А.А. Васин, В.В. Морозов - Введение в теорию игр с приложениями к экономике (1184512), страница 41
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� ñèë� ñâîéñòâ� D4 ôóíêöè� ñïðîñà, D−1 (V ) → ∞ ïð� V → 0 + . Ïîýòîì� ïð� äîñòàòî÷í� ìàëî� v a > 0 u a (0||v a ) = v a (D−1 (v a ) − c a ) > 0 = u a (0), ÷ò� ïðîòèâîðå÷è� îïðåäåëåíè� ñèòóàöè� ðàâíîâåñèÿ. Óñëîâè� 1)-3) ÿâëÿåòñ� íåîáõîäèìûì� óñëîâèÿì� äë� òî÷ê� ìàêñèìóì� v a äèôôåðåíöèðóåìî� ôóíêöè� ua (v||v a ) îäíî� ïåðåìåííî� v a í� îòðåçê� [0, V a ] (ñì.
çàäà÷� (19.4)). Âû÷èñëè� ïðîèçâîäíó� ôóíêöè� ïðèáûë� ï� îáúåì� âûïóñê� v a êà� ïðîèçâîäíó� ïðîèçâåäåíèÿ: !Xu0va (v) = D−1 v b − c a + v a /Ḋ(p(v)). (19.5) b∈A Óòâåðæäåíè� 19.1. Ïóñò� c1 ≤ c2 ≤ ... ≤ cm , ò.å. ïðåäïðèÿòè� óïîðÿäî÷åí� ï� âîçðàñòàíè� óäåëüíû� ñåáåñòîèìîñòåé, � ôóíêöè� ñïðîñ� D(p) − äèôôåðåíöèðóåìà� � óáûâàþùàÿ. Òîãä� íàéäåòñ� òàêî� ïðåäïðèÿòè� k , ÷ò� � ðàâíîâåñè� ï� Íýø� v a > 0, a = 1, 2, ..., k, v a = 0, a = k + 1, ..., m, ïðè÷å� ck+1 > ck . Äîêàçàòåëüñòâî. 1) Ïóñò� v − ðàâíîâåñè� ï� Íýøó.
Ï� ëåìì� 19.1 v =6 0. Âîçüìå� k òàêî� k , ÷ò� v > 0. Ïîêàæåì, ÷ò� òîãä� 199ÃËÀÂ� IV. ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�D−1 Pv b − ck > 0. Cîãëàñí� ëåìì� 19.1 è� v k > 0 ñëåäóå� b∈A u0vk (v) ≥ 0. È� ôîðìóë� (19.5) ïîëó÷àå� X −1 Dv b − c k ≥ −v k /Ḋ(p(v)) > 0, b∈A ïîñêîëüê� Ḋ(p(v)) < 0. 2) Ïîêàæåì, ÷ò� v a > 0 ïð� a = 1, ..., k −1. Äîïóñòè� î� ïðîòèâíîãî, ÷ò� v a = 0 äë� íåêîòîðîã� a ∈ {1, ..., k − 1}. Òîãä� X � X � u0va (v) = D−1 v b − c a + v a /Ḋ(p(v)) = D−1 v b − c a ≥ b∈Ab∈A ≥ D−1 Xv b − c k > 0, b∈A òà� êà� ca ≤ ck ïð� a < k.
Ï� ëåìì� 19.1 è� u0va (v) > 0 cëåäóå� v a > 0.Ïîëó÷èë� ïðîòèâîðå÷è� � ïðåäïîëîæåíèåì, ÷ò� v a = 0. Ñëåäîâàòåëüíî, v a > 0 ïð� a = 1, ..., k − 1. 3) Òàêè� îáðàçîì, íàéäåòñ� òàêî� ìàêñèìàëüíî� k , ÷ò� v a > 0, a = 1, 2, ..., k v a = 0, a = k + 1, ..., m. Íàêîíåö, ïîêàæåì, ÷ò� ck+1 > ck .
