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midtrmsol_a

# midtrmsol_a - 1 Maxcfw_qi NPV = Maxcfw_qi L = F.O.C L =...

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1. ( 29 0,1 i for subject to ) ( ) ( i r) (1 1 NPV 0 1 0 } { = = - + = = S q q C q B Max i i i i q i ( 29 - + - = = + 1 0 0 r) (1 1 } , { S ) ( ) ( L i i i i i q q q C q B Max i l l F.O.C 0 0 )) ( ' ) ( ' ( 1 1 0 ) ( ' ) ( ' 1 0 0 1 1 1 0 0 0 = - - = = - - + = = - - = q q S L q C q B r q L q C q B q L l l l (1) From which we get the following important inter-temporal equilibrium conditions 0 1 0 1 1 0 0 ) ( ' ) ( ' 1 1 ) ( ' ) ( ' S q q q C q B r q C q B = + - + = - (2) Perfect Competition In perfect competition 1 , 0 for ) ( ' = = i q B P i i Since we know that 1 , 0 for 5 . 1000 = - = i q P i i , 1 , 0 for 400 ) ( ' = = i q C i , and % 10 = r , then (2) becomes 1000 ) 5 . 0 600 ( 1 . 1 1 5 . 0 600 1 0 1 0 = + - = - q q q q (3) The solution to the above simultaneous system of equation is 66 . 466 * 33 . 533 * 1 0 = = q q To find the market price in each period just substitute q into the demand equations. 66 . 766 ) 66 . 466 ( 5 . 1000 *) ( 5 . 0 1000 33 . 733 ) 33 . 533 ( 5 . 1000 *) ( 5 . 0 1000 1 1 0 0 = - = - = = - = - = q P q P To find l use the first equation of the F.O.C system (1) 33 . 333 400 33 . 733 ) ' *) ( ' *) ( ' *) ( ' 0 0 0 0 0 0 = - = - = - = - = q C p q C P q C q B l

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Monopolist Under the monopolist the marginal benefit is not equal to the price. Competitive Market: B(q) = pq (price is given) B’(q) = p Monopolist: B(q) =p(q)q (price is affected by monopolist’s output) B’(q) = p(q) + p’(q)q Therefore, in this case we have the following: % 10 2 , 1 for 400 ) ( ' 2 , 1 for 1000 ) ( ' = = = = - = r i q C i q q B i i i So (2) becomes 1000 ) 600 ( 1 . 1 1 600 1 0 1 0 = + - = - q q q q (4) The solution to the above simultaneous system of equation is 23 . 495 76 . 504 1 0 = = m m q q To find the market price in each period just substitute q into the demand equations. 38 . 752 ) 23 . 495 ( 5 . 1000 ) ( 5 . 0 1000 62 . 747 ) 76 . 504 ( 5 . 1000 ) ( 5 . 0 1000 1 1 0 0 = - = - = = - = - = m m q P q P To find m l use the first equation of the F.O.C system (1) 23 . 95 400 76 . 504 1000 ) ( ' 1000 *) ( ' *) ( ' 0 0 0 0 = - - = - - = - = m m m q C q q C q B l Open Access
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