90732
International Comparison Program
[04.02]
PRELIMINARY AND INCOMPLETE
Consumer price indexes,
purchasing power parity
exchange rates, and updating
Angus Deaton
7th Technical Advisory Group Meeting
September 17-18, 2012
Washington DC
Consumer price indexes, purchasing power parity exchange rates, and updating
DRAFT
Angus Deaton
Princeton, May 2012
I am grateful to Mark Aguiar and to Oleg Itskhoki for extremely helpful and patient discussions. They
bear no responsibility for errors or misunderstandings.
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Abstract
PPPs are calculated at each round of the ICP, using multilateral, symmetric, and transitive price indexes.
From one round to the next, the PPPs change in a way that, at least in the last two rounds, has been
consistently different from what would be predicted from the changes in domestic consumer price
indexes. Between rounds of the ICP, the World Bank and the Penn World Table use local CPIs to update
the PPP and when the 2005 ICP results became available, there was a large change in many PPPs for
2005 compared with earlier published estimates for the same year, typically with the price levels in poor
countries revised upwards relative to rich countries. The same happened for the 1993 round of the ICP
relative to the 1985 baseline. It is generally understood that changes in PPPs will not be the same as
changes in relative CPIs, even in the absence of a multitude of practical issues such as the treatment of the
trade balance, the different commodity lists for CPIs and PPPs, differential hedonic quality adjustment in
different countries, and changes in procedures from one round to the next. I focus here on the difference
between changes in PPPs and changes in CPIs that comes from the difference in their weights; the
symmetric indexes that must be used for PPPs, even for two countries, use both countries’ weights, while
within-country indexes, such as CPIs, use only the weights for one country. I use the data from the 1993
and 2005 rounds to show that the discrepancy is both substantial and systematic, and will generally lead
to an inter-round upward revision of the PPPs for poor relative to rich countries. Even so, the effect,
which is very small within OECD countries, is not as large as the actual revision in 2005. I make some
suggestions for updating, and clarify the relationship with the Balassa-Samuelson effect and the possible
use of the Balassa-Samuelson effect for PPP updating between rounds. Much remains to be understood.
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1. CPIs and PPPs in theory
Imagine an ideal world in which the prices collected for domestic CPIs and the ICP are the same, there
are no differences across countries in hedonic quality adjustment, and there are no changes in procedures
in the ICP from one round to the next. Even in this ideal case, there will generally inconsistencies
between changes in CPIs and changes in PPPs from one round to the next, see for example Locker and
Faerberg (1984), Dalgaard and Sorensen (2002), Hill (2004), Rhoades (2004), Biggieri and Laureti
(2011), and McCarthy (2011). This paper explores one of the reasons for these inconsistencies.
To fix ideas consider a two country case where preferences are identical and homothetic. Write
the common cost (expenditure) function as
c(u , p ) = ua ( p ) (1)
where a ( p ) is a linearly homogeneous scalar function of the price vector p. If the two countries price
vectors are p1 and p2 , the rate of change of the domestic CPIs are given by
d ln Pi = d ln a ( pi ) (2)
for i=1, 2. The log of the PPP for 2 relative to (base country) 1 is
=
ln PPP2 ln a ( p2 ) − ln a ( p1 ) (3)
From (2) and (3), we have consistency between PPP updating and CPI updating, i.e.
PPP2 d ln P2 − d ln P
d ln = 1 (4)
which is what everyone would like to happen.
But (4) is very special. In particular, it stops working as soon as we have non-homothetic tastes,
even if we maintain identical tastes across countries. In general, we can write, in place of (2),
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∂ ln c(u , pi )
=d ln Pi ∑
=
∂ ln p
n
d ln p in si′d ln pi (5)
in
For budget shares sin for good n in country i. The differential rate of price inflation can then be written
d ln P2 − d ln P=
1 s2′d ln p2 − s1′d ln p1 (6)
The PPP index, if it is to satisfy symmetry, must involve both sets of budget shares. The algebra is simple
if I use the Törnqvist index, so that
0.5( s2 + s1 )′(ln p2 − ln p1 )
ln PPP2 = (7)
Comparing (6) and (7), we have, instead of the simple (4),
d ln PPP2 = (d ln P2 − d ln P ′
1 ) − 0.5( s2 − s1 ) ( d ln p2 + d ln p1 ) (8)
so that, without homotheticity, the rate of growth of the PPP is not the same thing as the differential rate
of change of the two consumer price indexes. If the second term is important, CPI updating of PPPs will
fail, not because of errors and differences in procedures, but because the two concepts are distinct. This
must be the case if tastes are non-homothetic, and if the PPP uses an index that is symmetric across
countries, thus ruling out the use of only one set of weights.
