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First order condition

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Consider the problem of a social planner who wants to maximize utilitarian welfare in an exchange economy with two people. His objective is thus
maxu1 + u2 ,where u1 and u2 are the utilities of the two individuals. Both have the same utility overconsumption c, so ui = u(ci ) = ln ci , i=1,2 (ln stands for the natural logarithm). The total available quantity of the consumption good is X, so that c1 + c2 = X .
1. Show (preferably analytically) that the planner will allocate half of the available
consumption good to each individual.
2. Briefly explain why you generally would not expect this result if the utility functions of the two individuals were different.

What are the steps to solve this analytically?

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Solution Summary

First order condition is clearly expressed in this solution. Natural logarithms are given.

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1. Show (preferably analytically) that the planner will allocate half of the available consumption good to each individual.
Let U = social utility, then the objective function becomes
max U = U1 + U2
U = ln(C1) + ln(C2)

Then the marginal utility of C1 and C2 would be:
MU1 = dU/dC1 = 1/C1
MU2 = dU/dC2 = 1/C2

According to the first order condition, U is maximized when ...

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