Constrained optimisation with transfer I have been stuck on this question for about two days and can find no way out (apologies if the question seems really simple as I haven't started university yet). I would strongly prefer it if this can be solved using the Lagrange multiplier. Thanks.

There are two individuals, A and B, in an economy. Each derives utility from his
consumption, C, and the fraction of his time spent on leisure, l, according to the
utility function:
U = ln(C) + ln(l)
However, A is made very unhappy if B’s consumption falls below 1 unit, and he
makes a transfer, G, to ensure that it does not. B has no concern for A. A faces a
wage rate of 10 per period, and B a wage rate of 1 per period.
(a) For what fraction of the time does each work, and how large is the transfer G?
I'll guess that, since you're not yet in college but you're doing something this "advanced," you're not doing a homework problem, and that you're asking for a hint.

Since B doesn't care about A, you should be able to find B's consumption $(C_B, l_B)$ using standard method (maximizing utility w.r.t. budget constraint, etc.)

Now there are two cases.


$C_B \geq 1$. Then what would the problem for A turn out to be?
$C_B &lt; 1$. Then think about how much should A transfer to B? Should he transfer nothing? 0.1? Up until what point would the increase in the transfer stop making A better off?