On the optimal path of an economy with capital and non-renewable resource inputs, and constant returns output of consumption and investment, the rate of exogenous technical progress in net national product equals the rate of progress in (gross) production, divided by one minus the production elasticity of the resource flow.exogenous technical progress, net national product, non-renewable resources

John C. V. Pezzey

English

Research Papers in Economics

Measuring Technical Progress in
Gross and Net Products
JOHN PEZZEY
EEN0202Measuring Technical Progress
in Gross and Net Products
John C. V. Pezzey
Centre for Resource and Environmental Studies
Australian National University
Canberra, ACT 0200, Australia
Tel/fax +61 2 6125 4143/0757
E-mail pezzey@cres.anu.edu.au
Website http://cres.anu.edu.au/~pezzey
and
Visiting Fellow, Dept of Economics, University of Bath, BA2 7AY, U.K.
Draft of 21 January 2002
Abstract On the optimal path of an economy with capital and non-renewable resource
inputs, and constant returns output of consumption and investment, the rate of exogenous
technical progress in net national product equals the rate of progress in (gross) production,
divided by one minus the production elasticity of the resource flow.
Keywords. Exogenous technical progress, net national product, non-renewable resources
JEL classification: O30, Q301. The result
Consider a closed, smooth, representative-agent economy with a stock of
human-made capital K(t) at time t, a stock of a non-renewable resource S(t), and
an aggregate production function F(K,R,t) that depends on the capital stock,
resource depletion flow R(t), and time (that is, it exhibits exogenous technical
progress). Production is divided, with constant returns to scale but not
necessarily linearly, into investment K(t) and consumption C(t). So we can
write, using overdots to denote total time derivatives and subscripts to denote
partial derivatives as usual,
K = J[F(K,R,t),C]; K(0) = K0 >0 ; JF >0 ,JC < 0 [1]
where J[.] is itself a constant returns function of F and C, hence
J = FJF + CJC. [2]
The dynamics of the non-renewable resource are given by
S =− R; S(0) = S0 > 0. [3]
The economy chooses consumption and resource flow over time to maximise,
using some discount factor φ (t), the present discounted value of U(C), the
instantaneous utility of consumption (with UC >0 ,UCC < 0):
Max ∫ 0
∞ φ (t)U[C(t)]dt; φ (t)>0 ,φ (t)<0 ,φ (0) = 1, limt→∞ φ (t)=0 . [ 4 ]
C,R
We assume the convergence of the integral, and the existence and uniqueness
of a solution to [4], called the optimal path.
The current value Hamiltonian of [4] is
H = U(C)+µ
KK + µ
SS = U(C)+µ
KJ[F(K,R,t),C]−µ
SR [5]
The maximum principle for an interior solution gives
HC = UC + µ
KJC =0 ⇒ µ
K =− UC/JC [6]
HR = µ
KJFFR − µ
S =0 ⇒ µ
S = µ
KJFFR =− UCJFFR/JC [7]
Net national product (NNP) is defined as
Y := C +( µ
K/UC)K +( µ
S/UC)S, [8]
= C − J/JC −( JFFR/JC)(−R) by [1], [3], [6] and [7],
=( F−FRR)JF /( − JC) by [2],
so Y = (1−ε R)FJF /( − JC), [9]
where ε R := FR/(F/R) is the production elasticity of the resource flow.
A fuller writing of [8] is
Y(C,K,S,R,t)=C +( µ
K/UC)J[F(K,R,t),C]−( µ
S/UC)R
so the exogenous effect of time on NNP is
Yt(C,K,S,R,t)=( µ
K/UC)JFFt = JFFt /( − JC) by [6] [10]
1Dividing [10] by [9] and rearranging gives our result: the rate of exogenous
technical progress in net national product is the rate of progress in (gross)
production, divided by one minus the production elasticity of the resource flow:
Yt/Y =( Ft/F)/( 1 −ε R). [11]
Intuitively, the pure time effect in F(K,R,t) gives exogenous progress only
in producing C and K, but not in the resource rent (µ
S/UC)R. So the progress
rate in making NNP = C + µ
KK − µ
SR is higher, by a factor 1/(1−ε R) that shows
the importance of the resource in production. If one relaxes the simplifying
conditions assumed here (for example, by allowing multiple capitals or
resources, or resource renewal), then the result obtained is usually not analytic,
and if analytic not as neat as [11]. But the conclusion remains that technical
progress in (gross) production generally does not occur at the same rate as
technical progress in NNP on an optimal path.
2. The connection to Weitzman’s technical progress premium
There is a connection between result [11] and the "technical progress
premium" (TPP), defined by Weitzman (1997) as:
TPP(0) := [Ψ (0)/Y(0)] − 1, [12]
where Ψ (t) is variously called "consumption NNP" (Asheim 1997);
"sustainability"(byWeitzman);and"wealth-equivalentincome"(Asheim2000),
the term we will use here. It is the consumption level which, if held constant
forever, gives the same present discounted value of consumption as actual
consumption on the optimal path. If the economy maximises welfare as in [4]
and the interest rate is a constant r, it equals
Ψ (t)=r∫ t
∞ C(s)e
−r(s−t)ds. [13]
Weitzman (1997) showed that in this case, if there is exogenous technical
progress (a pure effect of time) in producing NNP, then Y must be adjusted
upwards to give an accurate measure of wealth-equivalent income Ψ , and the
TPP in [12] that makes this adjustment is:
TPP(0) = χ ( 0 )/[ r − g(0)], where [14]
g(0) := ∫ 0
∞ Y(t)e
−rtdt / ∫ 0
∞ Y(t)e
−rtdt, and [15]
χ (0) := ∫ 0
∞ Yt(t)e
−rtdt / ∫ 0
∞ Y(t)e
−rtdt = Yt(0)/Y(0) if Yt/Y is constant. [16]
Here, g(0) is the average future growth rate of NNP; while Weitzman describes
χ (0) as "...the average future growth rate of the ‘residual’, which captures the
pure effect of time alone on enhancement of productive capacity" (p7), or the
"annual growth rate of total factor productivity" (p11).
The connection with our result is that one cannot replace Yt and Y in
expression [16] for χ with figures for Ft and F, which are relevant to total
2production of consumption and investment goods, but leave out the effect of
resource depletion. Otherwise, from [11], the TPP will be underestimated.
One can get a rough idea of the significance of this theoretical result (but
only by entering the realm of empirical controversy about long run production
functions) by assuming a Cobb-Douglas form of F(.) with exponential technical
progress, following Stiglitz (1974) except with a constant labour supply
implicitly assumed:
F(K,R,t)=K
α R
β e
ν t,0 < α , β , α +β≤ 1; ν > 0. [17]
Then, independently of the precise form of the consumption-investment frontier
corresponding to J(F,C) in [1],
1 result [11] becomes
Yt/Y = ν /(1−β ). [18]
So using just the technical progress parameter ν from the (gross) production
function [17] − which would be the rough effect of using data in our model for
conventionally measured NNP = C+K, rather than for "green" NNP =
C+K−FRR − would cause an underestimate of the TPP by a factor (1−β ).
According to Weitzman (1997, p11), this result would have no empirical
significance in the United States now because "the depletion of natural capital
like subsoil minerals, forests, or topsoil...is currently a neglible fraction of
national product" (that is, any β -like parameter is tiny). But the relevance of
such data, and the production functions for which they are estimated, to other
countries and for many decades into the future can be questioned. In some
cases, β -like parameters may be significant, and serious underestimates of the
TPP may happen. Somewhat paradoxically, the more important that non-
renewable resources are in production, the more important it is then to include
exogenous technical progress in national accounts, in order to find a more
accurate measure of wealth-equivalent income.
References
Asheim, Geir B. (1997). "Adjusting green NNP to measure sustainability."
Scandinavian Journal of Economics, Vol 99 No 3, 355-370.
Asheim, Geir B. (2000). "Green national accounting: why and how?" Environment
and Development Economics, Vol 5, 25-48.
Stiglitz, Joseph E. (1974). "Growth with exhaustible natural resources: Efficient and
optimal growth paths." Review of Economic Studies, Symposium on the
Economics of Exhaustible Resources, 123-137.
Weitzman, Martin L. (1976). "On the welfare significance of national product in a
dynamic economy." Quarterly Journal of Economics, Vol 90, 156-162.
Weitzman, Martin L. (1997). "Sustainability and technical progress." Scandinavian
Journal of Economics, Vol 99 No 1, 1-13.
1. For example, one could have K = F−C, as in Stiglitz (1974) and most economic
growth theory; or K =( F
η −C
η )
1/η , η > 1, which would be a specific example of the
strictly concave trade-off between consumption C and investment K assumed by
Weitzman (1976, p160).
3

Adjusting green NNP to measure sustainability.&quot;

For example, one could have K = F−C, as in Stiglitz

Green national accounting: why and how?&quot;

Growth with exhaustible natural resources: Efficient and optimal growth paths.&quot;

On the welfare significance of national product in a dynamic economy.&quot;

Sustainability and technical progress.&quot;

Measuring Technical Progress in Gross and Net Products

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Measuring Technical Progress in Gross and Net Products

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