Capacity Limitation Contracts (CLCs)

• Capacity Limitation Contracts (CLCs) are frameworks where restricted capacity is central, affecting incentive design, performance obligations, and operational control in various domains.
• They employ methods like hidden-action principal-agent models, contingent penalties, and renewal mechanisms, effectively scaling output to adjust incentive intensity.
• CLC applications span high-tech supply chain inducement, congestion management in power systems, and smart-contract SLAs in telecoms, though enforceability and practicality remain key challenges.

Capacity limitation contracts (CLCs) appear in the literature as contract structures in which limited capacity, or the consequences of limited capacity, is central to incentive design, performance obligations, or operational control. In the works considered here, the term is used in distinct but related senses: as a hidden-action principal-agent problem with a capacity constraint on the agent’s effort cost; as a supply-chain mechanism for inducing non-verifiable supplier capacity; and as a congestion-management product that gives a system operator the right to reduce the available network capacity of a connected party. A closely related precursor arises in blockchain-enabled service agreements for Dense Small-Cell-as-a-Service, where throughput commitments, traffic-based payment, and penalties for under-delivery implement capacity-related service limits without defining “CLCs” as a formal concept (Clark, 2024, Meijer et al., 2021, Holst et al., 18 Sep 2025, Pascale et al., 2017).

1. Semantic scope and research domains

The literature does not present a single standardized definition of CLCs. Instead, it uses the capacity-limitation idea in different analytical and operational environments. In principal-agent theory, the constraint is imposed on the agent’s feasible effort distributions through a bound on effort cost. In supply-chain contracting, the central issue is how an OEM induces adequate supplier capacity when capacity is non-verifiable ex ante. In power-system congestion management, the contract is a bilateral instrument that limits the available network capacity of a congestion service provider. In decentralized telecommunications, a smart-contract SLA operationalizes capacity-related service obligations through throughput commitments and penalties, but the paper is explicit that it is a practical precursor rather than a formal CLC framework (Pascale et al., 2017).

Domain	Contractual mechanism	Capacity role
Principal-agent	Pareto-optimal contracting with $c(p)\le k$	Capacity constrains effort cost
High-tech supply chain	Contingent penalty or contingent renewal	Capacity $x$ is induced under demand uncertainty
Power systems	Bilateral CLC with RC coordination	SO reduces available network capacity
Dense SCaaS	Smart-contract SLA on Ethereum	Throughput commitments and traffic-based payment

This heterogeneity matters because identical terminology does not imply identical primitives. A CLC may be an abstract incentive contract, a renewal-based supply agreement, a day-ahead network-flexibility product, or an operational SLA template. The common element is that contractual performance depends on constrained capacity, either through a hard feasibility condition, an investment decision, or a right of operational curtailment.

2. Principal-agent formulation under capacity-constrained effort

In the hidden-action principal-agent model, output depends on a finite state space $\Omega$, with output function $y:\Omega\to\mathbb{R}$. The agent chooses a feasible distribution $p\in\Delta(\Omega)$ over states, and the principal offers a verifiable state-contingent contract $b:\Omega\to\mathbb{R}$. Agent utility is

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$

while principal utility is

$$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$

Contracts must belong to a feasible set $B$, distributions must belong to a feasible set $D$, and the capacity constraint is imposed by requiring

$x$0

The agent’s incentive problem is therefore

$x$1

with participation

$x$2

These are the basic ingredients of the paper’s formal CLC environment (Clark, 2024).

The paper’s central theorem states that adding a capacity constraint to the hidden-action principal-agent problem yields the same set of Pareto optimal contracts as an otherwise identical unconstrained problem in which output is uniformly scaled down by a constant factor $x$3. The perturbed problem replaces $x$4 by $x$5, so the principal’s objective becomes

$x$6

If a Pareto-optimal contract in the original problem binds the capacity constraint, it can be recovered as a Pareto-optimal contract in a scaled-output problem; conversely, Pareto-optimal contracts of the scaled problem correspond back to Pareto optima of the original problem. The scaling factor $x$7 is increasing in the agent’s capacity $x$8 (Clark, 2024).

The significance of the result is that the capacity constraint changes the intensity of incentives rather than the qualitative structure of optimal contracting. The paper’s wording is that a binding capacity constraint “does not fundamentally change the shape of the optimal contract; it just reduces how much ‘effective output’ the principal needs to use to provide incentives.” This makes CLCs analytically tractable: one can study the unconstrained benchmark and then interpret capacity limitation as a uniform contraction of effective output.

The capital-structure example sharpens that point. If unconstrained Pareto-optimal contracts are debt-like,

$x$9

then with capacity constraints an optimal contract becomes

$\Omega$0

The principal’s payoff can then be written as

$\Omega$1

In the paper’s interpretation, the capacity-constrained firm issues debt of face value $\Omega$2 plus a fraction $\Omega$3 of equity, and as capacity increases, $\Omega$4 rises so the firm issues more debt and less equity.