Åñë� ck+1 = ck , ò� X � X � 0−1 b k+1 k+1−1 uvk+1 (v) = Dv − c + v /Ḋ(p(v)) = Dv b − c k > 0. b∈Ab∈A Ï� ëåìì� 19.1 è� u0vk+1 (v) > 0 cëåäóå� v k+1 > 0 (ïðîòèâîðå÷èå).Ïîëó÷åííû� ðåçóëüòà� èìåå� ïðîñòî� ýêîíîìè÷åñêè� ñìûñë: åñë� íåêîòîðîì� ïðåäïðèÿòè� âûãîäí� ïðîèçâîäèò� òîâà� ï� äàííî� öåíå, ò� ïðåäïðèÿòè� � ìåíüøå� óäåëüíî� ñåáåñòîèìîñòü� òå� áîëå� âûãîäí� ïðîèçâîäèò� ýòî� òîâàð. Ñëó÷à� � ðàâíûì� óäåëüíûì� ñåáåñòîèìîñòÿì� Ðàññìîòðè� ñëó÷àé, êîãä� óäåëüíû� ñåáåñòîèìîñò� ca âñå� ïðåäïðèÿòè� îäèíàêîâ� � ðàâí� c. È� óòâåðæäåíè� 19.1 ñëåäóåò, ÷ò� òîãä� � ðàâíîâåñè� ï� Íýø� v a > 0 ïð� âñå� a ∈ A, ò.å.
êàæäû� ïðîèçâîäèòåë� âûïóñêàå� ïîëîæèòåëüíî� êîëè÷åñòâ� òîâàðà. Ðàññìîòðè� ñíà÷àë� ìîäåë� áå� îãðàíè÷åíè� í� îáúåì� âûïóñêà. Áóäå� èñêàò� ðåøåíè� çàäà÷� (19.4), ñ÷èòà� V a = ∞ (ïðîèçâîäñòâåííû� 200 19. Ìîäåë� îëèãîïîëè�ìîùíîñò� ïðåäïðèÿòè� í� îãðàíè÷åíû). Òîãä� � òî÷ê� ìàêñèìóì� çàäà÷� (19.4) âñ� ïðîèçâîäíû� u0va (v) = 0 � äë� ïîèñê� ðàâíîâåñè� ï� Íýø� ïîëó÷àå� ñëåäóþùó� ñèñòåì� óðàâíåíè� îòíîñèòåëüí� íåèçâåñòíû� v a , a = 1, ..., m: X 0−1 uva (v) = Dv b − c + v a /Ḋ(p(v)) = 0, a ∈ A. (19.6) b∈A Ñóììèðóå� óðàâíåíè� (19.6) ï� âñå� a ∈ A. Ïîëó÷àå� X � X � � −1 aa−1 mDv − mc + v /Ḋ Dv a= 0.
a∈Aa∈A(19.7) a∈A Âûðàæåíè� (19.7) ïðåäñòàâëÿå� ñîáî� óðàâíåíè� îòíîñèòåëüí� îäíî� íåèçP a âåñòíî� âåëè÷èí� v . Ðàçðåøè� äàííî� óðàâíåíè� � ïîäñòàâè� íàéa∈A äåííó� âåëè÷èí� � óðàâíåíè� (19.6), íàéäå� çíà÷åíè� v a , a ∈ A, êîòîðûå, î÷åâèäíî, îäèíàêîâ� äë� âñå� a. Ïðîäåìîíñòðèðóå� äåéñòâè� ýòîã� àëãîðèòì� êîíêðåòíî� P í� P a ôóíêöè� −1añïðîñà. Ïóñò� D(p) = K/p. Òîãä� p(v) = D ( v ) = K/ v .
Ñèñòåì� a∈A(19.6) � ýòî� ñëó÷à� ïðèíèìàå� âè� K/ � v a − c − v a K/ a∈AXv a 2 = 0, a ∈ A, a∈A (19.6 0 ) a∈A � óðàâíåíè� (19.7) ïðåîáðàçóåòñ� � âèäó: XXmK/ v a − mc − K/ v a = 0. a∈AÈ� (19.7 0 ) ïîëó÷àåì, ÷ò� (19.7 0 ) a∈A Pv a = (m − 1)K/(mc). Ïîäñòàâëÿ� ýò� a∈A âûðàæåíè� � (19.6 0 ), íàõîäè� ðàâíîâåñè� ï� Íýø� äë� ìîäåë� Êóðí� def v a = v ∗ = K(m − 1) ∀ a ∈ A. cm2(19.8)Äë� ìîäåë� � îãðàíè÷åíèÿì� í� îáúåì� âûïóñê� îòìåòè� äâ� ñëó÷àÿ.
à) v ∗ ≤ min V b . Òîãä� ðåøåíè� çàäà÷� � îãðàíè÷åíèÿì� í� îáúåì� b∈A âûïóñê� ñîâïàäàå� � ðåøåíèå� òî� æ� çàäà÷� áå� ó÷åò� îãðàíè÷åíèé, 201ÃËÀÂ� IV. ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�òà� êà� v ∗ − ðåøåíè� çàäà÷� îïòèìèçàöè� ï� áîëå� øèðîêîì� ìíîæåñòâ� � � ò� æ� âðåì� ÿâëÿåòñ� äîïóñòèìû� îáúåìî� âûïóñê� äë� çàäà÷� � îãðàíè÷åíèÿìè.
