# (p.195) Appendix A Supply-Chain Management in a Predatory Environment

# (p.195) Appendix A Supply-Chain Management in a Predatory Environment

Supply Chain Management (SCM) is typically conceived as the process of integrating three components of the supply chain, or production network: supply, production, and distribution. It is intuitively obvious that in the long-run, the amount of finished goods a firm distributes cannot exceed the amount it produces, nor can the amount produced exceed the amount sourced. Expressed symbolically:

where *S* is the amount of material successfully supplied to the production process, *P* is the amount of material successfully processed during production, and *D* is the amount successfully distributed as final goods. Furthermore, it is intuitively clear that the long-run Pareto optimal solution for a profit-maximizing firm will be to set all components at par with one another:

In this way, the firm is neither sourcing more than it can process, nor processing more than it can distribute. This is an important, if obvious, point, when revenue is only generated upon distribution.

If we now assume for the sake of simplicity that the amounts of materials successfully supplied, processed and distributed are linear functions of combined material and labor inputs ($s,p,d$ respectively, which firms choose) and production coefficients ($\sigma ,\pi ,\delta $ respectively, which firms are assumed for the moment to take), then we may define the Pareto optimal equation in 8 as:

Unconstrained, this equation simply tells us that in Pareto optimality, $\left(s,p,d\right)$ vary with the inverse of $(\sigma ,\pi ,\delta ).$

If we now consider the productivity coefficient to be the multiplicative product of a technological parameter measuring productivity and a measure of freedom from the risk of predation (${{\rm A}}_{i}$ and ${\rho}_{i}$, where $0\le {\rho}_{i}\le 1$),^{1} we get the following equation:

If we now allow that firms recognize that dispersed economic activity decreases predation, we can see that firms do not take $\left(\sigma ,\pi ,\delta \right)$ unequivocally, but rather can disperse their activities spatially and temporally to raise the production coefficients.^{2} Alternatively, the firm can choose to shift its expenditure on capital and/or labor. To see this, we can reasonably posit that the chance that an input will escape predation may be expressed as an increasing function of the measure of dispersal of economic activity ($\left(G,H,I\right)$ for activities$\left(S,P,D\right)$ respectively) and a decreasing function of the efforts to predate or to “tax” goods by rebels, government or civilians (${r}_{i}$):

where $\left(0,0,0\right)\le \left(G,H,I\right)\le \left(1,1,1\right)$ and $s+p+d+G+H+I\le 1$.

Given the form of (11), the value of ${\rho}_{i}$ is bounded between zero and unity. Since we have established that $\left(G,H,I\right)$ are associated with opportunity costs and diseconomies of scale, they can in essence be treated factors of production. Treating ${r}_{i}$ as a constant and abbreviating the production functions as $f\left(s,G\right)$, $f\left(p,H\right)$, and $f\left(d,I\right)$, we might construct cost equations to be minimized:

In these cost equations, $\left(\phi ,\chi ,\psi \right)$ represent the marginal costs of the supply chain inputs (e.g., wages and rents in the case of labor and capital), whereas$\left(\gamma ,\eta ,\iota \right)$ represent the marginal costs of economic activity dispersal in each component. Given the constraints, we can write the Lagrangian function for the supply component as follows:

which will have the first order conditions:

Equations A.8c–A.8f simply represent the production constraints. A.8a and A.8b can be rearranged to show that the minimum cost will occur when the marginal cost of output due to increasing inputs and dispersal are at parity:

The same operation can be performed for the other two supply chain components to yield:

These in turn can be rearranged to imply that the summed marginal costs of output for inputs in all supply chain components is equal to zero at optimality:

It may also be helpful to recall Equations 14a and 14b in their expanded forms:

and

In words, greater investment in sourcing production factors (capital and labor) may be precipitated by (1) increasing marginal input costs, (2) rising technological sourcing productivity, (3) greater investment in dispersal, (4) falling efforts to predate goods, (5) greater importance of the disjuncture between supply and production, and supply and distribution, and (6) falling importance of the budget constraint. Likewise, greater investment in dispersal may be associated with (1) greater marginal dispersal costs, (p.198) (2) falling investment (or inability to invest further) in inputs, (3) falling technological productivity, (4) rising efforts to predate goods, (5) greater importance of the disjuncture between supply and production, and (6) falling importance of the disjuncture between supply and distribution and of the budget constraint. Notice, then, that rising predation in the supply chain is inversely related to investment in the supply process (e.g., hiring truck drivers). Therefore, the rate at which predation decreases such investment diminishes as predation rises. That is, the response is sharpest at the onset of predation, less as it worsens. The effect of predation on dispersal investment is more equivocal. Dispersal investment (e.g., paying more petty traders) will tend to grow with more predation (again diminishing at higher levels of predation) if it is more important to maintain the sync between supply and distribution than it is between supply and production. However, dispersal investment will tend to shrink with predation if it is more important to main the sync between supply and production than between supply and distribution. The former would tend to be the case when the inputs are relatively valuable, the latter when the inputs are relatively low-value.

## Notes:

(^{1})
The described function for each supply chain component now begins to resemble a Cobb-Douglass production function of the form $Y=A{L}^{\alpha}{K}^{\beta}$, except for the addition of ${\rho}_{i}$ to model the risk of predated inputs, the combination of the labor and capital terms in one, and the exclusion of the input elasticities as exponents.

(^{2})
Firms also recognize that choosing high-value inputs in any process heightens the risk of predation, and so may try to substitute low-value inputs into each process.