А.А. Васин, В.В. Морозов - Введение в теорию игр с приложениями к экономике (1184512), страница 38
Текст из файла (страница 38)
È� ïðèâåäåííû� âûø� óòâåðæäåíè� ìîæí� ñäåëàò� âûâîä, ÷ò� ìíîæåñòâ� ðàâíîâåñíû� öå� − ýò� ëèá� òî÷ê� (åäèíñòâåííà� öåí� p̃), ëèá� ñóùåñòâóå� îòðåçî� ðàâíîâåñíû� öå� [p̃1 , p̃2 ]. Çàìåòèì, ÷ò� ñëó÷àè, ïðîèëëþñòðèðîâàííû� í� ðèñóíêà� 16.6-7, íåòèïè÷í� � íîñÿ� âûðîæäåííû� õàðàêòåð, òà� êà� ïð� ìàëî� âîçìóùåíè� ïàðàìåòðî� ìîäåë� ãðàôèê� ñìåùàþòñ� � ïåðåñå÷åíè� ï� öåëîì� îòðåçê� ïðåâðàùàåòñ� � ïåðåñå÷åíè� � òî÷êå.
Ïð� ýòî� âîçìîæí� îäí� è� èçîáðàæåííû� í� ñëåäóþùå� ðèñ. 16.8 ñèòóàöèé: V6V 6ṼṼ-p̃-pp̃p Ðèñ. 16.8 � òèïè÷íû� ñëó÷àÿ� (èëè, èíûì� ñëîâàìè, � óñëîâèÿ� îáùåã� ïîëî185ÃËÀÂ� IV. ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�æåíèÿ) ðàâíîâåñíà� öåí� � ðàâíîâåñíû� îáúå� îïðåäåëÿþòñ� åäèíñòâåííû� îáðàçîì, ÷ò� � ïðåäïîëàãàåòñ� � äàëüíåéøåì. 17. Ìîíîïîëèçèðîâàííû� ðûíî� Ðàññìîòðè� ñèòóàöèþ, êîãä� í� ðûíê� ïðèñóòñòâóå� ëèø� îäí� ôèðìàïðîèçâîäèòåëü. Êà� � ðàíüøå, ïðîèçâîäèòåë� õàðàêòåðèçóåòñ� ñåáåñòîèìîñòü� òîâàðà, ò.å.
ôóíêöèå� èçäåðæå� C(V ), êîòîðà� ïîêàçûâàåò, ñêîëüê� äåíå� íåîáõîäèì� çàòðàòèòü, ÷òîá� ïðîèçâåñò� òîâà� � îáúåì� V . Ïîòðåáèòåë� í� ýòî� ðûíê� ïîëàãàþòñ� ìåëêèìè. È� ïîâåäåíè� õàðàêòåðèçóåòñ� ñóììàðíî� ôóíêöèå� ñïðîñ� D(p), ïîêàçûâàþùåé, êàêî� îáúå� òîâàð� áóäå� êóïëå� ïð� çàäàííî� öåí� p. Ôèðìà-ìîíîïîëèñ� óñòàíàâëèâàå� öåí� í� òîâà� p � îáúå� åã� ïðîèçâîäñòâ� V .
Òàêè� îáðàçîì, ñòðàòåãèå� ìîíîïîëè� ÿâëÿåòñ� ïàð� (p, V ). Ïîèñ� îïòèìàëüíî� ñòðàòåãè� ìîíîïîëè� Îáñóäè� çàäà÷� ïîèñê� îïòèìàëüíî� ñòðàòåãè� ìîíîïîëèè. Îí� ñîñòîè� � âûáîð� öåí� í� òîâà� p∗ � îáúåì� ïðîèçâîäñòâ� V ∗ , ìàêñèìèçèðóþùè� ïðèáûë� ìîíîïîëèè.