Equation (8) also says that if we want to update PPPs in order to get as close as possible to the
PPP that would be calculated directly, we need to make an attempt to calculate the second term on the
right hand side of (8); at the end, I shall discuss how this might be done. Updating using only the CPIs
will, in general, not give the correct answer.
Equation (8) also allows us to say something about the discrepancy between the growth of the
PPP and of the differential CPIs. The major difference between poor countries and rich countries in
budget shares is that poor countries have higher shares on food, and lower shares on other things,
including services. Food is a largely tradable good and so is relatively expensive, while many services—
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most notably housing, or local labor intensive goods—are largely non-tradable and so are relatively
cheap. Over time, as poorer countries grow richer, the parities the cheaper services can be expected to rise
relative to the parities of the more expensive foods. This will induce a negative correlation between the
differences in budget shares and the rates of change of the parities; the goods that are relatively heavily
consumed in poor countries are those whose parities are rising least rapidly. If this is the case, the last
term on the right hand side of (8) will be positive, so that the PPPs will rise more rapidly than the
differential rate of increase of the consumer price index. This is what happened between the 1985 and
1993 rounds of the ICP, and again between the 1993 and 2005 rounds, so that the explanation for the
discrepancy is at least in the right direction. Of course, there were many other things going on too, and we
so far know nothing about the size of the effect. That is the task of the next section.
Note finally that the changing relative prices of tradables and non-tradables recalls the Balassa-
Samuelson theorem, and indeed the stories have similarities. But they are not the same thing. The
Balassa-Samuelson theorem explains the divergence between the rate of change of the market exchange
rate and the differential rate of change of the CPIs whereas here, we are concerned with the divergence
between the purchasing power parity exchange rate and the differential rate of change of the CPIs. In
most accounts of Balassa Samuelson, this last difference is absent by assumption. I shall return to this
issue in the final section on updating.
2. Empirical evidence
The ICP 2005 provides data on expenditures and parities (prices relative to the US) for basic headings of
consumption, investment, government expenditure, and the foreign balance. I use only the consumption
data, and aggregate the categories to match the more aggregated data from the 1993−96 round which
distinguish only 26 items of consumption. Expenditures (and shares) are aggregated by summation; for
parities, I use the budget shares within groups to obtain weighted averages.
In order to abstract from multilateral index issues, as well as the full complexity of calculating the
PPPs as done by ICP2005, I look only at bilateral Törnqvist indexes with the US as base. The correlation
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between these short-cut consumption PPPs (bilateral, intransitive, and using aggregated basic heads) and
the actual consumption PPP from 2005 is a remarkable 0.9992 in logs for the 100 countries represented in
both rounds and is still 0.9763 for the consumption prices (consumption PPP divided by the exchange
rates). The scatter is shown in Figure 1; the outlier at the top right is Zimbabwe while those at the bottom
left are Tajikistan and Kyrgyzstan. While the correlation is not perfect, the bilateral indexes allow me to
use the formulas from the previous section without modification.
Figure 1: Consumption prices in 2005: actual and bilateral approximation
I calculate the second term on the right-hand side of (8)—the excess of the change in the log PPP
over the change in the log ratio of the two CPIs—taking the 1993 shares as the baseline shares, and the
parities from 1993 and 2005 for the prices. Note that I do not have price levels for either round, only
parities relative to the US, but (8) can be written in this form. Updating the notation, I calculate
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∆ i =−0.5( si93 − sUSA
93
)′(ln π i05 − ln π i93 ) (9)
where ∆ i is the calculated discrepancy for country i, the superscripts 93 and 05 index the two ICP rounds,
and π is vector of parities.
Figure 2: Contribution to PPP growth of weighting difference versus lnGDP
Figure 2 shows the plot of (9) against the logarithm of GDP per capita in 2005 international
dollars. As anticipated, the calculated correction is larger the poorer the country; the correlation is −0.39.
If the correction is ignored, we will be progressively understating international consumption inequality,
only to see a jump at the time a new ICP round becomes available. The discrepancy is essentially zero for
the rich countries while for poor countries, e.g. those with log per capita GDP less than 8, the average
discrepancy is about 9 percent. For those countries, we could expect the PPP to rise by 9 percent even if
their CPI rose at exactly the same rate as the CPI of the US. As we shall see, this is not large relative to
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the revisions that actually took place, but it is a correction that is surely worth having if it can be
calculated in advance.
