ACCOUNTING FOR THE ENVIRONMENT IN THE SOLOW GROWTH MODEL. Lendel K. Narine

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1 1 ACCOUNTING FOR THE ENVIRONMENT IN THE SOLOW GROWTH MODEL Lendel K. Narine A popular definition of sustainability given by Solow (1991) was an obligation to conduct ourselves so that we leave the future the option or the capacity to be as well off as we are. In this context, any loss of natural capital must be balanced out by the formation of new capital of equal or greater value. Firstly, this study attempts to estimate Trinidad s current rates of saving, population growth, environmental resource extraction, reproducible capital depreciation and total factor productivity (TFP). Following this, the aim is to simulate a balanced growth path according to the Solow model s definition of a steady state by estimating the ideal: rate of growth in saving rate of population growth non-renewable resource extraction rate rate of growth in reproducible capital accumulation and, rate of TFP growth Furthermore, the difference between current growth rates and ideal growth rates will be compared to determine the velocity of convergence to a steady state as defined by the Solow growth model. Ideal rates of growth will be used to inform national environmental policies. Theoretical framework Starting with a simple production function where aggregate production (GNP) is a function of reproducible capital (K) (as expressed in Dasgupta, 2009), labor supply (L), non-reproducible capital i.e. purely environmental resource (), and a parameter for total factor productivity (TFP = A) on the right hand side of the production function. We assume that the size of the economy does not affect per worker output and capital (K, ) exhibits near perfect substitution (capital accumulation (investments, i) can offset depreciation in both K and ) and initially, population growth is held constant: Y = A f (K, L, ) Eq. 1 Where: K is a numerical index of the economy s set of reproducible capital L is the country s total labour supply is a numerical index of the economy s set of non renewable natural resource capital, (purely environmental) that is used in production A is a parameter that represents total factor productivity, an exogenous parameter adjusted to include the value share factors of K, L and Y denotes the economy s aggregate output, or GNP

2 2 APPENDIX A The inclusion of the Environment in TFP growth (A) Kolstad (2011) noted that TFP is the rate of growth in quality adjusted output less the rate of growth in quality adjusted inputs. TFP is a parameter (A) on the right hand side of the production function and changes as technology changes, holding inputs constant. Therefore, TFP is a coefficient for technological progress or technical efficiency. A allows output to expand without additional inputs (K, L, ). Starting with the production function in Eq. 1, we assume that environmental capital () is a factor input into the production of output Y in year 1: Y = A f (L, K, ) Eq. 2 Then, output in year 2: Y + ΔY = A + ΔA f (L + ΔL, K + ΔK, + Δ) Eq. 3 We assume competitive markets and efficient environmental regulations (Pigouvian fees): Marginal damage = Marginal cost and P = MC; P i is the price of each input, so that; Y = MP L P L + MP K P K + MP P Eq. 4 Subtracting eq. 2 from eq. 3 yields: ΔY = ΔA f (L, K, E) + MP L.ΔL + MP K.ΔK + MP.Δ Eq. 5 To express eq.5 directly in terms of per unit change, divide eq. 5 by eq. 2: ΔY = ΔA + L MP L Y A Y ΔL + K MP K L Y ΔK + MP K Y Δ Eq. 6 Eq. 6 expresses the shares of each input factor. From eq.6, the value share factor i can be derived indirectly (in terms of input prices) by substituting eq.4 in eq.6 L MP L Y = P L L K MP K Y E MP Y = = P L L+P K K+P P K K P L L+P K K+P E E P P L L+P K K+P Eq.7 Eq.8 Eq. 9 Eq. 7-9 are the value share or weights of each production factor to TFP. Let S i denote the weight of each factor: i MP i = S Y i Eq. 10 From eq. 10, the rate of change in output (productivity growth) can be expressed as: ΔY = ΔA + S ΔL Y A L + S ΔK L K + S Δ K Eq. 11 S is the value share or importance of non-renewable natural resource (purely environmental factor) to output (GNP) and Δ is the rate of change in the extraction rate of natural resource. From eq. 11, TFP growth (A) can be expressed as: A ΔA = ΔY S ΔL A Y L S ΔK L K S Δ K Eq. 12 TFP growth rate is the rate of change in output (GNP) minus the rate of change in factor inputs weighted by their respective share values (S i ).

3 3 Measuring Solow s steady state growth with a technology-adjusted production function Starting with eq. 1: Y = A f (K, L, ) The per worker production function can be expressed as: y = A f (k, ke) Eq. 13 Where: y (per worker income/per capital GNP) = Y L, k (per worker capital input) = K L, ke (per worker environmental capital input) = L Per worker income (y) is a result of consumption (c) and investment (i): y = c + i Eq. 14 It is assumed that people save a fraction of their income and consume the remainder: c = (1 s) y 0 s 1 Eq. 15 Substituting eq. 15 in eq. 14, y = (1 s) y + i Eq. 16 Then, i = y - (1 s) y i = y y + sy i = sy Eq. 17 Therefore, investment is equal to saving as a fraction of income. Substituting for 13 in eq. 17 yields: i = A.s f (k, ke) Eq. 18 Recall, capital accumulation (investment, i), can offset depreciation in K and. From eq. 18, it is established that investments (i) should equal to savings (s) as a fraction of income (y) where income is a function of reproducible and non reproducible capital [f (k, ke)]. If f (k, ke) is defined as some level of old capital stock, then investment in the addition of new capital (k and only k) should ideally equal old capital (k, ke) subject to some rate of saving (0 s 1). In this case, the addition of new capital (k) exactly offsets old capital (k, ke) at some rate of s. This can be viewed as a very preliminary steady state with adjustment made only for the savings rate. Given the effect of savings on capital in the steady state, the effect of capital depreciation in k and ke can be added. Depreciation in reproducible capital k, δ is included as a constant rate (0 δ 1) to old capital stock (δk) and depreciation in non-renewable resource capital ke, is the extraction rate (0 β 1; this will gravitate to 0 in an intergenerational context) to existing environmental capital stock (βke). Capital (k, ke) in year t n will depreciation at an annual rate δ and β respectively in t 1+n. If in the steady state capital (k, ke) is unchanged, then an identified rate of saving (s) and depreciation (δ and β) will allow for capital in year t 1+n to equal capital in year t (Δ k = 0): Δ k = i δk βke

