# What is the difference between solving for utility maximization separately or aggregately?

I derived the Walrasian equilibrium for an economy of two consumers with their respective utility functions (u1, u2) and initial endowments but later on I am asked to maximize (u1+u2).

Does anyone know how this changes the problem ? Like, what is the difference in both approaches when it comes to the optimal allocations ?

## 2 Answers

Under what constraints is $$u_1+u_2$$ maximized? The exercise may be about the First Welfare Theorem: a competitive equilibrium is Pareto efficient. In this case, maximizing the sum may be designed to highlight that the equilibrium you found also maximizes the sum of utilities.

In general, the maximum of a sum of functions need not equal the maximum of any of these functions.

• The constraints are their budget constraints and then we have the economy resource constraints for each of the two goods. The utilities have the form aln(goodi)+bln(goodi) where i=1,2 for the two agents. I set up a new lagrangian with u1+u2 -lambda1(bc1) -lambda2 (bc2) but obviusly i get the same result as before and no new hindsight.. Thank you, Sander. – ceins Nov 11 '18 at 12:02

The second problem is probably for comparison - it is the problem of a benevolent social planner who can freely allocate resources between the two agents.