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

To determine whether a currency is over-valued or under-valued we need a benchmark that provides the true value of a currency should be. 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. This benchmark also provides a baseline forecast for 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, and could be used as a discount rate for the repatriation of earnings or dividends from an investment. Similarly, deviations from PPP could affect the profitability of firms, particularly where they may need to set international prices, or where they are engaging in long-term international contracts, hedging cash flows of foreign operations, etc. It is also worth noting that PPP is also particularly useful in various developmental studies as it is used to assess the cost-of-living in different countries.

As we will see when we look at the data, PPP does not hold very well in the short run. And the deviations from the theory are sometimes so large that some economists dismiss the potential practical applications of this 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 over the long run. In addition, it also appears to hold over the short term in some countries, where the relative importance of inflation dominates overall economic activity.

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

## 1.1 The General Idea of Purchasing Power

Purchasing power parity involves comparing the purchasing power of a currency within a country to the purchasing power of that money when spent in a different country. When we convert from monetary magnitudes into units of purchasing power, we need to convert the nominal units into real units. To move from a nominal to real value we would need to make use of an appropriate price index.

Such a nominal price index could measure the monetary value 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 individual goods and services in the consumption bundle. The percentage change in the price level is termed inflation, such that, when countries price level is rising, it will face inflationary pressure.

In South Africa, this data is collected by Statstics South Africa (StatsSA), and they follow the practices of the United Nations Statistical Division (UNSD) standard for classifying household expenditure on goods and services. This standard is termed the Classification of Individual Consumption by Purpose (COICOP) and it facilitates international comparisons, for those countries that follow this practice. Note that the compilation of these monthly indices is a significant task. Between January 2008 to November 2019, StatsSA collected data, mainly through fieldworkers, on 34,075 different commodities from 5,505 outlets that arise in 85 different areas.

Once we have an estimate for the price index, we then need to calculate what a unit of currency will actually buy in a particular country. To do this, they take the reciprocal, or inverse, of the price level, which may be used to measure the purchasing power of the currency. Therefore, if the price level is 100, at a point in time, then we can purchase 1/100 of a consumption bundle with a unit of currency. If the price level increases to 120, then we would only be able to buy 1/120 of a bundle of goods for a unit of currency.

Unfortunately, governments do not usually provide information on the actual consumer price levels, which is not surprising given the arduous data collection process. Instead they usually provide information on price indices, which represent the ratio of a price level at one point in time relative to the price level in a designated base year. Note that the base value for the consumption price index may change from time-to-time so when making comparisons over time, you need to ensure that you are using a common base period.

## 1.2 Measuring the Price Index and Inflation

In what follows, and only for notational purposes, we are going to associate the price index with the currency of a country. Strictly speaking the price level should be associated with a particular country and not a country. Hence, for South Africa, we can write the consumer price index as

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

where $$i$$ represents a category of a commodity, which itself is an index. By way of example, Beer is a separate category, for which an index is provided to summarise price movements for all the different brands and quantities that are sold. It represents about 2% of household expenditure in South Africa, as measured by either the quarterly Income and Expenditure Survey (IES) or the more recently from the Living Conditions Survey (LCS). Therefore, it has a weight of 0.02. In the above calculation, $$w_{i}$$ represents the weight or consumption share of commodity $$i$$, while $$P_{t, i, \mathsf{R}}$$ represents the price index for category $$i$$ at time $$t$$. The broad consumption price index is then $$P_{t,\mathsf{R}}$$,which is the weighted average of the price of the $$N$$ different goods and services.

