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SUPPLY CHAIN GAMES: OPERATIONS MANAGEMENT AND RISK VALUATION phần 6

Bằng chứng: Hãy xem xét những giải pháp sau đây để các biến nhà nước, hợp tác nhà nước và quyết định:dàng quan sát thấy các giải pháp này đáp ứng các điều kiện tối ưu Hơn nữa, giải pháp này là luôn luôn khả thi nếu điều kiện | 252 4 MODELING IN AN INTERTEMPORAL FRAMEWORK Equilibrium Consider now the supplier s problem. Applying the first-order optimality condition to the supplier s objective function 4.136 we find that the optimal wholesale price w is defined by the equation w - c X X 0 0. 4.168 Z1 Then with respect to Proposition 4.27 and A tj 1 a t dt this implies 0 thatX 0 bA t1 and equation 4.168 transforms into bdA t1 bdt1 w - cs 12 - bA tj w - cs pX1 a t1 - ốA tj 0 dw dw where t1 is determined by cs i h -h yHI At -h dt c . 0 A t 4.169 TT . . . . . hr Using implicit differentiation of 4.169 and the fact that H ố h - h we find that dt1 dw 1 4.170 i h - h- at lA d J0 A t which implies that the greater the wholesale price the earlier the manufacturer will start using his in-house capacity. Moreover this also means that a solution w which satisfies the optimality condition w - c ẺÉka f - bA f 0 is greater than cs. ds Let tj A- 0 I then equation 4.170 takes the following form b dt1 dw 1 A-1 Xr2 Õ ỉ .- b 14-1Ã X 0 V X 0 I h - h Hw a A I M dt 0 A t I b A t which by substituting into the first-order optimality condition results in 4.171 4.4 INTERTEMPORAL SUBCONTRACTING COMPETITION 253 -1Ã X 0 Ô w - cs ba A 4 I ---777 ----------------- f b -------------- X 0 0. Af X 0 Ô 1 b Ù .- b 1ÃX 0 Ô X 0 I h h a A I w dt J0 A t f b A t We thus conclude with the following proposition. Proposition 4.31. Let all conditions of Proposition 4.27 be met be H z defined by 4.171 f-1- 0 and Ỗ and Ằ satisfy the following equations z 4fi 3 Ằ i h h-Hf -h dt c J0 A t 4.172 n- cs ba A- A b i h h- Xb_ 0 A t 3 0. 4.173 Wv dt A t If n - c f lbdy a t1 -1 a t1 1 -1 a t1 ba t1 0 4.174 f dw dw dw then the wholesale price ws Ả c the manufacture s advance order X 0 Ỗ and production policy us t 0 for 0 t t1 us t ba t for t1 t t2 us t 0 for t2 t T constitute the unique Stackelberg equilibrium in the differential production balancing game. Proof To prove the proposition it is sufficient to verify the secondorder optimality .