The Optimal Choice of Monetary Instruments The Poole Model

Transcription

1 The Optimal Choice of Monetary Instruments The Poole Model Vivaldo M. Mendes ISCTE Lison University Institute 06 Novemer 2013 (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

2 Summary 1 Tools, targets and goals 2 A simple model with demand shocks (the Poole model) 3 Extending the model: monetary ase as potential instrument 4 Poole s model with supply shocks (not covered, lack of time) (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

3 Tools, targets and goals Tools, targets and goals (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

4 Tools, targets and goals The temporal steps of monetary policy 1 Policy tools 2 Operating targets (instruments) 3 Intermediate targets 4 Final goals (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

5 Tools, targets and goals The temporal steps of monetary policy 1 The CB uses (controls) operating targets in order to a ect the intermediate targets. 2 Which operating target is etter:money supply? Short term interest rates? 3 Answer: it depends upon the kind of shocks that a ect the economy: 1 Aggregate demand shocks 2 Money demand shocks 3 Supply shocks 4 Major result: relative variances of macroeconomic shocks matter for optimal choice of instrument William Poole (1970). "Optimal choice of monetary policy instruments in a simple stochastic macro model," Sta Studies 57, Federal Reserve Board, Washington. (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

6 The Taylor rule Tools, targets and goals A famous example of an operating target rule (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

7 The ECB example Tools, targets and goals (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

8 Tools, targets and goals (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

9 What is the New Keynesian Model? ECB key interest rates (Vivaldo M. Mendes) Novemer / 3

10 Only demand shocks A simple model with demand shocks (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

11 Only demand shocks The asic equations of the model 1 Consider an economy with an IS LM model with Rational Expectations 2 The IS function tells us that y t = (i t E t π t+1 ) + z t (1) Positive shocks on the IS (z t ) oost output at given interest rates. 3 The LM curve tells us that real money demand (m t p t ) is dependent upon output and the interest rate m t p t {z } log values = y t ai t + v t (2) 4 Positive shocks on the LM (v t ) oost money demand at given interest rates and output. (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

12 Only demand shocks Basic assumptions 1 In the old IS LM framework, prices and expected in ation are kept as constant. So P t = 1, then ln P t p t = 0 E t π t+1 = 0 2 The policy maker cannot oserve the shocks. It can set either the nominal interest rate (i) or the money supply (m). 3 The prolem of the policy maker is to choose etween these two instruments to minimize the variance of output min E[y t ] 2, or min E[y t y ] 2 4 What is est as instrument? Money supply (m) or (i)? (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

13 Solving the model Only demand shocks 1 With E t π t+1 = 0, and p t = 0, The IS and LM eq can e written as y t = i t + z t m t = y t ai t + v t 2 If there were no shocks, the solutions would e (no suscripts for simplicity) y = i = a + m and i 1 = a + m 3 The ojective of the CB is to minimize min E[y t y ] 2 (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

14 Only demand shocks The CB controls (pegs) the interest rate 1 The ojective of the CB is min E[y t y ] 2 2 Therefore, the CB sets i = i, and output is given y y = i + z = y + z 3 Then, it is immediate to see that E[y y ] 2 i = Var(z) {z } σ 2 z (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

15 Only demand shocks The CB controls (pegs) the money stock (m) 1 Now the CB instead pegs the money stock (m = m ). 2 From the IS we know that y = 3 And from the LM we know that i + z 4 Then we have i = 1 (y m + v) a y = a (y m + v) + z... = y a + v + a a + z 5 Then, it is immediate to see that 2 E[y y ] 2 m = σ 2 v + a + a a + 2 σ 2 z (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

16 Only demand shocks Comparing the instruments The comparison of oth instruments is easy E[y y ] 2 i = E[y y ] 2 m 2 σ 2 z = σ 2 v + a + (a + ) 2 σ 2 z = 2 σ 2 v + a 2 σ 2 z... (2a + 2 )σ 2 z = 2 σ 2 v σ 2 z = 2a + {z } φ<1 σ 2 v a 2 σ 2 z a + (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

17 Only demand shocks Comparing the instruments: summary E[y y ] 2 i = E[y y ] 2 m σ 2 z = 2a + {z } φ<1 1 Hence, choose an interest rate targeting procedure whenever there is 1 relatively high money demand volatility 2 relatively low aggregate demand volatility 2 In general targeting the interest rate is preferale if IS shocks are small, money demand is not sensitive to the interest rates (a is small), output is sensitive to the interest rate in IS ( is large): σ 2 v (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

