Nordhaus’s Programming Model of Energy Futures Revisited

Abstract

We conduct a small-scale linear programming simulation of an energy future based on the work of Nordhaus. We are able to link our quantity model (a primal model) quite precisely to its price problem (a dual model). Our amended Nordhaus formulation, with our present-value adjustments, has a dual (price) program that solves problems under correct dynamic efficiency conditions, standard in economics. We present, then, a corrected Nordhaus template, a linear program, suitable for simulating energy futures with good-quality data. We elevate the trajectory analysis of resource prices to a central role in the analysis of energy futures.

1. Introduction

We report on a flaw in Nordhaus’s 1973 linear programming model of an energy future. Our contribution is a solved linear program, an illustrative example and mini-version of Nordhaus’s model, with which we can exposit the flaw and provide a simple work-around. Nordhaus perceived that with a simple relabeling of the variables in the transportation linear program (see [1], pp. 16–17), he had an intertemporal linear programming (LP) model that he could use to simulate a future of energy prices, and specific drawdowns of estimated stocks of the energy supply. His projections of population (demanders of energy), costs of energy extraction and delivery, and interest rates were exogenous, and his model solved for the depletions of stocks that yielded flows of energy to an economy period by period into the future. The Hotelling model [2] in the economics of stock depletion was the basis of his orderly stock drawdowns. (Hotelling derived an extraction rule that maximizes the present value of social welfare. His rule defines stock “usage” for complicated economic models. In such complicated models, his rule is referred to as a dynamic efficiency condition. Departures from Hotelling’s extraction rule, due to market “distortions”, are taken up in [3,4,5].)

The Herfindahl model [6] links a series of Hotelling depletion models into correct textbook trajectories. The simulation of the energy future rests then on well-known textbook economic principles. The climax of Nordhaus’s empirical work was an energy future emerging as the solution of a very large LP, involving the outcome of the minimization of the present value of the costs of extraction and delivery of flows from stocks. Given his favored energy future from his trials with his large LP, Nordhaus could then compare current real world energy prices in 1970 with prices from the model. He discovered that the current price of energy from the model was lower than the observed energy price in Connecticut in 1970.

This wedge in energy price inspired Nordhaus to speculate why real-world data were at odds with those from the energy future generated from his LP. He concluded that the energy future from the model was based on undistorted or free market principles, while the real-world setting was reflecting departures from free market principles, particularly oligopoly in energy supply. Nordhaus also speculated that the real world was dealing with uncertainty, whereas the Hotelling–Herfindahl model was based on complete certainty for all variables. Central to our contribution here is the statement that a misspecification of his LP left Nordhaus solving a dual problem with defective (non-market) shadow prices, and as a consequence, we are forced to remain agnostic on his comparison of “model prices” and “reality prices”.

Interesting inferences from Nordhaus’s amended LP are thus yet to be made. His empirical work was based on a huge array of data on future populations, the sizes of various stocks for the energy supply, and future costs for the extraction and delivery of energy flows. He also made use of a series of regions for his world’s future. Any researcher interested in pursuing his sort of empirical work would face the massive task of data accumulation.

In Ref. [7], Nordhaus devoted Chapter 1 to reflections on Hotelling valuations or prices and then turned to the econometrics of future energy demands. He never returned to work with linear programming. Interest in linear programming and energy futures was taken up by the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria (see [8]), and in many articles by Alan Manne (e.g., [9]).

There is activity today in the intertemporal linear programming of energy future models (see, for example, [10,11,12]), but no one is directly asking “the Nordhaus question”: namely, how distant are today’s energy prices in the real world from textbook or “free market prices”. Today’s emphasis is on constructing planning frameworks for future energy transitions.

