Guns and Butter: Stability and Robustness

Gyoung-Gyu Choi

doi:10.1515/ajle-2020-0006

Article

Guns and Butter: Stability and Robustness

Gyoung-Gyu Choi

Published/Copyright: April 23, 2020

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From the journal Asian Journal of Law and Economics Volume 11 Issue 1

abstract

The paper considers the problem of allocating resources to economic goods and military goods (protecting property rights) in a contest model under an anarchic system which can be defined by the absence of a common superior authority to each state. I characterize the equilibrium and discuss its stability and robustness. The main finding is that if economic goods of the two states are complementary with Cobb-Douglas utility characteristics, the essential factor that affects a state’s resource allocation is its own maximum military level of production out of its resource endowment. In this sense, there is a unique stable equilibrium. This equilibrium is robust with the sequence of the states’ decisions insofar as the second mover’s observability is imperfect even slightly.

Keywords: contest; defense economics; guns and butter; international security; strategic complementarity

JEL Classification: D72

Acknowledgement

I am very grateful to my advisors, Paul Krugman, Bruce Bruno de Mesquita, and Daniel Diermeier for extremely helpful comments. My gratitude also goes to Robert Powell for the comments in the early development of the research idea. This paper is an extended version of the paper entitled “Economic Interdependence and the Optimal Resource Allocation” that was written in Korean.

Appendix

Proof of Proposition 1:

(i) Reaction Curve The general form of the reaction function is derived as follows:

m i = − 3 m j 2 + B 2 b i ,

where B=bimj(9bimj+8ri). The signs of the first and the second derivatives of the reaction curve are

∂ m i ∂ m j = − 3 2 + 9 b i m j + 4 r i 2 B > 0 ,

∂ 2 m i ∂ m j 2 = − 8 b i r i 2 B 3 < 0.

This implies that the reaction curves are increasing with decreasing rates.

(ii) Existence Note that limmj→∞∂mi∂mj=0 by L’Hospital’s rule. Therefore, the two reaction curves must intersect and only once from (i).

(iii) Stability Since two concave reaction curves intersect at the point with upward slopes and S₁’s reaction curve intersects with S₂’s one from below:

| d m 2 d m 1 | < 1 | d m 1 d m 2 | ⟹ | ∂ m 2 ∂ m 1 | ⋅ | ∂ m 1 ∂ m 2 | < 1.

Since the absolute value of the product of two slopes at the intersection is less than 1, the equilibrium is stable.

It is clear that the origin is trivially an equilibrium, because (m1∗,m2∗)=(0,0) lies on both reaction curves. However, it is not stable, because limmj→0∂mi∂mj=−32+limmj→09bimj+4ri2B=∞>1 if r_i > 0. This completes the proof that the unique stable equilibrium is the nontrivial one. ⊡

Proof of Proposition 2:

From the general form of the reaction function given by (6), the forms and the signs of the first derivatives of the reaction function with respect to each parameter are derived as

∂ m i ∂ r i = 2 m j B > 0 , ∂ m i ∂ b i = − 2 m j r i b i B < 0.

⊡

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Published Online: 2020-04-23

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Keywords for this article

contest; defense economics; guns and butter; international security; strategic complementarity