Short-Term Exchange Rate Behavior

Let us summarize a bit what we have done so far in this course. One of the main goals in this course is to construct at a model that allow us to determine the current spot exchange rates. To do this, we combined the money market equilibrium model along with the Uncovered Interest Parity model to study short and long-term effects of temporary and permanent changes in monetary policy and macroeconomic variables on the spot exchange rate. In order for the Uncovered Interest Parity to hold, we need three assumptions: 1) risk neutral investors, 2) market efficiency, and 3) no arbitrage. The first two assumptions made forward contracts redundant and the last assumption ensures that the parity holds. The money market equilibrium model is simply supply equals demand for cash or money. The key insight from the Uncovered Interest Parity is that expectations of long-term movements of the exchange rate matter in the short term. Additionally, because of sticky prices in the short-term,we observe short-term responses in interest rate and exchange rates from permanent money base changes.

Shot Exchange Rate Changes and Price Level Changes USD/DEM

This overshooting is observable in the month-to-month fluctuations of the spot exchange rate. Let’s look at the exchange rate between the USD and the German Deutsche Mark. This graph is the monthly change in the stop exchange rate for the U.S. Dollar to the German Deutsche Mark and the monthly change in the Consumer Price Index between the U.S. and Germany. Our time period ranges from 1987 to 2002. Vertical axis is just 1 month changes in USD to Deutsche Mark price ratio. The change in spot exchange rate is quite volatile despite very little month-to-month changes in the CPI ratio. So, the overshooting prediction from our model is consistent with empirical data.

Long-Term Exchange Rate Behavior

Our next task in this course, is to build a framework for forecasting future exchange rates in the long term. In the long-term, prices are fully flexible so temporary money supply changes doesn’t matter. However, even though real money supply is unchanged in the long-run, the central bank can still affect the future spot exchange rate by shifting expectations through permanent changes in the money supply or through permanent changes in relative prices. If you recall last week, we concluded that the exchange rate can trace both the domestic and foreign price levels through changes in the future exchange rate expectations. Given any exchange rate, a higher price level makes exports relatively expensive for foreigners and imports relatively cheap for domestic residents. Domestic residents will sell the domestic currency for foreign currencies to import goods and so the exchange rate depreciates until it makes up for the rise in the price level.This demand and supply logic also works for foreign price levels. A higher foreign price level will cause foreigners to import domestic goods so they will bid up the price of the domestic currency. This will cause the nominal exchange rate to appreciate in the long term. Our goal now is to add these long-run relative price features to our existing asset approach model of exchange rates. Specifically, we want to now build a model to understand permanent monetary policies impacting relative prices between countries. This will require us to introduce the Purchasing Power Parity theory. Remember, we are now working in the long-run, so prices are assumed to be fully flexible.

Nominal Exchange Rate Changes and Price Level Changes USD/DEM

Rather than look at changes in relative price levels, let us just look at the relative price levels for the the nominal exchange rate between USD and the German Deutsche Mark and the CPI ratio between US and Germany. We have a longer history now to give us a bit of perspective on the trends between the nominal exchange rate and the CPI ratio. While there is still a lot of short-run month-to-month changes in the exchange rate, the long-run trend is positive, which is roughly the same slope as the CPI ratio. We want a model to explain these similar trends and to ultimately help us forecast nominal exchange rates in the long-term.

The Exchange Rate in the Long Term

To reiterate, the goal is to understand the long-term behavior of the exchange rate and its relationship with relative price levels between countries. To link ourselves to price levels, we have to extend our model to include the commodities market (or real goods). I will begin by introducing three no-arbitrage conditions for international commodity markets:

1. The Law of One Price: This is the most restrictive price parity, which essentially states that identical goods should have identical prices in every country through the exchange rates. This, as you all know, is always violated for very obvious reasons such as different input or labor costs. A McDonald’s Big Mac tastes the same everywhere, but labor and input costs vary from country to country, which can explain the price differences after converting prices to the same currency.
2. Absolute Purchasing Power Parity: This parity is less restrictive than the Law of One Price since it only requires that the weighted average price of a basket of similar goods should be the same in every country.
3. Relative Purchasing Power Parity: Finally, this parity is even weaker because this parity only requires that the average prices changes for the weighted basket of similar goods be the same across countries when converted to the same currency.

I will explain each of these price parities separately.

