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.

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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