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monetary theory and expectations

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T ake-Home 1. by Jorge Rojas Problem. Consider the following money demand function, where all variables are in natural loga- rithms. m t − p t = − (E t  p t+1 − p t ) ∀t =0, 1, 2, 3, (1) E t  p t+1 = p t+1 (perfect foresigh t) (2) where m t is the log of the money supply, p t is the log of the price level at t, and E t  p t+1 the public’s expectation of p t+1 formed at time t. (a) Suppose that m t = 10 for t = 0, 1, 2, 3, . Using lag operators compute the equilibrium value for p t for t =0, 1, 2, 3, 4, 5. We know that m t = 10 ∀t. From equations (1) and (2) we obtain as follows: m t − p t = − (E t  p t+1 − p t ) m t − p t = − (  p t+1 − p t ) m t − p t = − p t+1 + p t m t + p t+1 = 2 p t ⇒ p t = 1 2 p t+1 + 1 2 m t (3) We can see fr om equation (3 ) that p t will conve rge smo ot hly to a cer ta in valu e p ∗ through time because the coefficien ts for p t+1 and m t of the diff ere nce equ at ion are smaller than 1 and greater than zero. We can write equation (3) in its general form as:  p t = φ p t+1 + ρ m t (4) where φ = 1 2 and ρ = 1 2 . We can rearrange equation (4) as follows: ⇒ p t − φ p t+1 = ρ m t  p t − φ L −1  p t = ρ m t 1 − φ L −1  p t = ρ m t ⇒ p t = ρ (1 − φ L −1 ) m t (5) 1

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