Çàäà÷� ðåøàåòñ� ïð� ñëåäóþùè� îãðàíè÷åíèÿõ: öåí� íåîòðèöàòåëüíà, � îáúå� âûïóñê� íåîòðèöàòåëå� � í� ïðåâîñõîäè� ñïðîñ� (ïðîèçâîäèòåë� íå� ñìûñë� ïðîèçâîäèò� áîëüø� òîâàðà, ÷å� ïîòðåáèòåë� ãîòîâ� êóïèòü). Ôîðìàëüí� ýò� çàäà÷� ñâîäèòñ� � íàõîæäåíè� îïòèìàëüíî� ñòðàòåãè� (p∗ , V ∗ ) ∈ Arg max (p,V ): p≥0, 0≤V ≤D(p) (pV − C(V )). (17.1) Áóäå� çàïèñûâàò� ôóíêöè� ñïðîñ� � âèä� D(p) = [D− (p), D+ (p)], ãä� D− (p), D+ (p) − íèæíÿ� � âåðõíÿ� ãðàíèö� äë� D(p).
Àíàëîãè÷íà� çàïèñ� áóäå� èñïîëüçîâàòüñ� � äë� ôóíêöè� ïðåäëîæåíèÿ: S(p) = [S − (p), S + (p)]. Ðåøåíè� îïòèìèçàöèîííî� çàäà÷� (17.1) ïðîâåäå� � äâ� ýòàïà. Ýòà� 1. Îïòèìèçàöè� ï� îáúåì� ïð� ôèêñèðîâàííî� öåí� Ôèêñèðóå� ëþáó� öåí� p � îïðåäåëè� îïòèìàëüíî� çíà÷åíè� V ∗ (p). Çàìåòèì, ÷ò� äë� ìîíîïîëèçèðîâàííîã� ðûíê� ìîæí� òà� æå, êà� � äë� êîíêóðåíòíîã� ðûíêà, ââåñò� ôîðìàëüí� ôóíêöè� ïðåäëîæåíè� S(p) = Arg max(pV − C(V )) � íàéò� ðàâíîâåñíó� öåí� p̃, òàêóþ, ÷ò� V ≥0 S(p̃) ∩ D(p̃) 6= ∅. 186 17.
Ìîíîïîëèçèðîâàííû� ðûíî�Óòâåðæäåíè� 17.1. Åñë� p̃ −ðàâíîâåñíà� öåíà, ò�⎧ p < p,˜⎨ S(p), ∗+V (p) ∈ S(p) ∩ [0, D (p)], p = p,˜⎩ +D (p), p > p.˜Äîêàçàòåëüñòâî. Ïóñò� p < p.˜ Çàìåòèì, ÷ò� åñë� îòáðîñèò� � çàäà÷� (17.1) îãðàíè÷åíè� í� îáúåì, ò� å� ðåøåíè� ïð� ôèêñèðîâàííî� p ñîâïàäàå� ñ� çíà÷åíèå� ôóíêöè� ïðåäëîæåíè� S(p) � òî÷ê� p. Ñëåäîâàòåëüíî, òà� êà� S(p) ÿâëÿåòñ� ðåøåíèå� çàäà÷� îïòèìèçàöè� í� áîëå� øèðîêî� ìíîæåñòâå, ò� çíà÷åíè� îïòèìèçèðóåìîã� ôóíêöèîíàë� � S(p) í� ìåíüøå, ÷å� � òî÷ê� V ∗ (p). Îäíàê� ïð� p < p̃ ôóíêöè� ïðåäëîæåíè� S(p) óäîâëåòâîðÿå� îãðàíè÷åíè� í� îáúåì, ò.å. S + (p) < D− (p). Ýò� óñëîâè� âûïîëíÿåòñÿ, òà� êà� ôóíêöè� S(p) ìîíîòîíí� í� óáûâàåò, � D(p) ìîíîòîíí� í� âîçðàñòàå� è, ñëåäîâàòåëüíî, äë� ëþáîã� p < p̃ ðàçíîñò� D(p) − S(p) > 0 (ñì.