Review: Domestic Price Level in the Long Term

But first, let me review some macroeconomic definitions used to study domestic price levels in the long term:

1. Quantity Theory of Money: You should view this equation as a tautology. For a given period of time, M^S is the money supply, v is the velocity of money, P is the average price level associated with transactions for the economy, and Y is an aggregate measure of real goods produced. Velocity of money is number of transactions per domestic currency per given period. The left side of the equation is equal to the amount of the money supply that is used for real transactions. The right side of the equation is aggregate nominal expenditures of the entire economy. In the long term, for a given national income and velocity of money, the real money supply is assumed to be a constant or more specifically equal to real money demand. Note that the velocity of money may depend on R. If the v and Y^GNP are fixed, doubling M^S will double P.
2. Weighted average Price level P: The weighted average price is defined as a weighted sum of commodities 1 through N in a predefined basket. Alpha_1 through Alpha_N are commodity-basket weights. Think of these weights as the proportion of the aggregate household’s income used to consume the goods in a basket. These goods could be groceries, rent, etc. We will use CPI to as a measure for P.

So given this definition of P, we should have some idea what inflation is then. Inflation is the percent change in P or the average change in prices for each good given that weights are fixed.

Long-Term Exchange Rate Behavior

Lastly, in order to help express the concept of Purchasing Power Parity, we now will revisit the concept of the real exchange rate. Recall that the real exchange rate q is defined simply as the ratio of foreign price levels in domestic currency units to the domestic price level. E is the spot exchange rate where the numerator is the domestic currency and the denominator is the foreign currency. P^star is the CPI for foreign households. P is the CPI for domestic households. Q is in terms of goods so it is unitless. Strictly speaking, it is just a relative price ratio. A low q means that the exchange rate is very appreciated in real terms. This means that foreign goods are cheap relative to domestic goods. A high q means the exchange rate is very depreciated in real terms. Foreign goods are very expensive for domestic residents.

The Law of One Price

Purchasing Power Parity has 3 levels of strictness in its no arbitrage assumption. The Law of One Price is probably the most intuitive, but it actually has the most strict assumption that is violated almost in every instance. The Law of One Price states that identical goods should sell at the same price in different countries when expressed in the same currency units. Formally, for each good i, the domestic price is equal to the exchange rate multiplied by the foreign price. For example, if the USD to GBP exchange rate is 1.5, a pair of jeans that costs 20 GBP in the U.K. should sell at 30 USD in the U.S. The reason this parity is so strict is because this exchange rate must hold for all commodity prices: cars, clothing, food, cell phones, Big Macs, etc. A good example of where the Law of One price fails is the price difference of the iphone by country. The price of the iphone varies wildly from country to country. In fact, as far as I know, people make a lot of money going to the U.S., buying iphones, bringing them back to China and selling them for profit. The reason why the Law of One Price is violated so often is due to high fixed costs such as trade barriers or transportation costs. The Law of One Price can also fail if markets are inefficient. If a firm has a monopoly over the cell phone market in a particular country, they can charge higher prices.

Absolute Purchasing Power Parity

The no arbitrage assumption of the Absolute Purchasing Power Parity is much weaker. Rather than have the no arbitrage assumption hold for every commodity, the Absolute Purchasing Power Parity only needs the no arbitrage assumption to hold for identical commodity baskets. That means using the same basket goods with the same weights, the weighted average of the domestic prices is equal to the weighted average of foreign prices. Intuitively, you are allowing prices to violate to violate the Law of One Price, but the Law of One Price should hold on average. The no arbitrage assumption of the Absolute Purchasing Power Parity is therefore that the real exchange rate is equal to 1 or that the relative price levels between countries is the same.

Relative Purchasing Power Parity

Lastly, the weakest version of the Purchasing Power Parity is the Relative Purchasing Power Parity, which only assumes that relative price of identical commodity baskets in different countries remain constant when expressed in the same currency units. In other words, prices for each commodity can be different across countries and the average price levels can also be different across countries, relative Purchasing Power Parity can still hold if q doesn’t change over time. Expressed in terms of percentage change, (q^e - q)/q must equal 0 for relative PPP to hold. This means that q^e must equal q or that future expectations of the real exchange rate is equal to its current levels. The first two versions of the Purchasing Power Parity are too restrictive for modelling purposes. Therefore, we will work with the relative PPP from now on.

Aside: Calculating Changes in Variables

Before moving on to the main insight from combining the long-run model we learned in the last few weeks with the relative PPP model of the commodities market, I have to review some math so that we can express products and ratios of variables as sums and differences of marginal changes in these variables. Of course, these “rule of changes” identities are approximations, but for our purposes to make the math more tractable, we will assume they are equivalent. The trick is to realize that the natural logarithm can transform variables from products to sums and differences. Once we take the natural logarithm on both sides, we can differentiate each variable separately with respect to time to get marginal changes in these variables. Just remember these two rules.

