @**LU@|�L? �?_�Ui_ M) ^W� �?i |h���@* i^���*�Mh��4 �t �?_�Ui_ M) tL4i ^f 5 'f @?_ |�i L|�ih L?i M) tL4i ^� 5 '�� �| ^f �i �@�ioE^f� ' �� @?_ tL RE^f� 5 � fc u�** h@|�L?�?} L? |�i t�TT*) @?_ ?L h@|�L?�?} L? |�i _i4@?_��| ^� �i �@�i oE^�� ' � @?_ tL RE^�� 5 � �c ?L h@|�L?�?} L? t�TT*)c @?_ u�** h@|�L?�?} L?_i4@?_� Oi?Uic @| Rf ' RE^f� |�i Th�Uit ��|��? � U@??L| Mi _iUhi@ti_ u�h|�ih �? |�i _�hiU�|�L? ��c ��i� �� �t �? |�i ?Lh4@* UL?i CERf�c @?_ |��t �iU|Lh �� _i|ih4�?it |�i h@|�L?�?}tU�i4i E�fc �f� ��|� �f ' f? @?_ �f � �� 5�4�*@h*)c @| R� ' RE^��c |�i Th�Uit ��|��? �

�S

Page 21: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

U@??L| Mi �?Uhi@ti_ u�h|�ih �? |�i _�hiU|�L? �c @?_ |��t �iU|Lh _i|ih4�?it |�i u�** _i4@?_h@|�L?�?} tU�i4i E��c ��� ��ihi �� ' f? @?_ �� ' �2�� � ?L?�|h���@* "�, �t �?_�Ui_ M)@?) ^ 5 � ?L| �? 'f Lh '�� Wu ^ 5 �c |�i? RE^� ' ^ �t @ @*h@t�@? i^��*�Mh��4 Th�Ui �iU|LhcoE^� ' f?c @?_ |�i �?_�Ui_ h@|�L?�?} tU�i4i }��i? M) E�c �� ' E��c�� �t ?L| M�?_�?}c ��i�?L UL?t�4ih �t h@|�L?i_ �? ��t _i4@?_ Lh t�TT*) Lu @?) Lu |�i }LL_t� Wu ^ �t ?L| �? � @?_tL RE^� 9' ^c |�i? RE^� *�it L? |�i ML�?_@h) Lu |�i ti| � Lu @_4�tt�M*i Th�Uitc |�i �iU|Lh^�RE^� *�it �? |�i }h@_�i?| CERE^�� @| RE^� |L � @?_ tL |�i Th�Uit @| RE^� U@??L| Mi 4L�i_u�h|�ih �? |�i _�hiU|�L? ^�RE^� ��|�L�| *i@��?} � � i @hi ?L� hi@_) |L ThL�i A�iLhi4 ��2��hLLu Lu A�iLhi4 ��2A@!i |�i u�?U|�L? s G'$ � � W+? �T�MU6 [ � _i�?i_ M)sE^� ' ERE^�c oE^�c %�E^�c � � � c %6E^��c ^ 5 '��L?t�_ih |�i UL??iU|i_ ti| � Lu 3ihL TL�?|t Lu 5 @t _i�?i_ �? A�iLhi4 ��H� W| uL**L�tM) A�iLhi4 ��S |�@| @?) TL�?| �? � �?_�Uit @ "�, Lu H � 5�?Ui s �t UL?|�?�L�t @?_ ��t UL??iU|i_ M) A�iLhi4 ��Hc �| uL**L�t |�@| sE�� �t UL??iU|i_� �) A�iLhi4 ��.c �| uL*�*L�t |�@| sE^� �t @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 ��i?i�ih ^ 5 'f @?_ sE^� �t@ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4 ��i?i�ih ^ 5 '�� 5�?Ui � �?|ihtiU|t ML|� 'f@?_ '�c sE�� UL?|@�?t ML|� @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_ @ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4� "�,�#�O@��?} ThL�i_ |�@| |�ihi i �t|t @ UL??iU|i_ ti| Lu "�,t UL?|@�?�?} ML|� @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_ @ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4c �| �t i@t) |L t�L�|�@| |��t ti| UL?|@�?t ti�ih@* i*i4i?|t �?_�U�?} "�,t ��|� tL4i tTiU��U ThLTih|�it� A�itiThLTih|�it @hi 4Lt| i@t�*) t|@|i_ �? |ih4t Lu |�i T@h@4i|h�3@|�L? M) |�i ti| 'c ��|� i@U�^ 5 ' �?_�U�?} @ Th�Ui RE^�c @ h@|�L?�?} �iU|Lh oE^�c @?_ @? @**LU@|�L? E%�E^�c � � � c %6E^���6�ht| Lu @**c ?L|�Ui |�@| uLh @?) ^f 5 � _ 'f �| �L*_t |�@| oE^f� ' �� @?_ |�@| uLh@?) ^� 5 � _'� �| �L*_t |�@| oE^�� ' �� A�ihiuLhic t�?Ui � UL??iU|t 'f @?_ '� @?_ o �t @UL?|�?�L�t u�?U|�L? L? 'c uLh @?) @ Th�Lh� U�Lti? & 5 U? |�ihi i �t|t @ ^E&� 5 � t@|�tu)�?}o&E^E&�� ' f� 5�U� @ ^E&� �?_�Uit @ h@|�L?�?} tU�i4i E�c �� ��|� �& ' ��& @?_ �& ' �&�A�i TL�?| ^E&� �? ' |�ihiuLhi �?_�Uit @ "�, @| ���U� ?L UL?t�4ih �t Mi�?} h@|�L?i_ �?UL44L_�|) &� i U@** t�U� @? i^��*�Mh��4 @ }i?ih@*�3i_ #hi3i i^��*�Mh��4 ��|� hitTiU||L UL44L_�|) &��LhL**@h) ��b6Lh @?) & 5 U?c |�ihi i �t|t @ }i?ih@*�3i_ #hi3i i^��*�Mh��4 ��|� hitTiU| |L UL44L_�|) &�A�i ULhL**@h) �4T*�it |�@| �u UL44L_�|) & �t |�i ?�4ih@�hic |�ihi i �t|t @ "�, @| ���U� |�i?�4ih@�hi �t ?L| Mi�?} h@|�L?i_� 5�4�*@h*)c |�ihi i �t|t @ ^3 5 � t@|�tu)�?} 4@ � o�E^3� ' f�

�.

