# Finding Pareto efficiency with negative externalities

Student S has one hobby of listening to music. The noise of the speaker can produce noise up to 100 decibels (measured by D). Her money is measured by Ms . Her utility function is Us = 10D1/2 + Ms

Student F has a utility function of UF = 10(100-D)1/2 + MF . MF is Student F’s money and D is the decibels of Student S’s speaker.

What is a Pareto-Efficient Level? (We also learnt in the previous part that Student S has no money, we do not know about Student F)

I am aware that Pareto-Efficiency means we can’t make one student better off without making another student worse off, but since one likes loud music and one does not like any music. Isn’t any level of noise Pareto-Efficient?

The common method, as I'm sure you have learnt, is the comparison of the two persons' Marginal Rates of Substitution, in your example: $$MRS_F = MRS_S$$, as given by the Edgeworth Box: