PLEASE RESPOND TO ALL PARTS OF THE QUESTION. SEE IMAGES FOR Parts A, B, C, and D

Question

PLEASE RESPOND TO ALL PARTS OF THE QUESTION. SEE IMAGES FOR Parts A, B, C, and D!! Thanks.

Assume that Ben and Jerry are neighbors. They live deep in the woods of Vermont, far away from other people. They each have an income of $mi (mB for Ben and mJ for Jerry). Their utility functions are:

Ben’s utility: uB(G, cB) = ?ln(G) + ln(cB)

Jerry’s utility: uJ(G, cJ) = ?ln(G) + ln(cJ)

G is a public good (fireworks, mosquito control, snow plowing for the remote road that they both use…). The total amount of the public good that gets provided is sum of the amount that Ben and Jerry each provide:

G=gB+gJ, where gi is the amount of the public good provided by person i has left over after whatever they have spent on the public good. So ci is measure in dollars and the price of ci is $1.

Let the price of the public good by $p. The private good, ci, is just the amount of money that person i has left over after whatever they have spent on the public good. So ci is measured in dollars and the price of ci is $1. Person i’s budget constraint is:

ci+pgi=mi

This problem asks you to think about the spending that each person will do on the public good (gi). As usual, we will assume that each person chooses ci and gi to maximize his utility subject to his budget constraint. Assume that Ben and Jerry choose gi simultaneously, without knowing what the other will choose.

a) Show that Ben’s utility function can be re-written as: uB(G, cB) = ?ln(gB + gJ) + ln(mB -pgB)

3. [15 points] This problem is set up very similarly to Problem 1. But in this problem, "good 1" is a public good rather than a private good. Assume that Ben and Jerry are neighhers. They live deep in the woods of Vermont, far away from any other people. They each have an income of Sm YmB for Ben and mJ for Jerry). Their utility functions are: Ben's utility: u(G, ch) Bln(G)+ In(c) Jerry's utility u'(G, c) yIn(G)+ In(e) G is a public good (fireworks, mosquito control, snow plowing for the remote road that they both use...). The total amount of the public good that gets provided is sum of the amount that Ben and Jerry each provide: G = g, g, where gi is the amount of the public good provided by person i. Let the price of the public good by $p. Ben Terry The private good, c', is just the amount of money that person i has left over after whatever they have spent on the public good. So ci is measured in dollars and the price ofe is $1. Ci Person i's budget constraint is: Ci + pg' = m' This problem asks you to think about the spending that each person will do on the public good (g). As usual, we will assume that each person chooses c' and g to maximize his utility subject to his budget constraint. Assume that Ben and Jerry choose g' simultaneously, without knowing what the other will choose. 3a. [1 point] Show that Ben's utility function can be re-written as:

Explanation / Answer

a) Person i’s budget constraint is:ci+pgi=mi so cB = mB - pgB it is given that G=gB+gJ

and it is given that Ben’s utility: uB(G, cB) = ln(G) + ln(cB) now substituting the above mention equations we get

Ben’s utility: uB(G, cB) = ln(gB+gJ) + ln(mB-pgB)

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