The Math Behind Fair Electricity Prices Finally Adds Up

The Hidden Tax on Clean Energy

Deep in the machinery of how electricity gets priced, there is a paradox that costs consumers billions of dollars annually and distorts investment decisions in clean energy. It's not a carbon tax or a subsidy—it's a side payment called "uplift," and understanding how to eliminate it is the subject of a new paper that bridges two separate branches of optimization theory in a way that no one expected.

When a power plant fires up to keep the lights on, it earns money by selling electricity at the market price. But here's the catch: that market price is often lower than what the plant actually spent to turn on its turbines and start generating. The gap is covered by uplift—extra payments that the plant receives simply because the market's uniform price couldn't pay its bills. These payments are substantial. In U.S. wholesale electricity markets, uplift costs have run into hundreds of millions of dollars per year, passed on to consumers not as transparent line items but buried in the price of electricity itself.

The problem is fundamental. Electricity markets solve a logistical nightmare called unit commitment—deciding which generators should turn on, at what times, and how much power each should produce. Every generator must decide, yes or no, whether to commit to running. That binary choice makes the problem "non-convex," which in the mathematics of optimization is a polite way of saying it behaves very badly. With non-convexity in the mix, economists have known since the 1990s that no single uniform electricity price can simultaneously clear the market efficiently and ensure every generator recovers its costs. Something has to give. Either the price clears the market and leaves some generators in the red, or it pays everyone fairly and creates inefficiencies.

For decades, the standard workaround has been to ignore the non-convexity, pay the uplift, and hope the gap isn't too large. But a paper by Madhusudan Ghosh, Antoine Lesage-Landry, and Joshua Adam Taylor, posted to arXiv in July 2026, now shows that two existing approaches to this problem—one that calculates prices by relaxing the binary constraints, and another that works through the dual of an exact reformulation—are actually mathematically equivalent under everyday conditions. The finding is both a theoretical triumph and a practical roadmap for making electricity markets fairer and more efficient.

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The Science

The paper's starting point is a puzzle that has nagged power systems economists for two decades. Wholesale electricity markets in the United States and Europe rely on uniform locational marginal prices (LMPs)—a single price per location per time period that all generators receive for the power they sell and all consumers pay for the power they buy. These prices are elegant, transparent, and computationally tractable. But they break down when generators have to make discrete on/off decisions with startup costs, minimum run times, and minimum down times—all the operational constraints that govern how real power plants actually work.

The commitment decision is binary: a generator is either committed or not. There is no fractional commitment, no partial startup. When economists first articulated the problem in the 1990s and early 2000s, they identified two families of solutions. The first, convex hull pricing (CHP), calculates a price by taking the convex hull relaxation of the unit commitment problem—essentially, pretending the binary constraints are continuous, solving that simpler problem, and then reading the price off the dual variables. The idea, introduced by William Hogan and Brian Ring in 2003 and formalized by Paul Gribik, Hogan, and Susan Pope in 2007, was to find the uniform price that minimizes total uplift while still clearing the market efficiently. If you must pay uplift, you might as well pay as little of it as possible.

The second family of solutions emerged more recently from a different direction. In 2026, Chenhua Guo, Merve Bodur, and Joshua Taylor introduced copositive duality pricing (CDP), which takes the unit commitment problem, reformulates it as a completely positive program—a convex optimization problem in a higher-dimensional matrix space—and then extracts prices from the dual of that reformulation. The reformulation is exact: Burer's theorem, published in 2009, guarantees that the lifted problem has the same optimal value as the original unit commitment problem. From that dual, you can read off a set of prices. The two approaches started from completely different premises and used completely different tools, and for years there was no reason to think they should give the same answer.

Ghosh, Lesage- Landry, and Taylor's contribution is to define the centralized convex hull price—specifically, the convex hull price computed over the joint feasible set of the unit commitment problem, not the per-generator convexification that had become standard in the literature—and then prove that, under non-degeneracy, it coincides with the marginal copositive duality price. They do this by showing that both prices are subgradients of a common value function: the function that maps a demand vector to the minimum cost of serving it. Where that function is differentiable, the two prices are identical. Where it is not differentiable—at the demand levels where a generator with a positive startup cost must be brought online—the two prices may differ, but they share a common subgradient, like two different paths to the same peak.

