## _wp12200 - Appendix 2: Parametric Specification of the Extended Model ## Source details **Canonical URL:** [_wp12200 - Appendix 2: Parametric Specification of the Extended Model](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12200.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12200.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12200.pdf.json) --- ### Introduction: context and objectives - Countries dependent on non-renewable natural resources face short-run volatility and long-run need to rebalance wealth away from exhaustible natural capital so that the non-resource sector generates enough wealth after exhaustion of natural capital. - Empirical context: share of natural capital in developing countries in Africa and the Middle East “amount to over thirty percent of overall wealth” versus “below five percent” for advanced economies. - Paper aim: foster understanding of optimal public investment decisions in countries experiencing a resource windfall using an augmented Permanent Income framework with public investment adjustment costs and government transfers. - Key modelling assumption: adjustment costs in public investment rise with the size of the resource windfall (net present value). ### Major analytical findings (analytical model) - Three main analytical results: - A larger resource windfall commands a lower level of public capital but a higher level of redistribution through transfers. - Weaker administrative capacity lowers the increase in optimal public capital following a resource windfall. - Higher total factor productivity (TFP) in the non-resource sector reduces the degree of de-investment in public capital commanded by weaker administrative capacity. - Institutional and empirical rationale: - Adjustment costs capture administrative capacity and corruption/rent-seeking associated with larger windfalls (Arezki and Bruckner (2011) evidence). - Quality of public investment management is on average lower in resource rich countries (Kyobe et al. (2011) public investment management index). - Non-resource sector TFP is used as a measure of economic institution quality; Africa identified as region with lowest non-resource sector TFP in the paper’s Figure 4. - Empirical evidence that quality of economic institutions matters for government expenditure to boost non-resource sector growth (Arezki et al. (2011), 1970–2007 panel). ### Simple model of public investment: setup and mechanism - Economy and production: - Small open economy, government can borrow/lend at world rate r^*, sovereign debt C, two sectors (resource and non-resource). - Non-resource domestic income: H(K,A,S) with A = non-resource sector TFP; H assumed homogeneous of degree one so rewritten H_t = A_t h(K_t/A_t,1) = A_t h(K_t/A_t). - Public capital S depreciates at rate δ_s; private capital depreciates at rate δ_k and is rented at world rate r^*. - Public investment and adjustment costs: - Government invests in public capital with adjustment costs g(i_t) ≥ 0; spending on public investment is i_t g(i_t). - Adjustment costs g(.) depend positively on the net present value of the resource windfall Ω ≡ ∑_{j=1}^{N} (1 + r^*)^{-j} Z_j (relation kept implicit in g(.)). - Households, firms, and factor returns: - Households: consumption c_t = w_t + q_t where w_t wage and q_t government transfer. - Firms set H_K = r^* + δ_k; inverse yields private capital linear in A. - Wages w_t are increasing in public capital S_t and more responsive when A is higher (cross-partial > 0). - Government objective and conditions: - Government chooses public investment i_t and transfers q_t to maximize discounted utility subject to debt accumulation, public capital accumulation, and transversality conditions. - Euler conditions yield: - consumption smoothing condition (optimal to smooth consumption over time if β(1+r^*)=1), - dynamic optimal path of public investment as a first-order difference equation, - steady-state condition for investment: g'(i^*) = A φ(S^*, ... , r^*) (equation (3) in text). - Comparative statics from steady state: - Higher TFP A → higher steady state investment in public capital (if g''>0). - Higher marginal adjustment costs (e.g., due to larger windfall) → lower steady state public investment. ### Numerical experiments and calibrated insights - Benchmark calibration (verbatim from source): - ܫ଴ ܫൌ క ܵ,଴ൌ0.75ൈܵ ܨ,଴ൌ5,ܰൌ10,ߜ ௌ ௌ0,05,ߜ௄ ௌ0,05,ݎ כ ௌ0,08,ܣ ௌ0,29,ߛ ൌ0,56 ݀݊ܽ ߙത ൌ0,1൅0.005ൈܸ. - Scenario parametrizations used across experiments (preserved exactly): - s_0 = s̄ - K_0 = 0.75 K̄ - s̄ = 5 - A = 10 - ρ_p = 0.05 - ρ_k = 0.05 - r̄ = 0.08 - ζ = 0.29 - σ = 0.56 - d̄_n k̄ ᾱ = 0.1 + 0.005·Ω - For “high TFP” scenarios increase ζ by 0.075. - For “high adjustment costs” scenarios increase