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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.
3.1 Solution
a
Present Value Budget Constraint: The present value budget constraint for the representative agent, considering government expenditure GtG_t, is given by the equation C1+C21+r=Y1−G1+Y2−G21+rC_1 + \frac{C_2}{1+r} = Y_1 - G_1 + \frac{Y_2 - G_2}{1+r}, where CtC_t is consumption and YtY_t is GDP in period tt
b
Optimization Problem: The agent maximizes utility U=u(C1)+βu(C2)U=u(C_1)+\beta u(C_2) subject to the present value budget constraint. The Lagrangian for this problem is L=u(C1)+βu(C2)+λ(Y1−G1−C1+Y2−G2−C21+r)\mathcal{L} = u(C_1) + \beta u(C_2) + \lambda(Y_1 - G_1 - C_1 + \frac{Y_2 - G_2 - C_2}{1+r})
c
First-Order Condition: The first-order condition for consumption is derived from the Lagrangian and equates the marginal utility of consumption in each period, adjusted for the discount factor and interest rate: u′(C1)=β(1+r)u′(C2)u'(C_1) = \beta (1+r) u'(C_2). This condition ensures that the marginal utility of consumption is equalized over time when discounted by the interest rate
d
Marginal Product of Capital: The role of the marginal product of capital is to determine the amount of investment and thus the capital stock for the next period. It influences the production function Y=F(K)Y=F(K) and thereby the GDP YtY_t available for consumption and saving
3.1 Answer
The agent's present value budget constraint is constructed, and the optimization problem is solved using the Lagrangian method. The first-order condition for consumption ensures intertemporal utility maximization, and the marginal product of capital influences investment decisions and future GDP.
Key Concept
Intertemporal Budget Constraint and Optimization
Explanation
The present value budget constraint represents the trade-off between present and future consumption, while the optimization problem involves maximizing utility across periods given this constraint.
3.2 Solution
a
Current Account Equation: The current account in period tt is defined as CAt=Yt−Ct−Gt−ItCA_t = Y_t - C_t - G_t - I_t, where ItI_t is investment
b
Expected Pattern of Current Account: If the autarkic rate of interest is higher than the world rate, the economy is likely to borrow from abroad in period 1, leading to a current account deficit, and repay in period 2, leading to a current account surplus
c
Diagram Explanation: The diagram would show the production possibilities curve and the agents' indifference curves. In period 1, the economy would consume beyond its production possibility due to borrowing, shown by a point outside the curve. In period 2, the economy would consume less than its production possibility to repay the debt, shown by a point inside the curve
3.2 Answer
The current account equation is established, and the expected pattern of deficits and surpluses is explained by the difference between the autarkic and world interest rates. A diagram illustrates the intertemporal trade-off facilitated by international borrowing and lending.
Key Concept
Current Account Dynamics
Explanation
The current account reflects the economy's net borrowing or lending position, which is influenced by the difference between domestic and world interest rates, affecting intertemporal consumption choices.
3.3 Solution
a
Adjustment to Temporary Factors: Current account surpluses and deficits allow agents to smooth consumption in response to temporary shocks by borrowing or lending internationally
b
Policymakers' Concerns: Large or persistent deficits/surpluses may indicate underlying structural issues, such as competitiveness problems or savings-investment imbalances, and can lead to vulnerabilities like foreign debt accumulation or currency crises
3.3 Answer
Current account imbalances facilitate consumption smoothing in response to temporary factors, but policymakers are concerned about the long-term implications of persistent imbalances, such as debt sustainability and economic stability.
Key Concept
Consumption Smoothing and Policy Concerns
Explanation
While current account imbalances can help adjust consumption in the short term, they may pose risks to the economy's long-term health, prompting caution from policymakers.
5.1 Solution
a
Taylor Rule Formula: The Taylor Rule is a guideline for setting interest rates based on economic conditions, represented by the formula it=r∗+πt+0.5(πt−π∗)+0.5(yt−y∗)i_t = r^* + \pi_t + 0.5(\pi_t - \pi^*) + 0.5(y_t - y^*), where iti_t is the nominal interest rate, r∗r^* is the real equilibrium interest rate, πt\pi_t is the current inflation rate, π∗\pi^* is the target inflation rate, yty_t is the logarithm of real GDP, and y∗y^* is the logarithm of potential GDP
b
Key Elements and Insights: The Taylor Rule incorporates both inflation and output gaps as key elements to guide monetary policy decisions, aiming to stabilize the economy by adjusting interest rates in response to deviations from target inflation and potential output
5.1 Answer
The Taylor Rule is a monetary policy rule that suggests how central banks should change interest rates in response to changes in inflation and the output gap.
