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Optimal Pricing of Cloud Services: Committed Spend under Demand Uncertainty

Dirk Bergemann, Michael C. Wang

TL;DR

This paper addresses pricing under demand uncertainty by modeling a monopolist service provider who offers contracts before demand is realized, while buyers receive noisy signals about their future demand. It derives an optimal dynamic mechanism using a Myersonian framework for a nonlinear, multi-unit setting, showing that higher-signal buyers receive better usage terms but must commit more upfront, and demonstrates that the mechanism can be implemented via a two-part tariff or a committed spend contract. The authors extend the base model to analyze frictions such as capital costs and the presence of liquid spot markets, showing that capital constraints favor backloaded, committed-spend arrangements and that spot markets discipline long-horizon pricing and can alter allocation distortions. The results have practical implications for cloud computing and SaaS contracts, informing policy discussions on competition, efficiency, and financing of long-term commitment agreements in digital services.

Abstract

We consider a seller who offers services to a buyer with multi-unit demand. Prior to the realization of demand, the buyer receives a noisy signal of their future demand, and the seller can design contracts based on the reported value of this signal. Thus, the buyer can contract with the service provider for an unknown level of future consumption, such as in the market for cloud computing resources or software services. We characterize the optimal dynamic contract, extending the classic sequential screening framework to a nonlinear and multi-unit setting. The optimal mechanism gives discounts to buyers who report higher signals, but in exchange they must provide larger fixed payments. We then describe how the optimal mechanism can be implemented by two common forms of contracts observed in practice, the two-part tariff and the committed spend contract. Finally, we use extensions of our base model to shed light on policy-focused questions, such as analyzing how the optimal contract changes when the buyer faces commitment costs, or when there are liquid spot markets.

Optimal Pricing of Cloud Services: Committed Spend under Demand Uncertainty

TL;DR

This paper addresses pricing under demand uncertainty by modeling a monopolist service provider who offers contracts before demand is realized, while buyers receive noisy signals about their future demand. It derives an optimal dynamic mechanism using a Myersonian framework for a nonlinear, multi-unit setting, showing that higher-signal buyers receive better usage terms but must commit more upfront, and demonstrates that the mechanism can be implemented via a two-part tariff or a committed spend contract. The authors extend the base model to analyze frictions such as capital costs and the presence of liquid spot markets, showing that capital constraints favor backloaded, committed-spend arrangements and that spot markets discipline long-horizon pricing and can alter allocation distortions. The results have practical implications for cloud computing and SaaS contracts, informing policy discussions on competition, efficiency, and financing of long-term commitment agreements in digital services.

Abstract

We consider a seller who offers services to a buyer with multi-unit demand. Prior to the realization of demand, the buyer receives a noisy signal of their future demand, and the seller can design contracts based on the reported value of this signal. Thus, the buyer can contract with the service provider for an unknown level of future consumption, such as in the market for cloud computing resources or software services. We characterize the optimal dynamic contract, extending the classic sequential screening framework to a nonlinear and multi-unit setting. The optimal mechanism gives discounts to buyers who report higher signals, but in exchange they must provide larger fixed payments. We then describe how the optimal mechanism can be implemented by two common forms of contracts observed in practice, the two-part tariff and the committed spend contract. Finally, we use extensions of our base model to shed light on policy-focused questions, such as analyzing how the optimal contract changes when the buyer faces commitment costs, or when there are liquid spot markets.

Paper Structure

This paper contains 12 sections, 4 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Literature
  4. Model
  5. Optimal Dynamic Mechanism
  6. Dynamic Direct Mechanism
  7. Two-Part Tariff
  8. Committed Spend Contract
  9. Contract Frictions
  10. Cost of Committed Capital
  11. Sequential Screening in the Presence of Spot Markets
  12. Conclusion

Key Result

Theorem 1

Under Assumptions asu:fosd-asu:reg, the mechanism characterized by eq:opt-q-eq:opt-ir is optimal.

Figures (3)

  • Figure 1: Optimal Mechanism for Example \ref{['ex:1']} ($q$ solid, $t$ dashed).
  • Figure 2: $t_0({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$ and Unit Price for Example \ref{['ex:1']} when $c=1$ and ${ \if@compatibility \mathchar"010B {} \mathchar"010B } = \frac{1}{2}$.
  • Figure 3: $B({ \if@compatibility \mathchar"0112 {} \mathchar"0112 })$ and Unit Price for Example \ref{['ex:1']} when $c = 1$ and ${ \if@compatibility \mathchar"010B {} \mathchar"010B } = \frac{1}{2}$.

Theorems & Definitions (10)

  • Theorem 1: Optimal Dynamic Direct Mechanism
  • proof
  • Example 1
  • Definition 1: Two-Part Tariff
  • Definition 2: Committed Spend Contract
  • Proposition 2: Guaranteed Positive Quantity
  • proof
  • Proposition 3: Optimal Contract with Commitment Cost
  • Proposition 4: Optimal Contract with Spot Market
  • proof