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Published in: Public Choice 3-4/2020

23-07-2019

Trade and the predatory state: Ricardian exchange with armed competition for resources—a diagrammatic exposition

Author: Martin C. McGuire

Published in: Public Choice | Issue 3-4/2020

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Abstract

Armed conflicts—especially wars—between/among nation-states, considered as monolithic wholes, surely reduce the mutual benefit they enjoy from trade. That common sense is virtually self-evident. Wars destroy trust that is essential to trade, wars and preparations for war absorb resources otherwise available for productive investment and exchange. Wars destroy people, their capital, and land. Anticipation or fear of war distorts free exchange, causing nations to protect domestic production. Accordingly, I was surprised and puzzled, when attending a seminar by Professors Garfinkel and Syropoulos (Trading with the enemy, Memo of March 1, 2017, Department of Economics, University of California-Irvine, Irvine, CA) to learn of a rigorous mathematical model that entailed mixed motives in international systems that might lead at once to armed conflict among states which nevertheless simultaneously benefit from mutual trade. So here I develop a primitive Ricardian model to explore how incentives to trade interact with those of predation. Its primary purpose is heuristic: to assemble the components of Ricardo, present them in a manner that specifically incorporates opportunities for appropriation through armed conflict, and shows how the component working parts fit together. Importantly, the paper builds the simplest possible model for the case at hand. Methodologically, the steps are geometric/diagrammatic with only ancillary attention to mathematics. The aim is to construct and explore the simplest possible classical, old-fashioned model that could permit analysis of those trade-predation linkages that seem to permit so unexpected a conclusion.

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Appendix
Available only for authorised users
Footnotes
1
See Garfinkel and Syropoulos (2017). An updated version is Garfinkel et al. (2019).
 
2
Use of geometry to derive relationships (rather than merely illustrate conclusions already known from mathematics) has gone out of style in recent years. Still, geometry can yield insight that mathematics does not, as in physics or engineering. Geometry and visualization allow rough generalizations and skirt the constraints that specific algebraic formulas entail. Gordon Tullock’s “conflict success function” is one example of those constraints.
 
3
Since we will ignore the destruction that often marks conflict, the term “guns” should be taken figuratively. Of course, other factors than GB influence A’s choice of GA, but all are incorporated into the function f.
 
4
However, to find it one must wade through the elementary geometry. To follow the argument, the reader must be comfortable with Ricardo’s theory of comparative advantage, offer curves to represent demand/supply, and Edgeworth box diagrams with their relation to production possibility frontiers.
 
5
This insightful simplifying assumption is due to GS. It implies a degree of separation between predation and trade that otherwise would be lost in a muddle of counterbalancing interactions.
 
6
Effects of unequal disproportional factor endowments: If the endowments of K and L were not of the same proportion in all countries, then even though production functions for Z and for G were identical, disproportional endowments would lead to total world production inside the true world production possibilities frontier, or PPW, changing our analysis.
 
7
However, if inside of A the ownership of K and L is divided unequally among diverse groups, not everyone would benefit from such endless expansion; some will lose, but that is an issue ignored in this paper. For detailed analysis of this issue, the reader is referred to McGuire (1978) which is available on request.
 
8
The assumption that it is capital that can be added to the initial endowment simplifies welfare analysis here, since the construction could apply to any factor of production, including labor. Leaving the overall population constant, however, avoids one source of confusion over the meaning of utility as applied to an entire country when its population changes. A similar exercise to that pursued here could apply if the contested “foreign” resource was Labor. However, then the thorny question would arise of how to assess welfare when the number of people varies. We skirt that question here.
 
9
Figure 6a, b derive the rays pA and pB of Fig. 5b on the assumption that the underlying utility functions are linear homogeneous. First consider Fig. 6a. It erects several upward-shifting origins for B, i.e., OB0, OB1 and OB2, along the vertical upright through XMAXA. Each of these shifting origins generates its own value of ΖB. B’s resource constraint ππB is drawn from XMAXA. The same vertical line applies to all the upside-down B-opportunities irrespective of the B-origin. From each B-origin, draw the income-expenditure path (IEP) for pB to its intersection with ππB. Then draw in the A offer curve that intersects each of those junctures of IEPB and ππB, each OCA being drawn from point XMAXA. Each OCA curve implies its own A-origin, OA, and its own value of ZA. Enter corresponding values of ZA and ZB on the x and y-axes in Fig. 6b; connect the dots and label the locus pB. Using the same procedure, build the ΖA–ΖB locus for pA (not shown).
 
