2. [Externality in consumption.] Consider an economy with a continuum of consumers, and normalize the total mass of consumers to one. Each consumer has preferences given by u(c, l, c) = u(c, l) + v(c) where c and I are the individual's consumption and leisure, respectively, and c is average consumption across the population. Each individual is very small relative to the population and treats c as given. Assume that u(., .) has standard properties, and v(.) is strictly increasing. strictly concave and twice differentiable. There is an externality in consumption in that any individual is better off when others consume more. The production technology is given by y =n where y is output and n is the labor input. (a) Determine the Pareto optimum where all consumers consume the same quantities. (b) Define a competitive equilibrium, and show that it is not Pareto optimal. (c) Now, suppose that the government subsidizes each consumer's consump- tion. For each unit of consumption, a consumer receives s units of consump- tion from the government. The government finances subsidies by a lump-sum tax 7. That is, consumer's budget constraint is c= w(1-l) + sc -T. Show that, if the government sets the subsidy appropriately, then the competitive equilibrium is Pareto optimal. Determine the optimal subsidy, and explain your results