Why Compare Exchange Rates With Price Levels?
An exchange rate tells us how many units of one currency trade for another. By itself, it does not tell us whether a currency buys more or less than another currency in its own country. Absolute purchasing power parity (absolute PPP) supplies a benchmark: if comparable baskets cost the same after conversion, the exchange rate equals the ratio of their national-currency prices.
The benchmark is useful for organizing a price comparison, not for asserting that market exchange rates always equal the calculated rate. Taxes, transport costs, trade barriers, differences in consumption patterns, and goods that are not traded across borders can all separate observed prices and exchange rates from the simple PPP relationship.
Deriving the Implied Rate
The relationship follows from comparing the cost of the same basket in a common currency. At rate \(e\), one unit of foreign currency costs \(e\) domestic-currency units. A basket priced at \(P^*\) in foreign currency therefore costs \(eP^*\) in domestic currency after conversion. If absolute PPP holds, that converted cost equals the domestic basket price \(P\). Solving \(P=eP^*\) for \(e\) gives \(e_{\mathrm{PPP}}=P/P^*\).
This argument uses the law of one price: in the frictionless benchmark, the same good should have the same price in a common currency. Applying that idea to a basket requires the basket to be comparable across countries. The prices must be measured for the same quantities and quality of goods, and both price levels must refer to the same time period.
Use domestic currency per unit of foreign currency. This makes the implied rate \(P/P^*\).
Confirm that \(P\) and \(P^*\) price the same basket in their respective currencies.
Divide the domestic basket price by the foreign basket price.
Convert any market quote to the same orientation before calculating its deviation from \(e_{\mathrm{PPP}}\).
Worked Comparisons
Worked Example: Comparing a Market Quote With PPP
A comparable basket costs 240 units of domestic currency A in Country A and 160 units of foreign currency B in Country B. The observed exchange rate is 1.65 A per B. Calculate the PPP-implied rate, the percentage deviation of the observed rate from that benchmark, and the converted foreign basket cost. State the method and interpret the result.
Method. Use the domestic-per-foreign quote throughout. First calculate the implied rate; then compare the observed rate with it as a percentage of the implied rate. This avoids confusing the direction of the currency comparison with the direction of the price comparison.
The PPP-implied rate is $$ e_{\mathrm{PPP}}=\frac{P}{P^*}=\frac{240\ \mathrm{A}}{160\ \mathrm{B}}=1.50\ \mathrm{A/B}. $$
The observed rate is 1.65 A/B, so the difference is $$ 1.65-1.50=0.15\ \mathrm{A/B}. $$
As a percentage of the PPP-implied rate, the deviation is $$ \frac{1.65-1.50}{1.50}\times 100\%=10.00\%. $$
At the observed rate, the foreign basket costs $$ 160\ \mathrm{B}\times 1.65\ \mathrm{A/B}=264\ \mathrm{A}. $$ This is 24 A more than the domestic basket price of 240 A. The observed rate is 10.00% above the PPP benchmark, and the foreign basket is more expensive after conversion. Under this quote convention, the domestic currency is weaker than the PPP benchmark indicates. The basket-cost check agrees with the rate comparison.
Worked Example: Converting a Reverse-Quoted Rate
A basket costs 315 units of currency C in the domestic country and 250 units of currency D in the foreign country. A market screen quotes 0.76 D per C, the reverse of the domestic-per-foreign convention. Find the implied PPP rate, express the market quote in the required orientation, and calculate the percentage deviation. Give the deviation to two decimal places.
Method. Calculate \(e_{\mathrm{PPP}}\) as domestic currency per foreign currency, then invert the market quote. Comparing 0.76 D/C directly with a rate in C/D would be invalid because the units differ.
The PPP-implied rate is $$ e_{\mathrm{PPP}}=\frac{315\ \mathrm{C}}{250\ \mathrm{D}}=1.26\ \mathrm{C/D}. $$
The market quote is 0.76 D/C, so its reciprocal is $$ e_{\mathrm{market}}=\frac{1}{0.76}=1.315789\ldots\ \mathrm{C/D}. $$
The market rate exceeds the PPP rate by $$ 1.315789\ldots-1.26=0.055789\ldots\ \mathrm{C/D}. $$
Relative to the PPP benchmark, the percentage deviation is $$ \frac{1.315789\ldots-1.26}{1.26}\times 100\% =4.4277\ldots\%. $$
Rounded only at the final step, the observed rate is 4.43% above the PPP-implied rate. As a check, the foreign basket costs \(250\times 1.315789\ldots=328.9473\ldots\) C at the market rate, more than the domestic price of 315 C. The signs agree: the market quote, once correctly oriented, makes foreign currency more expensive than the PPP benchmark.
Interpreting Deviations Carefully
For an observed rate \(e_{\mathrm{obs}}\) expressed in domestic currency per foreign currency, a convenient measure of the deviation is $$ \text{percentage deviation from PPP} =\frac{e_{\mathrm{obs}}-e_{\mathrm{PPP}}}{e_{\mathrm{PPP}}}\times 100\%. $$ A positive value means the observed rate is above the PPP-implied rate; a negative value means it is below. This percentage describes the rate's distance from the PPP benchmark. It is not automatically the percentage by which either currency is “overvalued” or “undervalued” under every possible convention. State the quote convention and explain the comparison.
