Cheap or clean? How do you weigh cost against emissions?

Spot the plan nobody should pick, slide a carbon price, and trace the frontier with a solver

Lesson 12 of Energy Systems, Step by Step: compare plans for a small fictional town by cost and CO₂, find the dominated one, see how a carbon price picks a plan, trace the cost and CO₂ frontier with a solver and a CO₂ cap, and weigh who should choose the point.
  • Analyst · L2
  • Lesson 12 of 14
  • 15 min
  • After lesson 11

Six plans for Prairie Town: yearly estimates from seven representative weeks of 2025

Plan Cost a year CO₂ a year Note
A · Gas only $103.9M 815 kt cheapest at no carbon price
B · Gas + 300 MW wind $119.4M 521 kt wins between $53 and $64 a tonne
C · Gas + 500 MW solar $131.4M 488 kt not dominated, but no carbon price picks it
D · Gas + 600 MW wind $136.8M 249 kt wins between $64 and $166 a tonne
E · Gas + 800 MW solar $159.6M 433 kt dominated: D costs less and emits less
F · Wind, solar, battery and gas backup $159.1M 115 kt wins above $166 a tonne

Plan F: 600 MW wind, 200 MW solar, a 100 MW / 400 MWh battery, and 300 MW of combined-cycle gas plus 80 MW of peakers as backup.

With JavaScript on, you pick the dominated plan, slide a carbon price and trace the frontier with a solver.

Model: lesson 13's; it estimates each plan's year from seven representative weeks, and a full-year check is separate.
Data: real 2025 hourly shapes of the Southwest Power Pool grid (EIA-930).
Costs: EIA's and NREL's, dated in lesson 13's sources.
Price year: 2025 dollars throughout; NREL's 2024 battery costs are converted with BEA's GDP implicit price deflator (×1.0263).
Plans: illustrative, not recommendations; Prairie Town is fictional.

Need a refresher? Toolbox K3 (how a building cost becomes a cost a year).

What you just did, as diagrams

Dominated plan
Another plan costs no more and emits no more, and is better on one count: nobody should pick it.
Weighted sum
Score each plan by cost plus a carbon price times its CO₂; the lowest score wins.
Epsilon-constraint
Cap the CO₂ and find the cheapest plan under the cap; sweep the cap to trace the frontier.
Frontier
The plans that cannot get cheaper without getting dirtier, or cleaner without getting dearer.
Shadow price
Used as a carbon price, the cap's shadow price makes the cap's plan one of the cheapest choices.
Go deeper · AnalystThe limits of weighted sums
  • Dents. Plan C ($131.4M, 488 kt) lies above the straight line from B to D, so no carbon price picks it, though no plan beats it on both counts. Frontiers built from yes-or-no choices, such as building a plant or not, often have such dents.
  • Nearly straight pieces. Between 500 and 400 kt the slope barely changes ($43.4 to $44.0 a tonne), so prices from $43 to $45 move the plan from 541 kt to 389 kt.
  • Uneven spacing. Even steps in price give uneven steps along the frontier: up to $45 a tonne moves the plan from 792 kt to below 400 kt; from $45 to $220 it moves only to 100 kt.
  • Weights carry units. The weight in cost + w × CO₂ is a price in dollars per tonne; measure CO₂ in kilotonnes and the same number means something a thousand times larger.

Das and Dennis (1997) showed these drawbacks for weighted sums in general. The epsilon-constraint method avoids the first two by fixing one goal and optimizing the other.

Go deeper · ModelerThe cap's dual as a carbon price

The epsilon-constraint problem: the cheapest plan under a CO₂ cap.

\[\min_x \; c(x) \quad \text{subject to} \quad e(x) \le \varepsilon \;\; (\lambda), \;\; x \in X.\]

Symbol Meaning
\(x\), \(X\) a plan (what to build and how to run it), and every plan that meets the other rules
\(c(x)\), \(e(x)\) its yearly cost ($) and its CO₂ (tonnes)
\(\varepsilon\) the cap, in tonnes a year
\(\lambda\) the cap’s multiplier, or dual: its shadow price, in $ per tonne
\(\lambda^*\), \(c^*(\varepsilon)\) that multiplier at the best plan, and the best cost for cap \(\varepsilon\)
  • Duality. For a linear program, strong duality (a theorem that pairs every LP with a “dual” LP of prices) guarantees a \(\lambda^* \ge 0\) for which the cap’s best plan also minimizes the weighted sum \(c(x) + \lambda^* e(x)\). Other plans may tie with it, and a solver may return one of them.
  • Slope. Where the frontier is smooth, \(\lambda^* = -\,\mathrm{d}c^*/\mathrm{d}\varepsilon\); at a kink, \(\lambda^*\) lies between the slopes on either side.
  • Here. A 200 kt cap has \(\lambda^* = \$85.2\) per tonne; a $85.2 price with no cap lands at 199 kt and $129.4M.
  • Ties. On an exactly straight piece every plan ties at its slope; with integer decisions there is no such guarantee at all.

Mavrotas (2009) shows how to sweep \(\varepsilon\) efficiently. Next: A22 (five goals at once); M6 (carbon in models).

Check yourself

Myth or fact? “The cheapest plan is always the right choice.”

  • Myth
  • Fact
Answer

Myth. Here plan A stays the pick only while CO₂ is priced below $53 a tonne. The frontier shows what each tonne avoided costs; people decide what that is worth.

Myth or fact? “A carbon price and a CO₂ cap can lead to the same plan.”

  • Myth
  • Fact
Answer

Fact. A 200 kt cap has a shadow price of $85.2 a tonne; that price, with no cap, lands at 199 kt, next to the cap's plan.

Myth or fact? “Every plan that is not dominated can be picked by some carbon price.”

  • Myth
  • Fact
Answer

Myth. Plan C is not dominated, yet it sits in a dent between B and D, so no price picks it.

Plan B costs $119.4M a year and emits 521 kt; plan D costs $136.8M and emits 249 kt. Above what carbon price does D score lower than B?

  • About $64 a tonne
  • About $17 a tonne
  • About $273 a tonne
  • D never scores lower
Answer

About $64: D costs $17.4M more and emits 272 kt less, and 17.4 million ÷ 272,000 tonnes ≈ $64 a tonne.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
52.91 kg CO₂ released by burning one million Btu (MMBtu) of natural gas. EIA, CO₂ emissions coefficients · released September 2024
0.33 t/MWh CO₂ per MWh from a new combined-cycle plant (6,226 Btu per kWh); a peaker emits 0.48. EIA, AEO2026 assumptions, Table 3 (PDF) · April 2026
$3.85/MMBtu Average cost of natural gas to U.S. power plants in 2025. EIA, Electric Power Annual, Table 7.4 · 2025 data

Next on the starter path

Lesson 13 · What should Prairie Town build? Plan its power system

In the real world

Both routes exist: some places tax CO₂ (a price); others cap it and trade allowances, so the market finds the price. Planning studies trace frontiers like yours; lesson 13 lets you plan the town yourself.

Case file CF6 (average and marginal emission factors, and when moving electricity use to cleaner hours lowers CO₂).

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.