Optimum choice of an MNC: location and investment.

Ghosh, Dilip K. ; Ghosh, Shyamasri

The existing literature is quite voluminous on how one country engages in trade relations with another country, how international trade is mutually gainful to each other, what factors determine trade structures, and so on. There has been a substantial exploration on different aspects of multinational corporations engaged in production and transnational trade. Yet there exists an area which seems to have remained quite unexplored, and it is the question of optimum location of production of goods and services and the optimum marketable quantity of output and investment outlay. In this paper, we try to fill in this gap in the literature by way of determining optimum factory location for a multinational corporation (MNC) and ascertaining optimum investment outlay for such a firm.

Consider an MNC that produces only one good (x) by employing two mobile factors, labor (L) and capital (K), which are available at geographic locations of A and B, respectively. Let the market for the product X (which capital and labor can only jointly produce) be at location M. The location triangle ABM [ILLUSTRATION FOR FIGURE 1 OMITTED] depicts the locational situations of factor L at A, factor K at B and the final destination of product X at M. Production of X is a function of capital and labor as follows:

X = f(K,L), [f.sub.i] [greater than] 0, [f.sub.ii] [less than]0, for i = K,L; 0 = f(0,L) = f(K,0)

Let w = (given) price of L at its source A, and

r = (given) price of K at its source B.

Now the choice of the location of production (factory) can be anywhere on, inside or outside of, the locational triangle, but once the MNC's objective is to minimize cost (for a given output) and/or maximize output (for a given expenditure), the location choice is a matter of optimization. Assume, at this stage, that the MNC decides to build its factory away from the market be a given linear distance, say M[a.sub.0]. That means it can build his factory anywhere on the circle u[a.sub.0][n.sub.0][b.sub.0] (whose radius is M[a.sub.0]); but once the transport costs of factors are considered, it is obvious that it would want to locate its factory somewhere on the arc [a.sub.0] [n.sub.0] [b.sub.0]. (1) Let factor A[a.sub.0] distance = [d.sub.1] and B[a.sub.0] distance = [d.sub.2]. If the factory is built at point [a.sub.0], then cost for each unit of factor L and K at the factory location respectively are:

[w.sup.[a.sub.0]] = w + [d.sub.1] [T.sub.L] (1)

[r.sup.[a.sub.0]] = r + [d.sub.2] [T.sub.k] (2)

where [T.sub.L] and [T.sub.K] measure the transportation costs of each unit of labor and capital per unit distance, respectively. Similarly, one can easily ascertain the unit factor cost at location [b.sub.0] as follows:

[w.sup.[b.sub.0]] = r + [D.sub.2] [T.sub.K] (3)

[r.sup.[b.sub.0]] = r + [D.sub.2] [T.sub.K] (4)

where [D.sub.1] and [D.sub.2] are the measures of linear (mile) distances from A and B to [b.sub.0].

If the budgeted expenditure of the firm is [E.sub.0], then availability of the factor units are spelled out by the following equations for location [a.sub.0] and [b.sub.0], respectively:

L [multiplied by] [W.sup.[a.sub.0]] + K [multiplied by] [r.sup.[a.sub.0]] = [E.sub.0] (5)

L [multiplied by] [W.sup.[b.sub.0]] + K [multiplied by] [r.sup.[b.sub.0]] = [E.sub.0] (6)

These equations are plotted in Figure 2:Given the total expenditure [E.sub.0], the MNC can get and employ any capital labor combinations defined by coordinates on the line [[Alpha.sub.0] [[Epsilon].sub.0] if its factory is located at [a.sub.0] in Figure 1, and if [b.sub.0] happens to be the chosen factory location, its available inputs combinations are denoted by the line [[Gamma].sub.0] [[Beta].sub.0]. For obvious reasons, any input mix defined by [T.sub.0][[Epsilon].sub.0] and [T.sub.0] [[Gamma].sub.0] is noneconomic, and thus the kinked locus [[Alpha].sub.0] [T.sub.0] [[Beta].sub.0] is the meaningful iso-cost line for the MNC. Note we have so far considered only locations [a.sub.0] and [b.sub.0]. If we consider a location, say [n.sub.0] on the arc [a.sub.0] [b.sub.0], then iso-expenditure line [[Omega].sub.0] [[Delta].sub.0] and [[Alpha].sub.0] [G.sub.0] [H.sub.0] [[Beta].sub.0] becomes the meaningful iso-expenditure locus. Obviously, if all points along the arc [a.sub.0] [b.sub.0] are considered as the possible factory locations, their iso-outlay locus becomes convex-to-the-origin, signifying that only the end points [[Alpha].sub.0] and [[Beta].sub.0] will correspond to location [a.sub.0] and [b.sub.0] of Figure 1.

