2 fixed interest rate r r if the bank sets the rate

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determined by the interaction between money demand and supply. 2 Fixed interest rate: r = r ± If the bank sets the rate of interest, then the level of output is determined only by the IS equation. In this case, the bank will set r so that the expected level of output is Y f . Thus E [ Y ] = ° 0 + ° 1 r (since E [ u ] = 0 ), and thus the optimal interest rate r ° = Y f ± ° 0 ° 1 . Actual output is: Y = Y f + u (4) 3 Fixed money supply: M = M ± The alternative approach is for the Bank to °x the money supply to minimise its expected losses. First solve Y in terms of M : Y = 1 ° 1 ± 1 + ± 2 [ ° 0 ± 2 + ° 1 ( M ° ± 0 ) + ± 2 u ° ° 1 v ] : Next note that expected output is E [ Y ] = 1 ° 1 ± 1 + ± 2 [ ° 0 ± 2 + ° 1 ( M ° ± 0 )] : So to minimise expected losses the Central Bank will set M so that the expected level of output is equal to Y f . Hence: E [ Y ] = Y f = 1 ° 1 ± 1 + ± 2 [ ° 0 ± 2 + ° 1 ( M ° ° ± 0 )] or M ° = ± 0 ° ° 0 ± 2 ° 1 + ° 1 ± 1 + ± 2 ° 1 Y f . 3
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Therefore actual output for the M = M ° case is: Y = Y f + 1 ° 1 ± 1 + ± 2 [ ± 2 u ° ° 1 v ] : (5) 4 Comparing Losses To derive the loss from following the interest rate rule, we substitute equation (4) into the loss function (3): L r = E h ( Y ° Y f ) 2 i = E h ( Y f + u ° Y f ) 2 i = E ° u 2 ± = ² 2 u : Similarly to derive the expected loss from a money rule, we substitute equa- tion (5) into the loss function: L M = E h ( Y ° Y f ) 2 i = E " ² Y f + 1 ° 1 ± 1 + ± 2 [ ± 2 u ° ° 1 v ] ° Y f ³ 2 # = E " ² 1 ° 1 ± 1 + ± 2 [ ± 2 u ° ° 1 v ] ³ 2 # = E ´² 1 ( ° 1 ± 1 + ± 2 ) 2 ° ° 2 1 ² 2 v ° 2 ° 1 ± 2 ³² u ² v + ± 2 2 ² 2 u ± ³µ : The two policies can be compared by considering the ratio of the expected losses: ° = L M L r = 1 ( ° 1 ± 1 + ± 2 ) 2 ´ ° 2 1 ²
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