Purchasing power parity
Thepurchasing power parity (PPP) theory was developed by Gustav Cassel in 1920.  It is the method of using the long-run equilibrium exchange rate of two currencies to equalize the currencies' purchasing power. It is based on the law of one price, the idea that, in an efficient market, identical goods must have only one price.
Purchasing power parity is often called absolute purchasing power parity to distinguish it from a related theory  relative purchasing power parity, which predicts the relationship between the two countries' relative inflation rates and the change in the exchange rate of their currencies.
A purchasing power parity exchange rate equalizes the purchasing power of different currencies in their home countries for a given basket of goods. These special exchange rates are often used to compare the standards of living of two or more countries.  The adjustments are meant to give a better picture than comparing gross domestic products (GDP) using market exchange rates. This type of adjustment to an exchange rate is controversial because of the difficulties of finding comparable baskets of goods to compare purchasing power across countries.
Market exchange rates fluctuate widely, but many believe that PPP exchange rates reflect the long run equilibrium value.  The distortions caused by using market rates are accentuated because prices of non-traded goods and services are usually lower in poorer economies. For example, a U.S. dollar exchanged and spent in the People's Republic of China will buy much more than a dollar spent in the United States.
The differences between PPP and market exchange rates can be significant.  For example, the World Bank's World Development Indicators 2005 estimates that one United States dollar is equivalent to approximately 1.8 Chinese yuan by purchasing power parity in 2003. .  However, based on nominal exchange rates, one U.S. dollar is currently equal to 7.6 yuan. This discrepancy has large implications; for instance, GDP per capita in the People's Republic of China is about US$1,800 while on a PPP basis it is about US$7,204.  This is frequently used to assert that China is the world's second largest economy, but such a calculation would only be valid under the PPP theory.  At the other extreme, Japan's nominal GDP per capita is around US$37,600, but its PPP figure is only US$30,615.
Estimation of purchasing power parity is complicated by the fact that countries do not simply differ in a uniform price level; rather, the difference in food prices may be greater than the difference in housing prices, while also less than the difference in entertainment prices. People in different countries typically consume different baskets of goods. It is necessary to compare the cost of baskets of goods and services using a price index. This is a difficult task because purchasing patterns and even the goods available to purchase differ across countries. Thus, it is necessary to make adjustments for differences in the quality of goods and services. Additional statistical difficulties arise with multilateral comparisons when (as is usually the case) more than two countries are to be compared.
When PPP comparisons are to be made over some interval of time, proper account needs to be made of inflationary effects.

# Explanation
For a U.S. dollar to buy as much in the UK as in the U.S., as is assumed under the law of one price, the price of a basket of goods in pounds in the UK (denoted as: P) times the spot exchange rate (denoted as: $/) should equal the price of the same basket in the U.S. priced in dollars (denoted as: $P).
This implies that the exchange rate that equalizes the value of a dollar of purchasing power (the PPP exchange rate) is:
If the actual spot rate is greater, it suggests that the  is over-valued against the $.
If the actual spot rate is less, it suggest that the $ is over-valued against the .
For example if a "representative" consumption basket costs $1,500 in the U.S. and 1,000 in the UK the PPP exchange rate would be $1.50/.  If the actual spot rate was $1.80/ this would indicate that the pound is overvalued by 20%, or equivalently the dollar is undervalued by 16.7%.

## Relative PPP
Relative PPP relates the inflation rate (the change of price levels) in each country to the change in the market exchange rate.
\frac{S_t}{S_{t-1}}=\frac{P_{t}^*/P_{t-1}^*}{P_{t}/P_{t-1}},
where S_t is the spot rate in Foreign Currency/Domestic Currency and P_t is the price level in period t (foreign values are marked by an asterisk). This relation is necessary but not sufficient for absolute purchasing power parity.
According to this theory, the change in the exchange rate is determined by price level changes in both countries. For example, if prices in the United States rise by 3% and prices in the European Union rise by 1% the purchasing power of the USD should depreciate by 2% compared to the purchasing power of the EUR (equivalently the EUR will appreciate by about 2%)

## PPP equalization and the law of one price
The law of one price states that differing prices of a traded good will tend to equalize in the absence of tariffs, other barriers to trade and prohibitively high shipping rates. The law of one price can also be stated as: "In an efficient market all identical goods must have only one price."
The naive PPP hypothesis is that free trade of goods should revert exchange rates to their PPP values. However, econometric analysis rejects this hypothesis, and gives a better prediction of the PPP/exchange rate relationship (the CPI) based on relative GDPs. Neo-classical economics includes Balassa-Samuelson effect theory, which explains the PPP model adjustment giving the equilibrium CPIs.