Page 22: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

A��t i*i4i?| Lu � �?_�Uit @ "�, @| ���U� @| *i@t| L?i UL44L_�|) �t ?L| h@|�L?i_ @?_@** L|�ih UL44L_�|�it @hi h@|�L?i_ L?*) �? |�i�h t�TT*) @?_ |�ihiuLhi �t @ }i?ih@*�3i_t�TT*)�UL?t|h@�?i_ i^��*�Mh��4� 5�4�*@h*)c |�ihi i �t|t @ "�, @| ���U� UL?t�4iht @hiL?*) h@|�L?i_ �? |�i�h _i4@?_ @?_ @| *i@t| L?i UL44L_�|) �t ?L| h@|�L?i_c @ }i?ih@*�3i__i4@?_�UL?t|h@�?i_ i^��*�Mh��4� 5�U� @? i^��*�Mh��4 �t �?_�Ui_ M) @ TL�?| ^n �? � uLh���U� 4�?� o�E^n� ' f� �L|�Ui |�@| ML|� |)Tit Lu i^��*�Mh�@ @hi "�,t @| ���U� @| *i@t|L?i UL44L_�|) �t ?L| h@|�L?i_c M�| �| U@??L| Mi t@�_ �? @_�@?Ui ���U� UL44L_�|) �t ?L|h@|�L?i_��LhL**@h) ���fA�ihi i �t|t @ }i?ih@*�3i_ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_ |�ihi i �t|t @ }i?ih@*�3i__i4@?_�UL?t|h@�?i_ i^��*�Mh��4�e �TT*�U@|�L?t @?_ i |i?t�L?twi| �t UL?t�_ih |�i U@ti |�@| � �t @ M@** ��|� Ui?|ih @ }��i? Th�Ui t)t|i4 eR� `�|�L�| *LttLu }i?ih@*�|)c �i @tt�4i |�@| eR& : � uLh @?) & 5 U? @?_ |�@| |�i h@_��t Lu |�i M@** �t �c tL|�@| @?) Th�Ui �iU|Lh �? |�i ti| �t t|h�U|*) TLt�|��i� A�ihiuLhi |�i ti| Lu @_4�tt�M*i Th�Uit� �t }��i? M)

� ' iR 5 W+?mnR� eRn2 � �j��*i@h*)c |�i ti| ' �t @ M@** @hL�?_ eR ��|� h@_��t 2 @?_ uLh @?) ^ 5 ' �| �L*_t |�@| oE^� ' f?�u ^ 5 � @?_ oE^� ' >E^�ERE^� � eR� uLh tL4i TLt�|��i ?�4Mih >E^� ��i? ^ 5 ' q ���LhiL�ihc uLh @?) R 5 M?_E� �c �i �@�i |�@| CER� ' io 5 W+?mo ' bER� eR�c b � fj� A�i?ML|� ti|t 'f ' ieR � 28�82�j @?_ '� ' ieR n 28�82�j UL?|@�? L?i i*i4i?|� 6hL4 A�iLhi4��H �| uL**L�t |�@| |�ihi i �t|t @ UL??iU|i_ ti| Lu 3ihL TL�?|t Lu 5 UL??iU|�?} |�i �?�^�ii*i4i?| ^f ' eR� 28�82� Lu 'f ��|� |�i �?�^�i i*i4i?| ^� ' eRn 28�82� Lu '�� Oi?Uic |�ihii �t|t @ UL??iU|i_ ti| Lu "�,t UL??iU|�?} |�i �?�^�i |h���@* i^��*�Mh��4 ��|� u�** t�TT*)h@|�L?�?} @?_ Th�Ui �iU|Lh eR� �8�82� �?_�Ui_ M) ^f ��|� |�i �?�^�i |h���@* i^��*�Mh��4 ��|�u�** _i4@?_ h@|�L?�?} @?_ Th�Ui �iU|Lh eR n �8�82� �?_�Ui_ M) ^�� �UULh_�?} |L �LhL**@h)��b |��t UL??iU|i_ ti| UL?|@�?t @ }i?ih@*�3i_ #hi3i i^��*�Mh��4 ��|� hitTiU| |L UL44L_�|)& uLh @?) & 5 U?� W? U@ti |�i TL�?| ^E&� �?_�U�?} |��t i^��*�Mh��4 _Lit ?L| Mi*L?} |L �c �|4�t| �L*_ |�@| o&E^E&�� ' f @?_ oE^E&�� ' >&ERE^E&��� eR� uLh tL4i >& : fc �4T*)�?} |�@||�i Th�Ui R&E^E&�� Lu UL44L_�|) & 4�t| Mi ThiU�ti*) i^�@* |L eR&� W? U@ti ^E&� *�it �? �c^E&� �t @ @*h@t�@? Th�Ui t)t|i4� 5�44@h�3�?} �i �@�i |�i uL**L��?} ULhL**@h)��LhL**@h) e��5�TTLti � �t @ M@** Lu TLt�|��i Th�Uit ��|� Ui?|ih eR @?_ h@_��t �� A�i? |�ihi i �t| @

�H

Page 23: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

UL??iU|i_ ti| Lu "�,t UL??iU|�?} |�i �?�^�i |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 ��|�Th�Ui eR� �8�82� |L |�i �?�^�i |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4 ��|� Th�Ui eRn �8�82��A�i UL??iU|i_ ti| UL?|@�?t uLh i�ih) & 5 U? @ }i?ih@*�3i_ #hi3i i^��*�Mh��4 ��|� hitTiU| |L&� 5�U� @ #hi3i i^��*�Mh��4 �t @ `@*h@t�@? i^��*�Mh��4 Lh �@t R& ' eR&��t @ tiUL?_ i @4T*i �i UL?t�_ih |�i U@ti

