Chapter 6 Purchasing Power Parity

The Law of One Price (LoOP)

The law of one price says that identical goods should sell for the same price in two separate markets when there are no transportation costs and no differential taxes applied in the two markets. Consider the following information about movie video tapes sold in the U.S. and Mexican markets.

The dollar price of videos sold in Mexico can be calculated by dividing the video price in pesos by the spot exchange rate as shown:

$$\frac{P_{\text{p}}^v}{E_{\text{p/\$}}}\left[\frac{\text{peso/video}}{\text{peso/\$}}=\frac{\text{peso}}{\text{video}}\times \frac{\$}{\text{peso}}=\frac{\$}{\text{video}}\right]=\frac{150}{10}=\text{\$15/video}\text{.}$$

To see why the peso price is divided by the exchange rate rather than multiplied, notice the conversion of units shown in the brackets. If the law of one price held, then the dollar price in Mexico should match the price in the United States. Since the dollar price of the video is less than the dollar price in the United States, the law of one price does not hold in this circumstance.

The next question to ask is what might happen as a result of the discrepancy in prices. Well, as long as there are no costs incurred to transport the goods, there is a profit-making opportunity through trade. For example, U.S. travelers in Mexico who recognize that identical video titles are selling there for 25 percent less might buy videos in Mexico and bring them back to the United States to sell. This is an example of “goods arbitrage.” An arbitrage opportunity arises whenever one can buy something at a low price in one location, resell it at a higher price, and thus make a profit.

Using basic supply and demand theory, the increase in demand for videos in Mexico would push up the price of videos. The increase in supply of videos on the U.S. market would force the price down in the United States. In the end, the price of videos in Mexico may rise to, say, p180 while the price of videos in the United States may fall to $18. At these new prices the law of one price holds since

$$\frac{P_{\text{p}}^v}{E_{\text{p/\$}}}=\frac{180}{10}=\$18=P_{\text{\$}}^v\text{.}$$

The idea in the law of one price is that identical goods selling in an integrated market in which there are no transportation costs, no differential taxes or subsidies, and no tariffs or other trade barriers should sell at identical prices. If different prices prevailed, then there would be profit-making opportunities by buying the good in the low price market and reselling it in the high price market. If entrepreneurs took advantage of this arbitrage opportunity, then the prices would converge to equality.

Of course, for many reasons the law of one price does not hold even between markets within a country. The price of beer, gasoline, and stereos will likely be different in New York City and in Los Angeles. The price of these items will also be different in other countries when converted at current exchange rates. The simple reason for the discrepancies is that there are costs to transport goods between locations, there are different taxes applied in different states and different countries, nontradable input prices may vary, and people do not have perfect information about the prices of goods in all markets at all times. Thus to refer to this as an economic “law” does seem to exaggerate its validity.

From LoOP to PPP

The purchasing power parity theory is really just the law of one price applied in the aggregate but with a slight twist added. If it makes sense from the law of one price that identical goods should sell for identical prices in different markets, then the law ought to hold for all identical goods sold in both markets.

First, let’s define the variable CB_$ to represent the cost of a basket of goods in the United States denominated in dollars. For simplicity we could imagine using the same basket of goods used in the construction of the U.S. consumer price index (CPI_$). The consumer price index (CPI)An index that measures the average level of prices of goods and services in an economy relative to a base year. uses a market basket of goods that are purchased by an average household during a specified period. The basket is determined by surveying the quantity of different items purchased by many different households. One can then determine, on average, how many units of bread, milk, cheese, rent, electricity, and so on are purchased by the typical household. You might imagine it’s as if all products are purchased in a grocery store with items being placed in a basket before the purchase is made. CB_$ then represents the dollar cost of purchasing all the items in the market basket. We will similarly define CB_p to be the cost of a market basket of goods in Mexico denominated in pesos.