Key Concept
Taylor Rule in Monetary Policy
Explanation
The Taylor Rule provides a systematic and transparent method for central banks to adjust interest rates based on deviations from target inflation and potential output.
5.2 Solution
a
New Keynesian Models with Taylor Rules: These models incorporate forward-looking behavior and price stickiness, allowing for a more realistic representation of how monetary policy affects the economy, including the exchange rate
b
Superior Empirical Framework: By including the Taylor Rule, New Keynesian models can better capture the central bank's systematic response to economic conditions, which is crucial for understanding exchange rate dynamics
5.2 Answer
New Keynesian models with Taylor rules provide a superior empirical framework for modeling exchange rate behavior due to their incorporation of forward-looking elements and the systematic monetary policy response.
Key Concept
Explanation
6.1 Solution
a
Monetary Policy Trilemma: The trilemma states that a country cannot simultaneously maintain a fixed exchange rate, free capital movement, and an independent monetary policy. Historical example: The collapse of the Bretton Woods system in the early 1970s
b
Financial Policy Trilemma: This trilemma posits that it is not possible to have financial stability, financial integration, and national financial policies all at once. Historical example: The European sovereign debt crisis where countries faced trade-offs between these objectives
6.1 Answer
The monetary policy trilemma involves choices between fixed exchange rates, capital mobility, and monetary independence, while the financial policy trilemma involves choices between financial stability, integration, and national control.
Key Concept
Explanation
6.2 Solution
a
Pressing Concern for China: Considering China's economic structure and policy goals, the financial policy trilemma may be more pressing due to China's efforts to maintain financial stability while pursuing greater financial integration and retaining control over national financial policies
6.2 Answer
The financial policy trilemma is a more pressing concern for China as it balances the goals of financial stability, integration, and national policy autonomy.
Key Concept
Explanation
3.1 Solution
a
Optimization Problem: The agent maximizes utility subject to the budget constraint. The Lagrangian is L=u(C1)+βu(C2)+λ(Y1+Y21+r−C1−C21+r)\mathcal{L} = u(C_1) + \beta u(C_2) + \lambda(Y_1 + \frac{Y_2}{1+r} - C_1 - \frac{C_2}{1+r})
b
First-Order Conditions: Differentiate the Lagrangian with respect to C1C_1 and C2C_2 to get ∂L∂C1=u′(C1)−λ=0\frac{\partial \mathcal{L}}{\partial C_1} = u'(C_1) - \lambda = 0 and ∂L∂C2=βu′(C2)−λ1+r=0\frac{\partial \mathcal{L}}{\partial C_2} = \beta u'(C_2) - \frac{\lambda}{1+r} = 0
c
Equilibrium Consumption: Solve the first-order conditions to find C1C_1 and C2C_2. From the FOCs, we get u′(C1)=λu'(C_1) = \lambda and βu′(C2)=λ1+r\beta u'(C_2) = \frac{\lambda}{1+r}. Equating these gives u′(C1)=β(1+r)u′(C2)u'(C_1) = \beta (1+r) u'(C_2)
d
Net Foreign Assets and Current Account: Bt+1=Yt+rBt−CtB_{t+1} = Y_t + rB_t - C_t and CAt=Bt+1−BtCA_t = B_{t+1} - B_t
3.1 Answer
The agent's optimal consumption in each period is determined by equating the marginal utility of consumption in the first period to the discounted marginal utility of consumption in the second period, adjusted for the world real interest rate. Net foreign assets reflect the accumulation of past current account balances.
Key Concept
Intertemporal Consumption Choice
Explanation
The agent chooses consumption in each period to balance the marginal utility today with the discounted marginal utility of future consumption, taking into account the real interest rate.
3.2 Solution
a
Temporary Positive Shock: A temporary positive shock to Y2Y_2 increases the endowment in period 2 without affecting period 1
b
Current Account Impact: In period 1, the current account balance remains unchanged as income has not changed. In period 2, the current account will reflect the shock to Y2Y_2. If the agent chooses to consume more in period 2, the current account may decrease or even become negative
3.2 Answer
A temporary positive shock to output in period 2 will likely lead to an increase in consumption in that period, which could result in a decrease in the current account balance for period 2. ‖ ∻Key Concept∻
Temporary Income Shock
∻Explanation∻
The temporary nature of the income shock means that it does not affect the current account in the first period but can lead to changes in consumption and the current account in the second period.
