GDP valued at purchasing power parity prices is widely treated as a measure of production, even though it is calculated on the expenditure side of the national accounts. This paper shows that GDP and GDE (or GDI)are not generally equal (although they are if they are measured in transaction prices). It suggests that we should relabel the measure as GDI at purchasing power parity prices which is actually being what is measured or, better still, measure GNI.
Introduction: Repricing GDP
It is an elementary truism that nominal Gross Domestic Product can be measured on the production side (that is in terms of the products of firms) and on the expenditure side (that is in terms of the final purchases of spenders) and that the two aggregates are exactly equal to one another (although in practice there will be a measurement error – the ‘statistical discrepancy’).
This equality arises from the properties of the relationships between the products and the prices on the two sides. However this mathematical congruency does not apply when a different set of prices is applied to the production and expenditure sides. That means that GDP at these prices is not necessarily equal to GDI at these prices, even if the prices are consistent.
Economists apply different prices from those in which the actual transactions take place. Over time they want to compare for volume (or real or constant price) GDP where the effect of the changing price level is eliminated. Between countries they want PPP-adjusted GDP which use common prices to the production.
It has long been known that through time, the application of different prices from the actual transaction ones, results in estimates of the two GDP sides which are not conceptually equal particularly where there is a change in the terms of trade. The SNA recognises this by identifying two volume measures:
Constant price GDP measured on the production side is called RGDP or Real Gross Domestic Product;
Constant price GDP measured on the expenditure side is called RGDI, or Real Gross Domestic Income.
Neither measure is to be preferred over the other. Rather they have different purposes. RGDP indicates what is occurring on the production side of the economy, while RGDI is a measure of the resulting spending power. A lift, say, in the terms of trade, means that the domestic spending power increases more than production, because the (exported) products are able to purchase more imports and hence give the purchasers more purchasing.
However the problem of the divergence, once non transaction prices are used is not confined to this case. Conceptually, PPP-adjustment has broadly the same mathematical structure, that is it is the application of another set of prices to the two sides of GDP, although in this case the prices come from the same time but a different country, rather than a different time and the same country.
This paper provides a rigorous formulation of the phenomenon. It does so by separating out the production side and its prices and products from the expenditure side and its prices and products. Thus butter in a shop appears on the expenditure side as a single item, but on the production side it appears as the result of the activities of a chain of firms: the farm produces the milk, the dairy factory turns it into butter, the transport system distributes it and the shop adds a retail margin. This chain (for every expenditure item) is characterised by a matrix Γ . The analysis shows that a divergence between RGDP and RGDI arises when a new set of prices arise where, as is likely, a different Γ matrix applies.
This exercise is done initially for a closed economy and then generalised to an open one, where the terms of trade effect becomes evident as a part of the effect. However, further terms arises, reflecting the distinction between product prices of international tradeables which are assumed to be the same in all economies (other than the scaling effect of the exchange rate). The analysis incorporates a Λ matrix which converts the price of the goods at the border to the domestic product price, the difference reflecting such things as protection (such as tariffs) on imports and subsidies (and other assistance) on exports.
The formal model shows that GDP is no longer equal to GDI except in special circumstances typically involving particular conditions on the and, where applicable, the matrices. (The analysis also looks at the effects on internal indirect taxes and subsidies.)
We now set out the findings mathematically.
The Formal Model for a Closed Economy
In an economy firms produce products (which are measured in the production side of the economy) the quantities of which in a period are represented by a (1 x n) column vector P where n is the number of products.
These products get transformed into expenditure items (which are measured on the expenditure side of the economy) the quantities of which in a period are represented by (1 x m) column vector E where m is the number of expenditure items (and n is not generally equal to m).
The relation between P and E is given by
(1) P =Γ.E,
where Γ is an (m x n) matrix.
The prices of the goods and services are given by a (1 x n) column vector pp, and the prices of the expenditure items are given by a (1 x m) column vector pe. It follows from 1 (and various routine economic assumptions) that
(2) pe’ = pp’.Γ
Nominal GDP and GNE is given by
(3) GDP = pp’.P
(4) GDE = pe’.E
Substitution from (1), (2), (3), (4) gives
(5) GDE = pe’.E =( pp’.Γ ).E = pp’.(Γ .E) = pp’.P = GDP
So GDE = GDP
Now suppose another set of prices are applied. The prices might be from another year of the closed economy (as a part of constructing a constant price series), or from another country (as a part of constructing a PPP adjusted measure). Call these new prices pp* and pe*, and the equivalent of equation 2 is
(6=2*) pe*’ = pp*’.Γ *
(7=5*) GDE* = pe*’.E =( pp*’.Γ *).E = pp*’Γ.(Γ.E) + pp*’.(Γ * – Γ).E
= GDP* + pp*’.( Γ* – Γ ).E
So generally, GDE* = GDP* only if (Γ* – Γ) = 0.
It is usual to assume for constant price comparisons through time that for practical purposes
(8) (Γ* – Γ) almost equals 0,
so in such cases GDE* almost equals GDP* in a closed economy.
In the case of PPP comparisons, the assumption in equation (8) appears to be less true.
Diversion: An IllustrationΓ
The following is a very simple illustration of the result in the previous section.
Suppose an economy consists of two input goods ig1 and ig2 and a final good fg, each unit of which is composed of one unit of each production good. We suppose the economy produces one hundred units of the final good.
In the notation of the previous section
Γ = (1, 1)
E = (100)
From which it follows that
P = Γ.E = (100, 100),
so the economy produces one unit of each input good.
Suppose the price of each input good is $1. Then
pp = ($1, $1)
pe’ = pp’. Γ = ($2)
So GDI = $200 and GDP = $200.
We can summarise the economy with the following simple tabulation:
First Year (1)
|Item||Final good||Input good 1||Input good 2|
GDI = $200;
GDP = $100 +$100 = $200.
Now suppose in the following year the , that is the way final goods are composed of input goods changes, so that while it now takes 1 unit of ig1 but only half a unit of ig2 to produce 1 unit of fg. (For instance suppose the second input good might be transportation, a new method of transportation or a new route is found, which reduces the required input). Again assume final production is 100 units.
Denoting second year variables by an #,
Γ# = (1, ½)
E# = (1)
From which it follows that
P# = Γ#.E# = (1, ½)
Suppose the price of each input good remains at $1. (There are numerous reasons why despite the productivity gain from input good 2 its price does not fall by the same extent.)
pp# = ($1, $1)
pe#’ = pp#’’.Γ# = ($1½)
So GDI# = $150 and GDP# = $150.
The next year is tabulated as
Next Year (2)
|Item||Final good||Input good 1||Input good 2|
GDI + $150;
GDP = $100 +$50 = $150.
Now apply year 2 prices to year 1 production.
GDI (or GNE) in year one valued at year two prices is one hundred units of the final good times the year two price of $1½ = $150.
However GDP in year one valued at year two prices is one unit of production good 1 valued at $1 and one unit of production good 2 valued at $1 or $2.
Thus valued in year 2 prices, year 1 GDP does not equal year 1 GDI.
The tabulation is
Year (1) at Year (2) prices
|Item||Final good||Input good 1||Input good 2|
GDI + $150;
GDP = $100 +$100 = $200.
Which illustrates the general principle, that under a different, but consistent, price regime GDI valued at these prices need not equal GDP valued at these prices.
The Formal Model for an Open Economy (through time)
Suppose the economy has the same variables as in the closed economy, plus the additional opportunity of importing and exporting products. The quantities internationally traded are represented by a (1 x n) column vector T. Elements in the vector may be positive (in which case the product is imported), zero (in which case it is a not traded), or negative (in which case the product is exported).
In the following we shall assume that
(9) pp’.T = 0,
that is there current external account is in balance, and so GDE stills equals GDP.
The relation between P and E is given by
(10) P + T = .E,
it being unnecessary to identify, for these purposes, what determines T.
Substitution using equations (1), (2), (3), (4), (9) and (10) gives
(11) GDE = pe’.E = ( pp’.Γ).E = pp’.(Γ.E) = pp’.(P – T) = GDP – pp’.T = GDP.
so GDE = GDP
Now suppose another set of prices, pe*, are applied as previously. In which case, using (2) and (10):
(12) GDE* = pe*’.E = (pp*’.Γ*).E = pp*’Γ( .E) + pp*’.(Γ* – Γ).E
= GDP* – pp*’.T + pp*’.(Γ* – Γ).E.
So even if Γ* = Γ, then generally, GDE* = GDP* only if pp*’.T = 0,
That pp’.T = 0 provides no guarantee that pp*’.T = 0. In practice pp*’.T may differ greatly from zero for a country which experiences significant terms of trade changes. As a result the constant price GDP series can show a different pattern depending on whether it is measured on the product or the expenditure side. Hence the distinction which the SNA recognises.
Diversion The New Zealand Experience with the Terms of Trade
The problem arises because the goods and services consumed in an open economy differ from that which is produced because some of the domestic production is exchanged for foreign production, – exported to in exchange for imports. The ratio of the exchange values can vary, and that leads to the difference when, for instance, constant price GDP estimates are made over time. This exchange ratio can be measured as the ratio of export prices over import prices. Note that the exchange rate does not directly influence the terms of trade ratio, providing the prices are measured either in the local currency or the international currency.
