Deepractice Agent Cloud Production Function
From MaaS to AaaS
How much money invested in AI yields how much value?
This is the question almost every decision-maker is asking.
In February 2025, Yale University's Cowles Foundation published a paper: The Economics of Large Language Models (Bergemann, Bonatti, Smolin). The paper proposed a production function to measure the value of tokens and fine-tuning.
But this framework has a problem: it was designed for MaaS (Model as a Service)—focusing on model-layer value.
The 2025 market reality is: value creation is shifting from the model layer to the agent layer.
We need a new framework to measure the value of AaaS (Agent as a Service).
This is the Deepractice Cloud Production Function.
The Formula
Value = x^α × y^β × (b+z)^γ × (1+a)^δ
Four variables:
| Variable | Meaning |
|---|---|
| x | Input tokens (content given to AI) |
| y | Output tokens (content generated by AI) |
| b+z | Model quality (base model + fine-tuning) |
| 1+a | Agent investment |
Four exponents (α, β, γ, δ) represent the elasticity of each input—the greater the elasticity, the more valuable the investment.
How did this formula come about? What's it good for? Read on.
Two Core Principles
Principle One: Diminishing Returns
Q: Does giving AI more tokens lead to better results?
Yes. But not linearly.
100 tokens → 200 tokens, noticeable improvement. 1000 tokens → 2000 tokens, improvement is less obvious.
This is called diminishing marginal returns. The more you invest, the less additional effect each increment provides.
How to express this mathematically? Exponent < 1.
Value = Token^0.5
Quadruple the tokens, only double the value.
It's not just tokens. Models work the same way:
- GPT-3.5 → GPT-4, big improvement
- GPT-4 → GPT-5, improvement not as significant
Agents too:
- No Agent → Simple Agent, big improvement
- Simple Agent → Complex Agent, improvement diminishes
All investments have diminishing returns.
Principle Two: Multiplication, Not Addition
Q: Can tokens, models, and agents be calculated separately?
No. They have a multiplicative relationship, not additive.
- Poor model → nothing else matters
- No tokens → even the best model is useless
- No agent → efficiency can't improve
Any factor = 0, total value = 0.
This explains why some companies buy the most expensive models but get poor results—their agent layer is 0.
Breaking Down Each Term
Token Terms: x^α × y^β
Why separate input and output?
Because they have different costs.
| Type | Cost |
|---|---|
| Input tokens | Cheap |
| Output tokens | 3-5x more expensive |
For input, AI just needs to "understand." For output, AI needs to "generate," which requires much more computation.
Different scenarios have vastly different input/output ratios:
| Scenario | Input | Output |
|---|---|---|
| Long document summary | Large | Small |
| Creative writing | Small | Large |
| Conversational Q&A | Medium | Medium |
This means: different business scenarios have completely different token cost structures.
Model Term: (b+z)^γ
b is base model quality, z is fine-tuning investment.
Key insight: γ is already very small in 2025.
What does this mean?
According to IBM and multiple consulting firms, current market realities are:
- General-purpose models are already very capable
- Fine-tuning requires massive compute resources and specialized teams
- The industry direction is Prompt Engineering + RAG, not Fine-tuning
When γ is small, z's contribution is negligible, b dominates.
In plain language: just pick a good model and don't bother with fine-tuning.
This is why everyone is doing prompt engineering, not fine-tuning.
Agent Term: (1+a)^δ — This is the Key
This term is our extension.
The Yale paper focused on the "fine-tuning" layer. But the 2025 market reality is: value creation has shifted to the agent layer.
Why (1+a) instead of just a?
- a = 0 (no agent) → (1+0)^δ = 1 → base value unchanged
- a > 0 (with agent) → (1+a)^δ > 1 → value is amplified
Agents aren't additive, they're multiplicative. They amplify the value of all previous investments.
According to McKinsey's 2025 report, 85% of organizations have integrated AI Agents into at least one workflow. MarketsandMarkets predicts the AI Agent market will grow from $7.8 billion in 2025 to $52.6 billion by 2030.
This growth rate far exceeds the model and token layers.
δ is getting larger. This is the core battleground for AI value today.
Agent investment a is itself a complex system.
Prompt engineering, tool use, RAG, MCP, human services, server resources, sandbox environments... How do these elements combine? What are the marginal returns of each? What's the optimal investment ratio?
This is the deep logic of the AaaS industry. We'll expand on this in future articles.
Strategic Implications of the Formula
Elasticity Parameters = Market Value Distribution
| Parameter | If Large | If Small |
|---|---|---|
| α (input tokens) | Input matters a lot | Input not critical |
| β (output tokens) | Output matters a lot | Output not critical |
| γ (fine-tuning) | Fine-tuning is valuable | Fine-tuning is pointless |
| δ (agent) | Agents are valuable | Agents don't matter |
2025 Market State
- α, β: Medium. Tokens have value, but prices keep dropping.
- γ: Very small. Fine-tuning is nearly pointless, general models are strong enough.
- δ: Very large. Agents are the value core.
What Does This Mean?
If you're an investor:
- Stop investing in fine-tuning companies
- Focus on agent-layer value creation
If you're a business decision-maker:
- Token costs will keep dropping, don't stress about it
- Pick a good model, then focus on the agent layer
If you're an entrepreneur:
- The model layer is already a giants' game
- The agent layer is where the opportunity lies
Summary
Value = x^α × y^β × (b+z)^γ × (1+a)^δ
─── ─── ─────── ───────
Input Output Model Agent
Three principles:
- Every term has diminishing returns (exponent < 1)
- Multiplicative relationship, none can be zero
- Elasticity magnitude reflects market value distribution
2025 market reality:
- Tokens are getting cheaper (α, β medium)
- Fine-tuning is pointless (γ very small)
- Agents are the value core (δ very large)
The essence of Deepractice Cloud: we sell (1+a)^δ — the value of AaaS.
Theoretical Foundations
This formula is extended from the Yale University Cowles Foundation 2025 paper The Economics of Large Language Models (Bergemann, Bonatti, Smolin). The original paper proposed a production function framework for tokens and fine-tuning; we added the agent layer.
References:
- Bergemann, Bonatti, Smolin (2025). The Economics of Large Language Models: Token Allocation, Fine-Tuning, and Optimal Pricing. arXiv: https://arxiv.org/abs/2502.07736
- McKinsey (2025). The state of AI in 2025: Agents, innovation, and transformation
- MarketsandMarkets. AI Agents Market worth $52.62 billion by 2030