Suppose that two individuals, Jon and David, form a community and  would  like to construct a communal fort that would protect them from  attacks.  They both consume good X , a private good, and the protection of the fort, P . One unit of good X costs 1 unit of currency, and one unit of P costs 2 units of currency. Both Jon and David have an income of 100 and a utility function of the form: U = log( X i) + 2 × log( PJ + PD ) The budget constraint for each is given by: Xi + 2 × Pi = 100 (a)  Find the amount of protection Jon will provide as a function of  how  much David provides, and explain why the relationship is the way it  is. (b) How much protection P will be privately provided in this case? (c)  Explain the economic intuition behind this amount, and compare  it to  the socially optimal amount without solving for the socially  optimal  amount. .

1. Suppose that two individuals, Jon and David, form a community
and would like to construct a communal fort that would protect
them from attacks. They both consume good
X
, a private good, and the protection of the fort,
P
. One unit of good
X
costs 1 unit of currency, and one unit of
P
costs 2 units of currency. Both Jon and David have an income
of 100 and a utility function of the form:
U
= log(
X
i) + 2 × log(
PJ
+
PD
)
The budget constraint for each is given by:
Xi
+ 2 ×
Pi
= 100
(a) Find the amount of protection Jon will provide as a function
of how much David provides, and explain why the relationship
is the way it is.

2. (b) How much protection
P
will be privately provided in this case?
(c) Explain the economic intuition behind this amount, and
compare it to the socially optimal amount without solving for
the socially optimal amount.