Now consider two different pictures exhibited by Figure 3 and Figure 4. Figure 3 portrays the fact that [a.sub.0] and [b.sub.0] are the only two possible locations whereas Figure 4 considers every possible location from [a.sub.0] to [b.sub.0] on arc [.sub.0] [b.sub.0].

Now the problem is to determine the optimum location under the assumed constraints. Since the tangency of the iso-outlay locus with one of the iso-quants defines the optimum decision, in this illustrative case (in [ILLUSTRATION FOR FIGURE 3 OMITTED]), [Z.sub.0] is the equilibrium point, and since it lies on [[Alpha].sub.0] [T.sub.0] line, [a.sub.0] is the optimum factory location under the given situation. Had the firm chosen location [b.sub.0] instead, to attain the same output - that is to land on the same iso-quant [IQ.sub.0] on which [Z.sub.0] is located - the firm would have incurred an extra cost worth the value of HG units of labor at its unit cost of [w.sup.[b.sub.0]]. The inspection of Figure 4 reveals that convex-to-the-origin iso-outlay locus [[Alpha].sub.0] [m.sub.0] [[Beta] .sub.0] is tangent to iso-quant [IQ.sub.0] at [m.sub.0], which means that the optimum location is neither [a.sub.0] nor [b.sub.0], but at point no on the open interval of arc [a.sub.0][b.sub.0]. At this point it should be pointed out that if the iso-quant is enveloped by convex-to-the-origin iso-outlay locus, interior location on the arc [a.sub.0][b.sub.0] is the optimum choice; but if the reverse condition is true - (that is, if the iso-outlay locus is enveloped by the iso-quant), then tangency rule fails and the corner solution will be optimal, which, means either [a.sub.0] or [b.sub.0], depending upon the structure of iso-quant map, will be the optimum location.

So, what transpires from all the discussion up to this point is the following: if total outlay and the distance of production location from the product market are given, then optimum choice of location is a matter of standard optimization rule for a firm. But if the MNC wants to determine the optimum distance of its factory from the product market it means then, in terms of the location triangle, that the firm likes to determine whether to stay on arc [a.sub.0][b.sub.0] or on arc [a.sub.1][b.sub.1] or arc [a.sub.2][b.sub.2], and so on.

Everything else remaining unchanged, if the MNC moves the factory away from the market and toward the input sources, it would certainly save on input costs. So, if the factory is located on the arc [a.sub.1][b.sub.1] (in [ILLUSTRATION FOR FIGURE 1 OMITTED]), unit costs of labor and capital would decrease since the transportation cost would be less since:

[[w.sup.a].sub.1] [less than] [[w.sup.a].sub.0], [[w.sup.b].sub.1] [less than] [[w.sup.b].sub.0]

[[r.sup.a].sub.1] [less than] [[r.sup.a].sub.0], [[r.sup.b].sub.1] [less than] [[r.sup.b].sub.0]

and so on. These input cost reductions then translate into the parallel outward shift of the iso-outlay (for a given initial budgeted expenditure [E.sub.0]), and that means an increase in optimum output. Figure 5 depicts this new outcome. Here [[Alpha].sub.0][T.sub.0][[Beta].sub.0] is the original iso-outlay line, as depicted in Figure 3, when locations [a.sub.0] and [b.sub.0] are the only possible choices. [[Alpha].sub.1][T.sub.1][[Beta].sub.1], [[Alpha].sub.2][T.sub.2][[Beta].sub.2] are the iso-outlay loci if the MNC picks up ([a.sub.1], [b.sub.1]) and ([a.sub.2], [b.sub.2]) as locations on arc [a.sub.1][b.sub.1] and are [a.sub.2][b.sub.2], respectively. [Mathematical Expression Omitted] is the iso-outlay locus if the MNC builds its factory at either A or B.(2) Note now that closer the factory is to the input sources higher the level of optimum output for a given outlay, and the maximum output that can be produced optimally for a given outlay E, is by locating its factory at either A or B - and that means by shifting its outlay locus to [[Alpha].sub.0][T.sub.0][[Beta].sub.0] in the illustrative case of Figure 5. By this maximum possible outward shift of the iso-outlay locus the firm lands on [Mathematical Expression Omitted] at [Mathematical Expression Omitted] (which denotes, [Mathematical Expression Omitted] units, say, 100 units of the product). It is now established that by moving its factory toward input sources the MNC can increase the maximum attainable output, and this continuously rising output hits the ceiling when the firm optimally chooses one of the input sources. Upon a close inspection of Figure 5 one easily finds, in this illustrative diagram, that B is the best factory location.

