Social objectives in general equilibrium

I consider an exchange economy in which each agent’s preferences are given by Ui = ui + θF , where ui is a standard utility function, F is a social objective function and θ is the weight F receives. Both F and θ are common to all individuals. I show that F ’s equilibrium value may be a decreasing function of θ. I also show that if F is a social welfare function whose argument are the ui’s, then the economy’s equilibria are independent of θ.