Question

1. College Station has two residents: Ben and Gerry. 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 = 10 x log(x) + 5x log(F), where F = Fg + Fois the total amount of fireworks, and Fo, Fa are Ben's and Gerry's contribution to fireworks. Ben and Gerry each have an income of $100, and the price of both the private good and fireworks $1. Their utility functions imply that their marginal utilities (they have the same marginal utilities because they have the same utility functions) are: Fireworks: MW-5/(F,+F) Private Goods: M x =10/X = 10/(100-F) where the į and j superscript is į = Ben, Gerry. Partial Solution: Step 1: Set up the maximization problem For Ben: Max U= 10 x log(X.) + 5 xlog(Fo+FG) subject to the budget constraint: 100 - 2.XatleEx=X+F. (since the prices of X and Fare both 1.) Step 2: Since X. - 100-F, holds from the budget constraint, we can plug that into the utility function: Max Ug = 10 x log(100-F.) +5 xlog(F. + Fo) with respect to F, taking Gerry's fireworks spending as given (so Ben also gets utility from Fo but cannot control it) Step 3: Take the derivative with respect to F, and set it to 0: (5/(100- Fx))*(-1) Chain rule example: the derivative of 5 x log(100-F) with respect to Fois Remember that the derivative of log(x) is 1/x له المليلة → (10/(100-F.))*(-1) (5/F.+F.X(1) = 0 Solving for F, using a bit of algebra, we get: → 500-5F, -10F, +10F Fo= 33.33-2/3F, is Ben's condition for utility maximization Since Gerry's maximization problem is the same as Ben's, we can just reverse the subscripts and characterize Gerry's condition for utility maximization to be F-33.33-2/3 F. Okay, now from here, you should be able to calculate: a. Without government intervpntion, how many fireworks will Ben provide?

Answer 1

2 individuals Pen and Gerry 2 goods: Private goods and Fireworks. ů = 10x lg (XB) + 5 g (Fet FG) Ug = 10 e lo (xa) + 5lg (Fat FB) F = Fg + Foi PE FB & Ppx = mo Fg + g = 100 P& FG & Peixo - ma > Fon + Xo = loo 0 Fg + XB - 100 Maxination problem for Ben L = Up - ( Fp + Xo =100) Maximise up with respect to budget constraint Up = 10 x lg (100 - Fg) + 5 Lg FB + FG) aug = -10 + 5 =0 OFS (100-Fg) Foz + FG To F+For 100-FE 500 - Stp = 10 FB + 10 Fr. 5 F8 + 10 For = 500. Pultiply too neg 10 Fot Since Gerry's maxionisation problem is same, SUG = -10 + s =0 dFG 100-FG Fot to 5 FG + 10 Fe = 500. - 20 F8 + 1o Fq = 1000 - 5 FG + roF G =500 - 15 Fp = 500 Jos Scanned with For = 33.33, = F = 33.33 le trents alimeniffer

a) Without government intervention, Gary will provide 33 fireworks

b) Without government intervention, Ben will provide 33 fireworks.

c) On total they will provide 66 fireworks.

d) The socially optimum level is again Fb=33.33 and Fg= 33.33

This can be checked by substituting MRS values in the following equation

MRSB + MRSG = 1

Ми Fot for MU 10o-Fo - MRS MR & MRS MOS a MU - MUG FB+ FG Fat FG too-FB 100- For 100-Fp. + 100 - Fg 1 2 (FB + FG) 200 – (FR+ Fa) = 2 ( FotFQ) 200 = FBT FG .= 66.66 Scanned with CamScanner