Figure 3: Actual revisions in log PPP (horizontal axis) versus predictions from weight
differences
Figure 3 shows a plot of the actual revisions—the log of the ratio of the 2005 PPP for consump-
tion after to before the ICP 2005 results—against the discrepancies calculated here and shown in Figure 2.
Although the correlation is positive, it is not significantly different from zero, and more importantly, the
size of the actual discrepancy dwarfs that of the predicted discrepancy. Both are small for the rich
countries—recall that the OECD results are updated in between rounds—but if we again look at countries
whose log GDP in 2005 is less than 8, the actual average adjustment was 45 percent upwards, more than
nine times the predicted upward adjustment of 9 percent.
Why do the formulas here predict so little of the adjustment? There are reasons that are internal to
the current analysis as well as reasons that are external, of which the latter seem likely to be more
important. Of the internal reasons, I have approximated the continuous derivatives of equation (8) by the
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discrete difference in (9). The discrete difference is taken over more than ten years, and the approximation
might well be poor. I have also had to aggregate the basic heads for 2005 to match 1993, and I am using
only bilateral Törnqvist indexes, not the full multilateral indexes in the ICPs own calculations. Perhaps
more importantly, I am not allowing for any difference between the domestic weights that go into the
CPIs and the domestic weights that are used in the ICP. The external reasons cover all of the other reasons
why the consumption PPP for 2005 differed from the consumption PPP for 1993, all of the changes in
procedure, the hedonic quality corrections to some CPIs not others, the differences in the definitions of
commodities from one round to the next, and general measurement errors, particularly in the very weak
1993 round. (Note that the trade balance—a major source of adjustment in the PPPs for GDP—plays no
role in the consumption PPPs.) That the net effect of all of these adjustments should be large is not a
surprise. What remains unexplained is why the direction should be so strongly positive for poor countries.
Greater hedonic adjustment in the US goes in the wrong direction, because it would lead to an
overstatement in poor countries CPI relative to the US CPI, so that the poor country price levels would
tend to fall when, at each ICP round, only strictly comparable goods are compared. The much more
precise definition of commodities in 2005 over 1993 probably increased the price levels in poor countries
relative to rich countries, but does not explain why there was a similar effect between the 1985 and 1993
rounds. Nor do the results in Deaton (2010) support the contention that the collection of data on rare and
expensive Western goods played much of a role in explaining the high price levels in poor countries. The
correction discussed here is in the right direction, but is small relative to the actual discrepancy in 2005.
3. How to update between rounds and the Balassa-Samuelson theorem
The correction developed here, equation (8), seems to be worth using in updating PPPs for the period
between each round and the advent of the next. Current practice in the Bank is to update the PPPs from
the previous round using the differential rate of change of the country’s CPI and that of the US, which
amounts to using the first term on the right hand side of (8) but ignoring the second. The second term is
perhaps more difficult to evaluate than the first, but it still seems practicable. The weights are available
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from the ICP itself, and the prices for updating are required only at the basic heading level. They could
come either from the implicit price deflators of consumption in the national accounts, or from some
disaggregation of CPI data. If, as I suspect, the main effect is coming from food versus other items, it
would be possible to get a useful figure with only the very rough disaggregation that is likely to be
routinely available.
Note that my claim here is very limited, only that the correction seems worth doing on its own
account, and not that it will solve the mass upward revision of poor country price levels that has occurred
in the last two rounds of the ICP.
Ravallion (2010) has suggested that updating be done using what he refers to as a dynamic
version of the Penn effect, and provides empirical results that support this method over updating using the
differential CPIs. I discuss this briefly, and elucidate its links to the updating procedure suggested here.
A classic treatment of the Balassa-Samuelson (BS) effect can be found in Chapter 4 of the text by
Obsfeldt and Rogoff (1996). BS is about the discrepancy between the rate of change of the exchange
rate, on the one hand, and the differential rates of change of the CPIs on the other. The basic idea is that,
apart from high frequency fluctuations, and in the absence of tariffs and transport costs, the exchange rate
should equalize the prices of traded goods. If the CPI consisted entirely of traded goods, then the ratios of
CPIs should move with the exchange rate. However, the CPI also contains non-traded goods, whose
parities are unconstrained by trade. In consequence the ratio of CPIs for two countries can move
independently of the exchange rate. If the relative prices of non-traded to traded goods are lower in poor
countries, and if this is to do with poor countries being poor, then economic growth in poor countries that
is higher than in rich countries should work to eliminate the BS effect, so that ratio of a poor country’s
CPI to a rich country’s CPI should increase faster than their exchange rate increases, so that the poor
countries should experience real exchange rate appreciation.
