The South African Reserve Bank Governor, Lesetja Kganyago, noted in an interview with Bloomberg on 5 October 2016 that “At the moment we could say that there is undervaluation of the rand, by how much we don’t know because there are so many moving parts.” This comment relates to the important topic of exchange rate determination, for which you obviously need a benchmark that provides the correct value of a currency. One popular benchmark model is provided by the purchasing power parity (PPP) concept, which links exchange rates to the prices of goods in different countries.1

Why should you study the theory of purchasing power parity? First, PPP provides a baseline forecast of future exchange rates that is usually considered whenever it is necessary to forecast future cash flows in different currencies, especially when inflation rates differ across these countries. Consequently, PPP plays a fundamental role in corporate decision making, such as the international location of manufacturing plants, and other international capital budgeting issues. Second, understanding the theory of purchasing power parity is important because deviations from PPP significantly affect the profitability of firms. For example, pricing products internationally, analyzing long-term international contracts, hedging the cash flows of an ongoing international operation, and evaluating the performance of foreign subsidiaries all require an analysis in terms of deviations from PPP. Third, PPP is particularly useful in assessing cost-of-living differences across countries. If you are going to work in a different country, and your salary is denominated in a foreign currency, you would like to know what standard of living you will experience.

As we will see when we look at the data, PPP does not hold very well in the short run. The deviations from the theory are sometimes so large that some economists dismiss the theory, at least as far as the determination of exchange rates is concerned. Nevertheless, for the world’s major currencies, we will also see that PPP has some validity in the long run. It even works reasonably well over shorter horizons, whenever inflation dominates the economic environment.

Because purchasing power parity involves comparing the purchasing power of a money within a country to the purchasing power of that money when spent in a different country, we need to examine how to measure these purchasing powers. When economists convert from monetary magnitudes into units of purchasing power, they say they are converting from nominal units into real units. This chapter also introduces the real exchange rate. You will see that deviations from PPP can also be described as fluctuations in real exchange rates.

To understand these ideas, we first need to discuss price levels and price indexes.

# 1 Price Levels, Price Indexes, and the Purchasing Power of a Currency

## 1.1 The General Idea of Purchasing Power

Economists usually measure the purchasing power of a country’s currency in two steps:

1. First, economists calculate the monetary value, or nominal price, of a typical bundle of consumption goods in a country. We call this the price of the country’s consumption bundle, and it represents the country’s price level. Specifically, the price level is the weighted average of the nominal prices of the goods and services consumed in the economy. The weights of the goods and services usually represent the percentage shares of the goods and services in the consumption bundle. That is, if shoes constitute 1% of the typical consumer’s budget, the price of shoes receives a weight of 0.01 in constructing the weighted average of all prices. When the price level of an economy is rising, inflation is occurring. Conversely, when the price level is falling, deflation is occurring.
2. Second, economists figure out what the purchasing power of the country’s money is - that is, what a unit of currency will actually buy, given the price level in the country. To do this, they take the reciprocal, or inverse, of the price level. Taking the reciprocal of the price level gives the purchasing power of the currency. The purchasing power measures the amount of goods that can be purchased per unit of currency.

## 1.2 Calculating the Price Level

Rather than associate the price level with a country, for notational purposes, we associate the price level with the currency of a country. Hence, for South Africa, we can write the price level as

$\begin{eqnarray*} P_{t, \mathsf{R}} =\sum_{i=1}^{N} w_{i} P_{t, i, \mathsf{R}} \end{eqnarray*}$

where $$P_{t, i, \mathsf{R}}$$ represents the rand price of good $$i$$ at time $$t$$, $$w_{i}$$ represents the weight or consumption share of good $$i$$, and $$P_{t,\mathsf{R}}$$ is the rand price level, the weighted average of the rand prices of the $$N$$ different goods and services.

## 3.2 Why Violations of the Law of One Price Occur

No good or service will literally always satisfy the law of one price. Nevertheless, obvious violations of the law of one price do not necessarily represent unexploited profit opportunities. Why might the prices of goods and services deviate from the law of one price?

### 3.2.1 Tariffs and Quotas

One obvious reason for violations of the law of one price is because countries impose different tariffs on imports, taxes and/or subsidies on exports, quotas on imports and exports, and other non-tariff barriers to trade. Governments often tax international shipments of goods at their borders to generate revenue, and, more likely, to protect their industries.2

For example, Malaysian tariffs on imported fully assembled cars range from 75% on cars with less than 1,800-cc engines to 105% on cars with greater than 3,000-cc engines. These tariffs protect the Malaysian national car companies, Proton and Perodua, from foreign competition and allow those automakers to enjoy a market share of over 50% in Malaysia.

If we measure prices of goods in different currencies with these taxes incorporated into the prices, there will be deviations from the law of one price. For example, with a 100% tariff on imported cars, we should expect the domestic price of imported cars to be twice the world price, where the world price is the exchange rate multiplied by the foreign currency price of the cars. Average tariff rates in many developed countries are quite low, but they are generally much higher in emerging markets. For example, Canada’s average rate is 6.5%, Japan’s is 5.4%, and the U.S. average is 3.5%, whereas Brazil’s is 31.4%, Mexico’s is 36.1%, and India’s is 49%. China is anomalous among emerging markets, with an average tariff of only 10%. Its tariffs are also quite uniform across product categories. Its highest average tariff is 27.4% on sugars and confectionery. In most other countries, there is great dispersion across product categories. For example, Canadian tariffs on clothing average 17.2%, whereas its average tariff on dairy products is 179.7%. Japan has average tariffs of 86.3% on cereals and preparations and 134.7% on dairy products. The average U.S. tariff is 11.4% on clothing and 20.8% on dairy products. Mexico’s highest average tariff is 119.4% on sugars and confectionery, whereas India’s highest rate is 168.9% on oilseeds, fats, and oils.

### 3.2.2 Transaction Costs That Prevent Trade

In theory, all goods and services can potentially be traded across countries, but when transaction costs in international markets are prohibitively large, goods become non-traded. The quintessential example of a non-traded good is a haircut. If the rand price of the euro multiplied by the euro price of Italian haircuts is lower than the rand price of haircuts in South Africa, you might consider getting your hair cut by an Italian barber. But the transaction costs of doing so are simply prohibitive. The true economic cost of the Italian haircut must include the cost of the trip to Italy. Given that this cost is high, when you are at home, you get your hair cut locally, and when you are in a foreign country and need a haircut, you pay the foreign currency price of haircuts. This foreign currency price multiplied by the domestic currency price of foreign currency might be very different from the domestic currency price of your usual haircut.

Notice that a haircut is a service performed by an individual; it is not a commodity that can be shipped from place to place. Of course, if the law of one price for services is violated in one direction by a large enough magnitude for a sufficiently long time, suppliers of these services will migrate from one country to another. If giving haircuts provides a higher real income in South Africa than it does in Italy, for example, barbers will move from Italy to South Africa. But migration is a slow way to equalize wages across countries.

Thus, if wages are not equalized by international trade, we should expect some violations of the law of one price even for traded retail goods because the sale of a retail good in a particular country always involves a certain amount of service. The goods must be shipped to retail outlets, and the retailer must hire someone to sell the goods. Because these services cannot be exported or imported, there can be differences in the prices of retail goods that arise purely from the fact that the purchase of the goods involves the purchase of some non-traded services.

### 3.2.3 Speculation and Contracts

Another reason for deviations from the law of one price in the goods market is that it is often difficult to find a buyer for a particular good at a point in time. In addition, because it takes time to ship goods between countries, a speculative element is introduced into the goods market arbitrage transaction. You may think or expect that you will be able to sell the goods for a profit in a particular country after buying them in a different country, but only if you are able to contract with a buyer at a specified price when you initially purchase the goods will you be sure to earn an arbitrage profit. If no contractual relationship is possible, there is a potential risk that either the market price for the commodity in the country of sale or the exchange rate between the two currencies may change. In such a circumstance, you are speculating that you will make a profit, and the transaction is risky. It is no longer an arbitrage. Of course, many companies sign long-term contracts with suppliers, and one of the parties necessarily bears the foreign exchange risk. Fixed price contracts imply that retail prices will adjust slowly to changes in exchange rates, leading to deviations from the law of one price.

### 3.2.4 Non-Competitive Markets

Deviations from the law of one price also arise when goods are sold in non-competitive markets. Under pure competition, individual buyers and sellers of goods do not influence the prices of the goods. In the absence of pure competition, though, firms may be able to effectively segment markets in different countries. This allows firms to charge different prices in different countries, a practice that is called pricing to market. Segmenting markets is especially easy if the goods are marketed through dealerships established in foreign countries. For example, when the dollar was very strong in the mid-1980s, the dollar prices of European luxury cars in the United States were much higher than the dollar values of the foreign currencies multiplied by the foreign currency prices of the cars in the countries of production. In other words, you could travel abroad, convert your dollars to a foreign currency, and purchase a foreign car much more cheaply than you could purchase the same car in the United States.

