The idea of purchasing power parity (PPP) is quite simple. A menu of common prices is used to determine at what exchange rate the same basket of goods costs the same amount in all countries. This exchange rate $\mathit{\epsilon}$ is called purchasing power parity or parity exchange rate.

$${P}_{\text{goodsbasket}}^{\mathit{USA}}=\mathit{\epsilon}\cdot {P}_{\text{goodsbasket}}^{\mathit{Germany}}\iff \mathit{\epsilon}=\frac{{P}_{\text{goodsbasket}}^{\mathit{USA}}}{{P}_{\text{goodsbasket}}^{\mathit{Germany}}}$$ |

Thus, for the exchange rate $\mathit{\epsilon}$, purchasing power parity applies, i.e. $\mathit{\epsilon}$ is a value for the nominal exchange rate, so that

- 1$ in the USA buys the same basket of goods as
- 1$, converted into Euro at the rate $\mathit{\epsilon}$, buys in Germany.

So on average, goods cost the same in both countries if the nominal exchange rate is $\mathit{\epsilon}$.

The purchasing power parity represents a natural equilibrium level for the exchange rate. The law of uniform price says that (under certain conditions, such as the absence of transport- and trade- costs) a good has the same price everywhere, otherwise arbitrage would occur.

What does this mean? Let’s imagine we have a good that is sold cheaper in country A than in country B. Then traders would buy the good cheaply in country A and sell it more expensively in country B and thus make a profit. This would have three effects on (1) the market in A, (2) the market in (B) and (3) the currency market. All three effects work towards an alignment of prices.

- In A the demand for the good increases because the traders buy it in addition to the local demand. Thus, the price increases.
- In B, the supply of the good increases because the traders offer it in addition to the local supply. Thus, the price decreases.
- For their transaction, the traders need currency A and receive currency B when they sell. So they have to exchange currency B for A. This means that the supply of B increases and the demand for A increases. As a result, the exchange rate increases and currency A appreciates (becomes more expensive).

Thus, the prices converge: the cheap good in A becomes more expensive, the expensive good in B becomes cheaper and from the point of view of country A, the increase in currency value makes the good in B even cheaper (or from the point of view of country B, the good in A becomes more expensive because the currency of B is now worth less). This process continues until the prices have converged.

Although the arbitrage idea behind the purchasing power parity theory is very simple, clear and intuitive, empirical evidence often provides very weak results. The PPP is only considered a long-term relationship, since the underlying assumptions are violated several times:

- Price rigidities: In reality, prices do not adjust instantly and completely.
- Transportation costs: Arbitrage opportunities are limited by transportation costs, since the transportation of real goods over spatial distances and obstacles is costly.
- Non-tradable goods: There are goods and services that by their nature cannot be traded – for example, a haircut or a hot lunch – or that cannot be traded due to legal regulations – such as certain weapons.
- Different baskets of goods, monopolistic or oligopolistic practices, demand preferences of relevant dimensions: All of these have an influence on price formation and can possibly distort it.

Another important limiting factor is the volume of capital-market-induced currency transfers. This is many times higher than the goods-induced volume and thus, at least in the short term, has a stronger influence on exchange rate developments than PPPs. This means that there can be systematically frequent and larger short-term deviations from the purchasing power parity, especially in the case of freely floating exchange rates.

Logs and differences

The use of purchasing power parity in macroeconomic models requires the presentation in a formal way. In addition, we always consider the representation in level values as well as in logarithmic values. Purchasing power parity means that the nominal exchange rate exactly compensates for the difference in price levels:

$$P=S{P}^{\ast}$$or in log form

$$p=s+{p}^{\ast},$$where lower case letters denote logarithmic variables. The real exchange rate is:

$$Q=\frac{S{P}^{\ast}}{P}\text{or}q=s+{p}^{\ast}-p.$$With purchasing power parity, of course, the following applies:

$$Q=1\text{or}\mathrm{log}\left(Q\right)=q=0$$

If you now look at the change compared to the previous year (here we limit ourselves to the log display, as this is simpler), you will see that

$${p}_{t}-{p}_{t-1}={s}_{t}-{s}_{t-1}+{p}_{t}^{\ast}-{p}_{t-1}^{\ast}\text{orsimpler}d{p}_{t}=d{s}_{t}+d{p}_{t}^{\ast},$$ where $d{X}_{t}$ is the difference
of the variable $X$ to the
previous period. If $x$ is a
logarithmic value, then $d{x}_{t}$
represents the rate of change of the level value
$X$.^{3}
$d{p}_{t}$ is thus the domestic
inflation rate ${\pi}_{t}$,
$d{p}_{t}^{\ast}$, correspondingly, the
foreign inflation rate, and $d{s}_{t}$
is the depreciation rate of the exchange rate. So, we get

Thus, the rate of depreciation corresponds exactly to the difference between the inflation rates.

The formalization of transaction costs for level values happens by means of a factor $K$, which enters the model as a surcharge on the foreign prices $P=\mathit{KS}{P}^{\ast}$ and becomes as a log value an additive term $q=k+s+{p}^{\ast}-p$.

Tradable and non-tradable goods

If non-tradable goods are explicitly included, the price index must reflect them.

$$P={P}_{T}^{c}\cdot {P}_{\mathit{NT}}^{1-c}\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}\text{orinlogform:}\phantom{\rule{0.33em}{0ex}}\phantom{\rule{0.33em}{0ex}}p=c{p}_{T}+\left(1-c\right)\cdot {p}_{\mathit{NT}}$$ |

^{3}$d{x}_{t}={x}_{t}-{x}_{t-1}=\mathrm{log}{X}_{t}-\mathrm{log}{X}_{t-1}=\mathrm{log}\frac{{X}_{t}}{{X}_{t-1}}\approx \frac{{X}_{t}}{{X}_{t-1}}-1=\frac{{X}_{t}-{X}_{t}}{{X}_{t-1}}$ is the
absolute change of $X$
relative to the initial value.

(c) by Christian Bauer

Prof. Dr. Christian Bauer

Chair of monetary economics

Trier University

D-54296 Trier

Tel.: +49 (0)651/201-2743

E-mail: Bauer@uni-trier.de

URL: https://www.cbauer.de