Value Function

I once sat in a pricing meeting where two people looked at the exact same data and walked away with completely different conclusions. The data analyst said, "Price increases linearly with feature count." The economist said, "Price increases with the log of feature count because marginal value diminishes."

They were looking at the same relationship. But they were using different value functions—different mathematical models of how value relates to price.

A value function is the invisible logic underneath every pricing decision you make.

Definition

A Value Function is the mathematical relationship between inputs (features, company size, usage volume, time value) and the economic value a customer receives. It describes how value compounds, diminishes, or scales as customer circumstances change.

In practice, it answers: "If we add one more feature, or one more user, or one more integration, how much additional value does the customer gain?"

The answer determines your pricing model, your expansion strategy, and ultimately your revenue ceiling.

Why Value Functions Matter

Most pricing is intuitive. "We charge $99/month because our competitor charges $99/month." Or "We charge per user because SaaS does." Neither is based on an actual understanding of how value scales.

But value doesn't scale predictably. It follows patterns. Understanding the pattern is the difference between leaving money on the table and capturing it.

Linear Value Functions

The function: Value = a + b(x)

For every additional unit of input, you get a proportional increase in value.

Example: A project management tool charges per user. The 10th user creates approximately the same value as the 1st user (they all get to manage projects, track tasks, etc.). Marginal value stays constant.

If each user creates $50/month in value, the value function for 100 users is $5,000/month in total value. Your pricing should scale linearly (charge per user) because value does.

This is why Per-User Pricing is common in team collaboration tools. The value function is linear.

Logarithmic Value Functions

The function: Value = a + b × ln(x)

For each additional unit of input, the value gain gets smaller. Big differences early, then flattening out.

Example: A data analytics platform adds more integrations. The 1st integration (SQL database) is worth $500/month to you in time saved. The 2nd integration (Salesforce) is worth $400/month (you're already getting insights; this adds some but not transformatively). The 10th integration (obscure CRM) is worth $20/month because you're already getting most insights from other integrations.

Logarithmic value functions mean marginal value diminishes. If you price linearly (same price per integration), you'll be overcharging customers for later integrations and leaving value on the table early on.

The better pricing model: tiered feature bundling (get all integrations for one price) or feature-based pricing where advanced integrations cost more.

Power Law Value Functions

The function: Value = a × x^b

Value scales exponentially based on scale. Small increases create huge value gains.

Example: A network platform (like LinkedIn or Slack) where value increases exponentially with user count. With 10 users, you can reach 10 people. With 100 users, you can reach 100 people. But the value of that network isn't linear—it's exponential because you can create 100-to-100 connections.

Metcalfe's Law states that network value is proportional to the square of the number of users. This is a power law.

For power law value functions, pricing should reflect exponential value creation. As customers grow (more users, more teams), pricing should increase at an accelerating rate, not linearly. This justifies why some networks use Expansion-Led Growth models with aggressive expansion pricing.

Diminishing Return Value Functions

The function: Value = a - b × e^(-cx)

Early inputs create massive value. Later inputs create minimal additional value. Eventually, returns plateau.

Example: A learning management platform for enterprises. The first module (compliance training) saves 20 hours per employee per year. The second module (professional development) saves 15 hours. The tenth module saves 2 hours. After you've optimized the core training workflows, additional modules don't move the needle.

For diminishing return functions, pricing should NOT scale with the number of modules. Better to offer a flat-rate "all modules" plan because the 10th module isn't worth much, and the customer won't pay for it linearly.

Categorical Value Functions

The function: Value jumps at thresholds

Value increases in steps, not continuously.

Example: A compliance tool. At $0 users, it has $0 value (useless). At 50+ users, it becomes valuable (covers regulatory thresholds). At 500+ users, the value jumps again (requires dedicated compliance role). At 5,000+ users, value jumps again (requires compliance infrastructure).

For categorical value functions, pricing should have cliffs. Free tier (under 50 users). Paid tier (50-500 users). Enterprise tier (500+ users). Each tier reflects a different value reality, not a continuous scale.

How to Identify Your Value Function

You won't have perfect data, but you can estimate:

Method 1: Customer Interviews (Qualitative)

Ask 20 customers: "If we added X (feature/user/integration), how much additional value would you gain?"

Listen for patterns. Are they saying "same as before" (linear)? "A bit less than before" (logarithmic)? "A lot more because everything multiplies" (power law)?

Method 2: Usage Data Analysis (Quantitative)

Track outcome metrics (revenue saved, time saved, productivity gain) against input variables (feature count, user count, integration count).

Plot them. Do they line up straight (linear)? Do they curve (logarithmic)? Do they spike then plateau (diminishing)?

Method 3: Competitive Analysis

What do competitors charge as they scale? If they charge linearly per user (Slack), their value function assumption is linear. If they jump prices at thresholds (Salesforce), they assume categorical value.