3. Supply-chain capacity inducement through penalties and renewal

In the high-tech OEM–supplier setting, a single OEM designs and sells a high-tech system, and a single critical supplier produces a generation-specific component. Demand for each generation is

$\Omega$5

where $\Omega$6 is base demand known in advance and $\Omega$7 is additional uncertain demand realized after launch. The supplier chooses capacity $\Omega$8 before demand is realized; capacity is non-verifiable ex ante, generation-specific, and cannot be reused across product generations. This generates incentive misalignment: the OEM wants ample capacity to avoid shortages, while the supplier bears capacity cost $\Omega$9 and, under a standard wholesale contract, tends to underinvest (Meijer et al., 2021).

Under centralized control, total expected profit is

$y:\Omega\to\mathbb{R}$0

and the first-best capacity is

$y:\Omega\to\mathbb{R}$1

with

$y:\Omega\to\mathbb{R}$2

A standard wholesale price contract induces supplier profit

$y:\Omega\to\mathbb{R}$3

and optimal supplier capacity

$y:\Omega\to\mathbb{R}$4

Coordination requires $y:\Omega\to\mathbb{R}$5, which holds only if $y:\Omega\to\mathbb{R}$6. The paper states that coordination is therefore possible but not acceptable to the OEM, since the OEM’s margin becomes zero, and that the standard wholesale contract is inefficient because $y:\Omega\to\mathbb{R}$7 (Meijer et al., 2021).

The contingent penalty contract augments the wholesale price with a penalty if realized demand exceeds capacity. The OEM sets $y:\Omega\to\mathbb{R}$8, where $y:\Omega\to\mathbb{R}$9 is the wholesale price and $p\in\Delta(\Omega)$0 is a lump-sum penalty payable if $p\in\Delta(\Omega)$1. Supplier profit becomes

$p\in\Delta(\Omega)$2

Optimal capacity is

$p\in\Delta(\Omega)$3

and the coordination condition is

$p\in\Delta(\Omega)$4

The paper identifies the optimal coordinating contract

$p\in\Delta(\Omega)$5

$p\in\Delta(\Omega)$6

under which

$p\in\Delta(\Omega)$7

The paper therefore concludes that the contingent penalty contract coordinates the supply chain and allows the OEM to extract the entire surplus in the single-generation case. It also analyzes a per-unit shortfall penalty variant with the same coordinating structure. Its practical limitation is that the resulting penalties may be too large to enforce, especially when $p\in\Delta(\Omega)$8 is large (Meijer et al., 2021).

The contingent renewal contract is the paper’s multi-generation alternative when direct penalties are too harsh. The OEM offers a wholesale price $p\in\Delta(\Omega)$9, the supplier invests capacity $b:\Omega\to\mathbb{R}$0 each generation, and the OEM renews the contract for the next generation only when the supplier can fulfill current demand. Renewal probability is endogenous: $b:\Omega\to\mathbb{R}$1 Non-renewal operates as an implicit penalty because the supplier risks losing future profit if it underinvests. The supplier’s discounted NPV becomes

$b:\Omega\to\mathbb{R}$2

and optimal capacity under contingent renewal is

$b:\Omega\to\mathbb{R}$3

where $b:\Omega\to\mathbb{R}$4 is the Lambert $b:\Omega\to\mathbb{R}$5 function. The paper shows that $b:\Omega\to\mathbb{R}$6 is increasing in $b:\Omega\to\mathbb{R}$7, that $b:\Omega\to\mathbb{R}$8, and that higher base demand $b:\Omega\to\mathbb{R}$9 increases the value of renewal and thus increases capacity investment (Meijer et al., 2021).

Coordination is achieved by choosing

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$0

which yields

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$1

Unlike the standard wholesale case, the coordinating price satisfies $$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$2. However, the OEM cannot extract the full surplus. With

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$3

and

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$4

the paper states

$$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$5

This limitation is structural: the supplier must retain enough expected continuation value to make capacity investment worthwhile.

4. Congestion management as a risk-aware CLC portfolio

In electric-power congestion management, a capacity limitation contract is a bilateral contract in which the system operator obtains the right to reduce the available network capacity of a connected party, the congestion service provider, for a future delivery day $$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$6. It is a capacity-based product: it sets a minimum available capacity for the CSP’s portfolio, is usually activated day-ahead, typically by 8:00 on day $$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$7, and compensation can be either a fixed price per MW of reduced capacity or a payment linked to the CSP’s missed day-ahead income. In the paper’s application, if the CLC is activated, the CSP must limit the charging power of its EV fleet, thereby reducing the probability that EV load will overload a constrained asset (Holst et al., 18 Sep 2025).

