Suppose that we have numerical representations (V1, ... , VI) for the preferences (≻i) of our I consumers, and we are using a weighted utilitarian social welfare functional for some given collection of weights (ai) to make social choices. Show that we can change to the (equal-weighted) utilitarian social welfare functional without changing the choices that are made by changing at the same time the numerical representations of the preferences (≻i).s

Suppose that we have numerical representations (V₁, ... , V_I) for the preferences (≻_i) of our I consumers, and we are using a weighted utilitarian social welfare functional for some given collection of weights (a_i) to make social choices. Show that we can change to the (equal-weighted) utilitarian social welfare functional without changing the choices that are made by changing at the same time the numerical representations of the preferences (≻_i).s