Early on in his research, Nordhaus realized that he could add transportation costs for flows from supply stocks to demand stocks. We observe that the presence of exogenous transportation costs leads to an appropriate spatial price equilibrium in the dual problem (Ref. [13] deals with exhaustible resource analysis with a regional dimension incorporated into the framework). Though we deal with a specific seven-period framework (a “back-of-envelope” or mini-model), it is straightforward to carry out the analysis with more periods and more exhaustible resource stocks. Our numerical work involves developing a Herfindahl-type solution as a primal LP, and we report on its dual or price LP. (The Herfindahl model, as in [6], is a string of linked Hotelling models, each with an unchanging unit cost of extraction, with distinct costs rising over time. There is the issue of correctly linking consecutive Hotelling models across time.) This motivates our choice of parameter values, namely declining future values of demands, period by period, and unit costs for outflows from specific stocks rising across periods into the future. The unit costs are exogenous, and we observe that the solution to the quantity linear program involves drawing from stocks with lower unit costs before drawing from stocks with higher unit costs. We also fix our exogenous values of stock supplies so that only the last values, for fusion power, receive a price of zero in the dual program. (We consider solutions with zero resource prices to be less interesting than those with positive prices. Hence, we are careful in our selection of values for our exogenous demands and stock sizes.) We also set our transportation costs at zero in our empirical work, simply to allow us to focus on the emergent prices or flows from stocks. We are aware that our basic model, like that of Nordhaus, readily provides solutions that are quite different from the Herfindahl types, particularly when positive transportation costs are placed in the objective function of the quantity or primal LP.

2. Primal (Quantity) Program

We turn to an example with seven periods (with seven demand levels, dated) and three supply stocks [14] (pp. 545–546) for a linear program (LP) of an energy future. This is a linear program with 21 quantity flow variables to solve for. In accordance with linear programming science, the dual price problem is an LP with 21 price relationships (21 inequalities) to deal with in a problem about maximizing dollar value. We solve four LPs: a quantity and price pair with the original Nordhaus formulation in [14] and a quantity and price pair of the amended Nordhaus formulation. The two quantity problems turn out to have the same solution, while the dual of the amended Nordhaus problem exhibits valuation relationships of a Hotelling sort (dynamic efficiency in stock drawdown). We find, then, for our choice of parameters in our example that the original formulation possesses unsatisfactory price relations in the dual linear program.

We amend Nordhaus’s formulation in a routine way and observe that the new price (dual) LP exhibits appropriate dynamic efficiency conditions (Hotelling valuation conditions). The amended Nordhaus problem is constructed from the basic Nordhaus problem by weighting each demand inequality, from both sides, with the appropriate present-value weight. These weights are new parameters to the basic Nordhaus model, and they appear of course in the dual program. Our four solved LPs are presented simply as illustrative of Nordhaus’s original model, free of transportation costs, and of our amended formulation. Since they are textbook illustrations, there is no sense that we are performing economics or energy analysis closely connected to the real world. Nordhaus, on the other hand, was calibrating a model with parameters that he considered good approximations to prospective real-world values.

We also observe that the presence of exogenous transportation costs leads to an appropriate spatial price equilibrium in the dual problem. Though we deal with a specific seven-period framework, it is straightforward to conduct the analysis with more periods and more exhaustible resource stocks. Our numerical work involves developing a Herfindahl-type solution as primal and dual linear programs. This motivates our choice of parameter values, namely declining values of demands, period by period, and unit costs for outflows from specific stocks rising across periods. The unit costs are exogenous, and we observe that the solution to the quantity linear program involves drawing from stocks with lower unit costs before drawing from stocks with higher unit costs. We also fix our exogenous values of stock supplies so that only the last values, for fusion power, receive a price of zero in the dual. (We consider solutions with zero resource prices to be less interesting than those with positive prices.) We also set our transportation costs to zero in our empirical work, simply to allow us to focus on the emergent prices for flows from stocks. We are aware that our basic model, like that of Nordhaus, readily admits solutions that are quite different from the Herfindahl types.