� ' iR 5 W+?n m R� � R� � R�c � 5 Uc Z�ERU� � R� � Z�ERU�c � 5 aj E2���ihi U � U? �t @ ti| Lu �?_i UL44L_�|�it @?_ a ' U?qU �t |�i ti| Lu �?_i i_ UL44L_�|�it��|� Th�Uit ML�?_i_ uhL4 Mi*L� @?_ @ML�i M) �?_i u�?U|�L?t _iTi?_�?} L? |�i Th�Uit Lu|�i UL44L_�|�it �? U� A��t ti| }i?ih@*�3it |�i ti| Lu @_4�tt�M*i Th�Uit UL?t�_ihi_ �? #i�i3@?_ #hi3i E�bHe� @?_ t@|�t�it |�i tL�U@**i_ ?L?U�hU�*@h�|) UL?_�|�L? Lu i__iTL�* E�bH.�ct@)�?} |�@| |�i Th�Uit 4@) ?L| Mi �?_i i_ _�hiU|*) Lh �?_�hiU|*) M) |�i4ti*�it� i @tt�4i|�@| uLh @?) � 5 U �| �L*_t |�@| f R� � R� 4 @?_ |�@| uLh @?) � 5 a c |�i *L�ihML�?_ u�?U|�L? Z� �t UL?�i c |�i �TTih ML�?_ u�?U|�L? Z� �t UL?U@�i @?_ |�@| uLh @**ui@t�M*i RU �| �L*_t |�@| f Z�ERU� � Z�ERU� 4c tL |�@| � t@|�t�it �tt�4T|�L? �� AL}��i @ U�@h@U|ih�3@|�L? Lu @ "�,c uLh t�4T*�U�|) �i @tt�4i |�@| @** �?_i u�?U|�L?t @hiUL?|�?�L�t*) _�gihi?|�@M*i�5�TTLti ^ 5 ' �?_�Uit @ "�, @?_ |��t 5E^� ' f?� 5�?Ui oE^� 5 CERE^��c �i �@�i|�@| RE^� � oE^� ' 4@ RM� R � oE^�� 6hL4 |�i �ht|�Lh_ih k��?�A�U!ih UL?_�|�L?t �| |�i?uL**L�t |�@| |�ihi i �t|t ?L??i}@|��i ?�4Miht b� @?_ b� c � 5 U?c t�U� |�@|

o�E^� ' b� � b� n [�Ma b�YZ�ERUE^��YR� �[

�Ma b�YZ�ERUE^��YR� c � 5 Uc E��o�E^� ' b� � b� c � 5 ac Ee�

@?_b�ER�E^�� R�� ' fc b�ER� � R�E^�� ' fc b�b� ' fc � 5 Uc ED�b�ER�E^�� Z�ERUE^��� ' fc b�EZ�ERUE^��� R�E^�� ' fc b�b� ' fc � 5 a� ES�

+iU@** |�@| �? @ "�, �L?_�|�L? E���� Lu #i�?�|�L? 2�� �t t@|�t�i_c M�| ?L| ?iUitt@h�*)�L?_�|�L? E��� Lu #i�?�|�L? 2��� 5Lc ��i? @ UL44L_�|) �t h@|�L?i_ �? �|t t�TT*) E_i4@?_�c�|t Th�Ui �t ?L| ?iUitt@h�*) L? �|t *L�ih E�TTih� ML�?_� AL 4@!i |��t 4Lhi ThiU�tic �ht||@!i @? �?_i i_ UL44L_�|) � 5 a � 6hL4 Ee� @?_ |�i *@t| UL4T*i4i?|@h�|) UL?_�|�L? �?ES� �| uL**L�t |�@|b� ' �o�E^� : f �u o�E^� fc @?_ b� ' o�E^� : f �u o�E^� : f�

�b

Page 24: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

W| uL**L�t ��|� |�i �ht| |�L UL4T*i4i?|@h�|) UL?_�|�L?t �? ES� |�@| R�E^� ' Z�ERUE^�� �uUL44L_�|) � 5 a �t h@|�L?i_ �? �|t t�TT*) @?_ |�@| R�E^� ' Z�ERUE^�� �u UL44L_�|) � 5 a�t h@|�L?i_ �? �|t _i4@?_c ��i� �? U@ti Lu t�TT*) E_i4@?_� h@|�L?�?} |�i Th�Ui Lu @? �?_i i_UL44L_�|) �t L? �|t *L�ih E�TTih� ML�?_ }��i? |�i Th�Uit RUE^� Lu |�i �?_i UL44L_�|�it�6Lh @? �?_i UL44L_�|) � 5 Uc �? }i?ih@* t�U� @ U*i@h *�?! _Lit ?L| i �t|� OL�i�ihct�TTLti |�@| @** �?_i u�?U|�L?t @hi 4L?L|L?�U@**) �?Uhi@t�?} �? RU @?_ @** |�i _ih��@|��it@hi TLt�|��i� �L�c �u R&E^� : Z&ERUE^�� uLh @** & 5 a c |�i? b& ' f uLh @** & 5 a @?_ �|uL**L�t uhL4 i^�@|�L? E�� |�@| uLh @** � 5 U �| �L*_to�E^� ' b� � b� �[

&Ma b& YZ&ERUE^��YR� � b� � b�c@?_ |��t b� : f �u o�E^� : f� Oi?Uic �u |�i Th�Uit Lu |�i �?_i i_ UL44L_�|�it @hi @**@ML�i |�i�h *L�ih ML�?_c h@|�L?�?} L? |�i _i4@?_ Lu @? �?_i UL44L_�|) U@? L?*) LUU�h��i? �|t Th�Ui �t L? �|t �TTih ML�?_� �*i@h*)c ��i? |�i Th�Ui Lu @? �?_i i_ UL44L_�|) �tL? �|t *L�ih ML�?_c _i4@?_ h@|�L?�?} Lu @? �?_i UL44L_�|) 4@) LUU�hc U@�ti_ M) |�iu@U| |�@| |�i Th�Uit Lu |�i �?_i i_ UL44L_�|�it U@??L| Mi _iUhi@ti_ u�h|�ih� 5�4�*@h*)c �u|�i Th�Uit Lu |�i �?_i i_ UL44L_�|�it @hi @** Mi*L� |�i�h �TTih ML�?_c h@|�L?�?} L? |�it�TT*) Lu @? �?_i UL44L_�|) U@? L?*) LUU�h ��i? �|t Th�Ui �t L? �|t *L�ih ML�?_� W?T@h|�U�*@hc @| @ |h���@* t�TT*)�UL?t|h@�?i_ E_i4@?_�UL?t|h@�?i_� i^��*�Mh��4 |�i Th�Uit Lu@** |�i �?_i i_ UL44L_�|�it @hi L? |�i�h *L�ih E�TTih� ML�?_ }��i? RUE^�c @?_ |��t @*tL|�i Th�Uit Lu @** |�i �?_i UL44L_�|�it @hi L? |�i�h *L�ih E�TTih� ML�?_t� 5�44@h�3�?}�i �@�i |�i uL**L��?} ULhL**@h)��LhL**@h) e�2wi| � Mi }��i? @t �? E2� @?_ *i| ^ 5 ' Mi t�U� |�@| 5E^� ' f?� A�i?c uLh @** � 5 ac

R�E^� ' Z�ERUE^�� �u o�E^� f @?_ R�E^� ' Z�ERUE^�� �u o�E^� : f�`�i? @** �?_i u�?U|�L?t @hi 4L?L|L?�U@**) �?Uhi@t�?}c |�i?c uLh @** � 5 Uc

