The purchasing power parity (PPP) is a non-arbitrage condition. It states that a basket of identical goods should have the same price, adjusted for exchange rate, across different countries. Although this idea has a history dating back at least to scholars at the University of Salamanca in the 15th and 16th centuries, the term purchasing power parity was first used by Gustav Cassel (1922, 1928), co-founder of the Swedish School of Economics together with Knut Wicksell and David Davidson. In particular, Cassel studied the gold parities of the currencies of WWI participating countries with the specific purpose of estimating the equilibrium exchange rates (Colombo and Lossani, 2003).

1. Purchasing power parity (PPP) and the law of one price (LOOP

The PPP is a generalization of the law of one price (LOOP). The LOOP says that the same good sold in different countries should have the same price when expressed in common currency. In short: if one unit of domestic currency could buy a larger amount of good x than it would be affordable within the national economy, then it would be profitable to buy x abroad and sell it inside, making profits on the difference between the prices. This arbitrage (repeated over time by multiple subjects) implies the convergence of international prices of the goods considered. In the real world, the PPP is not always verified. However, this theory provides a benchmark for monitoring exchange rate movements. For example, suppose that, based on the comparison of prices for the U.S.A. and the Eurozone, the exchange rate consistent with the PPP is equal to 0.85 (euro for one dollar), but in reality the exchange rate determined on the Forex market is 0.75. In this case, it is possible to conclude that the euro is overvalued against the dollar. Over the next few paragraphs, we will have the opportunity to clarify what can determine this discrepancy and how the values of the PPP can be restored.

Formally, the PPP implies that:

-  is the price index for the domestic basket of goods ( is the weight of the j-th item in the domestic basket);
-  is the price index for the foreign basket of goods (is the weight of the i-th item in the foreign basket);
- St is the nominal exchange rate (NER) that indicates how much one unit of foreign currency is worth in terms of the domestic one.
Equation (1) defines the so-called absolute purchasing power parity to distinguish it from its "soft" version known as relative purchasing power parity. Indeed, the relative PPP is said to hold when the rate of depreciation of one currency with respect to another is proportional to the difference in terms of inflation rates. The Relative PPP can be defined as follows:
where c is a positive constant. 
Whether it is considered in its absolute version (1) or in its relative version (2), the PPP theory shows an interesting dynamic relationship between exchange rate and price level. By using the logarithms and differentiating with respect to time yields:

-  is the nominal depreciation rate;
-  is the domestic inflation rate;
-  is the foreign inflation rate.
From equation (3) it is easily seen that, if the PPP holds, the greater is the spread between domestic and foreign inflation rate, the higher the depreciation of the domestic currency will be.

2. Purchasing power parity (PPP) and the real exchange rate (RER)

Starting from the PPP, we can define the RER as a measure of the number of baskets of domestic goods necessary for the purchase of a basket of foreign goods, in formulas:
Looking at equation (4), it is clear that if the PPP is respected, and therefore the NER moves in the same direction of inflation rates differential, then the real exchange rate (RER) remains constant. If the price of the currency (represented by the NER) reflects the purchasing power, then the RER remains constant over time. In formulas, using logarithms and differentiating with respect to time:
Hence, if the PPP holds, then the RER is constant, so that movements in the real exchange rate represent deviations from the PPP.

3. The Big Mac Index 

An index widely used to determine whether a currency is appreciated or depreciate with respect to another
is the so-called Big Mac Index, based on prices of the famous sandwich. Published by The Economist since 1984, this index, based on the LOOP, is used to calculate the PPP for a large number of exchange rates using the Big Mac as reference good.For example, let's consider the table in figure 2 (source: Economist May 8th, 2009). From the comparison of the Big Mac prices in the U.S.A. and in the EU, it emerges that the European currency, traded at 0.96 against the dollar, is overestimated. Indeed, the (average) price of the sandwich in Europe (U.S. $ 4.38) is more than 24 higher than the price paid in the U.S.A. (US$ 3.54). 
As clearly shown by the figure, the index provided by The Economist indicates a significant misalignment of nominal exchange rates.

Data concerning the Big Mac Index suggest that the LOOP is not verified. This failure is due to the fact that both the LOOP and the PPP assume the existence of competitive markets and the absence of transaction costs. Thus, to evaluate properly the appreciation or depreciation of a currency, it would be necessary that the goods traded were produced by using only inputs that can be easily traded on international markets and that there were no trade barriers, transport costs and differences in tax regimes. The real world is, unfortunately, very different. Indeed, despite the fact that the weight of intermediate inputs has been significantly reduced through globalization, strong differences remain in the cost of those inputs that are not (easily) traded internationally - eg. labour and physical capital - and that play an important role also in the production of standardised goods, such as the Big Mac. While input prices traded on international markets (such as onion, for example) converge rapidly towards the theoretical values of the PPP, this is not the case for the cost of labour and other non-traded inputs that contribute significantly to the formation of the Big Mac price.
More generally, it is possible to identify four factors that can determine deviations from the PPP.
1) Transaction costs: the presence of transport costs and insurance costs, tariff and non-tariff barriers, etc. implies that the LOOP does not hold and reduces the chance of profit arising from international trade.
2) Differentiation: the LOOP, upon which the PPP is based, assumes that the good traded on different national markets is homogeneous. The real world instead provides many examples of differentiated goods. This differentiation prevents prices from equalising across countries.
3) Fixed capital formation: the presence of high fixed costs (distribution contracts) prevents the trader from taking advantage of arbitrage opportunities (Sarno and Taylor, 2002).
4) Non-traded goods: the LOOP implicitly assumes that all goods are tradable, but in fact a very significant part of the income is spent on the purchase of non-traded goods on international markets, or for which it is not possible to perform arbitrage operations (e.g. housing, medical services, education, etc.). 

