## 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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## Notes

- 1.
- 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 G

_{B}influence A’s choice of G_{A}, 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 PP

^{W}, 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 p

_{A}and p_{B}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., O^{0}_{B}, O^{1}_{B}and O^{2}_{B}, along the vertical upright through X^{A}_{MAX}. Each of these shifting origins generates its own value of Ζ_{B}. B’s resource constraint ππ_{B}is drawn from X^{A}_{MAX}. 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 p_{B}to its intersection with ππ_{B}. Then draw in the A offer curve that intersects each of those junctures of IEP_{B}and ππ_{B}, each OC_{A}being drawn from point X^{A}_{MAX}. Each OC_{A}curve implies its own A-origin, O_{A}, and its own value of Z_{A}. Enter corresponding values of Z_{A}and Z_{B}on the x and y-axes in Fig. 6b; connect the dots and label the locus p_{B}. Using the same procedure, build the Ζ_{A}–Ζ_{B}locus for p_{A}(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 p

_{W}= p_{B}), B’s welfare is unaffected by increases or decreases in A’s resources, Z_{A}. In that region, B trades as if in autarky. So, there, any value or utility contour is a horizontal line for B. As Z_{B}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 p

_{W}= p_{B}? 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 p_{W}= p_{A}, 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 p_{W}= p_{B}), we cannot extend line \( {\text{Z}}_{\text{B}}^{*} \) horizontally across that ray. The equal value line descends until ray p_{W}= p_{A}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, Z_{BV}, 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 (K

^{i}+**κ**)**/**L^{i}, not the original endowed proportions, K^{i}**/**L^{i}. - 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 L_{G}= μ[Κ_{G}], and K_{Z}= ϕ[L_{Z}]. Rotating**κ**(G) clockwise in the 4th quadrant produces a counterclockwise rotation of μ[Κ_{G}] in the 3rd quadrant, and a clockwise rotation of K_{Z}= ϕ[L_{Z}] in the 2nd quadrant, clearly leading to higher values for Z_{OPT}. - 18.
Here, I dodge the cause of changes in endowed resources Z

^{i}, whether flowing from proportional factor growth or not; the relation between changes in Z^{i}and Z^{N}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, Z

^{i}_{MAX}. With linear homogeneous production as assumed here, a proportional inflation of factors of production will increase Z^{i}_{MAX}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 Z^{N}, growth combined with trade is not in A’s interest as A’s welfare declines. Thus, at some value of endowed Z^{i}= Z^{i}_{O}, A’s optimal choice of G and, therefore, of maximal Z^{N}= Z^{N}_{O}, immiserization will begin to set in. For endowed Z^{i}exogenously greater than this Z^{i}_{O}, allocations to arms will decline to maintain the optimal/maximal Z^{N}_{O}that already has been achieved. Over some stretch of the x-axis, therefore, A—being vulnerable to immiserization as Z^{i}increases—will first begin to reduce allocations to G—maybe all the way down to nil. That choice maintains Z^{N}at Z^{N}_{O}. But after that point (i.e., after Z^{i}= Z^{i}_{O}), things become complicated; at some point as Z^{i}increases beyond Z^{i}_{ο}(lowering after-trade welfare), do allocations of Z to G kick back in again? Yes, possibly; once Z^{i}grows endogenously to a point, say Z^{i}= Z^{i}_{R}, where allocation to G can feasibly take A sufficiently beyond the range of immiserization, i.e., such that U(Z^{N}_{R}) > U(Z^{i}_{R}), 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 Z^{N}= f[Z^{i}, 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.
- 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 U

_{A}or of U_{B}situated between rays p_{A}and p_{B}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 U

_{B}is flatter than U_{A}is, but the statement remains valid if contour U_{B}is steeper than U_{A}. - 25.
But it could be true if A realized that a change in G

_{A}will affect not only Z_{A}directly, but also Z_{B}directly or indirectly. If A understood that relationship, then the country would see its unilateral power to move point t^{0}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 O

_{A}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(L

_{G}, K_{G}) and Z(L_{Z}, K_{Z}) are identical, the contract curve is a straight line connecting O_{Z}with O_{G}. - 30.
A benefit of geometric analysis is its flexibility. For example, how does the opportunity curve K

_{Z}= ϕ[L_{G}], derived for given endowments of L and K, change if those endowments are altered proportionally while the capital capture function**κ**=**κ**[G(L_{G}, K_{G})] is unchanged? As the diagram shows, that question is easily answered.