R�E^� ' R� �u o�E^� f @?_ uLh @** & 5 ac R&E^� Z&ERUE^��c@?_R�E^� ' R� �u o�E^� : f @?_ uLh @** & 5 ac R&E^� : Z&ERUE^���

�i | �i UL?t�_ih @? i |i?t�L? @**L��?} |�@| |�i ti| Lu @_4�tt�M*i Th�Uit �t �?ML�?_i_� it�L� |�@| |�ihi �t @? �?ML�?_i_ UL??iU|i_ ti| Lu "�,t ��|� @| *i@t| L?i Lu |�i Th�Uit}L�?} |L �?�?�|)� wi| � Mi @ UL?�i c U*Lti_c �?ML�?_i_ ti| Lu @_4�tt�M*i Th�Uit �? |�i�?|ih�Lh Lu W+?n @?_ _i�?i ' ' i^ 5 W+?mn^ � Rn2 � �c uLh tL4i R 5 �j� �L|�Ui |�@| 'f �t2f

Page 25: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

?L?�i4T|)c ���*i '� �t i4T|)� 6Lh ^ 5 'c *i| RE^�c oE^�c @?_ 5E^� Mi _i�?i_ @t MiuLhi� wi|�f ' S?&'��&^& uLh @?) ^ 5 'fc _i�?i uLh r � �f |�i ti| 'Er� M)'Er� ' i^ 5 'mS?&'��&^& � rjc@?_ uLh @?) r : �fc 'Er� ' i^ 5 'Er�mS?&'��&^& ' rj @t |�i �TTih ML�?_@h) Lu 'Er���TT*)�?} A�iLhi4 ��H |L 'Er� @?_ �t�?} |�i u@U| |�@| 'Er� _ff ' 'f @?_ 'Er� _f� ''Er�c �| uL**L�t |�@| |�ihi i �t|t @ UL??iU|i_ ti| Lu 3ihL TL�?|t Lu 5 �? 'Er� UL?|@�?�?} @TL�?| �? 'f @?_ @ TL�?| �? 'Er� uLh @?) r : rf� �*i@h*)c @ 3ihL TL�?| �? 'f �?_�Uit @}@�?@ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4� OL�i�ihc �? }i?ih@* @ 3ihL TL�?| �? |�i �TTihML�?_@h) 'Er� �t ?L| @ |h���@* _i4@?_�UL?t|h@�?i_ i^��*�Mh��4� A�i ?i | |�iLhi4 t@)t|�@| |�ihi i �t|t @? �?ML�?_i_ UL??iU|i_ ti| Lu 3ihL TL�?|t Lu 5 �? ' UL?|@�?�?} @ TL�?|�? 'f�A�iLhi4 e�� wi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc N@?_ `c @?_ *i| � Mi @ ?L?�i4T|)c UL?�i c U*Lti_c �?ML�?_i_ ti| �? |�i �?|ih�Lh Lu W+?n�A�i? |�ihi i �t|t @? �?ML�?_i_ UL??iU|i_ t�Mti| � Lu ' t�U� |�@| � _ 'f 9' > @?_ i@U�TL�?| �? � �t @ 3ihL TL�?| Lu |�i hi_�Ui_ i Uitt _i4@?_ u�?U|�L? 5��hLLu6Lh r : rfc *i| �Er� Mi @ UL??iU|i_ ti| Lu 3ihL TL�?|t Lu 5 �? 'Er� UL?|@�?�?} @ TL�?| �? 'f@?_ @ TL�?| �? 'Er�� A@!i � ' 'f ^ E^r:rf�Er��� 5�?Ui @?) TL�?| �? 'f �t @ 3ihL TL�?| Lu5c |�i ti| � �t @? �?ML�?_i_ UL??iU|i_ ti| Lu 3ihL TL�?|t Lu 5 �? ' t�U� |�@| � _'f 9' >�"�,�#��t @ tTiU��U i @4T*i �i UL?t�_ihc uLh tL4i @hM�|h@h�*) t4@** 0 : fc |�i ti| � }��i? M)

� ' iR 5 W+?n mR� � 0c � 5�c [�MU R� ' �c R� � R� � R�c � 5 U ^ ajc E.�

��ihi � c U @?_ a uLh4 @ T@h|�|�L? Lu U?�2 A��t ti| }i?ih@*�3it ti|t Lu @_4�tt�M*i Th�UitUL?t�_ihi_ �? k�h3 E�bH2�c �@? _ih w@@? E�bHe� @?_ i__iTL�* E�bH.�c ��ihi� �t |�i ti|Lu ?�4ih@�hi UL44L_�|�it ��Lti Th�Uit U@? 4L�i uhii*) @ML�i 0c U �t @ ti| Lu �?_i UL4�4L_�|�it ��|� t�4 Lu |�i Th�Uit ?Lh4@*�3i_ |L L?ic @?_ a �t @ ti| Lu �?_i i_ UL44L_�|�it��|� *L�ih @?_ �TTih ML�?_t L? |�i Th�Uit� i @tt�4i |�@| uLh @?) � 5 U ^ a �| �L*_t|�@| f R� � R� 4 @?_ S�MU R� � S�MU R� � `�|�L�| *Ltt Lu }i?ih@*�|)c �i 4@)@tt�4i |�@| U ' i�c � � � c mUmj @?_ ��n� � �� c � ' �c � � � mUm � �� �LhiL�ihc uLh |�i t@!i Lut�4T*�U�|)c �i hit|h�U| L�hti*�it |L |�i }i?ih�U U@ti |�@| |�ihi �t ?L T@h|�|�L? U�c U2 Lu U5Iru vlpsolflw| zh wdnh sulfhv erxqghg dzd| iurp }hur1 Wkh dqdo|vlv fdq eh h{whqghg wr % @ 3 e|jhqhudol}lqj wkh frqfhsw ri frqvwudlqhg htxloleulxp wr txdvl0frqvwudlqhg htxloleulxp dv kdv ehhq glvfxvvhglq Nxu} +4<;5,1