Why can’t you arbitrage this situation? The problem is that automobile manufacturers typically only sell one car to an individual foreign buyer who then has to take receipt of the car in the foreign country. Many individuals did take advantage of this opportunity to purchase cars cheaply and simultaneously enjoyed vacations in the foreign countries.

Given such an apparent arbitrage opportunity, ideally you would like to make some real money by purchasing more than just one car: You would like to call the BMW factory in Germany, buy enough cars to establish a dealership in the United States, ship the cars to the United States, and sell the cars for less than their current dollar prices at established BMW dealers. Unfortunately, BMW’s managers will not be willing to sell you more than one car. The managers are happy with their current dealer network and with the profitability of their exports. If they wanted to sell more cars to Americans, they could open more dealerships or ship more cars to their existing U.S. dealers and charge lower dollar prices (versus selling cars to you in Germany so you could profit from the price difference).

### 3.2.5 Sticky Prices

The last reason that there may be observed deviations from the law of one price arises from the fact that the nominal, or money, prices of many goods are set by firms for various lengths of time. Unlike exchange rates and the prices of financial assets such as stocks and bonds, which change continuously, the nominal prices of many goods and services are not changed very often. Economists say the prices of such goods and services are “sticky”.

One reason for sticky prices was noted by Okun (1981), who distinguished between auction goods and customer goods. Auction goods are traded on organized exchanges and are homogeneous commodities, such as wheat, soybeans, gold, and oil. Customer goods are heterogeneous products that are highly differentiated and require marketing through established customer relations. Examples of customer goods include items from refrigerators to automobiles.

Auction goods should be expected to satisfy the law of one price much more consistently than customer goods. One reason has to do with the menu costs related to customer goods. Menu costs refer to the costs that a firm incurs in changing its prices. The classic example is a restaurant that must print up a new menu whenever the manager wants to change prices. If inflation is low, the restaurant may leave its prices unchanged for several months or even years, replacing the menus only as they become too dirty to use. But if inflation is high, the restaurant will find it optimal to print new prices weekly or even daily. If inflation is extreme enough, the restaurant could even adjust prices hourly on a chalkboard. The frequent adjustment of prices due to inflation is costly to consumers, who have no idea from one time to the next how much a particular item will cost.

Menu costs are ubiquitous. They arise whenever the marketing of a good requires the producer or retailer of the good to provide price information to potential customers in advance of the sale of the good, as in customer goods. Whenever a good is sufficiently complex that buyers would like to be able to do comparison shopping, retailers find it in their interests to set prices in advance and to leave their prices fixed for some period of time. Hence, changes in the exchange rate create deviations from the law of one price with regard to customer goods because firms do not continuously adjust the prices of their goods.

## 3.3 How Wide Is the Border?

Because of tariffs, non-competitive markets, sticky prices, and the other sources of deviations we just discussed, the prices of comparable goods differ across cities within a country as well as across countries. Broda and Weinstein (2008) use barcode data - that is, Universal Product Codes (UPCs) - to examine differences in prices of identical goods across cities, both within the United States and across the border in Canada for 2001 to 2004. UPCs provide a unique identifier for hundreds of thousands of different goods, and Broda and Weinstein (2008) can therefore be sure that they are comparing the exact same goods. Their first finding is that the composition of consumption varies systematically with distance and across borders. The share of common goods is 28% between New York and Philadelphia, whereas it is only 18% between New York and Los Angeles. In comparisons between U.S. and Canadian cities, the commonality in consumption bundles falls to 7.5%. Their second finding is that prices of the same good vary substantially across cities. The typical difference, measured as the standard deviation of log price differences, is 22.3% between U.S. cities and 18.7% between Canadian cities. When comparing prices across countries, the typical difference rises to 26.7%. Thus, borders matter, but perhaps less than others had thought.

Early research by Engel and Rogers (1996) examines the failure of the law of one price using U.S. and Canadian data for 23 North American cities and 14 disaggregated commodities, such as men’s and boy’s apparel, footwear, medical care, and other goods. Their statistical analysis indicates that a substantial amount of the variation in the relative prices of similar goods across cities is attributable to the distance between the cities. However, Engel and Rogers (1996) conclude that crossing a border between countries adds as much variability to the relative prices of similar goods as does adding 2,500 miles to the distance between two cities within the same country. Clearly, if Engel and Rogers (1996) are correct, borders between countries, and in particular, the change in currencies that occurs with crossing the border, matter a great deal. Broda and Weinstein (2008) take issue with this finding, arguing that the Engel and Rogers (1996) study, although it uses disaggregated commodities, still suffers from an aggregation bias. When Broda and Weinstein (2008) use individual prices and the Engel and Rogers (1996) methodology, they find that crossing the border adds between 36 and 106 miles to the distance between cities. When they aggregate their individual prices into price indexes, they find results similar to Engel and Rogers (1996).

One problem with the study by Broda and Weinstein (2008) is that its data come from an ACNielsen household survey so that the majority of the goods they examine are in the grocery, drug, and mass merchandise sectors. Thus, it is unclear how robust the results are to the major differentiated products like machine tools, refrigerators, and automobiles. A study of prices of televisions across European countries by Imbs et al. (2010) does find that identical televisions sell for different prices across the eurozone countries.

In the same way the deviations we just discussed affect the law of one price, they likewise affect PPP.

# 4 Describing Deviations from PPP

## 4.1 Overvaluations and Undervaluations of Currencies

Before we look at actual exchange rates and PPP predictions, we first need to discuss some additional terminology. A currency is said to be overvalued if its external purchasing power is greater than its internal purchasing power. An undervalued currency’s external purchasing power is less than its internal purchasing power. Because purchasing power parity makes one prediction for the actual exchange rate between two currencies, if currency $$A$$ is overvalued relative to currency $${B}$$, currency $${B}$$ must be undervalued relative to currency $$A$$.

An easy way to remember which currency is overvalued and which currency is under-valued is to add the phrase “on foreign exchange markets”" to the statement. For example, the rand is “overvalued on foreign exchange markets” if the rand’s external purchasing power is greater than its internal purchasing power.3

Overvaluation of the Rand Implies Undervaluation of the Pound

In this example, we check our ability to manipulate internal and external purchasing powers by verifying that if the rand is overvalued relative to the pound, as in the previous example, the pound must be undervalued relative to the rand.

Recall that the rand price level is R175,000 consumption bundle, the pound price level is £10,000 consumption bundle, and the exchange rate is GBPZAR 16.00. The statement that the rand is overvalued relative to the pound implies that the external purchasing power of the rand is greater than its internal purchasing power. As in the previous example, we calculate the external purchasing power of R1 million in the United Kingdom as

$\begin{eqnarray*} &&\mathsf{R}1,000,000 \times \frac{1}{\mathsf{GBPZAR } 16.00} \times \frac{1}{\text{£} 10,000 / \mathsf{consumption \; bundle}} \\ &&= 6.25 \; \mathsf{consumption \; bundles} \end{eqnarray*}$

This is larger than the internal purchasing power of R1 million in South Africa, which is

$\begin{eqnarray*} \mathsf{R}1,000,000 \times \frac{1}{\mathsf{R}175,000/ \mathsf{consumption \; bundle}} = 5.71 \; \mathsf{consumption \; bundles} \end{eqnarray*}$

Thus, the rand is overvalued on the foreign exchange market. Now, let’s look at the pound. Is the pound overvalued or undervalued on the foreign exchange market? The internal purchasing power of 1,000,000 is

$\begin{eqnarray*} \text{£} 1,000,000 \times \frac{1}{\text{£} 10,000/ \mathsf{consumption \; bundle}} = 100 \; \mathsf{consumption \; bundles} \end{eqnarray*}$

but the external purchasing power of the pound in South Africa is

$\begin{eqnarray*} &&\text{£} 1,000,000 \times \frac{\mathsf{GBPZAR } 16.00}{\text{£}} \times \frac{1}{\mathsf{R} 175,000 / \mathsf{consumption \; bundle}} \\ &&= 91.43 \;\mathsf{consumption \; bundles} \end{eqnarray*}$

Because the internal purchasing power of the pound is greater than its external purchasing power, the pound is undervalued on the foreign exchange market. Hence, the statement that the rand is overvalued relative to the pound is equivalent to the statment that the pound is undervalued relative to the rand.