10
Of course, if we were to include distributive effects of growth in the model, the scope for immiserization—in an extended sense of the word—may increase dramatically. But, here, distribution within countries is ignored.
 
11
Note that the superscripts “Small” and “Big” refer to the relative Z-size of A and B, not to the levels of utility.
 
12
This result seems not recognized in the literature on Ricardian trade.
 
13
Suppose I want to draw in constant value contours for Country B in Fig. 5b. Inside the wedge (to the left of the ray through the origin pW = pB), B’s welfare is unaffected by increases or decreases in A’s resources, ZA. In that region, B trades as if in autarky. So, there, any value or utility contour is a horizontal line for B. As ZB increases, it crosses higher and higher parallel and horizontal constant-value contours. Now extend one of those value contours horizontally, say \( {\text{Z}}_{{\mathbf{B}}}^{*} \).
Let that line \( {\text{Z}}_{\text{B}}^{*} \) continue into the middle wedge and then beyond into the wedge where A’s size governs and B obtains all gains from trade. Will that extended horizontal line represent an unchanging level of value for B as it did before it crossed the delimiting ray pW = pB? Well, obviously, not! In the middle region, B benefits not only from the autarchic value of \( {\text{Z}}_{\text{B}}^{*} \), but in addition B realizes some benefits of trade, as it shares them with A. The share going to B increases gradually as line \( {\text{Z}}_{\text{B}}^{*} \) continues across the middle wedge approaching the region where A’s size dominates. So, let line \( {\text{Z}}_{\text{B}}^{*} \) cross over that next ray where pW = pA, defining the wedge where B obtains all of the benefits from trade. Thus, to draw a constant value contour for B, call it \( {\text{Z}}_{{{\mathbf{B}}{\mathbf{V}}}}^{*} \), equivalent in value to line \( {\text{Z}}_{\text{B}}^{*} \) (to the left of ray pW = pB), we cannot extend line \( {\text{Z}}_{\text{B}}^{*} \) horizontally across that ray. The equal value line descends until ray pW = pA is reached, and then levels off horizontal again. Following this logic, in Fig. 16 (Appendix 2), I have drawn in a few equal value contours for B, ZBV, and have drawn in some equal value contours for A, following the same reasoning.
 
14
Let us consider the problem this way before asking, how should Country A allocate resources to guns, when those guns compete with another country? That is, as an introduction to the problem, first we assume no armed competition. Note that we assume that both Z and G are produced with the same technology using factors in the same proportions, and that those proportions are post-capture (Ki + κ)/Li, not the original endowed proportions, Ki/Li.
 
15
Shifting gears this way could be a source of confusion when aggregate outcomes are derived or compared without specific recognition of their mixed competitive-centralized origins.
 
16
One advantage of building G = γ(Z), i.e., the opportunity cost of guns, in this way is that the effects of diversity in the Z and G production functions is easily introduced—for instance, to analyze the consequences if, relative to Z, G is the more labor-intensive industry, or vice versa.
 
17
But first, note Fig. 17 (Appendix 3) illustrates that constant marginal returns to investment in guns κ = κ(G) in no way undermines the incentives to fight for additional capital. Figure 17 also implies how greater efficacy of guns in capturing capital will re-work curves LG = μ[ΚG], and KZ = ϕ[LZ]. Rotating κ(G) clockwise in the 4th quadrant produces a counterclockwise rotation of μ[ΚG] in the 3rd quadrant, and a clockwise rotation of KZ = ϕ[LZ] in the 2nd quadrant, clearly leading to higher values for ZOPT.
 
18
Here, I dodge the cause of changes in endowed resources Zi, whether flowing from proportional factor growth or not; the relation between changes in Zi and ZN also is ignored, so that the argument here is solely heuristic on both counts.
 
19
The correct measure of a country’s size in this context is, surely, ZMAXi. With linear homogeneous production as assumed here, a proportional inflation of factors of production will increase ZMAXi by the same proportion. Along any such ray, the marginal rate of substitution between L and K is constant, so “investment” in G will increase.
 