4. Empirical evidence

There is extensive empirical literature on the LOOP and the PPP. Generally, these econometric studies suggest a rejection of the LOOP for a wide range of goods. 
The PPP implies that the change in the exchange rate in a certain period of time offsets the difference between domestic and foreign inflation. This implies that the RER is mean reverting, or, in other words, that any shock – such as a sudden change of the NER – is absorbed over time. In formal terms, many studies tried to verify whether the relative price index followed the trend described by the equation:
or that described by a random walk
A systematic survey of the empirical studies on the PPP should be organized according to the econometric methodology applied.1
1. Papers based on co-integration analysis have provided heterogeneous results (Enders, 1988; Mark, 1990). By
using the so-called "two-step Engle-Granger method", a co-integration analysis is usually applied to the following equation:
if st, pt and  are integrated of order 1, then the relative PPP exists if is stationary. The absolute PPP requires also that the parameter restrictions = 1 and = -1 be satisfied.
Early co-integration studies have generally shown the absence of significant mean-reversion of the exchange rate towards the PPP for the recent floating rate experience, but they have found support for the reversion towards the PPP for the inter-war float and for high inflation countries (Sarno and Taylor, 2002).
2. Another approach to test the RER mean reversion hypothesis is provided by the Augmented Dickey-Fuller (ADF) test for a unit root, which is generally based on a regression:
where  is a p-th order polynomial in the lag operator L and  is a white noise disturbance. Testing the null hypothesis that = 0 would imply no long-run equilibrium level for the RER. A further approach proposed by Meese and Rogoff (1988) tested the alternative hypothesis .
These empirical studies generally can not reject the random walk hypothesis for the RER
3. An alternative method to test the non-stationarity of the RER is based on the variance ratio test (VR) developed by Cochrane (1988).
According to this approach, the persistence of the RER is measured by using a simple non-parametric test, :
where k is a positive integer and var stands for variance.
If the RER follows a random walk, then the ratio (10) must be equal to 1; on the contrary, if the RER is mean reverting then 1. By using VR, Huizinga (1987) found that the permanent component of the RER is around 60% (Sarno and Taylor, 2002).
4. A way to circumvent the problems of low power shown by the standard unit-root test is to infer from the panel data. The first attempt in that direction is due to Hakkio (1984) who employed Generalized Least Square (GLS) on data from the UK, France, Germany and Japan. Also in this case, the random walk cannot be rejected.
In conclusion, three observations can be drawn from this brief review. Firstly, it is evident that the empirical results on the validity of the PPP depend substantially on the econometric technique used. Secondly, the literature provides empirical support to the PPP only in the long run, confirming the existence of a slow mean reversion of the RER. In the short term, the dynamics of the real exchange rate are linked to fluctuations of the NER and therefore they exhibit a high volatility. Finally, although many studies have highlighted the tendency of the real exchange rate to converge in the long run towards the values of the PPP, the low speed at which they converge after the shocks raises some theoretical doubts. In other words, even if in the long run each currency reflects its purchasing power, empirical evidence indicates that a period from 3 to 5 years may be required before the exchange rate converges to the equilibrium established by the PPP. Along this line, Rogoff (1996) pointed out that there is a purchasing power parity puzzle due to the contrast between the volatility of short term revealed by the exchange rates and the extreme slowness with which they adjust after shocks.
For further details, see Giovannetti (1992), Sarno and Taylor (2002) and Arese-Visconti (2002).

Arese-Visconti G., (2002), "Tassi di Cambio Reale: Teoria ed Evidenza Empirica," Econometrics Working Papers Archive, wp n. 2002/11, Università degli Studi di Firenze, Dipartimento di Statistica "G. Parenti".
Cassel G., (1922), Money and Foreign Exchange after 1914, Macmillan, London.
Cassel G., (1928), Post-War Monetary Stabilization, Columbia University Press, New York.
Cochrane J. (1988), "How Big is the Random Walk in GNP?", Journal of Political Economy, Vol. 96, pp. 893-920.
Colombo E. and Lossani M. (2003), Economia Monetaria Internazionale, Carocci editore, Roma.
Frenkel J.A. (1978), "Purchasing Power Parity: Doctrinal Perspectives and Evidence from the 1920s", Journal of International Economics, Vol. 8, pp.169-191.
Giovannetti G. (1992), "A Survey of Recent Empirical Tests of the Purchasing power parity Hypothesis," BNL Quarterly Review, Vol. 180, pp. 81-101.
Hakkio C.S. (1984), "A Re-examination of Purchasing Power Parity: a Multi-Country and Multi-Period Study", Journal of International Economics, Vol. 17, pp. 265-277.
Huizinga J. (1987), "An Empirical Investigation of the Long-Run Behaviour of the Real Exchange Rates," Carnegie-Rochester Conference Series on Public Policy, Vol. 27, pp. 149-214.
Mark N. (1990), "Real and Nominal Exchange Rates in the Long-Run: An Empirical Investigation," Journal of International Economics, Vol. 28, pp. 115-136.
Rodrik D., (2008), "The purchasing-power parity doctrine: A reappraisal" Brookings Papers on Economic Activity, Fall 2008, Vol.2, pp. 365-412.
Rogoff K., (1996), "The purchasing-power parity puzzle" Journal of Economic Literature, Vol 34, pp. 647-668.
Samuelson P.A. (1964), "Theoretical notes on trade problems", The Review of Economics and Statistics, Vol 46, pp.145-154.
Sarno L. and Taylor M.P. (2002), The Economics of exchange rate, Cambridge University Press, Cambridge, UK.
Editor: Lorenzo CARBONARI