Answer this question using calculus. Consider a version of the one period general equilibrium model with production in which the government produces a public good, G. The public good is something like national parks. It enters as an input into the utility function of the household. That is, the utility function is given by U(C, l, G), where all of the marginal utilities are positive. The government finances its spending via a proportional sales tax.

(a) Formally define a competitive equilibrium in this economy.

(b) Generate equations that characterize the equilibrium allocation and prices.

(c) Write down the government's optimization problem.

(d) Will the government optimally choose to provide the same level of the public good as would a social planner? Explain. (You can answer this problem in words only.)