Real Exchange Rates and International Competitiveness

Why Nominal Currency Movements Are Not Enough

A currency can depreciate without domestic goods becoming cheaper relative to foreign goods. If domestic prices rise rapidly at the same time, that inflation can offset some or all of the price advantage implied by the currency movement. To compare goods prices across countries, we therefore need to consider both the nominal exchange rate and the two countries’ price levels.

This comparison is useful when assessing price competitiveness: the relative price of domestic goods compared with foreign goods. It is a measure of relative prices, not a complete measure of a country’s ability to compete. Product quality, delivery reliability, productivity, and other factors also matter.

Definition: Let $E$ be the nominal exchange rate measured as units of domestic currency per unit of foreign currency. Let $P$ be the domestic price level and $P^*$ the foreign price level, each measured using a consistent price index. Define the real exchange rate as $$ q=\frac{E P^*}{P}. $$ Under this convention, $E P^*$ is the domestic-currency cost of the foreign price-level basket, and $P$ is the cost of the domestic basket. A rise in $q$ means foreign goods have become more expensive relative to domestic goods, so the domestic country’s measured price competitiveness has improved. A fall means the opposite.

The quotation convention matters. Here, an increase in $E$ means that more domestic currency is needed to buy one unit of foreign currency: the domestic currency has depreciated nominally. Because $q$ uses this convention, a rise in $q$ is called a real depreciation and indicates an improvement in relative domestic price competitiveness. A source using the reciprocal exchange rate may define its real exchange rate in the opposite direction; always check the quotation and formula before interpreting a change.

Price levels are commonly represented by index numbers. The numerical level of $q$ depends on how the price indexes are scaled, so it is usually safer to interpret a change in $q$ over time than to attach meaning to an isolated level. For comparisons over time, use consistent price-index series and the same exchange-rate convention at every date.

Deriving the Change in the Real Exchange Rate

The real exchange rate is a ratio of two relative costs. Between an initial date 0 and a later date 1, divide its value at date 1 by its value at date 0. This shows exactly how exchange-rate and price-level changes combine.

\frac{q_1}{q_0} = \frac{E_1}{E_0} \frac{P^*_1}{P^*_0} \frac{P_0}{P_1}

If domestic inflation over the period is $\pi_d$, so that $P_1/P_0=1+\pi_d$, and foreign inflation is $\pi_f$, so that $P^*_1/P^*_0=1+\pi_f$, then

\frac{q_1}{q_0} = \frac{E_1}{E_0}\frac{1+\pi_f}{1+\pi_d}.

This identity gives the exact change when inflation rates and the nominal exchange-rate change are measured over the same period. For small changes, the percentage change in $q$ is approximately the percentage change in $E$, plus foreign inflation, minus domestic inflation. That approximation is convenient, but the exact ratio should be used when changes are not small or the question asks for a precise result.

Why the formula holds: By definition, $q_1=E_1P^*_1/P_1$ and $q_0=E_0P^*_0/P_0$. Dividing the first expression by the second gives $$ \frac{q_1}{q_0} = \frac{E_1P^*_1/P_1}{E_0P^*_0/P_0} = \frac{E_1}{E_0}\frac{P^*_1}{P^*_0}\frac{P_0}{P_1}. $$ Substituting the price-level growth factors $1+\pi_f$ and $1+\pi_d$ yields the exact expression above. Thus, a nominal depreciation increases $q$ only to the extent that it is not offset by relatively faster domestic inflation.

Worked Applications

Worked Example: A Depreciation That Improves Price Competitiveness

A hypothetical country quotes its exchange rate in domestic currency per unit of foreign currency. Initially, $E_0=1.20$, $P_0=120$, and $P^*_0=100$. One year later, $E_1=1.32$, $P_1=126$, and $P^*_1=104$. Calculate the initial and later real exchange rates, then find the percentage change in $q$. Interpret the result. The data are index values and exchange-rate quotes for this example, not industry observations.

Method. Calculate $q=EP^*/P$ at each date, then compare the two values. This method avoids treating the nominal exchange-rate change as the whole competitiveness result.

At the initial date, $$ q_0=\frac{1.20(100)}{120}=\frac{120}{120}=1.0000. $$

At the later date, $$ q_1=\frac{1.32(104)}{126} =\frac{137.28}{126} =1.0895238\ldots $$

The percentage change is $$ \left(\frac{q_1}{q_0}-1\right)100\% = \left(\frac{1.0895238\ldots}{1.0000}-1\right)100\% =8.95238\ldots\%. $$ Rounded to four decimal places, $q_1=1.0895$ and the real exchange rate increased by $8.9524\%$. Since $q$ rose under the stated convention, foreign goods became more expensive relative to domestic goods, and measured domestic price competitiveness improved.

As a check, the nominal exchange rate rose by $1.32/1.20-1=10\%$, while domestic and foreign prices rose by $5\%$ and $4\%$, respectively. The nominal depreciation was partly offset by the price changes, so the real increase is less than the nominal increase.

Worked Example: Nominal Depreciation Does Not Guarantee a Real Depreciation

In a second hypothetical economy, $E_0=2.00$, $P_0=100$, and $P^*_0=100$. Later, $E_1=2.16$, $P_1=112$, and $P^*_1=103$. Find the percentage change in the real exchange rate and state whether measured price competitiveness improved. The exchange rate is again domestic currency per unit of foreign currency.

Method. Compute the real exchange rate at both dates. This directly compares the relative basket costs and avoids assuming that nominal depreciation necessarily improves competitiveness.

Initially, $$ q_0=\frac{2.00(100)}{100}=2.0000. $$

Later, $$ q_1=\frac{2.16(103)}{112} =\frac{222.48}{112} =1.9864286\ldots $$

The percentage change is $$ \left(\frac{1.9864286\ldots}{2.0000}-1\right)100\% =-0.678571\ldots\%. $$ Rounded to four decimal places, $q_1=1.9864$ and $q$ fell by $0.6786\%$. Under this convention, measured domestic price competitiveness worsened slightly, even though the nominal exchange rate rose by $8\%$.

The price movements explain the result: the domestic price level rose by $12\%$, while the foreign price level rose by $3\%$. Relatively faster domestic inflation more than offset the nominal depreciation. A useful check is the exact change factor: $$ \frac{q_1}{q_0} = 1.08\frac{1.03}{1.12} = 0.9932143\ldots, $$ which is less than 1, confirming the decrease.

Worked Example: Finding the Nominal Movement Needed to Offset Inflation

Suppose domestic inflation is $7\%$ and foreign inflation is $2\%$ over a year. Find the nominal exchange-rate change that would keep $q$ unchanged. Then suppose the exchange rate actually rises by $4\%$; calculate the resulting percentage change in $q$. Use the convention that $E$ is domestic currency per unit of foreign currency.

Method. Set the exact real-exchange-rate change factor equal to 1 to find the offsetting nominal change. Then substitute the actual exchange-rate change into the same formula. This avoids assuming the nominal change must equal the inflation difference exactly.

For $q$ to be unchanged, $q_1/q_0=1$. Therefore, $$ 1=\frac{E_1}{E_0}\frac{1.02}{1.07}, \qquad \frac{E_1}{E_0}=\frac{1.07}{1.02}=1.0490196\ldots $$

The required nominal exchange-rate increase is $1.0490196\ldots-1=0.0490196\ldots$, or $4.9020\%$ rounded to four decimal places. In this quotation, that is a nominal depreciation of the domestic currency.

If $E$ instead rises by $4\%$, the exact real exchange-rate factor is $$ \frac{q_1}{q_0} = 1.04\frac{1.02}{1.07} = \frac{1.0608}{1.07} =0.9914019\ldots $$

Thus, $q$ changes by $(0.9914019\ldots-1)100\%=-0.8598\%$, rounded to four decimal places. The actual depreciation is slightly smaller than the amount needed to offset the inflation difference, so measured domestic price competitiveness declines slightly.

Interpreting Competitiveness Carefully

A real exchange rate is a relative-price indicator. Under the convention used here, a rise in $q$ indicates that the foreign basket costs more relative to the domestic basket. That can support domestic producers’ price competitiveness, other things equal. A fall in $q$ indicates that domestic goods have become more expensive relative to foreign goods.

The phrase “other things equal” is important. Broad consumer price indexes may include many goods and services that are not traded internationally. The actual prices faced by exporters or import-competing producers can move differently from an aggregate price index. Competitiveness can also change through productivity, product quality, transport costs, taxes, or differences in product mix. A real exchange-rate movement therefore informs a comparison of measured prices; by itself, it does not prove that exports or output will rise.