Now if the law of one price holds for each individual item in the market basket, then it should hold for the market baskets as well. In other words,

$$\frac{P_{\text{p}}^v}{E_{\text{p/\$}}}=P_{\text{\$}}^v\Rightarrow \frac{C{B}_{\text{p}}}{E_{\text{p/\$}}}=C{B}_{\text{\$}}\text{.}$$

Rewriting the right-hand side equation allows us to put the relationship in the form commonly used to describe absolute purchasing power parity, which is

$$E_{\text{p/\$}}^{\text{PPP}}=\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\text{.}$$

If this condition holds between two countries, then we would say PPP is satisfied. The condition says that the PPP exchange rateThe exchange rate that equalizes the cost of a market basket of goods between two countries. (pesos per dollar) will equal the ratio of the costs of the two market baskets of goods denominated in local currency units. Note that the reciprocal relationship $E_{\text{\$/p}}^{\text{PPP}}=\frac{C{B}_{\text{\$}}}{C{B}_{\text{p}}}$ is also valid.

Because the cost of a market basket of goods is used in the construction of the country’s consumer price index, PPP is often written as a relationship between the exchange rate and the country’s price indices. However, it is not possible merely to substitute the price index directly for the cost of the market basket used above. To see why, we will review the construction of the CPI in Chapter 6 "Purchasing Power Parity", Section 6.2 "The Consumer Price Index (CPI) and PPP".

PPP Using the CPI

The purchasing power parity relationship can be written using the CPI with some small adjustments. First, consider the following ratio of 2009 consumer price indices between Mexico and the United States:

$$\frac{CP{I}_{\text{p}}^{09}}{CP{I}_{\text{\$}}^{09}}=\frac{\raisebox{1ex}{$C{B}_{\text{p}}^{09}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{p}}^{08}$}\right.}{\raisebox{1ex}{$C{B}_{\text{\$}}^{09}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{\$}}^{08}$}\right.}=\frac{C{B}_{\text{p}}^{09}}{C{B}_{\text{\$}}^{09}}\frac{C{B}_{\text{\$}}^{08}}{C{B}_{\text{p}}^{08}}=\frac{\raisebox{1ex}{$C{B}_{\text{p}}^{09}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{\$}}^{09}$}\right.}{\raisebox{1ex}{$C{B}_{\text{p}}^{08}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{\$}}^{08}$}\right.}\text{.}$$

Given that the base year is 2008, the ratio is written in terms of the market basket costs on the right-hand side and then rewritten into another form. The far right-hand side expression now reflects the purchasing power parity exchange rates in 2009 divided by the PPP exchange rate in 2008, the base year. In other words,

$$\frac{CP{I}_{\text{p}}^{09}}{CP{I}_{\text{\$}}^{08}}=\frac{\raisebox{1ex}{$C{B}_{\text{p}}^{09}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{\$}}^{09}$}\right.}{\raisebox{1ex}{$C{B}_{\text{p}}^{08}$}\!\left/ \!\raisebox{-1ex}{$C{B}_{\text{\$}}^{09}$}\right.}=\frac{{}^{09}E{}_{\text{p/\$}}^{\text{PPP}}}{{}^{08}E{}_{\text{p/\$}}^{\text{PPP}}}\text{.}$$

So, in general, if you want to use the consumer price indices for two countries to derive the PPP exchange rate for 2009, you must apply the following formula, derived by rewriting the above as

$${}^{09}E{}_{\text{p/\$}}^{\text{PPP}}={}^{\text{base}}E{}_{\text{p/\$}}^{\text{PPP}}x\frac{CP{I}_{\text{p}}^{09}}{CP{I}_{\text{\$}}^{09}}\text{,}$$

where ${}^{\text{base}}E{}_{\text{p/\$}}^{\text{PPP}}$ represents the PPP exchange rate that prevails in the base year between the two countries. Note that in order for this formula to work correctly, the CPIs in both countries must share the same base year. If they did not, a more complex formula would need to be derived.

