pareto optimal theory pure exchange economy example

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Unformatted text preview: umer B's marginal rate of substitution as: MRSA = MUXA MUYA = __5__ 2 At the consumer equilibrium, the equation consumer A needs to satisfy is: MRSA = _5_ = PX 2 PY (1B) So now that we have the equilibrium price ratio. Since only the ratio matters we can use the relationship PX = PY in consumer B's demands below (we can derive consumer A's demands from consumer B's...more in a second) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = 2X + 4Y1/2 B = (XB , YB) = (3,14) We can figure out consumer A's marginal rate of substitution as: MRSB = MUXB MUYB = ___2___ 2Y-1/2 = YB 176 At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB 1/2 = PX PY YB = PX2 PY2 Subbing (1) into (2) we get: PX XB + PY YB = MB PX XB + PX2 = MB PY XB = MB - PX PX PY (3) (1) (2) and we know that MB = 3 PX + 14 PY is the endowment income of consumer B, so we sub this in for the MB in (4) to get: XB = 3 PX +14 PY - PX PX PY XB = 3 + _14 PY__ - PX PY PX XB = 3 + _14PY__ - PX PY PX = 3 + 28_ 5 / 2 5 = 3 + 56 / 10 25 / 10 XB* = 6.1 YA = PX2 PY2 YA = 52 22 YB* = 6.25 177 (4) XA = X XB* XA* = 8.9 YA = Y YB* YA* = 8.75 This example illustrates that the individual with a linear indifference curve (consumer A) will dictate the price ratio, while the individual with the non-linear indifference curve will dictate the final demands in the pure exchange economy. Now, what about the Edgeworth Box for this problem? Recall, in the first example (2 Cobb-Douglas consumers) we derived the contract curve in terms of YA mostly because this gives us a familiar reference point in our usual north-east quadrant space. With quasi-linear preferences, the contract curve will be the level of consumption of the quasi-linear good for the individual that has the quasi-linear preferences. So, in the example above consumer B is quasi-linear in...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.

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