# A geometric method for analyzing many-firm or many-period problems in micro theory.

1. Introduction

In this paper, we propose a simple geometric device that has the potential of becoming an important teaching aid in analyzing various problems in microeconomic theory dealing with either many firms or many time periods. We choose two areas to illustrate the method. These are of a monopolist selling in two different markets separated by some distance, and of a monopolist producing an exhaustible resource with a finite horizon. The problem facing them is one of optimal allocation among markets separated by space (in the former) and time (in the latter).

2. Monopoly and Transport Costs

For geometric purposes, we assume linear demand curves, constant transportation costs (to the consumer) and constant average (equal to marginal) costs of production. Let demand in the two markets be (1) [p.sub.1] = a - [bq.sub.1] (2) [p.sub.2] = c - [dq.sub.2] where a, b, c and d are all positive. It is also assumed that a > c and b > d. We further denote marginal cost of production by m and the per unit transportation charge by t. The geometric intuition comes naturally once we analyze the case for t = 0. Clearly, in such a case, the monopolist cannot maintain market separation. Thus the problem reduces to that of a single price monopoly. Aggregating the demand curves, we obtain (3a) [Mathematical Expression Omitted] and (3b) [Mathematical Expression Omitted] where we denote the aggregate market quantities by the capital letters. This is represented graphically in Figure 1.

Naturally, the problem is somewhat trivial when the final outcome is such that one of the two markets demands a zero output. Hence, we focus on the segment of the market demand to the right of the kink. In other words, from now on we deal with the demand curve expressed by (3b). For algebraic simplicity let us also redefine the intercept and the slope such that (3b) is now written as (4) P = [Alpha] - [Beta] Q At t = 0, the profit maximizing monopolist sets MR = MC. From (4), (5) MR = [Alpha] - 2 [Beta] Q and, setting this equal to m (the marginal cost), we have (6a) [Q.sub.*] = [Alpha] - m/2 [Beta] (6b) [P.sub.*] = [Alpha] + m/2 where the asterisks denote optimal values.[1] A graphical representation is immediate. This is shown in Fig. 1 (iv).

Now, letting t > 0, we note that in the absence of resale prevention mechanisms, market equilibrium requires (7) [p.sub.1] = [p.sub.2] + t Profit maximization would still require MR = MC as above. Hence the profit maximizing output level cannot change. The allocation between the two markets will however change. Equilibrium condition (7), in light of (1) and (2), may be rewritten as (7a) [Mathematical Expression Omitted] The other equilibrium condition, (6a), may also be restated in terms of [q.sub.1] and [q.sub.2]: (7b) [Mathematical Expression Omitted]

We thus have two equations given by (7a) and (7b) in two unknowns, [q.sub.1] and [q.sub.2]. Graphically the solution is shown in Figure 2. Notice that the intercept on the [q.sub.2]-axis of equation (7a) may either be positive or negative, depending on the magnitude of the parameters {a, c, t}. We have drawn such that this is negative. The two equations, one being positively and the other negatively sloped, will generally intersect.

It is now evident that the previous solution (equations (6a) and (6b) is a special case of the above for t = 0 (i.e., [p.sub.1] = [p.sub.2]), and a similar graphical procedure follows in determining the allocation of output in the two markets.

The actual calculation of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] is left to the student.

3. Monopoly and Exhaustible Resources

The simplest kind of a problem in the area of exhaustible resources is that of a monopolist deciding on how to allocate production (and sale) of a known (fixed) quantity of the resource over time. The traditional answer has been that the monopolist should set marginal revenue in excess of marginal costs in order to capture the future "scarcity" value of the resource. It is thus of interest to see how this comes about and differs from the competitive solution. Such problems, it appears to us, are beyond the grasp of intermediate level theory students. Since this is a less familiar problem at the textbook level, we offer a brief outline of the general nature of the problem before taking up with the geometry.

Mathematically, using a two-period model, we can write the general problem as follows:

Maximize (8) [Mathematical Expression Omitted] where (9) [Mathematical Expression Omitted] Here V denotes the present value of profits, n for profits, i the rate of interest, and the rest of the notation corresponds to that of the last section, mutatis mutandis. Note that the marginal costs of extraction are being assumed constant in each period (not necessarily constant over time, however). This is merely for algebraic simplicity.

Substituting from the constraints, the problem can be set up where solving for [q.sub.1], given Q[bar], automatically solves the entire problem. Thus, the optimality condition is (10) [Mathematical Expression Omitted] where [r.sub.j] (j = 1, 2) denotes the marginal revenue in each period. The profit maximizing rule then is to:

allocate production such that the difference

between marginal revenue and marginal

costs, appropriately discounted, be the

same in all periods. Several special cases seem to be of interest. a) [m.sub.T] = [m.sub.1] [(1 + i).sup.T-1], i.e., real costs grow at

the rate of interest. This requires setting (10a) [Mathematical Expression Omitted]

i.e., marginal revenues, suitably

discounted, to be the same in all periods.

Interestingly enough, the same conclusion

applies where production costs are

negligible, [M.sub.1] = [M.sub.2] = . . . = [M.sub.T] = 0 b) [M.sub.T] = . . . = [M.sub.2] = [M.sub.1] (i.e., real production

costs remain constant over time). Here we

have (10b) [Mathematical Expression Omitted]

i.e., present marginal revenues exceed

future (discounted) marginal revenues. The

logic of this is straightforward. With real

extraction costs constant over time,

present value of profits is greater, ceteris

paribus, with conservation.

These are simple extensions of the well-known Hotelling results (obtained for competition) to the case of monopoly. Recently, Stiglitz has analyzed the problem in greater detail (1, 655-661).

To facilitate a geometric exposition, we linearize the above as follows. Without confusion we may use the same notation as in Section 2. [p.sub.1] = a -- [bq.sub.1] [p.sub.2] = c -- [dq.sub.2], (11) a > c > 0; b > d > 0 Focussing attention on case (a) discussed above, namely, real costs growing at the rate i, the optimal output allocation over time is given by the following conditions: (12a) [Mathematical Expression Omitted] and (12b) [Mathematical Expression Omitted] Graphically, we have the representation given in Figure 3.

Again, the actual computation of [Mathematical Expression Omitted] is left to the student as an exercise. Also, note that the case (b), where real costs remain constant over time, may also be analyzed in an identical manner.

4. Conclusion

We have shown that for linear demand and cost curves, otherwise complex problems of intermediate level microeconomic theory can be represented diagrammatically following the procedure outlined here. By way of examples we focussed on a problem of a monopolist with two markets (with transport costs as the sole reason for market segmentation), and another of output allocation over time of a monopolist operating in the market for exhaustible resources. Needless to say, one can think of many other problems that would lend themselves to a similar treatment.

Notes

(1)These quantities can be easily translated back to the original parameters of the demand functions. In other words, Q* = (a/b + c/d)/2 - (m)/ (2(b + d)), and P* = {a/b + c/d) (b + d) + m}/2.

References

Stiglitz, J. E., (1976), "Monopoly and the Rate of Extraction of Exhaustible Resources", American Economic Review, 66(4), September, 655-661. Syed M. Ahsan Professor of Economics, Concordia University
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