PPP Equilibrium Story

To see why the PPP relationship represents an equilibrium, we need to tell an equilibrium story. An equilibrium story in an economic model is an explanation of how the behavior of individuals will cause the equilibrium condition to be satisfied. The equilibrium condition is the PPP equation written as

$$E_{\text{p/\$}}^{\text{PPP}}=\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\text{.}$$

The endogenous variable in the PPP theory is the exchange rate. Thus we need to explain why the exchange rate will change if it is not in equilibrium. In general there are always two versions of an equilibrium story, one in which the endogenous variable (E_p/$ here) is too high and one in which it is too low.

PPP equilibrium story 1. Let’s consider the case in which the exchange rate is too low to be in equilibrium. This means that

$$E_{\text{p/\$}}^{}<\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\Rightarrow C{B}_{\text{\$}}{E}_{\text{p/\$}}<C{B}_{\text{p}}\text{,}$$

where E_p/$ is the exchange rate that prevails on the spot market. Since it is less than the ratio of the market basket costs in Mexico and the United States, it is also less than the PPP exchange rate. The right side of the expression is rewritten to show that the cost of a market basket in the United States evaluated in pesos (i.e., CB_$E_p/$) is less than the cost of the market basket in Mexico also evaluated in pesos. Thus it is cheaper to buy the basket in the United States, or in other words, it is more profitable to sell items in the market basket in Mexico.

The PPP theory now suggests that the cheaper basket in the United States will lead to an increase in demand for goods in the U.S. market basket by Mexico. As a consequence, it will increase the demand for U.S. dollars on the foreign exchange (Forex) market. Dollars are needed because purchases of U.S. goods require U.S. dollars. Alternatively, U.S. exporters will realize that goods sold in the United States can be sold at a higher price in Mexico. If these goods are sold in pesos, the U.S. exporters will want to convert the proceeds back to dollars. Thus there is an increase in U.S. dollar demand (by Mexican importers) and an increase in peso supply (by U.S. exporters) on the Forex. This effect is represented by a rightward shift in the U.S. dollar demand curve in Figure 6.1 "Forex Adjustment When ". At the same time, U.S. consumers will reduce their demand for the pricier Mexican goods. This will reduce the supply of dollars (in exchange for pesos) on the Forex, which is represented by a leftward shift in the U.S. dollar supply curve in the Forex market.

Both the shift in demand and supply will cause an increase in the value of the dollar and thus the exchange rate (E_p/$) will rise. As long as the U.S. market basket remains cheaper, excess demand for the dollar will persist and the exchange rate will continue to rise. The pressure for change ceases once the exchange rate rises enough to equalize the cost of market baskets between the two countries and PPP holds.

PPP equilibrium story 2. Now let’s consider the other equilibrium story (i.e., the case in which the exchange rate is too high to be in equilibrium). This implies that

$$E_{\text{p/\$}}>\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\Rightarrow C{B}_{\text{\$}}{E}_{\text{p/\$}}>C{B}_{\text{p}}\text{.}$$

The left-side expression says that the spot exchange rate is greater than the ratio of the costs of market baskets between Mexico and the United States. In other words, the exchange rate is above the PPP exchange rate. The right-side expression says that the cost of a U.S. market basket, converted to pesos at the current exchange rate, is greater than the cost of a Mexican market basket in pesos. Thus, on average, U.S. goods are relatively more expensive while Mexican goods are relatively cheaper.

The price discrepancies should lead consumers in the United States or importing firms to purchase less expensive goods in Mexico. To do so, they will raise the supply of dollars in the Forex in exchange for pesos. Thus the supply curve of dollars will shift to the right as shown in Figure 6.2 "Forex Adjustment When ". At the same time, Mexican consumers would refrain from purchasing the more expensive U.S. goods. This would lead to a reduction in demand for dollars in exchange for pesos on the Forex. Hence, the demand curve for dollars shifts to the left. Due to the demand decrease and the supply increase, the exchange rate (E_p/$) falls. This means that the dollar depreciates and the peso appreciates.

Extra demand for pesos will continue as long as goods and services remain cheaper in Mexico. However, as the peso appreciates (the dollar depreciates), the cost of Mexican goods rises relative to U.S. goods. The process ceases once the PPP exchange rate is reached and market baskets cost the same in both markets.

