Managing finances in a startup is quite a tricky business. The concept of burn rate, or how quickly you spend your funds available, is a key aspect of success for startups. In the initial stages of running a startup, stakeholders keep a close eye on how fast or how slow funds are spent.

As engineers, there is often an attraction among us to doing things mathematically. This post is a crude attempt to define the burn rate in mathematical terms, and suggest ways to optimise burn rate in a rather mathematical fashion.

The first part of this challenge is establishing the Mathematical Model, that defines our problem at hand.

To develop our mathematical model, we consider a startup that receives an angel fund (A). The fund is allocated to various expense categories, including Marketing (M), Salaries (S), setting up Office and Furniture (F), and Licenses (L) for software and packages. The burn rate (Burn) can be expressed as Burn = M + S + F + L. This burn rate represents the monthly expenditure that is subtracted from the Angel Fund (A).

Once the mathematical models are established, we would need certain Optimization Strategies. The following are some paths that we could take:

a. Least Expense: Minimizing the burn rate allows startups to extend their runway and maximize the time they can operate before securing additional funding. To achieve this, the objective is to minimize Burn while still meeting necessary operational requirements.

b. Most Marketing and Most Burn: In certain cases, a startup may prioritize aggressive marketing to gain market share or achieve rapid growth. This optimization strategy aims to maximize Burn by allocating a larger portion of funds to Marketing (M), while keeping the other components (S, F, L) within acceptable limits.

c. Best Office and Licenses with Optimal Burn: For startups focused on building a strong infrastructure and acquiring essential software licenses, this strategy aims to optimize Burn while ensuring the maximum allocation to Office and Furniture (F) and Licenses (L). The objective is to strike a balance between these expenses and the remaining components (M, S) to maintain a sustainable burn rate.

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After we decide, what are we optimizing for, the real fun starts. We would need to find out methods to set up our relations, equations, and then start to solve them and weigh them.

Since, there are many moving parts in our problem, with funds that seem to change shape over time, we will opt for Differential Equations for Optimization. Differential equations help us to put the rate of change of each portion of fund over time, and then help us to develop some relationships among our variables.

Differential equations can be employed to optimize burn rate and the allocation of funds among different components. By formulating appropriate constraints and objective functions, we can use mathematical optimization techniques to find the optimal values for each variable. Techniques like gradient descent or mathematical programming can be explored to solve these equations and determine the ideal allocation.

Let us work this out with a practical example:

Let us consider a scenario with $1 million in Angel funds. We can start by setting the limits for each component based on industry standards or specific requirements. For instance, we might allocate 30% to Marketing (M), 40% to Salaries (S), 20% to Office and Furniture (F), and 10% to Licenses (L).

Using differential equations, we can then optimize these allocations while considering the objective of the chosen strategy (least expense, most marketing and burn, or best office and licenses). By solving the equations, we will obtain the optimal values for each component that will result in an optimized burn rate.

# Mathematical Thinking

To optimize the burn rate and allocation of funds among the different components (M, S, F, L), we can formulate a system of differential equations. Let’s denote the allocations as follows:

The angel fund remaining at time t will be denoted as A(t). We can express the burn rate (Burn) as:

Burn(t) = M(t) + S(t) + F(t) + L(t)

To optimize the burn rate, we need to consider the constraints and objectives of each optimization strategy. Let’s explore the differential equations for each strategy:

- Least Expense Strategy: Objective: Minimize Burn while meeting necessary operational requirements.

The differential equations for this strategy can be formulated as follows:

These equations indicate that the allocations for each component remain constant over time, resulting in the least expense scenario.

- Most Marketing and Most Burn Strategy: Objective: Maximize Burn by allocating a larger portion of funds to Marketing (M).

The differential equations for this strategy can be formulated as follows:

In this case, the allocation rates for each component (M, S, F, L) are proportional to the remaining angel funds A(t) with coefficients k1, k2, k3, and k4 determining the relative weights.

- Best Office and Licenses with Optimal Burn Strategy: Objective: Optimize Burn while maximizing the allocation to Office and Furniture (F) and Licenses (L).

