Inter-jurisdictional Spillovers:

It has already been noted that there may be external benefits for neighbouring localities as a result of any one jurisdiction's  expenditures. The fact that the spillovers are external benefits implies that the local authority responsible for such activity takes no account of it in its decision-making. If region R undertakes programmes that create spillovers for region S, then, the optimal decision is to set the provision of that good at the quantity where

MB_r  + MB_s  = MC

Where MB_r  = the marginal benefit of residents of R

MB_s  = the marginal benefit of residents of S

MC = the marginal cost of the programme

However, if the local authority does not include the spillover benefits in its decision- making calculations, it will provide only to the point where

MB_r  = MC

In such circumstances a subsidy granted at the per-unit rate of MB_s  / (MB_r  + MB_s ) would lead the authority to the optimal point.  The maraginal cost of expansion to R would then be set equal to MC[1 - MB_s  / (MB_r  + MB_s )]  and the authority would increase its  provision of this good. Matching grants are better in this context because, in altering the price, they stimulate a greater provision of the good at a lower cost to the grant donor. The fact that the donor is better off, because the  recipient demands  more of the  good, makes a grant that changes prices better than one that simply changes income for the done.