Adjustment to Price Level Changes under PPP

In the PPP theory, exchange rate changes are induced by changes in relative price levels between two countries. This is true because the quantities of the goods are always presumed to remain fixed in the market baskets. Therefore, the only way that the cost of the basket can change is if the goods’ prices change. Since price level changes represent inflation rates, this means that differential inflation rates will induce exchange rate changes according to the theory.

If we imagine that a country begins with PPP, then the inequality given in equilibrium story 1, $C{B}_{\text{\$}}{E}_{\text{p/\$}}<C{B}_{\text{p}}\text{,}$ can arise if the price level rises in Mexico (peso inflation), if the price level falls in the United States (dollar deflation), or if Mexican inflation is more rapid than U.S. inflation. According to the theory, the behavior of importers and exporters would now induce a dollar appreciation and a peso depreciation. In summary, an increase in Mexican prices relative to the change in U.S. prices (i.e., more rapid inflation in Mexico than in the United States) will cause the dollar to appreciate and the peso to depreciate according to the purchasing power parity theory.

Similarly, if a country begins with PPP, then the inequality given in equilibrium story 2, $C{B}_{\text{\$}}{E}_{\text{p/\$}}>C{B}_{\text{p}}\text{,}$ can arise if the price level rises in the United States (dollar inflation), the price level falls in Mexico (peso deflation), or if U.S. inflation is more rapid than Mexican inflation. In this case, the inequality would affect the behavior of importers and exporters and induce a dollar depreciation and peso appreciation. In summary, more rapid inflation in the United States would cause the dollar to depreciate while the peso would appreciate.

Problems with the PPP Theory

The main problem with the purchasing power parity (PPP) theory is that the PPP condition is rarely satisfied within a country. There are quite a few reasons that can explain this and so, given the logic of the theory, which makes sense, economists have been reluctant to discard the theory on the basis of lack of supporting evidence. Below we consider some of the reasons PPP may not hold.

Transportation costs and trade restrictions. Since the PPP theory is derived from the law of one price, the same assumptions are needed for both theories. The law of one price assumes that there are no transportation costs and no differential taxes applied between the two markets. These mean that there can be no tariffs on imports or other types of restrictions on trade. Since transport costs and trade restrictions do exist in the real world, this would tend to drive prices for similar goods apart. Transport costs should make a good cheaper in the exporting market and more expensive in the importing market. Similarly, an import tariff would drive a wedge between the prices of an identical good in two trading countries’ markets, raising it in the import market relative to the export market price. Thus the greater transportation costs and trade restrictions are between countries, the less likely for the costs of market baskets to be equalized.

Costs of nontradable inputs. Many items that are homogeneous nevertheless sell for different prices because they require a nontradable input in the production process. As an example, consider why the price of a McDonald’s Big Mac hamburger sold in downtown New York City is higher than the price of the same product in the New York suburbs. Because the rent for restaurant space is much higher in the city center, the restaurant will pass along its higher costs in the form of higher prices. Substitute products in the city center (other fast food restaurants) will face the same high rental costs and thus will charge higher prices as well. Because it would be impractical (i.e., costly) to produce the burgers at a cheaper suburban location and then transport them for sale in the city, competition would not drive the prices together in the two locations.

Perfect information. The law of one price assumes that individuals have good, even perfect, information about the prices of goods in other markets. Only with this knowledge will profit seekers begin to export goods to the high price market and import goods from the low-priced market. Consider a case in which there is imperfect information. Perhaps some price deviations are known to traders but other deviations are not known, or maybe only a small group of traders know about a price discrepancy and that group is unable to achieve the scale of trade needed to equalize the prices for that product. (Perhaps they face capital constraints and can’t borrow enough money to finance the scale of trade needed to equalize prices.) In either case, traders without information about price differences will not respond to the profit opportunities and thus prices will not be equalized. Thus the law of one price may not hold for some products, which would imply that PPP would not hold either.

