# Why is the derivative of this SWF non-positive?

This question is related to the development of The Joint Giving Theorem (by S. Kolm).

There are two types of agents: benevolent and beneficiaries.

Benevolents' preferences are represented by utilities: $u^i=u^i(x_i,x,c_i,g_i,c_{-i},g_{-i})$

Where $x_i=X_i-g_i-t_i$ is final wealth, $X_i$ initial wealth, $g_i$ some private gift made to the poor, $t_i$ is the transfer to the public sector (who then gives it to the poor). $x$ is the final wealth of the beneficiaries. $c_i=g_i+t_i$ is total contribution of agent $i$. Subscripts $~_{-i}$ denote variables of other benevolents.

Beneficiary's preferences are represented by an increasing ordinal utility function $u=u(x)$.

The assumptions are (subscripts mean derivatives): $u^i_{x_i}>0,u^i_{x}\geq 0, u^i_{c_i}\geq 0, u^i_{g_i}\geq u^i_{c_j} \leq 0$

The theorem then says (I quote):

Pareto efficiency for this society of potential givers and receivers implies that there exist coefficients $\lambda_i >0$ such that $U=\sum \lambda_j u^j +u$ is maximal (without loss of generality). Public policy chooses taxes $t_i$. When it implements a Pareto efficient social state, this choice maximizes such a function $U$. This implies, for tax $t_i$ :

$\lambda_i \cdot (-u^i_{x_i}+u^i_{x}+u^i_{c_i})+\sum_{j\neq i} \lambda_j \cdot (u^j_{x}+u^j_{c_i})+u'\leq 0$

with $=0$ if $t_i>0$ and $\leq 0$ if $t_i=0$.

My question is: Why is the derivative of $U$ with respect to taxes $t$ non positive? More precisely, why is it non-positive if taxes are zero?

This looks like a simple first order condition from constrained optimization. If the maximum is interior, i.e. if $t_i>0$, then the first derivative must be zero. If the maximum is on the boundary, i.e. if $t_i=0$, then the first derivative must be $\leq 0$. If it were $>0$, then we can't have an optimum, because then the value of $U$ could be raised by increasing $t_i$. It can be negative, though, assuming that $t_i$ cannot be negative.