Problem Set #4 Revised: April 13, 2007

Transcription

1 Global Economy Chris Edmond Problem Set #4 Revised: April 13, 2007 Before attempting this problem set, you might like to read over the lecture notes on Business Cycle Indicators, on Money and Inflation, on the Cagan model, and on Monetary Policy. A. Term structure and economic conditions How well is the economy doing? One source of information is the forward rate curve. Suppose current forward rates f m and averages over the last 25 years f m were given by: Maturity (m) Forward Rate (f m 1 ) Mean Forward Rate (f m 1 ) Maturity is measured in years and forward rates are annually compounded percentages. 1. Compute zero-coupon yields y m and bond prices p m for maturities between 1 and 5 years (5 points). 2. Use the forward rate curve to construct forecasts of future 1-year yields. How do you see the 1-year rate evolving over the next four years? What does this suggest for output growth? (15 points). Answer. This is pretty much straight from the notes. 1. The bond prices, zero coupon yields, etc, are given in the following table. Maturity Bond Price Yield Risk Premium Future Short Rate The flat to decreasing path of expected future short rates suggests modest to below-average growth of output.

2 Problem Set #4 2 B. Quantity theory of money You have been told the following data for a small Latin American economy. The money supply this year, 2007, is M = 100 billion pesos while nominal consumption is P C = 600 billion pesos. Your best estimate of the growth rate of real consumption between this year and next is γ C = 0.03 or 3%. Now: 1. Use this data to provide an estimate of the velocity of money for this year (5 points). 2. To finance the government s fiscal deficit, the central bank has been ordered to print an extra 10 billion pesos to be in circulation next year (so the money supply will increase to 110 billion pesos). Use the quantity theory of money to make a forecast of the inflation rate between this year and the next (10 points). 3. What would happen to your forecast of inflation if velocity was not in fact constant between this year and next? To be specific, suppose velocity increases by γ V = 0.02 or 2%. What would the true inflation rate be? (5 points). Answer. 1. Velocity is defined by the exchange equation, namely: MV = P C or So velocity is 6 per year. V = P C M = 600 billion pesos 100 billion pesos = 6 2. Using the exchange equation, the annual inflation rate is equivalent to: π γ P = γ M + γ V γ C But according to the quantity theory of money, velocity should be approximately constant (γ V = 0) so inflation is just the excess of money growth over real consumption growth: π = γ M γ C Now the annual growth rate of the money supply is ( ) 110 γ M = ln

3 Problem Set #4 3 Our best estimate of real consumption growth is γ C = 0.03, so our forecast of the annual inflation rate is: π = 0.07 or about 7%. 3. If velocity was not constant but instead grows by γ V = 0.02, then our estimate of inflation will be biased down. To be precise, actual inflation would be π = γ M + γ V γ C = 0.09 or about 9% instead of 7%. C. Cagan model Consider a version of the Cagan model where price dynamics are given by p t = 1 3 p t m t 2 3 m t 1 where p and m denote the log price level and the log money supply respectively. Government policy sets the path of the log money supply. 1. Suppose that in the long run government policy is in steady state, m t = m t 1 = m. What does the steady state price level p equal? Does the quantity theory of money hold in this long run? Why or why not? (10 points) 2. Now suppose that the government monetizes a deficit so that the log money supply increases. To be specific, suppose that the economy is in steady state at time t = 0 with initial money supply m 0 = m = 0 and with initial price level p 0 = p. At time t = 1 the money supply is increased by 100% so that at t = 1 we have m 1 = 1.0. Thereafter, the money supply stays fixed at m t = 1.0 for t = 2, 3,... Use the equation for price dynamics to calculate p t for t = 1,..., 10. Do prices eventually settle down to a new steady state level? If so, what is that level? Also use the exchange equation to calculate velocity. How does velocity respond to the monetization? [Hint: in the Cagan model, log consumption is c = 0.] (20 points) Answer.

4 Problem Set # Since the money supply is in steady state, m t = m t 1 = m, the steady state price level for which p t = p t 1 = p has to solve p = 1 3 p m 2 3 m so p = m Since log prices are proportional to log money, the quantity theory of money holds in this steady state. An increase in the long run supply of money will result in a proportional increase in the price level (or, equivalently, if the money supply grows by x % then inflation will be x % in the long run too). 2. Here are the calculations: t m t p t π t = p t p t 1 v t = p t m t As the log money supply settles down to the new steady state of m = 1, so too does the log price level settle down to a new steady state of p = 1. In the short run, inflation is even higher than 100% before there is a period of deflation (negative inflation) as the price level stabilizes. Log velocity increases above its long run level of zero as the increased money supply is passed around at a higher rate. So in the short run, money growth and inflation are not equal. But as the price level stabilizes, velocity returns to its steady state value of v = p m = 0 (which is the same as the original steady state). Once again, the quantity theory of money holds in the long run. D. Taylor rule You work for a central bank that sets nominal interest rates according to the Taylor rule i t = π t (π t π) (y t y t )

5 Problem Set #4 5 where π t is the inflation rate, π is the central bank s inflation target, y t is log real GDP and y t is the central bank s best estimate of trend real GDP. Suppose that the inflation target is 2% and the research department forecasts that the next four annual inflation rates and log levels of real GDP will be π t y t Moreover, the research department estimates that trend output is currently equal to y 0 = 0 and real GDP is expected to grow at a constant annual 2% for the foreseeable future. Now: 1. Calculate the expected gaps between output and trend output. Use these expected gaps to forecast the central bank s setting of the nominal interest rate over the next four years. Does the nominal interest rate rise or fall over the next few years? Why or why not? (10 points) 2. Debate inside the bank begins to rage between believers in a new economy who argue that trend real GDP growth has accelerated to 4% a year and inflation hawks who argue (i) not only that trend real GDP growth has not accelerated (and so remains at 2% annual) but also (ii) that the bank should target a lower level of inflation, 1.5% annual. Calculate the nominal interest rate settings that these two opposing sides would want to implement. Explain the differences (if any) between the nominal interest rate settings these two groups would choose (20 points). Answer. 1. Since trend real GDP growth is 2% annual we have y t = 0.02t + y 0 = 0.02t for t = 1, 2, 3, 4. Using this, the output gaps and nominal interest settings are: y t i t where, for example, i 1 = π (π 1 π) (y 1 y 1 ) or i 1 = ( ) + 1 ( ) = The nominal interest rate rises over the next two years as both output and inflation deviate above trend/target. Then as inflation begins to fall again, so does the nominal interest rate.

6 Problem Set # For believers in the new economy we have: y t i t and for inflation hawks the only difference as compared to the benchmark case in question 1. above is that the inflation target π = 0.015, not This means the inflation hawks always want nominal interest rates higher by ( )/2 = , or 25 basis points, over the benchmark nominal interest rate: y t i t Clearly, the new economy types feel that the true output gap is smaller since they believe trend output growth is faster (at 4% annual instead of 2% annual), so they want to implement nominal interest rates significantly lower than either the benchmark or the inflation hawks. In all cases, however, the nominal interest rate first increases over the next two years and then declines. So the qualitative pattern of nominal interest rates is always the same. But the factions differ in terms of how much they want to increase interest rates at first and then how much they want to reduce them further out. c 2007 NYU Stern School of Business