Two mariners end up on a island, with 1800 pounds of food to share, i.e. $F1 + F2 = 1800$.
I'm expected to maximize the social welfare function given by $W=U1^{0.25}*U2^{0.75}$ where $U1=\sqrt{F1}$ and $U2=0.5\sqrt{F2}$
I tried to use the fact that W was a Cobb-Douglas function, but I found the constraint to be equal to $U1^{2}*4U2^{2}=1800$. Hence I don't know how to move on from there.
$MP1=(1/4)*(U2/U1)^{0.75}$ & $MP2=(3/4)*(U1/U2)^{0.25}$
Thanks for your help
This problem can be approached using the method of Lagrange's multipliers. First, we express the welfare function through $F_1$ and $F_2$ as
$W(F_1,F_2) = \bigg(\frac{1}{2}\bigg)^{3/4}(F_1F_2^3)^{1/8}$.
Then, we can write the function to be maximized as
$L(F_1,F_2) = W(F_1,F_2) + \lambda(F_1+F_2-1800)$.
Next, we find partial derivatives
$\frac{\partial L}{\partial F_1} = \bigg(\frac{1}{2}\bigg)^{3/4}\frac{1}{8}(F_1^{-7}F_2^3)^{1/8} + \lambda= 0$;
$\frac{\partial L}{\partial F_2} = \bigg(\frac{1}{2}\bigg)^{3/4}\frac{3}{8}(F_1F_2^{-5})^{1/8} + \lambda = 0$;
$\frac{\partial L}{\partial \lambda} = F_1+F_2 + 1800 = 0$.
Now, the first two equations yield, after eliminating $\lambda$
$F_2 = 3F_1$,
which, in combination with the last derivative, gives $F_1 = 450$ and $F_2 = 1350$.
Hope this helps.