The paper studies CLCs jointly with redispatch contracts (RCs). An RC is an energy-based product that forces a CSP to participate in the Dutch redispatch market GOPACS and is activated intraday after the day-ahead market closes. The CSP submits a sell order, the SO matches it with counter-bids from outside the congested area, and the matched redispatch reduces net load in the congested zone without affecting system balance. GOPACS is modeled as a continuous market with pay-as-bid pricing; orders are limit orders, partial matching is possible, the market remains open until gate closure 45 minutes before delivery, and the SO can activate RCs at various times $$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$8 (Holst et al., 18 Sep 2025).

The paper’s conceptual contribution is to treat CLCs and RCs as instruments with different risk profiles rather than as substitutes. CLCs are earlier, capacity-based, more exposed to EV forecasting uncertainty, less exposed to market liquidity risk, and can remove price risk for the SO through a fixed contractual price. RCs are later, energy-based, better informed by improved forecasts, exposed to redispatch price and liquidity risk, and able to correct early forecast errors from the CLC stage. The central question is therefore how much of each instrument should be procured and when RCs should be activated (Holst et al., 18 Sep 2025).

The optimization model is a chance-constrained two-stage stochastic mixed-integer linear program. Stage 1 activates CLCs before day-ahead closure; Stage 2 activates RCs at time $$\mathbb{E}_p[b]-c(p)=\sum_{\omega\in\Omega}p(\omega)b(\omega)-c(p),$$9 after some uncertainty has been revealed. The objective minimizes CLC costs plus expected RC costs: $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$0 and, in scenario form,

$$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$1

Redispatch cost is defined as

$$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$2

Reliability is imposed through chance constraints requiring EV load not to exceed the asset capacity limit $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$3 with probability greater than $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$4: $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$5

$$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$6

The paper approximates these with CVaR-based linear constraints using auxiliary variables $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$7 and $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$8: $$\mathbb{E}_p[y-b]=\sum_{\omega\in\Omega}p(\omega)\bigl(y(\omega)-b(\omega)\bigr).$$9

$B$0

$B$1

$B$2

$B$3

The paper notes that this approximation is linear and tractable but can be conservative, potentially causing overprocurement.

The EV fleet is modeled with an aggregated virtual battery representation from Tang et al., summarized by stochastic parameters $B$4, $B$5, and $B$6. For $B$7, charging power under CLC is

$B$8

For $B$9, post-redispatch charging power is linked to earlier charging prognosis through

$D$0

Because the $D$1 operator is nonconvex, the paper reformulates it with binary variables and big-$D$2 constraints (Holst et al., 18 Sep 2025).

The uncertainty model has two independent sources. EV charging uncertainty is generated by training a random forest model on UK domestic charging data, analyzing forecast errors on a validation set, modeling error dynamics with a VAR(1) process, and generating scenarios for fleets of 2,500, 10,000, and 25,000 EVs. Redispatch market uncertainty is estimated from Dutch GOPACS order book data from ETPA covering May 2023–June 2024, with total available buy volume and buy price estimated for each trading window starting at $D$3, using filters to avoid manipulation and double counting, VWAP aggregation, Gaussian or Student-$D$4 copulas, and fast-forward scenario reduction (Holst et al., 18 Sep 2025).

The main result is that the optimal policy usually combines both instruments. For $D$5, CLCs are preferred because the minimum redispatch bid size of $D$6 MW is large relative to expected congestion, making RCs prone to overprocurement. For $D$7, RCs become more useful and a more balanced CLC/RC mix is optimal. For $D$8, RC use becomes limited by market liquidity risk, conservative CVaR constraints further reduce RC activation, and the optimal RC activation time is around 11:00 on day $D$9. Later RC activation usually improves forecast quality, but if activation is too late the trading window shrinks, liquidity risk rises, and RC effectiveness falls. The paper therefore concludes that a mixed strategy is generally best, but the timing and sizing of RCs must be adapted to fleet size, forecast uncertainty, and redispatch market liquidity (Holst et al., 18 Sep 2025).

5. Smart-contract SLA precursor in Dense Small-Cell-as-a-Service

The Dense Small-Cell-as-a-Service paper is not a formal CLC paper, but it provides an operational example of a capacity-constrained service contract in a decentralized telecom setting. Its motivation is to make Small-Cell-as-a-Service feasible at much smaller scale, potentially down to individual homes or retail venues. The paper argues that current SCaaS arrangements are mainly negotiated as long-term bilateral business deals between Mobile Network Operators and Small Cell Providers, creating cost, complexity, and scalability barriers. It proposes smart contracts to implement simple but effective Service Level Agreements between SCPs and MNOs, with Ethereum as the reference blockchain platform (Pascale et al., 2017).

