International Macro Finance

Transcription

1 International Macro Finance Economies as Dynamic Systems Francesco Franco Nova SBE February 21, 2013 Francesco Franco International Macro Finance 1/39

2 Flashback Mundell-Fleming MF on the whiteboard Francesco Franco International Macro Finance 2/39

3 Ingredients Buidling Blocks We have so far derived and studied three fundamental equations that summarize part the system (variables are in natural log) 1 Money Supply (set by monetary authorities): m t 2 UIP (Short run arbitrage by financial markets): i t+1 = i ú t+1 + se t+1 s t 3 PPP: p t = p ú t + s t 1 Absolute: arbitrage by international goods markets 2 real exchange rate: t = p ú t + s t p t deviations from PPP Francesco Franco International Macro Finance 3/39

4 Closure To close the model we need to add a money demand (explain it: transaction/liquidity) m d t = p t + y t i t+1 You can also close the model with an interest rate rule (Taylor Rule, instead of controling money supply) where fi t = p t+1 p t i t+1 = TR(fi t, y t ) Francesco Franco International Macro Finance 4/39

5 Closure Consider the variables of the model with PPP holding and fixed output (high frequency) home endogenous home exogenous foreign exogenous s m i ú i y p ú p (a) Money demand home endogenous home exogenous foreign exogenous s i i ú m y p ú p (b) Taylor Rule Table: Understanding what you determine in your model Francesco Franco International Macro Finance 5/39

6 From Static to Dynamic Start with a static model, assuming CB controls money supply, s is flex, PPP holds and output is exogenous ȳ: i = i ú s = m p ú ȳ + i ú p = p ú + s If s is fixed m becomes endogenous: loss of monetary authority Francesco Franco International Macro Finance 6/39

7 From Static to Dynamic The static model gives the steady state of the economic system. A state in which the variables of the model do not change. It can be interpreted as the state to which the economy tends (Long Run) We are interested in dynamics: how we are approaching the Long Run (transitional dynamics) Francesco Franco International Macro Finance 7/39

8 Systems of linear di erence equations First-order The variable z t follows a first-order di erence equation z t = az t 1 + m t (1) where m t is an exogenous function of time. A solution of this equation is to express z t as a function of the current, future, and lagged values of m t as well as some intial value of z 0.Try backward substitution Francesco Franco International Macro Finance 8/39

9 Systems of linear di erence equations The stable case: a < 1 Define the lag operator L by Lx t = x t 1 hence the lag operator maps a variable to its value in the previous period. We can write equation (1) as (1 al) z t = m t Francesco Franco International Macro Finance 9/39

10 Systems of linear di erence equations The stable case: a < 1 Define the inverse of the (1 al) as (1 al) 1 = 1 + al + a 2 L an operator that converges because a < 1. Obviously (1 al) 1 (1 al) =1 Francesco Franco International Macro Finance 10/39

11 Systems of linear di erence equations The stable case: a < 1 Now write z t =(1 al) 1 (1 al) z t =(1 al) 1 m t with (1 al) 1 m t = m t + am t 1 + a 2 m t A solution is therefore tÿ z t = a t s m s s= Œ Francesco Franco International Macro Finance 11/39

12 Systems of linear di erence equations The stable case: a < 1 It is important to notice that it is not the only solution. If b 0 is a constant then adding the term b 0 a t yields another solution. Therefore we refer to z t = tÿ s= Œ a t s m s + b 0 a t as the general solution Francesco Franco International Macro Finance 12/39

13 Systems of linear di erence equations The stable case: a < 1 The arbitrary constant b 0 is determined by an initial condition on the variable z t. Suppose we know z 0 then the general solution is correct only if 0ÿ z 0 = a s m s + b 0 holds, that is, if s= Œ b 0 = z 0 0ÿ s= Œ a s m s Francesco Franco International Macro Finance 13/39

14 Systems of linear di erence equations The stable case: a < 1 Substitute in the general solution corresponding to the initial value z t = tÿ a t s m s + z 0 a t s=1 gives the particular equation Francesco Franco International Macro Finance 14/39

15 Systems of linear di erence equations The unstable case: a > 1 Unstable roots usually govern the behavior of forward-looking economic variables, such as equity prices. Define the lead operator Fx t = x t+1 forward by one period our equation z t+1 = az t + m t+1 and isolate z t z t = 1 a z t+1 1 a m t+1 Francesco Franco International Macro Finance 15/39

16 Systems of linear di erence equations The unstable case: a > 1 Now apply our forward operator: 31 1 a F 4 z t = 1 a Fm t and given - 1 a - < 1 we can define the inverse operator 11 1 a F 2 1 Francesco Franco International Macro Finance 16/39

17 Systems of linear di erence equations The unstable case: a > 1 As in the stable case a solution is therefore z t = a F a F z t = a F 1 a Fm t z t = Œÿ s=t a 4 s t m s Francesco Franco International Macro Finance 17/39

18 Systems of linear di erence equations The unstable case: a > 1 As before we find a general solution by adding the term b 0 a t z t = Œÿ s=t a 4 s t m s + b 0 a t now you can determine b 0 using an initial condition on z 0,however any choice for b 0 other than b 0 = 0 would lead to z t exploding. Under the assumption that agents are not willing to participate in an unstable economy (see Shiller, 1978), the markets will determine z 0 such that b 0 = 0. Francesco Franco International Macro Finance 18/39

19 Systems of linear di erence equations First-order vector systems In our models we have systems of variables. If we interpret z as a vector and the parameter a as a conformable matrix we can apply the same method. The only question is to determine which variables are driven by unstable dynamics and which variables by stable dynamics. Consider the system C z1t z 2t D = A C z1t 1 z 2t 1 D + C m1t m 2t C D a11 a where A = 12 is non singular. Now we have to a 21 a 22 transform the system into an unstable and stable part. D Francesco Franco International Macro Finance 19/39

20 Systems of linear di erence equations First-order vector systems Any square matrix can be decomposed into AE = E C D e1 e where E = 2 is a matrix containing the eigenvectors and 1 1 C D 1 0 = is a matrix containing the eigenvalues of A. 0 2 Thus A = E E 1 Francesco Franco International Macro Finance 20/39

21 Systems of linear di erence equations First-order vector systems Now use the decomposition C z1t z 2t D = E E 1 C z1t 1 z 2t 1 D + C m1t m 2t D and premultiply by E 1 E 1 C z1t z 2t D = E 1 C z1t 1 z 2t 1 D + E 1 C m1t m 2t D Francesco Franco International Macro Finance 21/39

22 Systems of linear di erence equations First-order vector systems Define the transformed vectors z Õ t = E 1 z t and m Õ t = E 1 m t. The last matrix equation becomes C z Õ 1t z Õ 2t D = C z Õ 1t 1 z Õ 2t 1 D + C m Õ 1t m Õ 2t where because is diagonal we have expressed the system in terms of two variables with non interacting dynamics. Each of these can be solved separately using methods described above and the solution of the original variables can be recovered by applying the reverse transformation z t = Ez Õ t D Francesco Franco International Macro Finance 22/39

23 AsimpleMonetaryModelofExchangeRates Consider a small, open economy in which real output is exogenous the demand for money is given by mt d = p t + y t i t+1 PPP holds p t = pt ú + s t UIP holds i t+1 = it+1 ú + E t s t+1 s t Francesco Franco International Macro Finance 23/39

24 AsimpleMonetaryModelofExchangeRates Substitute PPP and the uncovered interest parity into the money demand (with m d = m) and get! mt y t + i ú t+1 p ú t " st = ((E t s t+1 s t ) which we now know how to solve s t = Œÿ s=t 3 4 s t ) E t ms y s + is+1 ú p ú * s 1 + Francesco Franco International Macro Finance 24/39

25 AsimpleMonetaryModelofExchangeRates In this monetary model, raising the path of the home money supply raises domestic price level and forces s up through the PPP. This is a depreciation of the home currency against foreign currency. nominal exchange rate must be viewed as an asset price the data are not very kind to this monetary model (outside hyperinflationary environment) Francesco Franco International Macro Finance 25/39

26 Areviewofthedata The most di cult task in international finance is to build a bridge between the real economy and its monetary side can actual economies be usefully characterized by flexible-price models? any casual look at international data shows that price rigidities are relevant the exchange rates are an order of magnitude more volatile than CPIs Francesco Franco International Macro Finance 26/39