ðèñ. 17.1). Ñëåäîâàòåëüíî, � ýòî� ñëó÷à� V ∗ (p) ∈ S(p). pV − C(V )V 66D(p)S(p)bb-p̃p-D(p)S(p)Ðèñ. 17.1 VÐèñ. 17.2 Ïóñò� p = p.˜ � ýòî� ñëó÷à� íàä� âçÿò� îïòèìàëüíû� îáúå� ïðîèçâîäñòâà, í� òàêîé, êîòîðû� ïîòðåáèòåë� � ñîñòîÿíè� êóïèòü. Ñëåäîâàòåëüíî, � ýòî� ñëó÷à� V ∗ (p̃) ∈ S(p̃) ∩ [0, D+ (p̃)]. Íàêîíåö, ïóñò� p > p.˜ Îïòèìèçèðóåìà� � çàäà÷� (17.1) ôóíêöè� ïðåäñòàâëÿå� ñîáî� ðàçíîñò� ëèíåéíî� � âûïóêëî� ôóíêöè� è, ñëåäîâàòåëüíî, ñàì� ÿâëÿåòñ� âîãíóòî� ôóíêöèåé. Ïðè÷åì, êà� áûë� çàìå÷åí� � ïðåäûäóùå� ïóíêò� äîêàçàòåëüñòâà, ìàêñèìó� å� äîñòèãàåòñ� � S(p).
Çíà÷èò, ðàññìàòðèâàåìà� ôóíêöè� âîçðàñòàå� í� îòðåçê� [0, S − (p)]. Ïð� p > p̃ � ñèë� îãðàíè÷åíè� í� îáúå� ïðåäëîæåíè� ìàêñèìàëüí� äîïóñòèìû� îáúåìî� ïðîèçâîäñòâ� áóäå� D+ (p), òà� êà� � ýòî� ñëó÷à� D+ (p) < S − (p) (ñì. ðèñ. 17.1 � ðèñ. 17.2). Ñëåäîâàòåëüíî, � ýòî� ñëó÷à� V ∗ (p) = D+ (p).
187 ÃËÀÂ� IV. ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�Ýòà� 2. Îïòèìèçàöè� ï� öåí� Èòàê, ì� îïðåäåëèë� îïòèìàëüíû� îáúå� äë� êàæäî� ôèêñèðîâàííî� öåíû. Òåïåð� íàéäå� îïòèìàëüíó� ìîíîïîëüíó� öåíó. Óòâåðæäåíè� 17.2. 1) Ìîíîïîëè� âñåãä� íàçíà÷àå� öåí� p∗ í� íèæ� ðàâíîâåñíîé, ò.å. p∗ ≥ p.˜2) Åñë� ôóíêöè� ñïðîñ� D(p) − îäíîçíà÷íà� � ãëàäêà� � îêðåñòíîñò� òî÷ê� p,˜ ò� ìîíîïîëè� íàçíà÷àå� öåí� p∗ âûø� ðàâíîâåñíîé, ò.å. p∗ > p.˜Äîêàçàòåëüñòâî. 1) Ñíà÷àë� ïîêàæåì, ÷ò� öåí� ìîíîïîëè� í� íèæå, ÷å� p̃.
Áåðå� ëþáó� öåí� p < p.˜ � ñèë� ïðåäûäóùåã� óòâåðæäåíè� 17.1 îïòèìàëüíî� çíà÷åíè� îáúåì� V ∗ (p) ∈ S(p). Ðàññìîòðè� àëüòåðíàòèâíó� ñòðàòåãè� (p̃, V ∗ (p)). Íà� èçâåñòíî, ÷ò� D(p̃) ∩ S(p̃) =6 ∅. Òîãä� è� ìîíîòîííîñò� ôóíêöè� ñïðîñ� � ïðåäëîæåíè� ñëåäóåò, ÷ò� D(p̃) ≥S(p) ∀ p < p.˜ Ñëåäîâàòåëüíî, ñòðàòåãè� (p̃, V ∗ (p)) äîïóñòèìà, ò.å. òî� æ� îáúå� òîâàð� V ∗ (p) ìîæå� áûò� ðåàëèçîâà� ï� áîëüøå� öåí� p̃. Çàìåòèì, ÷ò� èçäåðæê� ïð� ýòî� í� èçìåíÿòñÿ, òà� êà� îáúå� âûïóñê� îñòàëñ� òå� æå. Çíà÷è� îáùà� âûðó÷ê� ïðîèçâîäèòåë� î� ïðîäàæ� ï� öåí� p̃ áóäå� âûøå, ÷å� î� ïðîäàæ� ï� öåí� p < p̃. Èòàê, ìîíîïîëèñ� âñåãä� íàçíà÷àå� öåí� p∗ í� íèæ� p̃. 2) Ïîêàæåì, ÷ò� äë� îäíîçíà÷íî� � ãëàäêî� � îêðåñòíîñò� òî÷ê� p̃ôóíêöè� ñïðîñ� D(p) ìîíîïîëèñ� íàçíà÷è� öåí� âûø� p̃.