The contract template assumes that the SCP and MNO have already agreed to a template contract, perhaps via an online platform or a third-party register. The SLA includes QoS parameters, explicitly naming QCI-based service characterization: priority, packet delay budget, and packet loss rate. KPIs are used to measure performance, often per QCI. Capacity-related clauses enter through throughput commitments and traffic-based payment. Periodic payment is proportional to the amount of traffic served, measured by the MNO, with a price per kilobyte agreed in advance. The paper’s description identifies this as the strongest capacity-limitation element: compensation is tied to actual traffic delivered, and service is constrained by agreed throughput levels for each QCI (Pascale et al., 2017).

Enforcement is specified through an infraction procedure for throughput violations. A monitoring component in the MNO triggers a throughputBreach function when the SCP fails to meet the agreed average throughput for a given QCI. The function identifies the culprit SCP, the relevant QCI, and the deficit throughput relative to the agreed average. The penalty is a debit proportional to the difference between measured and agreed throughput, and this debit is subtracted from current or future credits owed to the SCP. The paper also introduces a “3-strike rule”: after three consecutive infractions, the SCP is removed from the register, effectively preventing further transactions with that MNO (Pascale et al., 2017).

Ethereum serves as the execution and enforcement layer. The smart contract is deployed on Ethereum, events such as PeriodicPayout and InsufficientThroughput are stored on-chain and can be observed asynchronously by the MNO or SCP, and payments follow the withdrawal pattern. The enforcement logic is therefore not left to off-chain negotiation: the blockchain provides immutable storage, automatic triggerable functions, and auditable records of compliance and breach. The paper also states that the code is public and immutable, which supports trust in a “trustless environment,” while warning that bugs and legal enforceability remain concerns (Pascale et al., 2017).

The paper does not provide formal equations, pseudocode, or LaTeX constraints defining capacity bounds. It does not set maximum user counts, explicit bandwidth ceilings, or a formal contract type dedicated to capacity limitation. Its contribution is instead to show how blockchain smart contracts can encode and enforce capacity-related service limits through throughput thresholds, QCI-specific performance expectations, traffic-based compensation, and penalties for under-delivery.

6. Comparative interpretation, limitations, and recurrent issues

Across these works, capacity limitation enters contracts through different legal and mathematical devices. In the principal-agent model, it is a feasibility condition $x$00 on the agent’s effort cost, with the main result that the constrained problem is equivalent to scaling output by $x$01 (Clark, 2024). In the high-tech supply chain, it is an induced investment variable $x$02 shaped either by explicit penalties or by continuation-value incentives under contingent renewal (Meijer et al., 2021). In congestion management, it is an operational right to reduce available network capacity, coordinated with redispatch as a two-stage stochastic control problem (Holst et al., 18 Sep 2025). In Dense SCaaS, it is expressed indirectly through throughput commitments, traffic-based payment, and performance-triggered debits within a smart-contract SLA (Pascale et al., 2017).

A recurrent limitation is enforceability. In the high-tech OEM–supplier setting, contingent penalties may be too large to enforce, especially when $x$03 is large (Meijer et al., 2021). In the blockchain SLA setting, public and immutable code supports trust, but bugs and legal enforceability remain concerns (Pascale et al., 2017). In congestion management, the CVaR approximation of chance constraints is linear and tractable but can be conservative, potentially causing overprocurement, and late redispatch activation faces shrinking trading windows and rising liquidity risk (Holst et al., 18 Sep 2025). In the principal-agent setting, the theorem shows that capacity limitation need not generate a wholly new contract form; it modifies incentive intensity through a uniform scaling of effective output (Clark, 2024).

A common misconception is that CLCs designate a single homogeneous contract class. The literature summarized here indicates otherwise. The same label covers a formal Pareto-frontier equivalence result, a multi-generation supply contract using non-renewal as an implicit penalty, and a day-ahead network-flexibility product. This suggests that “CLC” is best treated as a family resemblance centered on capacity-constrained performance and incentives rather than as a single canonical contract specification.

Another recurrent issue is the trade-off between firmness and adaptability. Direct penalties in supply chains are powerful but may be unrealistic; contingent renewal is more practical but cannot transfer the entire surplus to the OEM (Meijer et al., 2021). Day-ahead CLCs in congestion management provide early, relatively firm intervention but have limited ability to correct later forecast errors, so redispatch serves as recourse (Holst et al., 18 Sep 2025). Smart-contract SLAs make enforcement auditable and automatic, yet they remain dependent on measurable proxies such as throughput per QCI rather than on a complete formal grammar of capacity bounds (Pascale et al., 2017). In that sense, the literature consistently frames capacity limitation not simply as a restriction, but as a contracting problem about how to convert limited capacity into enforceable incentives, operational rights, or continuation values.