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

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.

Appendices

Appendix 1: Immiserizing growth illustrated

In Fig. 14, Country A, with resources Z⁰_A (not drawn), produces at X⁰_A at the corner of ππ⁰_A then exports good X to and imports good Y from B to enjoy utility level U⁰_A. Now let Z_A increase to Z¹_A (not drawn). Then as A will produce on ππ¹_A at X¹_A, its export equilibrium price and welfare outcome are seen to decline at the intersection of OC¹_A and OC¹_B—both drawn from the displaced x-intercept at X¹_A. Relationship U¹_A < U⁰_A shows immiserization.²⁶

Fig. 14

Immiserization: when country A grows its loss from trade exceeds gain from size

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.

Fig. 15

Trajectory of welfare and country size: ordinary vs. immiserizing growth

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.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 U⁰_A = h(Z_A, Z_B), showing one utility contour, say U_A = U⁰_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.

Fig. 16

Welfare contours depend on own and adversary's resources: alternative patterns

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, U⁰_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.²⁸

For U⁰_A to co-exist with p^A = p^W, A’s resources must increase, shifting say to Z¹_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 Ζ¹_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 Lⁱ − Kⁱ. 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.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 (Kⁱ + κ)/Lⁱ.

Fig. 17

Geometric construction/derivation of the payoff from allocations to guns

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 = G₁ and, therefore, of κ = κ₁, and for this value extend the box downward by that amount, generating a new (more “southern”) origin for Z and a new diagonal CC₁. Along the “old” CC_O, find the point for G₁ 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.³⁰ Next, for any value on that curve (labeled L_G = μ[Κ_G])—for instance, the point on CC₁—trace vertically up for that value of L_G to intersect with CC₁ (as copied/shifted to the second quadrant to originate at κ₁) and labeled there as ${\text{CC}}_{{\mathbf{1}}}^{*}$. That intersection gives the values of L¹_Z = (Lⁱ − L¹_G) and of K¹_Z = (Kⁱ + κ₁ − K¹_G) on the promised function K_Z = ϕ[L_Z]. Repeat for choices of G other than G₁. 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

$$ {\varvec{\upkappa}} = {\text{a}}\left[ {\text{G}} \right]^{{{\mathbf{1/2}}}} . $$

(5 repeated)

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

$$ {\text{G}} = {\varpi }{\text{L}}_{\text{G}}^{{{\mathbf{1/2}}}} {\text{K}}_{\text{G}}^{{{\mathbf{1/2}}}} $$

(22)

$$ {\text{Z}} = {\varpi }{\text{L}}_{\text{Z}}^{{{\mathbf{1/2}}}} {\text{K}}_{\text{Z}}^{{{\mathbf{1/2}}}} $$

(23)

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

$$ {\varvec{\upkappa}}\left[ {\text{G}} \right] =\upalpha ({\text{L}}_{\text{G}} )^{1/2} ({\text{K}}_{\text{G}} )^{1/2} $$

(24)

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

$$ \mathop {\text{Max:}}\limits_{{{\mathbf{L}}_{{\mathbf{G}}} ,{\mathbf{K}}_{{\mathbf{G}}} }} \quad {\text{Z}} = [({\text{L}}^{\text{i}} - {\text{L}}_{\text{G}} )^{1/2} ]\left[ {{\text{K}}^{\text{i}} - {\text{K}}_{\text{G}} +\upalpha\left\{ {({\text{L}}_{\text{G}} )^{{{\mathbf{1/2}}}} ({\text{K}}_{\text{G}} )^{{{\mathbf{1/2}}}} } \right\}} \right]^{{{\mathbf{1/2}}}} $$

(25)

The first-order conditions are

$$ {{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial K_{G} : - 1 + {\alpha \mathord{\left/ {\vphantom {\alpha {2(L}}} \right. \kern-0pt} {2(L}}}}} \right. \kern-0pt} {\partial K_{G} : - 1 + {\alpha \mathord{\left/ {\vphantom {\alpha {2(L}}} \right. \kern-0pt} {2(L}}}}_{G} )^{1/2} (K_{G} )^{ - 1/2} = 0 \to \sqrt {K_{G} /L_{G} } = {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2} $$

(26)

$$ {{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial L_{G} :}}} \right. \kern-0pt} {\partial L_{G} :}}\quad [K^{i} - K_{G} + \alpha (L_{G} )^{1/2} (K_{G} )^{1/2} ] = ({\alpha \mathord{\left/ {\vphantom {\alpha {2)}}} \right. \kern-0pt} {2)}}[{{K_{G} } \mathord{\left/ {\vphantom {{K_{G} } {L_{G} }}} \right. \kern-0pt} {L_{G} }}]^{1/2} [(L^{i} - L_{G} )] $$

(27)

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

$$ (K^{i} + K_{G} ) = (L^{i} - L_{G} )({{\alpha^{2} } \mathord{\left/ {\vphantom {{\alpha^{2} } 4}} \right. \kern-0pt} 4}):\quad or\quad 2K_{G} = \{ [({{\alpha^{2} } \mathord{\left/ {\vphantom {{\alpha^{2} } 4}} \right. \kern-0pt} 4})(L^{i} )] - K^{i} \} $$

(28)

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.

Table 2 Outcomes from optimal allocation of resources to capture additional capital, κ: when objective is maximization of net product Z[(L_G)^1/2(K_G)^1/2] and κ[G] = α(L_G)^1/2(K_G)^1/2

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).