# 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!

• As you said, it makes sense that the utility functions are defined positive. You have to justify, why it is make sense to you, that you can assume negative individual utilities. But you will have another problem. If the number of individuals are odd, then you will always have a negative NSWF. If the number of individuals are even, then you will always have a positive NSWF. – callculus Oct 6 '15 at 12:09
• There could be negaitve utility functions but in most cases with this kind of function, the marginal utility is positive and decreasing. For exemple ramsey model without discount rate is an exemple. you can find and example by this link : uhero.hawaii.edu/assets/WP_2013-9.pdf (by the way, the paper is published in Resource and Energy Economics, which is a high ranked journal.) – optimal control Oct 6 '15 at 12:24
• I disagree with @calculus: I don't see any particular reason why utility should be positive. In macroeconomics, most utility functions are negative-valued values for a set of (attainable) points. Just think of $log(c)$ – FooBar Oct 6 '15 at 12:29
• @FooBar You are right, it is often used. But the problem of changing sign is still there. – callculus Oct 6 '15 at 12:46
• @calculus which makes the question (on how to solve the problem) all the more interesting! :) – FooBar Oct 6 '15 at 12:50

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