# Big Mac Index
An entertaining example of one measure of PPP is the Big Mac Index popularised by The Economist, which looks at the prices of a Big Mac burger in McDonald's restaurants in different countries. If a Big Mac costs US$4 in the U.S. and GBP3 in Britain, the PPP exchange rate would be 3 for $4. The Big Mac Index is presumably useful because it is based on a well-known good whose final price, easily tracked in many countries, includes input costs from a wide range of sectors in the local economy, such as agricultural commodities (beef, bread, lettuce, tomatoes), labour (blue and white collar), advertising, rent and real estate costs, transportation, etc.  The Big Mac Index is innacurate in certain cases because of the different market conditions that exist in differing McDonald's locations.  For instance, a Big Mac sold in downtown Chicago is likely to be priced higher than one the same product sold just miles away in Wisconsin.  Such pricing differences existing in one country demonstrate the imperfection of the Big Mac Index.  In addition, in some emerging economies western fast food represents an expensive niche product price well above the price of traditional staples - i.e. the Big Mac is not a mainstream 'cheap' meal as it is in the west but a luxury import for the middle classes and foreigners. Although it is not perfect, the index still offers significant insight and an easy to understand example of PPP.

# iPod Index
The Australian securities firm CommSec introduced the iPod Index as a light-hearted method of measuring PPP.  Unlike the Big Mac, which is affected by local labour and transport costs, the iPod manufacturing costs are the same and the iPod is a tradable commodity.

# West and Central African Franc
In 2003, the U.S. Dollar bought on average about 550 CFA franc. Because of a difference in purchasing power within some of the regions using the CFA franc, their purchasing power parity exchange rate differed greatly (lower implies a stronger currency): Cameroon 240, Central African Republic 166, Chad 172, Republic of the Congo 677, Equatorial Guinea 114, Gabon 413, Benin 273, Burkina Faso 167.

# Need for PPP adjustments to GDP
Using market exchange rates to compare countries' standard of living or per capita Gross Domestic Product can give a very misleading picture. The exchange rate only reflects traded goods in contrast to non-traded ones. Also, currencies are traded for purposes other than trade in goods and services, e.g., to buy capital assets whose prices vary more than those of physical goods. Also, different interest rates, speculation, hedging or interventions by central banks can influence the foreign-exchange market.
The PPP method is used as an alternative.
For example, if the value of the Mexican peso falls by half compared to the U.S. dollar, the Mexican Gross Domestic Product measured in dollars will also halve. However, this exchange rate results from international trade and financial markets. It does not necessarily mean that Mexicans are half poorer; if incomes and prices measured in pesos stay the same, they will be no worse off assuming that imported goods are not essential to the quality of life of individuals. Measuring income in different countries using PPP exchange rates helps to avoid this problem.
PPP exchange rates are especially useful when official exchange rates are artificially manipulated by governments. Countries with strong government control of the economy sometimes enforce official exchange rates that make their own currency artificially strong. By contrast, the currency's black market exchange rate is artificially weak. In such cases a PPP exchange rate is likely the most realistic basis for economic comparison.

# Difficulties
The main reasons why PPP does not perfectly reflect standards of living are
- PPP numbers can vary with the specific basket of goods used, making it a rough estimate.
- Preferences and choices can vary from country to country. Goods then differ in their contribution to welfare.
- International competitiveness is mainly affected by the exchange rate and not by PPP.
- Differences in quality of goods are not sufficiently reflected in PPP.
- The price of a product within one currency-zone may vary drastically. For example, the average price of a can of soda is drastically lower in Lexington, Kentucky than in New York, New York. This is even more noticeable within the Eurozone.
- Imported goods are purchased at market exchange rates, and thus a country that has to import all of its food would appear better off than it actually is if the PPP is used as the measurement of well-being.
- There is no single global market, so "in an efficient market, identical goods must have only one price" is no better than a platitude. What we find in the real world is a multiplicity of markets. Prices are dependent upon the interplay between supply and demand, and these vary from time to time and place to place.
PPP calculations are often used to measure poverty rates. For problems with this methodology, see How Not To Count The Poor.