The cost of the energy provision being minimized (the present value of “extraction” and transportation over the seven periods) is

$$\begin{gathered} {j^0\left[c_O+{\tau }_{O,DO}\right]q_O\left(0\right)+j^0\left[c_U+{\tau }_{U,DO}\right]q_U\left(0\right)+j}^0\left[c_F+{\tau }_{F,DO}\right]q_F\left(0\right)] \\ +[j^1\left[c_O+{\tau }_{O,D1}\right]q_O\left(1\right)+j^1\left[c_U+{\tau }_{U,D1}\right]q_O\left(1\right)+\left[j^1\left[c_F+{\tau }_{F,D1}\right]q_F\left(1\right)\right] \\ +[j^2\left[c_O+{\tau }_{O,D2}\right]q_O\left(2\right)+j^2\left[c_U+{\tau }_{U,D2}\right]q_O\left(2\right)+\left[j^2\left[c_F+{\tau }_{F,D2}\right]q_F\left(2\right)\right] \\ +[j^3\left[c_O+{\tau }_{O,D3}\right]q_O\left(3\right)+j^3\left[c_U+{\tau }_{U,D3}\right]q_O\left(3\right)+\left[j^3\left[c_F+{\tau }_{F,D3}\right]q_F\left(3\right)\right] \\ +\left[j^4\left[c_O+{\tau }_{O,D4}\right]q_O\left(4\right)+j^4\left[c_U+{\tau }_{U,D4}\right]q_O\left(4\right)+j^4\left[c_F+{\tau }_{F,D4}\right]q_F\left(4\right)\right] \\ +[j^5\left[c_O+{\tau }_{O,D5}\right]q_O\left(5\right)+j^5\left[c_U+{\tau }_{U,D5}\right]q_O\left(5\right)+\left[j^5\left[c_F+{\tau }_{F,D5}\right]q_F\left(5\right)\right] \\ +\left[j^6\left[c_O+{\tau }_{O,D6}\right]q_O\left(6\right)+j^6\left[c_U+{\tau }_{U,D6}\right]q_O\left(6\right)+j^6\left[c_F+{\tau }_{F,D6}\right]q_F\left(6\right)\right] \end{gathered}$$

(1)

where c_O is the unchanging unit cost of energy provision from the oil stock, c_U is the unchanging unit cost of energy provision from the uranium stock, and c_F is the unchanging unit cost of energy provision from the large fusion power stock. The transportation costs make each demand a distinct place defined by the transportation costs from the three supply places. For example, ${\tau }_{U,D3}$ is the cost of transporting a unit of uranium stock from the given location of the uranium stock to the demand location defined by the demand level, D3. r is the interest rate or discount rate here. j = 1/[1 + r]. q_O(t), q_U(t), and q_F(t) are non-negative quantities of energy being solved for. Our positive discount rate contributes to our sequencing of energy source use. Distant future costs will have higher discount factors. We posit that exogenously given demands are declining over time as in a Hotelling or Herfindahl framework. As Nordhaus noted, the transportation costs make the setup an instance of a variant of the well-known transportation problem in linear programming. Of central interest to us is seeing how shadow prices in the dual problems relate to Hotelling valuation relations.

The demand constraints for the primal LP are

$$\begin{gathered} z^0\left[q_O\right(0)+q_U(0)+q_F(0\left)\right] \geqq z^0{D}_0 \\ {z}^1\left[q_O\right(1)+q_U(1)+q_F(1\left)\right] \geqq z^1{D}_1 \\ {z}^2\left[q_O\right(2)+q_U(2)+q_F(2\left)\right] \geqq z^2{D}_2 \\ {z}^3\left[q_O\right(3)+q_U(3)+q_F(3\left)\right] \geqq z^3{D}_3 \\ {z}^4\left[q_O\right(4)+q_U(4)+q_F(4\left)\right] \geqq z^4{D}_4 \\ {z}^5\left[q_O\right(5)+q_U(5)+q_F(5\left)\right] \geqq z^5{D}_5 \\ {z}^6\left[q_O\right(6)+q_U(6)+q_F(6\left)\right] \geqq z^6{D}_6 \end{gathered}$$

(2)

where D₀ is the exogenous demand in the first period, D₁ is the exogenous demand in the second period, etc. For z set at 1, we are dealing with the Nordhaus formulation of the quantity LP, and for z set exogenously at 1/[1 + r], we are dealing with the amended Nordhaus formulation. The amended Nordhaus problem is constructed from the basic Nordhaus problem by weighting each demand inequality, from both sides, with the appropriate present-value weight. Thus, for example, the fifth demand inequality receives z⁴ by multiplying terms on both sides of the basic demand inequality, for z = 1/[1 + r]. These weights on the demands in the primal or quantity problem are of course carried over to their correct places in the dual (price) problem, which is presented below.

