Coursenotes_ECON301

# L 1 production contract curve k 1 187 consumption

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Unformatted text preview: equation of the UPF at point A, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA 2XA)1/2 = XA (21/2) UB = (XB YB)1/2 = (XB 2XB)1/2 = XB (21/2) and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA (21/2) + XB (21/2) UA + UB = (XA + XB)(21/2) UA + UB = (1/3)(21/2) UA + UB = (21/2) 3 [b] At point B on the PPF, we have the following equations: XA + XB = 1/2 YA + YB = 1/2 YA = YB XA XB which can be solved for the equations of the contract curve YA = XA YB = XB To find the equation of the UPF at point B, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA XA)1/2 = XA UB = (XB YB)1/2 = (XB XB)1/2 = XB and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA + XB 189 UA + UB = X U A + UB = [c] At point C on the PPF, we have the following equations: XA + XB = 3/4 YA + YB = 1/4 YA = YB XA XB which can be solved for the equations of the contract curve YA = 1/3XA YB = 1/3XB To find the equation of the UPF at point C, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA 1/3XA)1/2 = XA (1/3)1/2 UB = (XB YB)1/2 = (XB 1/3XB)1/2 = XB (1/3)1/2 and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA (1/3)1/2 + XB (1/3)1/2 UA + UB = (XA + XB)(1/3)1/2 UA + UB = (3/4)(1/3)1/2 UA + UB = (3)1/2 4 190 Y 1 MRS = 2 2/3 MRS = 1 MRS = 1/3 PPF (slope = -1) 1/3 1 X OVERALL EFFICIENCY Among the three points A, B, and C on the PPF, only point B satisfies the overall efficiency condition. Point A Point B Point C MRS = 2 MRS = 1 MRT = 1 MRT = 1 MRS = 1 MRT = 1 3 That is, while production efficiency and consumption efficiency hold separately at all three of these points, only at point B are production and consumption efficiency compatible with each other (i.e. MRS = MRT = 1). In the diagram below, this overall efficiency condition can be seen as the UPF corresponding to point B is higher than that of the other two points. In fact, it can be shown that in this special case, the UPF at point B is also the Grand UPF. 191 UB 2/3 UPF (point B) = GRAND UPF UPF (point A) 3/4 UPF (point C) 3/4 2/3 UA SOCIAL WELFARE FUNCTIONS So how many solutions are there for the Pareto Optimal Production Economy? For each point on the PPF, we have to solve one pure exchange economy to get [1] one Edgeworth box,...
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