The proof hinges on a key decomposition of the copositive duality price. The authors show that the centralized copositive price at each time period $t$ takes the form:

$${\lambda }_{\text{CHP},t}={\lambda }_t^{\ast }+2d_t{\Lambda }_t^{\ast }$$

where ${\lambda }_t^{\ast }$ is the dual multiplier for the demand balance constraint—the standard demand price—and ${\Lambda }_t^{\ast }$ is a second multiplier that arises from the lifted, squared version of that constraint in the completely positive reformulation. The second term, $2d_t{\Lambda }_t^{\ast }$, is the "lifted component." It is present in the centralized price but vanishes in the standard decentralized convex hull price, because the decentralized version convexifies each generator's feasible set separately and never introduces the joint demand constraint in its squared form.

The numerical experiments use the modified Scarf example, a benchmark that dates to the foundational work of Hogan and Ring. The setup involves three generator types with different cost structures, startup costs, and capacity parameters, run across a demand sweep from 10 to 160 megawatts in 5-megawatt increments. The authors implemented all computations in Python using CVXPY, with Gurobi, MOSEK, and Clarabel as solvers.

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What They Found

The experiments confirm the theoretical equivalence and expose some important nuances.

At most demand levels, the three decentralized prices—the decentralized convex hull price, the decentralized copositive duality price, and the semidefinite restriction of the decentralized price—coincide precisely. They diverge only at a single transition point, around 35 megawatts, where the value function is nondifferentiable and the subgradient is not a singleton. This is exactly what the theory predicts: where the value function is smooth, there is one price; where it has a kink, multiple prices can coexist.

The relationship between centralized and decentralized pricing is more subtle. At higher demand levels, the centralized price runs slightly higher than the decentralized price. This reflects the presence of the lifted term $2d_t{\Lambda }_t^{\ast }$ in the centralized formulation. As demand rises, this term contributes additional price, pushing the centralized price above what the decentralized approach would recommend.

The uplift comparisons tell a striking story. The decentralized copositive price, despite being mathematically equivalent to the standard decentralized convex hull price, results in more total uplift than the centralized copositive price. But the authors are careful to note that this does not contradict the uplift-minimizing property of convex hull pricing. The decentralized convex hull price minimizes uplift over the class of uniform linear prices. The centralized price, by contrast, is nonlinear because ${\Lambda }_t^{\ast }$ is the multiplier of a squared constraint. Under the lost-opportunity-cost definition of uplift that the authors adopt, the standard decentralized convex hull price does indeed result in less uplift than the centralized price—but that centralized price is not a uniform linear price, so the comparison is apples and oranges.

The semidefinite restriction—the tractable approximation that replaces the intractable copositive cone with the more manageable cone of positive semidefinite matrices plus entrywise nonnegative matrices—introduces a measurable gap. The value of the semidefinite restriction is always less than or equal to the true value function, which means the SDP prices are not quite the same as the exact copositive prices. But the experiments suggest this gap is small in practice.

The comparison between centralized and decentralized CDP prices across the demand range. At higher demand levels, the centralized price runs slightly above the decentralized price—a consequence of the lifted term in the centralized formulation.

The three decentralized prices coincide at most demand levels. The divergence occurs only at commitment transitions, where the value function's subgradient is not unique and multiple prices can be consistent with the same market outcome.

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Why This Changes Things

The practical significance of this paper lies in what it does not require: it does not propose a new market design or call for new infrastructure. It takes two existing frameworks and shows they are, in the right formulation, the same thing. That unification matters for several reasons.

First, it settles a question that had been hanging over the literature. Convex hull pricing had been derived from one set of principles—relaxation, subgradients, value functions—while copositive duality pricing had been derived from an entirely different set—reformulation, conic duality, matrix-space geometry. That they converge is not an accident. It reflects a deeper structure in the unit commitment problem that both frameworks are, in their own language, describing correctly. The authors show that both prices are subgradients of the same value function $v(d)$, and where that function is differentiable, they collapse to a single price. This gives practitioners who have been using one framework or the other a new measure of confidence: they are effectively computing the same thing.

