请教Pareto Optimality和利润最大化问题

两个相邻的农场利用它们各自的水井在同一个含水层抽地下水。每个农场的抽水费用部分与邻居抽水的数量相关。农场1和农场2的抽水费用用如下公式给出，X1和X2分别为农场1和农场2的抽水量：
C1=4+2*X1+1/4*X1^2+d1*X2     0<d1<4
C2=4*X2+1/4*X2^2+d2*X1         0<d2<1.3

di 代表两个农场水井的相互作用的程度，这些因子每年在一定数值之间独立变化，但是他们的值将会在种植季节开始之前知道。production function如下

yi=16*Xi-1/2*Xi^2

价格为P=1/2

问题1：假设农场之间没有合作，计算地下水开采的最大利润和每个农场获得的最大利润
问题2：问题1中得到的结果是否economically efficient？ 并解释

大概需要用到偏导和拉格朗日乘数

请各位牛人帮忙，非常感谢

definitions of Pareto efficiency: An alloncation X is Pareto efficiency, iff there is no other feasible allocation X(hat).
So, PE  focus in x1,x2's efficiency.

顶一下，希望能有答案

有一点不理解的就是 什么是地下水开采最大利润？

这道题的本质应该是一个古诺，所以只要按古诺的解法，分别求出两家的反应函数就可以了。 第二问要画一下等利润线。

真让人失望

C1=4+2*X1+1/4*X1^2+d1*X2     0<d1<4
C2=4*X2+1/4*X2^2+d2*X1         0<d2<1.3
yi=16*Xi-1/2*Xi^2
price  P=1/2
(Q1)
profit=TR-TC=pYi-(c1+c2)=1/2(16*X1-1/2X1^2+16X2-1/2*X2^2)-(4+2*X1+1/4*X1^2+d1*X2+4*X2+1/4*X2^2+d2*X1 )
By 拉格朗日乘数 to find MAX profit for X1,X2...we need co-state.

(Q2)Pareto
product function(yi) for  X1=(16*X1-1/2X1^2)
subject to:
product function(yi) for  X2=(16*X2-1/2X2^2)
c1=(4+2*X1+1/4*X1^2+d1*X2 )
c2=4*X2+1/4*X2^2+d2*X1
We have one product function and 3 S.T, We need By 拉格朗日乘数 to find MRS of X1,X2 and  Let MRS1=MRS2 to find X1,X2

Check above Please!!
TKS!!

6# shui2
I am very sorry,  the answer is wrong
the right answer of the first question is

1.
a. each farm pursuit its own benefit maximization, use first order condition, get X1=6 and X2=4
b. maximum the joint benefit, R=P（y1+y2）—（C1+C2), compute the partial derivatives of X1 and X2, and then get X1=6-d2, X2=4-d1.


2. change the benefit or amount of water of X1 does not change the benefit of X2 or X1 can gain more benefit than X2' loss.

8# shui2
Sorry , the answer of the second question is right, but if you solve the equtions, the value of x1 or x2 is negative, but the teacher told us, we do not need to solve those equtions, just do some tricky analysis.

Anyway, thank you very much

The forum coins are yours.

Tks~~your coins!!

A perfectly competitive market have p=MR=MC and  Pareto efficiency characteristics.
But the 2nd problem can not satiated above perfectly competitive conditions.
So,not P.E