Comment on Problem 2 of hwk 8

Solution of problem 2 of hwk 8 is valid only for L=1.
When L>1, assume (to the contrary) that x_11 not equal to x_12.
Then from strict convexity of preference, we can say that
(x_11+x_12)/2 is strictly preferred to x_11 under type 1's preference.
(See MWG p.656 Prop. 18.B.2)

I thank Matsuda-san for pointing this out.