Answer 1.13. By Proposition 1.27, the following equation defines the set of P O. points Or Feasibility requires r2+a2 Letr≡ I and y≡r. Then above two equations imply y Therefore the set of P O allocations=f[(, y),(1-a,1-y)llx=y,x20 This set is the diagonal line in the following diagram Figure 4.4. P.O. Allocations Answer 1. 14. Denote g= guns, a=oil,b= butter, price of guns Pg, price of butter Po, price of oil Pr=l (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are consumer1:u1(g,b)=904606,g=2x,1=10. 2(G,b)=905605,g=3x,2=10 Firm ls problem: 丌1≡maxP9-x=max(2P-1)xAnswer 1.13. By Proposition 1.27, the following equation defines the set of P.O. points: 1/x1 1 1/x2 1 = 1/x1 2 1/x2 2 or x2 1 x1 1 = x2 2 x1 2 . Feasibility requires x1 1 + x1 2 = 1 and x2 1 + x2 2 = 1. Let x ≡ x1 1 and y ≡ x2 1. Then above two equations imply y x = 1 − y 1 − x =⇒ y = x. Therefore, the set of P.O. allocations =  [(x, y), (1 − x, 1 − y)] | x = y, x ≥ 0  . This set is the diagonal line in the following diagram. 1 2 y=x P.O. x y Figure 4.4. P.O. Allocations Answer 1.14. Denote g = guns, x = oil, b = butter, price of guns Pg, price of butter Pb, price of oil Px = 1 (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are: consumer 1: u1(g, b) = g0.4b0.6, g = 2x, x¯1 = 10. consumer 2: u2(g, b) = g0.5b0.5, g = 3x, x¯2 = 10. Firm 1’s problem: π1 ≡ maxx Pg g − x = maxx (2Pg − 1)x. 10