## 4.2 Predictions Based on Overvaluations and Undervaluations

The logic of overvaluations and undervaluations of currencies leads to predictions of currency depreciation or appreciation. If a currency is overvalued on foreign exchange markets, it must weaken, or suffer depreciation, on the foreign exchange markets if the exchange rate is to return to the prediction of PPP. This weakening, or depreciation, of the currency lowers its external purchasing power and returns the external purchasing power of the currency to its internal purchasing power. Conversely, a currency that is undervalued on foreign exchange markets must strengthen, or experience an appreciation, on foreign exchange markets if its external purchasing power is to increase to equal its internal purchasing power. Of course, apart from currency appreciations and depreciations, differences in the rates of inflation can also re-establish the PPP relationship.

## 4.3 The MacPPP Standard

One criticism of using CPI data is that the consumption bundles of the different countries are not the same. Fortunately, The Economist calculates implied PPP exchange rates for a large number of countries, using a bundle of goods that is the same around the world - namely, a McDonald’s Big Mac sandwich.

There are several advantages to using the Big Mac as an index of prices. First, McDonald’s strives to make the sandwich the same way in all its outlets. Just as with the consumer price level, there are particular weights that McDonald’splaces on each item in the Big Mac, and these weights are the same across countries. Specifically, the commodity bundle is “two all-beef patties, special sauce, lettuce, cheese, pickles, and onions on a sesame seed bun.” Second, McDonald’s uses local suppliers for the goods entering the index, which reduces the role of international transportation costs.

Each spring since 1986, The Economist has had its correspondents sample the prices of Big Macs in local currencies in a large number of countries. Implied PPP exchange rates for various currencies relative to the dollar are calculated by taking the ratio of the local currency price of the Big Mac to its average dollar price in four U.S. cities.

Although the Big Mac PPP standard, called MacPPP, may seem somewhat silly in light of the fact that one cannot transport fresh Big Macs across countries, the deviations of actual exchange rates from the implied PPP values are actually about the same size as those that arise using more conventional consumer price indexes. Also, the degree of overvaluation or undervaluation of particular currencies has been used by The Economist to make a few interesting predictions that have had some accuracy, as you will see.

Table 2 gives MacPPP values for 2016 from The Economist. The first column shows the prices of Big Macs in the local currencies of the countries in which they are sold. For example, the average price of a Big Mac in the United States was $5.04, whereas it cost ¥370.00 in Japan, and R30 in South Africa. The second column gives the dollar price of a Big Mac in the different countries calculated as the local currency price of a Big Mac divided by the exchange rate of local currency per dollar. This is the price that an American travelling in that country might calculate. Because the rand-dollar exchange rate was R14.27/$, the dollar cost of a Big Mac in South Africa was

$\begin{eqnarray*} \frac{(\mathsf{R}30.00/\mathsf{Big Mac})}{(\mathsf{R}14.27/{})}= \ 2.10 \; \mathsf{Big Mac} \end{eqnarray*}$

The most expensive Big Mac for a person paying in U.S. dollars was in Switzerland, where it cost $6.59. The cheapest Big Mac for a dollar purchaser was in Malaysia, where it cost only$1.99.

### 4.3.1 The Implied MacPPP Rates

The third column of Table 2 gives implied PPP exchange rates of the currency versus the dollar. This is the ratio of the local currency price of the Big Mac to the dollar price of the Big Mac in the United States. The fourth column provides the actual exchange rate measured in local currency per dollar. For Big Macs to satisfy the law of one price, implied PPP exchange rates in the third column should equal the actual exchange rates in the fourth column. The fact that they do not indicates that the local currencies are either overvalued or undervalued relative to the dollar.

Big Mac Prices
Exchange Rates
Local
% Under(-) / Over (+)
Currency Dollars PPP Actual Valuation against the dollar
United States dollar 5.04 5.04 1 1
South Africa rand 30 2.1 5.95 14.27 -58.3%
Australia dollar 5.75 4.3 1.14 1.34 -14.59%
Britain pound 2.99 3.94 0.59 0.76 -21.82%
Canada dollar 6 4.6 1.19 1.3 -8.64%
China yuan 18.6 2.79 3.69 6.68 -44.74%
Egypt pound 23 2.59 4.56 8.88 -48.61%
Euro area euro 3.82 4.21 0.76 0.91 -16.55%
Hungary forint 900 3.15 178.57 285.64 -37.48%
Indonesia rupiah 31000 2.36 6150.79 13112.5 -53.09%
Japan yen 370 3.47 73.41 106.73 -31.21%
Malaysia ringgit 8 1.99 1.59 4.03 -60.58%
Mexico peso 44 2.37 8.73 18.54 -52.9%
Norway kroner 46.8 5.51 9.29 8.49 9.33%
Poland zloty 9.6 2.42 1.9 3.97 -51.97%
Russia ruble 130 2.05 25.79 63.41 -59.32%
Saudi Arabia riyal 12 3.2 2.38 3.75 -36.52%
South Korea won 4400 3.86 873.02 1140.95 -23.48%
Switzerland franc 6.5 6.59 1.29 0.99 30.79%
Taiwan dollar 69 2.15 13.69 32.03 -57.26%
Thailand baht 119 3.4 23.61 34.97 -32.47%
Turkey lire 10.75 3.53 2.13 3.04 -29.91%
UAE dirham 13 3.54 2.58 3.67 -29.78%

Table 2: MacPPP in 2016

### 4.3.2 Overvaluations and Undervaluations

The fifth column presents the overvaluation or undervaluation of the local currency in percentage points defined as the percentage appreciation or depreciation of the dollar required to return the actual exchange rate to the implied PPP value. For example, the South African rand is 58.3% undervalued because with the actual exchange rate at R14.27/$, a 58.3% depreciation of the dollar versus the rand would be required to increase the exchange rate to the implied PPP value of R5.95/$. Similarly, the Swiss franc is 30.79% overvalued because with an actual exchange rate at CHF0.99/$, a 30.79% appreciation of the U.S. dollar relative to the Swiss franc would be required to increase the exchange rate to the implied PPP value of CHF1.29/$.

The average of the emerging market valuations relative to the dollar is about -25%, indicating that the average emerging market currency is 25% undervalued versus the dollar. These undervaluations are consistent with the fact that Big Macs also contain some labour, which is less expensive in emerging markets than in the United States. If we take the ratio of the local currency price of the Big Mac in Thailand to the price in Malaysia, we find the PPP prediction of the Thai baht price of the Malaysian ringgit, which is THB14.88/MYR. The actual exchange rate is THB8.68/MYR, implying that the ringgit is only 71% undervalued relative to the baht.

### 4.3.3 Predicting British Heartburn

At this point, you might be feeling that PPP would appear to be well off the mark. Before you decide that the theory is does not work well, it is important to realize that The Economist made surprisingly accurate predictions using its MacPPP standard.

For example, in April 1991, The Economist noted that the implied PPP of the Deutsche mark relative to the British pound was DEM2.58/£. However, the central parity of the two currencies in the European Exchange Rate Mechanism (ERM) was DEM2.95/£ when Britain entered the ERM in October 1990. Given this difference of more than 14% between the implied PPP and the central parity, The Economist noted that the pound was overvalued, and the Deutsche mark was undervalued. The Economist also suggested that the British Treasury would eventually get severe “heartburn” if it tried to defend the actual exchange rate rather than devalue the pound within the ERM.

The logic of the argument is as follows: the ERM required countries to buy their currencies with foreign currencies if the currency weakened by a certain amount relative to the central parity. The maximum deviation of the pound from its central parity with the DEM was DEM2.78/£ (6% below the central parity), which is substantially above the MacPPP value. Thus, if the pound began to weaken in the ERM to correct its overvaluation, the British Treasury would be forced to buy pounds with Deutsche marks. Given the limited amount of DEM that the Bank of England had in its international reserves, the market could force a devaluation of the pound by borrowing pounds and lending Deutsche marks. Investors would expect to profit from the devaluation because the pounds they would borrow would be easy to repay with the appreciated Deutsche marks they would own. The only way this would not occur would be if pound-denominated interest rates were increased sufficiently by the Bank of England to make it unattractive to borrow pounds and attractive for investors to hold pound-denominated assets.

### 5.2.2 Yen - Dollar

The data for the yendollar exchange rates in Table 4 differ somewhat from the previous ones. First, notice that the PPP line is upward sloping from 1973 to 1977, and then it is downward sloping thereafter. Because the PPP line corresponds to $$P(t, \yen)/ P (t,\)$$, the positive slope indicates that Japanese inflation was higher than U.S. inflation during the first part of the sample, whereas the negative slope of the ratio of the price levels indicates that Japanese inflation was lower than U.S. inflation during the second part of the sample.

