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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

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    $\begingroup$ Your question is not that clear. you ask why the utility of agents is maximised under a budget constraint ? $\endgroup$ – optimal control Dec 26 '17 at 9:35
  • $\begingroup$ @optimalcontrol See edits. $\endgroup$ – EconJohn Dec 26 '17 at 18:59
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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 $\max_x\pi(x)$.

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    $\begingroup$ It is perhaps worth pointing out that in the special case when $u$ is quasi-linear and it is linear in income spent on other goods then $\lambda = 1$. $\endgroup$ – Giskard Dec 26 '17 at 15:54
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    $\begingroup$ Strictly speaking neither problem is unconstrained, usually there are non-negativity constraints on $x$. $\endgroup$ – Giskard Dec 26 '17 at 15:54
  • $\begingroup$ Sorry, I was asking with regards to a function that does not utilize the Lagrangian. $\endgroup$ – EconJohn Dec 26 '17 at 18:56
  • $\begingroup$ @EconJohn: What's so special about the Lagrangean that's different from the "profit style" objective function? The Lagrangean can be decomposed into a "revenues" part and a "cost" part, just like the profit function. The method of solution is the same (taking FOCs and setting them equal to zero). $\endgroup$ – Herr K. Dec 27 '17 at 6:45
  • $\begingroup$ @HerrK. my issue is that we have to carry around an extra term $\lambda$. I know it eventually cancels out if the calculations are followed through properly. However since $\lambda$ has its own interpretation in consumer theory (i.e. marginal utility from income), we are adding an additional component that your profit function does not account for. $\endgroup$ – EconJohn Dec 27 '17 at 14:25
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Upon reviewing my copy of Microeconomic Analysis I found that he mentions an unconstrained optimization problem (more specifically a formula with the constraint included)faced by the consumer in the context of welfare analysis.

... Let $CS(x)=u(x)-px$ be the consumer's surplus associated with a given level of output; this measure the difference between the "total benefits" from the consumption of the x-good and the expenditure on x-good. Similarly let $PS(x)=px-c(x)$ be the profits or producer surplus eared by the representative firm.
  Then the maximization of total surplus intails: $$\max_x \ CS(x)+PS(x)=[u(x)-px]+[px-c(x)]$$ or $$\max_x \ u(x)-c(x)$$

based on this excerpt, I think its safe to say the consumers "profit function" is consumer surplus maximization problem as defined by:

$$\max_x \ CS(x)=u(x)-px$$

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