<p>This looks like a simple first order condition from constrained optimization. If the maximum is interior, i.e. if $t_i&gt;0$, 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 $&gt;0$, 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.</p>

CC BY-SA 3.0
2015-02-25T02:39:51.850Z
2015-02-25T02:39:51.850Z
Tilman Borgers
<p>This question is related to the development of <strong>The Joint Giving Theorem</strong> (by S. Kolm).</p>

<p>There are two types of agents: benevolent and beneficiaries.</p>

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

<p>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.</p>

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

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

<p>The theorem then says (I quote):</p>

<blockquote>
  <p>Pareto efficiency for this society of potential givers and receivers implies that there
  exist coefficients $\lambda_i &gt;0$ 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 $t_i$ :</p>
  
  <p>$\lambda_i \cdot (-u^i_{x_i}+u^i_{x}+u^i_{c_i})+\sum_{j\neq i} \lambda_j \cdot (u^j_{x}+u^j_{c_i})+u'\leq 0$</p>
  
  <p>with $=0$ if $t_i&gt;0$ and $\leq 0$ if $t_i=0$.</p>
</blockquote>

<p>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 <strong>zero</strong>?</p>

<mathematical-economics><social-welfare>
Why is the derivative of this SWF non-positive?
Ayar ʕʘ̅͜ʘ̅ʔ
# Why is the derivative of this SWF non-positive?

(Asked by: username_0 on 2015-02-24)

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**?

**Tags:** mathematical-economics, social-welfare


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## 1 answers

### Answer by username_1 on 2015-02-25. Score: 6

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.


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