# Lecture 9 The efficient and optimal use of non – renewable resources.

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Lecture 9 The efficient and optimal use of non – renewable resources

What are non – renewable resources? - fossil fuel energy supplies, such as oil, gas and coal; -minerals, such as copper and nickel. They are formed by geological processes over millions of years  once extracted cannot be immediately be renewed,  what is the optimal path for extraction? Are non-renewable resources exhaustible?

Take the example of Aluminium (year 1991, quantities in million of metric tons): Production:112 Reserves, Qty.:23 000 Reserves life, y.:222 World reserve base28 000 - base life270 Consumption:19 Resource potential:3 519 000 Base resource (crustal mass):1 990 000 000 000 Base resource: mass that is thought to exist in the earth crust. Resource potential: estimates of the upper limits on resource extraction possibilities given current and expected technologies. World reserve base: estimates of the upper bounds of resource stocks, that are economically recoverable under ‘reasonable expectations’. Reserves: quantities available under current costs and prices.

A non-renewable multi period resource model We write the current-valued Hamiltonian and get as the necessary conditions for the maximum social welfare: We start with the cake-eating model:

A non-renewable multi period resource model We now know the rate at which the resource net price (royalty) must rise. This does not fully characterize the solution to the optimizing problem: 1. We need to know the optimal initial value of the resource net price. 2. We need to know the extraction period T - the optimal value of T. 3. What is the optimal rate of resource extraction at each point in time? 4. What should be the values of P and R at the end of the extraction period?

A non-renewable multi period resource model We now know the rate at which the resource net price (royalty) must rise. This does not fully characterize the solution to the optimizing problem: 1. We need to know the optimal initial value of the resource net price. 2. We need to know the extraction period T - the optimal value of T. 3. What is the optimal rate of resource extraction at each point in time? 4. What should be the values of P and R at the end of the extraction period? We can’t answer these question without knowing the particular form of the resource demand function. Suppose the resource demand function is: P ( R = 0) = K. K is the so-called choke price for this resource. Demand is driven to zero, ‘choked off’ at this price.

U(R) = shaded area Ke -aR 0 P K RQuantity of resource extracted, R Figure 15.2 A resource demand curve, and the total utility from consuming a particular quantity of the resource (Perman et al.: page 513)

A non-renewable multi period resource model Given knowledge of: a particular resource demand function. Hotelling’s efficiency condition, an initial value for the resource stock, and and a final value for the resource stock, it is possible to obtain expressions for the optimal initial, interim, and final resource net price (royalty) and resource extraction rates. We know already: R T = 0, S T = 0. For demand to be zero at time T, the net price P must reach the choke price at time T. That is: P T = K. This implies:

A non-renewable multi period resource model We know: From that follows: The result gives an expression for the rate at which the resource should be extracted along the optimal path.

A non-renewable multi period resource model Substituting in the previous result: To find the optimal extraction period T, recall:

A non-renewable multi period resource model Using the result, we can find the initial royalty level P 0 : Considering: We find the price for the resource royalty at time t: The optimal initial extraction level is:

A non-renewable multi period resource model Optimality conditions for the multi-period model Initial (t=0)Interim (t=t)Final (t=T) Royalty, P Extraction, R Depletion time The same results are obtained for resource extraction in perfectly competitive markets. By assuming that the area under the demand function, the gross benefits, are quantities of utility, we impose the condition,  = r, the social discount rate equals the social consumption discount rate. Additionally, we assume the private market interest rate equals the social consumption discount rate.

Net price P t Time t 45º Time t T T Area = S = total resource stock RR0R0 P0P0 Demand P T = K Figure 15.3 A Graphical representation of solutions to the optimal resource depletion model (Perman et al.: page 517) RtRt PtPt 45 o

A non-renewable multi period resource model Results under a monopolistic market: The results are represented in the following figure. The key result is, that profit- maximizing extraction programmes will be different in perfectly competitive and monopolistic resource markets.

Net price P t Perfect competition Monopoly Time t 45º Time t TMTM T TTMTM Area = S RR0R0 R 0M P 0M P0P0 Demand P T = P T M = K Figure 15.4 A comparison of resource depletion in competitive and monopolistic markets (Perman et al.: page 519)

Extension of the non-renewable multi period resource model The following simplifying assumptions are made: utility discount rate and market interest rate are constant over time, fixed stock of known size non-renewable natural resource, demand curve is identical at each point in time, no taxation or subsidy is applied to the extraction or use of the resource, marginal extraction costs are constant, there is a fixed ‘choke price’ (backstop technology exists), no technological change occurs, no externalities are generated in the extraction or use of the resource.

P0P0 K P TTime AC B Figure 15.5 The effect of an increase in the interest rate on the optimal price of the non-renewable resource (Perman et al.: page 520)

Net price P t Time t 45º Time t T T’T’ T’T’ T R R’0R’0 R0R0 P 0 Demand K Figure 15.6 An increase in interest rates in a perfectly competitive market (Perman et al.: page 521) P’0P’0

Net price P t Time t 45º Time t T TT’T’ R R’0R’0 P 0 Demand K Figure 15.7 An increase in the resource stock (Perman et al.: page 521) P’0P’0 T’T’

PtPt t Net price path with no change in stocks Net price path with frequent new discoveries Figure 15.8 The effect of frequent new discoveries on the resource net price or royalty (Perman et al.: page 522)

Net price P t Time t 45º Time t T’T’ T’T’ T R R ’ 0 P ’ 0 K Figure 15.9 The effect of an increase in demand for the resource (Perman et al.: page 522) P’0P’0 T D D’D’ R0R0

Net price P t Figure 15.10 A fall in the price of a backstop technology (Perman et al.: page 523) Time t 45º Time t T T’T’ T’T’ T R R’0R’0 R0R0 P 0 K P’0P’0 D PBPB Backstop price fall

K T Time New net price Original gross price New gross price Original net price K T Time New net price Original gross price New gross price Original net price T’ (a) (b) Figure 15.11 (a) An increase in extraction costs: deducing the effects on gross and net prices; (b) An increase in extraction costs: actual effects on gross and net prices (Perman et al.: page 523) CLCL CHCH

Net price P t Time t 45º Time t T ’ T TT’T’ R R0R0 R’0R’0 P ’ 0 K Figure 15.12 A rise in extraction costs (Perman et al.: page 524) P0P0 New gross price path Original gross price path

Impact of taxes and subsisdies Notation: p t : net – price (royalty) at time t. P t : gross – price at time t. c t : extraction cost at time t, assumed to be constant. Effect of tax or subsidy on royalty p t : Hotelling’s efficiency rule continues to operate unchanged in the presence of a royalty tax (subsidy). It is simply a redistribution of the rent from the non-renewable resource to the government in the case of a tax.

Impact of taxes and subsidies Effect of tax or subsidy on revenue P t. Before tax (subsidy) Since c/(1-  )>c, an imposition of a revenue tax is equivalent to an increase in the resource extraction cost. Similarly, a subsidy is equivalent to a decrease in extraction costs. Tax: higher c => increase P 0, decrease growth of P => increase T. Subsidy: lower c => decrease P 0, increase growth of P => decrease T. After tax (subsidy):

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