Question

Consider a social welfare function (SWF) defined as \[ SWF=f(U_1,U_2,\ldots,U_m), \] where \(U_i\) is the utility function of the \(i^{th}\) person, given that there are \(m\) people in the society. Identify which one of the following statements is CORRECT.

• A. Jeremy Bentham and John Stuart Mill proposed \[ SWF=\sum_{i=1}^{m} w_iU_i, \] where \(w_i\) is the weight assigned to individual \(i\) for their unique characteristics and \[ w_1\neq w_2\neq \cdots \neq w_m \]
• B. Since SWF reflects value judgements, interpersonal comparisons can be made on scientific grounds.
• C. John Rawls proposed \[ SWF=\max\{U_1,U_2,\ldots,U_m\} \]
• D. If \[ SWF=\sum_{i=1}^{m}U_i, \] it is referred as the utilitarian function.

Hint:

Utilitarian social welfare maximizes the sum of utilities, whereas Rawlsian welfare focuses on improving the welfare of the least advantaged individual.

Answer 1

The Correct Option is D

Step 1: Recall the welfare function.
A social welfare function adds up the well being of all $m$ people into one score, $SWF=f(U_1,\ldots,U_m)$.

Step 2: Check the Bentham and Mill claim.
Classical utilitarians gave everyone equal weight, so their form is a plain sum, not a weighted sum with unequal weights. So option A is wrong.

Step 3: Check the scientific claim.
Comparing one person's utility with another's needs a value judgement. It cannot be done on purely scientific grounds. So option B is wrong.

Step 4: Check the Rawls claim.
Rawls used the maximin rule, lifting the worst off person. His function is $SWF=\min\{U_1,\ldots,U_m\}$, not the maximum. So option C is wrong.

Step 5: Check the last claim.
The utilitarian function is indeed the simple sum
\[ SWF=\sum_{i=1}^{m}U_i \]
So option D is correct.
\[ \boxed{SWF=\sum_{i=1}^{m}U_i \text{ is utilitarian}} \]