For PPP & LOOP: e = EP/P* = 1 So both are valid if goods in the US and EU cost the same in $ for example. But then, what exactly is the difference between both?
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PPP uses the price level, the price of the consumer basket. The price level can be identical without all prices being identical. An example:
Consider a consumer basket with just two goods and assume the weight of both goods in the basket is one half. If $p_1 = p_2 = 4$ then the price of the consumer basket is $$ P = \frac{1}{2} \cdot p_1 + \frac{1}{2} \cdot p_2 = 4. $$ However you get the same price level if you have $p_1 = 3, p_2 = 5$.
So you can have two countries with identical price levels but differing prices. The converse of course is not possible. Thus LOOP implies PPP but PPP does not imply LOOP.
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$\begingroup$ So as consequence: LOOP requires identical prices and price level and PPP just identical price levels? $\endgroup$ Commented Oct 28, 2016 at 17:20
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$\begingroup$ @visionInc Factoring in the exchange rate, yes. $\endgroup$– GiskardCommented Oct 29, 2016 at 7:03