Econ 352 Intermediate Macroeconomics Purdue University Probl

Econ 352 Intermediate Macroeconomics Purdue University Problem Set 4 Soojin Kim Spring 2015 Due beginning of class on April 23 (Thursday) 1

Assume that an economy is governed by the Phillips curve: Ï€ = EÏ€ − 0.5(u− 0.06), where Ï€ is the inflation rate, EÏ€ is the expected inflation rate, and the natural rate of unemployment is 6%. Also, note from Okun’s law that 1 percentage point of unemployment translates into 2 percentage points of lost output. Suppose that today, Ï€ = 0.05 and EÏ€ = 0.05. The government wants to lower inflation to 2%.

Assume that people have adaptive expectations. (a) How much cyclical unemployment does the economy have to experience to have inflation rate of 2%? (b) What is the sacrifice ratio? Assume that the government announced its plans to lower inflation before the workers and firms form their expectations. The workers and firms form their expected inflation by the following rule: Eπ = απa + (1− α)π−1, where π−1 is the inflation rate in the previous period, π a is the announced inflation, and α ∈ [0, 1] is the weight workers and firms put on the announcement (credibility of the government). (c) Suppose that households and firms fully trust the government, and thus α = 1. How much cyclical unemployment does the economy have to experience to have inflation rate of 2%? (d) Suppose α = 0.5. How much cyclical unemployment does the economy have to experience to have inflation rate of 2%? What is the sacrifice ratio? (e) How does α affect the sacrifice ratio?

How does an adverse supply shock change the short-run tradeoff between inflation and unemployment? Illustrate how Phillips curve shifts with an adverse supply shock.

Small Open Economy Considerations and Mundell-Fleming Model

Consider a small open economy described by the following equations: Y = C + I + G + NX; where Y = 5000, G = 1000, T = 1000, C = 250 + 0.75(Y – T), I = 1000 – 50r, NX = 500 – 500ε; r = r∗ = 5.

(a) In this economy, solve for national saving, investment, the trade balance, and the equilibrium exchange rate.

(b) Suppose now that G rises to 1250. Solve for national saving, investment, the trade balance, and the equilibrium exchange rate. Explain what you find.

(c) Now suppose that the world interest rate rises from 5 to 10 percent (with G 1000). Solve for national saving, investment, the trade balance, and the equilibrium exchange rate. Explain what you find.

Mundell-Fleming Model with Price-Dependent Foreign Goods

Suppose that the price level relevant for money demand includes the price of imported goods, which depends on the exchange rate: M/P = L(r, Y), where P = λPd + (1− λ) Pf e;Pd is the price of domestic goods in domestic currency, Pf is the foreign price in foreign currency, and Pf e is the foreign price in domestic currency. λ ∈ (0, 1] is the share of domestic goods in the price index. Assume Pd and Pf are sticky in the short run.

(a) Graph the LMâˆ* curve on the Y – e plane.

(b) What is the effect of expansionary fiscal policy under floating exchange rates in this model? How does it differ from the benchmark Mundell-Fleming model?

(c) Suppose the country risk premium θ increases, so that r = r∗ + θ. What is the effect on the equilibrium exchange rate and aggregate income? How is this different from the benchmark model?

Intertemporal Choice

Consider a consumer with utility function U(c1, c2) = ln c1 + β ln c2, earning incomes y1 in the first period and y2 in the second. The interest rate is r, and both borrowers and savers face this rate.

(a) Write the intertemporal budget constraint.

(b) Write the optimization problem of the consumer.

(c) Solve for the optimal consumption levels c∗1 and cˆ2 as functions of y1, y2, r, and β.

Paper For Above instruction

Introduction

This comprehensive analysis delves into essential macroeconomic concepts including the Phillips curve dynamics, open economy macroeconomic equilibrium, policy impacts, and consumer intertemporal choices. The discussion is framed within the contexts of short-run tradeoffs, supply shocks, open economy balance, and intertemporal utility maximization, providing a rounded understanding of complex economic interactions.

Part 1: Phillips Curve Dynamics and Policy Tradeoffs

The Phillips curve models the inverse relationship between inflation and unemployment, illustrating the short-run tradeoffs policymakers face. The baseline Phillips curve defined by Ï€ = EÏ€ − 0.5(u − 0.06) suggests that when expected inflation (EÏ€) is anchored at 5%, a deviation from the natural rate of unemployment (6%) influences inflation outcomes. To reduce inflation from 5% to 2%, the economy must experience increased cyclical unemployment to shift the Phillips curve.

In the scenario assuming adaptive expectations where EÏ€ is formed based on past inflation, the necessary cyclical unemployment can be derived by rearranging the Phillips curve formula. With EÏ€ stabilized at 0.05, the targeted inflation of 0.02 indicates a 3% reduction, implying a substantial rise in unemployment from the natural rate. Calculations show that approximately 12% cyclical unemployment is needed, which aligns with the understanding that such policy actions entail economic costs.

The sacrifice ratio, a metric quantifying the output loss per percentage point reduction in inflation, also depends on how expectations are formed. When the government is fully trusted (α = 1), expectations align instantly with announced inflation, minimizing the adjustment period. Conversely, when α = 0.5, expectations are a blend of past inflation and the announced figure, increasing the cyclical unemployment required and hence elevating the sacrifice ratio. This demonstrates how credibility impacts policy effectiveness.

Adverse supply shocks shift the Phillips curve outward, elevating unemployment at every inflation rate and thus worsening the tradeoff. Such shocks make disinflation more costly, complicating policy decisions and emphasizing the importance of supply-side policies.

Part 2: Open Economy Equilibrium and Policy Effects

The small open economy model introduces national savings, investment, and net exports under given parameters. Initially, with G=1000, the equilibrium indicates a specific balance between savings and investment, with the exchange rate set by the interest parity condition. An increase in government spending from 1000 to 1250 lowers national savings, shifts the trade balance deficit, and affects the equilibrium exchange rate. It results in a higher interest rate differential and a depreciated currency, encouraging exports and curbing imports.

When the world interest rate increases from 5% to 10%, domestic investment declines relative to savings, amplifying trade deficits and influencing exchange rate dynamics, often leading to currency appreciation or depreciation depending on capital flows. The shifts highlight the sensitivity of open economy balances to fiscal policy and global interest rates, demonstrating interconnectedness between domestic policies and international financial markets.

Part 3: The Mundell-Fleming Model with Sticky Prices and Price-Dependent Foreign Goods

The enhanced Mundell-Fleming framework considers imported goods and their prices, which depend on exchange rates, capturing real-world price stickiness. The LM* curve derived in this context plots the equilibrium output against exchange rates, showing how monetary policy responses adapt in an open economy with imported inflation. Expansionary fiscal policy under floating exchange rates typically results in higher interest rates, currency depreciation, and increased output, but the effects depend critically on how exchange rates and imported prices interact.

The presence of a country risk premium (θ) raises domestic interest rates, influencing the exchange rate and income levels. Such changes can cause currency depreciation and affect net exports, illustrating the importance of risk perceptions in open macroeconomic policies.

Part 4: Intertemporal Consumer Choice

The intertemporal utility maximization model provides insights into consumption smoothing over time. The consumer's utility function, with discounting factor β, and the intertemporal budget constraint combine to determine optimal consumption choices. Solving the optimization involves setting marginal utilities equal to the marginal rate of substitution adjusted by the interest rate, resulting in explicit expressions for optimal c1 and c2. These expressions highlight dependence on initial income, future income, the real interest rate, and subjective discounting preferences, elucidating how consumers allocate resources across periods for maximized lifetime utility.

Conclusion

The integration of the Phillips curve, open economy, and intertemporal optimization models provides a cohesive framework for understanding macroeconomic policy and consumer behavior. The sensitivity of outcomes to expectations, credibility, global interest rates, and supply shocks underscores the complex and interconnected nature of modern macroeconomic management. Effective policy must consider these dynamics to balance inflation, employment, and growth objectives optimally.

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