added:a new look at new zealand's growth …...added:a new look at new zealand's growth...
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Added:A New Look at New Zealand's Growth Performance
By Ulrich Kohli
WORKING PAPER(2003/01)
ISSN 13 29 12 70ISBN 0 7334 2051 6
CENTRE FOR APPLIEDECONOMIC RESEARCH
Terms of Trade, Real GDP, andReal Value Added:
A New Look at New Zealand’sGrowth Performance
Ulrich Kohli ∗
November 2002
Abstract
This paper shows that the conventional measure of real GDP underesti-mates the growth in real value added when the terms of trade improve.Thus, in New Zealand, where the terms of trade have been improving overthe past 15 years, real GDP has underestimated the country’s real growthperformance by nearly 0.4% per year on average. Our analysis has a solidtheoretical foundation, being based on the GDP-function approach to mod-elling the production sector of an open economy.
JEL Classification : O11, O41, C43, F11Keywords : real GDP, terms of trade, real value added, economic growth, indexnumbers
∗Chief Economist, Swiss National Bank, P.O. Box 2800, CH-8022 Zurich, Switzerland.Phone: +41-1-631-3233/34; fax: +41-1-631-3188; e-mail: [email protected]; home page:www.unige.ch/ses/ecopo/kohli/kohli.html. I am grateful to Michael Oliver of Statistics NewZealand for his advice concerning the data. I also wish to thank Kevin J. Fox, Nick Oulton,Mary Louise O’Dowd, and Jo Paisley for their comments on an earlier draft of this paper. Theyare obviously not responsible for any errors or omissions.
Preamble
Once upon a time there was an island somewhere in the South Seas. The nativesspent most of their time fishing. Real GDP in this country was equal to their fishcatch. The central bank, by carefully managing the supply of cowry shells, wasable to maintain a constant price level. Setting the price level to unity, nominalGDP was thus equal to real GDP.
The natives’ fishing boats needed a fair bit of maintenance. This was quitetedious work. Over time, the islanders realised that some of them had a compar-ative advantage at fixing boats, while others were relatively better at going out tosea. They thus decided that the natural born ship builders would stay on shore,while the others would do the fishing. The catch would then be shared betweenthem all. This reorganisation of production, which involved specialisation withtrade, was similar to a technological progress. It led to an increase in the totalfish catch, i.e. an increase in real (and nominal) GDP.
The islanders had the choice between two fishing spots, the East Bay and theWest Bay. Sometimes the fish were more plentiful in one bay, sometimes in theother bay. Every morning, the fishermen would randomly select a fishing site.Even though their daily catch fluctuated a fair bit, depending on whether or notthey were lucky enough to have chosen the right spot, their annual catch wasremarkably stable. Eventually the tribe’s elders realised that the fish were mostlyin the East Bay when the sky was cloudy, and in the West Bay when it wassunny. Using this new knowledge, the fishermen managed to increase their catchsubstantially. Thus, this new technological advance led to a further increase inreal (and nominal) GDP.
The ocean was actually populated by two types of fish, blue fish and red fish.They were equally easy (or difficult) to catch. The blue fish tended to be foundon the north side of the East Bay or of the West Bay, depending on the weather,whereas the red fish seemed to prefer the south side. The islanders were perfectlyindifferent between the two kinds of fish. Both fish were therefore selling forthe same price, and the composition of production and consumption was totallyrandom.
The islanders knew that they were not alone in the world. There were manyother, much larger, islands just beyond the horizon, with very similar economies.The only difference was that in the rest of the world the inhabitants had a verystrong preference for blue fish, so much so that the price of blue fish was fourtimes that of red fish. Needless to say, the fishermen in the home country consid-ered going abroad to trade blue fish for red fish, but unfortunately this was noteconomically feasible, for about half the cargo of fish would rot during a crossing.By shipping one ton of blue fish, only one half would arrive unspoilt. This half-ton of blue fish could then be exchanged for two tons of red fish, half of which
1
would rot on the voyage back. Even though the other transportation costs wereinsignificant, the operation made no economic sense. Until the day the islandersdiscovered refrigeration. From that day on, the fishermen specialised in catchingblue fish, and by the time they came back to their home port, they were carryingfour times the amount of red fish. Real (and nominal) GDP thus quadrupledthanks to this tremendous technological progress.
Let us go back a few steps, and assume for a moment that the price of blue fishwas the same as the price of red fish, at home and abroad, and, moreover, thatrefrigeration had been around for a long time. Since relative prices would thenbe the same in both regions, there would be no incentive to trade, even in theabsence of transportation costs. Assume next that, for whatever reason, the priceof blue fish quadrupled in the rest of the world. Trade suddenly becomes highlyprofitable for the home country. The fishermen harvest blue fish exclusively, andby the time they return to port, they carry four times the amount of red fish.The improvement in the terms of trade has led to a spectacular increase in realvalue added. What will the island’s national accountant have to say about this?Although nominal GDP has quadrupled, he or she will argue that real GDP hasnot changed. It therefore must be the GDP deflator that has been multiplied byfour, irrespective of the fact that the domestic price of fish has remained constant.
Even though the consequences of the improvement in the terms of trade hereare perfectly analogous to those following the discovery of refrigeration in the pre-vious example, the treatment of these two events by the national accounts differsradically. The former is viewed as a price effect, while the latter is viewed as areal effect. The national accounts take a very narrow view of production whenit comes to international trade, even though domestic specialisation and domes-tic trade enter the calculation of real value added. Domestic production involvesthe transformation of inputs into outputs. International trade involves additionaltransformations. By excluding terms-of-trade effects from the calculation of aneconomy’s real value added, one omits an important source of transformation.In most industrialised nations, services, rather than manufacturing, account forthe bulk of economic activity. Many services, such as wholesaling, retailing, andbanking, involve trades. The trader’s markup is a measure of his or her valueadded. It makes little sense to handle domestic and foreign trades differently, justas it makes little sense to treat terms-of-trade effects and technological changesunevenly. The asymmetrical treatment of terms-of-trade improvements and tech-nological progress is all the more problematic as in many cases it may not bepossible to distinguish between the two. A drop in the price of imports could becaused equally well by an improvement in the terms of trade or by technologicalprogress in the shipping industry.
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1 Introduction
As shown by Figure 1, New Zealand’s terms of trade have improved by approxi-mately 24% over the past 15 years.1 They have increased very strongly in the late1980s, only to fall back in the early 1990s, before resuming their increase, at amore moderate pace, for the remainder of the decade. This average improvementin the terms of trade should have been a blessing for New Zealand. As any studentof international trade theory knows, this should have led to an increase in realincome and welfare. Yet, it can be shown that this improvement in New Zealand’sterms of trade has, ceteris paribus, reduced its real GDP as it is conventionallymeasured.