The differential equations for this strategy can be formulated as follows:

Here, the allocation rates for Marketing (M) and Salaries (S) are proportional to the current burn rate (Burn), while the allocation rates for Office and Furniture (F) and Licenses (L) are proportional to the remaining funds after deducting the burn rate (A(t) – Burn(t)).

By solving these systems of differential equations using appropriate mathematical techniques such as gradient descent or numerical methods, the optimal values for each component allocation can be obtained.

Note: The coefficients k1, k2, k3, and k4 need to be determined based on the specific objectives and constraints of the startup. Adjusting these coefficients will affect the relative importance and allocation of funds to each component.

## Solving Strategies

There are several numerical methods that can be employed to solve the system of differential equations and obtain optimal solutions for the allocation of funds. Here are a few commonly used strategies:

Euler’s Method: Euler’s method is a simple numerical method for solving ordinary differential equations. It approximates the derivatives by using finite differences and iteratively updates the values of the variables. Although Euler’s method may not be as accurate as other methods, it provides a basic approach for solving the equations and obtaining approximate solutions

Runge-Kutta Methods: Runge-Kutta methods are a family of numerical methods that provide more accurate solutions compared to Euler’s method. The most commonly used variant is the fourth-order Runge-Kutta method (RK4), which involves multiple iterations to refine the estimates of the variables. RK4 is known for its good balance between accuracy and computational efficiency

Finite Difference Methods: Finite difference methods discretize the continuous differential equations by approximating the derivatives with finite difference formulas. This approach converts the differential equations into a system of algebraic equations, which can then be solved using numerical linear algebra techniques such as Gaussian elimination or iterative methods like the Jacobi method or the Gauss-Seidel method

Optimization Algorithms: Instead of directly solving the differential equations, optimization algorithms can be employed to find the optimal values for the variables that satisfy the given objectives and constraints. Techniques like gradient descent, genetic algorithms, or particle swarm optimization can be used to iteratively search for the optimal solution by adjusting the variable values based on their impact on the objective function.

When applying these numerical methods, it is important to carefully define the initial conditions, constraints, and objective functions specific to the startup’s requirements. The choice of the most suitable method depends on the complexity of the system and the desired level of accuracy. It is often helpful to experiment with different methods and compare their results to ensure the reliability of the solutions obtained.

The creation and application of a mathematical model for startup economics can provide several benefits and help founders optimize their operations in the following ways:

Financial Planning: A mathematical model allows founders to have a systematic approach to financial planning. By considering various factors such as expenses, burn rate, and revenue, founders can gain insights into the financial health of their startup. This helps in setting realistic financial goals, estimating runway, and making informed decisions regarding fund allocation

Resource Allocation: The model enables founders to optimize the allocation of resources, such as funds, human capital, and infrastructure. By analyzing the impact of different allocations on the burn rate, founders can make data-driven decisions regarding marketing budgets, salary structures, office investments, and software licenses. This helps in utilizing resources efficiently and maximizing the value generated from each investment

Risk Management: By simulating different scenarios and analyzing the outcomes using the mathematical model, founders can identify potential risks and mitigate them proactively. The model allows for sensitivity analysis, enabling founders to assess the impact of changes in variables and external factors on the burn rate and overall financial stability. This helps in identifying potential bottlenecks, optimizing risk mitigation strategies, and ensuring long-term sustainability

Strategic Decision-Making: Mathematical thinking and optimization techniques provide a structured framework for strategic decision-making. By formulating and solving optimization problems, founders can identify the most favorable strategies for their startup based on specific objectives, constraints, and market conditions. This includes determining the optimal mix of expenses, prioritizing marketing efforts, and aligning resource allocation with the startup’s growth plans

Scalability and Growth: The mathematical model can be utilized to assess the scalability and growth potential of the startup. By incorporating growth assumptions and analyzing the resulting financial projections, founders can make informed decisions about when and how to scale their operations. The model helps in understanding the financial implications of expansion and optimizing growth strategies to achieve sustainable and profitable growth.

Overall, the mathematical model provides a quantitative framework that enhances the decision-making process for founders. It helps in reducing uncertainty, increasing financial discipline, and aligning actions with long-term objectives. By leveraging mathematical thinking and optimization, founders can optimize their startup’s performance, improve financial stability, and increase the likelihood of success in a competitive business environment.

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