È� óòâåðæäåíè� 17.1 ñëåäóåò, ÷ò� V ∗ (p) = D(p) ïð� p > p̃, � ïð� p = p̃ ìàêñèìàëüí� âîçìîæíû� îáúå� V ∗ (p) ∈ S(p̃) ∩ [0, D(p̃)] ðàâå� D(p̃). Ñëåäîâàòåëüíî, çàäà÷� îïòèìèçàöè� ïðèáûë� ìîíîïîëèñò� ïð� p ≥ p̃ ïðèíèìàå� âè� max(pD(p) − C(D(p))). p≥˜pÂû÷èñëè� ïðîèçâîäíó� ôóíêöè� ïðèáûë� W (p) = pD(p) − C(D(p)) Ẇ (p) = D(p) + Ḋ(p)(p − Ċ(D(p))). (17.2) Ïð� p = p̃ å� çíà÷åíè� ðàâí� Ẇ (p̃) = D(p̃) + Ḋ(p̃)(˜p − Ċ(D(p̃))). Ïîñêîëüê� D(p̃) ∈ S(p̃), � S(p̃) ÿâëÿåòñ� ðåøåíèå� çàäà÷� îïòèìèçàöèè, ò� p̃ = Ċ(D(p̃)) � âòîðî� ñëàãàåìî� âûðàæåíè� (17.2) ðàâí� íóëþ. Îòêóä� ñëåäóåò, ÷ò� Ẇ (p̃) = D(p̃) > 0.
Ì� ïîêàçàëè, ÷ò� ôóíêöè� ïðèáûë� W (p) âîçðàñòàå� � òî÷ê� p̃. Çíà÷èò, ìîíîïîëèñ� íàçíà÷è� öåí� âûø� öåí� êîíêóðåíòíîã� ðàâíîâåñèÿ. Óïðàæíåíè� 17.1. Ïóñò� ôóíêöè� èçäåðæå� ôèðìû-ìîíîïîëèñò� C(V ) ðàâí� ôóíêöè� C a (V ) è� ïðèìåð� 16.1, � ôóíêöè� ñïðîñ� D(p) = 3/p2 . Íàéäèò� îïòèìàëüíó� ìîíîïîëüíó� öåí� p∗ . 188 17. Ìîíîïîëèçèðîâàííû� ðûíî�Çàìå÷àíèå. Íàñêîëüê� ìîíîïîëèñ� çàâûñè� öåí� ï� ñðàâíåíè� � öåíî� êîíêóðåíòíîã� ðàâíîâåñèÿ, áóäå� çàâèñåò� î� êîíêðåòíîã� âèä� ôóíêöè� ñïðîñà, òî÷íåå, î� ñêîðîñò� å� óáûâàíè� ïîñë� ðàâíîâåñíî� öåíû. Åñë� ïðîèñõîäè� ðåçêî� óáûâàíèå, ò� ìîíîïîëüíà� öåí� áóäå� ìàë� îòëè÷àòüñ� î� êîíêóðåíòíîé, åñë� æ� óáûâàíè� ôóíêöè� ñïðîñ� ïðîèñõîäè� ìåäëåííî, ò� ðàçíèö� ìåæä� öåíîé, íàçíà÷åííî� ìîíîïîëèñòîì, � ðàâíîâåñíî� áóäå� çíà÷èòåëüíîé. Îïðåäåëåíèå. Ôóíêöè� ñïðîñ� D(p) íàçûâàåòñ� ìåäëåíí� óáûâàþùå� í� îòðåçê� [p1 , p2 ], åñë� p2 D(p2 ) ≥ pD(p) ∀ p ∈ [p1 , p2 ], ò.å.
