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The real exchange rate and PPP
The board at the bureau de change does not say whether a country is expensive: here is the number that does.
q = e × P / P* (= e / e_PPP)Problem / motivation
In late July 2026, €1 buys about $1.14. That number says how many dollars you get — it says NOTHING about the question that really matters to an exporter or a tourist: is France dear or cheap seen from the United States?
To know that, one piece of information is missing: PRICES on both sides. A euro at $1.14 does not mean the same thing if a basket of shopping costs €100 here and $120 there, or if it costs €100 and $100. The bureau de change's rate — the NOMINAL rate — only becomes interpretable once related to prices. The number that does that job is called the REAL EXCHANGE RATE, and this course builds it step by step.
A convention first, without which the whole reasoning inverts: we write the rate like the airport board, in dollars per euro (€1 = $1.14), what professionals call a DIRECT quotation — the same as the course “What is an exchange rate?”. Remember the rule: when that number RISES, the euro appreciates. We shall see that an appreciating currency makes the country dearer for its foreign customers — and that this makes winners as much as losers.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Take an identical basket on both sides: it costs P = €100 in the euro area and P* = $120 in the United States, with e = 1.14 $/€. Comparing €100 with $120 as they stand is impossible — a common currency is needed first. Let's convert the European basket into dollars: €100 × 1.14 $/€ = $114. That is what “our” basket costs an American. Compare it with theirs: 114 / 120 = 0.95. This number is the REAL EXCHANGE RATE, written q. It is 0.95: our products are 5% cheaper than theirs. Remember the reading rule, and never invert it — **q above 1: we are the dearer ones** (the euro is overvalued in real terms, and our exporters suffer); **q below 1: we are the cheaper ones** (favourable price competitiveness).
What we have just done is written q = e × P / P*. Check that it is properly built by following the units: e is in dollars per euro, P in euros, so e × P is in dollars — the European basket expressed in dollars. Dividing by P*, also in dollars, the units cancel: q is a PURE NUMBER, unit-free, which is essential since we are comparing two prices. (Careful if you meet the formula elsewhere: many textbooks quote indirect, in euros per dollar, and then write q = e × P*/P. Both are correct, each with its own quotation; mixing them inverts every conclusion.) And read the direction: if the euro appreciates, e rises, so q rises — our products become dearer for foreigners. A strong currency does indeed penalise exporters. One reflex to defuse straight away, because it catches almost everyone out: “a strong euro means more dollars per euro, so the exporter earns more”. No — they sell in dollars and convert back into euros, and each dollar collected then brings them LESS. Above all, their American customer has to hand over more dollars for the same product, and will therefore buy fewer of them.
Now ask the reverse question: what exchange rate would make the two baskets exactly equivalent? We need e × 100 = 120, so e = 120/100 = 1.20 $/€. That rate which equalises purchasing power is called the PURCHASING POWER PARITY RATE, written e_PPP, and it is simply P*/P. Now take the formula and substitute: q = e × P/P* = e ÷ (P*/P) = **e / e_PPP**. So the real rate is nothing other than the observed rate related to the rate that would equalise prices. That finally gives a precise meaning to an expression heard everywhere: “the currency is overvalued by 8%” means exactly that q = 1.08. Here, e = 1.14 against e_PPP = 1.20: the euro is undervalued by 5%, which is indeed our q = 0.95.
ABSOLUTE PPP asserts that q should equal 1: the same basket, the same price everywhere once converted. It follows from the law of one price (assumption 3) applied to every good in the basket — and it is massively refuted, as we shall see why. RELATIVE PPP is more modest: it only asserts that q stays CONSTANT, which requires the nominal rate to offset the inflation gap. If our prices rise by 3% and American prices by 1%, the euro must depreciate for q not to move. By how much? As an approximation, by the gap: 1 − 3 = −2%. Exactly: 1.01/1.03 − 1 = −1.94%. Keep both — the approximation to reason quickly, the exact one to compute correctly, the gap between them growing with inflation. Click each term:
Solving / calculation
Three computations: the euro's position today, the same thing by the Big Mac method, then the silent drift of the poll. The baskets and the Big Mac prices are working examples (surveys vary from one to the next); the exchange rate, however, is real.