Ñëåäîâàòåëüíî, � ýòî� ñëó÷à� v a çàäàåòñ� ôîðìóëî� (19.8). á) v ∗ > min V b . Óïîðÿäî÷è� ïðîèçâîäèòåëå� ï� óáûâàíè� ìàêñèìàëüb∈A íû� îáúåìî� âûïóñêà: V 1 ≥ V 2 ≥ ... ≥ V m . Òîãä� íàéäåòñ� òàêî� ïðîèçâîäèòåë� k, ÷ò� � v ∗ , a ≤ k, v a =(19.9)V a , a > k. Àëãîðèò� ïîèñê� ðàâíîâåñè� ï� Íýø� âèä� (19.9) Ôèêñèðóå� íåêîòîðî� k .
Òîãäà, ïåðåïèñà� (19.6) äë� ýòîã� ñëó÷àÿ, ïîëó÷àå� óðàâíåíè� äë� íàõîæäåíè� v ∗ (k): K/ m Xm� X2 V a + kv ∗ − c − v ∗ K/ V a + kv ∗ = 0, a=k+1ãä� âûðàæåíè� (19.6 00 ) a=k+1 mPV a ïîëàãàåòñ� ðàâíû� íóëþ.
a=m+1 Íà÷èíàå� ïîèñ� ðàâíîâåñè� ï� Íýø� � k = m. Íàõîäè� v ∗ (m) � ïðîâåðÿå� óñëîâè� v ∗ (m) ≤ V m . Åñë� îí� âûïîëíÿåòñÿ, ò� íàéäåííà� ñèòóàöè� v ÿâëÿåòñ� ðàâíîâåñèå� ï� Íýøó. Èíà÷� áåðå� k = m − 1 � ò.ä. � ðåçóëüòàò� ç� êîíå÷íî� ÷èñë� øàãî� íàéäå� ðàâíîâåñè� ï� Íýø� äë� ýòîã� ñëó÷àÿ. Ñðàâíåíè� ðàâíîâåñè� ï� Íýø� � ï� Âàëüðàñ� äë� ìîäåë� Êóðí� Ïóñò� D(p) = K/p, V a ≡ V, ca ≡ c. � ýòî� ñëó÷à� ðàâíîâåñè� ï� Íýø� v � öåí� p∗ îïðåäåëÿåòñ� ï� ôîðìóëà� (v ∗ = K(m − 1)/(cm2 ), åñë� v ∗ ≤ V, a v =∀ a ∈ A,V,åñë� v ∗ > V, (K/(mv ∗ ) = cm/(m − 1), åñë� v ∗ ≤ V, p∗ =K/(mV ), åñë� v ∗ > V. ×òîá� íàéò� ðàâíîâåñè� ï� Âàëüðàñó, íàä� íàéò� ïåðåñå÷åíè� ôóíêöè� ñïðîñ� � ïðåäëîæåíèÿ, ïðè÷å� 202 19.
Ìîäåë� îëèãîïîëè�⎧ p < c,⎨ 0, S(p) = [0, mV ], p = c,⎩mV, p > c.Åñë� K/c ≤ mV , ò� p̃ = c (ñì. ðèñ. 19.1), èíà÷� p̃ = K/(mV ) (ñì. ðèñ. 19.2). á) V 6a) V 6D(p)D(p)S(p)mVS(p)mVp̃ = c p∗ =cmm−1 -pccmm−1Ðèñ. 19.1 p̃ = p∗ =KmV-pÐèñ. 19.2 � èòîã� ïîëó÷àå� ñëåäóþùè� ðåçóëüòàò.Óòâåðæäåíè� 19.2.
Äë� äàííî� ìîäåë� âîçìîæí� ñëåäóþùè� ñîî�íîøåíè� ðàâíîâåñè� ï� Íýø� � ï� Âàëüðàñó. 1) Åñë� V ≥ K/(cm), ò� p∗ = cm/(m − 1), v ∗ = K(m − 1)/(cm2 ), p̃ = c, ṽ a = K/(cm), , ò.å. ðàâíîâåñíà� öåí� äë� îëèãîïîëè� p∗ ïðåâûøàå� öåí� êîíêóðåíòíîã� ðàâíîâåñè� p̃ � m/(m − 1) ðà� (ðèñ.
19.1). 2) Åñë� K/(cm) > V > K(m − 1)/(cm2 ), ò� p∗ = cm/(m − 1), v ∗ = K(m − 1)/(cm2 ), p̃ = K/mV, ṽ a = V , ò.å. öåí� îòëè÷àþòñ� � ìåíüøå� ñòåïåíè. 3) Ïð� V ≤ K(m − 1)/(cm2 ) öåí� � îáúåì� ñîâïàäàþò: p∗ = p̃ = K/(mV ), v ∗ = ṽ a = V. (ñì. ðèñ.