Two points should be noted at this stage: First, the MNC can maximally produce, as in Figure 6, [Mathematical Expression Omitted] units for the fixed total outlay, and that means marginal cost of production up to [Mathematical Expression Omitted] units is zero. Secondly, one should recognize that by moving the factory closer and closer to input source(s), the MNC makes the distance of the market longer and longer from production location and that means the delivery price of the product at the market makes the good dearer and dearer. Demand for the product the product has been ignored up to this point, and it is high time that this dimension is highlighted and profit-maximization principle is invoked to determine the optimal choice of location.

Assume that the product sells in imperfectly competitive market and hence demand curve for the good is negatively sloped, as in Figure 7, IJ is the demand curve and IR the corresponding marginal revenue curve. Replot the marginal cost of production from Figure 6 on Figure 7, and it is immediately clear that OR units are the optimum level since at this level profit-maximizing conditions:

(i) marginal cost = marginal revenue

(ii) slope of marginal cost [greater than] slope of marginal revenue

are satisfied. RP is the optimal price and total profit is the area IOR less given initial outlay. If the inventory build-up is immaterial in our case, it is clear then that when the iso-quant that denotes the OR units of output is optimally attained we ascertain automatically the optimum location for the factory. Let, in Figure 5, [IQ.sub.2] denote OR units of output, and it is clear then [Z.sub.2] is the equilibrium input mix and the factory location is [a.sub.2] on the location triangle ABM in Figure 1. Note here that we have factored in both the demand condition and production condition against the backdrop of optimization. A few points are worth considering now. If [Mathematical Expression Omitted] level of output is less than OR level in Figure 7, then [Mathematical Expression Omitted] is the maximal output available under the given outlay constraint, and as Figure 5 portrays, B is the best location.

So far every decision is optimal with the exception of the total outlay which is exogenously given, and the result is quite consistent with the findings of Predohl [6], Weber [7], Moses [5], Isard [4], Hoover [2a], [2b] and Hoover and Garriani [3]. It is instructive therefore that we allow this to be a variable and determine its optimum magnitude as well. Assume that the multinational firm can spend as much money as it chooses it to, but it will choose only that amount which yields the maximum possible profit. With this new dimension and old technique, Figure 8 (which closely resembles [ILLUSTRATION FOR FIGURE 5 OMITTED]) is drawn. Here the iso-outlay loci [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], ..., [Mathematical Expression Omitted] correspond to total outlay of [E.sub.0], [E.sub.1], [E.sub.2] ..., [E.sub.n] (w here [E.sub.0][less than][E.sub.1][less than][E.sub.2][less than]...[less than][E.sub.n]), [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], ..., [Mathematical Expression Omitted] correspond to location point [a.sub.0] and [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], ..., [Mathematical Expression Omitted] similarly correspond to location [b.sub.0] of Figure 1. As per Figure 8, optimum location is always at [a.sub.0] corresponding to higher and higher levels of total outlay until it reaches [E.sub.n] for which the location switching takes place - [b.sub.0] becomes then the optimum location. A careful scrutiny of Figure 8 yields one important economic fact which is the following: if production function is linear homogeneous or at least homothetic - which implies that expansion path is linear-location switching can never occur. If [a.sub.0] happens to be the optimum location to begin with, [a.sub.0] will remain optimum location no matter how much the expenditure changes.

So far in the face of changing amounts of total outlay location consideration has been either point [a.sub.0] or [b.sub.0] (of [ILLUSTRATION FOR FIGURE 1 OMITTED]). But, as already pointed out, location can be anywhere on arcs such as [a.sub.1][b.sub.1], [a.sub.2][b.sub.2], and so on. If the total outlay is [E.sub.0] and the firm chooses to locate the factory at say, [a.sub.2] or [b.sub.2], as in Figure 1, then its iso-outlay locus moves out bodily from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] and, as one can notice, optimum is attained when [Iq.sub.0] is tangent to [Mathematical Expression Omitted]. Obviously, [Iq.sub.0] denotes a higher level of output than [IQ.sub.0] represents. Similarly, for every higher amount of total outlay the factory closer to input sources produces more output, which is exhibited clearly by Figures 9(a) and 9(b). Than means, if [E.sub.1] is the chosen outlay, then the factory at [a.sub.0], [a.sub.1], [a.sub.2] ..., A produces [E.sub.1][V.sub.0], [E.sub.1][V.sub.1], [E.sub.1][V.sub.2], ..., [E.sub.1][V.sub.A], levels of output, respectively, as in Figure 9(a). For a different outlay, output levels of the factory at those locations can be read off in the similar way. By the same way one can measure the output levels for the factory at locations [b.sub.0], [b.sub.1], [b.sub.2] ..., B at any outlay from Figure 9(b).

