homework4_soln

homework4_soln - Economics 150 Homework 4 Answers Spring...

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Economics 150, Homework 4 Answers Spring 2008 Question 4.1 When the price of bananas is p b = 1 and the price of coconuts are p c , then the value of both agents’ endowments are m = p b 1 + p c c = 1 + p c c. Since both agents have Cobb-Douglas utility functions we know immediately what their demands are. For Friday, the demands are x F b = αm p b = α (1 + p c c ) x F c = (1 - α ) m p c = (1 - α )(1 + p c c ) p c and similarly, we know that the demands for Robinson are x R b = (1 - α )(1 + p c c ) x R c = α (1 + p c c ) p c . Question 4.2 To have an equilibrium we must have that supply equals demand. That is that the sum of Friday and Robinson’s demands for bananas and coconuts must be equal to the total amount in the economy. So the condition we must check is that x F b + x R b = 2 x F c + x R c = 2 c Substituting from the demand functions in [4.1], this is x F b + x R b = (1 + p c c ) = 2 x F c + x R c = (1 + p c c ) p c = 2 c 1
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Dividing the two equations gives the result that p c = 1 c . Plugging this back into the demands we have that Friday’s demands are x F b = 2 α x F c = 2(1 - α ) c and Robinson’s demands are x R b = 2(1 - α ) x R c = 2 αc. We can verify that, for example, the second market clearing condition is also satisfied, so x F c + x R c = (1 - α )(1 + p c c ) p c + α (1 + p c c ) p c = (1 + p c c ) p c = 2 . Question 4.3 When c = 1 the problem is completely symmetric with respect to bananas and coconuts, so the prices of the two goods are the same ( p c = 1) and we have x F b = x R c and x F c = x R b . As c increases, the share of their wealth (the value of their endowment) that Robinson and Friday spend on consumption of bananas remains unchanged. Since the total amount of bananas in the economy is unchanged, this means that the wealth of the two agents must be unchanged. So we must have that p c decreases as the endowments of coconuts increases, and in particular the price must be p c = 1 c . Since the wealth of both agents is unchanged, but the price of coconuts decreases, as c increases both agents increase their consumption of coconuts proportionally.
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