Typically, the ratio of the two price levels or indices is multiplied by 100 to describe the percentage change in the average of all nominal prices. This would reflect the amount of inflation that in a country has been experienced between two periods of time. For example, the rand price index in year $$t+k$$ with year $$t$$ as a base year may be used to calculate the amount of inflation, which we denote $$\pi_{t+k,\mathsf{R}}$$, over this period:

$\begin{eqnarray*} \pi_{t+k,\mathsf{R}}= \left( \frac{P_{t+k,\mathsf{R}}}{P_{t,\mathsf{R}}} \right) \times 100 &=& \left(\frac{\sum_{i=1}^{N}w_{i}P_{t+k,i,\mathsf{R}}}{\sum_{i=1}^{N}w{}_{i}P_{t,i,\mathsf{R}}}\right) \times 100 \end{eqnarray*}$

If the price index today is 115, then we know that prices are 15% higher than what they were in the base year. We could then conclude that the cost of living has increased by 15%, as we would need to increase our expenditure by 15%, to purchase the same consumption bundle.

United
United
South
Year States Canada France Germany Italy Japan Kingdom Africa
1960 27.6 24.6 17.2 39.4 9.8 21.2 13.2 15.7
1970 36.1 32.3 25.2 50.9 14 36.9 19.6 19.9
1980 76.5 69.7 63.3 82.6 51 87.2 70.7 52.9
1985 100 100 100 100 100 100 100 100
1990 121.4 124.1 116.3 107.7 131.2 107 133.4 209.4
1995 141.7 139.2 129.9 126.2 168.6 113.5 158.4 367.5
2000 159 150 138 133.9 188.3 115.2 179.9 507.3
2005 179.4 167.8 151.8 144.9 212 112.4 202.1 656.5
2008 197.8 179 161.1 154.5 227.8 114.3 219.3 790.6
2009 197.1 179.5 161.2 155 229.5 112.7 224 863.9
2010 199.3 181.1 162.6 155.8 231.3 111.7 228.2 916.8

Table 1: Price Indexes for the G7 Countries and South Africa, 1960-2010

Table 1 provides some information on consumer price indexes for the G7 countries. In addition, we’ve also include data for South Africa over the period 1960 to 2010. This data would allow us to consider the different inflationary experiences in each of these countries over time. Note that between 1960 and 1985, Italy and the United Kingdom experienced the two largest changes in their price indices. However, since 1985 the deterioration in buying power within South Africa is almost four times that of the United Kingdom. In addition, we also note that Japan has hardly experienced any inflation over this period of twenty-five years.

Calculating an Annual Rate of Inflation

Note that when using a common base year, the ratio of the two price indexes measures 1 plus the rate of inflation. This would be equivalent to the change in the respective price levels:

$\begin{eqnarray*} \frac{P_{t+1}}{P_{t}}=\big[1+\pi_{t+1}\big] \end{eqnarray*}$ where $$\pi_{t+1}\equiv\frac{P_{t+1}-P_{t}}{P_{t}}$$.

Now, let’s use the data from the above table to determine the South African rate of inflation between 2008 and 2009. The values of the price indexes for 2008 and 2009 were 863.9 and 790.6, respectively. We can then find the percentage rate of inflation by subtracting 1 from the ratio of the price indexes, before we multiply the result by 100:

$\begin{eqnarray*} \pi_{2008:2009, \mathsf{R}} = \left( \frac{863.9}{790.6}-1 \right) \times 100=9.271\% \end{eqnarray*}$

Calculating the Cumulative Rate of Inflation

To detrive the average annual rate of inflation between 1985 and 2010 for South Africa, note that the price index moved from 100 to 916.8. This implies that the average rand prices of goods and services in 2010 were 816.8% higher than were the prices in 1985. Hence, over the 25 year period, prices increased at a compound annual rate of inflation of 9.267%, as is shown in the following calculation:

$\begin{eqnarray*} \left( \frac{916.8}{100} \right)^{1/25}=1.09267 \end{eqnarray*}$

Now that we know how to construct a price index, which may be used to calculate inflation, we can discuss concepts relating to the purchasing power of a domestic currency. Note that the units of the internal purchasing power of a rand are the amount of goods and services that can be purchased with a rand in South Africa. Previously we noted that the amount of goods that correspond to the purchasing power of 1 rand is measured by taking the reciprocal of the S.A. price level. Since the units of the S.A. price level are rands per S.A. consumption bundle, the units of purchasing power (the reciprocal of the price level) are S.A. consumption bundles per rand.