- Our basket, seen from New York
€100 × 1.14 $/€$114 — against $120 for the American basket - The real exchange rate
q = 114 / 120q = 0.95: our products are 5% cheaper - The rate that would equalise prices
e_PPP = P*/P = 120/1001.20 $/€ — against 1.14 observed - Check of the identity q = e / e_PPP
1.14 / 1.200.95 — the same number, by another route - The same method on a one-good basket (Big Mac)
e_PPP = $5.95 / €5.70≈ 1.044 $/€ - The Big Mac's verdict: the euro looks OVERvalued
1.14 / 1.044 − 1≈ + 9% — the opposite conclusion to the full basket - The poll's drift: 5 years at 3% against 1%
(1.01/1.03)⁵ − 1− 9.3%: the euro ought to depreciate by that much - If it does not move (monetary union, frozen rate)
q is multiplied by (1.03/1.01)⁵ = 1.103: 0.95 → 1.05from cheaper, we become dearer - ⚠️ Two numbers not to be confused
price drift + 10.3% · depreciation that would offset it − 9.3%rising by 10.3% then falling by 9.3% does bring you back to the start
Three lessons. (1) The nominal rate says nothing on its own: at an unchanged 1.14 $/€, price competitiveness worsens by 10.3% in five years through inflation alone. This is the central mechanism of competitiveness debates inside the euro area, where devaluation is no longer possible (interpretation, point 3). (2) The verdict depends on the BASKET: the full basket says the euro is undervalued by 5%, the Big Mac says it is overvalued by 9%. This is no contradiction but a lesson — a one-good basket, made largely of rent and local wages, mainly measures NON-tradable costs (limits, card 1). (3) “Overvalued” only means something relative to e_PPP, never in the absolute: the identity q = e/e_PPP turns an opaque formula into a readable sentence.
q = e × P / P*, with e_PPP = P*/P; relative PPP: the rate should vary by (1+π*)/(1+π) − 1Set the nominal rate and the two price levels: the simulator deduces the PPP rate, the real rate and the competitiveness gap. Then let an inflation gap run for a few years WITHOUT touching the nominal rate — the situation of a euro area country — and watch q drift on its own.
Economic interpretation
Three readings: why PPP fails without being wrong, who gains from a strong currency, and what happens when devaluation is no longer possible.
The law of one price assumes you can buy where it is cheap in order to resell elsewhere. But an enormous share of what we consume cannot be shipped: a rent, a haircut, a nursery place, a restaurant meal. On these NON-TRADABLE goods, no arbitrage can equalise prices, and their weight alone is enough to make absolute PPP fail. Better still, the gap is SYSTEMATIC, and that is the Balassa-Samuelson effect: in a rich country, the high productivity of exporting industry pulls wages up across the WHOLE economy, including at the hairdresser's, whose productivity has not moved at all. Services there are therefore structurally dear — hence rich countries durably “overvalued” with respect to PPP, with nothing bringing them back towards 1. The empirical verdict: absolute PPP is rejected, relative PPP holds in the long run but slowly — the literature puts the half-life of deviations at between three and five years, that is the time it takes for a deviation from PPP to shrink by half (the reference survey: Kenneth Rogoff, 1996).
An appreciating euro pushes q up: our products become dearer for our foreign customers, and our exporters lose markets — that is half the story, the half you hear. The other half: everything we IMPORT becomes cheaper. Energy, raw materials, consumer goods, but also the components our own factories need. A strong currency is therefore a transfer: it costs the sectors exposed to international competition, and it benefits households' purchasing power and the firms that import. Add one decisive nuance: competitiveness is not only about price. NON-PRICE competitiveness — quality, market segment, innovation, delivery times, after-sales service — explains a good part of trade positions, and it is what lets an expensive country go on selling (course “Comparative advantage”).
Two words not to be confused. A DEPRECIATION is a fall in the rate caused by the market; a DEVALUATION is a fall decided by the authorities, which presupposes a fixed exchange rate regime. In the euro area, neither is available against partners inside the area: there is no longer a bilateral nominal rate to move. The real rate, however, goes on living through prices — that is the whole poll. A country whose costs drift can then only correct them by acting on them directly, which is called an INTERNAL DEVALUATION: wage restraint, lower charges, productivity gains. Finally, the effect of a change in the real rate on the trade balance is neither immediate nor guaranteed. Not immediate: after a depreciation, contracts under way and buying habits mean QUANTITIES take months to move, while the import bill grows heavier at once — so the balance starts by worsening before recovering, and the curve it draws gave the J-CURVE its name. Not guaranteed either: the eventual recovery assumes quantities end up reacting strongly enough to prices, a precise requirement called the Marshall-Lerner condition; if it is not met, a weaker currency worsens the balance durably. Both mechanisms are the ones the course “The trade balance” announced.
Limits / critiques
Exercises
A basket costs €200 in the euro area and $220 in the United States. The rate is 1.10 $/€. What is the real exchange rate q (two decimals)?
The PPP rate is 1.20 $/€ and the observed rate 1.32 $/€. By how much is the euro overvalued (in %)?
Inflation reaches 4% in the euro area and 1% in the United States. According to relative PPP, by how much must the euro move over the year for the real rate to stay constant (in %, one decimal)?
True or false: since arbitrage equalises prices, the real exchange rate quickly returns to 1.
Same situation (basket at €200 and $220). What is the PPP rate, e_PPP, in dollars per euro (two decimals)?
A basket is worth €200 in the euro area and $220 in the United States. With prices unchanged on both sides, the euro goes from $1.10 to $1.25. What happens to euro area exporters' price competitiveness?
A Big Mac costs €5.00 in the euro area and $6.00 in the United States. What exchange rate would equalise the two prices (in $/€)?
True or false: a rich country whose services are expensive necessarily has an abnormally overvalued currency, which ought to be corrected.