Arrow’s Impossibility & Validity of Social Welfare and Pareto Improvement Analysis Arrow’s Impossibility states it is impossible to formulate a social ordering (Social Welfare Function) without violating some desirable conditions ( one of the “No dictatorship”, “ Pareto Efficiency”,  “Independence of Irrelevant Alternatives”, “Unrestricted Domain” and “Social Ordering”). However, there are many (prestigious) academic papers assuming some (utilitarian) Social Welfare Function,the most popular Welfare Function in related literature, to draw some conclusions, For example “Redistribution through Markets” and &quot;Monetary Policy with Heterogeneous Agents:
Insights from TANK models&quot;.
I'm wondering, having Arrow impossibility in mind (There is no valid Social Welfare Function), how one can wholeheartedly believe social welfare improvement claims? and why famous economists and Journal referees think that such analysis are justifiable?
Would you mind helping me? Maybe I did not get implications of Arrow impossibility Theorem.
Let $X$ be the nonempty set of alternatives, $\mathcal{P}_X$ the set of preference relations on $X$ and $N=\{1,\ldots,n\}$ a finite set of agents. Under the universal domain condition, Arrow's theorem concerns functions from the set $\mathcal{P}_X^N=\underbrace{\mathcal{P}_X\times \mathcal{P}_X\cdots\times\mathcal{P}_X}_{n\text{ times}}$ to $\mathcal{P}_X$. Utilitarianism does not just depend on preferences alone, and can therefore not be formulated in this setting.
Still, one can formulate Arrow's theorem in terms of utility functions. So let $\mathcal{U}_X$ be the space of real-valued functions on $X$, interpreted as &quot;utility functions.&quot; I'll explain the scare-quotes later. We are now looking at functions from $\mathcal{U}_X^N$ to $\mathcal{P}_X$. Since utility functions determine preferences, we can do now the same things we could do before and more.
Now there are two ways to formulate Arrow's condition of independence of irrelevant alternatives in the new setting; adapting the other conditions is straightforward.
The first form is:
IIA1: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA1 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)\geq u_i(y)$ if and only if $u_i'(x)\geq u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
Note that IIA1 has implicitly two components. The first says that the social ranking of two alternatives depends only on the utility values for these two alternatives. The second says that only the preference ranking matters, but not the &quot;intensity.&quot; If we only keep the first, we get
IIA2: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA2 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)-u_i(y)=u_i'(x)-u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
IIA2 is actually weaker than requiring that the utility for both alternatives are the same in both profiles; only the differences need to be the same.
Now, with the other axioms suitably reformulated, IIA1 implies that all such functions $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ must be dictatorial, while IIA2 is compatible with utilitarianism.
Now, so far this was a purely mathematical issue. In particular, we treated utility functions as being somehow objective things and not just representations of preferences. They also measure well-being in a way that is comparable across agents. So utilitarianism uses information that is not included in purely positive models of economic behavior, in which only preferences matter. Now, one can discuss how sensible it is to compare the well-being of different agents and what information one needs for certain SWFs. There has been a research program starting in the late 1970s of discussing the exact informational assumptions one uses for such comparisons of well-being. A nice introduction to the topic is given in the 1998 book &quot;Theories of distributive justice&quot; by John E. Roemer. Roemer directly relates all this to Arrow's theorem, so his book will give you a very extensive answer.