The difference has long been understood in New Zealand, which is a small open multi-sectoral economy much prone to changes in its terms of trade as the following Chart shows.
There can be substantial changes in the terms of trade both on a year to year basis and secularly.
The issue was important practically in the 1950s and 1960s when the Court of Arbitration made a General Wage Order, which changed all wage rates across the economy. There was no express reference to the terms of trade in the law guiding the court but it was required to take into consideration ‘any increase or decrease in productivity and in the volume and value of production in the primary and secondary industries of New Zealand’ (as well as changes in consumer prices). How then to measure productivity? To simplify, it could be measured as volume GDP per unit of labour input, but that makes the different measures of volume became significant. In any case workers were keen to a share any benefit form a rise in export prices, while businesses were keen to share their fall.
The resolution was to have two measures of volume GDP. Today they would be called RGDP and RGDI, although the latter was then called ‘effective GDP’. RGDP was calculated on the production side, indicating what had been produced after adjusting for price changes. RGDI, calculated on the expenditure side, was a measure of the purchasing power the production generated. It differed from RGDP by valuing exports in terms of the imports they would purchase. The relationship between them was
RGDI = RGDP + (value of exports)/(import price index)
RGDI = RGDP + (volume of exports)x(terms of trade)
= RGDP*(1 + (volume of exports/RGDP)x(terms of trade)). 
In an economy with exports as a low proportion of output, and not prone to major changes in its terms of trade, the difference between RGDP and RGDI would be small. But in an economy of New Zealand’s characteristics the effect is not insignificant as the following Chart of the ratio of RGDI to RGDP shows.
The overall pattern is a downward trend in the ratio, implying that RGDI has grown more slowly that RGDP by about .1 percent a year. The downward trend reflects that New Zealand had faced deteriorating terms of trade in the post-war era. (There were two main drivers. The largest traditional export, wool, was undercut by synthetics, and the other two traditional exports, meat and dairy products – still the largest good exports today – are subject to widespread international protectionism including restrictions of access in affluent markets and dumping by producers in those affluent markets into third markets.) The effect is that RGDI has grown less than RGDP in the post 1950 era by about 5.5 percent.
There is considerable variation around this postwar decline, again reflecting swings in the terms of trade. The Standard Deviation of the year to year changes is 2.0 percent compared to an average annual change in RDGP and RGDI of 1.5 and 1.4 percent respectively. In about half the years the growth of RGDP and RGDI diverged by more than their trend growth rate.
Thus for New Zealand the distinction between RGDP and RGDI is important in the short run and the long run. The production story is quite different from the income/expenditure story.
The Formal Model for a Open Economy (cross national (PPP) comparisons)
For comparisons of open economies, we need to represent prices of tradeable products at the border. The (1 x n) price vector is pb.
Equation (9) is now replaced by
(13) pb’.T = 0,
Additionally there are international prices, represented by pi, where
(14) pb = e.pi, and e is the exchange rate. 
Border prices do not always equal domestic prices. The simplest case is when there is a tariff (on an import) or subsidy (on an export), although the generalisation to a tariff/subsidy equivalent is not difficult. We characterise the relationship between pb and pp as follows:
(15) pp = .pb,
pp’ = pb’.Λ (since is symmetrical).
where Λ is a n-square diagonal matrix (all off-diagonal elements are zero) in which the elements on the diagonal are
1+t if the product is an import (where t is the tariff rate)
1+s if the product is an export (where s is the export subsidy)
1 if the product is a non-tradeable.
As demonstrated in equation (11), GDE = GDP in the prices of the day, it follows from (15) and (5) that
(16) GDP = pb’.Λ.P = GDE.
Applying a set of prices from another country, as before, the result is as for (12) but applying (15):
(17) GDE* = GDP* – pp*’.T + pp*’.(Λ* – Λ).E.
= GDP* – pb*’. *.T + pp*’.(Λ* – Λ).E.
= GDP* – pb*’.(Λ*-I).T + pp*’.(Λ* – Λ).E,
noting that from (14)
pb* = e*.pi = (e*/e).pb so that pb*’.T = (e*/e).pb’.T = 0.
So even were Γ* – Γ = 0, GDE* would not equal GDP*, unless Λ*=I, that is there were no tariffs or export subsidies in the economy whose expenditure prices are being used.
The parallel with the constant price distinction between GDE and GDP should not go unnoticed. It suggests that
PPP-adjusted GDP measured on the production side is analogous to GDP, and should be called “PPP-adjusted RGDP”;
PPP-adjusted GDP measured on the expenditure side is analogous to GDI, and should be called “PPP-adjusted RGDI”;
It is to be noted that the standard estimates of PPP-adjusted GDP are measured on the expenditure side and so are more analogous to Gross Domestic Income (RGDI). RGDP needs to be adjusted for the border interventions, characterised by ( *-I).
