Here in Longmont, Colorado the birthplace of the F.I.R.E. movement and home to “The Leisure Engineer” himself, Mr. Money Mustache there’s definitely a vibe. A good vibe, but one that’s often misunderstood.
Financial independence and early retirement, as Mr. Mustache will often argue, isn’t just about quitting your job it’s about sustainability. His secret sauce? A deep personal preference (a utility function, if you will) for simplicity, a spouse with similar preferences, and both have extraordinary skillsets that functions as a powerful substitute for wages. Those skillsets are crucial to their ability to maintain such incredibly low levels of consumption—something that most people cannot easily replicate.
But your secret sauce? It might look completely different.
Economists have understood this for decades they call it household equilibrium. And it’s not just about income minus expenses. Psychological and occupational factors play just as critical a role. The way you value work and leisure, your perceived worth in the labor market, and even your intrinsic satisfaction with consumption vs. savings these are the real drivers of financial success.
The Mr. Money Mustache household has optimized its equation. But what does your version look like?
Here’s the Reality
A dual-income household doesn’t just need a budget. It needs a model one that balances work, leisure, consumption, and savings in a way that maximizes total lifetime well-being. That’s not just about spreadsheets; it’s about Lagrangian optimization the real mathematics behind making choices under constraints. I don’t expect you to sit down and solve it, but I do want you to see what’s missing when financial planning is reduced to “spend less, save more.”
There’s so much more to personal finance than simple arithmetic. In fact, if your financial advisor isn’t helping you explore your own household equilibrium, then you aren’t really talking about your personal finances at all. Because there’s nothing personal about taking your bills, subtracting them from your income, and dumping the rest into an index fund.
The Real Math of Household Equilibrium
At equilibrium, a household maximizes its utility U(x1, x2, l) subject to its budget constraint:
Where
x1 ,x2 = quantities of two goods consumed
p1 ,p2 = prices of those goods
T = total available time (24 hours in a day, or a working lifetime)
l = leisure time
w = wage rate (opportunity cost of leisure)
V = non-labor income (investments, inheritance, passive income, government transfers)
This means:
- You allocate income to goods
- You balance time spent working vs. leisure
- You consider opportunity cost (your wage)
To find the optimal allocation, we use the Lagrangian function:
Taking derivatives:
For each good 
For leisure 
For the budget constraint: 
From here, we solve for optimal consumption and labor-leisure trade-offs. The key condition for equilibrium:
This tells us:
How much you should work vs. relax
How much you should spend vs. save
Why you might be overworking or under-saving
This is why budgeting advice from gurus fails. They don’t tell you that the answer isn’t “just spend less.” It’s about your personal utility function. It’s about how you balance work and spending over a lifetime.
I’m not afraid of this math. This is what I do. And if you want real financial advice—not just personal finance soundbites—I’m here for it.
If you skipped right through the above equations, I don’t blame you. All of those variables don’t really mean anything without real values, and I’ll provide those later in a real life example using two separate budget items: food and entertainment. The complexity involved in that simple example will surprise you and should leave you wondering if you have ever truly budgeted properly in your entirely life. If you consider budgeting to be a simple act of subtracting expenses from income, then sure, I guess you could say you have done some budgeting before. However, what you may have never actually attempted is to achieve household equilibrium given your own utility function and that of the other members of your household.
Example: Finding Household Equilibrium with Food & Entertainment
Imagine a dual-income couple trying to decide how to balance their spending on food and entertainment, while also choosing how much to work vs. enjoy leisure time.
This is a simplified example for demonstration purposes only.
Household details:
- They earn $10 per hour and have 16 hours per day available (8 hours sleep is non-negotiable).
- They spend money on food (x1) and entertainment (x2).
- Food costs $5 per unit (p1=5), and entertainment costs $10 per unit (p2=10).
- They have $50 in passive income (maybe rental or investments).
Step 1: Budget Constraint
where:
- T = 16 (total hours available),
- l is leisure hours,
- w = 10 (wage per hour),
- V = 50 (passive income)
If they choose 8 hours of leisure (sleep), their budget becomes:
Step 2: Utility Function (What Makes Them Happy?) They get satisfaction from:
- Food (x1),
- Entertainment (x2),
- Leisure (l).
A simple utility function could be:
This function means:
- They like food and entertainment together (not just one).
- They like leisure, but also need money to afford stuff.
Step 3: Solve for Optimal Spending We set up the Lagrangian:
Taking derivatives and setting equal to zero:
Step 4: Solve for
and
Dividing the first two conditions:
Rearrange:
Now substitute into the budget constraint:
Using:
Final Answer:
- Buy 13 units of food ($65).
- Buy 6.5 units of entertainment ($65).
- Work 8 hours, enjoy 8 hours of leisure.
This balances their budget AND maximizes happiness, NOT just minimizing spending like personal finance gurus say.
Visualizing in 3D
Now let me graph this equilibrium so you can see where it lands!
Here’s the 3D visualization of the household equilibrium for food and entertainment with a fixed 8 hours of leisure.
Key Insights from the Graph:
- The utility surface (viridis color) represents different levels of satisfaction from consuming food (x1) and entertainment (x2).
- The red budget line shows the affordable combinations of food and entertainment, given a wage, prices, and passive income.
- The black dot marks the equilibrium point where the household maximizes utility (13 units of food, 6.5 units of entertainment) while staying within budget.
As financial advisors, we’ve been trained to use complex financial planning software that attempts to calculate something akin to household equilibrium using dozens of inputs in order to more closely model the decision making realities of each client’s household. In reality, what it mostly does is determine the net present value of future expenses like vacations in retirement and then spit out a probability of success tied to a household’s ability to cover those costs. The issue with that probability of success metric is a whole article in itself, but let’s just say it’s often misunderstood.
That said, these programs are still the best tools available for combining the needs and preferences of an entire household. Unfortunately, as with most good things, they’ve been co-opted by large banks and investment firms whose primary goal is sales and asset accumulation. Your parents’ financial advisor (and probably yours) likely lured them in with a free financial plan—one of those bloated, 30-to-50-page monstrosities stuffed with way too much information. They probably get a new one every year, only to either toss it in a drawer or let it gather dust in a binder. Meanwhile, their advisor keeps charging outrageous fees to manage their investments, barely adjusting the portfolio as it grows, yet skimming 1% to 2% annually.
These firms rely on a key human flaw: object impermanence. Once your money is transferred into an investment account, fees are silently deducted from your balance. You won’t see them coming out of your checking account. They won’t hit your credit card. Instead, they’re buried on page 29 of that 50-page report, buried under an obscure label on a spreadsheet designed to be just confusing enough to keep you from asking questions.
But here’s the good news: the financial services world is shifting. Clients especially younger, more informed ones are asking harder questions. They want transparency. They want real value, not just a slick sales pitch. Advisors like myself are responding by flipping the traditional model on its head.
Instead of hiding fees, we charge transparently often flat fees or hourly rates. Instead of using financial planning software as a sales gimmick, we use it to help clients make real, long-term decisions. Instead of silently collecting a cut while the market does its thing, we actively demonstrate our value through tax strategies, investment planning, and estate moves that actually save our clients money over time.
Because financial planning shouldn’t be about extracting value from clients. It should be about creating it.