How to set up a Lagrangian for a crypto mining profit maximization problem

Key takeaway: This is a standard constrained optimization setup dressed in a modern story. The economics is simple: choose hours for each activity, respect the time limit, and maximize profit given prices and production functions. The real work is translating the narrative into symbols cleanly.

Let

• $x_1\ge 0$ = hours spent mining Bitcoin,
• $x_2\ge 0$ = hours spent mining Ethereum.

The outputs are given as $\sqrt{x_1}$ and $\sqrt{x_2}$, and the prices are $p_1$ and $p_2$.

1) Write profit

Since electricity and other operating costs are zero, profit equals revenue:

$\pi(x_1,x_2)=p_1\sqrt{x_1}+p_2\sqrt{x_2}.$

2) Write the time constraint

The computers may work for at most 15 hours per day:

$x_1+x_2\le 15.$

At the optimum, because more hours increase output and there is no cost of hours directly, the full time constraint will typically bind:

$x_1+x_2=15.$

3) Form the Lagrangian

Using multiplier $\lambda$ for the constraint,

$\mathcal{L}(x_1,x_2,\lambda)=p_1\sqrt{x_1}+p_2\sqrt{x_2}+\lambda(15-x_1-x_2).$

4) First-order conditions

Differentiate with respect to each choice variable:

$\frac{\partial \mathcal{L}}{\partial x_1}=\frac{p_1}{2\sqrt{x_1}}-\lambda=0,$

$\frac{\partial \mathcal{L}}{\partial x_2}=\frac{p_2}{2\sqrt{x_2}}-\lambda=0,$

$\frac{\partial \mathcal{L}}{\partial \lambda}=15-x_1-x_2=0.$

These conditions characterize the optimum.

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Pitfalls the pros know 👇 A common error is to forget the constraint is an inequality and then write a Lagrangian without checking whether it binds. Here, since output is increasing in both variables, the solution uses all 15 hours. Another trap is putting the prices inside the square root or treating output as cost. The problem asks for profit maximization, so revenue goes in the objective and the zero cost assumption keeps the setup clean.

What if the problem changes? If the firm had a fixed hourly cost $c$ for using the computers, the objective would become $p_1\sqrt{x_1}+p_2\sqrt{x_2}-c(x_1+x_2)$. If the 15-hour limit were replaced by $x_1+x_2\le T$, the same Lagrangian form would apply with $T$ in place of 15. If short-selling or negative hours were allowed, which they are not here, the problem would no longer make economic sense.

Tags: Lagrangian, constrained optimization, profit maximization