Modeling Revenue and Cost: Gradient Series Applications for Tech Products

Hello! Welcome back to your course on Engineering Economics.

In our last few lessons, you've built a solid foundation by learning to work with uniform series, arithmetic gradients (linear change), and geometric gradients (percentage change). We've focused on the mechanics of calculating their present values.

Today, we transition from mechanics to application. Your goal for this lesson is to model revenue growth and cost escalation scenarios for technology products using gradient series. We'll take the tools you've learned and apply them to build realistic financial models, which is central to evaluating the kinds of technology projects and business investments you manage. This lesson directly addresses your interest in applying these concepts to software development and business decisions.

Recap: Choosing the Right Gradient

Before we build models, let's quickly recall the difference between our two types of gradients:

Arithmetic Gradient: Cash flows change by a constant amount ( $G$ ) each period.
- Think: A service contract for your office that increases by a flat $100 each year.
Geometric Gradient: Cash flows change by a constant percentage ( $g$ ) each period.
- Think: Revenue from a SaaS product growing by 5% month-over-month as your user base expands.

The key to modeling is choosing the gradient that best represents the real-world behavior you're analyzing.

Modeling Revenue Growth with Geometric Gradients

In the tech industry, revenue rarely grows linearly. It's often driven by user adoption, network effects, or compounding marketing efforts, all of which are better modeled by percentage growth. This makes the geometric gradient our primary tool for modeling revenue.

Let's imagine you are launching a new API service. You might project:

First-year revenue ( $A_{1}$ ): $50,000, based on an initial set of clients.
Growth rate ( $g$ ): 15% per year, as word spreads and more developers integrate the API.
Analysis period ( $n$ ): 5 years.
Interest rate ( $i$ ): 12% (your company's MARR).

To find the present worth of this revenue stream, you would use the geometric gradient formula, $P_{g} = A_{1} [\frac{1 - ( 1 + g ) ^{n} ( 1 + i ) ^{- n}}{i - g}]$ , because the growth is based on a percentage.

To see this in action, let's watch a worked example that is very similar to this scenario.

Geometric Gradient - Engineering Economics Lightboard

The 'Engineering Economics Guy' video we used last time contains an excellent example of calculating the present value of a geometrically growing revenue stream. Let's focus on that specific problem now.

Please watch from 08:56 to 14:35. Pay attention to how the monthly revenue, monthly growth rate, and monthly interest rate are identified and used together to find the present value of the entire revenue stream.

This example of modeling monthly revenue is directly applicable to many software businesses you've managed, where Monthly Recurring Revenue (MRR) is a key metric and month-over-month growth ( $g$ ) is the target.

Modeling Cost Escalation

Costs can behave in different ways, so you might use either type of gradient.

Scenario 1: Linearly Escalating Costs (Arithmetic Gradient)

Sometimes, costs increase by a predictable, fixed amount. This is common with multi-year service or maintenance contracts where the provider specifies a fixed annual price hike.

For example, imagine you purchase a new server for your on-premise infrastructure.

First-year maintenance cost: $2,000.
Annual increase: The contract states the cost will increase by $250 each year. This is your arithmetic gradient, $G$ .

To find the present worth of this cost stream, you would decompose it into two parts:

A uniform series (annuity) representing the base cost of $2,000 per year.
An arithmetic gradient series of $G = $250$ , starting in year 2.

The following video provides a perfect, step-by-step walkthrough of this exact type of problem.

Arithmetic Gradient - Engineering Economics Lightboard

This video from the 'Engineering Economics Guy' breaks down an arithmetic gradient problem involving car maintenance costs. The logic is identical to modeling a server maintenance contract.

Watch from the beginning to 11:37. Focus on how the cash flow is broken down into a base annuity (A) and a pure gradient (G), and how the present worth of each part is calculated and then summed.

Scenario 2: Exponentially Escalating Costs (Geometric Gradient)

Other costs, especially those tied to inflation or usage, are better modeled with a geometric gradient. For example:

Cloud computing costs might increase by a percentage ( $g$ ) that's tied to your user growth.
Employee salaries or rental costs might be projected to increase by a certain percentage each year to account for inflation.

In these cases, you would use the same geometric gradient formula we used for revenue, but the cash flows would be negative (costs).

The textbook below provides detailed examples for both arithmetic and geometric gradients that you can refer to for further study.

Chapter 1

This chapter from an engineering economy textbook covers both types of gradients with worked examples. It's a useful reference for the concepts we're applying today.

You don't need to read this now, but you can bookmark sections 3 ('Arithmetic Gradient Factors') and 4 ('Geometric Gradient Series Factors') for future reference. Example 2-4 models revenue growth, and Example 2-5 models escalating costs.

Test your understanding!

You are developing a new software platform. You need to decide whether to build a custom logging component or use a third-party service.

The third-party service has the following cost structure over the next 4 years:

Year 1 Cost: $10,000
Annual Cost Increase: The provider has announced a fixed price increase of $2,000 per year.
Your company's MARR: 10%

What is the Present Worth (PW) of the cost of this service over the 4-year period? Which type of gradient applies here?

Show answer

This is an arithmetic gradient problem because the cost increases by a constant amount ($2,000) each year.

We break the problem into two parts:

A uniform series (annuity) of $A = $10, 000$ for 4 years.
An arithmetic gradient of $G = $2, 000$ for 4 years.

The total present worth ( $P_{T}$ ) is the sum of the present worth of the annuity ( $P_{A}$ ) and the present worth of the gradient ( $P_{G}$ ).

$P_{T} = P_{A} + P_{G}$
$P_{T} = A (P / A, i, n) + G (P / G, i, n)$

Using $i = 10%$ and $n = 4$ :

From interest tables, $(P / A, 10%, 4) = 3.1699$
From interest tables, $(P / G, 10%, 4) = 4.3781$

Now, plug in the values:
$P_{T} = $10, 000 (3.1699) + $2, 000 (4.3781)$
$P_{T} = $31, 699 + $8, 756.20$
$P_{T} = $40, 455.20$

The present worth of the cost for this third-party service over four years is $40,455.20. You would compare this value to the present worth of the cost of building and maintaining your own custom component to make a decision.

Conclusion

Today, you've taken the theoretical concepts of gradients and put them to practical use, learning how to select the right tool to model different financial scenarios in a technology context.

Key Takeaways:
- Geometric gradients are your go-to for modeling scenarios with percentage-based changes, like revenue growth from a scaling product or inflation-adjusted costs.
- Arithmetic gradients are best for modeling scenarios with fixed-amount changes, like a service contract with a pre-defined annual cost increase.
- To analyze a project, you often need to decompose its cash flows into different components (e.g., a base annuity and a gradient) and find the total present worth by summing the present worth of each component.

Preview of the Next Lesson

We've now seen how to model revenue and cost streams individually. The next logical step is to combine them into a single, comprehensive analysis. In our next lesson, we will focus on how to evaluate cash flow streams composed of combined uniform and gradient series. This will enable you to analyze more complex and realistic projects that involve multiple types of simultaneous cash flows.

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