Non-Renewable Resources and Hotelling Rule and Optimal Extraction

Introduction to Non-Renewable Resource Economics

• Production Framework:     * In a simplified economy producing a single good ($Q$), output can be either consumed or invested.     * Output is determined by the input of a non-renewable resource ($R$) and the capital ($K$) applied to production.     * General form: $Q = Q(K, R)$.
• Functional Relationships:     * Cobb-Douglas (CD) Form: $Q = AK^{\alpha}R^{\beta}$.         * $A$ represents factor productivity.         * 1 > \alpha > 0 and 1 > \beta > 0, such that $\alpha + \beta = 1$.         * $\alpha$ and $\beta$ represent the capital shares of $K$ and $R$ respectively.     * Constant Elasticity of Substitution (CES) Form: $Q = A(\alpha K^{-\theta} + \beta R^{-\theta})^{-\frac{\epsilon}{\theta}}$.         * Parameters include $A$, $\alpha$, $\beta$, \epsilon > 0, and $\alpha + \beta = 1$.         * -1 < \theta \neq 0, where $\theta$ is the substitution parameter.         * $\epsilon$ represents the degree of homogeneity of the production function.
• Essentiality of Resources ($R$):     * A resource is essential if production is impossible without it, defined as $Q(K, R=0) = 0$.     * CD Form: Both $R$ and $K$ are essential; production stops ($Q = 0$) if the ratios fall to zero.     * CES Form:         * If $\theta < 0$, no input is essential.         * If $\theta > 0$, all inputs are essential.

Feasibility of Sustainability

• Requirements for Assessing Feasibility:     1. Criterion of Sustainability: Use a conventional criterion of maintaining non-declining per capita consumption over an indefinite timeframe.     2. Conditions of Material Transformation: Understanding what can be obtained in the future based on population, technology, culture, and politics, simplified through the production function.
• Case Studies in Sustainability:     * Case A (Cobb-Douglas): Produced under competitive markets with constant returns to scale ($Q = K^{\alpha}R^{\beta}$ where $\alpha + \beta = 1$).         * Sustainability is feasible if population and technology are constant and the capital share is larger than the resource share (\alpha > \beta).         * As $R$ diminishes, $K$ must increase to substitute for it to keep $Q$ constant.     * Case B (CES): Produced under competitive markets with constant returns to scale.         * Elasticity of substitution: $\sigma = \frac{1}{1 + \theta}$.         * Sustainability is feasible if $\sigma \geq 1$.         * If \sigma > 1, the resource is not essential, allowing indefinite production of positive output even with finite resources.         * If $\sigma = 1$, CES collapses into the CD form (Case A).     * Case C (Backstop Technology): Occurs when a backstop technology is permanently available.         * Natural resources are not essential because a new "$R$" can always emerge.         * Sustainability is always feasible here.
• Core Takeaways:     * Sustainability requires high substitutability between capital and resources, a high rate of technical progress, or a permanent backstop technology.

Substitution Mechanisms and Perspectives

• Economic Optimism: Economists generally believe the magnitude of substitution is high due to innovation and the ability to economize on fossil fuels or minerals.
• Dasgupta (1993) Innovation Mechanisms:     1. Innovations allowing a given resource to be used for a specific purpose.     2. Development of new materials.     3. Technological developments increasing extraction productivity.     4. Scientific discoveries making exploration activities cheaper.     5. Technical developments increasing efficiency in resource use.     6. Techniques for exploiting low-grade but abundant deposits.     7. Developments in recycling techniques that lower costs and raise effective stock size.     8. Substitution of low-grade resource reserves for high-grade deposits.     9. Substitution of fixed manufacturing capital for vanishing resources.
• Ecological/Environmental Concerns:     * Limited substitution possibilities exist between resources and reproducible inputs.     * Argument that reproducible inputs cannot replace all functions of nature (e.g., regenerated timber in a closed system).     * Concerns regarding "space" rather than just "time" (uneven population leveling).
• Oil Production Context:     * Historical data (2010–2018) shows a shift in production. Conventional oil production has faced competition from Shale Oil production.     * Spot Price WTI (Cushing, Okla.) fluctuated significantly, peaking around $105.79$ dollars/barrel before dropping to roughly $30.32$ dollars/barrel around 2016.

Resource Reserves and Life Expectancy (1991 Data)

• Aluminium: Production: $112.22 \times 10^6\,\text{metric tons}$; Reserves: $23,000 \times 10^6\,\text{metric tons}$; Reserve Life: $222\,\text{years}$. Base resource (crustal mass): $1.99 \times 10^{12}\,\text{metric tons}$.
• Iron Ore: Production: $929.75 \times 10^6\,\text{metric tons}$; Reserves: $150,000 \times 10^6\,\text{metric tons}$; Reserve Life: $161\,\text{years}$.
• Copper: Production: $9.29 \times 10^6\,\text{metric tons}$; Reserves: $310 \times 10^6\,\text{metric tons}$; Reserve Life: $33\,\text{years}$.
• Zinc: Production: $7.137 \times 10^6\,\text{metric tons}$; Reserves: $140 \times 10^6\,\text{metric tons}$; Reserve Life: $20\,\text{years}$.
• Lead: Production: $3.424 \times 10^6\,\text{metric tons}$; Reserves: $63 \times 10^6\,\text{metric tons}$; Reserve Life: $18\,\text{years}$.
• Platinum: Production: $0.0003 \times 10^6\,\text{metric tons}$; Reserves: $0.37 \times 10^6\,\text{metric tons}$; Reserve Life: $124\,\text{years}$.

Simple Two-Period Model of Optimal Extraction

• Goal: Obtain dynamic efficiency between present and future use by maximizing the Present Value ($PV$) of net social benefits.
• Assumptions:     * Two periods ($t_0, t_1$).     * Known, fixed stock size: $\bar{S}$.     * Inverse Demand Function: $P_t = a - bR_t$.         * $P$ is price, $R$ is quantity.         * Price impacts quantity in a standard demand function ($Q=f(P)$), but quantity impacts price in the inverse demand function ($P=f^{-1}(Q)$).
• Gross Social Benefit ($B$):     * $B(R_t) = \int_0^{R_t} (a - bR) dR = aR_t - \frac{b}{2} R_t^2$.
• Extraction Costs:     * $c$ is the constant marginal cost of extraction ($c \geq 0$).     * Total extraction cost: $C_t = cR_t$.
• Net Social Benefit ($NSB$):     * $NSB_t = B_t - C_t$.     * $NSB(R_t) = aR_t - \frac{b}{2} R_t^2 - cR_t$.

Socially Optimal Extraction Policy

• Social Welfare Function ($W$):     * $W = NSB_0 + \frac{NSB_1}{1 + p}$, where $p$ is the social discount rate.
• Constraints:     * Fixed initial stock $\bar{S}$.     * Exhaustibility constraint: $R_0 + R_1 = \bar{S}$.
• Optimization via Lagrange Multiplier Method:     * Lagrangian Function ($L$): $L = (aR_0 - \frac{b}{2} R_0^2 - cR_0) + \frac{aR_1 - \frac{b}{2} R_1^2 - cR_1}{1 + p} - \lambda(\bar{S} - R_0 - R_1)$.     * First-order conditions:         1. $\frac{\partial L}{\partial R_0} = a - bR_0 - c + \lambda = 0$         2. $\frac{\partial L}{\partial R_1} = \frac{a - bR_1 - c}{1 + p} + \lambda = 0$     * Implication: $a - bR_0 - c = \frac{a - bR_1 - c}{1 + p}$.     * In terms of net prices (resource rent/royalty): $P_0 - c = \frac{P_1 - c}{1 + p}$.

Hotelling’s Rule

• Definition: Proposed by Harold Hotelling (1931), it states that the net price of a non-renewable resource must rise at the rate of interest (or discount rate).
• Logic: A resource owner can keep resources in the ground or extract and sell them to invest the proceeds. In a competitive market with profit-maximizing owners, the price must rise at the discount rate to prevent reallocation between periods.
• Mathematical Form:     * $p = \frac{(P_1 - c) - (P_0 - c)}{P_0 - c}$.     * General form: $\sigma = \frac{m_t - m_{t-1}}{m_{t-1}}$, where $\sigma$ is the market interest rate.
• Optimal Program Summary:     * Satisfies the demand functions for both periods.     * Satisfies the exhaustion constraint: $R_0 + R_1 = \bar{S} - S^{<em>}$ ($S^{</em>}$ being remaining stock if any).     * Satisfies the efficiency condition: $P_0 - c = \frac{P_1 - c}{1 + p}$.

Multi-Period Continuous Time Model

• Framework: Moves from discrete time to a continuous scale.
• Social Utility ($U$):     * $U(R) = \int_0^R P(R) dR$.     * Marginal social utility: $\frac{\partial U}{\partial R} = P(R)$.
• Welfare Function:     * $W = \int_0^T U(R_t) e^{-\rho t} dt$, where $\rho$ is the instantaneous social utility discount rate.
• Constraints:     * $\int_0^T R_t dt = \bar{S}$.     * Stock depletion rate: $\dot{S}_t = -R_t$.
• The Cake-Eating Problem:     * Deciding how much to consume today versus leaving for the future.     * Trade-offs:         1. Delaying consumption is costly (discount factor).         2. Delaying consumption is attractive due to diminishing marginal utility (concave function $U$).     * Efficiency Condition:         * Welfare is maximized when discounted marginal utility is constant over time: $\frac{\partial U}{\partial R} e^{-\rho t} = \text{constant}$.         * $P_t e^{-\rho t} = P_0$.         * This implies the Hotelling rule in continuous form: $\frac{\dot{P}_t}{P_t} = \rho$.

Optimal Solution Parameters

• Required Data for Solution:     1. Optimal initial net price ($P_0$).     2. Optimal extraction period ($T$).     3. Optimal extraction rate at each point in time ($R_t$).     4. Values of $R$ and $S$ at the end of the horizon ($t=T$).
• Specific Demand Form: Typically assumed non-linear for accuracy: $P(R) = Ke^{-aR}$.
• Boundary Conditions:     * Stock goes to zero at the exact time dependency and extraction go to zero ($S_T = 0$, $R_T = 0$).     * Resource stock and extraction remain positive across the active time horizon.