Calculating the Purchasing Power of R1,000,000

Suppose the price level in South Africa is R175,000 for the average consumption bundle. What is the purchasing power of R1,000,000?

The purchasing power of 1 rand is (1/ R175,000), so the purchasing power of R1,000,000 is

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

In other words, R1,000,000 is enough to purchase 66.67 consumption bundles.

The units of the external purchasing power of a rand are the amount of goods and services outside South Africa that can be purchased with a rand, say, in the United Kingdom. Therefore, calculating the external purchasing power of a rand in the United Kingdom involves two steps. First, it is necessary to purchase some amount of pounds with South African rand. Second, it is necessary to examine the purchasing power of those pounds in the United Kingdom.

For a given spot exchange rate of $$S_{t,\mathsf{R}/\text{£}}$$ rands per pound, it would imply that one rand buys $$1/S_{t,\mathsf{R}/\text{£}}$$ pounds. Now we know that the purchasing power of the pound may be measured by taking the reciprocal of the price level in the United Kingdom, $$1/PB_{t,\text{£}}$$, which represents the number of consumption bundles that can be bought per pound in the United Kingdom. Therefore, the external purchasing power of the rand in the United Kingdom may be derived from:

$\begin{eqnarray*} \frac{1}{S_{t,{\mathsf{R}}/\text{£}}} \times \frac{1}{PB_{t,\text{£}}} \end{eqnarray*}$

Note that this expression could be written out as:

$\begin{eqnarray*} \frac{\mathsf{Pounds}}{\mathsf{Rands}} \times \frac{\mathsf{UK \; consumption \; bundles}}{\mathsf{Pound}} = \frac{\mathsf{UK \; consumption \; bundles}}{\mathsf{Rands}} \end{eqnarray*}$

which clearly displays the concept of the external purchasing power of a rand in the United Kingdom. Now that we can calculate the purchasing power of the rand in two countries, we can examine what happens when we equate the two.

# 2 Absolute Purchasing Power Parity

## 2.1 The Theory of Absolute Purchasing Power Parity

One version of PPP, called absolute purchasing power parity, states that the exchange rate will adjust to equalise the internal and external purchasing powers of a currency. To investigate how this process may arise, note that the internal purchasing power is derived from the reciprocal of the price level, and the external purchasing power is derived by first exchanging the domestic currency into foreign currency, on the foreign exchange market, before we then calculate the purchasing power of that amount of foreign currency in the foreign country. Hence, the prediction of absolute PPP for the rand-pound exchange rate is found by equating the internal purchasing power of a rand with the external purchasing power of the same currency:

$\begin{eqnarray} \frac{1}{PB_{t,\mathsf{R}}}=\frac{1}{S^\mathsf{PPP}_{t,{\mathsf{R}}/\text{£}}} \times \frac{1}{PB_{t,\text{£}}} \tag{2.1} \end{eqnarray}$

where $$S^{\mathsf{PPP}}_{t,\mathsf{R}/\text{£}}$$ signifies the rand-pound exchange rate that satisfies the PPP relation. By solving (2.1) for $${S}^{\mathsf{PPP}}_{t,\mathsf{R}/\text{£}}$$, we find

$\begin{eqnarray} S^{\mathsf{PPP}}_{t,\mathsf{R}/\text{£}}=\frac{PB_{t,\mathsf{R}}}{PB_{t,\text{£}}} \tag{2.2} \end{eqnarray}$

You could think of this postulate for absolute PPP as a theory that makes a prediction about what the exchange rate should be given the respective price levels in the two countries. Equation (2.2) predicts that the rand-pound exchange rate should be equal to the ratio of the price level in the South Africa to the price level in the United Kingdom. The key here is that differences in prices across countries should be reflected in the relative price of the currencies - that is, in the exchange rate. In subsequent parts of this lecture, we investigate the relative merits of this postulate, by comparing actual exchange rates to the predictions that are provided by PPP. However, before we do that, we are firstly going to explore the foundations of the theory of absolute PPP in greater detail.