An improvement in the terms of trade means that, for a given trade balanceposition, the country can either import more for what it exports, or export lessfor what it imports. Simply put, one gets more for less. This is almost identical toa technological improvement. Both events imply an increase in real value addedand in economic welfare.2 Yet, the national accounts treat these two phenomenavery differently. A technological change is considered as a real phenomenon, and,for given factor endowments, it leads to an increase in the conventional measureof real GDP. An improvement in the terms of trade, on the other hand, is treatedas a price phenomenon. It enters the calculation of the GDP deflator, with anegative weight. A drop in the price of imports, for instance, will, other thingsequal, lead to an increase in the GDP price deflator. Nominal GDP will increaseas well, but the conventional measure of real GDP will fall, even though the dropin import prices must unambiguously increase real value added, real income, andwelfare.3
A change in the price of traded goods relative to the price of domestic sales isliable to trigger welfare effects as well. Consider an equiproportionate increase inthe prices of imports and exports, so that the terms of trade remain unchanged.If trade is balanced, the extra cost of imports is exactly offset by the additionalexport revenues. In case of a trade deficit, however, the higher traded good priceswill make the country worse off, while the reverse is true in the case of a tradesurplus. This effect is typically buried in the GDP price deflator, although it isa real effect. Figure 2 shows the path of the price of exports relative to the price
1The terms of trade are obtained by dividing the export deflator by the import deflator. Allyears refer to the years ending on March 31.
2Although real value added and economic welfare are clearly very different concepts, it is thecase that an increase in real value added allows, other things equal, for an increase in welfare.
3Although most statisticians are well aware of the difference between real GDP and realdomestic income, this distinction is generally overlooked in practice. Thus, Prescott (2002),who describes New Zealand as a depressed economy, focuses exclusively on real GDP and nevereven mentions terms-of-trade changes.
3
of domestic sales for the period 1986–2001. One sees that this price was far fromconstant. In fact, it fell quite steeply until 1998, before recovering somewhat.4
2 A Simple Description of the Aggregate Tech-
nology
Throughout this paper, we treat imports as an input to the technology. Thistreatment is consistent with the fact that most of world trade is in raw materialsand intermediate products, and that even most so-called “finished” products muststill transit through the production sector — where they are combined with do-mestic value added — before reaching final demand. Furthermore, this treatmentof imports is consistent with the national accounts framework where imports aretreated as a negative output.
We begin with a very simple description of the aggregate technology, similarto the one used by Kohli (1983). Thus, we assume that gross output is producedby a domestic composite factor of production (an aggregate of labour and capital)and imports. For the time being, we assume a single output that can be eitherabsorbed at home or exported to the rest of the world to pay for imports. Inaccordance with standard trade theory, we view the endowment of the compositefactor and the prices of goods as exogenous. We denote the quantity of importsat time t by qM,t, the endowment of the domestic aggregate factor by xt, and thequantity of gross output by qY,t. The technology can be described by the followingaggregate production function:
qY,t = f(qM,t, xt). (1)
We assume that f(·) is twice continuously differentiable, increasing, linearly ho-mogeneous, and quasi-concave. The competitive equilibrium can be describedas the solution of maximising GDP, subject to the technology, the domestic fac-tor endowments, and the prices of goods. This leads to the following first-ordercondition:
fM(qM,t, xt) =pM,t
pY,t
, (2)
where fM(·) ≡ ∂f(·)/∂qM,t, pM,t is the price of imports, and pY,t is the price ofoutput. Condition (2) can be solved for qM,t to yield the import demand function:
qM,t = qM(pM,t, pY,t, xt). (3)
4Throughout this paper the price of domestic sales (pS) is computed as a Tornqvist index ofthe deflators of consumption expenditures, investment, and government purchases.
4
Quasi-concavity and linear homogeneity of the production function imply that thedemand for imports is decreasing in its own price; linear homogeneity implies thatthe demand for imports is homogeneous of degree one in xt; profit maximisation,finally, guarantees that the demand for imports is homogeneous of degree zero inprices. Assuming that the aggregate domestic factor is mobile between firms, itscompetitive rental price (wt) will be equal to the value of its marginal product:
wt = pY,tfx(qM,t, xt)
= pY,tfx[qM(pM,t, pY,t, xt), xt]
= w(pM,t, pY,t, xt), (4)
where, naturally, fx ≡ ∂f(·)/∂xt.As an alternative to the production function, we can also use the country’s
GNP/GDP function to describe its technology.5 This will prove very useful lateron when we generalise the model and increase the number of inputs and outputs.The GDP function is defined as follows:
π(pM,t, pY,t, xt) ≡ maxqM,t,qY,t
{pY,tqY,t − pM,tqM,t : qY,t = f(qM,t, xt)}, (5)
for pM,t, pY,t > 0, xt ≥ 0. As shown by Diewert (1974), given the assumptionsmade on f(·), this function is linearly homogeneous and convex in prices, linearlyhomogeneous in the endowment of the domestic factor, nondecreasing in pY,t andnonincreasing in pM,t. Moreover, Hotelling’s Lemma implies that import demandfunction (3) can be obtained directly by differentiation of the GDP function:
qM(pM,t, pY,t, xt) = −∂π(·)∂pM,t
. (6)
Similarly, the supply of gross output can be obtained as:
qY (pM,t, pY,t, xt) =∂π(·)∂pY,t
, (7)
whereas the competitive return to the domestic factor is given by:
w(pM,t, pY,t, xt) =∂π(·)∂xt
. (8)
Before we proceed, it is useful to define some additional concepts. NominalGDP measures the value of all final goods and services produced during a given
5See Kohli (1978, 1991), Woodland (1982).
5
period of time. It is given by the value of the GDP function; in view of (5) it canthus be defined as:
πt ≡ pY,tqY,t − pM,tqM,t. (9)
Next, we define real value added (yt) — or, alternatively, real net output — asnominal GDP deflated by the price of output:6
yt ≡ πt
pY,t
=pY,tqY,t − pM,tqM,t
pY,t
= qY,t − pM,t
pY,t
qM,t. (10)
From the linear homogeneity of the production function, one can see that:
pY,tqY,t − pM,tqM,t = wtxt. (11)
Hence, real value added in this model is equal to real income, i.e. nominal incomedeflated by the price of output:
yt =wtxt
pY,t
. (12)
The conventional measure of real GDP (qLt ) is defined as follows:7
qLt ≡ qY,t − qM,t. (13)
This last definition is of course well known to anyone familiar with the nationalaccounts, where real GDP is defined as the constant-dollar value of gross outputminus the constant-dollar value of imports. It is common practice, finally, todivide nominal GDP by real GDP to obtain an average price of GDP, also knownas the GDP deflator (pP
t ):
pPt ≡ πt
qLt
=1
(1 + sM,t)p−1Y,t − sM,tp
−1M,t
, (14)
where sM,t is the GDP share of imports: sM,t ≡ pM,tqM,t/πt. It is well known that,given that real GDP as defined by (13) is a Laspeyres quantity index, the implicitGDP deflator as given by (14) has the Paasche form.
6All price and quantity indices are defined relative to a base period, for which prices aretypically normalised to one.
7Statistics New Zealand has recently switched from a direct (fixed-based) to a chainedLaspeyres index of real GDP. This means that expression (13) is only valid for consecutiveperiods: it is as if one renormalised prices every period. Comparisons over extended periods oftime can be made by compounding the yearly changes.
