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