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## Acknowledgements

I thank Prof. Charles Anderton for his insightful review and comments, and record my debt to the work of Garfinkel and Syropoulos (2017) on which this paper is based. Attending a seminar by Professors Michelle Garfinkel and Chris Syropolous (GS hereafter) I saw that the mixed motives in international systems may lead at once to armed conflict among states which nevertheless simultaneously benefit from mutual trade. Here I build on their idea as derived from a simple extension of the most primitive economic explanation for why nations trade—David Ricardo’s theory. See Garfinkel and Syropolous, “Trading with the Enemy” 3-4-2017, GPACS Workshop, UC-Irvine 4-23-2017. I have also benefited greatly from the comments of two anonymous referees, especially from the insightful comments and meticulous review of the primary referee.

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## Appendices

### Appendix 1: Immiserizing growth illustrated

In Fig. 14, Country A, with resources Z^{0}_{A} (not drawn), produces at X^{0}_{A} at the corner of ππ^{0}_{A} then exports good X to and imports good Y from B to enjoy utility level U^{0}_{A}. Now let Z_{A} increase to Z^{1}_{A} (not drawn). Then as A will produce on ππ^{1}_{A} at X^{1}_{A}, its export equilibrium price and welfare outcome are seen to decline at the intersection of OC^{1}_{A} and OC^{1}_{B}—both drawn from the displaced x-intercept at X^{1}_{A}. Relationship U^{1}_{A} < U^{0}_{A} shows immiserization.^{Footnote 26}

A more schematic picture of the effects of immiserization on trade is then given in Fig. 15 which plots a country’s utility versus its resource size, for the two cases. Without immiserization, as a country grows while others do not, it will benefit from trade. In the beginning when it is a small country and captures all the gains from trade, until as it grows to a crossover point, where it must share those gains as it becomes no longer “small”. Once it reaches that cutoff, welfare gains from greater Z slow, until reaching the status of a “large country”, after which it captures none of the benefits from trade.

On the other hand, a country that experiences immiserization over a range would have a different utility versus resource profile more like the broken line in Fig. 15 where, after a range of normal benefit from growth with trade, a point is reached at which further growth reduces utility. Figure 15 shows that, with an assumption that once immiserization kicks in, it gets worse until a minimum is reached and then increases till Z attains the size where the country trades as if under autarky.^{Footnote 27}

### Appendix 2: Construction of welfare contours as functions of resources of both trading partners

The argument that used Fig. 5 about relative country size stated that for some ratios of resources—on or above along the steeper ray in Fig. 16a—that B’s internal price ratio establishes the world price, that A captures all of the benefits from trade, and that further increases in Z_{B} will have no effect on A’s utility. Figure 16a repeats Fig. 5b, but tentatively adds three versions of a plot of U^{0}_{A} = h(Z_{A}, Z_{B}), showing one utility contour, say U_{A} = U^{0}_{A}. To the left of the critical ratio, the contour is vertical; increases in Z_{B} are of no benefit to A. The alternative, “candidate” utility contours pass through points t^{a}, t and t^{c}, respectively. They all pass through point “s” on the ray p_{B.}