If the market exchange rate is denoted by X, then the rate of change of the BS effect can be
written as
10
=
∆ BS (d ln P2 − d ln P 1 ) − d ln X (10)
Suppose that country 2 is the relatively poor country—think China—and 1 is the relatively wealthy
baseline country—USA. In PPP parlance, (10) is the rate of growth of the Chinese price level, the PPP
divided by the exchange rate; in international macro parlance, this is real exchange rate appreciation. If
productivity growth in China is relatively rapidly, the relative price of its non-tradable goods will be
rising, and its CPI will rise relative to the exchange rate, and ∆ BS will be positive. The international
macro literature is rarely concerned with the level of the price level—the ratio of the PPP to the exchange
rate—but instead focuses on its rate of change, which is the right hand side of (10). (This is also why the
PPP and macro literatures rarely make contact. The former—the ICP—focuses mostly on the construction
and measurement of the levels of PPPs at a moment in time. In the macro literature, with no need to
measure levels, the rate of change of the PPP is simply assumed to be the relative rate of change of the
two CPIs. In terms of (8), the left hand side is equal to the first term on the right hand side, and the last
term does not exist. No consideration to the fact that the levels of actual PPPs, as measured by the ICP,
use different weights from the domestic CPIs.
What is called the Penn effect is the empirical finding, in accord with BS, that price levels are
systematically lower in poorer countries, and what Ravallion calls the dynamic Penn effect is the link
between the rate of growth of price levels and the rate of growth of per capita GDP. Write this
∆ BS = θ 0 + θ1d ln Y (11)
where Y is per capita GDP. If (11) is substituted into (10), we have
= (d ln P2 − d ln P=
d ln PPP 1) d ln X − θ 0 − θ1d ln Y (11)
This equation is the basis for the alternative updating scheme that Ravallion proposes. The parameters θ
can be estimated from the levels version of (10) using the current round of the ICP, and the change in the
PPPs going forward calculated according to the last term in (11), ignoring the first term—the relative rate
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of growth of the CPIs—which is the basis for current practice. Ravallion provides empirical evidence that
the second term does better—gets closer to the 2005 PPPs from the 1993 PPPs—than does the middle
term, and recommends replacing the adjustment procedure.
I can see both advantages and disadvantages to Ravallion’s proposal. On the plus side it is
superior empirical performance, at least for 2005 ICP from 1993. On the negative side, as (11) makes
clear, and as Ravallion also notes, updating with the CPI—the middle term of (11)—should give the same
answer. That it does not means that there is something unexplained, and unless we know why the CPIs
and the PPPs have diverged in the past, it is hard to be sure that they will continue to do so in the future,
or if so, by how much. This would be less of an issue if we could be sure that the Balassa-Samuelson
regularity is indeed a regularity, meaning that we can be confident that the parameters of equation (10)
will remain stable over time, and that their values estimated in the cross-section will be valid for updating
over time. The Balassa-Samuelson effect is driven by differential rates of productivity growth. The
validity of (10) requires that there is a stable relationship between per capita GDP and productivity
growth, for which there seems no solid foundation. The Penn effect regressions in 1993 and 2005 have
slopes that are significantly different from one another, though the differences are not sufficiently large to
invalidate updating using the 1993 parameters. It would be good if these issues were better understood
than is currently the case.
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References
Biggieri, Luigi, and Tiziana Laureti, 2011, “Understanding changes in PPPs over time,” ICP TAG
Working paper, October.
Deaton, Angus, 2010, “Price indexes, inequality, and the measurement of world poverty,” American
Economic Review, 100(1), 5−34.
Dalgaard, Esben, and Henrik Sejerbo Sorensen, 2002, “Consistency between PPP benchmarks and
national price and volume indexes,” OECD meeting of national accounts experts, working paper.
Hill, Robert J., 2004, “Constructing price indexes across space and time: the case of the European
Union,” American Economic Review, 94(5), 1379−1410.
Locker, H. Krijsnse, and H. D. Faerber, 1984, “Space and time comparisons of purchasing power parities
and real values,” Review of Income and Wealth, 30(2), 53−83.
McCarthy, Paul, 2011, “Extrapolating PPPs and comparing ICP benchmark results,” ICP TAG working
paper, October.
Obstfeld, Maurice, and Kenneth S. Rogoff, 1996, Foundations of international macroeconomics, MIT
Press.
Ravallion, Martin, 2010, “Price levels and economic growth: making sense of the PPP changes between
ICP rounds,” World Bank Policy Research Working Paper No. 5229, March.
Rhoades, Darryl, 2011, “Towards a methodology for analyzing the coherence between PPP and SNA
based estimate of real GDP,” processed.
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