Aligning Pricing Model to Value Function

This is the critical connection. Your pricing model should reflect your value function. Misalignment leaves money on the table or causes churn.

If your value function is logarithmic but you charge per-feature linearly, you're overcharging customers for advanced features they don't want.

If your value function is power law but you charge flat-rate, you're leaving massive expansion revenue on the table when customers grow.

Real Example: Salesforce's Value Function Strategy

Salesforce recognized that their value function is complex:

1. Linear component: More users = more pipeline visibility = more sales control. Per-user pricing captures this.
2. Power law component: A 50-user Salesforce deployment creates more than 5x value of a 10-user deployment because you can analyze patterns, run complex forecasting, build playbooks. The network of users creates exponential value.
3. Categorical component: There's a threshold where Salesforce shifts from "sales tool" to "organizational infrastructure." Below 50 users, it's a nice-to-have. Above 50 users, it becomes essential.

Salesforce's pricing reflects all three:

• Per-user pricing (linear)
• Expansion-tier premiums (power law—they increase per-user cost as you scale)
• Minimum user counts for Enterprise edition (categorical threshold)

They're not pricing to one value function. They're pricing to the complex reality of how their product creates value.

Value Functions and Freemium Models

Freemium works when your value function has a natural threshold:

• Free tier: Users get basic value (logarithmic tail—additional features barely matter)
• Paid tier: Users unlock the step function (the categorical jump where value multiplies)

Notion's freemium model works because:

• Free tier: Personal note-taking (diminishing returns—you don't need much)
• Paid tier: Team collaboration (power law—value multiplies with more users and shared workspaces)

The free tier keeps them interested. The paid tier captures the value they'd actually miss if they switched.

Value Functions and Expansion Revenue

Understanding your value function is critical for Expansion Revenue:

If value is linear, expansion happens one unit at a time (add a user, charge for a user). Predictable, steady expansion.

If value is power law, expansion happens exponentially. A customer growing from 100 users to 500 users creates much more than 5x additional value. You can justify expansion pricing that's more aggressive than your base price.

If value is logarithmic, customers won't pay for expansion until they've exhausted your core offering. Your expansion strategy should focus on deepening usage of what they already have, not upselling new features.

Strategic Implications

Understanding value functions changes strategy:

1. Pricing transparency: If you understand your value function, you can explain your pricing rationally. "We charge per user because value scales linearly with team size." Customers get it. Negotiations become about value, not price.

2. Product roadmap prioritization: Features that move the value function upward (create power law value) should be prioritized over features that extend the tail (diminishing return value).

3. Segment strategy: Different customer segments have different value functions. A 10-person startup and a 1,000-person enterprise don't have the same value function for your product. Segment your pricing.

4. Expansion planning: If your value function is power law (most SaaS), expansion revenue should be your dominant growth lever. Build the function to capture it.

FAQs

Q: Can one product have multiple value functions?

A: Yes. Salesforce has linear (per-user), power law (team network effects), and categorical (threshold for enterprise use). The art of pricing is recognizing all components and designing a model that captures all of them.

Q: How do I test my value function hypothesis?

A: Run pricing experiments. Charge different prices to different customer segments and track adoption, churn, and expansion. The segment that adopts and expands most heavily is closest to your actual value function.

Q: Does value function change over time?

A: Yes. As competitors enter, value functions shift. As customers mature, their value function for your product might shift. Revisit your assumptions annually.

Q: What if I don't know my value function?

A: Start with qualitative customer interviews. Ask directly: "If we added X, how much additional value?" The patterns you hear will suggest your value function shape. Then validate with data.

Q: Is value function the same as Pricing Psychology?

A: No. Value function is objective (how value actually scales). Pricing psychology is subjective (how customers perceive value). Both matter. Ideally, they align.

Q: How does Willingness to Pay relate to value function?

A: Willingness to pay is how much a customer will actually pay (limited by budget). Value function is how much value they actually receive (unlimited). Pricing strategy bridges the gap: charge enough to capture value, but not so much you exceed willingness to pay.

Q: Can you have a negative value function?

A: In rare cases, yes. If adding more users decreases value per user (chaos, information overload), you have a negative marginal value function. This is rare in SaaS and usually suggests the product doesn't scale well.

Q: How does AI/ML affect value functions?

A: Dramatically. ML products often have power law value functions because insights improve with more data. Pricing models for AI/ML products should reflect this compounding value. Many AI SaaS companies underprice because they don't recognize their power law value function.

Sources & References

See also: Pricing Strategy, Per-User Pricing, Expansion Revenue, Willingness to Pay, Value Proposition Design, Freemium Models, Expansion-Led Growth, Pricing Psychology, Total Cost of Ownership, Feature Adoption

Written by Conan Pesci | April 6, 2026