18 Monetary ase as potential instrument Extending the model: monetary ase as potential instrument (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

19 Monetary ase as potential instrument Monetary ase as an instrument 1 Central anks control the money ase, ut not M (for example, M3) as in the model 2 Extend the model with endogenous money supply (M): h t is the the monetary ase. m t = h t + ei t + ω t (3) 3 Notice that the money multiplier is given y (expressed in logs) m t h t = ei t + ω t ω t is mean-zero money multiplier shock. 4 Money multiplier is increasing in interest rate (anks want to lend more/consumers want to hold less cash) (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

20 Monetary ase as potential instrument Solving the model 1 The three fundamental equations are now 2 The logic now is y t = i t + z t (IS) m t = y t ai t + v t (LM) m t = h t + ei t + ω t (M Multiplier) y depends on i i depends on m m depends on h 3 The ojective is to express y, i, m as determined only y h (which is controlled y the CB) (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

21 Monetary ase as potential instrument Solving the model (cont.) 1 If there were no shocks, the solutions would e (no suscripts for simplicity, see Appendix 1 for details) y = i = i = 1 a + + e h m = h + ei = a + a + + e h 2 The ojective of the CB is to minimize min E[y t y ] 2 a + + e h (4) (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

22 Monetary ase as potential instrument The CB controls (pegs) the interest rate 1 The same results as aove 2 The solution is totally given y the IS function. 3 The CB sets i = i, and output is given y y = i + z = y + z 4 Then, it is immediate to see that E[y y ] 2 i = Var(z) {z } σ 2 z (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

23 Monetary ase as potential instrument The CB controls (pegs) the monetary ase (h) 1 Now the CB pegs the monetary ase (h = h ). 2 Then, the Money Multiplier eq. comes as m = h + ei + ω 3 But from the LM eq. we know that m = y ai + v 4 Therefore, y equating oth previous eq. we get h + ei + ω = y ai + v 5 From where we can otain i = 1 a + e (y h + v ω) (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

24 Monetary ase as potential instrument The CB controls (pegs) the monetary ase (h) 1 Now, from the equilirium in the money market we know that i = 1 a + e (y h + v ω) 2 Then, we should use the IS function to solve the entire prolem y = i + z = a + e (y h + v ω) + z (5) 3 Solving for y, leads to (see Appendix 2 for details) y = y a + + e v + a + + e ω + 4 Then, it is immediate to see that E[y y ] 2 h = 2 σ 2 v + σ 2 ω + a + + e a + e a + + e z (6) a + e a + + e 2 σ 2 z (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

25 Monetary ase as potential instrument Comparing the instruments 1 Now it is easy to see which instrument is etter E[y y ] 2 i = E[y y ] 2 h σ 2 z = a + + e (a + + e) 2 σ 2 z = () 2 σ 2 v + σ 2 ω σ 2 z =... 2 (a + e) + {z } φ<1 2 σ 2 v + σ 2 ω + + (a + e) 2 σ 2 z σ 2 v + σ 2 ω a + e a + + e 2 σ 2 z 2 Hence, choose an interest rate targeting procedure whenever there is 1 relatively high money demand volatility 2 relatively high money multiplier volatility 3 relatively low aggregate demand volatility (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

26 Introducing supply shocks Extending the model: introducing supply shocks No time availale for covering this point. Read pages in the iliography if you are curious aout this point. But reading is not compulsory (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

27 Appendix 1 Introducing supply shocks 1 The 3 eq. with no shocks are 2 Using the LM and the MM eq., we get i = 1 (y h) a + e 3 Now insert this result into the IS eq. y = i (IS) m = y t ai (LM) m = h + ei (MM) y = i = a + + e h 4 Then, it is easy to write ack m and i also as functions of h i = 1 a + + e h, m = h + ei = a + a + + e h. (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27

28 Appendix 2 Introducing supply shocks 1 How to get from eq. (5) to eq. (6)? From eq. (5), we have y = 2 Solving for y, leads to y(1 + a + e ) = a + e h y = a + e (y h + v ω) + z a + + e h 3 Now, using the result in eq. (4) 4 We arrive at eq. (6) y = y y = a + e v + a + e ω + z a + + e v + a + + e h a + + e ω + a + + e v + a + + e ω + a + e a + + e z a + e a + + e z (Vivaldo M. Mendes) The Poole Model 06 Novemer / 27