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VEE Economics · Exchange Rates and Open Economies · Tutorial 889 of 998

Deriving Relative Purchasing Power Parity

Derive the exact relative PPP relationship and use domestic and foreign inflation to predict exchange-rate changes under a clearly stated quotation convention.

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What You'll Learn

  • Define the exchange-rate quotation used in relative purchasing power parity
  • Derive the exact relationship between price-level changes and exchange-rate changes
  • Distinguish the exact inflation ratio from the inflation-differential approximation
  • Apply relative PPP to compounded inflation over multiple years
  • Infer an inflation rate from an observed exchange-rate change
  • Avoid quote-direction and percentage-point calculation errors

Why Relative Purchasing Power Parity Matters

Inflation changes the prices of goods at home and abroad. If the domestic price level rises faster than the foreign price level, purchasing-power parity predicts that the exchange rate will adjust so that the foreign currency costs more in domestic currency. Relative purchasing power parity (relative PPP) expresses this prediction as a relationship between the rates of change in price levels and the percentage change in the exchange rate.

The key is to keep the quotation convention fixed. In this tutorial, the exchange rate is the number of units of home currency per one unit of foreign currency. With that convention, a rise in the exchange rate means the foreign currency appreciates against the home currency. Reversing the quotation reverses the direction of the prediction.

Definition: Let \(E_t\) be units of home currency per unit of foreign currency at time \(t\), and let \(P_{H,t}\) and \(P_{F,t}\) be the home and foreign price levels. Relative purchasing power parity states that the percentage change in \(E\) over a period equals the ratio of the home price-level growth factor to the foreign price-level growth factor, less one: $$ \frac{E_1}{E_0}=\frac{P_{H,1}/P_{H,0}}{P_{F,1}/P_{F,0}}. $$

Absolute PPP, covered in the earlier material on purchasing power parity, relates the exchange rate to the ratio of price levels at a particular date. Relative PPP focuses instead on how those quantities change between two dates. It does not require the exchange rate to equal the price-level ratio at the starting date; it requires the exchange rate and relative prices to move consistently over the period.

Deriving the Exact Relationship

Suppose absolute PPP holds at both the start and end of a period, using comparable price indexes for the two countries. Under the stated quotation convention, it gives \(E_0=P_{H,0}/P_{F,0}\) and \(E_1=P_{H,1}/P_{F,1}\). Dividing the second equation by the first yields the relative PPP formula.

1
Write the absolute PPP relationship at each date.
At time 0, \(E_0=P_{H,0}/P_{F,0}\); at time 1, \(E_1=P_{H,1}/P_{F,1}\).
2
Divide the ending equation by the starting equation.
This gives \(E_1/E_0=(P_{H,1}/P_{F,1})/(P_{H,0}/P_{F,0})\).
3
Rearrange into growth factors.
Reordering the numerator and denominator gives \(E_1/E_0=(P_{H,1}/P_{H,0})/(P_{F,1}/P_{F,0})\).
4
Express price changes as inflation.
If home and foreign inflation over the period is \(\pi_H\) and \(\pi_F\), respectively, their price-level growth factors are \(1+\pi_H\) and \(1+\pi_F\).

Therefore the exact finite-period relationship is:

$$ \frac{E_1}{E_0}=\frac{1+\pi_H}{1+\pi_F} \qquad\text{and}\qquad \frac{E_1-E_0}{E_0} =\frac{1+\pi_H}{1+\pi_F}-1 =\frac{\pi_H-\pi_F}{1+\pi_F}. $$

The second expression is the exact percentage change in the exchange rate, written as a decimal. Multiply it by 100 to express the change as a percentage. This formula shows why the inflation differential is not, in general, exactly equal to the exchange-rate percentage change: the exact difference is divided by \(1+\pi_F\).

Approximation: When inflation rates are modest, relative PPP is often summarized as \(\%\Delta E\approx\pi_H-\pi_F\). This is an approximation to the exact formula, not an identity. The approximation is more accurate when the inflation rates are small; for large rates or a request for an exact result, use \((1+\pi_H)/(1+\pi_F)-1\).

For example, if home inflation is 10% and foreign inflation is 2%, the simple differential is 8 percentage points. The exact predicted exchange-rate change is \(1.10/1.02-1=0.078431\), or 7.8431%. The approximation is useful for a quick interpretation, but it should not replace the exact calculation when precision matters.

Worked Applications

Worked Example: Applying Annual Inflation to Price Indexes

A home-country price index rises from 200 to 210 over one year. The foreign price index rises from 160 to 164 over the same year. The exchange rate at the start is 1.20 units of home currency per unit of foreign currency. Under relative PPP, find the ending exchange rate and its percentage change. Assume the indexes are comparable and that relative PPP applies over this period.

Method. First calculate each country’s inflation from its own price index. Then use the exact relative PPP ratio. This avoids comparing the indexes’ levels directly, even though their starting values differ.

Home inflation is \((210-200)/200=10/200=0.05\), or 5%. Foreign inflation is \((164-160)/160=4/160=0.025\), or 2.5%.

The exchange-rate growth factor is:

$$ \frac{1.05}{1.025} =\frac{42}{41} =1.02439024\ldots $$

Thus the predicted percentage change is \(1.02439024\ldots-1=0.02439024\ldots\), or an increase of 2.4390% to four decimal places. The ending rate is \(1.20(42/41)=50.4/41=1.22926829\ldots\), or 1.2293 home-currency units per foreign-currency unit to four decimal places.

Check. The home inflation rate exceeds foreign inflation, so the home-currency cost of foreign currency rises under this quotation. The calculated increase has the predicted direction. A common mistake is to divide the ending price indexes, \(210/164\), and treat that as the exchange rate; relative PPP predicts how the exchange rate changes, so the starting exchange rate must also be used.

Worked Example: Compounding an Inflation Differential Over Three Years

A home-per-foreign exchange rate begins at 0.90. Home inflation is 2% per year for three years, while foreign inflation is 1% per year. Assume each annual rate is constant, prices compound annually, and relative PPP holds in each year. Find the exact three-year percentage change and the ending rate.

Method. Calculate the three-year price growth factor in each country. Because the annual rates apply for three years, each gross annual rate is raised to the third power.

The home price-level growth factor is \(1.02^3=1.061208\). The foreign price-level growth factor is \(1.01^3=1.030301\). Therefore:

$$ \frac{E_3}{E_0} =\frac{1.02^3}{1.01^3} =\frac{1.061208}{1.030301} =1.0299980297\ldots $$

The exact three-year percentage change is \((1.0299980297\ldots-1)\times100=2.99980297\ldots\%\), an increase of 2.9998% to four decimal places. The ending exchange rate is \(0.90(1.0299980297\ldots)=0.9269982267\ldots\), or 0.9270 to four decimal places.

Check. Home prices rise faster, so the home-per-foreign rate increases; the answer has that direction. The annual inflation differential is one percentage point, but multiplying it by three gives only a rough three-year estimate. It does not produce the exact compounded percentage change. The exact result is an increase of 2.9998%, not 3.0000% to four decimal places.

Worked Example: Inferring Foreign Inflation From an Exchange-Rate Change

A home-per-foreign exchange rate rises from 0.80 to 0.84 over one year. Home inflation during the year is 8%. Assuming relative PPP holds, infer the foreign inflation rate. Treat the observed exchange-rate change and the stated inflation rate as exact.

Method. First calculate the exchange-rate growth factor. Then solve the exact relative PPP equation for the unknown foreign gross inflation factor.

The exchange-rate growth factor is \(0.84/0.80=1.05\), which is a 5% increase. Relative PPP gives \(1.05=1.08/(1+\pi_F)\). Rearranging:

$$ 1+\pi_F=\frac{1.08}{1.05} =1.0285714286\ldots, \qquad \pi_F=0.0285714286\ldots $$

The implied foreign inflation rate is 2.8571% to four decimal places. The home inflation rate is higher than the inferred foreign rate, consistent with the exchange rate rising under the home-per-foreign quotation.

