PURCHASING POWER PARITY 
In international comparisons of productivity, living standards and other similar indicators we face many difficulties. Not the least of them is the appropriate choice of exchange rates. Suppose, for example, that we want to compare the levels of development by comparing US GDP per capita with that of the Czech Republic. Naturally, we would take the aggregate GDP of each country and divide it by the country's population. This would give us, however, incomparable figures, because the US GDP per capita would be expressed in dollars, while the Czech would be expressed in Czech crowns (Kc). The Czech figures must be divided by the Kc/$ exchange rate to make the two figures comparable. 
What exchange rate to choose? During communist years in Czechoslovakia you would have encountered several quite distinct exchange rates: official, trade, tourist, black market, and in fact many variants of each of the above. Neither of them seems to be appropriate for our purpose. Official and tourist exchange rates may be arbitrarily set by the government, foreign trade and black market exchange rate may be distorted by administrative restrictions, state monopoly on foreign trade, various subsidies and surcharges etc. The similar problems appear in making comparisons with many less developed countries. Even for the developed countries for which we may have today a single market exchange rate we may be reluctant to use it. The market exchange rates are frequently influenced by random shocks and changes in economic policies and as a result they may fluctuate quite widely over relatively short periods. The use of the market exchange rates sometimes gives sudden changes in GDP per capita that are not accompanied by the corresponding changes in productivity or living standards. 
The purchasing power parity (PPP) exchange rate, which is usually considered to be the best for the purpose, is defined as the amount of foreign currency that would allow you to buy the same quantity of goods in foreign country which you can buy for one dollar in the USA. For example, if the PPP exchange rate is 20 Kc per $1 it means that for 200 crowns you can buy the same amount of goods in Prague as you can buy for 10 dollars in Boston. However, this is not so simple. If the relative prices in the two countries are not identical the PPP exchange rate depends on the basket of goods you want to buy. This is the same difficulty as the so called 'index numbers problem' related to Laspeyres and Paasche formulas of price and quantity indexes. In fact the PPP exchange rate is a kind of spatial price index. 
Let us illustrate our difficulty using the following example: 
Suppose that each of the four friends buys only two kinds of goods: beer and coffee, under a CobbDouglas utility function 
(1)  U = B^{a }C^{b}  
Maximization of CobbDouglas utility, subject to budget constraint results in the following demand functions:  
(2)  B = aI/P_{B} C = bI/P_{C}  
 
Bob and Pavel have identical tastes. They strongly prefer beer to coffee. Barbara and Petra have also similar although not identical tastes, they like more coffee then their boyfriends. Let us assume that these preferences are represented by the following numerical values of coefficients a and b:  
 
Suppose that the prices of beer and coffee in Boston and Prague are as follows:  
 
Bob and Barbara spend $10 each per day in Boston. Pavel and Petra spend in Prague 100 Kc each day. Applying the demand functions (2) they choose the following optimal consumption patterns. 

In Prague Bob and Barbara continue to consume the same quantities of beer and coffee as they used to consume in Boston. Similarly Pavel and Petra continued to consume in Boston as they did consume in Prague. 
To satisfy his daily craving for beer and coffee that costs him $10 in Boston Bob needs 2x20 Kc + 4x40 Kc = 200 Kc in Prague. He concludes that the PPP exchange rate is 20 Czech crowns per dollar and demands that exchange rate from the Czech friends. 
Pavel who has identical taste as Bob is offended by the greed of his American friend. He spends only 100 Kc daily in Prague on three beers and one cup of coffee and needed 3x$3 + 1x$1 = $10 in Boston to buy the same. Naturally, he wants to give to Americans only 10 Kc for each dollar. 
Petra, on the other hand thinks that 20 Kc per dollar is a fair exchange rate because she spends 100 Kc in Prague on 1 beer and 2 cups of coffee which costs $5 in Boston. 
Now enters Barbara and tells them that Bob should have asked for 30 Kc per dollar. That is because her daily consumption of 1 beer and 7 cups of coffee costs 300 Kc in Prague but only $10 in Boston. 
Apparently our friends did not learn much of microeconomics, otherwise they would have known that consumers should change their consumption patterns when relative prices change. The true PPP exchange rate should be based on unchanged level of utility and not on unchanged consumer basket. It would still vary with utility function, but at least Bob and Pavel should agree on the exchange rate, because they have identical preferences. 
The correct formula for the PPP exchange rate should be  
(3)  e = E(P_{BP} , P_{CP} , U)/E(P_{BB} , P_{CB} , U)  

The expenditure function for CD utility is  
(4)  E = U P_{B}^{a} P_{C}^{b }/ a^{a} b^{b}  
Using (4) in (3) we get  
(5)  e = (P_{BP} /P_{BB})^{a} (P_{CP} /P_{CB})^{b} = (20/3)^{a} 40^{b}  
Now substituting personal parameters a and b into (5) we obtain individual purchasing power parity exchange rates for our friends:  
 
What have we learned from this example? An important lesson: the PPP exchange rate may look differently to different persons and it may also look differently depending from which side are you looking. PPP is not unique, it depends on relative prices and on the composition of consumer basket.  
The theoretical or true ‘equal utility’ PPP exchange rate is impractical. We would not know whose preferences to take even if it would be possible to get information on personal utility functions. Also investment goods and net exports that make significant shares of GDP would create problems for us. Instead we usually look at the observed commodity structure of GDP in the two countries, that is we take the average ‘basket of goods’ and evaluate it at prices of the two countries. Because prices of the two countries are denominated in different currency units, we get the exchange rate as a result.  
Continuing in our simplified example: the two Bostonians consume 3 glasses of beer and 11 cups of coffee, while the two Pragers consume 4 glasses of beer and 3 cups of coffee. Multiplying these quantities by prices we get four alternative expenditure aggregates. 
 
EBB EPP EBP EPB are Boston quantities at Prague prices  
In other words EBB is the cost of the Bostonian basket in Boston, EPB is the cost the Bostonian basket in Prague, EBP is the cost of the Prague basket in Boston and EPP is the cost of the Prague basket in Prague.  
Now we can construct three types of PPP exchange rates  
1) PPP exchange rate measured on Bostonian basket  
2) PPP exchange rate measured on Prague basket  
3) Geometric mean of the two
 
Looking from Boston e_{B} is Laspeyres index e_{P} is Paasche index and e_{F} is Fisher’s ideal index. It is apparent that if countries determine their baskets rationally e_{B} is always greater than e_{P}. This implies that PPP exchange rate measured as a simple weighted mean of prices is always more favorable to the country whose consumption basket is taken for weights. Fisher’s ideal index may be, therefore, the most impartial measure of the PPP exchange rate. It is also closest to the ‘true utility based’ PPP exchange rate based on the assumption that all people in the country have identical utility functions. 
Last update Tuesday, August 24, 2004
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