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Answer to Question #60082 in Microeconomics for nazneen

Question #60082
You are asked to solve a public good problem with
2 agents. Agent i has utility function a, log (g1 + g2) + x, where g, is the amount of the public good contributed by agent i and xi = amount of private consumption of agent i. The budget constraint of agent i is x, + gi = wi where w, is the initial endowment of agent i. Assuming wi > ai for each agent,
(a)compute the Pareto — optimal level of public good provision for this economy.
(b)Describe the voluntary contribution equilibria for different values of a1 and a2.
Expert's answer
Agent 1 has utility function a1 = log (g1 + g2) + x1, where g, is the amount of the public good and x1 = amount of private consumption.
The budget constraint of agent 1 is x1 + g1 = w1.
(a) the Pareto — optimal level of public good provision for this economy is in the point, where w = a, so:
x1 + g1 = log (g1 + g2) + x1,
log(g1 + g2) = g1.
(b) The voluntary contribution equilibria for different values of a1 and a2 will be different, because the higher is the utility, the lower is the amount of contribution.

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