6
3 Graphical Analysis
Figure 3 represents the production function in (qM , qY ) space.8 For given x, thefunction is increasing and concave in qM . It need not go through the origin:the distance OA represents output in autarky. If the relative price of imports(pM/pY ) — the inverse of the terms of trade — is given by the slope of line BC,then GDP is maximised at point C where the marginal product of imports isequal to its relative price. Imports are given by the distance OD, gross output isequal to OE . Real income of the domestic composite factor is equal to OB. Thiscan also be interpreted at domestic real value added or real net output. Underbalanced trade, exports are equal to BE and domestic absorption amounts to OB.Absorption exceeds OB if there is a trade deficit, or falls short of it if there is asurplus. Assuming that all prices are normalised to one initially, real GDP too isgiven by the distance OB. Thus, this far, we can interpret real GDP as a measureof real net output, real value added, and real income. Note that the distance ABcan be interpreted as the gains from trade.
Consider now the effect of an improvement in the country’s terms of trade, i.e.a reduction in the relative price of imports. Assume that the new terms of tradeare given by the slope of B′C′. The equilibrium point moves to the northeast, fromC to C ′. Imports increase from OD to OD′, and gross output goes up, from OE toOE ′. Real value added unambiguously increases, from OB to OB′.9 What aboutreal GDP? It can easily be seen that real GDP decreases, from OB to OF : F isthe intercept of the line through C′ with a negative unit slope; this slope is minusone because real GDP — as shown by (13) — is evaluated at base-period prices.10
The intuitive explanation of this phenomenon is as follows. When import pricesfall, the country can afford to import more. Yet, as shown by (13), real GDP isobtained by subtracting imports valued at their base-period prices. By failing totake into account the lower price of imports, one ends up subtracting too much.
Another way to look at the problem is to consider the effect of a change inthe terms of trade on the GDP deflator. As shown by (14), a drop in the price ofimports leads, ceteris paribus, to an increase in the GDP deflator, even though noprice actually increases. This shows that, contrary to common belief, the GDPdeflator is a poor index of the general price level. Indeed, the drop in the price ofimports has no inflationary effects, quite the contrary. Consequently, if the GDP
8The time subscripts are omitted in this section and the next for more clarity.9It is interesting to note that this outcome does not depend on the position of the trade
account. That is, since exports are assumed, for the time being, to be perfect substitutes forgoods intended for domestic use, the effect of a change in the terms of trade on real value addedis independent of the level of exports.
10In fact, as noted by Kohli (1983), if the terms-of-trade improvement were sufficiently large,real GDP could become negative!
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deflator overestimates the price of value added, real GDP must underestimate itsquantity.
If the terms of trade were to worsen, imports, gross output, and real valueadded would fall. So would real GDP. However, real GDP would tend to under-estimate the reduction in real value added. While real GDP will tend to under-estimate the negative effect of a deterioration in the terms of trade, the situationis much more serious in the case of an improvement in the terms of trade, sincereal GDP will move in the wrong direction.
While our analysis is based on the GDP-function approach to modelling im-ports and exports, the same type of bias could be exposed if one used insteadthe Heckscher-Ohlin-Samuelson model of international trade where imports aretreated as final goods. An improvement in the terms of trade relative to the baseperiod, while unambiguously increasing real income, would lead to a reduction inreal GDP, i.e. total output minus imports valued at base period prices.11
It might be useful to emphasise here that the fact that an improvement inthe terms of trade leads to a reduction in the Laspeyres index of real GDP hasnothing to do with chaining or the absence of it. The example illustrated byFigure 3 refers to two states — or two periods — only. Hence, there can beno difference between chained and direct indices. Statistics New Zealand hasrecently moved from a direct (fixed-based) Laspeyres measure of real GDP to achained Laspeyres index. Although chained indices are to be preferred to directones in almost every respect, this switch does not address the problem identifiedin this paper. The reason why real GDP drops in our example has to do withthe functional form that is being used. The Laspeyres quantity index tends tounderestimate the aggregate quantity in the context of production theory, exceptin the extreme cases of linear and Leontief aggregator functions. It only providesa linear approximation to what is shown in Figure 3 to be a nonlinear productionfunction. If, instead of the Laspeyres index, one used a quantity index that isexact for the depicted production function,12 the index of real GDP would remainconstant. However, it would still fail to pick up the increase in real value addedthat results from the drop in import prices.13
11See Kohli (2002) for additional details.12Thus, the Tornqvist index is exact for the translog functional form, whereas the Fisher
index is exact for the square-rooted quadratic; see Diewert (1976).13The United Sates currently uses a chained Fisher index of real GDP. An improvement in
the terms of trade is therefore likely to have little impact on the index of real GDP, althoughreal value added must unambiguously increase.
8
4 Numerical Example: The Cobb-Douglas
Functional Form
How large is the bias illustrated in Figure 3? Some back-of-the-envelope calcula-tions using New Zealand data can be based on the assumption that the technologyis Cobb-Douglas. Assume that the production function is given by:
f(qM , x) = γqαMx1−α. (15)
The GDP function then is as follows:
π(pM , pY , x) = δp1−βM pβ
Y x, (16)
where δ ≡ (1 − α)αα/(1−α)γ1/(1−α) and β ≡ 1/(1 − α).14 The demand for im-ports, supply of gross output, and domestic factor rental price are obtained bydifferentiation:
qM = −δ(1 − β)p−βM pβ
Y x (17)
qY = δβp1−βM pβ−1
Y x (18)
w = δp1−βM pβ
Y . (19)
Consider now the effect of a large change in the price of imports, from pM0 to pM1 .The changes in the quantity of imports, gross output, and the domestic factorrental price are as follows:
∆qM = −δ(1 − β)pβY x(p−β
M1− p−β
M0) (20)
∆qY = δβpβ−1Y x(p1−β
M1− p1−β
M0) (21)
∆w = δpβY (p1−β
M1− p1−β
M0), (22)
where ∆ is the first-difference operator. Assume that pY and pM are normalizedto one initially; (20)–(22) then become:
∆qM = −δ(1 − β)x(p−βM1
− 1) (23)
∆qY = δβx(p1−βM1
− 1) (24)
∆w = δ(p1−βM1
− 1). (25)
One notes from (25) that an increase (decrease) in the price of imports unam-biguously reduces (increases) the return to the domestic factor. Hence, for givenfactor endowments and a given technology, an improvement in the terms of trademust increase real national income, real net output and real value added.
14See Kohli (1991), page 40.
9
The change in real GDP, on the other hand, can be calculated as:
∆qL = ∆qY − ∆qM
= δx[βp1−βM1
+ (1 − β)p−βM1
− 1]. (26)
Using 1986 New Zealand figures, and normalising all 1986 prices to unity, wefind that β = 1 + qM/qL is approximately equal to 1.34. We can interpret δ asthe 1986 return to the domestic factor (w), and δx as 1986 real GDP (qL). From1986 to 2001, the New Zealand terms of trade improved by about 24%; hence,we can set pM1 to 0.81. Using these figures, we find, on the basis of (26), that a24% improvement in the terms of trade reduces real GDP by about 1.1%, while,judging from (25), it increases real value added by about 7.4%. Thus, as a firstapproximation, we find that the change in real GDP underestimates the increasein real value added that took place between 1986 and 2001 by about 8.5%.
