# When GDP and GDE Are Not Equal

Keywords: Statistics;

Introduction

It is an elementary truism of economics that Gross Domestic Product can be measured on the production side (that is in terms of the products of firms) and the expenditure side (that is in terms of the final purchases of the products) and the two aggregates are exactly equal to one another (although in practice there will be a measurement error, called the ‘statistical discrepancy’).

This equality arises from the properties of the relationships between the products and the prices on the two sides. However for a number of reasons, economists apply different prices to those in which the actual transactions take place. One situation is for volume (or real or constant price) GDP where the effect of the changing price level is allowed for by using the same set of prices for GDP in each year. Another is PPP-adjusted GDP which is used for comparisons to be made between different countries by applying common prices to the production.

It has long been known that, in a particular situation, the application of different prices from the actual transaction ones, results in estimates of the two GDP sides which are not exactly equal. That situation is 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;
and
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.

Less well known, or rather implicitly well-known but rarely of much interest, is that there is index number problems in constant price GDP series, as the relative balance of the products which are the base weights of the GDP index change so that all the usual problems of index construction apply. This is not a major problem in the short run, because the changes are not generally great. But in the long run – as illustrated by the arrival of new products – the issue is complicated.

This paper provides a rigorous formulation of this phenomenon. It does so be carefully 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 for its costs. This chain (for every expenditure item) is characterised by a matrix Γ and 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 Γ effect.

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.

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.

In both cases the mathematics shows that RGDP is no longer equal to RGDI except in very special circumstances typically involving particular conditions on the Γ and, where applicable, the Λ matrices.

As already mentioned, 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.

Although not developed in this paper, the adjustment to get from PPP-adjusted RGDI to PPP-adjusted RGDP is relatively straight forward, although more data onerous than the terms of trade adjustment for time comparisons. It is conjectured the adjustment is likely to be significant for economies with a substantial proportion of their exports are subject to high degrees of intervention – such as agricultural products.

What to do about the Γ effect is more demanding. It probably involves estimating it to some degree, or deriving the RGDP estimates directly by the application of production side prices, although this is likely to be very data onerous. It is also possible that the Γ and Λ effect may explain some of the inconsistencies of projections of PPP-adjusted GDP through time.

The rest of this paper is not mathematically undemanding. Corrections and presentational improvements would be appreciated.

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 ≠ m, generally).

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

(3) GDP = pp’.P
And
(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). We call these new prices pp* and pe*, and the equivalent of equation 2 is

(6=2*) pe*’ = pp*’.Γ*

Now

(7=5*) GDE* = pe*’.E =( pp*’.Γ*).E = pp*’.(Γ.E) + pp*’.(Γ* – Γ).E
= GDP + pp*’.(Γ* – Γ).E

So generally, GDE* = GDP* only if (Γ* – Γ) = 0.

It is standard to assume for practical purposes that

(8) (Γ* – Γ) approximately equals 0,

for constant price comparisons through recent time, so in such cases GDE* approximately equals GDP* in a closed economy.

In the case of PPP comparisons, the assumption in equation (8) appears to be less true, perhaps considerably less true.

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 = Γ.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. As a result the SNA has adopted a convention that

Constant price GDP measured on the production side is called RGDP;
while
Constant price GDP measured on the expenditure side is called RGDI.

The Formal Model for a Open Economy (economy (PPP) comparisons)

For open economy 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.

(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.)

Border prices do not always equal domestic prices. The simplest case is when there is a tariff (on an export) or subsidy (on an import), although the generalisation to a tariff/subsidy equivalent is not difficult. We characterise the relationship between pb and pp as follows:

(15) pp = Λ.pb,
and also
pp’ = pb’.Λ (since Λ is symmetrical).

where Λ is a n-square diagonal matrix (all off-diagonal elements are zero) in which the elements on the diagonal
= 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*’.(Γ* – Γ).E.
= GDP* + pb*’.Λ*.T + pp*’.(Γ* – Γ).E.
= GDP* – pb*’.(Λ*-I).T + pp*’.(Γ* – Γ).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”;
while
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 (from OECD) estimates of PPP-adjusted GDP are measured on the expenditure side and so are more analogous to Gross Domestic Income (RGDI). They would need to be adjusted for the border interventions, characterised by (Λ*-I), to be better measures of GDP, that is the value of production in some international PPP prices, although that Γ* ≠ Γ presents a more unresolvable complication.

Conclusion

While, as equation (5) reports, GDP valued on the production side is exactly equal to GDP valued on the expenditure side (although there may be a statistical discrepancy when practically measured) this conceptual equivalence does not apply in other instances.

Constant Price GDP Estimates Through Time

As equation (7) reports, the constant price GDP estimates on the production and expenditure sides diverge through time in a closed economy. The reason for this is that the relationship between the production side prices and the expenditure side prices (characterised by the Γ matrix in the equation) changes over time. However the practical discrepancy may be small in the short run.

As equation (12) reports, the constant price GDP estimates on the production and expenditure sides diverge through time in an open economy. This is additional to the Γ matrix effect in a closed economy. The new effect arises because of the impact of the terms of trade on the traded product sector. This second effect is not small for a country which experiences large swings in its terms of trade, and is recognised in the SNA by describing the constant price estimates on the production side as RGDP and on the expenditure side as RGDI.