10 marks Find the equilibrium values of inflation and the output gap and the

# 10 marks find the equilibrium values of inflation and

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[10 marks] Find the equilibrium values of inflation, 𝜋 𝑡 , and the output gap, 𝑥 𝑡 , and the nominal interest rate, 𝑖 𝑡 , needed to implement the optimal policy. (c) [10 marks] Explain intuitively why inflation and the output gap move in opposite directions following a cost-push shock, ? 𝑡 , but do not respond at all to the demand shock, ? 𝑡 . (d) [10 marks] In the remainder of the question, assume ? 𝑡 = 0 for simplicity. Now suppose the central bank is worried about excessive interest rate fluctuations. It chooses to minimize the new loss function: 𝐿 = 1 2 (𝜋 𝑡 2 + 𝑖 𝑡 2 ) By using the IS curve equation, eliminate 𝑖 𝑡 from the loss function to write it in terms of 𝜋 𝑡 and 𝑥 𝑡 . Now repeat part (a), and sketch a diagram plotting the new first-
2 order condition together with the Phillips curve. What now happens to 𝜋 𝑡 and 𝑥 𝑡 following a shock to demand, ? 𝑡 ? Explain your answer. (e) [10 marks] Suppose the central bank uses the original loss function involving 𝜋 𝑡 and 𝑥 𝑡 , but now the interest rate directly affects firms’ costs, so that the Phillips curve becomes: 𝜋 𝑡 = β𝐸 𝑡 [𝜋 𝑡+1 ] + 2𝑥 𝑡 + 𝑖 𝑡 . Using the IS curve, eliminate 𝑖 𝑡 from the Phillips curve to write it in terms of only 𝜋 𝑡 and 𝑥 𝑡 . What will happen to 𝜋 𝑡 and 𝑥 𝑡 after a demand shock, ? 𝑡 ? Explain your answer. 2. Optimal monetary policy Consider the following New Keynesian model: 𝜋 𝑡 = 𝐸 𝑡 [𝜋 𝑡+1 ] + 𝑥 𝑡 + ? 𝑡 𝑥 𝑡 = 𝐸 𝑡 [𝑥 𝑡+1 ] – (𝑖 𝑡 – 𝐸 𝑡 [𝜋 𝑡+1 ]) + ?

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