Lecture 9: Exchange rates

Transcription

1 BURNABY SIMON FRASER UNIVERSITY BRITISH COLUMBIA Paul Klein Office: WMC 3635 Phone: (778) paul klein URL: Economics 305 Intermediate Macroeconomic Theory Fall 2013 Lecture 9: Exchange rates 1 The long run In the long run, it seems that the theory of purchasing power parity (PPP) is a pretty good predictor of exchange rates. Countries with higher inflation rates should see their currencies depreciate. Consider Turkey versus the U.S. until The logic is simple. PPP says P = EP so we should have E = P P. Actually, PPP theory is a bit more flexible than that. When it is tested on empirical data, the version of it being tested is not that strict. It does not insist that a kg of coffee costs the same in Vancouver as pounds of coffee in Seattle, when converted to a common currency. What it does claim is that the degree to which Seattle coffee is more or less expensive than Vancouver coffee should not change over time. This is the version of the theory that we tested when we looked at Turkey and the United States. 1

2 If the money supply per head grows at the same rate, inflation rate differentials and hence exchange rate changes will reflect changes in relative real output per head. Consider Norway versus Sweden. Again the logic is quite simple. Let s say the demand for money is given by M/N P = qy/n where q is some constant which is presumably a function of the nominal interest rate and hence of the inflation rate and, similarly, M /N P = q Y /N Now let M/N = M /N and suppose q = q ; both reflect the assumption of similar monetary policies. Finally, invoke PPP. We conclude that E = Y /N Y/N. However, neither of these approaches is very successful in accounting for U.S./Canada exchange rates. 2 The short run Why are exchange rates so volatile? It seems that real exchange rates (EP /P ) fluctuate more than relative prices across regions within a currency area. Dornbusch (1976) has an interesting explanation. Its flavour is decidedly Keynesian (it features price stickiness and an LM curve relationship) but it also has modern features; in particular, rational expectations. It is chock-full of interesting economic ideas, most of which are probably wrong but instructive. Dornbusch s starting point is the notion that, in the long run, the exchange rate is determined in goods markets, and it is whatever it needs to be to ensure purchasing power parity (PPP). In the short run, however, the exchange rate is determined in asset markets, and the exchange rate is whatever it needs to be to make sure that the expected rate of appreciation of (say) the Canadian dollar vis-à-vis (say) the U.S. dollar is equal to the U.S./Canada interest rate differential (uncovered interest rate parity). 2

3 i(t) i i(t) t 0 t Figure 1: The nominal interest rate The fundamental assumptions are (1) exchange rates can change instantaneously and (2) goods prices take time to adjust. A simplified version of the model can be found in Heffernan and Sinclair (1990), pages and it goes as follows. All the variables except the (nominal) interest rate i are in logs. m(t) p(t) = hi(t) (1) i(t) = i + ė(t) (2) ṗ(t) = a[e(t) p(t)] (3) where m is the log money supply, p is the log price level, i is the domestic nominal interest rate, i is the foreign nominal interest rate (which is assumed to be fixed), h > 0 is a fixed parameter, e is the log exchange rate (the price of one unit of foreign currency in terms of the domestic currency), and a > 0 is another fixed parameter. Notice that a fall in e represents an appreciation of the domestic currency vis-à-vis the foreign currency and vice versa of course. 3

4 m(t) t 0 t Figure 2: The log money supply Equation (1) is an LM curve relationship: it expresses a liquidity effect for which there is strong evidence in the data. Not only that, but it is pretty plausible if you think about it. The nominal interest rate is the opportunity cost of holding money. Meanwhile, m p is the real value of the stock money. As the nominal interest rate goes up, people are prepared to hold less money in real terms; this is what (1) says. Equation (2) describes an uncovered interest parity (UIP) relationship. The expected (and, under rational expecations, also actual) rate of return on holding domestic bonds should be the same as on holding foreign bonds, when converted into a common currency. An expected depreciation must be compensated for by an interest rate premium. Equation (3) says that the inflation rate is an increasing function of deviations from PPP. When PPP holds, we have e = p (by normalization). When the domestic currency is weak (e is high relative to p), this pushes up domestic prices, but, by the assumption of slow adjustment, not immediately. Prices don t jump in response to a weak currency, but inflation does. Objections: there is zero empirical support for UIP and not much for PPP. Notice that the version of PPP that we need here is rather weak. The long run equilibrium condition e = p is 4