5 Generalisation
The model of the previous section was rather restrictive. Fortunately, it can easilybe generalised to allow for technological change and to allow for many inputs andmany outputs. In what follows, we assume two outputs, domestic sales (S) andexports (X), as well as two primary inputs, labour (L) and capital (K).15 Primaryinput and output (including import) quantities at time t are denoted by xj,t andqi,t, respectively, with prices wj,t and pi,t (j ∈ {L,K}, i ∈ {S,X,M}). It isconvenient to describe the country’s technology by the GDP function that is nowdefined as:16
π(pS,t, pX,t, pM,t, xL,t, xK,t, t) ≡ {max pS,tqS,t + pX,tqX,t − pM,tqM,t :
(qS,t, qX,t, qM,t, xL,t, xK,t, t) ∈ Tt}, (27)
where Tt is the production possibilities set at time t; it is assumed to be a con-vex cone. The GDP function is linearly homogeneous and convex in prices, andlinearly homogeneous and concave in input quantities.
It is well known that the profit-maximising output supply and import demandfunctions can be obtained by differentiation:17
qi = ±∂π(·)∂pi,t
= qi(pS,t, pX,t, pM,t, xL,t, xK,t, t), i ∈ {S,X,M}, (28)
15Domestic sales are an aggregate of consumption expenditures, investment, and governmentpurchases.
16See Kohli (1978, 1991), and Woodland (1982).17Again, see Kohli (1978, 1991) and Woodland (1982).
10
where the minus sign applies to imports. Moreover, assuming that the domesticfactors are mobile between firms, the derivatives with respect to the fixed inputquantities yields the competitive domestic factor rental prices:
wj =∂π(·)∂xj
= wj(pS,t, pX,t, pM,t, xL,t, xK,t, t), j ∈ {K,L}. (29)
In what follows, it will be convenient to define gt as the inverse terms oftrade, and ht as the relative price of exports where domestic sales are used as thenumeraire:
gt ≡ pM,t
pX,t
(30)
ht ≡ pX,t
pS,t
. (31)
GDP function (27) can then be rewritten as:
π(pS,t, htpS,t, htgtpS,t, xL,t, xK,t, t) ≡ ψ(pS,t, ht, gt, xL,t, xK,t, t). (32)
It follows from the properties of π(·) that GDP function ψ(·) is linearly homoge-neous in pS,t. Moreover, one can see from (28)–(29) and (32) that:
∂ψ(·)∂pS,t
= qS,t + htqX,t − htgtqM,t (33)
∂ψ(·)∂ht
= pS,t(qX,t − gtqM,t) (34)
∂ψ(·)∂gt
= −pX,tqM,t (35)
∂ψ(·)∂xj,t
= wj,t, j ∈ {L,K} (36)
∂ψ(·)∂t
=∂π(·)∂t
. (37)
As shown by Diewert and Morrison (1986) in their path-breaking article, theGDP function is a convenient analytical tool to identify the GDP effect of tech-nological progress. The following index indicates the GDP impact of the passageof time, holding all output prices and domestic factor endowments constant:
RLt,t−1 ≡
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1). (38)
Note that all output prices and domestic input quantities were held constantat their values in period t − 1. RL
t,t−1 thus has a Laspeyres form, so to speak.
11
Alternatively one could have frozen output prices and fixed input quantities attheir period-t values to obtain the following Paasche-like index of the GDP effectof technological progress:
RPt,t−1 ≡
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t, ht, gt, xL,t, xK,t, t − 1). (39)
Diewert and Morrison (1986) recommend taking the geometric average of the twoindices just defined. This yields the following Fisher-like index of the GDP effectof technological progress:
Rt,t−1 ≡√
RLt,t−1 RP
t,t−1. (40)
The GDP impact of changes in domestic factor endowments can be defined ina similar way:18
XL,t,t−1 ≡√√√√ ψ(pS,t−1, ht−1, gt−1, xL,t, xK,t−1, t − 1)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t, ht, gt, xL,t−1, xK,t, t),
(41)
XK,t,t−1 ≡√√√√ ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t, t − 1)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t, ht, gt, xL,t, xK,t−1, t).
(42)Next, we can define the following GDP terms-of-trade effect:
Gt,t−1 ≡√√√√ ψ(pS,t−1, ht−1, gt, xL,t−1, xK,t−1, t − 1)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t, ht, gt−1, xL,t, xK,t, t),
(43)and the GDP trade-balance effect:
Ht,t−1 ≡√√√√ ψ(pS,t−1, ht, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t, ht−1, gt, xL,t, xK,t, t).
(44)All five effects just defined are real, and thus they contribute to explaining
changes in the country’s real value added. To square things off, we finally definethe GDP effect of domestic price changes:
PS,t,t−1 ≡√√√√ ψ(pS,t, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t−1, ht, gt, xL,t, xK,t, t).
(45)
18See Kohli(1990).
12
6 Measurement
Assume that the GDP function has the following translog form:19
ln πt = α0 +∑
i
αi ln pi,t +∑j
βj ln xj,t +1
2
∑i
∑h
γih ln pi,t ln ph,t +
1
2
∑j
∑k
φjk ln xj,t ln xk,t +∑
i
∑j
δij ln pi,t ln xj,t +
∑i
δiT ln pi,tt +∑j
φjT ln xj,tt + βT t +1
2φTT t2, (46)
i, h ∈ {S,X,M}; j, k ∈ {L,K},where
∑αi = 1,
∑βj = 1, γih = γhi, φjk = φkj,
∑γih = 0,
∑φjk = 0,
∑i δij = 0,∑
j δij = 0,∑
δit = 0, and∑
φjt = 0.We show in the Appendix that if GDP function π(·) is translog, then GDP
function ψ(·) defined by (32) is translog as well. That is, ψ(·) then provides aflexible representation of the country’s technology.
If estimates of the translog GDP function were available, it would be a simplematter to calculate the various effects defined in the previous section.20 However,it turns out that as long as the true GDP function is translog, all these effects canbe calculated from the data alone; that is, without needing to know the values ofthe parameters of the GDP function. Thus, following in the footsteps of Diewertand Morrison (1986), one can show that:21
Rt,t−1 =Πt,t−1
Pt,t−1 Xt,t−1
, (47)
where Πt,t−1 is the growth factor of nominal GDP, Pt,t−1 is a Tornqvist priceindex of the GDP output components, and Xt,t−1 is a Tornqvist quantity indexof domestic factor endowments:
Πt,t−1 ≡ ψ(pS,t, ht, gt, xL,t, xK,t, t)
ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
=pS,tqS,t + pX,tqX,t − pM,tqM,t
pS,t−1qS,t−1 + pX,t−1qX,t−1 − pM,t−1qM,t−1
(48)
19See Christensen, Jorgenson and Lau (1973) and Diewert (1974). See Kohli (1978) for anearly empirical estimation of this function. Estimates for New Zealand can be found in Fox,Kohli, and Warren (2002).
20See Kohli (1990, 1991) for such an econometric approach.21See the Appendix for a derivation of (47) and (51)–(54).
