The will be the same in every country.

The Purchasing Power Parity (PPP) is
undoubtedly one of the most tested theory and an important approach to the
theory of exchange rate determination. The law of one price is at the core of
the PPP condition, it states that:

        

(1)

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Where  is the price level in the domestic country
relating to the good or service number ,  is the foreign price level relating to the
good or service number  and  the nominal spot exchange rate. Equation 1 essentially
says that the same good should have the same price in different countries if
the prices are expressed in the same currency. The rational for the law of one
price condition to hold is that arbitrage will take place if price differences
for perfect substitutes are observed. We assume that equation 1 holds with
respect to all goods and services consumed in the two countries.

By assuming that the weights used in
compiling the respective price indices are identical, then a general equation is
derived where equation 1 applies equally to the general price levels and not to
individual goods or services:

 

(2)

Equation 2 is known as the absolute
PPP, it states that the general level of prices, when converted to a common
currency, will be the same in every country. The PPP in its absolute form is
the application of the law of one price to the aggregate level.

Solving for S:

 

(3)

The nominal exchange rate between
two currencies is, according to the PPP, equal to the ratio of aggregate price
levels between the two countries. Consequently, the purchasing power of a unit
of one currency would be the same in both economies.

But for this equation to hold, three
conditions need to be met:

The goods of each country must be freely tradable on the
international market.The price index for each of the two countries must be comprised of
the same basket of goods.All of the prices need to be indexed to the same year.

In its weaker form, called the
relative PPP, equation 2 is reformulated in terms of rate of growth by taking
the logs of the values and their derivatives:

 

(4)

Equation 4 says that the home
country’s inflation rate, , is equal to
the sum of the foregin inflation rate  and the rate of currency depreciation, .

It can be rearranged to:

 

(5)

Thus, equation 5 shows the
relationship between inflation rate and depreciation: the higher inflation, the
lower the value of money. In a two-country context, the more rapidly prices
rise in the home country relative to the foreign country, the more rapidly the
domestic currency loses value relative to the foreign country; that is the more
rapidly its exchange rate depreciates. So the relative PPP sees exchange rates
as a compensation for inflation differentials.

Distinguishing the absolute PPP from
its relative form has important consequences when it comes to test the PPP
hypothesis: using the relative PPP avoids the difficulties of choosing a base
for the price index.

The extent to which the PPP condition
may hold can be immediately objected by the presence, for instance, of
transaction costs – arising from taxes, tariffs, transportation costs…- that
would inevitably make the PPP equation impossible to hold. If for instance
transportation costs were to be observed, then equation 1 would not hold since
transportation costs would need to be added. Another theory that directly explains
deviation from the PPP is the presence of non-tradeable goods (Balassa, 1964;
Samuelson, 1964). Because no arbitrage opportunity can arise with the presence
of non-traded goods in different countries.

Also, in theory, it appears that the
PPP is essentially a long-term hypothesis: in the short term, it does not make
much sense as prices are sticky while nominal exchange rates can jump up and
down. For example, a temporary money supply expansion will temporarily
depreciate the country’s exchange rate leaving both the domestic and foreign
price levels unchanged. Therefore, the real exchange rate will temporarily
depreciate, decreasing the domestic price level with respect to the foreign
price level, both expressed in the same currency: this means that in the short
run PPP will not hold. This reasoning is confirmed empirically by Kim and Lima
(2010): one needs to differentiate short-term from long-term when testing the
PPP since prices cannot adjust in the short term. Deviations from the PPP in
the short-run are also confirmed by the observed short-term high volatility of
real exchange rates and their high level of persistence (Rogoff, 1996).

For the PPP to hold, a constant real
exchange rate much be observed, which is not the case in reality. This is why
most empirical studies focus on the long-term PPP. It allows for short-run
deviations of the PPP arising from the above reasons. But while the invalidity
of the short-term PPP is commonly admitted, the debate is more vivid when it
comes to admit whether the PPP, in some forms, holds in the long-run. The empirical
evidences found on the long-term PPP depend very much on the methodology
chosen.

The early empirical testings of the
PPP were usually based on estimates of equations of the form:

 

(6)

where  is the error term, and where the absolute PPP
would depict  et .

Frenkel (1978) does find convincing
estimates of  and  on high inflation countries, suggesting that the
PPP explains exchange rates in the long-run. But the estimates are supportive
of the PPP only in hyperinflationary economies, in countries facing normal
inflation, the PPP is rejected. Also, Frenkel does not test for stationarity: this
critic founds the core of the next empirical tests of the PPP.

To counter this caveat, empirical
tests started to concentrate on whether the exchange rate itself is stationary,
thus validating the long-run PPP, or whether it follows a unit root process;
i.e. the exchange rate follow a random walk pattern. Fleissig and Straaus (2000)
are unable to reject the stationarity hypothesis for six price indices
differing in their traded and non-traded components. They thus validate the
long-term PPP and the Balassa-Samuelson effect since stationarity is higher for
price indices with largely proportion of traded goods.

The literature also looked for
evidence of co-integration between prices and real exchange rates. This kind of
regression thus tests whether there is a mean reverting pattern depicted by
exchange rates, hence testing the long-run PPP. Taylor (1988) for example
rejects the unit root hypothesis for five major exchange rates: the PPP
hypothesis does not fit the fact, even in the long-run. Even when taking into
consideration transaction costs and measurement errors, he fails at finding a
long-run proportionality between exchange rates and relative prices. Enders
(1988) uses both ARIMA and cointegration and finds mixed evidence of PPP: depending
on the countries tested, he either rejects the stationarity hypothesis or is
unable to reject it.

It seems that the difference in data
used to assess the PPP has an important impact on the results. Some argue that
data sample should be very large to allow mean-reversion. Frankel (1986) is
able, by using 116 year of data on the US-UK exchange rate, to reject the
random walk hypothesis and to find a statistically significant speed of
adjustment to the PPP: 5 years. The PPP may have to be tested over very long
periods. A similar long-period test is conducted by Lothian and Taylor (1996).
They use two centuries annual data for dollar-sterling and franc-sterling real
exchange rates and find that they are both significantly mean-reverting. It
appears however that the mean-reversion is a relatively slow process: 3 years
for franc-sterling and 6 years for dollar-sterling.

Overall, the PPP is both
theoretically and empirically believed not to hold in the short-run since
prices are sticky. Many real world factors also prevent the PPP from holding: The
PPP cannot hold when obstacles to free trade, such as tariffs or transaction
costs, possibly in combination with monopolistic limits to free competition,
prevent the law of one price for one or many goods to hold. Also, the PPP
cannot hold if different basket of goods are used across countries to measure  and . The presence
of non-tradable goods also prevent the PPP from holding. All in all, obstacles
to the PPP are such that the relationship never really holds: even Frenkel’s
estimates (1978) are not perfectly validating the PPP. The empirical results on
the long-run PPP are mixed, the main obstacles to consensus on the matter are
more of an econometrical nature, notably on the data used. The poor performance
of the PPP empirically does not necessarily mean that the PPP is worthless, it
is a useful approximation of the ongoing relationship between price levels and
exchange rates.