Nash social welfare function with negative exponential utilities I got one question about the Nash-SWF. Typically it is defined as the product of individual utilities, ie. 
$$
NSWF:=u_1(x_1) \cdot u_2(x_2) \cdot u_3(x_3) \cdot ...
$$
For this to make sense, individual utilities are restricted to always being positive. Is there a way to adjust the Nash-SWF to work for utility fcts that are always negative, like $-e^{-ax}$? Meaning all individuals have the same utility fct. which is $-e^{-ax}$.

Thanks a lot!
I completely agree with @FooBar. Of course the utility function can be negative in some of its range. Conceptually, one could invoke the cardinality vs. the ordinality argument as a response to your question. I would think that your problem is as follows:

$$
\
max\, SWF=u(x_{1})\times u(x_{2})\times...\times u(x_{N})
\
 s.t
\
f(x_{1}....x_{N})\leq g
\
$$

The constraint(s) can be some resource constraint. All that matters here is that the SWF satisfies the normal properties associated with the existence of a maximum on the constraint set (You can invoke the assumptions of the Weierstrass Theorem here). Moreover, you should check Second Order Sufficient Conditions (in this case the negative definiteness of the Bordered Hessian) for a maximum. The value of the welfare function by itself is irrelevant as long as it is a maximum on the constraint set.