Now consider the combination at point “s” in Fig. 16a where A is small, B is large and world price p^{W} = p^{B}—B’s internal cost-ratio. An equivalent point in a Ricardo trading box like Fig. 14, with corresponding values of Z_{A} and Z_{B}, together with the relevant offer curves, will support “s”. Next ask, what configuration of the Ricardo trading box would yield the same utility for A? That, is, U^{0}_{A} if starting from point s, A grew to become the “large” country while B did not grow and thus became the small country; in that case world prices come to be governed by A’s interior price (p^{A} = p^{A}_{x}/p^{A}_{y}= p^{W} = world price) rather than by the internal price of B! Put differently, where in Fig. 14 would the point “t” on ray p^{A} be that provides A with the same level of utility as at point “s” on p^{B}? Figure 16a illustrates the answer, as derived from a trading box.^{Footnote 28}

For U^{0}_{A} to co-exist with p^{A} = p^{W}, A’s resources must increase, shifting say to Z^{1}_{A}, i.e., shifting the Z_{A} line to the right so that A’s offer curve drawn from the new x-intercept intersects A’s U^{o}_{A} at the slope of Ζ^{1}_{A}. Additionally, B’s offer curve must intersect A’s at the new tangency point, and doing so requires that Ζ_{B}, B’s resources, adjust maybe up, maybe down. Figure 16a depicts three possibilities, where it is B’s indifference map that drives the outcome by way of its offer curve. To represent constant utility in Fig. 16a, we then connect up points “s” and “t” or “t^{a}” or “t^{c}”, with straight lines or simple curves. (Immiserization potentials lurking in the preference maps of A and/or B probably would cause wiggles and slope reversals in such curves, but we leave that aside.)

The foregoing procedure allows us to combine the effects of comparative size and trade in a Ricardian world into a single measure. Any single contour in Fig. 16a is called “utility”, although maybe “resource equivalence” is a better term. Obviously, by repeating the just-summarized procedure, we can build a map of utility contours both for A and for B, so one could think of Fig. 16a as filled with such contours for both players. Omission of immiserization notwithstanding, we still have a rich banquet of possible combinations of utility maps for our two countries. Figure 16b, in each of its panels, depicts generic configurations, with radically diverse implications for countries’ incentives to fight and/or trade.

The first implication of these constructions is that if starting as a large, price-setting trader, a country declines in magnitude, its losses from diminished size can be offset partially or even entirely by gains in trade generated from its adversary/partner’s relative growth. The top two panels, I and II of Fig. 16b, illustrate that possibility. In panel II, a declining country “requires” only that its partner to grow at a fraction of its own decline to stay even (stay on the same utility contour). But in Panel I of Fig. 16b, for the benefits of trade to balance out B’s loss from a lesser size, A must grow by more than B’s decline. That potential can create novel dynamics for interactions between trading and fighting.

A second implication of Fig. 16a is displayed in panels III and IV of Fig. 16b and seems as remarkable as it looks logical. To see that imagine a country (A) so small that it has no impact on world price, which is determined entirely by the internal prices of its partner (B). Next, let the country grow to the point that it now dominates and so determines international prices. Then for some configurations of preference maps, B also may be allowed or required to grow without diluting A’s world dominance. That seems to be the message, encrypted in indifference maps like U^{c}_{A} of Fig. 16a, that have been extended or applied in Fig. 16b-III, b-IV. Such relationships are central to figure out how a country’s quest for gain from trade, and from productive size meld together.

### Appendix 3: Construction of opportunity curve K_{Z} = ϕ[L_{Z}], showing possibilities for armed capture of resources

Here, I provide a geometric construction of the opportunity frontier, K_{Z} = ϕ[L_{G}], to be drawn in the space of the inputs to the intermediate fungible good (L_{Z}, K_{Z}). This construction derives ϕ rather than merely illustrating, as did Fig. 7.