The Effect of a Sales Tax
Suppose that there is a sales tax on expenditure items so (2) becomes
(2s) pe’ = pp’.Γ.(I+ Σ)
Where Σ is a symmetric matric with the rates of sales tax on the diagonals (they may be negative if there is a subsidy), and zeros off diagonal. (Note Σ = Σ’.)
In which case (5) becomes
(5s) GDE = GDP + pe’.Γ.E
so now GDE in market prices equals GDP in basic (or factor) prices plus the second term of tax revenue, which is standard in SNA accounting.
The next equation adapts (7) when there is a sales tax to
(7s) GDI* = GDP* + pp*’.(Γ* – Γ).E + pe*’.(Σ*- Σ).E.
Thus there is a need for a further adjustment which represents the difference in the two sales tax regimes. Where Σ = Σ* as often happens through time, the adjustment is zero, but that is much less likely in cross-national comparisons.
The mathematics shows that RGDP is no longer equal to RGDI except in very special circumstances involving particular conditions on the and, where applicable, the and matrices. The implication for constant price GDP series through time is reasonably well known: that if RGDP is to be derived from RGDI, there has to be an allowance for the impact of terms of trade. Less well known is that there is a parallel effect for PPP-adjusted GDP.
In practice, the internationally accepted estimates of PPP-adjusted GDP are derived from the expenditure side, and correspond with RGDI. They do not, therefore, reflect the production side of the economy, even though that is often the way they are presented. The problem is likely to be most serious for small economies (because they contribute little to the PPP prices) and small open economies subject to substantial fluctuations in their terms of trade.
Also, where there is considerable market distortions in a country’s export markets – as for the agricultural goods New Zealand produces – there may be considerable divergence between GDI and GDP – perhaps in the order of 10 percent. Essentially protection against efficient agricultural producers reduces their income relative to their production, and raises the income relative to production of the inefficient protected users. The income transfer from this protection does not change productivity, of course, but it contaminates the method of using GDI to estimate GDP.
Towards a Resolution
Thus using GDI as an estimate of GDP is misleading. Consequentially any productivity comparisons between countries may be very misleading. Can anything be done?
The first strategy might be to directly estimate the Γ, Λ and Σ matrices and adjust GDI to GDP. In the case of Γ matrix that may prove quite challenging, because of the problem of the wholesale and retail trade sector, further discussed below. Estimating the Λ matrix may be more straightforward, not only because it has a simpler structure (off diagonal elements are zero), but because the tarrification of protection where there have been quotas and prohibitions gives direct estimates of the on diagonal elements. Estimating the Σ matrix may be similarly straight forward although even more tedious because it involves collection of an even larger data base. (An alternative might be to value before taxes.)
A second strategy would be estimate GDP at PPP prices directly on the production side. There has been an understandable reluctance to do this, because the required data base is much larger, and there is the fundamental problem of the treatment of the wholesale and retail trade sector as indicated by their being separated out in Input-Output Tables rather than being incorporated in the expenditure change. International comparisons of the domestic trade sector are notoriously difficult, something avoided (or suppressed) in the GDE estimates, but which would have to be confronted in a direct GDP estimate. Nevertheless in my view there is a case for doing direct production comparisons in those sectors where the method is reasonably tractable. I have done some rough ones for New Zealand, which leaves much of the non-government service sector as a residual. Comparing them with the official PPP adjusted GDP figures suggests that there may be something seriously wrong with the latter.
The third strategy would be to abandon the estimation of PPP adjusted GDP via the expenditure side, at least temporarily until the previous two strategies can be implemented. Instead the current estimates should be given their conceptually correct name, GDI, while discouraging their use for productivity comparisons.
I would go a step further, adjusting to GNI or, were it possible, to NNI (or National Income). I observe the World Bank prefers this notion. It is a better indicator of material welfare. It answers a different question to that which GDP does, but as this paper shows, the current measure of PPP adjusted GDP does not answer production questions either.
 See In Stormy Seas (p.93) for more details. The Court was also required to take into consideration ‘relative movement in the incomes of different sections of the community’ and ‘all other considerations that the court may deem relevant’, either of which could also refer to the terms of trade.
 The equation ignores the details of the price bases.
 The prices of non-tradeable products in pb present a problem since there is no international price. The following analysis could be done with partitioned vectors and matrices. Or the elements representing those prices could be set at infinity, reflecting the price at which the non-tradeables could be tradeable. However we shall put them as the local non-tradeable price which avoids both inelegant partitioning or calculations of the form (0 x ). We can do this, because the place where it matters is in the expression pb’.T, where the price element for a non tradeable product multiplies with a zero, since there is no trade in it.