## Range and quality of goods
The goods that the currency has the "power" to purchase are a basket of goods of different types:
- Local, non-tradable goods and services (like electric power) that are produced and sold domestically.
- Tradable goods such as non-perishable commodities that can be sold on the international market (e.g. diamonds).
The more a product falls into category 1 the further its price will be from the currency exchange rate. (Moving towards the PPP exchange rate.) Conversely, category 2 products tend to trade close to the currency exchange rate. (For more details of why, see: Penn effect).
More processed and expensive products are likely to be tradable, falling into the second category, and drifting from the PPP exchange rate to the currency exchange rate. Even if the PPP "value" of the Chinese currency is five times stronger than the currency exchange rate, it won't buy five times as much of internationally traded goods like steel, cars and microchips, but non-traded goods like housing, services ("haircuts"), and domestically produced rice. The relative price differential between tradables and non-tradables from high-income to low-income countries is a consequence of the Balassa-Samuelson effect, and gives a big cost advantage to labour intensive production of tradable goods in low income countries (like China), as against high income countries (like Switzerland). The corporate cost advantage is nothing more sophisticated than access to cheaper workers, but because the pay of those workers goes further in low-income countries than high, the relative pay differentials (inter-country) can be sustained for longer than would be the case otherwise. (This is another way of saying that the wage rate is based on average local productivity, and that this is below the per capita productivity that factories selling tradable goods to international markets can achieve. This is sometimes called exploitation.) An equivalent cost benefit comes from non-traded goods that can be sourced locally (nearer the PPP-exchange rate than the nominal exchange rate in which receipts are paid). These act as a relatively cheaper factor of production than is available to factories in richer countries.
PPP calculations tend to overemphasise the primary sectoral contribution and underemphasise the industrial and service sectoral contributions to the economy of a nation.

## Difficulties with PPP comparisons in welfare economics
While using PPP exchange rates for income comparison is an improvement over using market exchange rates, it is still imperfect, and comparisons using the PPP method can still be misleading.
Comparing standards of living using the PPP method implicitly assumes that the real value placed on goods is the same in different countries. In reality, what is considered a luxury in one culture could be considered a necessity in another. The PPP method does not account for this. (This is not primarily a flaw in the exchange rate methodology, as cultural and interpersonal differences in utility functions are a more fundamental microeconomic problem.)
A PPP exchange rate varies depending on the choice of goods used for the index (CPI). Hence, it is possible to deliberately or accidentally bias a PPP exchange rate by the choice of a bundle. Indeed, it may be hard to construct equivalent representative bundles for the consumption habits of very different societies. PPP could also have difficulty accounting for differences in quality between goods in one country and equivalent goods in another, see: consumer price index.
Differing levels of government involvement in social spheres further complicate development of good CPI baskets (and, consequently, PPP measurements). For example, in 1986, nominal GDP of the United States was almost 4 times larger than the nominal GDP of the Soviet Union (on a per capita basis). Direct comparison failed to capture, however, that the Soviet Union provided free secondary and higher education and free healthcare to all its citizens, whereas Americans had to pay for education and healthcare themselves. To properly account for differences in quality of life in this situation, the CPI basket would have to include these expenditures explicitly. More importantly, government subsidies can potentially have large effect on consumption levels (free higher education will result in more college graduates), making it difficult to choose weights for individual components of CPI.
Even if a good PPP is used, GDP per capita is still a measure of the economic output of the whole economy, not a direct measure of the mean or median person's quality of life. Other factors such as the standards of homes and schools, access to public services, the extent of pollution, and strength of consumer protection laws are hard to quantify and generally not fully reflected in the GDP. Even a PPP-adjusted measure of GDP per capita must be used with caution, for all the usual reasons that the GDP figure itself is limited (for instance, its inability to capture the surplus between subjective value and payment price).
For example, in 2002, the nominal GDP per capita in Japan was about US$40,000, while the equivalent PPP into a U.S. goods basket was estimated at $27,000. In the U.S., GDP per capita was about $36,400 (nominal and real if based on 2002 dollars). This means that the average U.S. citizen could enjoy slightly more consumption than the average Japanese (vastly more if private saving is removed from consumption income). However, it does not necessarily follow, that this implies a "higher standard of living" in the sense of "enjoying life" more; the U.S. has higher crime rates and less social cohesion than Japan, while Japan has much less physical space per person and arguably less individual freedom. Ultimately, the quality of life will depend on subjective judgement and individual preferences.
While per-capita income does not take into account inequalities in wealth distribution, neither does the PPP-scaled income.

## Clarification to PPP Numbers of the IMF
The GDP number for all reporting areas are one number in the reporting areas local currency. Therefore, in the local currency the PPP and market (or government) exchange rate is always 1.0 to its own currency, so the PPP and market exchange rate GDP number is always per definition the same for any duration of time, anytime, in that area's currency. The only time the PPP exchange rate and the market exchange rate can differ is when the GDP number is converted into another currency.
Only because of different base numbers (because of for example "current" or "constant" prices, or an annualized or averaged number) are the USD to USD PPP exchange rate not 1.0, see the IMF data here: . The PPP exchange rate is 1.023 from 1980 to 2002, and the "constant" and "current" price is the same in 2000, because that's the base year for the "constant" (inflation adjusted) currency.