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Econ 797 Lecture Arslan Razmi Fall 2016 This lecture is mostly based on Gandolfo(2004, chapters 10 and 11), Groth (2014, ch. 23), Blanchard and Fischer (1989, ch. 10), and Sarno and Taylor (2002)
Econ 797 Lecture 3 Outline The basic set-up Fixed exchange rates Flexible exchange rates Transitional dynamics and overshooting in a sticky price model
Econ 797 Lecture 4 1 Background and Overview [2], [1], [3], [4] Again, short-run (Keynesian) but now we introduce financial flows in a serious way. Since financial flows are influenced by expected returns on assets, we need to introduce asset markets. We will do this by incorporating the money market and the Uncovered Interest Parity (UIP) condition. Most of the previous assumptions remain. rigid prices (or exogenous constant inflation path) one good produced in the country, unemployed resources, small country imperfect substitution between domestic and foreign goods no international mobility of labor
Econ 797 Lecture 5 But now we add the free mobility of financial capital across international borders. The inclusion of financial flows coincides with the gradual liberalization of financial accounts in the industrialized, non-soviet bloc countries in the 1960s.
Econ 797 Lecture 6 2 The Static System Element 1: The goods market As before, let s start with the goods market equilibrium condition: y = c(y, α) + I(y, r e ) + g + tb(y, e, β) + ε D (1) As before, α covers shocks specific to consumption (including increases in lump-sum taxes, which is why I can write the consumption function with income, rather than disposable income, as an argument). To keep thing as simple as possible, I have specified a generalized demand shock ε D. Also, instead of defining exports and imports separately, I have specified a trade balance function.
Econ 797 Lecture 7 In addition, there is one new variable. Investment (I to differentiate it from the nominal interest rate, i) is now more realistically defined as a function of national income (as a proxy for aggregate demand), and the expected cost of investment. Conceptually one could think of this as capturing the assumption that the alternative to investing in real capital is holding long-term bonds. The expected return on long-term bonds is denoted by r e. As before, 0 < c y < 1 and 0 < i y < 1, while 0 < tb y < 1, tb e > 0 (i.e., the Marshall- Lerner condition is assumed to be satisfied), tb β = 1, while i r e < 0. The last partial differential reflects the expectation that a higher expected opportunity cost will discourage investment. Finally, recall that the expected real interest rate, r e, is the nominal interest rate i minus the expected rate of inflation π e (why?). But with a given exogenous rate of inflation, π, π = π e, and the expected real interest rate is simply the real interest rate, r.
Econ 797 Lecture 8 Before we move to the second element, let s repeat our practice from a previous lecture by consolidating the good market equilibrium condition and writing it in excess demand form: Goods market equilibrium (IS) A(y, i, α) + tb(y, e, β) + ε D y = 0 (2)
Econ 797 Lecture 9 Element 2: The money market Next, let s introduce the money market in its traditional form that incorporates the transactions and speculative (or portfolio) demand for money as an asset and medium of exchange. Holders of assets face a choice between money and (domestic and foreign) bonds The advantage of holding money is its low transaction cost (high degree of liquidity) The opportunity cost of holding money, of course, is the nominal return on short-term bonds. Why the nominal return? Because inflation erodes the value of money and other financial assets to the same extent (as a first approximation).
Econ 797 Lecture 10 Money demand is a function of income (transactions demand) and the nominal interest rate. The money market equilibrium condition, therefore, becomes: or, in excess supply form, M P = L(y, i) + ε L M P L(y, i) ε L = 0 (3) where, ε L is a catch all variable for money demand shocks, L y > 0 and L i < 0. A higher opportunity cost makes money less attractive (and short-term bonds more attractive).
Econ 797 Lecture 11 Element 3: Uncovered interest parity In addition to domestic (short-term) bonds, there are foreign (short-term) bonds. The nominal interest rate on the foreign bond, i, is exogenously given. Given free international mobility, and the assumption that domestic and foreign bonds are perfect substitutes (i.e., the choice between them depends only on expected returns, not on the volatility of returns), the following no-arbitrage condition should hold: i i ėe e = 0 (4) where, ėe e is the expected rate of depreciation of the currency (the expected percentage increase of the exchange rate). This specification assumes that the risk and liquidity characteristics of alternative assets can be ignored. Since the assets are short-term, we can also ignore inflation.
Econ 797 Lecture 12 If the domestic interest is below the foreign one, this must be because investors expect the domestic currency to appreciate (and thus compensate for the lower nominal interest rate). Talk a bit about the mechanism that ensures that UIP holds. We will assume very rapid adjustment in the financial markets so that UIP holds continuously.