Next, consider the demand condition for the product in the market. Let the demand for the product be represented by DD curve, as in Figures 9(c) and 9(d). Now read Figure 9(c) along with Figure 9(a), and Figure 9(d) along with Figure 9(b). In Figures 9(c) and 9(d), inverse price quantity relationship is portrayed.(3) Suppose now that [E.sub.1] is the chose amount of expenditure. For this expenditure, the firm can, as noted already, obtain [E.sub.1][V.sub.0], [E.sub.1][V.sub.1], [E.sub.1][V.sub.2], ..., [E.sub.1][V.sub.A] outputs; if it sells [E.sub.1][V.sub.0], market price is [OP.sub.0] and hence the corresponding revenue is ([E.sub.1][V.sub.0]) x ([OP.sub.0]) = [OR.sub.0], as in Figure 10. If, for the same expenditure [E.sub.1], the firm has a factory at, say, [a.sub.2], its total revenue is ([E.sub.1][V.sub.2]) x ([OP.sub.2]), which is to say, ON, in Figure 10. Through iterative process of computation one can easily ascertain the maximum possible total revenue that the MNC can generate at each level of expenditure. Let [OU.sub.A][R.sub.A] curve be the graphic representation of the maximum total revenue corresponding to each level of total outlay when the firm locates its factory on along the line MA of Figure 1. Exactly in the similar way can one draw [OU.sub.B][R.sub.B] curve that represents the maximum possible total revenue that the MNC generates by locating its production site along the line MB of Figure 1. Several other maximum possible total revenue schedules can be drawn corresponding to other points in the locational triangle; here such an exhaustive exercise is not made simply for keeping the geometry least cluttered.

Next, draw the 45 [degrees] -line OY - which is, for obvious reason, the total outlay line. Therefore, the vertical distance between [OU.sub.A][R.sub.A] and the 45 [degrees] -line measures the total net profit for the MNC at different outlay levels for revenue-maximizing factory locations along the MA line. The lower panel of Figure 10 depicts the total profit curve [OV.sub.a][[Pi].sub.a] that corresponds to [OU.sub.A][R.sub.A] curve, and similarly [OV.sub.b][[Pi].sub.b] is derived from [OU.sub.B][R.sub.B] curve. It is clear that [OV.sub.a] is the maximum possible profit for a factory location along the line MA and [OV.sub.b] is the maximum possible profit for the firm operating from a factory located somewhere between M and B. If [OV.sub.b] [greater than] [OV.sub.a], as in Figure 10, optimum outlay is [E.sub.2], and optimum location is somewhere on the line MB. Now examine Figure 9(b) and 9(d) and ascertain which output times the corresponding price yields [E.sub.2][U.sub.B] level of total revenue. When it is determined that for [E.sub.2] level of expenditure [E.sub.2][Z.sub.2] is produced at location [b.sub.2] and the price that can sell [E.sub.2][Z.sub.2] level output is [Mathematical Expression Omitted] and [Mathematical Expression Omitted], the entire quaesitum is solved. Now it is clearly and unquestionably established how much outlay should be optimally incurred, what the resultant total profit will be, what the market price, the quantity for sale at the market and the optimum choice of production location are.

Having ascertained all these, one should note that proportional transportation cost can be relaxed, and the availability of both inputs at each source such as A and B can be introduced. In fact it should be pointed out that such additional complications should not be too difficult to deal with. The more interesting extension, however, lies in exposing this analysis along that line. In a different paper [1], an attempt is made to explore that with a few more additional thoughts.

Notes

1. At any point such as U on the circle around M beyond the arc [a.sub.0][n.sub.0][b.sub.0], input cost will be higher because of longer hauling of inputs.

2. If A and B are not on the same radial distance from M, then the input source farther from M is ruled out as the possible optimum location.

3. Demand curve is normally negatively sloped; bur here, in this analytical context, it factors additionally the fact that further the market is from the factory location more transport cost is added on to the goods when it reaches consumers in the market which, in turn, means that consumer prices go up further for which demand goes down. Thus, the negativity of demand slope is further accentuated in this case. Demand curve DD here is, in some sense, net demand schedule that reflects optimum consumer choice at prices that have already absorbed transport costs.

References

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