The supply constraints for the primal LP are

q_O(0) + q_O(1) + q_O(2) + q_O(3) + q_O(4) + q_O(5) + q_O(6) ≤ S_O
q_U(0) + q_U(1) + q_U(2) + q_U(3) + q_U(4) + q_U(5) + q_U(6) ≤ S_U
q_F(0) + q_F(1) + q_F(2) + q_F(3) + q_F(4) + q_F(5) + q_F(6) ≤ S_F

(3)

The primal or quantity LP must solve for 21 non-negative q(t)s. S_O is the given stock of oil, in energy units. S_U is the initial given stock of uranium, in the same energy units. S_F is the stock of energy from fusion, taken here to be very large. We are approximating an infinitely large stock of energy from fusion power, available to the economy as a constant unit cost in the indefinite future. In our numerical example, the price p(t) per unit of q(t) will rise in the early periods to c_F, and c_F and D₄ will “prevail” as unchanging to the end date of our example. Our idea is to create a detailed numerical example with a solution that resembles the solution to a Herfindahl model. That is, our example will have early stages of declining quantities, produced by drawing down our finite stocks, with output prices, p(t)s, rising to the unit fusion cost, c_F. Supply from fusion power is referred to as “the backstop supply”. In “Clean Energy”, Arkolakis and Walsh [15] have energy produced early from hydrocarbon sources, while solar and wind power gradually replace the supply from hydrocarbon sources. Their framework includes multiple regions. Fressoz [16] presented evidence that energy from source A has not been succeeded or displaced by energy from source B in the historical record for “first world” countries. “Old” sources continue to be very active when “new” sources become active.

We proceed to set out a selection of parameter values and to solve our LP above twice, once with z = 1 and once with z = 1/[1 + r]. We solve numerically with the transportation parameters set at zero. Of interest is the numerical solution to each of our two quantity problems (one with z = 1 and one with z = 1/[1 + r]) being the same. This is not surprising given that the present-value parameters on demands are the same for both sides of an inequality in our amended Nordhaus formulation.

Complementary slackness indicates that if a supply inequality is solved as an inequality, its corresponding price in the dual program (our price problem) must be zero. (The materials collected together in Appendix A can be reviewed for the functioning of complementary slackness.) That is, something useful that ends up with excess supply in equilibrium receives a price of zero per unit. No demand will solve the problem above as an inequality because an excess quantity delivered to a demander would be placing unnecessary “extra cost” in the objective function. (This property holds for the transportation problem in linear programming. No demander receives a supply in the solution in excess of the given initial demand.) Complementary slackness is thus indicating that no price in the dual (price) LP corresponding to a particular demand in the quantity LP will automatically be zero in the price LP. It takes an “excess delivery” in the quantity LP in order for the corresponding price in the price LP to be zero.

We report the solutions now.

The parameters for the problems are j = 0.9 = 1/[1 + r]; (r = 1/9); c_O = 1:0; c_U = 1.2; c_F = 1.4; S_O = 9; S_U = 5; S_F = 20; D₀ = 5; D₁ = 4; D₂ = 3; D₃ = 2; D₄ = 1; D₅ = 1; and D₆ = 1. The stock quantity values were selected, given the demand parameters, so that neither value S_O nor S_U end up in the solution as excess values (as inequalities). The absence of an excess supply for oil or uranium allows us to rule out stock shadow prices of zero for S_O and S_U a priori in a numerical solution. Such zero-value solutions are perfectly acceptable in economics but are of little interest relative to cases with positive shadow prices for units of S_O and S_U. Recall we have assumed that all transportation cost terms, τs, are zero in our setup for numerical solving.