Econ 797 Lecture 13 Combining the three elements We now have all the ingredients that we need for the static version of the Mundell- Fleming model. Eqs. (2), (3), and (4) constitute a system of three equations. As we will see next, the endogenous variables depend on the exchange rate regime. Fixed: y, i, M Flexible: y, i, E
Econ 797 Lecture 14 2.1 Fixed exchange rates As we will see below, monetary policy is now aimed at defending the pre-announced exchange rate. Interest rates and money supply no longer under policy control. The central bank will carry out reserve transactions (foreign exchange for domestic money) to keep the exchange rate fixed. Eqs. (2), (3), and (4) determine the equilibrium values of y, i, and M.
Econ 797 Lecture 15 The model has a recursive structure. With a credibly fixed exchange rate, ė e = 0, so that equation (4) tells us that i = i. The goods market equilibrium condition, equation (2), then determines real output. Finally, the money supply, M, is at whatever level coincides with the given international interest rate [see equation (3)]. No monetary policy autonomy.
Econ 797 Lecture 16 Intuition Suppose the central bank desires to keep the interest rate below the international one. It could buy domestic bonds, for example, incipiently lowering the interest rate in order to stimulate the economy. Given a credibly fixed exchange rate, the returns to holding domestic assets are lower. Domestic and foreign investors will immediately shift toward foreign assets. The resulting excess supply (demand) of domestic (foreign) currency in the foreign exchange market will tend to weaken the domestic currency. In order to defend the exchange rate, the central bank will have to remove the excess supply of domestic currency by buying it in exchange for foreign currency. As the money supply declines, the interest rises. This will continue until the domestic interest rate is back to the international level. The initial increase in the money supply has been reversed!
Econ 797 Lecture 17 A monetary expansion with a fixed exchange rate
Econ 797 Lecture 18 Recall the central bank balance sheet in its boiled down form. Step 1: Monetary expansion Assets Liabilities Foreign exchange reserves Domestic monetary base Domestic securities Bank reserves
Econ 797 Lecture 19 Step 2: Retract step 1 by intervening in the FX market Assets Liabilities Foreign exchange reserves Domestic monetary base Domestic securities Bank reserves End result: No change in M1! Note, however, that this result holds in this extreme form only if investors are risk-neutral and domestic and foreign assets are perfect substitutes. We will come back to this issue later.
Econ 797 Lecture 20 Comparative statics Again, one can either solve by substitution or using matrix methods. This system is simple enough so that either method is equally convenient. For later set-ups, however, being familiar with matrix methods will pay off. Fiscal policy dy dg = 1 > 1 1 A y tb y dm dg = P L y 1 A y tb y > 0 Notice that fiscal policy is highly effective in changing output. This is because monetary policy accommodates fiscal policy in way so as to boost the initial impact on output. Put differently, the upward pressure on the interest rate due to fiscal policy forces the central bank to respond by increasing the money supply (buying foreign currency in exchange for domestic currency).
Econ 797 Lecture 21 Fiscal policy with a fixed exchange rate
Econ 797 Lecture 22 A money demand shock dy dε L = 0 dm dε L = P > 0 The money supply changes to match the price level as the central bank responds to an excess demand for money which puts incipient upward pressure on the interest rate and exchange rate by increasing the money supply (via accumulation of reserves).
Econ 797 Lecture 23 Exercise: Analyze the comparative statics, mathematically and graphically, of a decline in the foreign interest rate.
Econ 797 Lecture 24 2.2 Floating exchange rates Now e responds endogenously to imbalances in the FX market while the central bank does not intervene. High-powered money M is exogenous (successfully targeted by the CB). In reality most CBs target the interest rate in the very short run and the rate of inflation over an extended period. However, let s ignore this for now. The price level is fixed, as before. However, this assumption is more problematic now that e is endogenous. Exchange rate movements should affect the price level directly via import and export prices.
Econ 797 Lecture 25 We are going to assume for now that expectations too are unchanging (static), i.e., ė e = 0 Essentially they have settled at their steady state level. The system now becomes: A(y, i, α) + tb(y, e, β) + ε D y = 0 (5) M P L(y, i) ε L = 0 (6) i i = 0 (7)
Econ 797 Lecture 26 Again, the system is recursive. Eq. (7) determines i, which with static expectations, is given international interest rate. Eq. (6) determines y (instead of M as in the fixed case). As we will see, this renders fiscal policy less effective (in fact, completely ineffective in our case!). Finally, eq. (5) yields e (instead of y, as in the fixed case). Thus, the exchange rate is now determined in the goods market.
Econ 797 Lecture 27 Comparative statics Fiscal policy dy dg = 0 de dg = 1 < 0 tb e Fiscal policy is now ineffective. We move from point A to B and back to A. The reason is simple. With the money supply unchanged, money demand cannot change. Since the interest rate is tied down by international conditions, domestic output too is pinned down. The excess demand created by expansionary fiscal policy is crowded out instead by currency appreciation (which switches demand from domestic to foreign goods).