5 p(t) t 0 t Figure 3: The log price level just a normalization; the difference e p could converge to any constant and the model would work the same way. Thus our version of PPP doesn t say that a kilo of coffee should necessarily cost the same in Canada as in the United States when computed in a common currency. It merely says is that there should be no long run trend in the real exchange rate. Is there in fact? To sharpen the focus, let s fix a = 1 and h = 1/2. It might be interesting to investigate how the solution to the model depends on these parameters but that can wait. Solving out for i, our equations become { ṗ(t) = p(t) + e(t) ė(t) = 2p(t) i 2m. In a steady state, p = e = m + i /2. But what concerns us is what happens outside a steady state. Suppose the money supply suddenly increases. The long run price level and exchange rate will increase in the same proportion. But what happens in the short run? For the time being, p is fixed at a level below its new, higher, steady state value, i.e. p < m+i /2. This means that ė < 0. This means that the domestic currency must be appreciating (e must be falling). So it must jump to a level above its long run level. The exchange rate overshoots. We can illustrate this in a phase diagram (see below). We can also tell an economic story as follows. If the nominal money supply goes up and the 5

6 e(t) t 0 t Figure 4: The log exchange rate price level is fixed (for the time being), then the real money stock must go up. To persuade people to hold the stuff, the nominal interest rate must go down. If the nominal interest rate goes down, then it must go below the foreign nominal interest rate. To persuade people to hold domestic currency denominated bonds, people must expect the domestic currency to appreciate over time (e to go down). But then, by rational expectations, this must actually happen. What is the upshot of all this? Well, small changes in monetary policy can lead to large changes in the exchange rate if prices are sticky. Since prices are sticky, this means that there are large changes in real exchange rates too. 3 Forward exchange rates When laying out the Dornbusch model, we mentioned uncovered interest parity (UIP). What is covered interest parity? It is this equation: E f E = 1 + i (4) s 1 + i where E s (known as the spot exchange rate) is the price, in terms of domestic currency, of a unit of foreign currency for immediate delivery. On the other hand, E f (known as the forward exchange rate) is the price, in terms of domestic currency, of a unit of foreign currency 6

7 e(t) ṗ(t) = 0 p(t) ė(t) = 0 Figure 5: The dynamics of the model illustrated in a phase diagram for delivery at some fixed date in the future, say 3 months from now. Meanwhile i is the domestic (nominal) three-month interest rate and i is the foreign three-month interest rate. Of course there are (in principle) forward rates for every time horizon, not just three months. A forward contract, signed at date t, specifying that a currency exchange will take place at time t + T, where the quantities exchanged (and hence also the exchange rate) is determined at t is a very useful way of hedging against currency risk. Suppose you grow wheat in Manitoba (or you trade in Manitoba wheat) and export to England. (Canadian wheat is high in protein and consequently much sought after in the rest of the world.) You pay all your bills in Canada and so you are not the least interested in being exposed to the currency risk that would be involved if you were paid in pounds sterling. Your buyer doesn t particularly like currency risk either so he has insisted on paying you in pounds sterling when the harvest is ready and delivered at the port of Southampton. One thing you can do in this situation is to go to your bank and sign a forward contract. This 7

8 contract will stipulate: 1. Today, you pay nothing and receive nothing. 2. T days from now, you sell A pounds sterling at the exchange rate E f. This exchange rate is determined today. The question for the bank (and us) is what E f should be. It s tempting to think that this depends on market conditions, consumers risk attitudes etc. That turns out not to be true, except perhaps indirectly. There is a correct forward exchange rate that is fully determined by the spot exchange rate, the interest rate at home and abroad, and it is the formula we have seen already. What is the basis for this formula? It is the following. Suppose the bank, according to the contract, is obliged to exchange one pound for E f Canadian dollars. Let s say that T = 365 so that the forward contract has one year to maturity. To be 100 percent sure that it will be able to meet this obligation, it buys one pound now. No, wait, it doesn t do that: all it has to purchase now is 1/(1 + i ) pounds. It then invests this amount in an interest-bearing British savings instrument which at the end of the year will have grown to one pound. To finance this purchase of pounds, the bank borrows E s /(1 + i ) Canadian dollars. At the end of the year, the bank owes (1 + i)/(1 + i )E s Canadian dollars; it has to pay back its loan. Meanwhile, it receives E f Canadian dollars from the other party to the forward contract in return for giving him one pound. In Canadian dollar terms, the bank s profits are E f 1 + i 1 + i Es. For profits to be zero, these expressions should be equal, yielding our formula above. If the profits are not zero, the bank (or anyone) can make arbitrarily large profits by signing forward contracts involving large enough transactions in currencies. In that situation, we say that there is an arbitrage opportunity. If there is to be no such opportunity, our formula (4) must hold. And it does in the data. What Dornbusch s (1976) model assumes is that the expected future exchange rate equals the forward rate. This is plausible, but empirically false. 8

9 References Dornbusch, R. (1976, December). Expectations and exchange rate dynamics. Journal of Political Economy 84 (6), Heffernan, S. and P. J. N. Sinclair (1990). Modern International Economics. Basil Blackwell. 9