The correct answer is option 3, Vijay Kelkar.
Option 1, Vijay Tendulkar, is a renowned playwright and has no relation to the Finance Commission.
Option 2, M. Damodaran, is a prominent Indian bureaucrat and former Chairman of the Securities and Exchange Board of India (SEBI). However, he is not associated with the Finance Commission.
Option 3, Vijay Kelkar, is the correct answer. He served as the Chairman of the 13th Finance Commission. Vijay Kelkar is an Indian economist and a former bureaucrat who has held various important positions in the Government of India.
Option 4, Y.V. Reddy, is a retired Indian Administrative Services (IAS) officer who served as the Governor of the Reserve Bank of India (RBI). Although he has made significant contributions in his field, he is not the Chairman of the 13th Finance Commission.
In conclusion, the correct answer is option 3, Vijay Kelkar, as he served as the Chairman of the 13th Finance Commission.