Capítulo 2 La Cultura
4.5 Desafíos de la Medicina. Problemas del Ambiente
This section models potential spillovers from resource usage to the allocative behaviour of the household, re ecting the potential for environmental externalities to in uence demand for energy intensive services, for example by reducing heating requirements as a result of milder winters, or increasing the need for air conditioning due to hotter summers (see Appendix 5 for a survey of the literature on the potential magnitude of these eects).
To explore these issues, I adapt once more the basic neoclassical growth model with exhaustible resources, such that energy is consumed by the representative householder to provide heating services, for example subject to the following Cobb-Douglas utility function:
M ax
{c(t),E(t),χ(t)}∞t=0V = ˆ ∞
0
(e−it c(t)1−v − 1
1 − v + ((1 − χ (x (t))) E (t))1−w 1 − w
! +
λ(t)
Q(.) − c(t)N (t) − ˙K(t)
)dt (5.1)
where w is the preference parameter over directly consumed energy services; 1 − χ(.) determines the proportion of oil allocated to consumption, which is subject to the following postulated relationship:
χ (x (t)) = ζx(t) (5.2)
ζx(t) > (<)1determines the magnitude and sign of the allocative shift; and the revised production function is given by: Q(.) = A(t)K(t)a1N (t)a2(χ (x (t)) E (t))(1−a1−a2). All other terms are as previously de ned.
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Optimal oil allocation the balance between resources used in consumption and pro-duction satis es the following condition:
χ∗(x (t)) e−it((1 − χ∗(x (t))) E(t))−w = λ(t) (1 − a1− a2)Q(t)
E(t) (5.3)
which determines that, at the margin, the utility from oil allocated to consumption is equal to its shadow value in production.
Lemma 5.1: An interior solution requires that z = −i.
Dierentiating condition (5.3) with respect to time, and substituting for the time path of the marginal product of oil, yields the following expression for the growth rate of oil allocated to production:
ˆ
χ (1 + wΛ (x (t))) = z + i (5.4)
where Λ (x (t)) = 1−χ(x(t))χ(x(t)) represents the ratio of oil used in production to consumption in the presence of the allocative spillover. The stated condition holds by inspection.
If this knife-edge condition fails, the ratio of oil used in production and consumption is not stable over time (dΛ(xdt∗) 6= 0), leading to either economic collapse, or the production only representation outlined previously.
Lemma 5.2: Under stated assumptions, allocation of oil to the householder aects depletion rates as the sign of 1 − v. Furthermore, this condition also determines the in uence of the preference parameter, w.
Sketch proof. Steady state depletion, interest rates, consumption-capital and output growth are given by (see Appendix D for further details):
I (Λ (x∗)) x∗ = (v − 1) (h − n (1 − a1− a2)) +
wΛ (x∗) (1 − a1) + (v (1 − a1− a2) + a2) (1 + wΛ (x∗))
(i + z) (5.5)
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I (Λ (x∗)) r∗ = vh + n (a2+ (v − 1) m) + i second part of the result follows by envelope theorem.
Thus depletion rates fall with the allocation of oil resources to consumption due to the slower accumulation of productive capital. Moreover, for an increase in w, marginally utility falls more rapidly with higher oil consumption, raising incentives for the householder to preserve scant resources, thereby increasing the scale of the adjustment.
It follows that a shift in energy allocation away from household consumption has the potential to raise depletion and output growth: a 4 percent fall in the share of demand by households, for example which, as evidenced by the literature survey at Appendix 5 represents an indicative order of potential magnitude for the UK by around mid century
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corresponds with a 5 percent reduction in depletion rates for plausible assumptions over preferences and income shares to resources.65 66
Tax policy under a consumption spillover
Macroeconomic simulations of such allocative shifts in energy demand have not hitherto formally characterised the policy implications for energy taxation. I make an e ciency based argument here for dierential taxation of energy across consumptive and productive usages. However, and importantly, any welfare gains should be appraised against the potential costs in terms of revenue leakage and additional complexity in tax administration.
To illustrate this, consider the following revised decision of the representative household, assuming the imposition of the optimal oil excise, is given by:
M ax val-orem surcharge on oil used in production; and the redistribution of revenues satis es a balanced budget constraint given by: y(t)N(t) = Zx(t)χ (.) E (t).
Lemma 5.3: Optimal policy warrants an additional levy on energy used in
65Speci cally, v = w = 1.5, a1= 0.34, a2= 0.6 χ = 0.25, h = 2%, n = 1%.
66Simulations which incorporate the in uence of climate change on household energy demand into macroeconomic models without exhaustible resource inputs to production identify long terms output eects on the order of a few tenths of one percent of Gross Domestic Product (GDP) in magnitude (Aaheim et al. (2009), Bosello et al. (2009, 2007), Eboli et al.
(2010), Jorgensen et al. (2009)). The sign of such eects often dier across regions, being negative for hotter regions (see, for example, Aaheim et al. (2009); Eboli et al. (2010)), and is disputed for some important economies such as the US (contrast the ndings, for example, of Eboli et al. (2010) with those of Jorgensen et al. (2004)).
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production, Zχ to correct the distortion to the intra temporal allocation of oil resources, given by: Z(t)χ = 1 − ((1−χ(x∗))E(t))−w
((1−χ(¯x))E(t))−w. Sketch proof. See Appendix D.
Optimal tax policy thus warrants an additional measure to correct the distortion to the intra temporal resource allocation. Such dierential taxes on energy have been employed across a wide range of countries: in the UK, for example, diesel for agricultural uses has been subject to preferential excise treatment; while residential use of kerosene is more lightly taxed than aviation fuel in many developing countries.
However, these policies were originally designed with the objective of pursuing redis-tributive goals. Such arguments are generally considered weak in advanced countries where well developed social security systems oer alternative mechanisms for their achievement, which may be preferred both in theory and practice.
By contrast, this analysis presents an e ciency based rationale.67 However, one should be mindful of the potential costs in terms of revenue leakage and additional complexity in tax administration. One key issue in this latter regard, concerns the risk of tax leakage from imperfect market discrimination.
Practical eorts to mitigate these risks have sometimes relied on colouration of fuels for particular usages, which can, in theory at least, be monitored. However, such responses are inevitably imperfect and costly to implement, particularly in countries with weak tax administration.
Consumption tax rates: numerical results
Appendix D summarizes currently available evidence concerning the potential respon-siveness of energy demand to climatic conditions, and draws inference for potential
correc-67Diamond and Mirrlees (1971a, b), for example, show that, under certain key assumptions, it is preferable to tax nal consumers, since it avoids distorting production decisions (which serve to erode the tax base). Such theories have had a profound in uence on the evolution of modern tax systems, including growing international preferences for implementing a VAT. However, taxing externalities lies outside the general prescription that levies not be imposed on business inputs.
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tive tax policies in the UK. These studies dier importantly in terms of their projections.
However, in broad terms, they project that climate change has the potential to impact residential heating demand in the range +2% to -10% by mid century (and suggest that industrial energy demand in invariant to temperature changes).
Equating 1 − χ with the share of total energy use by households in the UK at 28 percent (DECC (2012)), Table 3 below provides a simple range of numerical results for the corrective tax on energy used in production under dierent assumptions over the size of the allocative shift and the preference parameter w. In the case of w = 1, for example, it suggests, for example, that a 4 percent shift in oil resources towards consumption implies an extra 3.1 percent ad valorem charge on oil allocated to production.