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Does printing a photo at a higher PPI use more or less ink, and why? Can this be calculated or estimated?

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It should neither use more, nor less. Ultimately, any image you print is going to be rasterized at the selected DPI setting in the printer driver. Most printers have a native maximum, such as 2880x1440 on an Epson, or 2400x1200 on a Canon. That is the DOTS Per Inch, which literally refers to the individual ink droplets laid down on the page. You could print a photo at 150ppi, 300ppi, or 600ppi on a Canon, or 180ppi, 360ppi, or 720ppi on an Epson, and the default native DPI will be used for all of them. You WILL likely notice the loss in resolution, but the difference in ink should be otherwise imperceptible.

It is possible to change how much ink is used. The maximum native DPI of a printer is often not the only option. Epson printers usually allow you to print as low as 720x720 dpi, and Canon printers often let you print as low as 600x600 dpi (maybe even less). A lower dpi will usually use less ink.

It is also possible to configure ink density on many of the prosumer and higher end ink jet printers. If you are a stickler for quality, then you will really want to tune your ink density for optimal quality, rather than minimal ink. My recommendation would be to not minimize the amount of ink used, and follow the appropriate instructions to optimize it for proper color reproduction instead.

Finally, if you are interested in calculating the cost of ink on a per-printed sheet basis, Red River paper has some good information:

Red River Paper: Cost of Printing Studies

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I believe you are trying to convince us that there is no diff, but I am trying to understand then, what you mean in paragraph 2 when you say, "A lower dpi will usually use less ink". –  Octopus Jul 24 '13 at 23:26
    
@Octopus Keep in mind that Pixel Per Inch (PPI) and Dots Per Inch (DPI) are not the same thing. –  Bart Arondson Jul 25 '13 at 0:02
    
Awesome - I read your question photo.stackexchange.com/questions/1715/… Since printing at a higher PPI results in more detail along with an increase in minimum possible viewing distance, if you wanted both those nice advantages, why do so many people print at lower PPIs like 240 or 300? When you say 'follow the appropriate instructions' would this be in the manual for my printer? I've got a PIXMA Pro-100 and there's a global density option, but nothing for individual colors that I can find –  DavyCrockett Jul 25 '13 at 0:04
    
Is density something to experiment with or? –  DavyCrockett Jul 25 '13 at 0:12
    
What exactly are your goals? Print quality is achieved via a delicate balance of a number of factors. Mess with one too much, and your print quality will drop... –  jrista Jul 25 '13 at 3:29

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