Second, the decomposition of the centralized copositive price into ${\lambda }_t^{\ast }+2d_t{\Lambda }_t^{\ast }$ is revealing. The term ${\lambda }_t^{\ast }$ is the familiar demand balance price—the shadow price of meeting one more megawatt of demand in period $t$. The term $2d_t{\Lambda }_t^{\ast }$ is something genuinely new, arising from the reformulation-linearization technique that lifts the linear constraints into their squared counterparts. This lifted term captures interactions between generators that the decentralized approach misses. It is the mathematical signature of the fact that the joint feasible set is not simply the product of individual feasible sets. In the decentralized approach, each generator is convexified independently, and the cross-generator structure of the unit commitment problem is lost. The centralized approach recovers it.

The practical implication is that the choice between centralized and decentralized convex hull pricing is not merely computational—it is structural. The centralized price, by correctly modeling the joint feasible set, produces a different price signal than the decentralized price. At low demand levels, this difference may be negligible. But as demand rises and generators approach their capacity limits, the lifted term $2d_t{\Lambda }_t^{\ast }$ becomes increasingly important. Accurate price signals at high demand levels matter for investment incentives, for demand response, and for the profitability of flexible generation—all of which are critical as grids integrate more variable renewable resources.

The paper's copositive formulation of the standard decentralized convex hull price is also a contribution to the literature. By recasting the decentralized price in the language of copositive duality, the authors show that it too arises from a dual of a completely positive program—each generator is lifted into its own matrix space, and the joint price is read from the demand balance multiplier. The cone constraints in this formulation separate across generators, which is why the lifted demand term disappears. The semidefinite restriction then provides a tractable approximation: replace the copositive cone with the cone of positive semidefinite matrices plus entrywise nonnegative matrices, and the problem becomes solvable with standard SDP solvers.

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What's Next

The gap introduced by the semidefinite restriction is small in the Scarf example, but the authors acknowledge that this gap could be larger in more complex market settings with many generators, more intricate operational constraints, and longer time horizons. Quantifying that gap across realistic market scenarios is an important direction for future work. If the SDP restriction performs well in large-scale problems, it could become a practical computational tool for actually computing convex hull prices in real markets—something that has historically been computationally challenging.

The Scarf example is a benchmark—three generators, a simple structure, clean transitions. Real wholesale markets operate thousands of generators across twenty-four or forty-eight hour time horizons, with thousands of constraints governing ramp rates, startup trajectories, reserve requirements, and transmission limits. Whether the equivalence result extends cleanly to these settings, or whether new complications arise from the additional structure, remains to be seen. The authors' non-degeneracy assumption—essentially, a condition that rules out pathological cases where the math gets messy—holds in practice for most realistic problems, but testing it at scale is a research program in itself.

Another open question concerns the relationship between centralized convex hull pricing and the various definitions of uplift that circulate in the literature. The paper shows that the centralized copositive price is not a uniform linear price, and therefore does not minimize uplift in the same sense as the decentralized convex hull price. But the centralized price may minimize a broader class of costs—perhaps the total cost of serving demand, including the opportunity costs that uplift is meant to compensate. Clarifying this relationship would sharpen the practical guidance for market designers.

The findings also invite a broader reflection on the role of mathematical structure in market design. Copositive programming has long been a theoretical curiosity—a way of representing combinatorial problems in continuous optimization language, useful for proofs but impractical for computation. The fact that it now appears in a practical pricing context suggests that the boundary between theoretical optimization and practical market design is more permeable than it once seemed. As the authors put it in the paper's conclusion, the copositive formulation provides a "unified perspective" on convex hull pricing. Whether that perspective leads to better prices in real markets remains to be demonstrated, but the theoretical foundation is now solid.

The Scarf example was introduced in 1953 by Herbert Scarf to study rationing and queueing problems in planned economies. It traveled through economics, operations research, and finally arrived at electricity market design as a testbed for the uplift problem. More than seventy years later, it is still revealing structure that the original designers could not have anticipated. The paradox of uplift—of paying generators extra simply because a uniform price cannot be fair—may not be solvable in full generality. But Ghosh, Lesage-Landry, and Taylor have shown that two roads to the same destination are, at least mathematically, the same road after all.