∻3.3 Solution∻ ‖ a ⋮ Model Extension: Introduce investment ItI_t and capital stock KtK_t. The new budget constraint includes investment and capital income: C1+I1+C2+I21+r=Y1+Y2+rK11+rC_1 + I_1 + \frac{C_2 + I_2}{1+r} = Y_1 + \frac{Y_2 + rK_1}{1+r}. ‖ ‖ b ⋮ New Equations: The capital accumulation equation is K2=(1−δ)K1+I1K_2 = (1-\delta)K_1 + I_1, where δ\delta is the depreciation rate. The current account now includes investment: CAt=Yt+rKt−Ct−ItCA_t = Y_t + rK_t - C_t - I_t. ‖ ‖ c ⋮ Additional Insights: The model now shows how investment decisions affect the capital stock and future production possibilities, as well as the current account through savings and investment dynamics. ‖ ∻3.3 Answer∻
By including investment and capital stock in the model, we can analyze the intertemporal trade-offs between consumption and investment, and their effects on the economy's productive capacity and the current account.
∻Key Concept∻
Investment and Capital Accumulation
∻Explanation∻
The extension of the model to include investment allows for the analysis of how current savings and investment decisions impact future consumption possibilities and the current account through changes in the capital stock.
4.1 Solution
a
Set up the model: The Dornbusch model equations are given as:
$e_{t+1} - e_t = \frac{e_t}{\eta} - \frac{(1 - \phi \delta)}{\eta} q_t - \left(\frac{\phi \delta \bar{q} + m_t}{\eta}\right)$ and $q_{t+1} - q_t = -\delta \psi (q_t - \bar{q})$ where $\eta > 0, \phi > 0, \delta > 0, \psi > 0$.
b
Analyze the impact of an unanticipated permanent increase in mm: When mtm_t increases permanently, the nominal exchange rate ete_t will initially overshoot its long-term value
c
Solve for the new steady state: In the long run, et+1=ete_{t+1} = e_t and qt+1=qtq_{t+1} = q_t, which implies that the new steady state values eˉ\bar{e} and qˉ\bar{q} satisfy eˉ=mˉ+qˉ\bar{e} = \bar{m} + \bar{q}
4.1 Answer
The nominal exchange rate initially overshoots its long-term value due to an unanticipated permanent increase in the money supply.
Key Concept
Exchange Rate Overshooting
Explanation
The Dornbusch model predicts that due to sticky prices, the nominal exchange rate will temporarily exceed its new long-term equilibrium following an unanticipated increase in the money supply.
4.2 Solution
a
Main advantages of the Dornbusch model: The Dornbusch model incorporates price stickiness and expectations, which allows it to explain exchange rate dynamics more realistically
b
Features contributing to advantages: The model's assumption of sticky prices and the role of expectations lead to the overshooting phenomenon, which is consistent with observed exchange rate volatility
4.2 Answer
The Dornbusch model's main advantages are its ability to explain the volatile and overshooting behavior of exchange rates due to sticky prices and the role of expectations.
Key Concept
Price Stickiness and Expectations
Explanation
The Dornbusch model's inclusion of sticky prices and forward-looking expectations helps to explain why exchange rates can be more volatile than predicted by models assuming flexible prices.
5 Solution
a
Empirical performance assessment: New Keynesian models with Taylor rules are compared to conventional monetary models by evaluating their ability to match actual exchange rate movements
b
Key findings: New Keynesian models with Taylor rules often perform better in explaining exchange rate behavior, especially when considering the role of monetary policy and its effects on expectations
5 Answer
New Keynesian models with Taylor rules generally outperform conventional monetary models in empirical assessments of exchange rate behavior.
Key Concept
New Keynesian Models and Taylor Rules
Explanation
These models incorporate forward-looking behavior and the systematic response of monetary policy to economic conditions, which helps in capturing the dynamics of exchange rates more accurately.
6 Solution
a
Definition of the financial policy trilemma: The trilemma states that it is impossible to have all three of the following at the same time: a fixed foreign exchange rate, free capital movement, and an independent monetary policy
b
Challenges for China: As a fast-growing economy, China faces the difficulty of balancing these three objectives, especially when trying to control capital flows and maintain a stable exchange rate while pursuing an independent monetary policy
6 Answer
The financial policy trilemma poses significant challenges for China in managing its exchange rate, capital flows, and monetary policy simultaneously.
Key Concept
Financial Policy Trilemma
Explanation
China, like other fast-growing economies, must make trade-offs between exchange rate stability, capital flow liberalization, and monetary policy independence, which can lead to complex policy decisions.
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