## 2.2 Goods Market Arbitrage

Suppose the internal purchasing power of the rand is less than its external purchasing power in a foreign country. How could you use this information to make a profit? Well, if the rand buys more goods abroad than it does at home, it ought to be possible to take some amount of rands, buy goods abroad, ship the goods to South Africa, and sell them for more rands than your original rand expenditure.

To demonstrate the benefits that may be derived from this arbitrage transaction, consider the following example.

A Goods Market Arbitrage

Suppose that the S.A. price level is R175,000 per consumption bundle and that the U.K. price level is $$10,000$$ consumption bundle. Let the exchange rate be GBPZAR 16.00. Rather than compute the purchasing power of R1, consider the internal and external purchasing powers of R1 million. As we saw earlier, the internal purchasing power of R1 million in the South Africa is

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

The external purchasing power of R1 million in the United Kingdom is found in two steps. First, convert the R1 million into pounds to get

$\begin{eqnarray*} \mathsf{R}1,000,000 \times \frac{1}{\mathsf{GBPZAR } 16.00} = \text{£}62,500 \end{eqnarray*}$

Then, find the purchasing power of 62,500 in the United Kingdom:

$\begin{eqnarray*} \text{£}62,500 \times \frac{1}{\text{£}10,000/\mathsf{consumption \; bundle}} = 6.25 \; \mathsf{consumption \; bundles} \end{eqnarray*}$

Since the external purchasing power of the rand in the United Kingdom is higher than the internal purchasing power of the rand in South Africa, we can profit by buying goods in the United Kingdom and shipping them to South Africa for resale. If we buy goods in the United Kingdom, we can purchase $$6.25$$ consumption bundles with our R1 million. If we sell the $$6.25$$ consumption bundles in South Africa at R175,000 consumption bundle, we will receive

$\begin{eqnarray*} (6.25 \; \mathsf{consumption \; bundles}) \times (\mathsf{R}175,000 / \mathsf{consumption \; bundle}) =\mathsf{R}1,093,750 \end{eqnarray*}$

Thus, by buying goods at low prices and selling goods at high prices, we have generated a 9.375% rate of return on our R1 million investment.

The above example demonstrates another way of looking at PPP. If absolute PPP holds, the costs of the consumption bundles in different countries are equal when expressed in a common currency. When absolute PPP does not hold, there is a potential opportunity for goods market arbitrage. Such goods market arbitrage would, of course, be subject to somewhat larger transaction costs than the financial arbitrages we discussed in previous lectures. For example, there would be transaction costs associated with the physical shipment of goods between countries. In addition, if you were looking to perform such activities, you would not want to buy an entire consumption bundle and would rather look to exploit the pricing differential that is associated with a single commodity or a limited range of goods.

# 3 The Law of One Price

## 3.1 The Perfect Market Ideal

If markets are competitive, we should not be able to make a profit from buying and reselling goods in different countries. In fact, if there were no transaction costs, arbitrage would drive the price of any good quoted in a common currency to be the same around the world. The law of one price states that the price of a good, when denominated in a particular currency, is the same, no matter where in the world the good is being sold. Therefore, PPP may be regarded as an extension of the law of one price, where instead of looking at a single good, PPP considers the price of a bundle of goods.

For example, in the absence of arbitrage possibilities, the dollar price of a barrel of oil should equal the dollar price of the British pound multiplied by the pound price of a barrel of oil:

$\begin{eqnarray*} \frac{\mathsf{Barrel \; of \; oil}}=\frac{\text{£}} \times \frac{\text{£}}{\mathsf{Barrel \; of \; oil}} \end{eqnarray*}$

### 4.3.4 The Econometric Evidence

More formal statistical studies by economists also support the usefulness of MacPPP. Cumby (1996) finds that deviations from MacPPP are temporary. After allowing for a constant deviation, he estimates that one-half of the deviation from parity disappears in 1 year. This evidence also indicates that both the exchange rate and the prices of the burgers are adjusting to eliminate the deviation. The prediction is that a 10% undervalued currency tends to appreciate over the next year by 3.5%. In addtion, Clements and Lan (2010) confirm that exchange rate forecasts using MacPPP have value, especially at 2- or 3-year horizons.