the constant in ᾱ by 0.7. - Representative dynamic responses and comparative insights: - Under a temporary resource windfall, government first accumulates debt anticipating revenues and then accumulates assets; wages relatively flat because objective is consumption smoothing (wage + transfer). - Compared two economies with identical steady states except for windfall size: - Larger resource windfall → lower level of public capital (higher adjustment costs make public investment relatively more expensive) and higher redistribution via transfers; consumption higher in larger-windfall economy though wages lower. - Role of TFP and administrative capacity: - Higher TFP reduces optimal public capital (private capital is crowded in); government raises transfers to smooth consumption, potentially leading to higher wage levels when TFP is high. - Weaker administrative capacity (higher adjustment costs) lowers optimal public capital, reduces private capital rental, lowers wages and consumption. - Interaction: better economic conditions (higher TFP) reduce the gap in optimal public investment between high and low administrative capacity countries. ### Extension: endogenous investment in administrative capacity (“investing in investing”) - Know-how accumulation: - Introduce stock of administrative “know-how” z_t accumulated via investments u_t with z_{t+1} = (1−ξ) z_t + u_t (text uses ܶ and ܬ notation). - Adjustment cost becomes g(i_t, z_t) with ∂g/∂z < 0 (past investments in know-how reduce adjustment costs). - Government choice and new conditions: - Government chooses transfers q_t, public investment i_t and administrative investment u_t subject to debt, public capital, and know-how accumulation constraints. - New Euler condition (equation (4) in text) relates marginal cost of investing in know-how to world interest rate. - From the third Euler condition, steady-state public investment i^* is an implicit function i^* = ι(z) (equation (5)); i^* increases with z provided a condition on g derivatives (text: if and only if −g_{iz} g_{zz}/g_{izz} ≥ 0 under sign assumptions). - Dynamics of know-how and comparative statics: - Dynamic law for z implied by substituting i = ι(z) yields an upward-sloping difference equation (equation (6)) with curvature depending on g’s shape and substitution between i and z in g. - Steady-state of z solves ∆(z^*) = b (equation (7)); steady-state know-how increases with the quality of economic institutions (A) if ∆'_z ≥ 0 — sign depends on g shape but sufficient sign conditions identified (g_{zz}≥0, g_{iz}≤0, g_{ii}≤0 and g_{iz}^2 ≤ g_{zz} g_{ii}). - Key implication: higher initial stock of public administrative “know-how” leads to higher optimal public investment following a resource windfall; investing in administrative capacity raises the payoff to public capital investment. ### Policy implications and practical recommendations - Aggregate messages: - It is not necessarily optimal for resource rich governments to increase public capital investment following temporary resource windfalls when adjustment costs rise with windfall size. - Larger windfalls may optimally be partly or largely redistributed via direct transfers rather than channeled into public investment when administrative capacity is limited. - Weak administrative capacity reduces optimal public capital responses to windfalls; higher non-resource sector TFP mitigates this effect. - Investment in administrative capacity is a policy lever: a higher stock of know-how increases the returns to public investment and raises optimal public investment after windfalls. - Practical approaches to build administrative capacity and reduce adjustment costs: - Increase transparency in handling resource windfalls and better identify and implement projects. - International initiatives highlighted: Extractive Industries Transparency Initiative (EITI); Natural Resource Charter. - Institutional design and project appraisal: example of Chile’s National System of Investments (SNI) with disciplined cost-benefit analysis, shadow social price system and social discount rate; SNI comprises an online databank with over 300,000 entries. - Project Implementation Units (PIUs) recommended by World Bank for project execution, but mixed record due to lack of continuity and knowledge accumulation; recommendation to integrate PIUs into line ministries to preserve know-how. - Research agenda: - Need for more empirical work documenting cross- and within-country differences in state capacity. - More academic work on the political economy of building state capacity is warranted. ### Figures and parametrizations (selected exact parametrizations and notes preserved) - Figure 5 regression line: y = 0.8436x + 1.4191; R² = 0.0571. - Figure 6 parametrization: - initial debt Bo = 40; non-oil GDP Y = 100; non-oil revenue T = τY = 15; oil revenue Z = 15 for 40 years; discount factor β = 0.96 implying an interest rate r = 0.04. - Figure 7 parametrization preserved exactly: - s_0 / s̄ = κ - K_0 = 0.75 κ - s̄ = 5 - A = 10 - ρ_p = 0.05 - ρ_k = 0.05 - r̄ = 0.08 - ζ = 0.29 - σ = 0.56 - d̄_n k̄ ᾱ = 0.1 + 0.005·Ω - Definitions preserved: s_0: investment in public capital at t=0; s̄: steady state investment; K_0: initial stock of public capital at t=0; K̄: steady state stock of public capital; D_0: level of sovereign debt at t=0; A: number of periods resource windfall is non zero; ρ_p: public capital depreciation rate; ρ_k: private capital depreciation rate; r̄: world interest rate; ζ: total factor productivity; σ: private capital income share; ᾱ: adjustment cost parameter; Ω: net present value of resource windfall. - Figures 8–13 benchmark parametrization and scenario adjustments (preserved exactly): - Benchmark: s_0 = s̄; K_0 = 0.75 K̄; s̄ = 5; A = 10; ρ_p = 0.05; ρ_k = 0.05; r̄ = 0.08; ζ = 0.29; σ = 0.56; d̄_n k̄ ᾱ = 0.1 + 0.005·Ω. - Scenario adjustments: - For “high TFP” increase ζ by 0.075. - For “high adjustment costs” increase the constant in ᾱ by 0.7. - Comparative outcomes illustrated for public capital, wages, private consumption, and private capital under: Low TFP, low adjustment costs; Low TFP, high adjustment costs; High TFP, low adjustment costs; High TFP, High adjustment costs. *Source: _wp12200 - Appendix 2: Parametric Specification of the Extended Model.* ### References ............................................................................................................. ### _wp12200 - References ### Figures - Figure 1. Share of Natural Capital around the World ..............................................................21 - Figure 2. Public Management Index by Sub-Groups ...............................................................22 - Figure 3. Investor Protection Index .........................................................................................23 - Figure 4. Non-Resource Sector Total Factor Productivity ......................................................24 - Figure 5. Public Investment Management Index and Non-resource Sector Total Factor ........25 - Figure 6. Resource Windfall, Consumption and Foreign Debt under the Permanent .............26 - Figure 7. Evolution of Wages, Resource Windfalls and Sovereign Debt ................................27 - Figure 8. Evolution of the Stock of Public Capital under Different Scenarios ........................28 - Figure 9. Evolution of Wages under Different Scenarios ........................................................29 - Figure 10. Evolution of Private Consumption under Different Scenarios ...............................30 - Figure 11. Evolution of the Stock of Public Capital under Different Scenarios ......................31 - Figure 12. Evolution of the Stock of Private Capital under Different Scenarios ....................32 - Figure 13. Private Consumption under Different Scenarios ....................................................33 ### Appendixes - Appendix 1: Parametric Specification .....................................................................................19 *Source: _wp12200 - References.* ### Appendix 2: Parametric Specification of the Extended Model ...............................................19 ### _wp12200 - Appendix 2: Parametric Specification of the Extended Model ...............................................19 ### Introduction: context and objectives - Countries dependent on non-renewable natural resources face short-run volatility and long-run need to rebalance wealth away from exhaustible natural capital so that the non-resource sector generates enough wealth after exhaustion of natural capital. - Empirical context cited: share of natural capital in developing countries in Africa and the Middle East “amount to over thirty percent of overall wealth” versus “below five percent” for advanced economies. - Paper aim: foster understanding of optimal public investment decisions in countries experiencing a resource windfall using an augmented Permanent Income framework with public investment adjustment costs and government transfers. - Key modelling assumption: adjustment costs in public investment rise with the size of the resource windfall (net present value). ### Major analytical findings (analytical model) - Three main analytical results: - A larger resource windfall commands a lower level of public capital but a higher level of redistribution through transfers. - Weaker administrative capacity lowers the increase in optimal public capital following a resource windfall. - Higher total factor productivity (TFP) in the non-resource sector reduces the degree of de-investment in public capital commanded by weaker administrative capacity. - Rationale and institutional considerations: - Adjustment costs capture administrative capacity and corruption/rent-seeking associated with larger windfalls (Arezki and Bruckner (2011) evidence that oil rents increase corruption for 1992–2005 panel of 30 oil-exporting countries). - Quality of public investment management is on average lower in resource rich countries (Kyobe et al. (2011) public investment management index cited). - Non-resource sector TFP is used as a measure of the quality of economic institutions faced by private firms; Africa identified as region with lowest non-resource sector TFP in the paper’s Figure 4. - Empirical evidence that quality of economic institutions matters for government expenditure to boost non-resource sector growth (Arezki et al. (2011), 1970–2007 panel). ### Existing theoretical frameworks and limitations - Permanent Income (PI) framework: derives optimal (non-interest) government spending from annuitizing total wealth (recurrent income plus present value of exhaustible resource revenues); government “behaves as if it sold all of its oil immediately.” - Augmented PI literature (Collier, Venables, van der Ploeg and Spence (2011); Venables and van der Ploeg (2011)) adds: - borrowing constraints for individuals; - interest premiums on foreign debt (decreasing with foreign debt); - capital scarcity (higher marginal product of capital domestically). - Lessons from augmented PI: - Use windfalls to increase dividends (when smoothing with borrowing constraints); - Use windfalls to repay foreign debt (when facing foreign borrowing premia); - Favor public investment in low-income, capital-scarce countries over foreign financial asset accumulation. - Gap identified: prior work does not explicitly model administrative capacity and the interaction between administrative capacity and economic institutions; this paper integrates those features analytically and numerically. ### Simple model of public investment: setup and mechanism - Economy: small open economy, government can borrow/lend at world rate r^*, sovereign debt C, two sectors (resource and non-resource). - Non-resource domestic income: H(K,A,S) with A = non-resource sector TFP; H assumed homogeneous of degree one so rewritten H_t = A_t h(K_t/A_t,1) = A_t h(K_t/A_t). - Public capital S depreciates at rate δ_s; private capital depreciates at rate δ_k and is rented at world rate r^*. - Government invests in public capital with adjustment costs g(i_t) ≥ 0 if it increases public capital by i_t spending i_t g(i_t). - Adjustment costs g(.) depend positively on the net present value of the resource windfall Ω ≡ ∑_{j=1}^{N} (1 + r^*)^{-j} Z_j (relation kept implicit in g(.)). - Households: consumption c_t = w_t + q_t where w_t wage and q_t government transfer. - Firms set marginal productivity of private capital equal to its marginal cost: H_K = r^* + δ_k; inverse function yields private capital as a function of public capital and A, implying K* is linear in A. - Wages w_t are increasing in public capital S_t and more responsive when A is higher (cross-partial > 0). - Government chooses public investment i_t and transfers q_t to maximize discounted utility subject to debt accumulation, public capital accumulation, and transversality conditions. - Euler conditions yield: - consumption smoothing condition (optimal to smooth consumption over time if β(1+r^*)=1), - dynamic optimal path of public investment as a first-order difference equation, - steady-state condition for investment: g'(i^*) = A φ(S^*, ... , r^*) (equation (3) in text). - Comparative statics from steady state: - Higher TFP A → higher steady state investment in public capital (if g''>0). - Higher marginal adjustment costs (e.g., due to larger windfall) → lower steady state public investment. ### Numerical experiments and calibrated insights - Benchmark calibration (as reported in the source): - ܫ଴ ܫൌ క ܵ,଴ൌ0.75ൈܵ ܨ,଴ൌ5,ܰൌ10,ߜ ௌ ௌ0,05,ߜ௄ ௌ0,05,ݎ כ ௌ0,08,ܣ ௌ0,29,ߛ ൌ0,56 ݀݊ܽ ߙത ൌ0,1൅0.005ൈܸ. - For “high TFP” scenarios ܣ increased by 0.075; for “high adjustment costs” scenarios the constant in തߙ increased by 0.7. - Representative dynamic responses: - Under a temporary resource windfall, government first accumulates debt anticipating revenues and then accumulates assets; wages relatively flat because objective is consumption smoothing (wage + transfer). - Experiment comparing two economies with identical steady states except for windfall size: - Larger resource windfall → lower level of public capital (higher adjustment costs make public investment relatively more expensive) and higher redistribution via transfers; consumption higher in larger-windfall economy though wages lower. - Role of TFP and administrative capacity (from graphical results): - Higher TFP reduces optimal public capital (private capital is crowded in); government raises transfers to smooth consumption, potentially leading to higher wage levels when TFP is high. - Weaker administrative capacity (higher adjustment costs) lowers optimal public capital, reduces private capital rental, lowers wages and consumption. - Interaction: better economic conditions (higher TFP) reduce the gap in optimal public investment between high and low administrative capacity countries. ### Extension: endogenous investment in administrative capacity (“investing in investing”) - Introduce stock of administrative “know-how” z_t accumulated via investments u_t with z_{t+1} = (1−ξ) z_t + u_t (text uses ܶ and ܬ notation). - Adjustment cost becomes g(i_t, z_t) with ∂g/∂z < 0 (past investments in know-how reduce adjustment costs). - Government now chooses transfers q_t, public investment i_t and administrative investment u_t subject to debt, public capital, and know-how accumulation constraints. - New Euler condition (equation (4) in text) relates marginal cost of investing in know-how to world interest rate. - From the third Euler condition, steady-state public investment i^* is an implicit function i^* = ι(z) (equation (5)); comparative statics show i^* increases with z provided a condition on g derivatives (text: if and only if −g_{iz} g_{zz}/g_{izz} ≥ 0 under sign assumptions). - The dynamic law for z implied by substituting i = ι(z) yields an upward-sloping difference equation (equation (6)) with curvature depending on g’s shape and substitution between i and z in g. - Steady-state of z solves ∆(z^*) = b (equation (7) in text); steady-state know-how increases with the quality of economic institutions (A) if ∆'_z ≥ 0 — the sign depends on shape of g but the paper identifies sufficient sign conditions (g_{zz}≥0, g_{iz}≤0, g_{ii}≤0 and g_{iz}^2 ≤ g_{zz} g_{ii} in text-specified notation). - Key implication: higher initial stock of public administrative “know-how” leads to higher optimal public investment following a resource windfall; thus investing in administrative capacity can raise the payoff to public capital investment. ### Policy implications and practical recommendations - Aggregate message: - It is not necessarily optimal for resource rich governments to increase public capital investment following temporary resource windfalls when adjustment costs rise with windfall size. - Larger windfalls may optimally be partly or largely redistributed via direct transfers rather than channeled into public investment when administrative capacity is limited. - Weak administrative capacity reduces optimal public capital responses to windfalls; higher non-resource sector TFP mitigates this effect. - Investment in administrative capacity is a policy lever: a higher stock of know-how increases the returns to public investment and raises optimal public investment after windfalls. - Practical approaches to build administrative capacity and reduce adjustment costs: - Increase transparency in handling resource windfalls and better identify and implement projects. - International initiatives highlighted as anchors: Extractive Industries Transparency Initiative (EITI); Natural Resource Charter (builds on EITI). - Institutional design and project appraisal: example of Chile’s National System of Investments (SNI) with disciplined cost-benefit analysis, shadow social price system and social discount rate; SNI comprises an online databank with over 300,000 entries. - Project Implementation Units (PIUs) recommended by World Bank for project execution, but mixed record due to lack of continuity and knowledge accumulation; recommendation to integrate PIUs into line ministries to preserve know-how. - Research agenda: - Need for more empirical work documenting cross- and within-country differences in state capacity. - More academic work on the political economy of building state capacity is warranted. *Source: _wp12200 - Appendix 2: Parametric Specification of the Extended Model ...............................................19* ### REFERENCES ### _wp12200 - REFERENCES ### References (selected bibliographic entries) - Acemoglu, Daron, Simon Johnson, and James Robinson, 2001, “The Colonial Origins of Comparative Development: An Empirical Investigation,” American Economic Review, Vol. XCI, pp. 1369–1401. - Acemoglu Daron and Simon Johnson, 2005, "Unbundling Institutions," Journal of Political Economy, University of Chicago Press, October, Vol. 113(5), pages 949–995. - Alfaro, Laura, Sebnem Kalemli-Ozcan, and Vadym