请教两步收费制的一个问题

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收藏 2005-04-20

平狄克－中微－第三版－P324－No. 10
10. As the owner of the only tennis club in an isolated wealthy community, you must decide on membership dues and fees for court time. There are two types of tennis players. “Serious” players have demand Q1 = 6 - P 
where Q1 is court hours per week and P is the fee per hour for each individual player. There are also “occasional” players with demand Q2 = 3 - (1/2)P. 
Assume that there are 1,000 players of each type. You have plenty of courts, so that the marginal cost of court time is zero. You have fixed costs of $5,000 per week. Serious and occasional players look alike, so you must charge them the same prices.
A friend tells you that you could make greater profits by encouraging both
types of players to join. Is the friend right? What annual dues and court 
fees would maximize weekly profits? What would these profits be?
当有两类顾客时，俱乐部的拥有者通过将场地费定在高于边际成本和通过将会员费定在等于需求较小的消费者的余下的消费者剩余是利润最大化。会员费T等于去掉场地费以后的消费者:T=（Q2－0）*（6－P）*（1/2），其中，Q2=3－（1/2），故T=[3－（1/2）P]*（6－P）*（1/2）=P－3P+P2/4。。所有网球手的总会员费是2000*（9－3P+3P2/2）]。场地费收益等于P（Q1+Q2）=P[（6－P）*1000+3－P/2]*1000]=1000（9P－3P2/2）。这样总收益TR=[9－3P+P2/4]*2+（9P－3P2/2）]*1000=（18+3P－P2）*1000。边际成本为零，边际收益由总收益与边际曲线的切线斜率给出：⊿TR/⊿P（或者MR）=3000－2000P。令边际收益与边际成本相等则得价格为每小时1.5美圆。总收益等于20250美圆。总成本等于固定成本5000美圆。所以利润为每周15250美圆，这大于只有专业网球手才成为会员的每周13000美圆。
疑问是：MR（边际收益）的定义不是⊿TR/⊿Q吗，此处却变成了⊿TR/⊿P？
 应该是MR＝（⊿TR/⊿P）×（⊿P/⊿Q）；其中（⊿P/⊿Q1）＝－1；则MR1=（⊿TR/⊿P）×（⊿P/⊿Q1）＝9000－2000×Q1＝0，得Q1＝4.5；据Q1＝6－P, 得P＝1.5
 同理推得MR2＝（⊿TR/⊿P）×（⊿P/⊿Q2）＝18000－8000Q2＝0，得Q2=2.25,据Q2=3-(1/2)P, 得P=1.5。结果与上述答案一致。只是不清楚为什么MR＝⊿TR/⊿P