2.1. Numerical Solving

We employed Maple software for solving our LPs, a program identified in Maple as LPSolver. The solution to the quantity problem (z = 1) is

q_O(0) = 5; q_O(1) = 4; q_O(2) = 0; q_O(3) = 0; q_O(4) = 0; q_O(5) = 0; q_O(6) = 0;

q_U(0) = 0; q_U(1) = 0; q_U(2) = 3; q_U(3) = 2; q_U(4) = 0; q_U(5) = 0; q_U(6) = 0;

q_F(0) = 0; q_F(1) = 0; q_F(2) = 0; q_F(3) = 0; q_F(4) = 1; q_F(5) = 1; q_F(6) = 1.

The objective function is solved with a value of 15.754844.

For the amended quantity problem (z = 1/[1 + r] = 0.9), the solution is identical.

More on the amended problem is discussed below.

2.2. The Dual or Price LP

We turn to the dual or price LPs. The problem involves finding non-negative prices (ps and λs) that maximize

p(0)D₀ + zp(1)D₁ + z²p(2)D₂ + z³p(3)D₃ + z⁴p(4)D₄ + z⁵p(5)D₅ + z⁶p(6)D₆ − λ_OS_O − λ_US_U − λ_FS_F

(4)

The p(t)s correspond to the exogenous demands in the primal problem, period by period, and the λs are the shadow prices for the units of the respective supply stocks. The 21 constraints for our dual problem are

$$\begin{array}{l}{z}^{0}p\left(0\right)-{\lambda }_O\leqq j^0\left[c_O+{\tau }_{O,D0}\right],z^{0}p\left(0\right)-{\lambda }_U\leqq j^0\left[c_U+{\tau }_{U,D0}\right],\\ z^{0}p\left(0\right)-{\lambda }_F\leqq j^0\left[{c_F+\tau }_{F,D0}\right],\\ z^{1}p\left(1\right)-{\lambda }_O\leqq j^1\left[c_O+{\tau }_{O,D1}\right],z^{1}p\left(1\right)-{\lambda }_U\leqq j^1\left[c_U+{\tau }_{U,D1}\right],\\ z^{1}p\left(1\right)-{\lambda }_F\leqq j^1\left[c_F+{\tau }_{F,D1}\right],\\ z^{2}p\left(2\right)-{\lambda }_O\leqq j^2\left[c_O+{\tau }_{O,D2}\right],z^{2}p\left(2\right)-{\lambda }_U\leqq j^2\left[{c_U+\tau }_{U,D2}\right],\\ z^{2}p\left(2\right)-{\lambda }_F\leqq j^2\left[c_F+{\tau }_{F,D2}\right],\\ z^{3}p\left(3\right)-{\lambda }_O\leqq j^3\left[c_O+{\tau }_{O,D3}\right],z^{3}p\left(3\right)-{\lambda }_U\leqq j^3\left[c_U+{\tau }_{U,D3}\right],\\ z^{3}p\left(3\right)-{\lambda }_F\leqq j^3\left[c_F+{\tau }_{F,D3}\right],\\ z^{4}p\left(4\right)-{\lambda }_O\leqq j^4\left[{c_O+\tau }_{O,D4}\right],z^{4}p\left(4\right)-{\lambda }_U\leqq j^4\left[c_U+{\tau }_{U,D4}\right],\\ z^{4}p\left(4\right)-{\lambda }_F\leqq j^4\left[{c_F+\tau }_{F,D4}\right],\\ z^{5}p\left(5\right)-{\lambda }_O\leqq j^5\left[c_O+{\tau }_{O,D5}\right],z^{5}p\left(5\right)-{\lambda }_U\leqq j^5\left[c_U+{\tau }_{U,D5}\right],\\ z^{5}p\left(5\right)-{\lambda }_F\leqq j^5\left[c_F+{\tau }_{F,D5}\right],\\ z^{6}p\left(6\right)-{\lambda }_O\leqq j^6\left[c_O+{\tau }_{O,D6}\right],z^{6}p\left(6\right)-{\lambda }_U\leqq j^6\left[c_U+{\tau }_{U,D6}\right],\\ z^{6}p\left(6\right)-{\lambda }_F\leqq j^6\left[{c_F+\tau }_{F,D6}\right].\end{array}$$

(5)

These 21 inequalities “connect with” the 21 q(t) variables in the primal or quantity LP (q(t) with p(t), etc.). Each of these inequalities reduces to one of two alternative forms, depending on the assumed value for z.

For the case of z = 1/[1 + r], the solution to the dual problem must satisfy, for example, ${\left[{\displaystyle \frac{1}{\left[1+r\right]}}\right]}^4[p\left(4\right)-\left[{c_O+\tau }_{O,D4}\right]]\leqq {\lambda }_O$. When this inequality is solved as an equality, the discounted rent ${\left[{\displaystyle \frac{1}{\left[1+r\right]}}\right]}^4[p\left(4\right)-\left[{c_O+\tau }_{O,D4}\right]]$ equals the price of a unit of stock, namely ${\lambda }_O$. This is an instance of Hotelling’s r-percent rule: discounted surplus equals the shadow price of a unit of stock $S_O$.

And for the case of z = 1, the solution must conform with $p\left(4\right)-{\lambda }_O\leqq \left[{\left[{\displaystyle \frac{1}{\left[1+r\right]}}\right]}^4{{(c}_O+\tau }_{O,D4})\right].$ This is not a potential Hotelling relation. This is the heart of the Nordhaus flaw. The condition z = 1 leads to the wrong equilibrium price relationships in the dual problem. The only case for the problem exhibiting the appropriate economic valuations for a market economy is when z = 1/[1 + r]. Our amended Nordhaus problem (with z = 1/[1 + r]) exhibits the correct efficiency conditions for stock depletion. The original Nordhaus problem, with z = 1, fails to exhibit the correct efficiency conditions for stock depletion (i.e., the correct Hotelling r-percent conditions).

2.3. Transportation Costs

(1) The transportation cost terms are interesting because they are associated with a spatial price equilibrium. We deal with the case of z = 1/[1 + r]: Let us assume that the solution values for p(4) and ${\lambda }_O$ satisfy the Hotelling resource rent condition, ${\left[{\displaystyle \frac{1}{[1+r]}}\right]}^4\left[p\left(4\right)-{\tau }_{O,D4}-c_O\right]\leqq {\lambda }_O$. This means that supply stock O is shipping some stock to demander D₄, located away from the site of stock O at transportation cost ${\tau }_{O,D4}$ per unit shipped, and p(4) is the delivery price per unit at the site of D₄. $\left[p\left(4\right)-{\tau }_{O,D4}-c_O\right]$ is the rent on the unit received at D₄, and ${\left[{\displaystyle \frac{1}{[1+r]}}\right]}^4\left[p\left(4\right)-{\tau }_{O,D4}-c_O\right]$ is the present value of rent. The present value of rent is ${\lambda }_O$.

We rewrite the condition as $p\left(4\right)-{\tau }_{O,D4}={[1+r]}^4$ ${\lambda }_O$ + $c_O$. This reads as “the delivery price, the net transportation cost per unit, equals the supply cost per unit of $S_O$”. Transportation costs open up natural wedges between local supply costs and delivery price.

When transportation costs are present, it becomes possible for the quantity demanded D₄ to be supplied by more than one supply stock. Delivery price p(4) will be the same for units shipped from more than one stock, but the structure of costs, including the ${\lambda }_i$ rent, can be different for units from different sites. We can infer that it is possible for more than one supply place for D₄ to make our model more complicated than is the case with each demand site provided for by a single stock site. The presence of positive transportation costs implies that there is a textbook transportation cost linear program operating simultaneously with the intertemporal model. The intertemporal model is of particular interest to us in our analysis.

(2) We noted above that Nordhaus presumably discovered that he could write down a textbook transportation cost linear program, introduced dates in an orderly way to the demand values, and presented value weights for the costs in the objective function to receive a very interesting intertemporal linear programming model. He then observed that he could introduce “geography” and transportation costs and still preserve the interesting intertemporal linear programming model. Our observation here is that he slipped up in this model-developing exercise and neglected to weight demands in the demand constraints with present-value weights. Once we introduce these present-value weights, we observe that correct economic valuation conditions emerge in the dual problem.

A simple re-statement is as follows: the “delivered” price per unit, p(4), differs from the local cost ${[1+r]}^4$ ${\lambda }_O$ + $c_O$ by the transportation cost per unit, ${\tau }_{O,D4}$. Such an expression of equilibrium is referred to as part of a “spatial price equilibrium”.

In a spatial price equilibrium, a delivery price for a unit differs from the local cost by the given transportation cost. In a linear programming setup, the delivered quantity can be zero because the transportation cost is “too high”. Thus, a price inequality that is solved as an equality in the dual problem is simultaneously satisfying a Hotelling valuation equilibrium, dealing with discounted resource rent, and a spatial price equilibrium (the delivery price covers the transportation cost).

Another perspective is the following: If one returns to the beginning of our analysis to the statement of the quantity LP and removes all terms of present-value weighting, one has a new LP that is an instance of the classic transportation problem set out in textbooks on linear programming. Nordhaus was clever in realizing that transportation costs and regional prices could be added to his resource depletion problem in a straightforward fashion. However, the simultaneous functioning of the resource depletion model and the transportation delivery model makes for a complicated basic model. Obtaining an easy interpretation of the outputs from the numerical runs of the full model becomes challenging. Nordhaus apparently lost his grip on Hotelling valuations when he took to dealing with large LPs that included transportation costs.

If inequality k in a primal LP or in its corresponding dual LP is solved as an inequality in a numerical run, the complementary slackness indicates that the corresponding variable in the companion LP must be solved as zero. In the Nordhaus problem and the amended Nordhaus problem, no demand will be solved as an inequality because excess delivery to a demander implies extra cost in the corresponding term in the objective function. Hence, demand conditions in the primal imply that no price in the dual problem will be zero, because of the complementary slackness. If a price inequality solved as an inequality, the corresponding quantity term would be zero. We know, however, that all q(t)s in the primal must be positive, given that the exogenous demand values are all positive. We have then weak evidence that no price relation above is solved as an inequality. We still have to consider the possibility that either ${\lambda }_O$ or ${\lambda }_U$ is zero in a numerical solution. However, in our numerical investigations, we used our demand values to set the stock quantities, $S_O$ and $S_U$, so that neither $S_O$ nor $S_U$ would end up as inequalities. Hence, we have weak evidence that neither ${\lambda }_O$ nor ${\lambda }_U$ will be zero in our numerical solution. Either ${\lambda }_O$ or ${\lambda }_U$ may still be solved as zero, but not because an inequality ends up being solved as an inequality.

3. Solving the Dual Problem

We have the same parameter values we specified above when we solved the primal problem numerically. We continue to take each transportation term as zero.

For z = 1, the solution to this dual LP is

p(0) = 1.9321; p(1) = 1.19322; p(2) = 1.10322; p(3) = 1.0060; p(4) = 0.91854; p(5) = 0.82668; p(6) = 0.74402; ${\lambda }_O$ = 0.29322; ${\lambda }_U$ = 0.13122; ${\lambda }_F$ = 0.0, and the objective function has value 15.7548434.

Observe that the prices are declining over time, which is a departure from what we expect for a standard stock drawdown problem. We infer that the formulation of the problem is not satisfactory because Hotelling dynamic efficiency conditions are not appearing in the solution to the problem.

We turn to the solution for the case z = 1/[1 + r]: p(0) = 1.29321; p(1) = 1.3257; p(2) = 1.362; p(3) = 1.38; p(4) = 1.4; p(5) = 1.4; p(6) = 1.4; ${\lambda }_O$ = 0.29321; ${\lambda }_U$ = 0.13122; ${\lambda }_F$ = 0.0. The objective function has value 15.7548434.

For this case, the prices are rising to the unit fusion cost, $c_F$. (Note that the stock prices are the same as those for z = 1, and(p(0) is also the same for the two problems). Our central observation is that our solution prices for the case z = 1/[1 + r] exhibit the Hotelling property, namely the appropriate discounted “price minus unit cost” equals the relevant stock price. That is, specifically, we observe the following in the solution data:

$\left[p\left(0\right)-c_O\right]$ = 0:2931, which is ${\lambda }_O$;

$\left[p\left(1\right)-c_O\right]\left[1/1+r\right]$= 0.2931;

$\left[p\left(2\right)-c_U\right]{\left[1/(1+r)\right]}^2$ = 0.13122, which is ${\lambda }_U$;

$\left[p\left(3\right)-c_U\right]{\left[1/(1+r)\right]}^3$ = 0.13122.

The LPs characterize an efficient dynamic economy only when the Hotelling property is part of the characterization. In our case, when z = 1/[1 + r], we infer that the solution when z = 1 is not a proper characterization of a solution for an efficient dynamic economy. (The LP and its solution for z = 1 is a completely satisfactory linear programming analysis but not a satisfactory “market economic” analysis.) Nordhaus’s LP model must be appropriately amended (have appropriate present-value weights in the demand inequalities) in order for a solved example to have the Hotelling property in inherent prices. These weights in the amended problem be carried over from the quantity LP to the dual or price problem and lead to the relevant inequalities exhibiting the Hotelling property.

Our solutions would not be much different if we set the transportation cost terms, the τs, to be very small and positive. However, if we set these terms to a similar order of magnitude as our $c_O, c_U,$ and $c_F$ terms, we would obtain solutions quite different from those of a Herfindahl case. Nordhaus’s preferred solution to his very large numerical model, reported on at length, exhibited a Herfindahl-like price path, declining back in time from the unit cost of fusion power prevailing in the distant future to an initial price of energy that was somewhat lower than the price of energy from hydrocarbons prevailing in 1970 in Connecticut. In other words, Nordhaus’s large published model possessed a Herfindahl-like solution. Exactly which key parameters drove the model to exhibit this Herfindahl sort of solution are unclear. Our model is much simpler, and one could generate many types of solutions by varying the selection of parameter values.

4. Concluding Remarks

Our amended version of Nordhaus’s linear program of an energy future leads to dual (price) linear programs that exhibit prices that satisfy dynamic efficiency conditions for exhaustible stock drawdowns (Hotelling shadow prices). This makes our amended version of Nordhaus’s model a part of the “free market” or textbook economics. We observe that our example of the dual problem regarding Nordhaus’s published model fails to exhibit “textbook” dynamic efficiency conditions (Hotelling relations) linked to “free market” economics, and this failure represents a severe shortcoming of his work on energy futures and linear programming.

Nordhaus’s amended model, however, constitutes a useful addition to the economics of exhaustible resource analysis and can serve as a template for analyzing energy futures. Nordhaus gave researchers a new framework for projecting future energy trajectories of quantities and prices. He chose to illustrate his innovations with a massive empirical example. This approach may or may not have attracted researchers to his new LP model. A team of researchers could recreate his empirical work with our amended Nordhaus model, but we are drawn to the idea of seeing energy futures investigated on smaller scales. There is always the danger of losing track of which variables are important in generating a future and which are not. The Nordhaus framework for the analysis of energy futures lives on in numerous empirical analyses of the present. For example, Ref. [11] contains an analysis of the transition problem from 2020 to 2050 in 5-year intervals with more than 700 technologies and 140 activities, and Ref. [12] compares the linear and integer programming analysis of long-term energy planning.