I doubt there will be a significant cost difference for the birdcage if the lanai was enclosed, however, from what I have seen most enclosures result in at least 1/3 reduction in the area this is open. Also, most enclosures seem to have a low wall, not sure if you can get an enclosure with the glass sliders all the way to the floor. The reason is that the lanai floors are NOT level as they are designed with a slope to drain rain water. IMHO, be sure you discuss this with a contractor before you make any decisions.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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