Showing the Utility Possibilities Frontier is downward sloping The slope of the UPF must be equal to the slope of the MRT and the MRS. Is it therefore enough to show that the slope of the MRS is negative?
The first statement does not seem to be true. The UPF exists in the space of utilities whereas the other curves (production, indifference) exist in the product space. As a result, the slope of the UPF may change if a monoton transformation is applied to one of the utility functions whereas the other curves and their slope will definitely not change.