Econ 797 Lecture 28 Fiscal expansion with a floating exchange rate
Econ 797 Lecture 29 Monetary policy dy dm = 1 > 0 P L y de dm = 1 A y tb y P L y tb e > 0 Monetary policy is now effective (the opposite result to that derived for fixed exchange rates!).
Econ 797 Lecture 30 Monetary expansion by the CB (say buying bonds) puts incipient downward pressure on the interest rate. As investors switch from domestic to foreign bonds, the resulting excess demand for foreign exchange causes currency depreciation. This depreciation, in turn, boosts demand for domestic goods, leading to higher output. How is a higher level of output possible given the money market balance constraint? Simple... the constraint has been relaxed by the monetary expansion creating room for output to expand. Note that fiscal expansion results in appreciation while monetary expansion leads to depreciation.
Econ 797 Lecture 31 MAJOR LESSON For small open economies, fixed exchange rates are better at stabilizing output if shocks are monetary, while flexible exchange rates are superior at dampening real demand shocks.
Econ 797 Lecture 32 US policy mix in the 1980s
Econ 797 Lecture 33 The Policy trilemma Argentina s currency board Greece and the Eurozone
Econ 797 Lecture 34 The unholy trinity!
Econ 797 Lecture 35 France (1968-69) Brewing revolution threatening the Gaullist government in 1968-69 Monetary policy constrained by Bretton Woods "Accord de Grenelle" with large (minimum and real) wage increases Swift deterioration of the current account The French government could have: (1) devalued, or (2) fiscally contracted Speculators (correctly) anticipated the former in spite of De Gaulle s promise to never devalue
Econ 797 Lecture 36 Fixed exchange rates provide a degree of certainty and reduce transaction costs but... There are serious credibility and time consistency problems. A fixed regime requires a consensus between a country and the mother currency country regarding the appropriate monetary stance. With the passage of time, conflicts of interest will inevitably rise (1960s Bretton Woods, 1990s EMS). Seen within a country, the optimal monetary policy today is not the optimal policy tomorrow. This sooner or later opens up the possibility of speculative runs.
Econ 797 Lecture 37 On the other hand... Post BW historical experience shows that flexible exchange rates can co-exist with large current account imbalances (e.g., the US experience) This may be partly because exchange rate flexibility may free up not only monetary policy but also fiscal policy space (especially if the capital account is relatively closed, so no crowding out through appreciation, and if policy makers believe that changes in currency values will take care of current account imbalances) It may also be because the increased exchange rate volatility associated with flexible exchange rates may make exporters and importers more reluctant to change behavior since they have doubts about the longevity of a given exchange rate change. This may be especially true if a country can issue debt in its own currency and domestic firms do not have large liabilities in foreign currency.
Econ 797 Lecture 38 EXERCISE (1) Work out the effects of a money demand shock in the case of a floating exchange rate. (2) Assume a floating exchange rate and regressive expectations rather than static expectations. That is, suppose that people expect the exchange rate to revert towards a value ē in the longer run. Thus, e e = ē. Derive the effects of fiscal and monetary policy under this alternative specification and compare these with the results based on static expectations.
Econ 797 Lecture 39 3 The dynamics Next, we explore the dynamics of the system. Recall that this gives us insights into the stability of the system and the adjustment mechanisms involved.
Econ 797 Lecture 40 3.1 Fixed exchange rates Since we are going to analyze dynamics, this is a good point to introduce: (1) a longterm asset (a bond or consol), and (2) rational expectations (or model consistent expectations). The latter help simplify our treatment of the dynamics The introduction of long-term bonds means that we need to consider inflation. To simplify, we will assume that these bonds are indexed in terms of domestic inflation. Moreover, the rate of inflation is identical in the home country and the rest of the world.
Econ 797 Lecture 41 I am going to suppress the time variable t to avoid clutter. But as we analyze the model, keep in mind that output y(t), the interest rate on bonds, R(t) and, later in the case of floating exchange rates, the exchange rate e(t) are all functions of time. The goods market is relatively slow-moving, so that we can specify an error correction mechanism for output: ẏ = dy/dt = λ[d(y, R, e) + g y]; y(0) > 0 given (8) where λ > 0 is a parameter that captures the speed of adjustment (or the speed at which any excess demand or supply is removed) Output is demand-driven. Firms respond to excess demand in the goods market by (slowly) increasing output. Since this change occurs gradually, the initial level of y determined by history..
Econ 797 Lecture 42 If the fixed exchange rate policy is credible (no expected large policy interventions, no large future cumulative current account deficits, no expected shortfall of FX reserves, etc.), then rational expectations imply that ė e = 0 t 0. Thus, the UIP condition reduces to: i(t) = i (9) The money market equilibrium condition is the same as before: m = M P = L(y, i) (10) Since asset markets adjust very quickly, this condition is assumed to continuously hold. As we see shortly, with a fixed exchange rate, m cntinues to be endogenous.
Econ 797 Lecture 43 The next few equations require more explanation. As derived in Section 4, the internal rate of return on a consol is the reciprocal of its real price q (which here is normalized to one unit of output per period). R = 1 q (11) The expected return from holding short-term bonds, r e, is the difference between the nominal interest rate, i ( i ), and expected inflation, π e ( P /P ). With rational expectations, π e = π so that r = r e. r = i π (12) Since we continue to assume an exogenous rate of inflation, P (t) = P (0)e πt.
Econ 797 Lecture 44 Finally, no-arbitrage ensures that the expected returns on the two assets equalize. With rational expectations, q e = q. Recall that the consol yields capital gains (or losses) as it is traded. r = 1 q + q q (13) The LHS represents the expected real rate of return on short-term bonds. The RHS captures the expected real rate of return on consols Notice that the absence of a risk premium implies that we are ignoring uncertainty (in a stochastic sense). The Appendix derives the solution to this first order differential equation.
Econ 797 Lecture 45 Now, differentiating eq. (11) w.r.t. t yields, q q = Ṙ R Recalling the UPIP condition (eq. (9)), employing the no-arbitrage condition (13) and the expression for the short-term real interest rate from (12), enables us finally to write down the equation of motion for the asset market: which is non-linear in the endogenous variable R. Ṙ = (R i + π)r (14)
Econ 797 Lecture 46 We now have a system of two differential equations (8) and (14) in two state variables, y and R. To refresh our memories, let s write them down again. ẏ = dy/dt = λ[d(y, R, e) + g y]; y(0) > 0 given Ṙ = (R i + π)r What about q and r? Equations (11) and (13) describe how these are adjusting in the background.
Econ 797 Lecture 47 The crucial thing to note here is that, once the system is in steady state with y and R constant, equations (11) and (13) tell us that r and q too are constant. This is good since we do not want endogenously moving parts in the steady state. As we see below, the fact that our small economy takes the nominal interest as given from the international bond markets means that the dynamics are similar to those of an economy where the CB sets the policy interest rate. Also, our endogenous variables are expressed in terms of deviations from their steady state values (although I now eschew using the tildas over variable symbols to simplify notation).
Econ 797 Lecture 48 Before we analyze the dynamics in detail, we need to linearize our system of differential equations. This means that our analysis will be local rather than global in nature. In matrix form: [ ẏ. R ] = [ ] [ ] λ(dy 1) λd R y 0 i π r (15) where, the derivatives have been evaluated at their steady state values (so, for example, R = i π is the economically interesting steady state value of R).
Econ 797 Lecture 49 By now we know how to derive the slopes of the isoclines. dr dy = 1 D y ẏ=0 D R < 0 (under the typical assumptions) dr dy = 0 r=0 Denoting the Jacobian matrix by A, T r = T r(a) = λ(d y 1) + i π Det = Det(A) = λ(d y 1)(i π) < 0 (since R = i π > 0). We have got a saddlepath equilibrium.
Econ 797 Lecture 50 Dynamics with a fixed exchange rate The dash line shows the effect of fiscal expansion.
Econ 797 Lecture 51 The phase arrows suggest that the horizontal isocline doubles up as the stable arm. Let s see if we can verify this. The first step is to find out the values of the eigenvalues (or characteristic roots). Since it is a 2 equation system, it has two roots, r 1,2. We could either use the system of equations (15) to calculate the roots via the quadratic formula or use the following formula (note: you should try both): r 1,2 = T r ± T r 2 4Det 2 = λ(d y 1), i π
Econ 797 Lecture 52 Now, we know that eigenroots are solutions to the system given by [A ri]ν = 0, where I is the identity matrix and ν is the eigenvector (which helps determine the the direction of the stable arm). So, for r 1 = λ(d y 1), the corresponding eigenvector ν i 1 is derived from: or, [ ] [ ] λ(dy 1) λ(d y 1) λd R ν 1 1 0 i π λ(d y 1) ν 2 1 = [ ] 0 0 [ ] [ ] 0 λdr ν 1 1 0 i π λ(d y 1) ν 2 1 = [ ] 0 0
Econ 797 Lecture 53 [ ] 1 Normalizing ν 1 1 to 1 gives us the eigenvector associated with r 1, which is 0 Repeating these steps for the second eigenvalue, r 2 = i π yields the second eigenvector, which after normalizing ν 1 1 [ ] [ ] ν 2 to 1 is, 2 1 ν 2 = 2 λ(1 D y)+i π λd R One can verify immediately that this eigenvector is associated with a positive slope.
Econ 797 Lecture 54 We now have our general solution: [ ] y R [ ] [ ] 1 = A 1 e r1t 1 + A 0 2 λ(1 D y)+i π λd R where A 1 and A 2 are constants of integration. Once again, it is important to recall that y and R are expressed here in deviation form. e r 2t
Econ 797 Lecture 55 Now suppose the system is away from the steady state. Unless it is on the stable arm at any given point in time, it will deviate for ever, and we get instability. With rational expectations, one would expect the system to be placed on the stable arm. This means that we need to "kill off" the positive root. This is achieved by setting A 2 =0.
Econ 797 Lecture 56 so that, [ ] y R = A 1 [ 1 0 ] e r 1t Now at time t = 0, this tells us that y = y(0) ȳ = A 1 and R = R(0) R = 0, where overbars denote steady state values. This latter expression confirms that, along the stable arm, R never deviates from its steady state value. This verifies our initial guess that the stable arm coincides with the Ṙ = 0 isocline.
Econ 797 Lecture 57 We finally have the equation for the stable arm of the saddlepath! [ ] y R [ ] 1 = [y(0) ȳ] 0 [ ] 1 = [y(0) ȳ] 0 e r 1t e λ(d y 1)t Now that we understand the basic dynamic properties of the system, we can carry out some thought experiments
Econ 797 Lecture 58 Expansionary fiscal policy Steady state changes dy dg = 1 1 D y dr dg = 0 Policy trilemma in action again, now with the long-term interest rates. Given π, R is a function of short-term interest rates, which are pinned down by exogenous international conditions. So it is not a surprise that dr dg = 0.
Econ 797 Lecture 59 The intuition is mostly the same as in the static case but now we can relate it to the transitional dynamics. A fiscal expansion immediately creates excess demand for domestic goods. Firms gradually increase output in response. As output rises, so does the real transactions demand for money. This creates an incipient tendency for the nominal interest rate to rise (equation (10)). Given the rate of inflation, this also translates into an incipient increase in the real short term interest rate (equation (12)) and, through the no-arbitrage condition, the consol rate to rise. In order to forestall the resulting appreciation of the currency, the CB has to step in by intervening in the FX market (i.e., buying FX reserves in exchange for domestic currency to satisfy the excess demand for the latter). 1 As the money supply increases, the interest rate remains constant at the international level, and r and R too remain unchanged. 1 Note that, now that we have saddlepath stability, the CB may also take these actions in order to target the interest rate and hence avoid instability.
Econ 797 Lecture 60 The time plots for fiscal policy with a fixed exchange rate
Econ 797 Lecture 61 EXERCISE: What is the effect of an increase in the international interest rate? Discuss both the effects on the steady state values of variables and the transitional dynamics.
Econ 797 Lecture 62 3.2 Floating exchange rates Since we incorporate the dynamics, we can now analyze the evolution of exchange rate expectations in interesting ways. Investigating floating exchange rates adds one more state variable the exchange rate to our analysis. Unlike output, which is a slow-moving variable, we will treat the exchange rate as a jump variable. This makes sense since the exchange rate is an asset price, and should, at least in theory, be a forward-looking variable. We adjust for the inclusion of another endogenous variable by simplifying in a different dimension, i.e., we now ignore the distinction between short-term and long-term bonds, so that both investment and money demand have essentially given an exogenous inflation rate the nominal interest rate as the argument. This nominal interest rate is still tied down by international conditions.
Econ 797 Lecture 63 As in the fixed exchange rate case, there are 3 basic building blocks: 1. Goods market equilibrium 2. Money market equilibrium 3. Foreign exchange market (UIP)
Econ 797 Lecture 64 As before, the goods market equilibrium condition is as follows: ẏ = dy/dt = λ[d(y, r e, e) + g y]; y(0) > 0 given (16) with the same partial signs as before. The only difference is aggregate demand is now a function of the short-term real interest rate rather than the long-term one. The money market equilibrium condition is still given by equation (10), although we now simplify notation by using m ( M/P ) to denote the real money supply. Then equation (10) can be re-written as: where i y = L y /L i > 0 and i m = 1/L i < 0. i = i(y, m) (17)
Econ 797 Lecture 65 The third and final basic element gives us the equation of motion for the exchange rate. or, assuming perfect foresight, i(t) = i + ėe (t) e(t) (18) ė = (i i )e (19) The steady state is defined by constant levels of output and the exchange rate. From eqs. (17) and (18), this means that the interest rate too is constant. Due to perfect foresight, r = r e and π = π e, so that the real interest rate, r = i π too is constant in the steady state. Indeed, we do not lose anything by assuming that P = P = 0, so that π = π = 0.
Econ 797 Lecture 66 The money market is taken to adjust rapidly so that it is always in equilibrium. This means that we can substitute equation (17) into the equations of motion for output and the exchange rate to obtain our system of first order, non-linear differential equations. ẏ = λ[d(y, i(y, m) π, e) + g y]; y(0) > 0 given (20) We have a system of two equations in y(t) and e(t). ė = [i(y, m) i ]e (21)
Econ 797 Lecture 67 One of the fixed points of the system is given by e(t) = 0. This is not economically interesting. The second fixed point involves i = i. Let s explore this... The Jacobian (evaluated at the fixed point) helps us establish crucial properties of the system: 2 A = [ λ(dy 1) + D r i y ] λd e i y e 0 T r(a) = λ(d y 1) + D r i y Det(A) = λd e i y e < 0 Another case of saddlepath (in)stability! 2 Again, keep in mind that both the endogenous variables are expressed in deviation form.
Econ 797 Lecture 68 The phase diagram in the flexible exchange rate case
Econ 797 Lecture 69 The saddlepath now appears to be downward-sloping. Let s verify this. The eigenvalues of the system are: r 1,2 = λ(1 D y D r i y ) ± λ 2 [(1 D y ) D r i y ] 2 + 4λi y ed e 2 The eigenvalue associated with the stable arm obviously is the negative one: r 1 = λ(1 D y D r i y ) λ 2 [(1 D y ) D r i y ] 2 + 4λi y ed e 2
Econ 797 Lecture 70 You can verify that the associated eigenvector is given by: [ 1 ] which is negatively-sloped. λ(1 D y D r i y ) [λ(1 D y D r i y ] 2 +4λi y ed e λd e
Econ 797 Lecture 71 More generally, the general solution to the system is given by: or, [ ] [ y ȳ = A e e 1 1 λ(1 D y D r i y )+r 1 λd e ] e r1t + A 2 [ 1 λ(1 D y D r i y )+r 2 λd e ] e r 2t [ ] y e = [ȳ ] [ + A 1 e 1 λ(1 D y D r i y )+r 1 λd e ] e r1t + A 2 [ 1 λ(1 D y D r i y )+r 2 λd e From here one could definitize the solution by determining the values of A 1 and A 2. ] e r 2t (22)
Econ 797 Lecture 72 Let s return to the previous phase diagram. Suppose we find the economy at t = 0 at a point like A, where y = y(0). Recall that y is a slow-moving (i.e., pre-determined) variable. If speculative exchange rate bubbles are assumed away, then e cannot be on an explosive or implosive path. This leaves the segment AE on the stable arm as the unique solution for t 0. Output is declining while the exchange rate is depreciating along the adjustment path (why? hint: there is excess supply in the goods market and expected depreciation of the currency).
Econ 797 Lecture 73 Before we carry out thought experiments, it will be useful in understanding the model to write down the steady state solutions to the dynamic system: 3 ȳ = D(ȳ, i π, ē) + g (23) m = L(ȳ, i ) (24) where overbars denote steady state values. Note tha, since i = i in the steady state, (UIP), the money market condition (equation (24)) is sufficient to pin down ȳ. Equation (23) then determines ē. The system is recursive in nature. 3 Note that these are not reduced form solutions since we have not expresssed the endogenous variables in terms purely of the exogenous variables and parameters.
Econ 797 Lecture 74 Monetary policy (Unanticipated rise in the real money supply) Suppose the CB buys bonds for money (i.e., carries out an open market operation) at time t 0. dȳ dm = 1 L y > 0 dē dm = ē dȳ ȳ dm = 1 D y 1 > 0 (25) D e L y The steady state multipliers are the same as the static ones. Not surprising!
Econ 797 Lecture 75 (Horizontal) Curve shifts (from eqs. (20) and (21)) ȳ m ẏ=0 = D ii m D 1 D y D i i y = i /L i 1 1 D y +D i L y /L i = ( ) 1 D > 0 y D L i +L y i ȳ m ė=0 = i m iy = 1/L i L y /L i = 1 L y > 0 Both curves shift rightward. Comparing the two expressions makes it obvious that the ė = 0 isocline shifts more in the horizontal direction. From equation (22), it is also clear that, since both ȳ and ē are higher, the saddlepath also shifts up and to the right.
Econ 797 Lecture 76 Unanticipated monetary expansion under a floating exchange rate regime
Econ 797 Lecture 77 Since y is slow-moving, eq. (24) tells us that i must jump down immediately to maintain (continuous) money market equilibrium. This prompts arbitrage. As portfolios shift toward foreign bonds, the currency depreciates (e jumps up). By how much does e jump? UIP dictates that the domestic interest rate can be lower than the international one only if there is a compensating expected appreciation. So the answer is that e must jump sufficiently so that the resulting expected appreciation equalizes expected returns at home and abroad.
Econ 797 Lecture 78 An expected appreciation, given rational expectations, translates into an actual appreciation as investors move to purchase domestic assets. Note that at a point like A in the previous phase diagram, which is the destination of the exchange rate jump, ė 0. The system is on the stable arm as the exchange rate depreciates, giving a boost to demand for domestic goods. Domestic output rises as a result.
Econ 797 Lecture 79 Overshooting A variable is said to overshoot if its initial reaction to a shock exceeds its long-run response. Notice that the exchange rate first jumps up to A, before appreciating to the new steady state. In other words, there is exchange rate overshooting. This phenomenon has been used to explain high exchange rate volatility in the post- Bretton Woods era. We will talk about overshooting in more detail later, but let s re-visit the intuition here.
Econ 797 Lecture 80
Econ 797 Lecture 81
Econ 797 Lecture 82 The intuition underlying the overshooting resultlet s take a closer look at the mechanics and their underlying intuition. Once a monetary expansion occurs: What will be the new steady state exchange rate expected by market participants? From the model (see equation (25), participants know that the new steady state exchange rate is going to be higher.
Econ 797 Lecture 83 Must there be a jump? First of all, notice that there has to be a jump. Suppose instead that e adjusted gradually instead. Then the outflow of capital caused by the lower interest rate will lead to expected depreciation, which means that foreign returns are now much higher on account of both lower i and ė e > 0. This means even greater outflows, speeding up the adjustment of e. Taken to the limit, this corresponds mathematically to an upward jump of e.
Econ 797 Lecture 84 Why must the initial depreciation be higher than that required to get to the steady state value? Or, alternatively, how large must the initial jump be? The answer is straightforward; it must be sufficiently large to ensure that the resulting expected appreciation equalizes expected returns on domestic and foreign assets (i.e., UIP holds) at the same time as convergence to the new equilibrium is ensured. This happens at point A. Ruling out bubbles, participants realize that any point above or below A represents arbitrage opportunities (since UIP doesn t hold in that instance). Exploiting these opportunities will instantaneously bring the system back to point A.
Econ 797 Lecture 85 Why does y (gradually) rise and e (gradually) appreciate after the initial jump? For t > t 0, the low interest rate and increased competitiveness (high e) creates excess demand for domestic goods. As y rises in response, money demand rises, so that i returns to its original value (see equation (20) and (18)). At this point, ė = 0 and i = i. Moreover, as i falls to its initial value, the currency appreciates and converges towards its new steady state value.
Econ 797 Lecture 86 Monetary expansion and overshooting: Time plots
Econ 797 Lecture 87 The Exchange Rate as a Forward-Looking Variable Given UIP an perfect foresight, the exchange rate today is a function of the future path of the interest rate. This can be seen from the UIP condition under the assumption of perfect foresight: ė(t) e(t) = i(t) i (26) This must hold at all times except for at points of discontinuity of e.
Econ 797 Lecture 88 As shown in the Appendix, the interest rate today is a weighted of all future interest rates. More specifically, the solution to the differential equation (26) is given by: e(τ) = e(t)e τ t [i(s) i ]ds = e(t)e τ t [ī i ](τ t) τ > t where ī is the mean of the interest rates between t and τ, i.e., ī τ i(s) t τ t ds Since the exchange rate is not pre-determined, it makes sense to write the solution in forward-looking form: e(t) = e(τ)e τ t [i(s) i ]ds = e(t)e τ t [ī i ](τ t) τ > t (27) e(τ) and i(s) are future values as seen from today.
Econ 797 Lecture 89 As expressed here, the exchange rate today is the expected future exchange rate discounted by the expected mean interest rate differential expected. Any new information regarding e will, for a given interest rate path, show up as an exchange rate change today. The solution (27) assumes that there is no jump between time t and τ. Arbitrage prevents any expected jump. So ex-post the solution is valid only if there are no actual jumps in e in the time period analyzed.
Econ 797 Lecture 90 Now let s return to the case of expansionary monetary policy and consider the path of the exchange rate over an infinite period. From (27), e(t) = lim e(τ)e τ t [i(s) i ]ds = ēe τ t [i(s) i ]ds > ē t As time proceeds, i(s) converges to i and e to ē from above (as shown in the time plot for e).
Econ 797 Lecture 91 An anticipated monetary expansion As our final thought experiment, suppose the CB credibly announces a discrete increase in the money supply at a future date t 1 (> t 0 ). The announcement will cause an immediate jump in e to a level such as that corresponding to point B in the next figure. Reason: Participants anticipate that the economy will be on its new saddlepath following time t 1.
Econ 797 Lecture 92 In terms of the underlying economics, participants realize that: (1) the increase in money supply will cause the domestic interest rate to be lower than i (2) the exchange rate must be such at time t 1 so as to be consistent with UIP (thanks to arbitrage). That is, the expected appreciation at that point should compensate for the interest rate differential after t 1.
Econ 797 Lecture 93 Under these circumstances, there must be excess supply of domestic bonds (and excess demand for foreign bonds) at the old exchange rate immediately after t 0. This is what makes the exchange rate jump immediately. After the jump, there is continuous depreciation (recall that arbitrage ensures that there cannot be any anticipated jumps; in particular there can be no jump at t 1 since no new information has become available).
Econ 797 Lecture 94 In the interval (t 0, t 1 ) the movement of e and y is governed by the old dynamics. This defines the jump from E to B. The economy follows a trajectory governed by the no arbitrage condition: i(y, m) = i + ė e Under perfect foresight the market mechanism selects the trajectory BC along which it takes exactly t 1 t o time units to get from B to C. It is in fact this requirement that determines the size of the jump EB.
Econ 797 Lecture 95 Aside: Suppose the jump is less than that to B. Then not only is there a longer road to be travelled to the new saddlepath but, since after the jump the system starts at a point closer to the (still operational) steady state point E, the initial speed of adjustment will be slower.
Econ 797 Lecture 96 The initial jump in e makes the economy more competitive, raising output and hence i. This creates an expected (and actual) depreciation of the currency. The process continues until the monetary expansion is actually carried out at t 1. Exactly at this point, the economy arrives at the new saddlepath at C. The actual rise in m then triggers the discrete fall in i to a level i < i.
Econ 797 Lecture 97 Note: Since right before t 1, i > i one might wonder whether the decline in i is necessarily large enough to ensure that i < i right after t 1. The answer is yes! This is because the dynamics between t 0 and t 1 ensure that y < ȳ, which means that i(y, m ) < i(ȳ, m ) [= i ] since i y > 0..
Econ 797 Lecture 98 Anticipated monetary expansion with a floating exchange rate
Econ 797 Lecture 99 Anticipated monetary expansion with floating exchange rates: Time plots
Econ 797 Lecture 100 Some shortcomings This can be seen as a short-run set-up with fixed prices. Analysis of longer-run dynamics would require inclusion of price adjustment, often done through a Phillips curve (or conflicting claims in heterodox models). Moreover, (positive or negative) investment will lead to changes in the capital stock, which has ramifications for output and the interest rate. Portfolio balance models address some of the resulting dynamics.
Econ 797 Lecture 101 4 Appendix Derivation of the internal rate of return on a consol The internal rate of return on a consol or the long-term interest rate is the discount rate, R, which transforms the payment stream on the consol into a present value equal to the market value of the consol at time t. That is, it is the value of R that satisfies the following condition (recall that the consol yields one unit of output per period): q(s) = (1)e R(s t) ds = e R(s t) t R = 1 R(t) From here it is trivial to derive equation (11). t
Econ 797 Lecture 102 The relationship between the market and fundamental values of a consol First, let s define the fundamental value of a consol; it is the present value of future payments from the consol using the (expected) future short-term interest rates as discount rates. 4 so that, from equation (11), q(t) = t (1)e s t r(τ)dτ ds (28) R(t) = 1 t (1)e s t r(τ)dτ ds From here, it can be shown that the long-term interest rate R(t) is a weighted average of expected future short-term rates, r(τ). I will skip that derivation here. In the absence of speculative bubbles, the no-arbitrage condition (13) is equivalent to 4 Notice that the discount rate being used is the short-term rate rather than the long-term rate that we used for calculating the internal rate of return on consols.
Econ 797 Lecture 103 saying that the market value of the consol equals its fundamental value. We will now establish this proposition.
Econ 797 Lecture 104 Solving the no-arbitrage condition The solution to the first order differential equation implied by the no-arbitrage condition can be derived as follows: First, let s write down equation (13) in a form that we may be more familiar with: q rq = 1 (29) This is a differential equation with a variable coefficient (r) and a constant term. As we know (see p. 485 of the Alpha Chiang and Wainwright text), the general solution to such a differential equation is given by: t t r(τ)dτ q = e 0 [ q 0 t t t = q t0 e r(τ)dτ t 0 t r(τ)dτ e 0 t 0 ] e s t r(τ)dτ 0 ds t t 0 e s t 0 r(τ)dτ ds
Econ 797 Lecture 105 Multiplying through by e t t 0 r(τ)dτ yields, qe t t 0 r(τ)dτ = q t0 t t 0 e s t r(τ)dτ 0 ds Finally, re-arranging and letting t approach infinity gives us: q t0 = lim qe t t r(τ)dτ 0 + t t 0 e s t r(τ)dτ 0 ds (30) The second term on the RHS expresses the fundamental value of the consol (recall equation (28)), while the first term captures the bubble term, i.e. the deviation of the market price from its fundamental value.
Econ 797 Lecture 106 Thus, in the absence of a speculative bubble, q t0 = which is the same as equation (28)! t 0 e s t r(τ)dτ 0 ds
Econ 797 Lecture 107 [1] Robert Mundell. The monetary dynamics of international adjustment under fixed and flexible exchange rates. Quarterly Journal of Economics, 74:227Ű57, 1960. [2] Marcus Fleming. Domestic financial policies under fixed and under floating exchange rates. Staff Papers 9, International Monetary Fund, 369Ű79, 1962. [3] Robert Mundell. Capital mobility and stabilization policy under fixed and flexible exchange rates. Canadian Journal of Economics and Political Science, 29:475 85, November 1963. [4] Rudiger Dornbusch. Expectations and exchange rate dynamics. Journal of Political Economy, 84:1161 76, 1976.