My setup (before and after a GPU upgrade):
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Before: AMD Phenom II X6 1055T, 2.8 GHz (up to 3.3 GHz turbo) 6-core processor, 8gb DDR3-1333 RAM, AMD Radeon HD6570 1 GB GPU. Ran base Elite just fine on medium/high settings, but barely. During Horizons Beta I had tons of lag around planets and got the BSOD twice when either landing and/or during orbital cruise around a planet.
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Now: All the same, except I updated the GPU to a 4GB ASUS R9 380 Strix GPU. Everything runs smooth as butter now, and even fixed issues I had previously chalked up to internet (i.e. really long transitions from SC to regular space on station approach, and long loading times when accessing starport services). EDIT: I also now run all settings on high/ultra, and while I do not have a FPS meter I have not noticed any lag issues.
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That being said, the minimum card mentioned by FDEV to run Horizons (I don't recall the model number off hand) was priced at about $80 USD at the store I bought my new GPU from (Fry's Electronics in California, USA). For reference the R9 380 I purchased was $239 USD plus I had to upgrade my PSU to handle it. So it shouldn't be too much of an issue to upgrade to at least minimum spec GPU (just make sure your current PSU can handle the card).