# Question about Social Welfare Function and Social Profile

What are the meanings of a social welfare function and social profile? How are they related?

Let $$X$$ be some space of "social outcomes". (Social outcomes could be anything. But if you want a concrete example, then suppose that social outcomes describe allocations of resources to individuals. If there are $$N$$ individuals, and there are $$M$$ distinct types of resources, the an allocation is an ($$N \times M$$)- dimensional vector, so in this case $$X$$ would be some subset of $$\mathbb{R}^{N\times M}$$.) A social welfare function is just a function $$W:X \rightarrow\mathbb{R}$$; heuristically, if $$x$$ is some element of $$X$$ (some "social outcome") then $$W(x)$$ measures the "total welfare" or "social value" or "overall desirability" (or whatever) of $$x$$. Thus, the benevolent social planner should aim for policies that maximize the value of $$W$$.
The key question, of course, is how to define $$W$$. One way to do this is to somehow build $$W$$ out of the preferences or utility functions of individuals. Suppose there are $$N$$ individuals, each of whom has a preference order $$\succeq_i$$ on $$X$$. This collection of preference orders $$(\succeq_1,\succeq_2,\ldots,\succeq_N)$$ is an (ordinal) social profile. (This partly answers your second question.) A big part of the modern theory of social choice and welfare is about defining systematic, principled ways of defining $$W$$ based on the profile $$(\succeq_1,\succeq_2,\ldots,\succeq_N)$$.
Typically, purely ordinal preference information about the individuals is not sufficient. (This is one way of reading Arrow's Impossibility Theorem --I won't get into the details here). So we might want to endow each individual with a utility function $$u_i:X \rightarrow \mathbb{R}$$ (which is more informative than just a preference order). A collection of utility functions $$(u_1,u_2,\ldots,u_N)$$ is a (utility) social profile. (This is the rest of the answer to your second question.) We then seek to construct $$W$$ out of $$(u_1,u_2,\ldots,u_N)$$.
For example, given a social profile of utility functions $$(u_1,u_2,\ldots,u_N)$$, we could define $$W(x):=u_1(x)+u_2(x)+\cdots+u_N(x)$$ for all $$x\in X$$ ---this is the utilitarian social welfare function. Or we could define $$W(x):=\min\{u_1(x),u_2(x),\ldots,u_N(x)\}$$ for all $$x\in X$$ ---this is the egalitarian (or Rawlsian) social welfare function. Or (assuming all utilities are strictly positive), we could define $$W(x):=u_1(x)\cdot u_2(x) \cdots u_N(x)$$ for all $$x\in X$$ ---this is the Nash social welfare function.