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homework1 2010 solutions

# homework1 2010 solutions - ~ 2 2 2 2 2 = n t U E Thus R cb...

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ECN/APEC 7240 Spring 2010 Homework 1 Solutions 1) a) The public knows x t , so the public is minimizing ]. ~ [ ) ( ] ) ~ [( 2 2 2 t t t t t t p z E w x w z x E L + - = - + = This has the first-order condition w t = x t . The Phillips curve then becomes t n t z U u θ - = . The central bank then has no control over unemployment and can just set its inflation target to minimize ]. ~ [ ] ~ [ 2 2 2 t t t z E x E + = π That is, it will set x t = 0. Thus, for the Ramsey equilibrium, we will have x t = w t = 0. b) Now suppose that the central bank acts after the public choose w t . Then the central bank’s loss function is ] ~ [ ) 1 ( )) ( ( ] )) ~ ( ( ) ~ [( 2 2 2 2 2 2 t t t n t t t t n t t p z E w x U x w z x U z x E L αθ θ α θ α + + - - + = - + - + + = This has the first-order condition 0 )) ( ( 2 2 = - - - = t t n t p w x U x dx dL θ αθ ) ( ) 1 ( 2 t n t w U x θ αθ αθ + = + . 1 ) ( 2 αθ θ αθ + + = t n t w U x As before, the public will set w t = x t . Thus, in equilibrium we will have 2 1 ) ( αθ θ αθ + + = t n t x U x t n t x U x 2 2 ) 1 ( αθ αθ αθ + = +

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ECN/APEC 7240 Spring 2010 n t t U x w αθ = = >0. c) Note that in both equilibria we have w t = x t , so u t = U n - θ z t , and . ] ~ [ ] ~ [ 2 2 2 2 2 θσ θ + = + = n t n t U z E U u E In the Ramsey equilibrium, π t = z t , so . ] ~ [ 2 2 σ π = t E Thus, L
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Unformatted text preview: ~ [ 2 2 2 2 2 + = n t U E Thus, R cb n N cb L U L + + + = 2 2 2 ) 1 ( ) 1 ( . d) Intuitively, one would expect that if the central bank has more information then it can achieve a better outcome. This is not the case, however, because the central bank can only achieve a better outcome by fooling the public. For the Phillips curve to give lower unemployment, expected inflation has to be positive. But since the public also has rational expectations, it sets wage inflation at the one value such that the central bank cannot improve unemployment by surprising the public with high inflation, and this value is positive....
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