The data on the ¥/$rate indicate that the dollar was undervalued in October 1978 by 39%, with the implied PPP rate at ¥253/$ and the actual rate at ¥182/$. By February 1985, the dollar was 26.4% overvalued. Once the dollar peaked in strength in 1985, though, it began to depreciate relative to the yen. At the end of the sample in January 2010, at a PPP value of ¥111.6/$, the dollar was undervalued relative to the yen by 20% because the actual exchange rate was ¥91.11/$. In other words, those converting dollars into yen for expenditures in Japan found that their purchasing power was quite a bit lower than they were used to in the United States. Figure 3: Actual JPY/USD and PPP Exchange Rates ### 5.2.3 Canadian Dollar - U.S. Dollar Figure 5 presents data for countries that share a common border, and here PPP works slightly better. The data for the Canadian dollar versus the U.S. dollar indicate that the maximal deviation from PPP was a 29.4% overvaluation of the U.S. dollar relative to the Canadian dollar in February 2002. The overall flatness of the PPP line indicates that although U.S. and Canadian inflation rates were not identical period by period, they averaged essentially the same value over the sample period. Thus, the nominal weakening of the Canadian dollar during the 1990s led directly to a deviation from PPP, but by June 2004, the Canadian dollar had strengthened to restore PPP. The subsequent strengthening of the Canadian dollar returned the currencies to parity, which implies a 10% undervaluation of the U.S. dollar. Figure 4: Actual CAD/USD and PPP Exchange Rates ### 5.2.4 South African Rand - U.S. Dollar Figure 2 shows the relationship between a developed and emerging market economy. In this case it would appear as if PPP does not provide a satisfactory explanation for the behaviour of the exchange rate, as the actual exchange rate for the South African rand deviates from the measure of PPP for a protracted period of time (i.e. more than twenty years). While it would appear that the two variables have followed a similar trend until the late 1990s, this is no longer the case over the latter subsample. Figure 5: Actual ZAR/USD and PPP Exchange Rates # 6 Explaining the Failure of Absolute PPP Figures 5 and 2 show that there are large, persistent deviations of actual exchange rates from the predictions of absolute PPP. Because PPP is ultimately based on the law of one price, we know that anything that causes deviations from it can also cause deviations from PPP. As we saw, the factors causing deviations from the law of one price are quite numerous, including tariffs, quotas, and transaction costs. But there are other factors that cause deviations from absolute PPP. ## 6.1 Changes in Relative Prices Changes in the relative prices of goods can cause deviations from PPP if price indices do not have the same weights across countries. To see this, suppose all goods are traded and assume that the prices of all goods satisfy the law of one price. Now, assume that tastes differ across countries so that expenditure shares on goods differ and let the price levels reflect the differences in consumption bundles. Typically, the residents of a country consume a larger share of the goods and services produced in that country than of imported goods and services. Consequently, the price indexes of each country will have a larger weight on goods produced at home and a smaller weight on imported goods. Changes in the relative prices will then lead to deviations from PPP. ### 6.1.1 A Burgers-and-Sushi World Consider a simple example of the problem of changes in relative prices. Suppose there are only two countries, the United States and Japan, and to keep things really simple, assume that people consume only two goods, hamburgers and sushi. Let the United States produce only hamburgers, with a dollar price of$10, and let Japan produce only sushi, with a yen price of ¥5,000. Assume the exchange rate is ¥100/$. The U.S. price level will put a weight of 60% on the dollar price of hamburgers because U.S. consumers prefer hamburgers to sushi and a weight of 40% on the dollar price of sushi (the yen price of sushi divided by the yen-dollar exchange rate). Thus, the U.S. price level will be $\begin{eqnarray*} P(t,)=0.60 \times \ 10 + 0.40 \times \frac{\yen 5,000}{\yen 100/\} = \26 \end{eqnarray*}$ Now, suppose the Japanese price level places a weight of 35% on the yen price of ham- burgers (the dollar price of hamburgers multiplied by the yen-dollar exchange rate) because Japanese prefer sushi and a weight of 65% on the yen price of the sushi. Thus, the Japanese price level will be $\begin{eqnarray*} P(t,\yen)\ = 0.35 \times (\yen 100/ \) \times \10 + 0.65 \times \yen 5,000=\yen 3,600 \end{eqnarray*}$ The ratio of the price level in Japan to the price level in the United States is $\begin{eqnarray*} \frac{P(t,\yen)}{P(t,)} = \frac{\yen 3,600}{\ 26} = \yen 138.5/ \ \end{eqnarray*}$ Thus, even though the law of one price is satisfied in each country, the dollar appears to be 38.5% undervalued on the foreign exchange market. The problem is the difference in consumption shares. You should convince yourself that if the consumption shares were the same in both countries and if the law of one price held, then PPP would be satisfied. It is now straightforward to understand how a change in relative prices can cause a change in the deviation between the exchange rate and measured PPP even though all goods are traded and all prices satisfy the law of one price. Suppose that there is a shift in demand away from U.S. hamburgers and toward Japanese sushi. With no changes in the supplies of the two goods, the relative price of sushi must rise both in the United States and in Japan. The increase in the relative price can be accomplished by an appreciation of the yen relative to the dollar, with no change in the dollar price of hamburgers and no change in the yen price of sushi. Suppose the yen appreciates to ¥90/$. With unchanged dollar prices of hamburgers and yen prices of sushi, the appreciation of the yen decreases the yen price of hamburgers in Japan and increases the dollar price of sushi, thereby making sushi relatively more expensive in both Japan and the United States. The U.S. price level will now be

$\begin{eqnarray*} P(t,)=0.60 \times \ 10 + 0.40 \times \frac{\yen 5,000}{\yen 90/ \} = \ 28.22 \end{eqnarray*}$

and the Japanese price level will now be

$\begin{eqnarray*} P(t, \yen)\ = 0.35 \times (\yen 90/\) \times \ 10 + 0.65 \times \yen 5,000 =\yen 3,565 \end{eqnarray*}$

The ratio of the price level in Japan to the price level in the United States is

$\begin{eqnarray*} \frac{P(t,\yen)}{P(t,)}=\frac{\yen 3,565}{\ 28.22}=\yen 126.33/\ \end{eqnarray*}$

Thus, even though the law of one price continues to be satisfied in each country, the dollar now appears to be 40.4% undervalued on the foreign exchange market because (126.33-90)/90=0.404. The shift in demand toward Japanese goods and away from U.S. goods causes the apparent undervaluation of the dollar to increase, but there is no opportunity for a goods market arbitrage.

Similar problems with absolute PPP arise when there are changes in the relative prices of traded and non-traded goods. Earlier in the chapter, we noted that when transaction costs are prohibitive, goods become non-traded. Since these goods are also included in the consumption bundles of individuals in the different countries, the prices of non-traded goods affect the price levels of the countries. Changes in the relative prices of traded and non-traded goods in two countries will cause deviations from absolute PPP that do not represent arbitrage opportunities.

### 6.2.1 Housing

Housing and other types of real estate are particularly important non-traded goods. If the price of housing in a country rises, with the price of other goods held constant, the relative price of housing rises, and the internal purchasing power of the country’s money falls. Nevertheless, there need be no effect on the exchange rate. Consequently, after an increase in the relative price of housing in a country, the currency of that country will appear more overvalued (or less undervalued) on foreign exchange markets than before the increase in housing prices.

### 6.2.2 Technological Change

Why would the relative prices of non-traded goods rise compared to traded goods? Differential rates of technological change, which are also called productivity improvements, provide one answer. As the personal computer industry has aptly demonstrated over the past 25 years, improvements in technology in a competitive market force the prices of PCs to fall rapidly over time. The same is true of goods in other markets. If technology increases faster in traded goods industries than in non-traded goods industries, which is reasonable to expect if non-traded goods are services, we would expect that the relative price of non-traded goods would rise over time. This effect, known as the Harrod-Balassa-Samuelson effect, can impart a systematic bias in PPP calculations.5

## 6.3 PPP Deviations and the Balance of Payments

Our last explanation for deviations from absolute PPP is that they arise as equilibrium changes in the relative prices of goods across countries in a process that involves the balance of payments. The balance of payments of a country represents the aggregate amounts of goods and services that are bought and sold between the residents of a country and the rest of the world. Hence, when a currency is overvalued relative to a PPP calculation, the external purchasing power of that currency increases, which shifts the nation’s expenditures from domestic to foreign goods. This weakens the competitive position of domestic firms relative to foreign firms.

# 7 Comparing Incomes Across Countries

Before we leave the subject of absolute PPP, we want to examine one particularly important use of PPP data: comparing nominal incomes across countries. Let’s consider an extended example to make things easier.

## 7.1 Comparing Incomes in New York and Tokyo

Suppose you are considering working in New York for Citigroup and have been offered $100,000 per year. Goldman Sachs has also offered you a job working in Japan for the next 2 years at ¥15,000,000 per year. Suppose you are indifferent between living in New York and living in Tokyo. Either sounds okay to you. The question then becomes, which job makes you better off financially - working in New York or Tokyo? ### 7.1.2 A Naive Calculation You might be tempted to make the decision by simply comparing the dollar value of the yen salary offer to the dollar salary of your New York offer by converting the yen salary into dollars at the current exchange rate. If the current exchange rate is ¥100/$, the ¥15,000,000 is worth $150,000. If you used this approach, you would accept the job offer to work in Japan. ### 7.1.3 Incorporating Purchasing Power By now, you should realize that this is a naive calculation because if you must live and work in Japan, you will not purchase goods with$150,000. You will spend your yen salary to purchase goods and services that are sold in Japan and priced in yen, just as you would spend your dollar salary in New York to buy goods and services that are priced in dollars. To do a proper salary comparison, you must determine the command over goods and services that you will have based on the purchasing powers of the nominal salaries in each country. If you knew the price level in the United States, $$P_{t,\}$$, you could divide your $100,000 salary offer by the price level to determine its command over goods and services. Similarly, if you knew the price level in Japan, $$P_{t,\yen}$$, you could divide your ¥15,000,000 salary by the Japanese price level to determine its command over goods and services in Japan. From a financial viewpoint, you would be indifferent between working in New York and working in Japan if the purchasing powers of your two salaries were the same - that is, if $\begin{eqnarray*} \frac{(\ 100,000 \; \mathsf{salary})}{P_{t,}} = \frac{(\yen 15,000,000 \; \mathsf{salary})}{P_{t,\yen}} \end{eqnarray*}$ ### 7.1.4 Working with the PPP Rate What if the prices levels are not available, but the PPP exchange rate is available? Multiplying on both sides of the previous equation by the price level in Japan gives $\begin{eqnarray*} (\ 100,000 \; \mathsf{salary}) \times \frac{P_{t,\yen}}{P_{t,}} =\yen 15,000,000 \; \mathsf{salary} \end{eqnarray*}$ This equation states that you would be indifferent between the two jobs if your dollar salary multiplied by the PPP exchange rate, $$[P_{t,\yen}/P_{t,\}]$$, equals your yen salary offer. Suppose the PPP exchange rate is ¥160/$. To achieve the same purchasing power in Japan as you would have in the United States, you need a salary of

$\begin{eqnarray*} (\yen 160/ \) \times \ 100,000 = \yen 16,000,000 \end{eqnarray*}$

But your offer is only ¥15,000,000.

Alternatively, if you divide your yen salary offer by the PPP exchange rate of yen per dollar, you get a dollar equivalent of your yen salary. Then, when you determine your command over goods and services by mentally dividing the dollar equivalent salary by the dollar price level, the resulting units are consumption bundles in Japan. The implied dollar salary is

$\begin{eqnarray*} \frac{\yen 15,000,000}{\yen 160/ \ }= \ 93,750 \end{eqnarray*}$

This calculation states that the purchasing power you would have in Japan from a ¥15,000,000 salary is equivalent to the purchasing power that you would have in the United States from a $93,750 salary. As you can see, if the PPP exchange rate were ¥160/$, you should turn down the offer to work in Japan or demand a higher yen salary.6

Given the occasional large percentage differences between actual exchange rates and implied PPP exchange rates that we saw in Figures 1 through 2 converting a foreign currency denominated salary into dollars using an actual exchange rate versus a PPP exchange rate will sometimes produce quite substantively different results. The numerical example in this section demonstrates that if the dollar is undervalued relative to the foreign currency, the dollar-equivalent salary of a foreign currency offer is lower when you use the PPP exchange rate rather than the actual exchange rate.

Conversely, whenever the dollar is overvalued relative to a foreign currency, converting a foreign currency salary into dollars with the actual exchange rate will result in a smaller dollar salary than if the PPP exchange rate were used. However, although your salary in dollars will seem low, the dollar prices of goods and services purchased in the country will also seem quite low relative to comparable items in the United States. In such cases, dividing by the implied PPP exchange rate again provides a better estimate of the standard of living that you will face in the country, were you to be stationed there and paid in the foreign currency. This is particularly important if you are considering job offers in emerging market countries, whose currencies often appear to be undervalued relative to the dollar.

## 7.2 Comparing GDPs Using PPP Exchange Rates

Table 3 presents a comparison of gross domestic product (GDP) per capita for selected countries, reported exchange rates in the first column and PPP exchange rates in the second column. The data is obtained from the World Economic Outlook, which is published by International Monetary Fund.

OECD Country
In U.S. Dollars, Based on
In U.S. Dollars, Based
Market Exchange Rates on PPP Exchange Rates
Australia 46777 55421
Austria 46913 51349
Belgium 43214 46366
Czech Republic 34139 23210
Denmark 46771 59999
Finland 41926 49897
France 41029 42473
Germany 47034 47786
Greece 26643 20572
Hungary 29439 17296
Iceland 49372 68794
Ireland 72000 76911
Italy 35115 33353
Japan 39795 41021
Korea 37542 31937
Luxembourg 95033 112846
Mexico 18435 9858
Netherlands 50878 53016
New Zealand 35965 41989
Norway 67021 79733
Poland 29474 15630
Portugal 28966 23311
Slovak Republic 32333 20155
South Africa 12109 6331
Spain 36278 30631
Sweden 47224 53004
Switzerland 57386 82412
Turkey 23922 8507
United Kingdom 40858 42310
United States 56566 64767

Table 3: GDP per Capita for selected countries using Exchange Rates and PPP values