13
Pt,t−1 ≡ exp
[∑i
1
2(si,t + si,t−1) ln
pi,t
pi,t−1
], i ∈ {S,X,M} (49)
Xt,t−1 ≡ exp
∑
j
1
2(sj,t + sj,t−1) ln
xj,t
xj,t−1
, j ∈ {L,K}, (50)
where si,t and sj,t are the GDP shares of output i and primary input j, respec-tievely: si,t ≡ pi,tqi,t/πt, sj,t ≡ wj,txj,t/πt, i ∈ {S,X,M}, j ∈ {L,K}. Similarly, itcan be shown that:22
Xj,t,t−1 = exp
[1
2(sj,t + sj,t−1) ln
xj,t
xj,t−1
], j ∈ {L,K}, (51)
Gt,t−1 = exp
[1
2(−sM,t − sM,t−1) ln
gt
gt−1
], (52)
Ht,t−1 = exp
[1
2(sB,t + sB,t−1) ln
ht
ht−1
], (53)
where sB,t ≡ sX,t − sM,t, and:
PS,t,t−1 =pS,t
pS,t−1
. (54)
Finally, we show in the Appendix that the six effects that we just definedtogether give a complete decomposition of the growth in nominal GDP:
Πt,t−1 = PS,t,t−1 × Ht,t−1 × Gt,t−1 × XL,t,t−1 × XK,t,t−1 × Rt,t−1. (55)
22Our measure of the terms-of-trade effect — see (52) below — is different from the oneproposed by Diewert and Morrison (1986), and which is defined in terms of GDP function (27)directly:
At,t−1 ≡√
π(pS,t−1, pX,t, pM,t, xL,t−1, xK,t−1, t − 1)π(pS,t−1, pX,t−1, pM,t−1, xL,t−1, xK,t−1, t − 1)
π(pS,t, pX,t, pM,t, xL,t, xK,t, t)π(pS,t, pX,t−1, pM,t−1, xL,t, xK,t, t)
.
Diewert and Morrison (1986) show that, if the true GDP function is translog, At,t−1 can becalculated as:
At,t−1 = exp
[∑i
12(si,t + si,t−1) ln
pi,t
pi,t−1
], i ∈ {X,M}.
There are, however, a couple of difficulties with At,t−1. Thus, if the prices of imports andexports increase in the same proportions (following a devaluation of the national currency, forinstance), At,t−1 will register a change unless trade happens to be balanced on average over thetwo periods, even though the terms of trade clearly do not change in such a case. Put differently,At,t−1,which is meant to measure a real effect, is generally not homogeneous of degree zero inprices. This implies that the element that is supposed to measure the contribution of prices inthe GDP growth decomposition will generally not be homogeneous of degree one in prices.
14
It is noteworthy that the product of the first three terms on the right-handside yields the Tornqvist price index defined by (49):
Pt,t−1 = PS,t,t−1 × Ht,t−1 × Gt,t−1. (56)
In other words, the remaining three terms together provide a measure of what canbe called an implicit Tornqvist index of real GDP:23
Qt,t−1 ≡ XL,t,t−1 × XK,t,t−1 × Rt,t−1 =Πt,t−1
Pt,t−1
. (57)
Such an implicit Tornqvist index of real GDP would be preferable to theLaspeyres index commonly used, in that it is a superlative index. Nevertheless,it excludes the terms-of-trade effect and the trade-balance effect that we haverepeatedly characterised as real — rather than price — effects. These considera-tions lead us to define the following implicit Tornqvist index of real value added(Yt,t−1) obtained by combining all five real effects contained in (55):
Yt,t−1 ≡ Ht,t−1 × Gt,t−1 × XL,t,t−1 × XK,t,t−1 × Rt,t−1
= Qt,t−1 × Ht,t−1 × Gt,t−1
=Πt,t−1
PS,t,t−1
. (58)
It is noteworthy that Yt,t−1, which measures the combined effect of five realforces, can be obtained simply by deflating the change in nominal GDP by theindex measuring the change in domestic prices; that is, without needing any dataon the inputs of labour and capital. This is quite useful given the difficulties thatone generally encounters when one tries to find adequate data on the prices andquantities of domestic primary inputs.
7 Evidence from New Zealand
We show in Figure 4 the yearly values of Gt,t−1 and Ht,t−1 for the period 1987to 2001. It can be seen that the terms-of-trade effect fluctuates a fair bit. Itis greater than one on average, although its contribution to GDP growth wasnegative in 1991-1992, and in 1998-2000. These were the years when the terms oftrade worsened. As to the trade-balance effect, it is very small for most years. Itcannot be ignored, however, if one wants the decomposition of GDP growth givenby (55) to hold exactly.
23See Kohli (1999).
15
Figure 5 shows the paths of our Tornqvist measure of real value added and ofthe official measure of real GDP. It can be seen that over the past 15 years realvalue added has increased more rapidly than the real GDP figures suggest. Thedifference between the two lines reflects mostly the positive average contributionof the terms-of-trade effect.24 From 1986 to 2001, real GDP has increased by justunder 39%. This translates into a yearly average growth rate of 2.22%. Realvalue added, on the other hand, has increased by a total of close to 47%, or, onaverage, by about 2.58% annually. Real GDP thus underestimated the growth inreal value added by close to 0.4% annually over the past 15 years. The cumulatedgrowth deficit adds up to about 5.43% over the period.25 Taking 2001 figures, thisamounts to about 5.7 billion dollars at 1995/96 prices.
8 Concluding Comments
One could object that even if it is true that real value added and real incomeincrease as a result of an improvement in the terms of trade, this is of limitedinterest since it does not create a single job. The reason why many economists areinterested in GDP figures is because an increase in real GDP is usually associatedwith the creation of jobs. Even if it were true that an improvement in the termsof trade does not create any jobs, this criticism is beside the point for severalreasons.26 For a start, as we have shown in Section 3, an improvement in theterms of trade leads to a reduction in real GDP, a reduction that is meaningless.Second, a technological change, which is integrated in the calculation of real GDP,leads to an increase in real GDP without necessarily creating any jobs either. Bothphenomena are perfectly similar, and there is no reason to treat them differently.In truth, if one were really interested in the demand for labour, it would be muchmore sensible to derive it from a GDP function such as (27), instead of relying ona very crude indicator such as real GDP.27 Finally, one ought to remember that
24It can be seen from Figure 4 that the contribution of the trade-balance effect is negligablein the case of New Zealand. Some of the difference is also imputable to the fact that the officialmeasure of real GDP is a Laspeyres chain index, whereas our calculations are based on theTornqvist aggregation, but this effect turns out to be very small as well.
25This is less than our back-of-the-envelope estimate of Section 4. This is because the Cobb-Douglas functional only provides a first-order approximation (in logarithms) to an arbitraryproduction function, whereas the translog gives a second-order approximation. Given that theterms-of-trade change that we are dealing with here is quite large, we should not be surprisedby this difference.
26Note that in the context of the GDP-function model, employment is exogenous; it is thereturn to labour that is endogenous.
27The inverse demand for labour (29) derived from the GDP-function model is much moresophisticated than most specifications commonly used in the literature, and it shows that achange in the terms of trade is likely to affect wages.
16
it is real income — and ultimately consumption — that generate utility, ratherthan work effort, which is usually considered to have a negative marginal utility.Work is a means to an income, it is not a goal in itself.
One could also argue that real GDP attempts to measure a country’s pro-duction effort — production requires hard work — and that there is little meritin experiencing an “effortless” improvement in the terms of trade. We wouldstrongly disagree with such a narrow view. While an improvement in the termsof trade may indeed be a purely exogenous event, foreign trade is an activitythat requires much effort. Importers and exporters must constantly scout worldmarkets in search of better opportunities, and domestic producers must positionthemselves to take advantage of existing comparative advantages, and always beon the lookout for new ones. Similar considerations apply to technological change.Technological progress too may be the outcome of chance, and it is certainly nottrue that every invention or innovation is the outcome of a systematic and tire-some research effort. There is therefore no reason to discriminate between thesetwo types of efforts on a a priori basis.
