Some analysts see the North Atlantic Treaty Organization (NATO) as a provider of a public good called "common security" to its members. Suppose we measure security in terms of amounts of military force, units of which we represent by the variable G. Units of this military good must be jointly consumed by three member governments, A, B, and C, which have the following respective utility functions over G and total dollars left over for spending on all other goods (x):

U^A(m, x) = 4ln(G) + x

U^B(m, x) = 32[ln(4G)] + x

U^C(m, x) = -G² + x

where ln() is the natural logarithm function, G is the amount of jointly consumed military goods, and x is dollars spent on all other goods. Note that in the above, government C attaches a negative utility to military goods (i.e., government C is a probably a very pacifistic government).

Because of how x is defined, the dollar price per unit of x is fixed at p_x = $1.

Given that military goods are jointly consumed, what must be the optimal amount of military force G that NATO should provide its members, and what price should it charge government C?