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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?

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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?
  • $C_B < 1$. 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?
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  • $\begingroup$ The problem, though, is that I am not supposed to find B's consumption using standard method. This is the wording of the part b of the same question: 'Suppose A is able to insist that B does not reduce his labour supply when he receives the transfer. How large should it be then, and how long should A work?' - suggesting that in part a, B does reduce his labour - I'm not sure then, how that works in terms of setting up equations? Thanks $\endgroup$ – Iso1510 Sep 27 '19 at 3:10

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