Why is incidence not included in social welfare maximization?

Question

I am very confused on why incidence is not included in social welfare maximization of one good. Typically, I see the optimization over price done something like this:

$C$ ~ production cost function
$U_i$ ~ consumer $i$'s utility from consumption of good $d_i$
$d_i$ ~ consumption of user $i$
$p$ ~ price

$\max -C(\sum_i d^*_i) + \sum U_i(d^*_i)$

s.t. $d^*_i = \arg\max{ U_i(d_i) - p\cdot d_i }$

There might be other components but this is the basic framework. What I don't understand is why it does not include incidence as below

$\max -C(\sum_i d^*_i) + \sum U_i(d^*_i) - d^*_i\cdot p$

s.t. $d^*_i = \arg\max{ U_i(d_i) - p\cdot d_i }$

Additionally, if consumers value incidence differently wouldn't this affect the role of incidence in the social welfare maximizers problem? Based on other reading it seems as though there is some equilibrium assumed with all other goods but I don't fully understand this.

I appreciate any help. Happy to clarify if my question is unclear.

Answer 1

If I understand correctly, this is because in standard consumer theory, consumers don't hold money. They just spend all of their budget in consumption good. This fact is already captured by $d$ following your notation.

In other words, money is "indirectly" valued by the indirect utility function $v(p,m)$. Therefore, as long as the shape of your utility function is determined, how you value the "incidence" (using your language) is determined. Technically speaking, the total payment $d_i^*\cdot p \equiv m_i$. In your objective function, you have $$U_i(d_i^*) \equiv v_i(p, d_i^*\cdot p)$$ which already reflect how the consumer values the "incidence."

If consumer's value has been accounted for, then $d_i^*\cdot p$ is simply a transfer from the consumers to the firms, which are, again, owned by the consumers. You are gonna ignore any transfers in the welfare function, to avoid double-counting.