Parsley and Wei (2007) study the components of the Big Mac Index and note that local labour costs account for 45.6% of its price. A later section addresses how such non-traded goods can affect PPP calculations. Parsley and Wei also find a very high correlation between PPPs calculated with Big Mac prices and those from CPI data, to which we now turn.

# 5 Exchange Rates and Absolute PPPs Using CPI Data

## 5.1 Interpreting the Charts

One disadvantage of the MacPPP analysis is its comparatively short time span because The Economist only started calculating MacPPP in 1986. Figures 1 through 2 present data for actual exchange rates and the predictions of absolute PPP calculated from consumer price indexes for several of the world’s major currencies. Note that in the case of the GBPUSD, the solid blue line represents the actual exchange rate, and the orange line is the implied exchange rate from the prediction of PPP. All this data is available from the IMF International Financial Statistics database.

### 5.1.1 Overvaluations and Undervaluations

When examining the deviations from PPP in Figures 1 through 2, it is important to remember how the respective exchange rates are quoted. For example, the pound and euro exchange rates are quoted directly as the amount of dollars it takes to purchase 1 pound or 1 euro, whereas the other exchange rates relative to the U.S. dollar are quoted indirectly as the amount of that currency that it takes to purchase 1 dollar. The PPP prediction for the dollar-pound exchange rate is therefore $$P_{t,\}/P_{t,\text{£}}$$, whereas the PPP predictions for the indirect quotes relative to the dollar are the ratios of the foreign price levels to the U.S. price level. Hence, the dollar is undervalued when the actual exchange rate $$S_{t,\/\text{£}}$$ is above the PPP prediction, $$P_{t,\}/P_{t,\text{£}}$$, because the dollar must strengthen relative to the pound if the undervaluation (on foreign exchange markets) is to be corrected. For the yen/dollar rate, the dollar is overvalued when the actual exchange rate, $$S_{t,\yen/\}$$, is above the PPP prediction, $$P_{t,\yen} / P_{t,\}$$, because the dollar must weaken relative to the yen if the overvaluation of the dollar (on foreign exchange markets) is to be corrected by a movement in the exchange rate.

### 5.1.2 Fixing When PPP Held

The data in Figures 1 through 2 begin in January 1973 and end in January 2020. Since the prices of goods are obtained as consumer price indexes rather than price levels, it is necessary to take a stand on when the actual exchange rate satisfied the PPP relationship in order for the units of the ratio of the prices to correspond to the units of the exchange rate. The data are plotted such that absolute PPP is assumed to have held on average during the decade of the 1980’s.

## 5.2 Analysing the Data

After looking at the data we could ascertain, how well or poorly the theory of absolute PPP holds up. Clearly, there are large and persistent deviations of actual exchange rates from the predictions of PPP, but this alone should not discredit the theory.

Figure 1: Actual USD/GBP and PPP Exchange Rates

The data for the $$\/\text{£}$$ rate in Figure 1 indicate that the pound was 30.2% overvalued in October 1980, but by February 1985, it was 43.8% undervalued.4. Because the ratio of the price levels in the two countries changed only slightly over this period, almost all of the change is due to the movement of the exchange rate from 2.40/£ to $1.10/£. Once the dollar peaked in strength in 1985, though, it began to depreciate, and by October 1990, the pound was again more than 25% overvalued relative to the dollar. Just prior to the beginning of the financial crisis in November 2007, the pound was 30.5% overvalued, and at the end of January 2010, the pound was 9.6% overvalued. ### 5.2.2 Yen - Dollar The data for the yen-dollar 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 2: 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 3: 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 4: 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. Since 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. However, there are other potential factors that may cause deviations from absolute PPP, which we will now investigate.

## 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 that are produced in that country (when compared to the consumptiom 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.