19.2). Òàêè� îáðàçîì, åñë� îãðàíè÷åíè� îáúåì� âûïóñê� íåñóùåñòâåíí� (V ≥ K/(cm)), ò� äë� ìàëû� m ïîëó÷àå� ñëåäóþùè� ñîîòíîøåíè� öåí: m = 2 ⇒ p̃ = p∗ /2; m = 3 ⇒ p̃ = 2p∗ /3; m = 4 ⇒ p̃ = 3p∗ /4.� aÒà� êà� p = K/ v , , ò� ñîîòíîøåíè� îáúåìî� îáðàòí� ïðîïîðöèîíàëüa∈A í� ñîîòíîøåíè� ñîîòâåòñòâóþùè� öåí, ò.å. v ∗ /ṽ = 1 − 1/m, � ÷àñòíîñòè, m = 2 ⇒ ṽ = 2v ∗ ; m = 3 ⇒ ṽ = 3v ∗ /2; m = 4 ⇒ ṽ = 4v ∗ /3.203ÃËÀÂ� IV.
ÂÂÅÄÅÍÈ� � ÌÀÒÅÌÀÒÈ×ÅÑÊÓ� ÝÊÎÍÎÌÈÊ�Ìîäåë� Êóðí� ïîçâîëÿå� îáîñíîâàò� ìåð� ï� àíòèìîíîïîëüíîì� ðåãóëèðîâàíèþ. È� àíàëèç� ýòî� ìîäåë� âûòåêàåò, ÷ò� åñë� í� ðûíê� äåéñòâóå� õîò� á� ÷åòûð� êîìïàíèè, ò� îòêëîíåíè� ï� îáúåì� î� ñîñòîÿíè� êîíêóðåíòíîã� ðàâíîâåñè� ï� Âàëüðàñ� ñîñòàâëÿå� í� áîëå� 25%, � îòêëîíåíè� ï� öåí� í� áîëå� 33%. Ïð� ýòî� îáúå� âûïóñê� í� êàæäî� è� ÷åòûðå� ïðåäïðèÿòèé, ïðèñóòñòâóþùè� í� ðûíêå, äîëæå� ñîñòàâëÿò� v ∗ = 3K/(16c) (ò.å. 3/16 î� ðàâíîâåñíîã� ï� Âàëüðàñ� îáùåã� îáúåì� ïðîèçâîäñòâà). Ñîãëàñí� àíòèìîíîïîëüíîì� çàêîíîäàòåëüñòâ� ÑØÀ, � ïðåäïðèÿòè� ìîãó� ïðèìåíÿòüñ� àíòèìîíîïîëüíû� ìåðû, åñë� îí� êîíòðîëèðóå� áîëå� 30% ðûíêà.
Èñõîä� è� ìîäåë� Êóðíî, òàêè� îáðàçî� îáåñïå÷èâàåòñ� îïðåäåëåííà� ñòåïåí� áëèçîñò� � êîíêóðåíòíîì� ðàâíîâåñèþ. Óïðàæíåíè� 19.1. Ïîêàæèòå, ÷ò� ðàâíîâåñè� ï� Íýø� äë� ìîäåë� Êóðí� í� ñóùåñòâóåò, åñë� D(p) = K/pα , 0 < α ≤ 1/m, ca ≡ c. Óïðàæíåíè� 19.2. Íàéäèò� ðàâíîâåñè� ï� Íýø� äë� ìîäåë� Êóðí� � ñðàâíèò� è� � êîíêóðåíòíû� ðàâíîâåñèå� � ñëåäóþùè� óñëîâèÿõ: 1. D(p) = K/pα , α > 1/m, ca ≡ c, V a ≡ V.2. D(p) = K/p, ca ≡ c, V 1 ≥ V 2 ≥ ... ≥ V m . 3. D(p) = K/p, c1 ≤ c2 ≤ ... ≤ cm , V a ≡ V ≥ K/c1 . Äîñòîèíñòâ� ìîäåë� Êóðíî: à) ïðîñò� äë� èññëåäîâàíèÿ; á) ñîãëàñóåòñ� � ìîäåëü� êîíêóðåíòíîã� ðàâíîâåñèÿ; â) óñëîâè� ñîâåðøåííî� êîíêóðåíöè� � îöåíê� îòêëîíåíè� î� íè� ëåãê� ôîðìàëèçóþòñÿ.
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