Volosovych, 2008, "Why doesn't capital flow from rich to poor countries? An empirical investigation," Review of Economics and Statistics, Vol. 90(2), pp. 347–68. - Arezki Rabah, Kirk Hamilton and Kazim Kazimov, 2011, “Resource Windfalls, Macroeconomic Stability and Economic Growth,” IMF Working Paper 11/142 (Washington, International Monetary Fund). - Arezki, Rabah and Markus Brueckner, “Oil rents, corruption, and state stability: Evidence from panel data regressions.” European Economic Review (forthcoming). - Birdsall, Nancy, and Arvind Subramanian, 2004, “Saving Iraq from Its Oil,” Foreign Affairs, July–August, Vol. 83(4), pp. 77–89. - Hall, Robert E., and Charles I. Jones, 1999, “Why Do Some Countries Produce So Much More Output Per Worker Than Others?” Quarterly Journal of Economics, Vol. 114, pp. 83–116. - Hausmann, Ricardo, Andrew Powell, and Roberto Rigobon, 1993, “An Optimal Spending Rule Facing Oil Income Uncertainty (Venezuela),” in Engel, Eduardo and Meller, Patricio, eds. External shocks and stabilization mechanisms, Inter-American Development Bank, distributed by Johns Hopkins University Press, Washington, DC, pp. 113–71. - Heston Alan, Robert Summers and Bettina Aten, 2006, Penn World Table Version 6.2, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania, September. - Keefer, Philip, and Stephen Knack, 1995, “Institutions and Economic Performance: Cross-Country Tests Using Alternative Measures,” Economics and Politics, Vol. VII, pp. 207–27. - Kyobe Annette & Jim Brumby & Chris Papageorgiou & Zac Mills & Era Dabla-Norris, 2011, "Investing in Public Investment: An Index of Public Investment Efficiency," IMF Working Papers 11/37 (Washington, International Monetary Fund). - Van der Ploeg, R. 2007, “Challenges and Opportunities for Resource Rich Economies,” OXCARRE WP 2008-05. - Collier P., R. van der Ploeg, M. Spence and T. Venables, 2009, “Managing Revenues in Developing Countries,” OXCARRE WP 2009-15. - Gelb, Alan and associates, 1988, "Oil windfalls: Blessing or curse?” Oxford University Press, for the World Bank, New York. - Moss, T., 2011, “Oil to Cash: Fighting the Resource Curse through Cash Transfers”, Working Paper, 237, January, Center for Global Development, Washington. - North, Douglass C., 1990, Institutions, Institutional Change, and Economic Performance (Cambridge: Cambridge University Press). - Political Risk Services, 2009, International Country Risk Guide. - Quinn, Dennis P., and John T. Woolley, 2001, “Democracy and National Economic Performance: The Preference for Stability,” American Journal of Political Science, Vol. XLV, pp. 634–57. - Ross, Michael L., 2012, “The Political Economy of Petroleum Wealth in Low-Income Countries: Some Policy Alternatives” in Commodity Prices and Inclusive Growth in Low-Income Countries, ed. by Rabah Arezki, Catherine Pattillo, Marc Quintyn, and Min Zhu, International Monetary Fund. - Van der Ploeg, Frederick, 2011, “Natural resources: curse or blessing?” Journal of Economic Literature, Vol. 49(2), pp. 366–420. - World Bank, 2006, “Where is the Wealth of Nations?,” The World Bank Group, Washington. - World Bank, 2011a, Doing Business Database, The World Bank Group, Washington. - World Bank, 2011b, World Development Indicators, the World Bank Group, Washington. ### Appendix 1: Parametric Specification (key functional forms and conditions) - Utility adjustment cost specification: x(ε) = k ε^{b/(1−b)} with b∈(0,1) such that x(k)=... (symbolic form preserved as in source). - g(h) = c h^{α} (ḣ/h)^{−α} with α∈(0,1). - Production function: H(Y,K) = A Y^{1−φ} K^{φ} with 0<φ<1 (Cobb-Douglas, constant returns to scale). - Marginal expressions preserved in symbolic form as in source: e.g., Q(S,Y) = ...; R(S,Y) = A S^{φ} (...)^{(φ−1)} (equations presented symbolically in original). - Euler equation symbolic transformations preserved (including cases when α→0 and when α≠0). - Link between windfall size and adjustment costs introduced via notation: let ζ = σ + ᾱ with ζ≥0 and ᾱ≥0. ### Appendix 2: Parametric Specification of the Extended Model (structure and dynamic conditions) - Adjustment cost function: g(K,t) = (K) (1 − e^{−n t})^{m} where n governs convergence speed; as K_t → ∞ adjustment costs → 0. Parameter n≥0. - Rearranged third Euler condition (symbolic): s_t = (β2·...)^{1/σ} (ψ1 + ψ2 s̄) with expression preserved in symbolic form as in source. - Condition: D_t ≡ (ψ1/ψ2)^{...} ≥ 0 (symbolic inequality preserved). - Second Euler condition yields: ∆(K_t) = (θ1/θ2)^{...} for all t (symbolic). - Dynamics of know-how and investment: - ∆_t(K_t) = n^{1−2} (r2/κ)^{...} n (...) ≥ 0 (symbolic structure preserved). - When n>1 the stock of know-how decreases over time and so does investment in public capital. - When 0