### 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.

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 Canada 44215 46419 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 obtain a better understanding of 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 South Africa is initially R175,000 per S.A. consumption bundle, while 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 experiencing a loss of 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 ensure that the loss of the pound’s external purchasing power is 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 that inflation lowers the purchasing power of money. Hence, 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 will be required to compensate for the differential rates of inflation. This would ensure 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. 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, which is usually the case when 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 at every point in time. Hence, the rate of change of the exchange rate does not equal the inflation differential between two currencies and may be regarded as an approximate relationship.

## 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 or indices:

$\begin{eqnarray} \text{RS}_{t,\mathsf{R} / \mathsf{€}} =\frac{S_{t,{{\mathsf{R}}}/ \mathsf{€}} \times P_{t,\mathsf{€}}}{P_{t,{{\mathsf{R}}}}} \tag{9.1} \end{eqnarray}$

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 may hold in this case, 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, while inflation in Europe is 3%, 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

$\begin{eqnarray*} S_{t,\mathsf{R} / \mathsf{€}} =\frac{\mathsf{R} 13.00/ \text{€} \times 1.08}{1.03}= \mathsf{R} 13.63 / \text{€} \end{eqnarray*}$

Hence the euro should strengthen by 4.89%. 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 then define the percentage rate of change in the real exchange rate as follows

$\begin{eqnarray} \text{rs}_{t+1,\mathsf{R} /\text{£}}=\frac{\text{RS}_{t+1,\mathsf{R} /\text{£}}-\text{RS}_{t,{{\mathsf{R}}}/\text{£}}}{\text{RS}_{t,{{\mathsf{R}}}/\text{£}}} \tag{9.2} \end{eqnarray}$

If the right-hand side of Equation (9.2) is positive, we have a real appreciation of the pound or a real 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 (and a real appreciation of the rand):

$\begin{eqnarray*} \text{rd}_{t+1,\mathsf{R} /\text{£}} = -\text{rs}_{t+1,\mathsf{R} /\text{£}}, \;\; \mathsf{if} \;\; \text{rs}_{t+1,\mathsf{R} /\text{£}}<0 \end{eqnarray*}$

Since 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):

$\begin{eqnarray} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]=\frac{[S_{t+1,\mathsf{R} /\text{£}} \times P_{t+1,\text{£}}/P_{t+1,\mathsf{R}}]}{[S_{t,{{\mathsf{R}}}/\text{£}} \times P_{t,\text{£}}/P_{t,{{\mathsf{R}}}}]} \tag{9.4} \end{eqnarray}$

After rearranging the exchange rate terms, the pound price-level terms, and the rand price-level terms together, we are able to produce the following expression:

$\begin{eqnarray*} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]= \frac{[S_{t+1,\mathsf{R} /\text{£}}/S_{t,{{\mathsf{R}}}/\text{£}}] \times [P_{t+1,\text{£}}/P_{t,\text{£}}]}{[P_{t+1,\mathsf{R}}/P_{t,{{\mathsf{R}}}}]} \end{eqnarray*}$

Then finally, by substituting the definitions of the ratios of variables at time $$t+1$$ into those at time $$t$$, we note that

$\begin{eqnarray} [1+\text{rs}_{t+1,\mathsf{R} /\text{£}}]=\frac{[1+s_{t+1,\mathsf{R} /\text{£}}] \times [1+\pi_{t+1,\text{£}}]}{[1+\pi_{t+1,\mathsf{R}}]} \tag{9.5} \end{eqnarray}$

Recall that the left-hand side of Equation (9.5) is 1 plus the percentage rate of change in the real rand-pound exchange rate, while the right-hand side equals 1 plus the percentage rate of change in 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

Since 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 keeps the real exchange rate constant, which 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

In much of what we have discussed, we have considered the behaviour of bilateral real exchange rates. To consider how an exchange rate performs relative to a basket of currencies, 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 since 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

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.

# 12 References

Placeholder

## 12.2 Interpreting the Charts–>

### 12.2.1 The Salary Offers–>

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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).↩︎