Other market participants. Notice that in the PPP equilibrium stories, it is the behavior of profit-seeking importers and exporters that forces the exchange rate to adjust to the PPP level. These activities would be recorded on the current account of a country’s balance of payments. Thus it is reasonable to say that the PPP theory is based on current account transactions. This contrasts with the interest rate parity theory in which the behavior of investors seeking the highest rates of return on investments motivates adjustments in the exchange rate. Since investors are trading assets, these transactions would appear on a country’s capital account of its balance of payments. Thus the interest rate parity theory is based on capital account transactions.

It is estimated that there are approximately $1–2 trillion dollars worth of currency exchanged every day on international foreign exchange (Forex) markets. That’s one-eighth of U.S. GDP, which is the value of production in the United States in an entire year. In addition, the $1–2 trillion estimate is made by counting only one side of each currency trade. Thus that’s an enormous amount of trade. If one considers the total amount of world trade each year and then divides by 365, one can get the average amount of goods and services traded daily. This number is less than $100 billion dollars. This means that the amount of daily currency transactions is more than ten times the amount of daily trade. This fact would seem to suggest that the primary effect on the daily exchange rate must be caused by the actions of investors rather than importers and exporters. Thus the participation of other traders in the Forex market, who are motivated by other concerns, may lead the exchange rate to a value that is not consistent with PPP.

Relative PPP

There is an alternative version of the PPP theory called the “relative PPP theoryA dynamic version of the PPP theory that relates currency appreciation or depreciation to differences in inflation rates between countries..” In essence this is a dynamic version of the absolute PPP theory. Since absolute PPP suggests that the exchange rate may respond to inflation, we can imagine that the exchange rate would change in a systematic way given that a continual change in the price level (inflation) is occurring.

In the relative PPP theory, exchange rate changes over time are assumed to be dependent on inflation rate differentials between countries according to the following formula:

$$\frac{E_{\text{p/\$}}^2-E_{\text{p/\$}}^1}{E_{\text{p/\$}}^1}={\pi }_{\text{p}}-{\pi }_{\text{\$}}\text{.}$$

Here the percentage change in the dollar value between the first period and the second period is given on the left side. The right side gives the differences in the inflation rates between Mexico and the United States that were evaluated over the same time period. The implication of relative PPP is that if the Mexican inflation rate exceeds the U.S. inflation rate, then the dollar will appreciate by that differential over the same period. The logic of this theory is the same as in absolute PPP. Importers and exporters respond to variations in the relative costs of market baskets so as to maintain the law of one price, at least on average. If prices continue to rise faster in Mexico than in the United States, for example, price differences between the two countries would grow and the only way to keep up with PPP is for the dollar to appreciate continually versus the peso.

Over- and Undervaluation with Respect to PPP

First let’s consider over- and undervaluation with respect to PPP. The PPP exchange rate is defined as the rate that equalizes the cost of a market basket of goods between two countries. The PPP exchange rate between the Mexican peso and the U.S. dollar would be written as

$$E_{\text{p/\$}}^{\text{PPP}}=\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\text{,}$$

which represents the PPP value of the U.S. dollar in terms of pesos.

If the U.S. dollar is overvalued with respect to the Mexican peso, then the spot exchange rate exceeds the PPP exchange rate:

$$E_{\text{p/\$}}>E_{\text{p/\$}}^{\text{PPP}}\text{.}$$

This will also mean the exchange rate exceeds the ratio of market basket costs:

$$E_{\text{p/\$}}>\frac{C{B}_{\text{p}}}{C{B}_{\text{\$}}}\text{;}$$

therefore, the following will hold:

$$E_{\text{p/\$}}C{B}_{\text{\$}}>C{B}_{\text{p}}\text{.}$$

The left side (LS) of this expression represents the cost of a U.S. market basket converted to pesos at the current spot exchange rate. The right side (RS) is the cost of the basket in Mexico also evaluated in pesos. Since LS > RS, goods and services cost more on average in the United States than in Mexico at the current exchange rate. Thus for the U.S. dollar to be overvalued with respect to the peso means that goods and services are relatively more expensive in the United States than in Mexico. Of course, it also implies that goods and services are relatively cheaper in Mexico.

