Begun in 1965, federal transfers to states to assist in providing health care for poor. The federal aid is a matching grant(what is the economic

Begun in 1965, federal transfers to states to assist in providing health care for poor. The federal aid is a matching grant(what is the economic rationale for that?)

Aj = mj Ej

Where Aj is aid per capita and Ej is expenditure per capita and

mj*=(1 - 0.45 (Vj2/Vnat2))

Vj= per capita income for the jth state

Vnat = per capita income for the nation

With

mj = 0.5 if mj*< 0.5

mj = mj* if 0.5< mj*< 0.83

mj = 0.83 if mj* > 0.83

What is missing in this formula?

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Devise a formula that is less blunt. What are the pros and cons of your formula.