Question

College Station has two residents: Barton and Jay. The 4th of July fireworks are funded from their individual contributions. They each have the same preferences over private goods (X) and total fireworks (F), represented by the utility function U = 10log(X) + 5log(F), where F = FB + FJ is the total amount of fireworks, and FB, FJ are Barton’s and Jay’s contribution to fireworks . Barton and Jay each have an income of $100, and the price of both the private good and fireworks is $1. Barton’s marginal utilities (It’s useful to remember that 1) dln(x)/dx = 1/x, and 2) they have the same marginal utilities because they have the same utility functions) are: Fireworks: MU B F = 5 FB+FJ Private Goods: MU B X = 10 XB = 10 (100−FB) 1 (flip the subscripts for Jay’s)

Okay, now from here, you should be able to calculate:
Without government intervention, how many fireworks will Barton provide?

Answer 1

Without government intervention, the number of fireworks Barton will provide is determined by equating his marginal utility per dollar spent on fireworks to his marginal utility per dollar spent on the private good, considering the budget constraint of $100. However, an exact amount cannot be provided without knowing Jay's contribution.

Step-by-step explanation:

To calculate how many fireworks Barton will provide without government intervention, we must find the point where Barton's marginal utility per dollar spent on fireworks equals his marginal utility per dollar spent on private goods (X). Given Barton's budget constraint of $100 and assuming the price of the private good and fireworks is $1 each, Barton will maximize utility by equating the ratio of marginal utilities with the ratio of prices, which is known as the optimal consumption bundle.

Since both goods cost the same, Barton will choose the quantity of each good where the marginal utility of spending an additional dollar on fireworks is equal to the marginal utility of spending an additional dollar on the private good. By plugging in the formulas for marginal utilities for Barton, we get:

MU BF = 5/(FB+FJ)

MU BX = 10/(100-FB)

Setting these equal to each other, as 5/(FB+FJ) = 10/(100-FB), we can solve for the amount of Barton's contribution to fireworks (FB). However, since we don't have the specific values for FJ (Jay's contribution), we cannot solve for an exact number for Barton's contribution. But we can express it in terms of Jay's contribution, which would guide us on the portion of income Barton would allocate to fireworks assuming Jay's contribution to fireworks is known.