27 Areviewofthedata Figure: Germany and the United States, exchange rate and prices Francesco Franco International Macro Finance 27/39

28 Areviewofthedata Figure: Germany and the United States, exchange rate and prices changes Francesco Franco International Macro Finance 28/39

29 The Mundell-Fleming-Dornbusch Model Building blocks UIP: i t+1 = i ú t+1 + s e t+1 s t (2) Money demand: PPP need not to hold m d t = p t + y t i t+1 (3) t = s t + p ú t p t (4) Francesco Franco International Macro Finance 29/39

30 The Mundell-Fleming-Dornbusch Model Building blocks Output is determined by demand (short run) which is an increasing funstion of the home real exchange rate y d t =ȳ + (e t + p ú p t ) (5) where ȳ is potential output and the equilibrium level of the real echange rate Francesco Franco International Macro Finance 30/39

31 The Mundell-Fleming-Dornbusch Model Building blocks Although asset markets clear at every moment output markets need not in the Dornbusch model. The empirical reality of sticky prices is captured by assuming that p is predetermined and responds only slowly to shocks. When price cannot move unanticipated shocks plainly can lead to excess supply or excess demand. Although p is predetermined it does adjust slowly p t+1 p t = Â 1 y d t 2 ȳ +( p t+1 p t ) where p t s t + p ú t. Assume and p ú are constant you get p t+1 p t = Â 1 y d t 2 ȳ +(s t+1 s t ) (6) Francesco Franco International Macro Finance 31/39

32 The Mundell-Fleming-Dornbusch Model System simplify the 5 equations system into 2 equations: use (4) and (5) to express (6) as next substitute (2)(5)(4) into (3) t+1 t = Â ( t ) s t+1 s t = s t (1 ) t ( + m t) where I have normalized i ú = p ú =ȳ = 0 Francesco Franco International Macro Finance 32/39

33 The Mundell-Fleming-Dornbusch Model Steady-State In steady state (Long Run) the variables are at rest therefore t+1 t = 0ands t+1 s t = 0: = and s =(1 ) +( + m) Francesco Franco International Macro Finance 33/39

34 The Mundell-Fleming-Dornbusch Model Steady-State Figure: The Dornbusch model if < 1 Francesco Franco International Macro Finance 34/39

35 The Mundell-Fleming-Dornbusch Model The saddle path The SS arrows show the path of the exchange rate which correspond to the solution where the explosive component has been canceld by choosing the right initial condition. In this case the market chooses s 0 to be on the stable solution. You can derive the solution using the methods illustrated above and get s t = m  ( t ) whcih is the equation describing SS (the saddle path) Francesco Franco International Macro Finance 35/39

36 The Mundell-Fleming-Dornbusch Model Graphical solution Consider an unanticipated increase in m to m Õ Start with the Long Run implications. From the steady state we see that s increases proportionally to the increase in m and that the real exchange rate remains at which implies an increase in p proportional to the increase in m (money neutrality). Basically in the Long Run we have a depreciation In the short run p cannot change. Using our normalization the initial price level p = m so when m increases to m Õ the price level remains at p. You need to jump on the new saddle path SS, both real exchange rate and nominal exchange rate depreciate. Notice that the short run nominal depreciation is larger that the long run: this is the overshooting result Francesco Franco International Macro Finance 36/39

37 The Mundell-Fleming-Dornbusch Model Unexpected permanent increase in m Figure: Exchange rate overshooting Francesco Franco International Macro Finance 37/39

38 The Mundell-Fleming-Dornbusch Model Overshooting Overshooting: the initial change in the nominal echange rate must be greater than the long run change. The result is driven by the rigidity in prices. It can explain part of the volatility observed in the exchange rate markets. Francesco Franco International Macro Finance 38/39

39 Readings M. Obstfeld and K. Rogo, Foundations of International Macroeconomics (MIT Press, 1996). Supplement C, Solving Systems of Linear Di erence Equations. Dornbusch, Rudiger, "Expectations and Exchange Rate Dynamics," Journal of Political Economy 84 (December 1976): * Rogo, Kenneth, "Dornbusch s Overshooting Model after 25 Years," IMF Sta Papers 49 (Special Issue 2002): 1-34 Francesco Franco International Macro Finance 39/39