A related check uses the real exchange rate, with the convention \(q=eP^*/P\). Here \(e\) is domestic currency per foreign currency, \(P^*\) is the foreign basket price, and \(P\) is the domestic basket price. At absolute PPP, the converted foreign basket and domestic basket cost the same, so \(q=1\). If \(q>1\), the converted foreign basket is more expensive; if \(q<1\), it is cheaper. This gives a direct price interpretation of the deviation.
Worked Example: Building Basket Prices From Quantities
A common hypothetical basket contains two units of grain, one transit pass, and three units of household fuel. In Country M, the respective prices are 18, 24, and 8 units of currency M. In Country N, they are 10, 15, and 5 units of currency N. The observed rate is 1.50 M/N. Calculate both basket costs, the PPP-implied rate, and the percentage deviation of the observed rate. Interpret the converted price comparison.
Method. Price the same quantities in each country before taking the ratio. This prevents the mistake of comparing single-item prices or baskets with different weights.
The Country M basket costs $$ P_M=2(18)+1(24)+3(8)=36+24+24=84\ \mathrm{M}. $$
The Country N basket costs $$ P_N^*=2(10)+1(15)+3(5)=20+15+15=50\ \mathrm{N}. $$
Thus the PPP-implied rate, in M per N, is $$ e_{\mathrm{PPP}}=\frac{84\ \mathrm{M}}{50\ \mathrm{N}}=1.68\ \mathrm{M/N}. $$
The observed rate is 1.50 M/N, so its percentage deviation is $$ \frac{1.50-1.68}{1.68}\times 100\% =\frac{-0.18}{1.68}\times 100\% =-10.714285\ldots\%. $$
The observed rate is therefore 10.71% below the PPP-implied rate, to two decimal places. At the observed rate, the foreign basket costs \(50\times1.50=75\) M, compared with 84 M for the domestic basket. The foreign basket is cheaper after conversion. The real-exchange-rate check is \(q=1.50(50)/84=75/84=0.892857\ldots<1\), which confirms the interpretation.
Assumptions and Limits of the Benchmark
Absolute PPP is clearest as a benchmark when the two prices refer to an identical basket, measured at the same time, and the exchange rate uses the stated currency orientation. Its simplest theoretical form also assumes that goods can be traded and that transport costs, tariffs, taxes, and other barriers do not prevent prices from being brought into line. If these conditions fail, different basket costs need not imply a calculation error.
In practice, national price measures may cover different goods, qualities, or expenditure weights. Many services are local rather than internationally traded, and price levels can differ because wages, rents, taxes, and distribution costs differ. A calculation using broad national price indexes should therefore be interpreted as a comparison under the chosen index measures, not as proof that every individual price obeys the law of one price.
There is also an important distinction between a price level and a normalized price index. If two indexes are each set to 100 in their own base year, dividing one index by the other does not necessarily produce an absolute PPP exchange rate: the index values record changes from separate reference points, not necessarily comparable prices for the same basket in common base conditions. The formula \(P/P^*\) requires price levels that are meaningfully comparable, such as the actual costs of the same specified basket.
Common Mistake: Mixing Quote Directions
Suppose the domestic basket costs 240 domestic units and the foreign basket costs 160 foreign units. The PPP rate in domestic currency per foreign currency is \(240/160=1.50\). A common wrong calculation uses \(160/240=0.667\) and compares it directly with an observed quote of 1.65 domestic per foreign. That reverses the units: 0.667 is foreign per domestic, while 1.65 is domestic per foreign.
The correction is to keep the orientation consistent. Either use \(1.50\) domestic per foreign and compare it with \(1.65\) domestic per foreign, or invert both rates and compare foreign per domestic. Never infer which currency is stronger from a numerical rate without first checking the quote units.
Check Your Understanding
For each problem, use the stated quote convention and compare the observed exchange rate with the absolute-PPP benchmark.
- A basket costs 198 units of domestic currency R and 150 units of foreign currency S. Find the PPP-implied rate in R/S. (Answer: 1.32 R/S)
- The prices of a comparable basket are 420 domestic units and 350 foreign units. The observed rate is 1.20 domestic per foreign. Find the percentage deviation from PPP and say whether the converted foreign basket is more or less expensive. (Answer: 0.00%; the baskets cost the same after conversion.)
- A domestic basket costs 280 units of currency U and a foreign basket costs 200 units of currency V. The market quote is 0.70 V/U. Convert it to U/V, find its percentage deviation from PPP to two decimal places, and interpret the basket comparison. (Answer: 1.4286 U/V; 2.04% above PPP; the foreign basket is more expensive after conversion.)
- A common basket has one device and two meal vouchers. In Country G, they cost 50 and 12 units of G; in Country H, they cost 40 and 10 units of H. The observed rate is 1.10 G/H. Calculate the two basket costs, the PPP rate, and the observed rate's percentage deviation to two decimal places. (Answer: 74 G and 60 H; 1.2333 G/H; −10.81%, so the foreign basket is cheaper at the observed rate.)
- A comparable basket costs 360 domestic units and 300 foreign units. An observed rate of 1.08 domestic per foreign is reported. Calculate the implied rate, the percentage deviation, and the real exchange rate \(q=eP^*/P\); state whether the foreign basket is dearer or cheaper after conversion. (Answer: 1.20 domestic per foreign; −10.00%; \(q=0.9000\); the foreign basket is cheaper.)