Check. Substituting back gives \(1.08/1.0285714286\ldots=1.05\), matching the observed exchange-rate growth factor. A common mistake is to subtract the 5% exchange-rate change from 8% home inflation and report 3% foreign inflation. That uses the approximate differential instead of solving the exact formula.

Reading the Result Correctly

The quotation convention determines how to interpret both the formula and its sign. With \(E\) measured in home currency per foreign currency, a positive percentage change in \(E\) means one unit of foreign currency buys more home currency. When home inflation exceeds foreign inflation, the exact ratio \((1+\pi_H)/(1+\pi_F)\) exceeds one, so \(E\) rises. When foreign inflation exceeds home inflation, it is below one, so \(E\) falls.

If instead the exchange rate is quoted in foreign currency per unit of home currency, it is the reciprocal of \(E\). Its relative PPP equation is correspondingly reversed:

$$ \frac{Q_1}{Q_0}=\frac{1+\pi_F}{1+\pi_H}, $$

where \(Q\) denotes foreign currency per unit of home currency. Do not use the home-per-foreign formula with a foreign-per-home quotation.

Relative PPP is a theoretical relationship, not a guarantee that market exchange rates will move exactly as predicted over every period. The derivation relies on comparable price indexes and the maintained PPP relationship across the dates being compared. Differences in the goods represented by indexes, barriers to trade, and departures from the model can make observed exchange-rate movements differ from the prediction. Empirical testing and practical limitations are considered separately in the material on testing purchasing power parity.

Common Exam Traps

  • Using the inflation difference as an exact answer. The exact percentage change is \((\pi_H-\pi_F)/(1+\pi_F)\); the simple difference is only an approximation.
  • Forgetting to compound. For constant annual inflation over \(n\) years, the price growth factors are \((1+\pi_H)^n\) and \((1+\pi_F)^n\), not \(1+n\pi_H\) and \(1+n\pi_F\) when an exact result is required.
  • Reversing the quotation. Home currency per foreign currency and foreign currency per home currency have reciprocal growth relationships.
  • Confusing percentage points with a percentage change. A difference between two inflation rates is measured in percentage points; the exchange-rate percentage change is calculated relative to the starting exchange rate.
Key takeaway: With \(E\) quoted as home currency per unit of foreign currency, relative PPP predicts \(E_1/E_0=(1+\pi_H)/(1+\pi_F)\). The inflation differential gives a useful approximation, but the ratio of gross inflation rates gives the exact finite-period result.

Check Your Understanding

For each problem, use the stated quotation and assume relative PPP applies over the specified period. Answers involving percentages are rounded to four decimal places.

  1. A home-per-foreign exchange rate is 2.00. Over one year, home inflation is 2% and foreign inflation is 5%. Find the predicted percentage change and ending exchange rate. (Answer: decrease of 2.8571%; ending rate 1.9429.)
  2. Home inflation is 3% per year and foreign inflation is 1% per year for two years, with annual compounding. Find the exact percentage change in a home-per-foreign exchange rate. (Answer: increase of 3.9996%.)
  3. Home and foreign price indexes change from 144 to 150 and from 120 to 126, respectively. If the starting home-per-foreign exchange rate is 1.10, find its predicted percentage change. (Answer: decrease of 0.7937%.)
  4. An exchange rate is quoted as foreign currency per unit of home currency. Home inflation is 4% and foreign inflation is 1% for one year. Find the predicted percentage change in this quoted rate. (Answer: decrease of 2.8846%.)
  5. A home-per-foreign exchange rate starts at 1.50. Home inflation is 2% per year and foreign inflation is 5% per year for two years. Find the ending rate and the percentage change. (Answer: ending rate 1.4155; decrease of 5.6327%.)

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