ñïðî� � äåíåæíî� âûðàæåíè� pD(p) äîñòèãàå� ìàêñèìóì� � òî÷ê� p2 . Ïðèìå� 17.1. Ïóñò� D(p) = K/p. Òîãä� ñïðî� � äåíåæíî� âûðàæåíè� ðàâå� ïîñòîÿííî� âåëè÷èí� K . Ñëåäîâàòåëüíî, òàêà� ôóíêöè� ÿâëÿåòñ� ìåäëåíí� óáûâàþùåé. � ýòîì� êëàññ� îòíîñÿòñ� òàêæ� ôóíêöè� âèä� D(p) = K/pα , 0 < α < 1.
Îïðåäåëåíèå. Äë� ãëàäêî� ôóíêöè� ñïðîñ� D(p) ýëàñòè÷íîñòü� íàçûâàåòñ� âåëè÷èí� e(D(p)) = |Ḋ(p)||dD(p)/D(p)|p = . D(p) dp/p Îí� ïîêàçûâàåò, í� ñêîëüê� ïðîöåíòî� èçìåíèòñ� îáúå� ñïðîñ� ïð� èçìåíåíè� öåí� í� îäè� ïðîöåíò. Îòìåòèì, ÷ò� åñë� ýëàñòè÷íîñò� e(D(p)) ≡ 1, ò� D(p) = K/p. Âûñîêîýëàñòè÷íû� ñïðî� (í� ïðåäìåò� ðîñêîøè) õàðàêòåðèçóåòñ� çíà÷åíèÿì� e(D(p)) > 1 , íèçê� ýëàñòè÷íû� ñïðî� (í� ïðåäìåò� ïåðâî� íåîáõîäèìîñòè) − çíà÷åíèÿì� e(D(p)) < 1.
� ýêîíîìè÷åñêî� ëèòåðàòóðå, êîãä� ãîâîðÿ� î� ýëàñòè÷íî� ñïðîñå, îáû÷í� ïîäðàçóìåâàþ� âûñîêîýëàñòè÷íû� ñïðîñ. Óòâåðæäåíè� 17.3. Åñë� ýëàñòè÷íîñò� ôóíêöè� ñïðîñ� e(D(p)) ≤ 1 í� îòðåçê� [p1 , p2 ], ò� D(p) ìåäëåíí� óáûâàå� í� äàííî� îòðåçêå. Óïðàæíåíè� 17.2. Äîêàæèò� óòâåðæäåíè� 17.3. ×ò� ìîæí� ñêàçàò� ïð� ïîâåäåíè� ìîíîïîëè� � ñëó÷àå, êîãä� ñïðî� ìåäëåíí� óáûâàåò? Óòâåðæäåíè� 17.4. Ïóñò� äë� íåêîòîðîã� çíà÷åíè� p > p̃ ôóíêöè� ñïðîñ� D(p) ìåäëåíí� óáûâàå� í� îòðåçê� [p̃, p]. Òîãä� äë� îïòèìàëüíî� ìîíîïîëüíî� öåí� âûïîëíåí� íåðàâåíñòâ� p∗ ≥ p. Ïîÿñíè� ñìûñ� óòâåðæäåíèÿ: ïîê� ýëàñòè÷íîñò� ìàëà, ìîíîïîëè� âûãîäí� óâåëè÷èâàò� öåíó.
189 ÃËÀÂ� IV. ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�Äîêàçàòåëüñòâî. Ïðåäïîëîæè� î� ïðîòèâíîãî, ÷ò� p∗ < p. Áóäå� ïðîäàâàò� òîâà� ï� öåí� p, ñîõðàíÿ� îáúå� âûðó÷ê� V = D(p∗ )p∗ /p. Òîãä� V < D(p∗ ), èçäåðæê� ñíèçÿòñÿ, � ïðèáûë� ñîîòâåòñòâåíí� âûðàñòåò. Îñòàåòñ� ïðîâåðèòü, ÷ò� ìîæí� ïðîäàò� îáúå� V ï� öåí� p. Äåéñòâèòåëüíî, è� óñëîâè� ìåäëåííîã� óáûâàíè� D(p) ≥ D(p∗ )p∗ /p, îòêóä� V ≤ D(p), ò.å.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
