456_an_4.pdf - ECON 456 Assignment 4 Answers Due in class Marks for each section are noted in the square parentheses There is a total of 100 marks

456_an_4.pdf - ECON 456 Assignment 4 Answers Due in class...

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ECON 456 Assignment 4 Answers Due: in class March 27, 2013 Marks for each section are noted in the square parentheses. There is a total of 100 marks. Question 1 [20] Suppose that future national income is uncertain. With probability p , future income will be ¯ y and, with probability 1 - p , future income will be y . Current income is y . The representative consumer has preferences given by: U = c - ( c 0 - c ) 2 where c is consumption in the current period and c 0 is consumption in the future period. Assume that the representative consumer has access to international capital markets at the world real interest rate, 1 + r . 1. Write down the expression for expected utility. EU = c - p ( c 0 y ) - c ) 2 - (1 - p )( c 0 (y ) - c ) 2 which we can re-write using the savings functions of the households as: EU = y - s - p y + s (2 + r ) - y ) 2 - (1 - p )(y + s (2 + r ) - y ) 2 ) where s is first period savings. 2. Solve for the optimal savings decision of the representative consumer. Here we need to solve ∂EU/∂s = 0. So, - 1 - 2 p (2 + r )(¯ y + s (2 + r ) - y ) - 2(1 - p )(2 + r )(y + s (2 + r ) - y ) = 0 Rearranging this yields: s * = 2 p (2 + r )(y - ¯ y ) + 2(2 + r )( y - y ) - 1 2(2 + r ) 2
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3. How does the optimal savings decision depend on r ? We need to evaluate ∂s * /∂ ( r ). To do this I rearrange slightly so that: s * = p (y - ¯ y ) + ( y - y ) (2 + r ) - 1 2(2 + r ) 2 The first part of the expression clearly falls as r increases. The second part increases as r increases. So the effect is non-linear: ∂s * ∂r = - p (y - ¯ y ) + ( y - y ) (2 + r ) 2 + 1 (2 + r ) 3 4. Suppose instead that there are two types of consumers and that one consumer
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