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.

for what fraction of the time does each work, and how large is the transfer g?

i would prefer it if this can be solved using the lagrange multiplier.