J 4. (22 pts total) 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 bejointly 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):11 gm, x) = |n(G) + x1] 931m. X) = 15[|n(s.l] + X'Il gm, X) = -l/a3+ X'Il where 0 is the natural logarithm function, G is the amount ofjointly 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).11 11 Because of how x is defined, the dollar price per unit ofx is fixed at px = $1.11 11 (a) (12 pts) Suppose that each government was to be charged the separate prices W and 99.3 respectively for units of G. Using the tangency requirement for utility maximization, derive expressions for their respective inverse demand functions for G (i.e., 535335 a function of G, pas a function of G, etc.) 11 (b) (3 pts) In the diagram below sketch and label all three ofthe inverse demand curves whose expressions you obtained in question (a) above: 1] Now suppose that as regards supplying military goods G, NATO believes it can provide units of G based on a long run total cost function of C(G) = (S/Agi (c) Given that military goods are jointly consumed, what must be the optimal amount of military force G that NATO should provide its members? (4 pts), and what price should it charge government C? (3 pts)11