In order to obtain our Tornqvist measure of real value added, we have useda Tornqvist price index to deflate nominal GDP. This choice was motivated bythe fact that the Tornqvist index is exact for the translog aggregator function.However, other, perhaps more familiar, price indices could be used. Thus, itwould make little numerical difference if pS were calculated as a Fisher index.One could even use a chained Paasche index. Thus, perhaps the simplest measureof real value added in the New Zealand case would be to deflate nominal GDPby the implicit deflator of Gross National Expenditure, an index that is readilyavailable from the New Zealand national accounts. We have verified that use ofthat measure for pS would not significantly alter our findings.
Our measure of real value added comes close to what the United States Bureauof Economic Analysis calls “Command-Basis GNP”.28 Command-basis GNP, orGDP, is a rather crude attempt to take into account the changing purchasingpower of exports when computing a country’s real value added. Command-basisGDP (qB,t) can be computed as:
qB,t ≡ qS,t +pX,tqX,t − pM,tqM,t
pM,t
= qLt + qX,t
(pX,t
pM,t
− 1
). (59)
Compared to the standard Laspeyres definition of real GDP, one sees that in(59) both components of the trade account are deflated by the same price index,namely the price of imports. While this correction goes in the right direction, thechoice of the import price index as the deflator is rather arbitrary. Why the priceof imports, instead of, say, the price of exports, or the GDP price deflator? To
28See Denison (1981).
17
the extent that trade is nearly balanced, the choice of the common deflator doesnot matter much, but our analysis based on the GDP-function model neverthelesssuggests that the proper deflator is the price of domestic sales.
Finally, we would like to emphasise that our argument that the conventionalmeasure of real GDP does not properly incorporate terms-of-trade effects shouldnot be viewed as a criticism of Statistics New Zealand. On the contrary, thisagency does a remarkable job with few resources. By recently switching to achained measure of real GDP, while most other industrialised countries are onlyjust contemplating such a move, Statistics New Zealand has demonstrated thatit is at the forefront of good international practices. The fact remains, however,that, except for the United States and its ad hoc measure of command-basis GNP,no country takes into account terms-of-trade effects when computing real valueadded. In view of the improvement in the terms of trade that New Zealand hasenjoyed over the past decade and a half, one must come to the conclusion thatthe recent New Zealand growth performance has been underestimated by officialreal GDP statistics.
18
Appendix
We begin by showing that if π(·) is translog, then ψ(·) is translog too. GDPfunction (46) can be rewritten as follows (we omit the time subscript for clarity):
ln π = α0 + αS ln pS + αX ln pX + αM ln pM +1
2γSS(ln pS)2 +
1
2γXX(ln pX)2 +
1
2γMM(ln pM)2 +
γSX ln pS ln pX + γSM ln pS ln pM + γXM ln pX ln pM +∑δSj ln pS ln xj +
∑δXj ln pX ln xj +
∑δMj ln pM ln xj +
δST ln pS t + δXT ln pX t + δMT ln pM t +∑βj ln xj +
1
2
∑ ∑φjk ln xj ln xk +
∑φjT ln xj t + βT t +
1
2φTT t2
= α0 + (αS + αX + αM) ln pS +
(αX + αM)(ln pX − ln pS) + αM(ln pM − ln pX) +1
2(γSS + γXX + γMM + 2γSX + 2γSM + 2γXM)(ln pS)2 +
1
2(γXX + γMM + 2γXM)(ln pX − ln pS)2 +
1
2γMM(ln pM − ln pX)2 +
(γXX + γMM + γSX + γSM + 2γXM) ln pS(ln pX − ln pS) +
(γMM + γSM + γXM) ln pS(ln pM − ln pX) +
(γMM + γXM)(ln pX − ln pS)(ln pM − ln pX) +
[(δSL + δXL + δML) ln xL + (δSK + δXK + δMK) ln xK ] ln pS +
[(δXL + δML) ln xL + (δXK + δMK) ln xK ](ln pX − ln pS) +
(δML ln xL + δMK ln xK)(ln pM − ln pX) + (δST + δXT + δMT ) ln pS t +
(δXT + δMT )(ln pX − ln pS) t + δMT (ln pM − ln pX) t +∑j
βj ln xj +1
2
∑ ∑φjk ln xj ln xk +
∑j
φjT ln xj t + βT t +1
2φTT t2
= α0 + aS ln pS + aH ln h + aG ln g +1
2cSS(ln pS)2 +
1
2cHH(ln h)2 +
1
2cGG(ln g)2 +
cSH ln pS ln h + cSG ln pS ln g + cHG ln h ln g +∑dSj ln pS ln xj +
∑dHj ln h ln xj +
∑dGj ln g ln xj +
dST ln pS t + dHT ln h t + dGT ln g t +∑
βj ln xj +
1
2
∑ ∑φjk ln xj ln xk +
∑φjT ln xj t + βT t +
1
2φTT t2, (60)
where aS = αS + αX + αM = 1, aH = αX + αM , aG = αM , cSS = γSS + γXX +γMM + 2γSX + 2γSM + 2γXM = 0, cHH = γXX + γMM + 2γXM , cGG = γMM , cSH =
19
γXX + γMM + γSX + γSM + 2γXM = 0, cSG = γMM + γSM + γXM = 0, cHG =γMM +γXM , dSj = δSj +δXj +δMj = 0, dHj = δXj +δMj, dGj = δMj, j ∈ {L,K, T}.This shows that ψ(·) is a translog function in pS, h, g, xL, xK and t. Moreover, asexpected, it is linearly homogeneous in pS.
