ECON 376

Widely applied GRAVITY MODEL: Reilly’s Law

for estimating the mile radius of the exclusive market area around a place.

Consider places i,j = A,B,C,D and E, with r = location of a customer along the road between places. The distances between these places:

	A	B	C	D	E
POPi	100	80	50	200	75
dAj	0	20	10	30	40

Reilly’s Law: m_i(j) =

POPj/POPA	1	0.8	0.5	2.0	0.75
Ö		0.89	0.7	1.41	.86
+1		1.89	1.7	2.41	1.86
dAj ∙/∙ = mA( j)		10.5	5.9	12.5	21.5

Thus, we can easily calculate the distances from place A, in the directions toward places B,C,D, and E, of the extent of A’s market area. All we need to know are the map distances between the places, and the populations. In the illustration, Place A’s market area is shaded yellow.

Deriving Reilly’s Law from the physical law of gravity:

Fundamentally: The gravitational attraction between two places is directly proportional to their masses and inversely proportional to the (square of the) distance between them:

1 = POP_i/POP_j x (d_rj/d_ri)²

where d_ri+d_rj=d_ij

We can express the distance from a customer at site r to place j (d_rj) as d_rj=d_ij-d_ri

Reilly’s Law will help us find the farthest site r, at distance m from place i at which a customer will shop at place i. So, we want to find the “d_ri*” which is the same thing as the radius of the market area around i, so we can label dij as m _i(j) . Rewriting the fundamental formula using m and d_ij -m in place of d_ri and d_rj, we have:

1 = (POP_i/POP_j)(d_ij –m)/m)²

rearranging this expression to solve for m, we have:

m_i(j) = Done!