Change under this convention	Relative-price interpretation	Measured domestic price competitiveness
$E$ rises, with prices fixed	Foreign goods become more expensive relative to domestic goods	Improves
$q$ rises	Foreign goods become more expensive relative to domestic goods	Improves
$q$ falls	Domestic goods become more expensive relative to foreign goods	Worsens
Domestic inflation exceeds foreign inflation	Pushes $q$ downward if the nominal exchange rate is unchanged	Worsens, other things equal

The direction of the inflation effect follows directly from $q=EP^*/P$. Holding $E$ and $P^*$ fixed, a higher $P$ increases the denominator and lowers $q$. Holding $E$ and $P$ fixed, a higher $P^*$ increases the numerator and raises $q$. These statements describe the formula’s effect with the other variables held fixed; in an actual economy, the variables may change together.

Common Exam Trap: Using Only the Nominal Exchange Rate

A frequent error is to observe that $E$ rose and conclude that real competitiveness improved by the same percentage. That conclusion ignores relative inflation. In the second worked example, one incorrect calculation would be: “$E$ rose by $8\%$, therefore $q$ rose by $8\%$.” This is wrong because it treats $P$ and $P^*$ as unchanged.

The correction is to use the full real-exchange-rate ratio: $$ \frac{q_1}{q_0} = \frac{E_1}{E_0}\frac{P^*_1/P^*_0}{P_1/P_0} = 1.08\frac{1.03}{1.12} =0.9932143\ldots. $$ Since this factor is below 1, $q$ fell by $0.6786\%$, rather than rising by $8\%$. Always identify the exchange-rate quotation, include both price-level changes, and interpret the direction only after calculating $q_1/q_0$.

Key takeaway: With $E$ quoted as domestic currency per unit of foreign currency, $q=EP^*/P$. A rise in $q$ indicates improved measured domestic price competitiveness, but the movement depends jointly on the nominal exchange rate and relative inflation. Nominal depreciation alone is not enough to determine the result.

Check Your Understanding

For each problem, use the convention that $E$ is units of domestic currency per unit of foreign currency. Interpret a rise in $q=EP^*/P$ as an improvement in measured domestic price competitiveness.

Initially $E_0=1.25$, $P_0=125$, and $P^*_0=100$. Later $E_1=1.30$, $P_1=130$, and $P^*_1=104$. Calculate $q$ at each date and its percentage change. (Answer: $q_0=1.0000$, $q_1=1.0400$; $q$ rises by $4.0000\%$, so measured domestic price competitiveness improves.)
An economy has $E_0=0.75$, $P_0=200$, and $P^*_0=160$. Later, $E_1=0.78$, $P_1=212$, and $P^*_1=164.8$. Calculate the two real exchange rates and the percentage change in $q$. (Answer: $q_0=0.6000$, $q_1=0.6063$; $q$ rises by $1.0566\%$, so measured domestic price competitiveness improves.)
Over a year, the domestic currency appreciates so $E_1/E_0=0.97$. Domestic inflation is $1\%$ and foreign inflation is $5\%$. Calculate the exact percentage change in $q$ and interpret it. (Answer: $q_1/q_0=0.97(1.05)/1.01=1.0084158\ldots$, so $q$ rises by $0.8416\%$ and measured domestic price competitiveness improves.)
Domestic inflation is $9\%$, foreign inflation is $4\%$, and the domestic currency depreciates so $E_1/E_0=1.06$. Calculate the exact percentage change in $q$. (Answer: $q_1/q_0=1.06(1.04)/1.09=1.0113761\ldots$, so $q$ rises by $1.1376\%$ and measured domestic price competitiveness improves.)
The nominal exchange rate rises by $2\%$, domestic inflation is $3\%$, and foreign inflation is $1\%$. Use the exact formula, not the small-change approximation, to find the percentage change in $q$. (Answer: $q_1/q_0=1.02(1.01)/1.03=1.0001942\ldots$, so $q$ rises by $0.0194\%$ and measured domestic price competitiveness improves very slightly.)

Change under this convention	Relative-price interpretation	Measured domestic price competitiveness
\(E\) rises, with prices fixed	Foreign goods become more expensive relative to domestic goods	Improves
\(q\) rises	Foreign goods become more expensive relative to domestic goods	Improves
\(q\) falls	Domestic goods become more expensive relative to foreign goods	Worsens
Domestic inflation exceeds foreign inflation	Pushes \(q\) downward if the nominal exchange rate is unchanged	Worsens, other things equal

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