Unconstrained Optimization:Why is there no "profit style" function in consumer theory? When being exposed to your equations for profit maximization you have an equation of:

$$\pi=pf(x)-c(x)$$

with this you can solve for the optimal input(s) $x$ from the first order conditions and some basic algebra.

I was wondering why in consumer theory we don't solve for the bundle that maximizes consumer surplus using a similar type of formula.

The motive behind this question is that if such a formula exists it would simplify the calculations for utility maximization in consumer theory. The profit function is convenient because it incorporates the constraint, however we can't seem to do this in consumer theory.

Why does such a function not exist?

Edit: I'm interested in understanding why we don't have a function which circumvents the use of the Lagrangian in consumer theory. I.e. Why don't we solve to maximize consumer surplus
The "profit" function for consumer choice does exist; it is the Lagrangean: 
\begin{equation}
\mathcal L(x;\lambda)=u(x)+\lambda[c-g(x)]\,,
\end{equation}
where utility $u(x)$ is to be maximized subject to budget constraint $c\ge g(x)$.

Note that $\max_x \mathcal L(x;\lambda)$ is an unconstrained optimization problem just like $\max_x\pi(x)$.