Knowledge that Transforms

DESCRIPTIONS of the shifting of demand and supply curves frequently are confusing. Some authors use the words up and down, others vertical and horizontal, while still a third group prefers right and left. Recently one writer' attempted to prove that a shift of the demand curve to the right graphically is the same as a shift upward, but he apparently failed to perceive that the demonstration was necessarily limited to the special case of a negatively sloping straight line.2 Under this particular restriction, clearly a shift upward a shift to the right, a shift downward-=a shift to the left.

IN THIS paper, an attempt is made to find a starting point for a comprehensive theory of urban land utilization and city growth in its broader aspects. The paper treats of the subject in mathematical terms, starting with certain assumptions as to the demands for land at various price levels in a growing city and proceeding to show the general characteristics of city development which result from the play of economic forces. The paper describes, in general terms, the city pattern and the spread of building development which result from the fact that each owner of land will, in most cases, seek to obtain the greatest possible value by the sale or utilization of his land. An increase of population brings about demands for additional land to serve as sites for additional commercial, industrial, residential, and social use buildings. Certain locations and sites are more desirable than others and, further, the intended use is a factor in determining the purchase price. For example, the prospective purchaser of a site for a legitimate theatre in the central business district of a city cannot pay so high a unit price for land as the prospective purchaser of a site for an office building. Starting with a tract of land, or an aggregate of parcels of land, all in the same ownership, and assuming that there is a constant annual demand for this land for many diverse uses and, therefore, at different price levels, it is shown that the owner will realize the maximum value by accepting off ers at all the prices from the highest down to a certain price, called the critical price, and by rejecting all offers at prices lower than this critical price. This means that he will realize a greater value by selling some land each year at each of the various price levels above a certain level than he will by selling it all at any one price, high or low, regardless of how slowly or rapidly he may be able to dispose of it. The critical price depends on the demand rates, the range of offered prices, and the original area involved. Methods are described in this paper for determining the critical price under various sets of conditions. It is also shown that, for a maximum value, the owner will reject, after a certain time, further demands at the critical price, and after a

1. General notions.-Assume that we have to do with a homogeneous population, whose distribution with respect to the individual (or per family) income x is described by the function y(x). For a price of a commodity A, the total expenditure of a person for this commodity is a function of his income WA(X), making the total expenditure of the population fxoo WA(x)y(x)dx, xO being the minimum income. It should be noted that (1) w is really a regression function, although for theoretical purposes we shall regard it as a function proper; (2) the commodity A may be a one-price commodity, such as crystalline sugar, or a manypriced one, such as cigarettes; in the latter case, when speaking of a given price, we really think of a price system. The function w, which we may call-according to Frisch's proposal -an Engel function, is, of course, limited in value by the inequality 0_ WA (x) <x. As x increases, WA(X) may increase in proportion to x, more slowly than x, or faster than x. Accordingly, we shall say that the elasticity of expenditure for A with respect to income is equal to, less, or greater than, unity, the elasticity being defined as follows: