For quantification of the pure energy cost of additional weight, I believe that effort is directly proportional to total weight of the loaded hiker, at least for loads that aren't too large (whatever that means).

I'm pretty geeky even for an engineer so tested this hypothesis a while ago with a light running pack and a heart rate monitor. I jogged the same short loop four times in succession, alternating between not carrying and carrying a small pack weighing 10% of my body weight. I don't have the numbers any more but the indicated calories for the runs were something like 200,225,210,235. This suggested to me a 10% cost for the 10% extra weight coupled with a fatigue factor indicating decreasing efficiency with time. A more applicable test for hikers would be to do one walk per day with or without a back and maybe use a more realistic backpacking load at hiking speed.

Physics-wise, walking and running are both controlled falling: you lift your body+load a few inches, lean forward, and let yourself fall back down that few inches while you move a foot forward and catch yourself. The energy cost to lift a known mass against gravity is pretty straightforward and dominates the total energy cost when using a normal gait. I believe that the DIFFERENTIAL energy cost of carrying additional weight has very little dependence on speed or incline within some "reasonable" range. This simplification probably breaks down with heavy loads, though, because there's additional cost in controlling the "fall" phase and excess weight throws off biomechanical efficiency.

As someone else pointed out, in the real world things like knee pain are the real limiting factors for big loads. Still, I bet that the weight percentage effort simplification is pretty accurate in predicting, say, the cost in miles/day of carrying a few extra pounds of gear over the same route. By this reasoning, a 100-pound hiker that's used to carrying a 20-pound pack 20 miles/day would only make it 18 miles with the same effort if she had to carry an extra 12 pounds of food. Alternately, she'd cover 24 files if she slack-packed. It also means that that 2-pound luxury item you're considering costs about a third of a mile or about 10 minutes of hiking when you think of it in terms of a percentage of body+pack weight.

Thoughts?