The logarithmic derivatives of ψ(·) with respect to its arguments are as follows:
∂ ln ψ(·)∂ ln pS
= aS + cSS ln pS + cSH ln h + cSG ln g + dSL ln xL + dSK ln xK + dST t
= 1 (61)
∂ ln ψ(·)∂ ln h
= aH + cSH ln pS + cHH ln h + cHG ln g + dHL ln xL + dHK ln xK + dHT t
= sB (62)
∂ ln ψ(·)∂ ln g
= aG + cSG ln pS + cHG ln h + cGG ln g + dGL ln xL + dGK ln xK + dGT t
= −sM (63)
∂ ln ψ(·)∂ ln xj
= βj + dSj ln pS + dHj ln h + dGj ln g + φjL ln xL + φjK ln xK + φjT t
= sj j ∈ {L,K} (64)
∂ ln ψ(·)∂t
= βT + dST ln pS + dHT ln h + dGT ln g + φLT ln xL + φKT ln xK + φTT t
≡ sT (65)
Next, it is useful to consider the change in the value of the GDP functionbetween consecutive periods:
ln Πt,t−1 = ln ψ(pS,t, ht, gt, xL,t, xK,t, t) − ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1)
= aS(ln pS,t − ln pS,t−1) + aH(ln ht − ln ht−1) + aG(ln gt − ln gt−1) +
βL(ln xL,t − ln xL,t−1) + βK(ln xK,t − ln xK,t−1) +
1
2cSS[(ln pS,t)
2 − (ln pS,t−1)2] +
1
2cHH [(ln ht)
2 − (ln ht−1)2] +
1
2cGG[(ln gt)
2 − (ln gt−1)2] +
cSH(ln pS,t ln ht − ln pS,t−1 ln ht−1) +
cSG(ln pS,t ln gt − ln pS,t−1 ln gt−1) +
cHG(ln ht ln gt − ln ht−1 ln gt−1) +
dSL(ln pS,t ln xL,t − ln pS,t−1 ln xL,t−1) +
dSK(ln pS,t ln xK,t − ln pS,t−1 ln xK,t−1) +
20
dHL(ln ht ln xL,t − ln ht−1 ln xL,t−1) +
dHK(ln ht ln xK,t − ln ht−1 ln xK,t−1) +
dGL(ln gt ln xL,t − ln gt−1 ln xL,t−1) +
dGK(ln gt ln xK,t − ln gt−1 ln xK,t−1) +
1
2φLL[(ln xL,t)
2 − (ln xL,t−1)2] +
1
2φKK [(ln xK,t)
2 − (ln xK,t−1)2] +
φLK(ln xL,t ln xK,t − ln xL,t−1 ln xK,t−1) +
dST [ln pS,tt − ln pS,t−1(t − 1)] +
dHT [ln htt − ln ht−1(t − 1)] +
dGT [ln gtt − ln gt−1(t − 1)] +
φLT [ln xL,tt − ln xL,t−1(t − 1)] + φKT [ln xK,tt − ln xK,t−1(t − 1)] +
βT +1
2φTT [t2 − (t − 1)2]
= aS(ln pS,t − ln pS,t−1) + aH(ln ht − ln ht−1) + aG(ln gt − ln gt−1) +
βL(ln xL,t − ln xL,t−1) + βK(ln xK,t − ln xK,t−1) +
1
2cSS(ln pS,t + ln pS,t−1)(ln pS,t − ln pS,t−1) +
1
2cHH(ln ht + ln ht−1)(ln ht − ln ht−1) +
1
2cGG(ln gt + ln gt−1)(ln gt − ln gt−1) +
1
2cSH(ln ht + ln ht−1)(ln pS,t − ln pS,t−1) +
1
2cSH(ln pS,t + ln pS,t−1)(ln ht − ln ht−1) +
1
2cSG(ln gt + ln gt−1)(ln pS,t − ln pS,t−1) +
1
2cSG(ln pS,t + ln pS,t−1)(ln gt − ln gt−1) +
1
2cHG(ln gt + ln gt−1)(ln ht − ln ht−1) +
1
2cHG(ln ht + ln ht−1)(ln gt − ln gt−1) +
1
2dSL(ln xL,t + ln xL,t−1)(ln pS,t − ln pS,t−1) +
1
2dSL(ln pS,t + ln pS,t−1)(ln xL,t − ln xL,t−1) +
1
2dSK(ln xK,t + ln xK,t−1)(ln pS,t − ln pS,t−1) +
21
1
2dSK(ln pS,t + ln pS,t−1)(ln xK,t − ln xK,t−1) +
1
2dHL(ln xL,t + ln xL,t−1)(ln ht − ln ht−1) +
1
2dHL(ln ht + ln ht−1)(ln xL,t − ln xL,t−1) +
1
2dHK(ln xK,t + ln xK,t−1)(ln ht − ln ht−1) +
1
2dHK(ln ht + ln ht−1)(ln xK,t − ln xK,t−1) +
1
2dGL(ln xL,t + ln xL,t−1)(ln gt − ln gt−1) +
1
2dGL(ln gt + ln gt−1)(ln xL,t − ln xL,t−1) +
1
2dGK(ln xK,t + ln xK,t−1)(ln gt − ln gt−1) +
1
2dGK(ln gt + ln gt−1)(ln xK,t − ln xK,t−1) +
1
2φLL(ln xL,t + ln xL,t−1)(ln xL,t − ln xL,t−1) +
1
2φKK(ln xK,t + ln xK,t−1)(ln xK,t − ln xK,t−1) +
1
2φLK(ln xK,t + ln xK,t−1)(ln xL,t − ln xL,t−1) +
1
2φLK(ln xL,t + ln xL,t−1)(ln xK,t − ln xK,t−1) +
1
2dST (2t − 1)(ln pS,t − ln pS,t−1) +
1
2dST (ln pS,t + ln pS,t−1) +
1
2dHT (2t − 1)(ln ht − ln ht−1) +
1
2dHT (ln ht + ln ht−1) +
1
2dGT (2t − 1)(ln gt − ln gt−1) +
1
2dGT (ln gt + ln gt−1) +
1
2φLT (2t − 1)(ln xL,t − ln xL,t−1) +
1
2φLT (ln xL,t + ln xL,t−1) +
1
2φKT (2t − 1)(ln xK,t − ln xK,t−1) +
1
2φKT (ln xK,t + ln xK,t−1) +
βT +1
2φTT (2t − 1)
= (ln pS,t − ln pS,t−1) +1
2(sB,t + sB,t−1)(ln ht − ln ht−1) +
1
2(−sM,t − sM,t−1)(ln gt − ln gt−1) +
22
1
2(sL,t + sL,t−1)(ln xL,t − ln xL,t−1) +
1
2(sK,t + sK,t−1)(ln xK,t − ln xK,t−1) +
1
2(sT,t + sT,t−1). (66)
We next provide the details of the derivation of expressions (47) and (51)–(54).To get (52), we begin with (43) that can be written as:
ln Gt,t−1 =1
2[ln ψ(pS,t−1, ht−1, gt, xL,t−1, xK,t−1, t − 1) +
ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −ln ψ(pS,t, ht, gt−1, xL,t, xK,t, t)]
= aG(ln gt − ln gt−1) +1
2cGG[(ln gt)
2 − (ln gt−1)2] +
1
2cSG(ln pS,t + ln pS,t−1)(ln gt − ln gt−1) +
1
2cHG(ln ht + ln ht−1)(ln gt − ln gt−1) +
1
2dGL(ln xL,t + ln xL,t−1)(ln gt − ln gt−1) +
1
2dGK(ln xK,t + ln xK,t−1)(ln gt − ln gt−1) +
1
2dGT (2t − 1)(ln gt − ln gt−1)
=1
2(−sM,t − sM,t−1)(ln gt − ln gt−1), (67)
Similarly, to prove the validity of (53), we begin with (44) that we can write as:
ln Ht,t−1 =1
2[ln ψ(pS,t−1, ht, gt−1, xL,t−1, xK,t−1, t − 1) +
ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −ln ψ(pS,t, ht−1, gt, xL,t, xK,t, t)]
= aH(ln ht − ln ht−1) +1
2cHH [(ln ht)
2 − (ln ht−1)2] +
1
2cSH(ln pS,t + ln pS,t−1)(ln ht − ln ht−1) +
1
2cHG(ln gt + ln gt−1)(ln ht − ln ht−1) +
23
1
2dHL(ln xL,t + ln xL,t−1)(ln ht − ln ht−1) +
1
2dHK(ln xK,t + ln xK,t−1)(ln ht − ln ht−1) +
1
2dHT (2t − 1)(ln ht − ln ht−1)
=1
2(sB,t + sB,t−1)(ln ht − ln ht−1). (68)
Next, using (45) as a starting point, we find:
ln PS,t,t−1 =1
2[ln ψ(pS,t, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) +
= ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −
= ln ψ(pS,t−1, ht, gt, xL,t, xK,t, t)]
= (ln pS,t − ln pS,t−1) +1
2cSS[(ln ht)
2 − (ln ht−1)2] +
1
2cSH(ln ht + ln ht−1)(ln pS,t − ln pS,t−1) +
1
2cSG(ln gt + ln gt−1)(ln pS,t − ln pS,t−1) +
1
2dSL(ln xL,t + ln xL,t−1)(ln pS,t − ln pS,t−1) +
1
2dSK(ln xK,t + ln xK,t−1)(ln pS,t − ln pS,t−1) +
1
2dST (2t − 1)(ln ht − ln ht−1)
= ln pS,t − ln pS,t−1, (69)
where we have taken account of the restrictions on aS, cSS, cSH , cSG, dSL, dSK , anddST .
As to the domestic factors, we get the following for labour:
ln XL,t,t−1 =1
2[ln ψ(pS,t−1, ht−1, gt−1, xL,t, xK,t−1, t − 1) +
= ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −
= ln ψ(pS,t, ht, gt, xL,t−1, xK,t, t)]
= βL(ln xL,t − ln xL,t−1) +1
2φLL[(ln xL,t)
2 − (ln xL,t−1)2] +
1
2φLK(ln xK,t + ln xK,t−1)(ln xL,t − ln xL,t−1) +
24
1
2dSL(ln pS,t + ln pS,t−1)(ln xL,t − ln xL,t−1) +
1
2dHL(ln ht + ln ht−1)(ln xL,t − ln xL,t−1) +
1
2dGL(ln gt + ln gt−1)(ln xL,t − ln xL,t−1) +
1
2φLT (2t − 1)(ln xL,t − ln xL,t−1)
=1
2(sL,t + sL,t−1)(ln xL,t − ln xL,t−1), (70)
and similarly for capital:
ln XK,t,t−1 =1
2[ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t, t − 1) +
= ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −
= ln ψ(pS,t, ht, gt, xL,t, xK,t−1, t)]
= βK(ln xK,t − ln xK,t−1) +1
2φKK [(ln xK,t)
2 − (ln xK,t−1)2] +
1
2φLK(ln xL,t + ln xL,t−1)(ln xK,t − ln xK,t−1) +
1
2dSK(ln pS,t + ln pS,t−1)(ln xK,t − ln xK,t−1) +
1
2dHK(ln ht + ln ht−1)(ln xK,t − ln xK,t−1) +
1
2dGK(ln gt + ln gt−1)(ln xK,t − ln xK,t−1) +
1
2φKT (2t − 1)(ln xK,t − ln xK,t−1)
=1
2(sK,t + sK,t−1)(ln xK,t − ln xK,t−1). (71)
For technological change, finally, we get the following:
ln Rt,t−1 =1
2[ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t) +
= ln ψ(pS,t, ht, gt, xL,t, xK,t, t) −ln ψ(pS,t−1, ht−1, gt−1, xL,t−1, xK,t−1, t − 1) −
= ln ψ(pS,t, ht, gt, xL,t, xK,t, t − 1)]
= βT +1
2φTT [t2 − (t − 1)2] +
1
2dST (ln pS,t + ln pS,t−1) +
1
2dHT (ln ht + ln ht−1) +
1
2dGT (ln gt + ln gt−1) +
25
1
2φLT (ln xL,t + ln xL,t−1) +
1
2φKT (ln xK,t + ln xK,t−1)
=1
2(sT,t + sT,t−1)
= ln Πt,t−1 − ln PS,t,t−1 − ln Ht,t−1 − ln Gt,t−1 − ln XL,t,t−1 − ln XK,t,t−1
= ln Πt,t−1 − ln Pt,t−1 − ln Xt,t−1, (72)
where we have made use of (65)–(71) and (49)–(50). Note that the second-lastline of (72) also provides the proof of (55).
26
ReferencesChristensen, Laurits R., Dale W. Jorgenson, and Lawrence J. Lau (1973) “Tran-scendental Logarithmic Production Frontiers”, Review of Economics and Statistics55, 28–45.
Denison, Edward F. (1981) “International Transactions in Measures of the Na-tion’s Production”, Survey of Current Business 61, 17–28.
Diewert, W. Erwin (1974) “Applications of Duality Theory”, in Michael D. In-triligator and David A. Kendrick (eds.) Frontiers of Quantitative Economics, Vol.2 (Amsterdam: North-Holland).
Diewert, W. Erwin (1976) “Exact and Superlative Index Numbers,” Journal ofEconometrics 4, 115–145.
Diewert, W. Erwin and Catherine J. Morrison (1986) “Adjusting Output andProductivity Indexes for Changes in the Terms of Trade”, Economic Journal 96,659–679.
Fox, Kevin J., Ulrich Kohli, and Ronald S. Warren Jr. (2002) “Accounting forGrowth and Output Gaps: Evidence from New Zealand”, Economic Record 78,312–326.
Kohli, Ulrich (1978) “A Gross National Product Function and the Derived De-mand for Imports and Supply of Exports”, Canadian Journal of Economics 11,167–182.
Kohli, Ulrich (1983) “Technology and the Demand for Imports”, Southern Eco-nomic Journal 50, 137–150.
Kohli, Ulrich (1990) “Growth Accounting in the Open Economy: Parametricand Nonparametric Estimates”, Journal of Economic and Social Measurement16, 125–136.
Kohli, Ulrich (1991) Technology, Duality, and Foreign Trade: The GNP Func-tion Approach to Modeling Imports and Exports (Ann Arbor, MI: University ofMichigan Press).
Kohli, Ulrich (1999) “An Implicit Tornqvist Index of Real GDP”, unpublished.
Kohli, Ulrich (2002) “Real GDP, Real Value Added, and Terms-of-Trade Changes”,unpublished.
Prescott, Edward C. (2002) “Prosperity and Depression”, American EconomicReview, Papers and Proceedings 92, 1–15.
Woodland, Alan D. (1982) International Trade and Resource Allocation (Amster-dam: North-Holland).
27
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1986 1988 1990 1992 1994 1996 1998 2000
Figure 1: Terms of Trade, 1986–2001
28
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1986 1988 1990 1992 1994 1996 1998 2000
Figure 2: Relative Price of Exports, 1986–2001
29
qM
qYf(qM , x)
O
AF
B
EB′
E ′
D D′
C
C′
Figure 3: Production Model — An improvement in the terms of trade reducesreal GDP from OB to OF
30
0.98
0.99
1.00
1.01
1.02
1.03
1988 1990 1992 1994 1996 1998 2000
Terms-of-Trade EffectTrade-Balance Effect
Figure 4: Terms-of-Trade and Trade-Balance Effects, 1987–2001
31
1.0
1.1
1.2
1.3
1.4
1.5
1986 1988 1990 1992 1994 1996 1998 2000
Tornqvist Real Value AddedReal GDP
Figure 5: Tornqvist Real Value Added and Real GDP, 1986–2001
32