Figure 17 begins with a variation on Fig. 7. The dimensions of the Edgeworth box give the original endowments L^{i} − K^{i}. Isoquants for provision of G are entered relative to origin O_{G} and are increasing to the southwest, as shown by their numbering along CC_{O}, the box diagonal that obtains before any acquisition of **κ**. The values of G so displayed also give the opportunity cost of G in terms of Z foregone for various K_{Z}/L_{Z} ratios.^{Footnote 29} For that reason, we label the G-isoquants also as C_{Z}(G)—“opportunity cost” of G in terms of Z. “Capital-capture opportunities” as a function of G, gives the productivity of G as the amount of K captured, or **κ** = f(G). It is as drawn to the right of the Edgeworth box and is linear for illustration. Now, as G increases along the right x-axis, **κ** is shown to increase measuring downwards; the new Edgeworth box expands downward in its vertical dimension so that the new box-diagonals, one for each value of **κ** and, therefore, of G show required ratios of (K^{i} + **κ**)/L^{i}.

Now to derive K_{Z} = ϕ[L_{Z}], that was merely illustrated in Fig. 7, consider the functions in parametric form, L_{G}(**κ**) and K_{G}(**κ**). Choose some value of G = G_{1} and, therefore, of **κ** = **κ**_{1}, and for this value extend the box downward by that amount, generating a new (more “southern”) origin for Z and a new diagonal CC_{1}. Along the “old” CC_{O}, find the point for G_{1} and its associated G-isoquant; then mark the intersection of that isoquant with that diagonal line CC_{O}. Repeat for all values of K, G and **κ**, then connect the dots to obtain L_{G} = μ[Κ_{G}]. Figure 17 shows the curve so produced.^{Footnote 30} Next, for any value on that curve (labeled L_{G} = μ[Κ_{G}])—for instance, the point on CC_{1}—trace vertically up for that value of L_{G} to intersect with CC_{1} (as copied/shifted to the second quadrant to originate at **κ**_{1}) and labeled there as \( {\text{CC}}_{{\mathbf{1}}}^{*} \). That intersection gives the values of L^{1}_{Z} = (L^{i} − L^{1}_{G}) and of K^{1}_{Z} = (K^{i} + **κ**_{1} − K^{1}_{G}) on the promised function K_{Z} = ϕ[L_{Z}]. Repeat for choices of G other than G_{1}. Figure 10 (second quadrant) shows the construction for points 1, 2 and 3.

### Appendix 4: An example calculated

The main text and especially Fig. 7a suggest diminishing marginal returns to Guns, say

Recall that G and Z have the same production function, say the constant-returns-to-scale functions

Here,
comporting with Fig. 17, I produce a simpler example for which **κ**[G] is linear. The linear specification shows that in utilizing force for resource capture, diminishing marginal returns are not necessary to limit the incentives for allocating resources to arms. Also, calculating optimal allocations is easier and more transparent with **κ**[G] linear. Specifically, I assume

to be inserted directly into the objective function, the maximand being the net production of Good Z.

The first-order conditions are

Then, substituting from (25) into (26) gives:

Table 2 displays variables of interest for this optimization, giving values that maximize Eq. (24), where the sole source of change between columns is difference in the “Productivity of Arms”, α.

Thus, this example confirms the elements derived from the earlier geometry. In particular, it illustrates how an optimal extension of endowments (captured at the expense of foregone intermediate good Z) requires factor intensity equalization across Z(L, K) and G(L, K). It also shows how diminishing marginal returns in the productivity of G are not necessary for an interior solution. Note also that the two calculated examples suggest that incentives to appropriate new capital through force may be rather sensitive to the efficacy of that force. In the example provided, merely doubling productivity coefficient α (from 2 to 4) increases the optimal capture of new capital by a factor of 7 (= 224/32).

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McGuire, M.C. Trade and the predatory state: Ricardian exchange with armed competition for resources—a diagrammatic exposition.
*Public Choice* **182, **459–494 (2020). https://doi.org/10.1007/s11127-019-00672-w

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### Keywords

- International trade
- Inter-state conflict
- War
- Resource-predation

### JEL Classification

- F11
- H56
- H87
- D74