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