4 4 Substituting for i from eq. 18 yields: Δ k = A.s f (k, ke) δk βke Eq. 19 In the steady state, Δ k = 0, manipulating eq. 19 gives; A.s f (k, ke) = δk + βke, or Eq. 20 sy = i = δk* + βke* Eq. 21 Eq. 20, 21 shows that in the steady state, investment is equal to old capital discounted by a depreciation rate δ plus existing environmental capital discounted by an extraction rate β. At k* and ke*, the total capital stock will remain unchanged over time. Also, over time, ke may be totally exhausted. At this point, we can express eq. 21 in an intergenerational context: If Y is to be sustainable when ke = 0, investments in k must counter this, placing additional pressure on k to maintain some level of welfare, let π denote the pressure on δk when ke is exhausted. In words, when environmental resources are totally exhausted, the economy requires greater reproducible capital to counter this loss in natural resources so output is sustained. Greater emphasis on reproducible capital will cause an increase in wear and tear of old capital k; this increase in wear and tear in capital is denoted as π. sy = i = δk* + βke* Over time, ke = 0 (β = 0), then greater pressure placed on δk sy = i = δ π k + β(0)* Over time, eq. 21 becomes sy = i = δ π k* Eq. 22 Eq. 22 suggests that when ke is exhausted, there would be an exponential increase in depreciation δ π. For sustainability, investments or capital accumulation (i) must be significant enough to offset this additional pressure for a steady state to exist. Adjusting for Population growth (n) Solow further states that a particular capital accumulation cannot explain sustained economic growth. An increase in population adds further pressure to capital stock of k and ke. We expect as the population increases, capital stock per worker (k, ke) decreases. This effect is comparative to that of depreciation so a modification of eq. 19 can be derived: Δ k = A.s f (k) (δ + n) k (β + n) ke Eq. 23 If the capital is to remain unchanged over time, with population growth included, eq. 22 becomes: sy = i = (δ + n) k + (β + n) ke Eq. 24 The savings rate (sy = i) must be enough to offset the effect of population growth and depreciation in k and ke. Maximizing consumption in the steady state Finally, as a requirement of sustainability, Solow s proposition of constant consumption despite declining resource flow (ke) through sufficient capital accumulation (i) (Solow, 1974) is

5 5 considered. This also satisfies Hick s principle of maximizing the minimum realized consumption level (Hicks, 1939). In this study, these requirements are met if the growth rates simulated allow the economy to gravitate towards the steady state with the highest level of consumption i.e. the golden rule level of capital. In this state, there is an optimal level of capital accumulation which will allow for intergenerational economic well-being. From eq. 14: If, y = c + i Then c = y i Eq. 25 Substituting for y and i from eq. 13 and eq. 24: c = A f (k, ke) - (δ + n) k - (β + n) ke Eq. 26 The total derivative of this function can be stated: DC = MP DY K + MP E δ + n + β + n Eq. 27 If the condition that maximizes consumption is given as DC = 0, then eq. 27 becomes; MP K + MP E = δ + n + β + n Eq. 28 Let the slope of the production function MP K + MP E = MP and total depreciation δ + n + β + n = П (total depreciation and population growth is a linear function with a constant slope of П) Then eq. 27 can be written as: MP = П Eq. 29 Eq. 29 suggests that the slope of the production function must equal to the slope of total depreciation in the steady state (Mankiw, 2007) i.e. the marginal depreciation rate of both reproducible and non reproducible capital plus population growth must be equal to the total marginal product of both reproducible and non reproducible capital. This study seeks to estimate the growth rates expressed in eq. 23 and 29. REFERENCES Dasgupta, P The welfare economic theory of green national accounts. Environmental and Resource Economics, Vol. 42 (2009): Hicks, John R Value and Capital: An Inquiry In to Some Fundamental Principles of Economic Theory. Oxford: Clarendon Press Kolstad, C. D Intermediate Environmental Economics. 2 nd edition. Oxford University Press: UK. Mankiw, G. N Macroeconomics. 6 th edition. Worth Publishers: USA Solow, R.M Sustainability: an economist s perspective, the eighteenth J. Seward Johnson lecture to the Marine Policy Center, Woods Hole Oceanographic Institution. In: Dorfman, R., Dorfman, N.S. (Eds.), Economics of the Environment: Selected Readings. Norton, New York, pp Solow, Robert M Intergenerational Equity and Exhaustible Resources. Review of Economic Studies, Symposium on the Economics of Exhaustible Resources. Edinburgh, Scotland, Longman Group Ltd. DY