ERE5: Efficient and optimal use of environmental resources A simple optimal depletion model Resource substitutability Static and dynamic efficiency Hotelling‘s rule Optimality An example Extraction costs Renewable resources Complications

Last week Efficiency and optimality Market failure & public policy Static efficiency Optimality Dynamic efficiency and optimality Market efficiency Market failure & public policy Externalities Public policy

Exhaustion, Production & Welfare One can construct production functions with various degrees of essentialness, but the question is of course empirical The question whether it matters that a resource gets exhausted depends on its substitutability; or, if you can‘t do without, don‘t lose it So far, we have not run out of anything essential, but that does not mean we won‘t Human ingenuity is the ultimate resource, but also works to create problems

K  = 0 0 <  <   =  R Substitution possibilities and the shapes of production function isoquants

Substitutability and Scarcity Feasibility of sustainable development depends on Substitutability Technical progress Backstop technology The magnitude of substitution possibilities Economists: relatively high Natural scientists and ecologists: limited However, it matters what services we look at

Optimal Resource Extraction - Discrete Time Social welfare function: Extraction: Investment: Production function:

Optimal Resource Extraction – Discrete Time (2)

Optimal Resource Extraction – Continuous Time Social welfare function: Extraction: Investment: Production function:

The Maximum Principle Objective function: Subject to: J depends on control variables (u), state variables (x), and time State variables describe the economy at any time; the equation of motion governs its evolution over time Control variables are time-dependent policy instruments To obtain the solution we construct a current value Hamiltonian

The Hamiltonian The Hamiltonian only contains the current state and controls; current optimality is a necessary condition for intertemporal optimality The co-state variables (l) secure intertemporal optimality; they are like Lagrange multipliers, indeed measure the shadow price FOC:

Optimal Resource Extraction - Continuous Time (2) Social welfare function: Equations of motion: Hamiltonian: Necessary conditions:

Static Efficiency Marginal utility of consumption equals the shadow price of capital Marginal product of the natural resource equals the shadow price of the resource stock

Dynamic Efficiency The growth rate of the shadow price of the resource equals the discount rate The return to capital equals the discount rate

Hotelling‘s Rule Dynamic efficiency required: The growth rate of the shadow price equals the discount rate An alternative interpretation: The discounted price is constant along an efficient resource extraction path Thus, environmental resources are like other assets

Growth Rate of Consumption The growth rate of consumption along the optimal time path: Since η>0, consumption grows if the marginal product of capital exceeds the discount rate The intuition:

Hotelling‘s rule and Optimality PtB = P0Bet PtA = P0Aet P0B P0A t

Extraction Costs Social welfare function: Constraints: Production function: Extraction costs:

Extraction Costs and Resource Stock Gt (for given value of Rt = ) (iii) (ii) (i) S0 Remaining resource stock, St

Extraction Costs –2 Hamiltonian: Necessary conditions:

Resource Price Net price = Gross price – marginal extraction cost Gross price = Marginal contribution to output, income (measured in utils) Net price = Marginal value of the resource in situ Net price = Rent = Royalty

Hotelling‘s Rule -2 The growth rate of the shadow price of the resource is lower if extraction costs rise with falling resource stocks The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the foregone increase in extraction costs

Graphical solution Net price Pt PT =K P0 Pt 45° T R R0 Time t Rt Demand P0 Pt 45° T R R0 Time t Rt Area = = total resource stock T Time t

Renewable Resources Social welfare function: Constraints: Production function:

Renewable Resources -2 Hamiltonian: Necessary conditions:

Hotelling‘s Rule -3 The growth rate of the shadow price of the resource is lower for renewable resources The discount rate equals the rate of return of holding the resource, which equals its price appreciation plus the increase in the resource growth

Complications The total stock is not known with certainty New discoveries increase the known stock There is a distinction between physical quantity and economically viable stock size Technical progress and R&D Heterogeneous quality Extraction costs differ Availability of backstop-technology