plifying assumption of a uniform transfer surface, it the physical weights of transferred inputs and ot be altered. In practice, this is often not true. For sed as metallic inputs, but it is possible to step up the use larger proportions of scrap at locations where it fact, there is at least some leeway for responding to The same principle also applies more broadly to sferable as well as transferable inputs and outputs be replaced by labor-saving equipment where it is stitutions of this sort, consider the locational triangle consider the decision of a locational unit with two erable output with a market located at M. To focus as given by limiting our consideration to locations I FIGURE 2-5: A Locational Triangle: Analysis of e same production technology is applicable at either the Location Decision with Variable Factor Proportions l the market, which we shall consider later The delivered price of a transferable input is its price at the source plus transfer charges. In the present example, there are two such inputs, xI and x2. Their delivered prices are respectively (1) where pi and p2 are the prices of each input at is source, and ri and r2 represent transfer rates per unit distance for these puts. The distance from each source to a particular location such as I or J is given by di and dz It is significant that the relative prices of the two inputs will not be the same at I as at J. Location I is closer than J to the source of XI, but farther away from the source of x2. So in terms of delivered prices, xI is relatively cheaper at I and X2 is relatively cheaper at J. The total outlay (tO) of the locational unit on transferable inputs is TO-p1XI p2X2 This equation may be reexpressed o inputs that could be bought are determined by outlay line. 13 the possible combinations of inputs xI and X2that sents the iso-outlay lines associated with locations with location I is represented by AA, and that olved in transporting input I to I rather than to J ce ratio determines the slope of the iso-outlay line an that of BB. Also, it is important to recognize put ratio(x1/xe). Movement out along such a ra Increasing lich transferable inputs are used, any ray could however, that if the firm chose to use the inpi utlay by producing at location I and accepting the plied by OR, location I would be efficient in this greater than that implied by OR is used, the unit be efficient and the unit would locate at j. The icably bound As decisions are made concerning ame time consider its locational alternatives. The line denoted by Qo in that figure is referred to as stitute between inputs in the production process. It B ation represented by the coordinates of a point ation that will minimize the total cost of inputs. In FIGURE 2-6: lso-Outlay Lines, Input Choice, and Location nted by OR"and this, in turn, implies location at We might characterize the outcome of the decision process in this example as a locational orientation towards the input x2 The word"orientation"is used in a somewhat less restrictive way here than in previous examples. Here, it is only meant to suggest that the outcome of the production-location decision is that the unit was drawn toward a location closer to x2 as a result of the nature of its production process and the structure of transfer rates While the problem analyzed above concerns a decision between two locations, it can be extended to include all possible12 roughly similar ideal weight. In the next chapter, when we drop the simplifying assumption of a uniform transfer surface, it will be possible to gain some additional perspective on rules of thumb about transfer orientation. 2.6 LOCATION AND THE THEORY OF PRODUCTION So far we have been assuming that for a particular economic activity the physical weights of transferred inputs and outputs were in fixed proportion; that is, the production recipe could not be altered. In practice, this is often not true. For example, in the steel industry, steel scrap and blast furnace iron are both used as metallic inputs, but it is possible to step up the proportion of scrap at times when scrap is cheap and to design furnaces to use larger proportions of scrap at locations where it is expected to be relatively cheap. In almost any manufacturing process, in fact, there is at least some leeway for responding to differences in relative cost of inputs and relative demand for outputs. The same principle also applies more broadly to nonmanufacturing activities, and it includes substitution among nontransferable as well as transferable inputs and outputs. Thus labor is likely to be more lavishly used where it is cheap, and to be replaced by labor-saving equipment where it is expensive. In order to explore some of the implications associated with input substitutions of this sort, consider the locational triangle presented in Figure 2-5.12 As in earlier examples, we shall once again consider the decision of a locational unit with two transferable inputs (x1 located at S1 and x2 located at S2) and one transferable output with a market located at M. To focus attention on the effects of input substitution, we shall take delivery costs as given by limiting our consideration to locations I and J, which are equidistant from the market, and we shall assume that the same production technology is applicable at either location. The arc IJ includes additional locations at that same distance from the market, which we shall consider later. The delivered price of a transferable input is its price at the source plus transfer charges. In the present example, there are two such inputs, x1 and x2. Their delivered prices are respectively p'1=p1 + r1d1 and (1) p'2=p2 + r2d2 where p1 and p2 are the prices of each input at is source, and r1 and r2 represent transfer rates per unit distance for these inputs. The distance from each source to a particular location such as I or J is given by d1 and d2. It is significant that the relative prices of the two inputs will not be the same at I as at J. Location I is closer than J to the source of x1, but farther away from the source of x2. So in terms of delivered prices, x1 is relatively cheaper at I and x2 is relatively cheaper at J. The total outlay (TO) of the locational unit on transferable inputs is TO=p'1x1 + p'2x2 (2) This equation may be reexpressed as x1=(TO / p'1) – (p'2 / p'1)x2 (3) For any given total outlay (TO), the possible combinations of the two inputs that could be bought are determined by equation (2), and these combinations can be plotted by equation (3) as an iso-outlay line.13 Locations I and J have different sets of delivered prices, and therefore the possible combinations of inputs x1 and x2that any given outlay TO can buy will vary according to location. Figure 2-6 presents the iso-outlay lines associated with locations I and J for a given total outlay and prices. The iso-outlay line associated with location I is represented by AA', and that associated with location J is represented by BB'. The shorter distance involved in transporting input 1 to I rather than to J implies that the price ratio (p'2/p'1) will be greater at location I. Since this price ratio determines the slope of the iso-outlay line (see equation (3) and footnote 13), we find that the slope of AA' is greater than that of BB'. Also, it is important to recognize that the slope of any ray from the origin, such as OR, defines a particular input ratio (x1/x2). Movement out along such a ray implies that more of each input is being used and that the rate of output must be increasing. Because we have relaxed the assumption restricting the ratio in which transferable inputs are used, any ray could potentially identify the input proportion used by the locational unit. Notice, however, that if the firm chose to use the input ratio identified with OR', it could produce more output for any given total outlay by producing at location I and accepting the iso-outlay line AA'. In fact, for any input ratio (x1/x2) greater than that implied by OR, location I would be efficient in this sense. By implication, if the production decision is such that an input ratio greater than that implied by OR is used, the unit would locate at I. Similarly, for any input ratio less than OR, BB' would be efficient and the unit would locate at J. The effective iso-outlay line is, therefore, represented by ACB’. The location decision and the production decision are therefore inextricably bound. As decisions are made concerning optimal input combination for a given level of output, the firm must at the same time consider its locational alternatives. The simultaneity of this process can be illustrated by reference to Figure 2-6. The line denoted by Q0 in that figure is referred to as an isoquant, or equal product curve, and characterizes the unit's ability to substitute between inputs in the production process. It indicates that the rate of output Q0 can be produced by every input combination represented by the coordinates of a point on that line. So for any specified output, there is a location and an input combination that will minimize the total cost of inputs. In our example, Q0 can be produced most efficiently at the input ratio represented by OR" and this, in turn, implies location at J.14 We might characterize the outcome of the decision process in this example as a locational orientation towards the input x2. The word "orientation'' is used in a somewhat less restrictive way here than in previous examples. Here, it is only meant to suggest that the outcome of the production-location decision is that the unit was drawn toward a location closer to x2 as a result of the nature of its production process and the structure of transfer rates. While the problem analyzed above concerns a decision between two locations, it can be extended to include all possible