Coursenotes_ECON301

This consumer has a utility function that allows us

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Unformatted text preview: subbing (3) into (2) as follows: PX XA + PX XA = MA 2 PX XA = MA XA = M A 2 PX (4) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + PY 2 PX XA = + _PY__ 4 PX (5) PY YA + PY YA = MA 2 PY YA = MA YA = M A 2 PY 100 (6) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (6) to get: YA = PX + PY 2 PY YA = + _PX__ 4 PY (7) 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 income that constrains their utility as follows: UB = UB(XB,YB) = X1/4Y3/4 MB = PX XB + PY YB = PX + PY We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.25 X-3/4Y3/4 0.75 X1/4Y-1/4 = YB 3XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX 3XB PY Rearranging (1B) we get: PX XB + PY YB = MB 3 XB PX = YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + 3 PX XB = MB 4 PX XB = MB 101 XB = MB 4 PX (4B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = PX + PY 4PX XB = 1/8 + _PY__ 8 PX (5B) 1/3 PY YB + PY YB = MB 4/3 PY YB = MB YB = 3MB 4 PY (6B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 3 ( PX + PY) 4 PY YB = 3/8 + _3PX__ 8 PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = + _PY__ + 1/8 + _PY__ 4 PX 8 PX = 3/8 + _3PY__ 8 PX and we know that the fixed supply of X in the economy is the total endowment of X... X = X 102 so, 3/8 + _3PY__ = 1 8 PX and solving the market equilibrium, we get the following equilibrium price ratio... _3PY__ = 5/8 8 PX or, 3PY = 5 PX _PX__ = __3__ 5 PY (8) We can do the same thing in the market for good Y. Okay, let's do it! Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, Y = YA + YB = + _PX__ + 3/8 + _3PX__ 8 PY 4...
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