A reformulation of classical location theory and its relation to rent theory

1. Chicago: University of Chicago Press, 1929.

2. Pick realized this, but he did not present the mathematics. He also saw that the Varignon frame is not limited to three-point problems.

3. E. M. HooverThe Location of Economic Activity (New York: McGraw-Hill, 1948). A brief summary is offered in my “Location Theory,” in J. R. P. Friedmann and W. Alonso,Regional Development and Planning (Cambridge: Massachusetts Institute of Technology Press, 1964).

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4. H. W. Kuhn and R. E. Kuenne, “An Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics,”Journal of Regional Science, Winter, 1962.

5. An interesting exposition of these concepts applied to geographic matters is found in W. Warntz, “The Topology of a Socio-Economic Terrain and Spatial Flows,”Papers of the Regional Science Association, XVII, 1966. Other treatments of the mathematical tools presented here may be found in standard texts of mathematics, particularly those for engineers, under sections dealing with analytic geometry and vector analysis.

6. For this reason, slope lines are also sometimes called integral lines, for they are the integral of the direction of the gradient. One could conceive of solving the Weber problem by integrating (∂K/∂y)/(∂K/∂x) in equations (1) to find the general equations of slope lines and then discovering low point by the intersection of any two slope lines, each defined by any point (x, y). Unfortunately, the same complexity that prevents the analytic solution of those equations places this integration beyond our reach.

7. I am reversing the sign. More accurately, they point directly away.

8. This will be the case for convex location polygons. In concave ones, the solution may be at one of the singular points even in the absence of a dominant weight.

9. If the projection of gradK _e, on the path of gradK _e is in the opposite direction of gradK _e, it is rotated 270°, and (x _e+1, y_e+1) will be found between (x, y _e) and (x _′e, y_e′).

10. W. Isard,Location and Space-Economy (Cambridge: The Technical Press of the Massachusetts Institute of Technology and New York City: John Wiley & Sons, Inc., 1956), p. 223.

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11. L. Moses, “Location and the Theory of Production,”Quarterly Journal of Economics, May, 1958, p. 265.

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12. The reader is reminded that for verbal simplicity we are disregarding the sign. To simplify the notation, we shall write ∂B/∂x for (∂B/∂s)(∂s/∂x). In polar terms, the pull would beQ{∂B/∂s,θ}. Of course, ∂B/∂s=[∂B/∂x ²+(∂B/∂y ²)]^1/2

13. See J. M. Henderson and R. E. Quandt,Microeconomic Theory (New York City: McGraw-Hill, 1958), pp. 74–5.

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14. Moses,op. cit.,, “Location and the Theory of Production,”Quarterly Journal of Economics, May, 1958, p. 265. arrives at a similar conclusion for a special case.

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15. Moses,Op. cit..

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16. Of course, or a horizontal demand curve, a firm with continuing economies of scale would produce to infinity.

17. One might imagine that one could solve for Q in equation (9) and substitute in equation (8), thus obtaining a grad G that considers the concurrent changes in (9). This would bedG/dx=∂G/∂x+(∂G/∂Q)(∂Q/∂x). But equation (9) has made ∂G/∂Q=0, and thereforedG/dx=∂G/∂x. In other words,∂G/∂Q=0, by definition, at all points on the scaoar surfaceG.

18. See. E. Chamberlin,the Theory of Monopolistic Competition (Cambridge: Harvard University Press, 1950).

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19. For instance, one text says, “As a provisional hypothesis we postulate that all prices are quoted f. o. b. factory by sellers ... this seems to be the customary assumption in economic analysis.” S. Weintraub,Price Theory (New York: Pitman Publishing Co.,

20. It is here that the assumption of linear demand simplifies the notation by permitting us to ignore the distinction between the elasticity as perceived by the customer, which is based onP _w, and that perceived by the producer, which is based onR.

21. It is interesting to note that market pulls as a whole are likely to be less than in the Weberian formulation or in the other cases considered here. This derives from the generally greater elasticity of demand curves in space, which will makeE greater than most, if not all,e _w's. For a recent discussion of this phenomenon of demand in space, see B. H. Stevens and C. P. Rydell, “Spatial Demand Theory and Monopoly Price Policy,”Papers of the Regional Science Association, XVII, 1966.

22. R. M. Haig. “Toward an Understanding ofthe Metropolis,”Quarterly Journal of Economics, May, 1926.

23. Location theory, in its more complex forms, has dealt with the competition of firms for markets and resources, as far as I know, rather than for the right to occupy land.

24. To avoid lengthening this paper, the definition and discussion of bid prices will be brief. A more thorough presentation of the basic concepts can be found in W. Alonso,Location and Land Use (Cambridge: Harvard University Press, 1964).

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25. For a more detailed exposition of this point, see W. Alonso,op. cit., pp. 106–16.

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26. This is done for the one-center case in W. Alonso,op. cit.. chap. 4. It can be easily extended to cases in which a family has several orientations such as various jobs of different members, shopping, recreation, in-laws, etc. Instead of the single all-purpose distance, the utility function considers the distances to each of these orientations, and the budget function includes travel costs functions to each. By manipulation of the differentials of the utility and budget functions, one arrives at an expression for the gradient of the bid-price surface as the vector sum of the locational pulls to each of these orientations divided by the quantity of land.

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