The only rules for the remaining $2,000 are that (1) the money must be spent only for the Medicaid beneficiary and (2) it may not exceed $2,000. However, with respect to the second rule, it may exceed $2,000 temporarily because income is not treated as an asset until the following month. For example, suppose a Medicaid beneficiary has $1,500 in the bank and on June 3rd she receives a Social Security payment of $800, bringing the bank balance up to $2,300. This is not a problem as long as the money is spent down so that by June 30th the balance is back under $2,000.
For information about Medicaid's asset rules, go here: https://www.elderlawanswers.com/medicaids-asset-rules-12016.