The last row indicates that the United States produced final goods and services that were worth $56,566 per person. When the currency of a country is stronger in foreign exchange markets than its PPP exchange rate, the dollar value of the country’s GDP per capita when measured by current exchange rates is larger than when measured by PPP exchange rates. Notice that the dollar value of South Africa’s GDP falls from$12,109 per capita in the first column to $6,331 in the second column. The fact that the euro strengthened considerably relative to the dollar between 2004 and 2008 and was overvalued relative to PPP leads the European countries to have higher incomes measured at actual exchange rates rather than in PPP over this period. Conversely, because non-traded goods are relatively inexpensive in emerging markets, their PPP exchange rates typically imply that their currencies are stronger versus the dollar than the actual exchange rates imply. Thus, the dollar value of the country’s GDP per capita when measured by PPP exchange rates is larger than when measured by actual exchange rates. The discussion in this section about comparing incomes across countries strongly suggests that the PPP exchange rates are the appropriate ones to use when comparing standards of living across countries. # 8 Relative Purchasing Power Parity The previous section discusses reasons why absolute PPP generally will not hold. In addition, the examples that have been provided demonstrate that currencies are often substantially undervalued and overvalued relative to the predictions of absolute PPP calculated using CPI data. Another form of PPP, called relative purchasing power parity, takes market imperfections into account, and it acknowledges that because of these imperfections, a consumption bundle will not necessarily have the same value from country to country. However, according to the theory of relative PPP, exchange rates adjust in response to differences in inflation rates across countries to leave the differences in purchasing power unchanged over time. If the percentage change in the exchange rate just offsets the differential rates of inflation, economists say that relative PPP is satisfied. To help you better understand these concepts, let’s begin with a numerical example. The Warranted Change in the Exchange Rate Suppose, as in the previous example, that the price level in the United States is initially R175,000/ S.A. consumption bundle, the price level in the United Kingdom is initially £10,000 / U.K. consumption bundle, and the exchange rate is R16.00/£. We determined that absolute PPP is violated. The pound is undervalued on foreign exchange markets because the implied PPP exchange rate of $\begin{eqnarray*} \frac{\mathsf{R}175,000}{\text{£}10,000}=\mathsf{R}17.50/\text{£} \end{eqnarray*}$ is not equal to the actual exchange rate. The pound would have to strengthen relative to the rand by 9.38% to correct its undervaluation because $\begin{eqnarray*} \frac{\mathsf{R}17.50/\text{£}}{\mathsf{R}16.00/\text{£}}=1.0938 \end{eqnarray*}$ Now, suppose that during the following year, the rate of S.A. inflation is 6%, and the rate of U.K. inflation is 4%. From the definition of inflation, we know that the new price level in South Africa is 6% higher: $\begin{eqnarray*} \mathsf{R}175,000 \times 1.06= \mathsf{R}185,500 \end{eqnarray*}$ and the new price level in the United Kingdom is 4% higher: $\begin{eqnarray*} \text{£}10,000 \times 1.04 = \text{£}10,400 \end{eqnarray*}$ Hence, the new implied PPP exchange rate is $\begin{eqnarray*} \frac{\mathsf{R}185,500}{\text{£}10,400}= \mathsf{R}17.84/\text{£} \end{eqnarray*}$ If the pound remains 9.38% undervalued on the foreign exchange market, as it was before, the pound must strengthen relative to the rand for relative PPP to be satisfied. The new exchange rate should equal $\begin{eqnarray*} S_{t+1,\mathsf{R} /\text{£}}=\frac{ \mathsf{R} 17.84/\text{£}}{1.0938}= \mathsf{R} 16.31/\text{£} \end{eqnarray*}$ This keeps the ratio of the PPP exchange rate to the actual exchange rate at 1.0938, as before. The pound appreciates relative to the rand by 1.92% because the actual exchange rate moves to R16.31/£ from R16.00/£, and $\begin{eqnarray*} \frac{\mathsf{R} 16.31/\text{£}}{\mathsf{R}16.00/\text{£}}=1.0912=1- (-0.192) \end{eqnarray*}$ Notice also that 1.0912 is the ratio of 1 plus the S.A. rate of inflation divided by 1 plus the U.K. rate of inflation because $\begin{eqnarray*} \frac{1.06}{1.04}=1.0192 \end{eqnarray*}$ Intuitively, the pound is losing purchasing power over goods and services due to U.K. inflation of 4% per year, and the rand is losing purchasing power over goods and services due to S.A. inflation of 6% per year. Hence, a 1.92% appreciation of the pound relative to the rand is therefore required to make the loss of the pound’s external purchasing power equal to the loss of its internal purchasing power. ## 8.1 A General Expression for Relative PPP The example in the preceding section demonstrates that relative PPP requires that 1 plus the rate of appreciation of the pound relative to the rand should equal 1 plus the rate of inflation in South Africa divided by 1 plus the rate of inflation in the United Kingdom. Relative PPP is derived from the following economic reasoning: Inflation lowers the purchasing power of money. If the amount of inflation in the foreign country differs from the inflation rate in the domestic country, a change in the nominal exchange rate to compensate for the differential rates of inflation is warranted so that the loss of internal purchasing power due to domestic inflation equals the loss of external purchasing power due to foreign inflation and the change in the exchange rate. If the change in the exchange rate satisfies this warranted change, relative PPP is satisfied.7 ### 8.1.1 A Symbolic Representation of Relative PPP In general symbolic terms, let $$s_{t+1,\mathsf{DC} / \mathsf{FC}}$$ denote the percentage rate of change of the domestic currency (denoted $$\mathsf{DC}$$) per unit of foreign currency (denoted $$\mathsf{FC}$$) from time $$t$$ to $$t+1$$, and let $$\pi_{t+1, \mathsf{DC}}$$ and $$\pi_{t+1,\mathsf{FC}}$$ represent the corresponding rates of domestic and foreign inflation, respectively; then relative PPP requires that $\begin{eqnarray} 1+s_{t+1,\mathsf{DC} / \mathsf{FC}}=\frac{1+\pi_{t+1,\mathsf{DC}}}{1+\pi_{t+1,\mathsf{FC}}} \tag{8.1} \end{eqnarray}$ If we subtract 1 from each side of Equation (8.1) and place terms over a common denominator, we get $\begin{eqnarray} s_{t+1,\mathsf{DC} / \mathsf{FC}} = \frac{\pi_{t+1,\mathsf{DC}}-\pi_{t+1,\mathsf{FC}}}{1+\pi_{t+1,\mathsf{FC}}} \tag{8.2} \end{eqnarray}$ Equation (8.2) states that the rate of appreciation of the foreign currency relative to the domestic currency is equal to the difference between the domestic rate of inflation and the foreign rate of inflation divided by 1 plus the foreign rate of inflation. Since $$[1+\pi_{t+1, \mathsf{FC}}]$$ is often close to 1 if the foreign inflation rate is low, some presentations of relative PPP ignore this term in the denominator of equation (8.2) and state that relative PPP requires equality between the rate of appreciation of the foreign currency relative to domestic currency and the difference between the domestic and foreign inflation rates. Equation (8.2) indicates that this statement is an approximation, albeit a pretty good one if the foreign inflation rate is small. Of course, since the graphs in the previous figures indicate that deviations from absolute PPP change over time, relative PPP also does not hold in the data. The rate of change of the exchange rate does not equal the inflation differential between two currencies. ## 8.2 Relative PPP with Continuously Compounded Rates of Change The discussion of relative PPP suggests ignoring the denominator of Equation (8.2) as a reasonable approximation. Note that if we measure the forward premium on the foreign currency and the domestic and foreign interest rates in continuously compounded terms, it is exactly correct to state that interest rate parity requires equality between the forward premium on the foreign currency and the interest differential between the domestic and foreign interest rates. Analogously, if we measure the rate of appreciation of the foreign currency relative to the domestic currency and the domestic and foreign inflation rates as continuously compounded rates of change, relative PPP requires equality between the rate of appreciation of the foreign currency and the difference between the domestic and foreign rates of inflation. We demonstrate this equality by using the dollar-pound exchange rate and the respective rates of inflation. If there are obstacles to international trade that prevent absolute PPP from holding, we can introduce a factor $$k$$ such that the internal purchasing power of the money equals $$k$$ times the external purchasing power of the money: $\begin{eqnarray} \frac{1}{P_{t,}}=k \times \frac{1}{S_{t,{} / \text{£}}} \times \frac{1}{P_{t,\text{£}}} \tag{8.3} \end{eqnarray}$ where $$S_{t, \ / \text{£}}$$ denotes the actual exchange rate and not the implied PPP value. By rearranging Equation (8.3), we have $\begin{eqnarray} \frac{S_{t,{} / \text{£}} \times P_{t,\text{£}}}{P_{t,{}}}=k \tag{8.4} \end{eqnarray}$ If the amount of overvaluation or undervaluation of the dollar relative to the pound is the same at time $$t+1$$, we have $\begin{eqnarray} \frac{S_{t+1, \ / \text{£}} \times P_{t+1,\text{£}}}{P_{t+1,\}}=k \tag{8.5} \end{eqnarray}$ Hence, the ratio of equation (8.4) to equation (8.5) is $\begin{eqnarray} \frac{S_{t+1, \ / \text{£}}}{S_{t,{} / \text{£}}} \times \frac{P_{t+1,\text{£}} / P_{t,\text{£}}}{P_{t+1, \} / P_{t,{}}}=1 \tag{8.6} \end{eqnarray}$ Now, if $$s_{t+1, \ / \text{£}}$$ denotes the continuously compounded rate of change of the dollar-pound exchange rate over the time interval from $$t$$ to $$t+1$$, then $$[S_{t+1,\ / \text{£}} / S_{t,\ / \text{£}}]= \exp[s_{t+1,\ / \text{£}}]$$. Similarly, let $$\pi_{t+1,\text{£}}$$ and $$\pi_{t+1,\}$$ now denote the continuously compounded rates of inflation over the time interval from $$t$$ to $$t+1$$ in the pound and dollar prices of goods, respectively. Then, $$P_{t+1,\text{£}} / P_{t,\text{£}}=\exp[\pi_{t+1,\text{£}}]$$, and $$P_{t+1,\} / P_{t,\}=\exp[\pi_{t+1, \}]$$. Substituting these exponential expressions into Equation (8.6) gives $\begin{eqnarray} \frac{\exp[s_{t+1, \ / \text{£}}] \times \exp[\pi_{t+1,\text{£}}]}{\exp[\pi_{t+1,\}]}=1 \tag{8.7} \end{eqnarray}$ If we apply the rules for taking natural logarithms to equation (8.7), we find $\begin{eqnarray*} s_{t+1,\ / \text{£}}+\pi_{t+1,\text{£}}-\pi_{t+1,\}=0 \end{eqnarray*}$ or, rearranging terms, we find $\begin{eqnarray} s_{t+1,\ / \text{£}}=\pi_{t+1,\} - \pi_{t+1,\text{£}} \tag{8.8} \end{eqnarray}$ Equation (8.8) expresses relative PPP in its continuously compounded version. The rate of appreciation of the pound versus the dollar equals the rate of dollar inflation minus the rate of pound inflation when all the rates of change are continuously compounded. # 9 The Real Exchange Rate While discussions of purchasing power parity have been around since the early twentieth century, the concept of the real exchange rate is much newer, as it entered the jargon of international finance in the late 1970’s. Nonetheless, the real exchange rate is important because it influences the competitiveness of firms. Here, we introduce the concept of the real exchange rate. ## 9.1 The Definition of the Real Exchange Rate The real exchange rate, say, of the rand relative to the euro, will be denoted $$\text{RS}_{t, {{\mathsf{R}}} / \mathsf{€}}$$. It is defined to be the nominal exchange rate multiplied by the ratio of the price levels: $\begin{eqnarray} \text{RS}_{t,\mathsf{R} / \mathsf{€}} =\frac{S_{t,{{\mathsf{R}}}/ \mathsf{€}} \times P_{t,\mathsf{€}}}{P_{t,{{\mathsf{R}}}}} \tag{9.1} \end{eqnarray}$ Notice that the real exchange rate would be 1 if absolute PPP held because the nominal exchange rate, $$S_{t,\mathsf{R} / \mathsf{€}}$$ would equal the ratio of the two price levels, $$P_{t,\mathsf{R}}/P_{t,\mathsf{€}}$$. Similarly, if absolute PPP is violated, the real exchange rate is not equal to 1. Also, the real exchange rate is constant if relative PPP holds, as we see in the next example. Since the real exchange rate is not equal to 1 in the following example, absolute PPP does not hold. However, relative PPP does hold in the following example, as the deviations from absolute PPP are constant in percentage terms. This keeps the real exchange rate constant. If deviations from absolute PPP vary over time, relative PPP does not hold, and the real exchange rate fluctuates. Essentially, the real exchange rate describes deviations from absolute PPP, and changes in the real exchange rate represent deviations from relative PPP. A Constant Real Exchange Rate Suppose that the S.A. price level is initially R150,000/ S.A. consumption bundle and the price level in Europe is initially €11,000/ European consumption bundle. With the nominal exchange rate equal to R13.00/€, the real exchange rate equals $\begin{eqnarray*} \text{RS}_{t,\mathsf{R} /\mathsf{€}} =\frac{\mathsf{R} 13.00/ \mathsf{€} \times \mathsf{€}11,000}{\mathsf{R} 150,000}=0.9533 \end{eqnarray*}$ Suppose that over the next year, there is 8% inflation in South Africa, there is 3% inflation in Europe, and the nominal exchange rate changes so that relative PPP is satisfied. Then, as Equation (8.1) indicates, the new nominal exchange rate should be $\begin{eqnarray*} S_{t,\mathsf{R} / \mathsf{€}} =\frac{\mathsf{R} 13.00/ \text{€} \times 1.08}{1.03}= \mathsf{R} 13.63 / \text{€} \end{eqnarray*}$ The euro strengthens by 4.85%. With 8% S.A. inflation, the new S.A. price level is R162,000=R150,000 $$\times$$ 1.08, and with 3% European inflation, the new European price level €11,330 = €11,000 $$\times$$ 1.03. The new real exchange rate is the same as it was before, because $\begin{eqnarray*} \text{RS}_{t+1,\mathsf{R} / \mathsf{€}} =\frac{\mathsf{R} 13.63 / \mathsf{€} \times \mathsf{€} 11,330}{\mathsf{R} 162,000} = 0.9533 \end{eqnarray*}$ ## 9.2 Real Appreciations and Real Depreciations Of course, when the concept of the real exchange rate took hold, people naturally began to refer to real appreciations and real depreciations of different currencies. The concepts of real appreciations and real depreciations are useful because they help us describe real exchange risk. Previously, we defined the percentage rate of change in the nominal exchange rate of the rand relative to the pound by $$s_{t+1,\mathsf{R} /\text{£}}=[S_{t+1,\mathsf{R} /\text{£}}-S_{t,\mathsf{R} /\text{£}}]/S_{t,\mathsf{R} /\text{£}}$$. If the percentage change in $$S_{t,\mathsf{R} /\text{£}}$$ was positive, we called it a nominal appreciation of the pound. We also defined a nominal appreciation of the pound by $$a_{t+1,\mathsf{R} /\text{£}}=s_{t+1,\mathsf{R} /\text{£}}$$, when $$s_{t+1,\mathsf{R} /\text{£}}>0$$. Similarly, we defined a nominal depreciation of the pound by $$d_{t+1,\mathsf{R} /\text{£}}=-s_{t+1,\mathsf{R} /\text{£}}$$, if $$s_{t+1, \mathsf{R}/\text{£}}<0$$. For example, if the percentage change in the rand-pound exchange rate was -5%, we said that the pound depreciated by 5%. ### 9.2.1 The Percentage Change in the Real Exchange Rate We can define the percentage rate of change in the real exchange rate by $\begin{eqnarray} \text{rs}_{t+1,\mathsf{R} /\text{£}}=\frac{\text{RS}_{t+1,\mathsf{R} /\text{£}}-\text{RS}_{t,{{\mathsf{R}}}/\text{£}}}{\text{RS}_{t,{{\mathsf{R}}}/\text{£}}} \tag{9.2} \end{eqnarray}$ If the right-hand side of Equation (9.2) is positive, we have a real appreciation of the pound (depreciation of the rand): $\begin{eqnarray*} \text{ra}_{t+1,\mathsf{R} /\text{£}}=\text{rs}_{t+1,\mathsf{R} /\text{£}}, \;\; \mathsf{if} \;\; \text{rs}_{t+1,\mathsf{R} /\text{£}}>0 \end{eqnarray*}$ and if the real exchange rate falls, we have a real depreciation of the pound (appreciation of the rand): $\begin{eqnarray*} \text{rd}_{t+1,\mathsf{R} /\text{£}} = -\text{rs}_{t+1,\mathsf{R} /\text{£}}, \;\; \mathsf{if} \;\; \text{rs}_{t+1,\mathsf{R} /\text{£}}<0 \end{eqnarray*}$ Because the ratio of the new real exchange rate to the old real exchange rate equals 1 plus the rate of change of the real exchange rate, we have $\begin{eqnarray} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]=\frac{\text{RS}_{t+1,\mathsf{R}/\text{£}}}{\text{RS}_{t,{{\mathsf{R}}}/\text{£}}} \tag{9.3} \end{eqnarray}$ To understand what leads to real appreciations and depreciations, we must substitute the definition of the real exchange rate from Equation (9.1) into Equation (9.3): $\begin{eqnarray} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]=\frac{[S_{t+1,\mathsf{R} /\text{£}} \times P_{t+1,\text{£}}/P_{t+1,\mathsf{R}}]}{[S_{t,{{\mathsf{R}}}/\text{£}} \times P_{t,\text{£}}/P_{t,{{\mathsf{R}}}}]} \tag{9.4} \end{eqnarray}$ Now, we group the exchange rate terms, the pound price-level terms, and the rand price-level terms together to get the following: $\begin{eqnarray*} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]= \frac{[S_{t+1,\mathsf{R} /\text{£}}/S_{t,{{\mathsf{R}}}/\text{£}}] \times [P_{t+1,\text{£}}/P_{t,\text{£}}]}{[P_{t+1,\mathsf{R}}/P_{t,{{\mathsf{R}}}}]} \end{eqnarray*}$ After substituting the definitions of the ratios of variables at time $$t+1$$ to those at time $$t$$, we find $\begin{eqnarray} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]=\frac{[1+s_{t+1,\mathsf{R} /\text{£}}] \times [1+\pi_{t+1,\text{£}}]}{[1+\pi_{t+1,\mathsf{R}}]} \tag{9.5} \end{eqnarray}$ The left-hand side of Equation (9.5) is 1 plus the percentage rate of change of the real rand-pound exchange rate. The right-hand side equals 1 plus the percentage rate of change of the nominal rand-pound exchange rate multiplied by 1 plus the U.K. rate of inflation, $$\pi_{t+1, \text{£}}$$, divided by 1 plus the S.A. rate of inflation, $$\pi_{t+1,\mathsf{R}}$$. ### 9.2.2 What leads to Real Appreciations and Depreciations Because the real exchange rate is composed of three variables that can all move simultaneously, many combinations of changes lead to a real appreciation of the pound. The three basic movements are as follows: 1. An increase in the nominal exchange rate, R/£, that is a nominal appreciation of the pound, holding the rand prices and pound prices of goods constant. 2. An increase in the pound prices of goods, holding the exchange rate and the rand prices of goods constant. 3. A decrease in the rand prices of S.A goods, holding the exchange rate and the pound prices of goods constant. Since relative PPP implies a constant real exchange rate, we know that $$\text{rs}_{t+1,\mathsf{R} /\text{£}}=0$$ in this case. We can therefore use this information to solve Equation (9.5) to find that the required percentage change in the nominal exchange rate that just keeps the real exchange rate constant is $\begin{eqnarray} [1+s_{t+1,\mathsf{R} /\text{£}}]=\frac{[1+\pi_{t+1,\mathsf{R}}]}{[1+\pi_{t+1,\text{£}}]} \tag{9.6} \end{eqnarray}$ Equation (9.6) provides the warranted percentage rate of change of the rand-pound exchange rate that leaves the real exchange rate unchanged. If the nominal appreciation is larger than the amount that is warranted by the right-hand side of Equation (9.6), there is a real appreciation of the pound. Conversely, if the actual rate of appreciation of the pound relative to the rand falls short of the warranted amount on the right-hand side of Equation (9.6), there is a real depreciation of the pound. Variable Real Exchange Rate When the real exchange rate was constant in the previous example, the annual S.A. rate of inflation was 8%, the annual European rate of inflation was 3%, and the rand-euro exchange rate offset the inflation differential, with the euro appreciating by 4.85%. Suppose that the euro actually appreciates in nominal terms by 2% relative to the rand during the year. Is this nominal appreciation of the euro associated with a real depreciation of the euro