20
Immiserization The effects of trade on the allocation to guns and benefits of ΔK = κ change, however, if A experiences “immiserizing” growth (not illustrated). Now a benevolent and informed decision-maker for Country A would see that over a range of values of ZN, growth combined with trade is not in A’s interest as A’s welfare declines. Thus, at some value of endowed Zi = ZOi, A’s optimal choice of G and, therefore, of maximal ZN = ZON, immiserization will begin to set in. For endowed Zi exogenously greater than this ZOi, allocations to arms will decline to maintain the optimal/maximal ZON that already has been achieved. Over some stretch of the x-axis, therefore, A—being vulnerable to immiserization as Zi increases—will first begin to reduce allocations to G—maybe all the way down to nil. That choice maintains ZN at ZON. But after that point (i.e., after Zi = ZOi), things become complicated; at some point as Zi increases beyond Zοi (lowering after-trade welfare), do allocations of Z to G kick back in again? Yes, possibly; once Zi grows endogenously to a point, say Zi = ZRi, where allocation to G can feasibly take A sufficiently beyond the range of immiserization, i.e., such that U(ZRN) > U(ZRi), allocation to G should be re-started. Thus, if immiserization reverses, then after some point the incentive to return to expansion by conquering ΔK can re-emerge. Once a value of ΔK is achievable that allows the country to “pull out” of its immiserizing dive, “investment” in Guns will resume. Such a purely conjectural value depends idiosyncratically on the ups and downs in welfare caused by immiserization as well as on the details of the functional from ZN = f[Zi, γ(G)]. But analysis of choice that may be involved in the management of immiserization goes well beyond the competitive-Nash behavior underlying the earlier parts of this essay, so it will not be addressed in detail. Those considerations highlight the fact that trade in this model results from competitive behavior, while predation results from centralized oligopolistic behavior.
 
21
As employed by Hirshleifer (1989, 1991), Skaperdas (1996) and others.
 
22
A disadvantage here is that describing conflict in our way is open-ended; the total amount of κ to be divided is not specified. More standard is to assume that for adjustments in forces by either adversary, a loss in κ by one is just balanced by the other’s gain. But our divergence from convention simplifies depicting adversarial and trade relations.
 
23
Immiserization likely will cause those segments of UA or of UB situated between rays pA and pB to wiggle up and down, or to display gaps indicating new candidate equilibria and conceivably instabilities in the balance between fighting and trading.
 
24
In Fig. 12, contour UB is flatter than UA is, but the statement remains valid if contour UB is steeper than UA.
 
25
But it could be true if A realized that a change in GA will affect not only ZA directly, but also ZB directly or indirectly. If A understood that relationship, then the country would see its unilateral power to move point t0 both north–south as well as east–west. However, imputing foresight and subtlety like that to our state actors exceeds the primitive Nash–Cournot posture we adopted above. So, here I avoid the more complex protocol. It could be relaxed, though, and would add insight to this paper, changing dramatically how the various configurations of Fig. 16c affect incentives and outcomes.
 
26
The figure is drawn with origin OA in a fixed spot to produce a fixed map of A isoquants relative to that origin.
 
27
The trajectory of the utility curve could follow many less-extreme paths, with immiserizeration merely a down-sloping wiggle. Thus, the case pictured is solely for illustration. Immiserization also raises numerous thorny questions of the incentives that a unified government would perceive as it recognized trade to be more and more punishing.
 
28
The details of the construction and derivation at hand are available on request.
 
29
Since functions G(LG, KG) and Z(LZ, KZ) are identical, the contract curve is a straight line connecting OZ with OG.
 
30
A benefit of geometric analysis is its flexibility. For example, how does the opportunity curve KZ = ϕ[LG], derived for given endowments of L and K, change if those endowments are altered proportionally while the capital capture function κ = κ[G(LG, KG)] is unchanged? As the diagram shows, that question is easily answered.
 
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Metadata
Title
Trade and the predatory state: Ricardian exchange with armed competition for resources—a diagrammatic exposition
Author
Martin C. McGuire
Publication date
23-07-2019
Publisher
Springer US
Published in
Public Choice / Issue 3-4/2020
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI
https://doi.org/10.1007/s11127-019-00672-w

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