The Monetary Approach to the Exchange Rate

So, we are still working in the framework of the Asset Approach of the Exchange Rate, which is just the Uncovered Interest Rate Parity. The Monetary Approach of the Exchange Rate adds the money market equilibrium in the determination of the nominal interest rate. Last week, our analysis primarily focuses on the short-run and we assumed the existence of sticky prices. We are now simply extending this model to work in the long run now and replacing stick prices with the relative Purchasing Power Parity. The relative PPP basically assumes that any deviations away from the Absolute PPP or the Law of One Price is okay so long as that deviation stays the same in the long run. Put in another way, we assume that the expected rate of depreciation in the real exchange rate is zero.

The Monetary Approach: UIP and Relative PPP

The main insight of the relative PPP is that for the long term, the expected rate of depreciation is equal to the difference between expected inflation at home and expected inflation abroad. This condition is derived from combining the definition of the real exchange rate and relative PPP assumption, and we do this in 2 steps.

1. We first apply the rule of changes to the real exchange rate definition.
2. Second, set the expected rate of depreciation equal to zero, as given in the Relative PPP assumption.

The first expression is the expected rate of depreciation of the nominal exchange rate. The second expression is the expected inflation abroad. The Third expression is the expected inflation at home.

Combining the Uncovered Interest Parity with Relative PPP, we arrive at what is called the Fisher Effect, which states that the interest rate differential between two countries must be equal to the expected inflation differential in the long run. Thus, the Fisher Effect gives central banks another tool to affect nominal exchange rates in the long run. Specifically, central banks can set long run inflation targets. I will expand on this intuition in the next slide.

The Monetary Approach: Money Supply Growth

The long run condition derived from the Relative PPP has another interpretation if we substitute price changes with the money market equilibrium condition. From the money market equilibrium we know that the price level is equal to money supply divided by real money demand. Again, using the rule of changes, the expected inflation can be rewritten as the difference between the expected money supply percentage change and the expected money demand percentage change. Thus combining the money market equilibrium model, the Uncovered Interest Parity model, and the Relative Purchasing Power Parity, we arrive that the expected nominal depreciation equals the difference in monetary tightness between two countries in the long-run.

Policy Change: Accelerated Money Supply Growth

So, the last expression from the previous gives us another interesting policy tool for central banks to make changes to the long run exchange rate. Rather than change the money supply sharply, one could simply change its level gradually over time through inflation targeting. Let us go over the case in which the central bank decides to accelerate the money supply growth. Suppose the money supply initially grows at a rate of pi, which is just the scheduled percent increase in the money supply level. Now, suppose the supply growth is accelerated to pi + delta pi. By the Fisher effect, the nominal interest rate must increase also by delta pi. From the money market equilibrium condition, the price level is equal to the money supply divided by real money demand or the liquidity preference. The liquidity preference is a function of nominal interest rates, so there will be a initial jump in prices to accommodate the immediate fall in the liquidity preference. M^S doesn’t jump since we are only adjusting its growth rate. P will then continue to rise at the rate of pi + delta pi. Thus, in the long run, the nominal exchange rate will trace the price level P.

Accelerated Money Growth: Inflation and Deflation Responses

Recall, that R = r + pi + delta pi. r is the real interest rate, pi is the inflation rate and delta pi is the additional speed of inflation. If Relative PPP holds, the real exchange rate is constant, which implies that nominal exchange rate must trace price levels in the long run.

Single Money Supply Shift: Price and Exchange Rate Responses

Here, R=r since pi is equal to 0. Assuming PPP holds, in both cases E will follow P.

Predictions of the Long-Term Model

Let us summarize the predictions we have made so far. From the Monetary Approach, we have concluded that an increase in money supply, either through large increases in the level or through gradual inflation targeting, will cause a proportional long-term deprecation of the domestic currency. The theme is that if relative PPP holds, then nominal exchange rate will follow the price level.

Because prices are flexible in the long run, a rise in domestic income will decrease the domestic price level (see P = M^S/L). Liquidity demand is higher and so prices fall because there’s fewer dollars in the market chasing more goods. Higher liquidity demand, all else equal, decreases inflation, which decreases the nominal interest rate from the Fisher Effect. Additionally, from the relative PPP, the domestic currency will appreciate.

