f Repeating our earlier analysis eliminating from IS and LM gives β γ β βγ 1 γ

F repeating our earlier analysis eliminating from is

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f. Repeating our earlier analysis, eliminating 𝑅 𝑡 from IS and LM gives: (β + γ) ? 𝑡 = β ( 𝑚 𝑡 − 𝑝 𝑡 ) + βγ𝐸 𝑡 [𝑝 𝑡+1 − 𝑝 𝑡 ] + γ𝑣 𝑡 (AD) And combining AS, AD and the policy rule now gives: (β + γ) α ( 𝑝 𝑡 − 𝐸 𝑡−1 [𝑝 𝑡 ] ) = β ( δ𝑣 𝑡−1 − 𝑝 𝑡 ) + βγ𝐸 𝑡 [𝑝 𝑡+1 − 𝑝 𝑡 ] + γ𝑣 𝑡 If we again conjecture that 𝑝 𝑡 = φ𝑣 𝑡 + ψ𝑣 𝑡−1 , then it follows that 𝐸 𝑡−1 [𝑝 𝑡 ] = ψ𝑣 𝑡−1 and 𝐸 𝑡 [𝑝 𝑡+1 ] = 𝐸 𝑡 [φ𝑣 𝑡+1 + ψ𝑣 𝑡 ] = ψ𝑣 𝑡 (note the change from our earlier result). Note also that 𝐸 𝑡 [𝑝 𝑡 ] = 𝑝 𝑡 . Hence (note the additional terms introduced by giving financial markets more information): [ (β + γ) α + β(1 + γ)](φ𝑣 𝑡 + ψ𝑣 𝑡−1 ) = (β + γ) αψ𝑣 𝑡−1 + βγψ𝑣 𝑡 + γ𝑣 𝑡 + βδ𝑣 𝑡−1 As before, this equation holds for all possible values of 𝑣 𝑡−1 and 𝑣 𝑡 . Hence, equating coefficients on the left and right-hand sides: 𝑣 𝑡−1 : [ (β + γ) α + β(1 + γ)]ψ = (β + γ) αψ + βδ ⇒ ψ = δ/(1 + γ) 𝑣 𝑡 : [ (β + γ) α + β(1 + γ)]φ = βγψ + γ ⇒ φ = γ(1 + βψ)/ [ ( β + γ )α + β(1 + γ)] = γ{1 + βδ/(1 + γ)}/ [ ( β + γ )α + β(1 + γ)] Note that φ now depends on the policy feedback parameter, δ . Finally, using AS gives: ? 𝑡 = α(𝑝 𝑡 − 𝐸 𝑡 [𝑝 𝑡 ]) = α(φ𝑣 𝑡 + ψ𝑣 𝑡−1 − ψ𝑣 𝑡−1 ) = αφ𝑣 𝑡
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3 Hence the choice of δ also affects the size of the response to the demand shock. In particular, if the policy maker sets δ = −(1 + γ)/β then φ = 0 and thus ? 𝑡 = 0 at all times. Hence the economy is perfectly stabilised, even though the policy maker cannot observe the current demand shock! This is a remarkable and surprising result. How does it come about? There are two key features. First, someone (agents in financial markets) does observe the current shock. Second, there is an intertemporal linkage between future policy and aggregate demand today. If it is a positive shock to demand today, then the policy maker will tighten policy tomorrow. But agents in financial markets seeing today’s shock can factor that in when forming their expectations, i.e. they will lower their expectations of inflation. That in turn raises the expected real interest rate and reduces demand today, so counteracting the original positive demand shock. This example shows that we do not need the policy maker to have an informational advantage to generate a role for policy. Other changes in the information structure can lead to a role for policy, though through more subtle channels than is the case when the policy maker has an informational advantage.
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