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Question 1 Consider the following three-equation log-linear, flexible-price mod...
May 6, 2024
1.1 Solution
a
Initial Equation: Start with the given money market equilibrium condition: mt−pt=−ηit+1+ϕytm_t - p_t = -\eta i_{t+1} + \phi y_t
b
Express ptp_t: Use the second equation to express ptp_t in terms of ete_t and pt∗p_t^*: pt=et+pt∗p_t = e_t + p_t^*
c
Substitute ptp_t: Replace ptp_t in the initial equation with the expression from step b: mt−(et+pt∗)=−ηit+1+ϕytm_t - (e_t + p_t^*) = -\eta i_{t+1} + \phi y_t
d
Rearrange for ete_t: Isolate ete_t to get et=mt+ηit+1−ϕyt−pt∗e_t = m_t + \eta i_{t+1} - \phi y_t - p_t^*. Define Ωt≡mt+ηit+1−ϕyt−pt∗\Omega_t \equiv m_t + \eta i_{t+1} - \phi y_t - p_t^*
e
Apply Law of Iterated Expectations: Use the law of iterated expectations to express ete_t in terms of future expectations of et+Te_{t+T} and Ωs\Omega_s: et=(η1+η)TEtet+T+11+η∑s=tT(η1+η)s−tEtΩse_t = \left(\frac{\eta}{1+\eta}\right)^T E_t e_{t+T} + \frac{1}{1+\eta} \sum_{s=t}^{T} \left(\frac{\eta}{1+\eta}\right)^{s-t} E_t \Omega_s
1.1 Answer
et=(η1+η)TEtet+T+11+η∑s=tT(η1+η)s−tEtΩse_t = \left(\frac{\eta}{1+\eta}\right)^T E_t e_{t+T} + \frac{1}{1+\eta} \sum_{s=t}^{T} \left(\frac{\eta}{1+\eta}\right)^{s-t} E_t \Omega_s
Key Concept
The law of iterated expectations allows us to express current exchange rates in terms of expected future values and current economic conditions.
Explanation
This approach shows how current exchange rates are influenced by expectations of future exchange rates and the current state of the economy, encapsulated in the Ωt\Omega_t term.
1.2 Solution
a
Monetary Policy Rule: Start with the given monetary policy rule: mt−mt−1=ρ(mt−1−mt−2)+εtm_t - m_{t-1} = \rho(m_{t-1} - m_{t-2}) + \varepsilon_t
b
Expectation of εt\varepsilon_t: Given that Et−1(εt)=0E_{t-1}(\varepsilon_t) = 0, we can ignore εt\varepsilon_t when taking expectations
c
Assume ηit+1+−ϕyt−p˙t=0\eta i_{t+1}^{+} - \phi y_t - \dot{p}_t = 0: This simplifies the expression for ete_t to et=mt−pt∗e_t = m_t - p_t^*
d
Substitute mtm_t: Replace mtm_t in the expression for ete_t using the monetary policy rule from step a: et=mt−pt∗=mt−1+ρ(mt−1−mt−2)−pt∗+εte_t = m_t - p_t^* = m_{t-1} + \rho(m_{t-1} - m_{t-2}) - p_t^* + \varepsilon_t
e
Simplify: Since Et−1(εt)=0E_{t-1}(\varepsilon_t) = 0, we can write et=mt+ηρ1+η−ηρ(mt−mt−1)e_t = m_t + \frac{\eta \rho}{1+\eta-\eta \rho}(m_t - m_{t-1})
1.2 Answer
et=mt+ηρ1+η−ηρ(mt−mt−1)e_t = m_t + \frac{\eta \rho}{1+\eta-\eta \rho}(m_t - m_{t-1})
Key Concept
The monetary policy rule can be used to express the exchange rate in terms of past money stock levels and policy parameters.
Explanation
This demonstrates how the exchange rate is affected by the central bank's monetary policy rule and the inertia in the money stock due to the parameter ρ\rho.
1.3 Solution
a
Forward-Looking Policy: Recognize that a forward-looking environment implies that current policy decisions are made with an eye on their future impact
b
Expectations and Policy: Understand that expectations play a crucial role in determining the effectiveness of policy measures
c
Credibility and Consistency: Acknowledge that policy credibility and consistency are essential for shaping expectations and achieving desired outcomes
1.3 Answer
Policymakers must consider the expectations and forward-looking behavior of economic agents when designing and implementing policy.
Key Concept
The importance of expectations in a forward-looking environment for macroeconomic policy.
Explanation
Considering forward-looking behavior is crucial for the success of monetary policy, as it influences how economic agents react to policy changes, thereby affecting the economy's overall performance.
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