Social objectives in general equilibrium

I consider an exchange economy in which each agent's preferences are given by U-i = u(i) + theta F, where u(i) is a standard utility function, F is a social objective function and 0 is the weight F receives. Both F and 0 are common to all individuals. I show that F's equilibrium value may be a decreasing function of theta. I also show that if F is a social welfare function whose arguments are the u(i)'s, then the economy's equilibria are independent of theta. (C) 2016 Elsevier B.V. All rights reserved.