2�

Page 26: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

t�U� |�@| S�MU� R� nS�MU2 R� ' �� A�i?c *i| �� 5 U Mi t�U� |�@| S���'� R� nS�U��'��n� R� �@?_ S��3��'� R� nS�U��'�� R� : ��5�TTLti ^ 5 ' �?_�Uit @ "�, @?_ |��t 5E^� ' f?� �}@�?c oE^� 5 CERE^�� �4T*�itRE^� � oE^� ' 4@ RM� R � oE^�� 6hL4 |�i �ht|�Lh_ih k��?�A�U!ih UL?_�|�L?t �| |�i? uL**L�t|�@| |�ihi i �t| ?L??i}@|��i ?�4Miht b� c � 5 �c ?L??i}@|��i ?�4Miht b�c b� c � 5 U ^ ac@?_ @ ?�4Mih bU t�U� |�@|o�E^� ' �b�c � 5�c EH�o�E^� ' bU � b� n b� c � 5 Uc Eb�o�E^� ' �b� n b� c � 5 ac E�f�

@?_ u�h|�ih4Lhi |�i uL**L��?} UL4T*i4i?|@h�|) UL?_�|�L?t �L*_b�ER�E^�� 0� ' fc � 5�c E���b�ER�E^�� R�� ' fc b�ER� � R�E^�� ' fc b�b� ' fc � 5 U ^ a� E�2�

�| @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 �i �@�i |�@| oE^� ' ��c tL |�@| b� ' �� @?_�i?Ui R�E^� ' 0 uLh @** � 5 � c b� ' �� c b� ' f @?_ �i?Ui R�E^� ' R� uLh @** � 5 a c ���*iuLh � 5 U �| �L*_tbU � b� n b� ' ����i U*@�4 |�@| bU ' ����c uLh @** � ' �c � � � c ���� �| �L*_t |�@| b� ' f @?_ b� ' ������ � fcb�� ' b�� ' fc @?_ uLh @** � ' �� n �c � � � c mUm �| �L*_t |�@| b� ' f @?_ b� ' �� � ��� � f� W||�i? uL**L�t �44i_�@|i*) |�@| Th�Uit Lu UL44L_�|�it � ' �c � � � c �� � � t@|�tu) R� ' R� c @?_Th�Uit Lu UL44L_�|�it � ' �� n �c � � � c mUm t@|�tu) R� ' R� � W?|��|��i*)c uhL4 |�i ThLTih|�it|�@| RE^� � oE^� ' 4@ RM� R � oE^�c oE^� ' �� @?_ S�MU R� ' �c �| uL**L�t |�@| |�i Th�UitLu |�i UL44L_�|�it ��|� |�i *@h}ih |L|@* i?_L�4i?|t @hi ti| L? |�i�h 4�?�4�4 *i�i*c @?_|�i Th�Uit Lu UL44L_�|�it ��|� |�i t4@**ih |L|@* i?_L�4i?|t L? |�i�h 4@ �4�4 *i�i*�AL uLh4@**) t�L� L�h U*@�4c UL?t�_ih @ UL44L_�|) & 5 i�c � � � c mUmj ' U t�U� |�@|R& R&E^� R&� 5�U� @ UL44L_�|) & i �t|tc t�?Ui �i UL?t�_ih |�i U@ti |�@| |�ihi �t?L T@h|�|�L? U�c U2 Lu U t�U� |�@| S�MU� R� n S�MU2 R� ' �� W| uL**L�t �44i_�@|i*) |�@|b& ' b& ' fc tL bU ' ��&� �L?t�_ih @?) UL44L_�|) � 5 i�c � � � c & � �j� A�i? �| uL**L�tuhL4 i^�@|�L? Eb� |�@| b� � b� ' bU � o�E^� ' bU n �� ' ��& n �� � fc tL |�@| b� ' f@?_ b� ' �& � �� � f� 5�4�*@h*)c uLh @?) UL44L_�|) � 5 i& n �c � � � c mUmj �| uL**L�t uhL4i^�@|�L? Eb� |�@| b��b� ' ��&n�� � fc tL |�@| b� ' f @?_ b� ' ��&n�� � f� AL}i|�ih��|� |�i UL4T*i4i?|@h�|) ThLTih|�it }��i? �? E�2� �| uL**L�t |�@|&3�[�'� R�E^� n R&E^� n �U�[