or a real appreciation? From Equation (9.6), we know that the warranted rate of appreciation of the euro relative to the rand is 4.85% because $\begin{eqnarray*} \frac{[1+\pi_{t+1,\mathsf{R}}]}{[1+\pi_{t+1,\mathsf{€}}]} = \frac{1.08}{1.03} = 1.0485 = 1-(-0.0485) \end{eqnarray*}$ Because the nominal rate of appreciation of the euro relative to the rand is only 2%, there has been a real depreciation of the euro. The new real exchange rate is now less than it was before. With the new nominal exchange rate of $\begin{eqnarray*} \mathsf{R}13.00/\mathsf{€} \times (1+0.02)= \mathsf{R}13.26/\mathsf{€} \end{eqnarray*}$ the new real exchange rate is $\begin{eqnarray*} \text{RS}_{t+1,\mathsf{R} / \mathsf{€} } =\frac{\mathsf{R}13.26/\mathsf{€} \times \mathsf{€}11,330}{\mathsf{R}162,000}=0.927 \end{eqnarray*}$ Since the old real exchange rate was 0.9533, there is a real depreciation of the euro, with an accompanying real appreciation of the rand. This is even though the rand depreciated relative to the euro in nominal terms. The nominal rand value of the euro just did not fall enough when compared to the respective rates of inflation of the two currencies. This is due to the fact that the euro only increased by 2% instead of the 4.85% that was warranted by the inflation differential. As a result, the euro actually decreased in real terms. Notice from Equation (9.5) that real appreciations and real depreciations can occur even if the nominal exchange rate does not change. If the exchange rate is fixed between two currencies, but the prices of goods measured in these currencies rise at different rates because of differences in inflation, the high-inflation country will experience a real appreciation of its currency, and the low-inflation country will experience a real depreciation. ## 9.3 Trade-Weighted Real Exchange Rates To this point, we have considered only bilateral real exchange rates. Many governments calculate a trade-weighted real exchange rate. The numerator of a trade-weighted real exchange rate contains the sum of the nominal exchange rates for different currencies multiplied by the price levels of different countries weighted by the proportion of trade conducted with that country. A trade-weighted real exchange rate makes good economic sense because a given currency rarely strengthens or weakens relative to all foreign currencies by the same amount, and real exchange rates are critical determinants of international trade. For example, if we are interested in describing the extent to which a depreciation of the domestic currency would affect a country’s trade balance, we must know how much trade the country is doing with other nations and how much the depreciation is increasing the relative prices of the goods of those countries. Hence this exchange rate is usually termed the relative exchange rate. # 10 Conclusion After considering the concept of purchasing power parity and how it relates to the real exchange rate, it was noted that absolute purchasing power parity suggests that the exchange rate adjusts to equate the internal purchasing power of a nation’s currency to the external purchasing power of that currency. In addition, while the internal purchasing power measures the amount of goods and services that a unit of currency can buy in the country that issues the money; the external purchasing power measures the amount of goods and services that a unit of money can buy in a foreign country after converting domestic currency into foreign currency. While considering the microeconomic evidence that relates to the theory of purchasing power parity we noted that the law of one price suggests that the price of a particular commodity should be the same no matter where in the world the good is sold. If markets are competitive and there are no transaction costs, goods market arbitrage would drive the price of the good quoted in common currency to be the same. Violations of the law of one price are caused by transaction costs, barriers to trade, government regulations and non-competitive markets. For non-tradable goods, the law of one price would seldom hold. Deviations from absolute purchasing power parity are large and persistent. For major currencies, deviations of 35% or more are not uncommon and such deviations often persist for five years or more. In the long-run the deviations from purchasing power tend to subside. Hence, when the external purchasing power is less than its internal purchasing power then a currency is said to be undervalued. The value of such a currency would be expected to appreciate in value to return to the value that is suggested by purchasing power parity. When using this benchmark, the rand has been undervalued for a considerable period of time. It was also noted that equilibrium changes in relative prices, especially between prices of traded and non-traded goods would explain some of the observed deviations from absolute purchasing power parity. The theory of relative purchasing power parity acknowledges that a consumption bundle will not necessarily be the same from country to country. However, exchange rates will adjust in response to differential inflation rates that may arise in different countries. Hence, when the percentage change in the nominal exchange rate exceeds the rate of change that is warranted by the differential in inflation rates there is a real appreciation in the value of the foreign currency and a real depreciation of the domestic currency. # 11 References Balassa, Bela. 1964. “The Purchasing Power Parity Doctrine: A Reappraisal.” Journal of Political Economy 72: 584–96. Broda, Christian, and David E. Weinstein. 2008. “Understanding International Price Differences Using Barcode Data.” National Bureau of Economic Research Working Paper 14017. National Bureau of Economic Research. Canzoneri, Matthew B., Robert E. Cumby, and Behzad Diba. 1999. “Relative Labor Productivity and the Real Exchange Rate in the Long Run: Evidence for a Panel of Oecd Countries.” Journal of International Economics 47: 245–66. Cassel, Gustav. 1916. “The Present Situation of the Foreign Exchanges.” Economic Journal 26: 319–23. ———. 1918. “Abnormal Deviations in International Exchanges.” Economic Journal 28: 413–15. Clements, Kenneth W., and Yihui Lan. 2010. “A New Approach to Forecasting Exchange Rates.” Journal of International Money and Finance 29: 1424–37. Cumby, Robert E. 1996. “Forecasting Exchange Rates and Relative Prices with the Hamburger Standard: Is What You Want What You Get with Mcparity?” National Bureau of Economic Research Working Paper 5675. National Bureau of Economic Research. Dornbusch, Rudiger. 1988. “Exchange Rates and Prices.” In, edited by R. Dornbusch. Cambridge, MA: MIT Press. Engel, Charles, and John Rogers. 1996. “How Wide Is the Border?” American Economic Review 86: 1112–25. Harrod, Roy. 1933. International Economics. London: Nisbet; Cambridge University Press. Imbs, Jean M., Haroon Mumtaz, Morten O. Ravn, and Héléne Rey. 2010. “One Tv, One Price?” Scandinavian Journal of Economics 112: 753–81. Lothian, James R., and Mark P. Taylor. 2008. “Real Exchange Rates over the Past Two Centuries: How Important Is the Harrod-Balassa-Samuelson Effect?” The Economic Journal 118: 1742–63. Okun, Arthur. 1981. Prices and Quantities. Washington, DC: Brookings Institution. Ong, Li Lian, and Jason D. Mitchell. 2000. “Professors and Hamburgers: An International Comparison of Real Academic Salaries.” Applied Economics 32: 869–76. Parsley, David C., and Shang-Jin Wei. 2007. “A Prism into the Ppp Puzzles: The Micro-Foundations of Big Mac Real Exchange Rates.” The Economic Journal 117: 1336–56. Samuelson, Paul. 1964. “Theoretical Notes on Trade Problems.” Review of Economics and Statistics 46: 145–54. 1. Dornbusch (1988) notes that the earliest references to the subject are from 16th century Spain and 16th century England. Swedish economist Cassel (1916) is generally credited with coining the name for the theory. 2. See http://tariffdata.wto.org for information on tariff rates in WTO member countries. 3. The terms overvalued and undervalued are also employed in discussions of the relationship of a particular exchange rate to other theories of exchange rate determination. An overvalued currency must weaken on the foreign exchange markets to return to the prediction of the theory, and an undervalued currency must strengthen. 4. The percentage overvaluation or undervaluation of the denominator currency is computed as the percentage change in the exchange rate that is required to return to the PPP value. For example, if the actual exchange rate is$1.50, and the PPP exchange rate is $1.80, the pound is 20% undervalued because the appreciation of the pound required to go from the actual exchange rate to the PPP exchange rate is [($1.80/£)/(\$1.50/£)-1]=20%

5. Harrod (1933), Balassa (1964), and Samuelson (1964) demonstrated that differential rates of technological change could produce systematic deviations from PPP. Canzoneri, Cumby, and Diba (1999) and Lothian and Taylor (2008) provide empirical support for the idea.

6. Ong and Mitchell (2000) use this approach with MacPPP rates to compare academic salaries across countries.

7. It was this formulation of the theory that Cassel (1918) called * purchasing power parity*. Cassel was writing about the re-establishment of exchange rates after World War I because foreign exchange markets had closed during the war. Prior to the war, the countries of the world were on the gold standard, and their exchange rates were fixed. Cassel wrote: The general inflation which has taken place during the war has lowered this purchasing power in all countries, though in a different degree, and the rate of exchange should accordingly be expected to deviate from their old parities in proportion to the inflation of each country. At every moment the real parity is represented by this quotient between the purchasing power of the money in one country and the other. I propose to call this parity “purchasing power parity” (p. 413).