Suppose we have an economy with two consumers, i = 1, 2, who have identical Cobb- Douglas preferences over their private goods consumption, xi, and a public good, g, that can be represented by the utility function
ui(xi,g)=3lnxi +2lng,leading to marginal utilities of MUx,i = 3 and MUg,i = 2.
xi g
The resources the economy has are the incomes of the two consumers, I1 and I2. For
simplicity, let I1 = I2 = 50, 000. Suppose further a simple production technology that permits members of the economy to transform 1 unit of the private good into 1 unit of the public good, that is, the price of the private good is px = 1 and the price of one unit of the public good is pg = 1.
(a) Consider that the two consumers are asked to voluntarily contribute to the provision of the public good. For that, suppose we asked each consumer i to choose an amount gi of the public good to provide from their income, knowing that the total amount of public good, g, will be a function of the joint contributions such that g = g1 + g2. Solve for the Nash equilibrium contributions. Show your workings, starting from each consumer’s utility maximization program. State the Samuelson condition for an efficient provision of public goods. Show whether your solution satisfies this condition.
What overall amount of public good would be provided in the efficient solution? Show your workings, starting from the optimization program of utilitarian welfare maximizing social planner.
