Modelling Conflictual Equilibrium

Each side \(i = 1,2\) divide its exogenously given resources \(R_i\) between productive effort and fighting effort.

\[\begin{equation} E_1 + F_1 = R_1\\ E_2 + F_2 = R_2 \label{eq:resource_partition} \end{equation}\]

Productive effort is summarised by an Aggregate Production Function (APF) showing how the productive efforts \(E_1\) and \(E_2\) combine to determine income \(I\) - the social total available between the two parties.

\[\begin{equation} I = A(E_1^{1/s} + E_2^{1/s})^s \label{eq:aggregate_production} \end{equation}\]

The type of production function is characterised by constant return to scale and constant elasticity of substitution. Parameter \(A\) is the total productivity index: as the overall yields of resource inputs rise over time, owing to technical progress, \(A\) increases. Parameter \(s\), which plays a crucial role in the analysis, is a complementary index: as nations become more closely and synergistically linked by international trade, for example, \(s\) rises. As can be seen in Figure 1 higher values of \(s\) are reflected in increasingly convex curvature of the productive isoquants. At the lower limits, when \(s=1\), the APF takes on a linear (additive) form.

⊕To calculate the isoquant for a given quantity of \(I\) we need to calculate \(E_2\) for different values of \(E_1\) holding \(s\) and \(A\) constant. We first remove the power and technology multiplier from both sides of the equation by division. \[(I/A)^{1/s} = (e_1^{1/s} + e_2^{1/s})\] We then rearrange for \(E_2^{1/s}\) \[E_2^{1/s} = (I/A)^{1/s} - e_1^{1/s}\] And multiply the power by \(s\). \[E_2 = (I/A^{1/s} - e_1^{1/s})^s\]

This type of production function is characterised by constant return to scale and constant elasticity of substitution.

The third element, the Contest Success Function (CSF) summarises the technology of the conflict whose inputs are the fighting efforts \(F_1\) and \(F_2\) and whose outputs are the distributive share \(p_1\) and \(p_2\) (where \(p_1 + p_2 = 1\)).

This assumes that outcome of the struggle depends on the ratio of parties’ conflictual efforts \(F_1\) and \(F2\) indexed by a single mass effect parameter \(m\):

\[\begin{equation} p1 = F_1^m/(F_1^m + F_2^m) \\ p2 = F_2^m/(F_1^m + F_2^m) \label{eq:contest_success} \end{equation}\]

As illustrated in Figure 2, the mass effect parameter \(m\) scales the decisiveness of conflict, that is, the degree to which a superior input ratio \(F_1/F_2\) translates into a superior proportionate success ratio \(p_1/p_2\).

Finally, there are Income Distribution Equations defining the achieved income levels \(I_1\) and \(I_2\):

\[\begin{equation} I_1 = p_1 I \\ I_2 = p_2 I \label{eq:income_distribution} \end{equation}\]

Equations \(\eqref{eq:contest_success}\) and \(\eqref{eq:income_distribution}\) together imply that all income falls into a common pool available for capture by either side.

The equation system \(\eqref{eq:resource_partition}\) through \(\eqref{eq:income_distribution}\) is illustrated by the four-way diagram of Figure 3. The upper-right quadrant shows the range of contender #1’s choices between productive effort \(E_1\) and conflictual effort \(F_1\) within his initial resource endowment \(R_1\). The diagonally opposite quadrant shows the corresponding options for contender #2. The upper-left quadrant shows how the respective fighting effort \(F_1\) and \(F_2\) determine \(p_1\), the share of aggregate income won by # 1, where \(p_2 \equiv 1 - p_1\) (The \(p_1\) contours are straight lines from the origins that follow from the assumption that distributive shares are functions only of the ratio \(F_1/F_2\)). Finally, the lower-right quadrants shows how productive efforts \(E_1\) and \(E_2\) combine to generate totals of income \(I\).

The dashed rectangle in Figure 3 illustrates one possible outcome of the postulated interaction: for given initial choice \(E_1F_1\) on the part of decision maker #1 and \(E_2F_2\) on the part of #2, the productive activity levels \(E_1\) and \(E_2\) determine aggregate income \(I\) while the conflictual commitments \(F_1\) and \(F_2\) determine the respective share \(p_1\) and \(p_2\). The dotted rectangle shows what happens when the two sides choose instead to devote more effort to fighting. As drawn here, the increase in \(F_1\) and \(F_2\) have cancelled one another out so that \(p_1\) and \(p_2\) remain unchanged. Thus, the only effect of symmetrically increased fighting efforts may be to reduce the amount of income to be divided.

Reaction Curves and Cournot Equilibrium

⊕Cournot competition is an economic model describing an industry structure in which rival companies offering an identical product compete on the amount of output they produce, independently and at the same time. It is named after its founder, French mathematician Augustin Cournot

One the assumption that the underlying strategic situation justifies the Cournot solution concept, the Reaction Curve \(RC_1\) and \(RC_2\) shows each side’s optimal fighting effort given the corresponding chose on the part of the opponent. The Cournot solution occurs at the intersection where each party’s decision is a best response to the opponent’s action.

Decision-maker # 1’s optimising problem can be expressed:

\[\begin{equation} \text{Maximise } I_1 = p_1(F_1|F_2) \times I(E_1|E_2) \text{ subject to } E_1 + F_1 = R_1 \label{eq:cournot} \end{equation}\]

⊕Set up the Lagrangian \[L = F_1^m/(F_1^m + F_2^m) * A*(E_1^{1/s} + E_2^{1/s})^s + \lambda*(R_1 - E_1 - F_1)\] Derive the first order conditions Take the ratio and substitute into the budget constraint and solve for \(I_1\).

and similarly for side #2. Using equations \(\eqref{eq:aggregate_production}\) and \(\eqref{eq:contest_success}\), by standard constrained optimisation techniques we can solve for the Reaction Curves \(RC_1\) and \(RC_2\):

\[\begin{equation} \frac{F_1E_1^{(1-s)/s}}{F_2^m} = \frac{m(E_1^{1/s} + E_2^{1-s})}{F_1^m + F_2 ^m} \\ \frac{F_2E_2^{(1-s)/s}}{F_1^m} = \frac{m(E_1^{1/s} + E_2^{1-s})}{F_1^m + F_2 ^m} \label{eq:reaction_curves} \end{equation}\]