From the Fisher Effect, we also know that increases in the long-term interest rate is associated with a depreciation of the domestic currency. Notice that this is the opposite prediction of the short-run effect of a monetary supply shock.

The Monetary Approach to the Exchange Rate

Let’s review the long run Monetary Approach model that we just built. The first two ingredients is what you’ve already seen in the short-run Monetary Approach model. We combine the Uncovered Interest Parity with the money market equilibrium. UIP say that that interest rate differentials between countries must be compensated by the expected depreciation rate of the domestic currency. The money market equilibrium states that real money supply must equal real money demand or the liquidity preference. Now, instead of having stick prices in the short run, we are replacing it with relative PPP, since in the long run, prices are fully flexible and doesn’t matter. All that matters is real goods. Relative PPP allows us to connect the commodities market to arrive at the Long-term Condition, which states that the expected depreciation rate must be equal to expected inflation differentials between two countries. This long-term condition only enters our short-run analysis through rational expectations.

The Monetary Approach to the Exchange Rate: Results

From these three ingredients, we arrive at the following predictions.

1. The long-term condition comes from Relative PPP.
2. The short-term condition from the money market equilibrium to help us understand how prices change in the short run.
3. Lastly, the Fisher Effect is derived from combining UIP and relative PPP.

Evidence of the Law of One Price and PPP

The next two sections of the lecture, I want to talk briefly about the empirical evidence regarding the law of one price and the purchasing power parities. Although intuitive, it is not surprising to see both concepts fail in real data. The Economist routinely publishes the Burger Index in order to compare purchasing power between countries. The products of Starbucks and McDonalds are very standardized and thus essentially identical and should sell at one price everywhere. However, this is obviously not true for many reasons already discussed such as market power and relative local input costs. Thus, short run deviations occur all the time.

The question is whether in the long run, does the real exchange rate become stable and revert to a constant long-term level? Statistically testing this requires lots of data. In fact, some economists believe you need more than 100 years to do any sort of credible tests.

Nominal Exchange Rate Moves and Price Level Changes USD/DEM

Even if you have a long history of data, if there are too many changes in the structure of markets, it becomes extremely difficult to argue if the reversions are really due to the purchasing power parity or to something in the short run. This chart, for example, between USD and the German Deutsche Mark, it is difficult to tell if the deviations away from long-run real exchange rate is due to violations of the relative PPP or because of the establishment of the European Union, which is a short-run event.

Evidence of the Law of One Price and PPP

Nonetheless, the general consensus is that the Law of One Price fails, and it fails even within countries. A good example of such a failure is gasoline or petro. Distance explains a large part in price differences. Deviations are most likely to do overshooting of of the nominal exchange rates resulting from sticky prices.

The relative purchasing power parity is satisfied in the long-term only really among industrialized countries. For other countries, I will present you a theory to explain why relative PPP fails through differences in productivity levels between countries for tradable and nontradable goods. Reversion of the real exchange rate to its long-term level is slow and jumpy.

Explanations for the Short-Term Failure of PPP

I present 3 obvious reasons why PPP can fail in the short run. The first is relative to price mark ups due to differences in market power across countries. If a firm has more market power in one country over the other, that firm can charge higher prices. If McDonald’s has more market power in one country, you can expect higher prices for the same Big Mac. Tariffs and trade barriers could also distort prices, if they are passed onto the customers. Similarly, transportation costs works the same way.
Nontrade goods and productivity shocks is a bit subtle, but is an interesting theory to explain why relative PPP fails or more specifically, why the real exchange rate can change in the short run.

Non-Traded Goods and Productivity Change

This theory relies on the premise that there are certain goods in an economy that cannot be traded. Things such as haircuts or massages can’t be traded and can only be consumed locally. If productivity in the tradable goods sector improves faster than in the non-traded good sector, the real exchange rate can theoretically appreciate. I will show you this through an example. Suppose there are only two goods in an economy: cars and non-traded goods. If the productivity of car production improves, the price of non-traded commodities increase. This is because car production produces more cars in the same amount of time. When all production workers exchange the extra output for non-traded goods, their price is bid up.

Examples: Productivity and Non-Traded Goods

Non-Traded Goods and Productivity Change

Real Exchange Rate USD/JPY

The Balassa-Samuelson Effect

The Monetary Approach Beyond PPP

Beyond PPP: Long-Term Exchange Rate Models

Beyond PPP: Real Interest Parity

The Long-Term Monetary Approach