A simple guide to judge whether a currency is overvalued is to consider it from the perspective of a tourist. When the U.S. dollar is overvalued, a U.S. tourist traveling to Mexico will find that many products seem cheaper than in the United States, after converting at the spot exchange rate. Thus an overvalued currency will buy more in other countries.

An undervalued currency works in the opposite direction. When the U.S. dollar is undervalued, the cost of a basket of goods in the United States is lower than the cost in Mexico when evaluated at the current exchange rate. To a U.S. tourist, Mexican goods and services would seem more expensive on average. Thus an undervalued currency will buy less in other countries.

Finally, if the U.S. dollar is overvalued with respect to the Mexican peso, it follows that the peso is undervalued with respect to the dollar. In this case, since the U.S. tourists would find Mexican goods comparatively cheap, Mexican tourists would find U.S. goods to be comparatively expensive. If the U.S. dollar were undervalued, then the peso would be overvalued.

Is overvaluation or undervaluation good or bad? That depends on what a person is trying to achieve. For example, if the U.S. dollar is overvalued with respect to the peso, then a U.S. tourist traveling to Mexico will be very happy. In fact, the more overvalued the dollar is, the better. However, for an exporter of U.S. goods to Mexico, its price in peso terms will be higher the more overvalued is the dollar. Thus an overvalued dollar will likely reduce sales and profits for these U.S. firms.

Over- and Undervaluation with Respect to Current Account Balance

The second way over- and undervaluation is sometimes applied is in comparison to an exchange rate presumed necessary to induce trade balance, or balance on the current account. If one imagines that a trade deficit, for example, arises primarily because a country imports too much or exports too little (rather than being driven by financial decisions tending to cause a financial account surplus), then one may also look for ways to either reduce imports or raise exports. A change in the exchange rate offers one viable method to affect trade flows.

Suppose the United States has a trade deficit (which it indeed has had for more than thirty years prior to 2010). If the U.S. dollar value were to fall—a dollar depreciation—then foreign goods would all become relatively more expensive to U.S. residents, tending to reduce U.S. imports. At the same time, a dollar depreciation would also cause U.S. goods to become relatively cheaper to foreign residents tending to raise U.S. exports.

Sometimes economists make numerical estimations as to how much the dollar value would have to fall to bring trade into balance. These estimations are enormously difficult to make for several reasons and should be interpreted and used with great caution, if at all. The primary reason is that many different factors on both the trade side and the financial side influence a country’s trade imbalance besides just the exchange rate. The exchange rate that balances trade would depend on the values taken by all the other factors that also influence the trade balance. Different values for all the other variables would mean a different exchange rate needed to balance trade. Thus there isn’t one exchange rate value that will balance trade. Instead, there is a different exchange rate value that will balance trade in each and every alternative circumstance. Indeed, even the current exchange rate—whatever that is—can balance trade if other factors change appropriately.

Despite these cautions, many observers will still contend that a country’s currency needs to depreciate by some percentage to eliminate a trade deficit, or needs to appreciate to eliminate a trade surplus. When it is believed a depreciation of the currency is needed to balance trade, they will say the currency is overvalued. When it is believed an appreciation of the currency is needed to balance trade, they will say the currency is undervalued. However, in a floating exchange rate system, it is hard to argue that the exchange rate is at the “wrong” value since—with competition in the market—it will always be at the rate that equalizes supply and demand. In other words, the “proper” value for the exchange rate can be said to be not the one that will satisfy PPP or not the one that will generate trade balance but rather whatever rate currently prevails. Under this notion, a currency can never be over- or undervalued in a floating exchange rate system. Instead, the spot exchange rate is always at the “proper” value.

In a fixed exchange rate system, a government can sometimes intervene to maintain an exchange rate that is very different from what would arise if allowed to float. In these cases, large trade surpluses can arise because the government maintains an artificially low value for its currency. Calls for a revaluation (appreciation) of the currency, to promote a reduction in a trade surplus, are somewhat more appropriate in these cases since the market does not determine the exchange rate. Similarly, large deficits could be reduced with a devaluation (depreciation) of the currency.