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.

Suppose that each government was to be charged the separate prices p_G^A, p_G^B, and p_G^C respectively for units of G. Using the tangency requirement for utility maximization, derive expressions for their respective inverse demand functions for G (i.e., p_G^A as a function of G, p_G^B as a function of G, etc.)