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5. (7 marks) There are two individuals who demand yl Z 0 and '92 2 0 units of output7 respectively. The total output demanded is y : yl + yg. To produce 3; units of output costs C(y), where C(O) : O and both C'(y) and C\"(y) are positive for all y 2 0. The costs are shared in proportion to the demands7 so person 1's cost is 1'1 : y10(y)/y and person 27s cost is $2 : y20(y)/y if y > 0. If y : 07 neither person pays anything. The individual utility functions are U1 ($1,311) : yl x1 and "MM, 312) : 3/2 x2. First suppose that the two individuals choose their demands independently and simultaneously. Let y: and y; be the amounts demanded by the two individuals in the Nash equilibrium to this problem. Assume that yf > 0 and y; > 0. Now suppose that the two individuals cooperate and choose their demands to maximize 15161317111) + u2(z2.y2) given the costsharing rule described above. Let "Q > 0 be the sum of the demands when the sum of the utilities is maximized. You may assume that the sum of utilities is positive when the sum of utilities is maximized. Show that y\" g Q. where 11* : yf + y'. Hint: As an intermediate step in your argument7 you should show that 17 > 0(3)). The assumptions on C(-) ensure that the secondorder conditions for the rele vant maximization problems are satised. You do not have to check second-order conditions