�'&n� R�E^� ' &3�[�'� R� n R&E^� n �U�[

�'&n� R� ' ��22

Page 27: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

A�ihiuLhic&[

�'� R� n �U�[�'&n� R� � �c

&3�[�'� R� n �U�[

�'& R� � �ctL & ' ��� 5�44@h�3�?}c @| @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4c |�i Th�Uit Lu @** |�i?�4ih@�hi @?_ �?_i i_ UL44L_�|�it @hi L? �|t *L�ih ML�?_c ���*i |�i Th�Ui Lu ThiU�ti*)L?i �?_i UL44L_�|) �t ?L| ML�?_i_ @?_ |�i Th�Uit Lu |�i �?_i UL44L_�|�it ��|� *@h}ih|L|@* i?_L�4i?|t @hi ti| L? |�i�h 4�?�4�4 *i�i*c @?_ |�i Th�Uit Lu |�i �?_i UL44L_�|�it��|� t4@**ih |L|@* i?_L�4i?|t L? |�i�h 4@ �4�4 *i�i*�6hL4 |�i i^�@|�L?t EH��E�2� �| uL**L�t |�@| @| @?) "�, �| �L*_t |�@| @ ?�4ih@�hiUL44L_�|) U@? L?*) Mi h@|�L?i_ �? �|t t�TT*)c ���U� L?*) LUU�ht ��i? �|t Th�Ui �t 0c ���*i@? �?_i i_ UL44L_�|) U@? L?*) Mi h@|�L?i_ �? �|t t�TT*) E_i4@?_� ��i? �|t Th�Ui �t L? �|t*L�ih E�TTih� ML�?_� 6Lh @? �?_i UL44L_�|) � 5 U h@|�L?�?} U@? @*tL LUU�h ��i? �|t Th�Ui�t ?L| M�?_�?}c _�i |L |�i hit|h�U|�L? |�@| |�i t�4 Lu |�i Th�Uit Lu |�i �?_i UL44L_�|�it �ti^�@* |L L?i� 6Lh @** UL44L_�|�it � 5 U ��|� ?L?�M�?_�?} Th�Uit �i �@�i |�@| o�E^� ' bU �A�ihiuLhic @** |�iti UL44L_�|�it @hi i�|�ih ?L| h@|�L?i_ EbU ' f�c Lh h@|�L?i_ ��|� |�it@4i @4L�?| L? i�|�ih |�i�h _i4@?_t EbU : f� Lh |�i�h t�TT*�it EbU f�� �| @ "�,�?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} S�MU ^� ' �c �| �L*_t |�@| bU ' f @?_ @ UL44L_�|) �? UU@? L?*) Mi h@|�L?i_ �? �|t t�TT*) E_i4@?_� �u �|t Th�Ui �t L? �|t *L�ih E�TTih� ML�?_� A�ii �t|i?Ui Lu t�U� @ TL�?| uL**L�t @t @ ULhL**@h) uhL4 |�i ?i | |�iLhi4�A�iLhi4 e�e wi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc Nc` @?_ *i| |�i ti| � Mi @t tTiU��i_ �? E.�� A�i? |�ihi i �t|t @? r : rf t�U� |�@| uLh @**3ihL TL�?|t ^ 5 ' Lu 5 ��|� S?&'� ^& � r �| �L*_t |�@| o�E^� ' �� uLh @** � 5 U ^ac R�E^� : 0@?_ o�E^� ' f uLh @** � 5���hLLu�6hL4 A�iLhi4 e�� �| uL**L�t |�@| |�ihi i �t|t @ ti^�i?Ui E^|�|MW� �?'c t@|�tu)�?}S?�'� ^|� � |@?_ 5E^|� ' f? uLh @** | 5 W�� 5�?Ui � @?_ |��t @*tL ' �t ML�?_i_ �? @?) � 5 U ^a c ��|�L�|*Ltt Lu }i?ih@*�|) �| uL**L�t |�@| ^|& $4 uLh tL4i & 5� � W| �t LM��L�t |�@| o&E^|� ' f� �L�c_i�?i U| ' i� 5 Umo�E^|� ��jc a | ' i� 5 a mo�E^|� ��j @?_ � | ' i� 5 � mR�E^|� ' 0j�A�i?c uLh @?) � 5 U ^ a ^� | �i �@�i |�@| R�E^|�R&E^|� }Lit |L 3ihL� �LhiL�ihc uLh @** � 5 U6 @?_� 5 U?c |�i _i4@?_ _��E^|� �t ML�?_i_ uhL4 @ML�i M) ��� n E�� � o�E^|��c ���*i o&E^|� ' f�4T*�it ?i| t�TT*) i^�@* |L ��& �t TLtt�M*i uLh UL44L_�|) &� 6Lh | t��U�i?|*) *@h}ic uLh@?) � 5 U6c uLh @?) � 5 U| ^ a | ^� |c �| uL**L�t uhL4 |�i 4L?L|L?�U�|) Lu |�i Thiuihi?Uit|�@| _��E^|� ' ��� n E�� � o�E^|�� : ���c UL?|h@_�U|�?} 5E^|� ' f?� Oi?Ui U| ^ a | ^� | ' >�

2�

Page 28: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

5Lc uLh | t��U�i?|*) *@h}i �| uL**L�t |�@| o�E^|� ' �� @?_ |�@| R�E^|� : 0 @?_ o�E^|� ' f uLh@** � 5� � "�,�#�A�i |�iLhi4 �4T*�it |�@| |�ihi i �t|t @ UL??iU|i_ ti| Lu "�,t UL?|@�?�?} @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_ @? i^��*�Mh��4 ��|� u�** h@|�L?�?} L? |�i _i4@?_ Lu |�i UL4�4L_�|�it �? U ^ a @?_ ?L h@|�L?�?} @| @** L? |�i UL44L_�|�it �? � ��|� Th�Uit @ML�i 0��*i@h*)c @| |�i *@||ih i^��*�Mh��4 �| 4�t| �L*_ |�@| uLh i�ih) UL?t�4ih |�i UL?t�4T|�L? Lu|�i UL44L_�|�it �? U ^ a �t i^�@* |L �|t �?�|�@* i?_L�4i?|t Lu |�iti UL44L_�|�it� Oi?Ui|�i Th�Uit Lu |�i UL44L_�|�it �?� @hi @*h@t�@? i^��*�Mh��4 Th�Uit uLh @ hi_�Ui_ iUL?L4)��|� L?*) UL44L_�|�it �? � @?_ Thiuihi?Uit L? |��t hi_�Ui_ UL44L_�|) tT@Ui LM|@�?i_uhL4 |�i Thiuihi?Uit �? H ��|� UL?t�4T|�L? Lu |�i UL44L_�|�it �? U ^a i^�@* |L |�i �?�|�@*i?_L�4i?|t� �| @?) ^f �?_�U�?} @ |h���@* i^��*�Mh��4 �i �@�i |�@| oE^f� ' ��c ���*i @|@?) ^ �?_�U�?} @? i^��*�Mh��4 ��|� u�** h@|�L?�?} L? |�i _i4@?_ Lu UL44L_�|�it �? U ^ a@?_ @*h@t�@? Th�Uit uLh |�i UL44L_�|�it �? � c �| �L*_t |�@| o�E^� ' �� uLh @** � 5 U ^ a c@?_ o�E^� ' f uLh @** � 5� � �Mtih�i |�@| o�E^f� ' ��� uLh @** � 5 U �4T*�it |�@| uLh tL4i>f : f[

�MU ^f� ' [�MUE>fo�E^f� n R�E^f�� ' �>f [�MU �� n[

�MU R�E^f� ' �>f [�MU �� n � �@?_ |�@| o�E^� ' �� uLh @** � 5 U �4T*�it |�@| uLh tL4i > : f[

�MU ^� ' [�MUE>o�E^� n R�E^�� ' >[

�MU �� n[�MU R�E^� ' >[

�MU �� n � : ��6hL4 |��t �i LM|@�? |�i uL**L��?} ULhL**@h)��LhL**@h) e�D wi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc Nc` @?_ *i| |�i ti| � Mi @t tTiU��i_ �? E.�� A�i? |�ihi i �t|t @ UL??iU|i_ ti| Lu "�,tcUL?|@�?�?} @ |h���@* t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 @?_ |�i uL**L��?} |)Tit Lu i^��*�Mh��4GE�� 6Lh @?) � 5 U ^ a c @? i^��*�Mh��4 �?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} o�E^� ' fc ��i� @}i?ih@*�3i_ #hi3i i^��*�Mh��4 ��|� hitTiU| |L UL44L_�|) � 5 U ^ a(E��� �? i^��*�Mh��4 �?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} o�E^� ' f uLh @** � 5� c ��i� @ }i?ih@*�3i_#hi3i i^��*�Mh��4 �? ���U� ?L UL44L_�|) � 5� �t h@|�L?i_(E���� �? i^��*�Mh��4 �?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} 4@ �MU? o�E^� ' fc ��i� @ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4c tL @| *i@t| L?i Lu |�i UL44L_�|�it �t �?UL?t|h@�?i_(E��� �? i^��*�Mh��4 �?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} 4@ �MU�a o�E^� ' fc ��i� @ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 �? ���U� @| *i@t| L?i Lu |�i UL44L_�|�it �? U ^ a �t �?UL?t|h@�?i_(E�� �? i^��*�Mh��4 �?_�Ui_ M) @ TL�?| ^ t@|�tu)�?} S�MU ^� ' �c ��i� @ UL44L_�|) �? U U@?L?*) Mi h@|�L?i_ L? �|t t�TT*) E_i4@?_� �u �|t Th�Ui �t L? �|t *L�ih E�TTih� ML�?_(E��� �? i^��*�Mh��4 ��|� u�** h@|�L?�?} L? |�i _i4@?_t L? |�i UL44L_�|�it �? U ^ a @?_`@*h@t�@? Th�Uit uLh |�i hi_�Ui_ iUL?L4) ��|� UL44L_�|�it �? � �

2e

Page 29: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

�| @? i^��*�Mh��4 Lu |)Ti E��� �| 4@) �L*_ uLh tL4i � 5� |�@| o�E^� f @?_ �i?Ui t�TT*)h@|�L?�?} LUU�ht L? tL4i Lu |�i UL44L_�|�it �? � � �| t�U� @? i^��*�Mh��4 �i �@�i |�@|uLh tL4i � 5 U ^ a c o�E^� ' f @?_ R�E^� � R� : f @?_ |��t R�E^�E��� � o�E^�� f� �L�ct�TTLti |�@| o�E^� f @?_ |��t R�E^� ' 0 uLh tL4i � 5 � � 6hL4 |�i 4L?L|L?�U�|) Lu |�iThiuihi?Uit �| |�i? uL**L�t uLh t��U�i?|*) t4@** 0 |�@| _��E^� ' ��� n E�� � o�E^�� : ��� uLh@** � 5 U6c UL?|h@_�U|�?} |�@| 5�E^� ' f� Oi?Ui �i 4�t| �@�i |�@| o�E^� ' f uLh @** � 5� c�4T*)�?} |�@| @** UL44L_�|�it �? � @hi �?UL?t|h@�?i_��LhL**@h) e�S wi| H ' Ei[ �c��c ��j6�'�c � � Mi @? iUL?L4) t@|�tu)�?} �tt�4T|�L?t jc Nc` @?_ *i| |�i ti| � Mi @t tTiU��i_ �? E.�� A�i? uLh 0 t��U�i?|*) U*Lti |L 3ihLc |�ihi i �t|t@ t�TT*)�UL?t|h@�?i_ i^��*�Mh��4 �? ���U� ?L?i Lu |�i UL44L_�|�it �? � @hi h@|�L?i_ @?_@| *i@t| L?i Lu |�i UL44L_�|�it �? U ^ a �t ?L| h@|�L?i_�A��t ULhL**@h) }i?ih@*�3it |�i hit�*|t Lu �@? _ih w@@? E�bHe� @?_ i__iTL�* E�bH.� LM�|@�?i_ uLh |�i U@ti ��|� L?i 4L?i) UL44L_�|) �? � @?_ ?L *L�ih @?_ �TTih ML�?_t uLh|�i �?_i UL44L_�|�it �? U� 6�?@**)c �u U^a ' >c |�i i^��*�Mh��4 Lu |)Ti E��� �t @ @*h@t�@?i^��*�Mh��4 uLh |�i iUL?L4) H ��|� Th�Ui ti| � ' iR 5 W+?n mR� � 0c � 5 U?j @?_ |��t �?|��t U@ti |�ihi �t @ UL??iU|i_ ti| Lu i^��*�Mh�@ UL??iU|�?} |�i |h���@* ?L�|h@_i u�**) t�TT*)UL?t|h@�?i_ i^��*�Mh��4 ��|� @ @*h@t�@? i^��*�Mh��4�+iuihi?Uitd�o k�a� �hhL� @?_ B� #iMhi� E�bDe�c , �t|i?Ui Lu @? i^��*�Mh��4 uLh @ UL4Ti|�|��iiUL?L4)c ,UL?L4i|h�U@c 22c 2SD�2bf�d2o a���� ��i?@tt) E�b.D�c �iL�ki)?it�@? _�ti^��*�Mh��4 |�iLh) �? @ 4L?i|@h) iUL?L4)c+i��i� Lu ,UL?L4�U 5|�_�it e2c Df��D2��d�o a���� ��i?@tt) E�bb��c �L?U*i@h�?} 4@h!i|tG 4�UhLiUL?L4�U UL?UiT|t @?_ 4@UhLiUL�?L4�U @TT*�U@|�L?tc aL�h?@* Lu ,UL?L4�U w�|ih@|�hi ��c .�2�.S��deo 6�,� �hL�_ih E�bSf�c �? UL?|�?��|) Lu � i_ TL�?|t �?_ih _iuLh4@|�L? Lu UL?|�?�L�t4@TT�?}tc 5�44@ �h@t�*�@ �@|�i4@|�U@ ec �H���b��dDo V�k� ��i||) @?_ ��+� �@)@! E�b.H�c #hi3i i^��*�Mh�@ uLh TL*)�i_h@* @?_ t|h�U|*) UL?�i Th�Ui ti|tc #�tU�tt�L? T@Tihc W?_�@? 5|@|�t|�U@* W?t|�|�|ic �i� #i*��c W?_�@�dSo �� ��|@??@c O� �hitc a� #hi3ic ��a�a� Oih�?}tc @?_ �� V�**@?@UU� E2ff��c �L?|�?�@ Lu�?_ihi4T*L)4i?| i^��*�Mh�@ hi�iU|�?} ULLh_�?@|�L? u@�*�hitc @*tL @| @*h@t�@? Th�Uitc�,A,�+ +iti@hU� �i4Lh@?_�4 f�%�fc N?��iht�|i�| �@@t|h�U�|c TT� �����

2D

Page 30: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

d.o ���� �L E�bHf�c A�i i?uLhUi4i?| Lu T�M*�U Th�Ui UL?|hL*tc aL�h?@* Lu �L*�|�U@* ,UL?�L4) HHc HH.�b�S�dHo B� #iMhi� E�bDb�c A�iLh) Lu V@*�ic v@*i N?��iht�|) �hittc �i� O@�i?c N5��dbo �� #i�i3 @?_ a�O� #hi3i E�bHe�c �? t�TT*)�UL?t|h@�?i_ i^��*�Mh�@c aL�h?@* Lu ,UL�?L4�U A�iLh) ��c �.2��H2�d�fo a�O� #hi3i E�b.D�c , �t|i?Ui Lu @? i U�@?}i iUL?L4) �?_ih Th�Ui h�}�_�|�itc W?|ih?@�|�L?@* ,UL?L4�U +i��i� �Sc ��f��2f�d��o a�O� #hi3i E�bb.�c @*h@t�ki)?it i^��*�Mh�@c ULLh_�?@|�L? @?_ 4@UhLiUL?L4�Utc ,��hLTi@? ,UL?L4�U +i��i� e�c �.�.��.S2�d�2o a�O� #hi3i E2ff��c �? |�i 4@UhLiUL?L4�Ut Lu �?Uih|@�?|) @?_ �?UL4T*i|i 4@h!i|tc+iU�ihU�it ,UL?L4�^�it _i wL��@�?c S.c ��2b�d��o a�O� #hi3i @?_ �� BL**�ih E�bb��c +�t! t�@h�?} L? |�i *@ML�h 4@h!i|c ,�hLTi@? ,UL�?L4�U +i��i�c �.c �eD.��eH2�d�eo V��� B�?tM�h}� @?_ w� V@? _ih Oi)_i? E�b.D�c �? i |i?_�?} |�i �i}�t�� @TThL@U�|L UL4T�|�?} i^��*�Mh�@G |�i U@ti Lu }L�ih?4i?| Th�Ui t�TTLh| TL*�U�itc aL�h?@* Lu,UL?L4�U A�iLh) eec �SH��.H�d�Do ��a�a� Oih�?}t E�bbS@�c 5|@|�U @?_ #)?@4�U �tTiU|t Lu Bi?ih@* ,^��*�Mh��4 A�iLh)ck*��ih �U@_i4�U ��M*�t�ihtc #Lh_hiU�|c A�i �i|�ih*@?_t�d�So ��a�a� Oih�?}t E�bbSM�c ,^��*�Mh��4 i �t|i?Ui hit�*|t uLh iUL?L4�it ��|� Th�Ui h�}�_��|�itc ,UL?L4�U A�iLh)c .c S��Hf�d�.o ��a�a� Oih�?}t E�bb.�c ,?_L}i?L�t*) _i|ih4�?i_ Th�Ui h�}�_�|�itc ,UL?L4�U A�iLh)c bce.��ebH�d�Ho ��a�a� Oih�?}t E�bbH�c �? |�i i �t|i?Ui Lu @ UL?|�?��4 Lu UL?t|h@�?i_ i^��*�Mh�@c aL�h�?@* Lu �@|�i4@|�U@* ,UL?L4�Utc �fc �L� �c 2D.�2.��d�bo ��a�a� Oih�?}t @?_ O��� �L*i4@hU�@!�t E2fff�c �@hi|L �4ThL��?} Th�Ui hi}�*@|�L?��i? |�i @tti| 4@h!i| �t �?UL4T*i|ic �,A,�+ +iti@hU� �i4Lh@?_�4 ff%�Sc N?���iht�|i�| �@@t|h�U�|�d2fo ��a�a� Oih�?}tc ��a�a� A@*4@?c @?_ ~� v@?} E�bbS�c A�i UL4T�|@|�L? Lu @ UL?|�?��4Lu UL?t|h@�?i_ i^��*�Mh�@c �@|�i4@|�Ut Lu �Tih@|�L?t +iti@hU�c 2�c S.D�SbS�

2S

Page 31: Quantity Constrained Equilibria · TI 2001-116/1 Tinbergen Institute Discussion Paper Quantity Constrained Equilibria P. Jean-Jacques Herings1 Gerard van der Laan2 Dolf Talman3 1

d2�o �� k�h3 E�bH2�c N?i4T*L)4i?| i^��*�Mh�@ �? @? iUL?L4) ��|� *�?!i_ Th�Uitc aL�h?@*Lu ,UL?L4�U A�iLh) 2Sc ��f��2��d22o B� �@? _ih w@@? E�bHf�c ,^��*�Mh��4 �?_ih h�}�_ Th�Uit ��|� UL4Ti?t@|�L? uLh |�iUL?t�4ihtc W?|ih?@|�L?@* ,UL?L4�U +i��i� 2�c D��.��d2�o B� �@? _ih w@@? E�bH2�c 5�4T*�U�@* @TThL �4@|�L? Lu �?i4T*L)4i?| i^��*�Mh�@c aL�h?@*Lu �@|�i4@|�U@* ,UL?L4�Ut bc H��b.�d2eo B� �@? _ih w@@? E�bHe�c 5�TT*)�UL?t|h@�?i_ � i_ Th�Ui i^��*�Mh�@ �? 4L?i|@h)iUL?L4�itc aL�h?@* Lu �@|�i4@|�U@* ,UL?L4�Ut ��c �.���H.�d2Do A�A� �}�)i? @?_ a� `�@**i) E�bHS�c ,^��*�Mh��4 �?_ih Th�Ui UL?|hL*t ��|� i?_L}i?L�t|h@?t@U|�L?t ULt|tc aL�h?@* Lu ,UL?L4�U A�iLh) �bc 2bf��ff�d2So A�A� �}�)i? @?_ a� `�@**i) E�bbf�c Bi?ih@* i^��*�Mh��4 @?@*)t�t Lu Th�Ui UL?|hL*tG @UL4T�@|�L?@* @TThL@U�c W?|ih?@|�L?@* ,UL?L4�U +i��i� ��c SS.�SHe�d2.o a� A��?t|h@ E2fff�c A�i i4ih}i?Ui Lu TL*�|�U@* M�t�?itt U)U*it �? @ |�L�tiU|Lh }i?ih@*i^��*�Mh��4 4L_i*c ,�hLTi@? aL�h?@* Lu �L*�|�U@* ,UL?L4)c �Sc Dfb�D�e�d2Ho v� vL�?it E�b.D�c �? |�i hL*i Lu 4L?i) �? |�i ThLUitt Lu i U�@?}i @?_ |�i i �t|i?UiLu @ ?L?� @*h@t�@? i^��*�Mh��4c +i��i� Lu ,UL?L4�U 5|�_�itc e2c eHb�Df��d2bo w� @*h@t E�H.e�c �,*�i4i?|t _< �iUL?L4�i TL*�|�^�i T�hic �LhM@3c w@�t@??ic 5��|3ih�*@?_�d�fo �� i__iTL�* E�bH.�c 5�TT*)�UL?t|h@�?i_ i^��*�Mh�@ �? iUL?L4�it ��|� �?_i i_ Th�UitcaL�h?@* Lu ,UL?L4�U A�iLh) e�c 2f��222�

2.