Often there are economies of scale for an activity such that one large factory can operate more economically than two factories half its size. This would suggest that there should be just one factory to serve the entire country. Occasionally this is the case, but usually the transportation costs of serving customers from all over the country from one single location outweigh the economies of scale in production. The optimum situation is for the market to be divided up into market areas such that the combined total of production cost and transportation costs are a minimum.
For example, suppose the relationship between average manufacturing cost and the level of output is as shown. This represents "economies of scale" because the higher the level of output the lower is the average manufacturing cost per unit. But manufacturing cost is not the only cost the company needs to consider. It also needs to take into account the cost of delivering the product to the customers. Some of the customers live close to the plant so transportation costs are lower and some live farther away where transportation. Generally the higher the output is, the larger is the market area served and the higher is the average transportation cost per unit. The average manufacturing cost line is downward-sloping to the right and the average transportation cost line is upward-sloping to the right so the average total (manufacturing plus transportation) declines to a minimum and then rise for larger output.
The level of output where average total (manufacturing plus transportation) cost is a minimum is the optimal level of output. Each level of output corresponds to a different market area size so the optimal level of output determines the optimal market area.
Consider a simple case of this problem. In the wheat-growing area of Kansas there is a need for storage facilities. If the area served is uniform and travel is unrestricted then the market areas would be hexagons. If travel is restricted to a rectilinear grid of roads then the market areas will be squares.
It will be more convenient mathematically for a preliminary analysis to take a market area as circular. The mathematical simplification makes it easier to gain insights into the problem.
Let R be the radius of the circular market area. The total area is
R2. If d is the density of quantity demanded per unit
area then the total quantity demanded is dR2. Most of
this demand is for the part of the circle toward the edge rather than
toward the center. The average customer lives two thirds of the way to
the edge of the market area. Therefore the transportation costs go up in
a larger market area not only because there are more customers but also
because the average customer lives farther away from the center.
Production costs are a function of the quantity demanded, let us say
where a is a parameter.
The average production costs, CM(Q)/Q, would then be:
Since Q=dR2,
dR2)2/3.If t is the transportation cost per unit distance, then the total transportion cost is
dR2)*(2R/3)*t
= (2)()td/3)R3.The total cost C is
dR2)2/3 + (2()td/3)R3,
and dividing this by the production dR2 gives the
average cost AC as
dR2)-1/3 or
AC = a(d)-1/3R-2/3
+ (2/3)tR.
If we choose R to minimize average cost the condition that must hold is
d)-1/3R-5/3 The optimum R is then such that
d)-1/3 = R5/3
R = (a/t)3/5(d)-1/5.
Qualitatively this says that as transportation cost t goes up the optimal market gets smaller. Also as the demand density d increases the optimal market area is smaller. Similarly if the productivity of the production process, as represented by the parameter a, is larger the optimal market area is larger.
The optimal production, i.e., production for the optimal market area,
Q =dR2 is then
)4/5.