Animal Disease and the Industrialization of Agriculture

Abstract

Descartes’ perspective that animals are machines, and perhaps little more, is a matter of great ethical disquiet in contemporary society (Cottingham 1978). Sweeping developments in the life sciences since about 1950 have provided technical insights on how to control life and growth in ways that have made the animal-as-machine analogy more real. The moral principles and economic tradeoffs at issue have become more clearly defined, in large part because production sciences and the systems they support demand clear definition of the production environment. Animal disease confounds control efforts, and also belies the attitude that an animal’s technical performance can be abstracted from its environs.

Appendix

Second-order conditions: Compute

$$ {T_{{QQ}}}( \cdot ) = \frac{{{{\bar{B}}_{{QQ}}}( \cdot )}}{{G( \cdot )}}\ >\ 0,\quad {T_{{zz}}}( \cdot ) = - \frac{{{G_{{zz}}}( \cdot )\bar{B}( \cdot )}}{{{{[G( \cdot )]}^2}}} \ >\ 0, $$

(5.29)

where optimality conditions ${T_Q}( \cdot ) = 0$ and ${T_z}( \cdot ) = 0$ have been applied. The cross-derivative for the average cost function is ${T_{{Qz}}}( \cdot ) = {\bar{J}_Q}( \cdot )r/G( \cdot )\ < \ 0$, where optimality condition (5.26a) and computation ${\bar{J}_Q}( \cdot ) = [Q{J_Q}( \cdot ) - J( \cdot )]/{Q^2}\mathop{ = }\limits^{\rm{sign}} {J_Q}( \cdot ) - \bar{J}( \cdot ) \ < \ 0$ have been applied. Finally,

$$ \Phi \equiv {T_{{QQ}}}( \cdot ){T_{{zz}}}( \cdot ) - {[{T_{{Qz}}}( \cdot )]^2} = - \overbrace{{\frac{{{G_{{zz}}}(z)\bar{B}( \cdot ){{\bar{B}}_{{QQ}}}( \cdot )}}{{{{[G( \cdot )]}^3}}}}}^{{ < 0}} - \overbrace{{\frac{{{{[{{\bar{J}}_Q}( \cdot )]}^2}{r^2}}}{{{{[G( \cdot )]}^2}}}}}^{{ > 0}}. $$

(5.30)

Convexity on $T( \cdot )$ requires $\Phi \ >\ 0$. If cost parameter $r \ > \ 0$ is small (i.e., the biosecurity innovation has been well developed) or economies of scale are small (i.e., $|{\bar{J}_Q}(Q)|$ is small) then $\Phi \ > \ 0$ is likely. Otherwise (5.30) may be violated, so that discontinuous scale and input responses result from a small parameter change. Henceforth we assume $\Phi\ > \ 0$.

Subsidy and innovation: Define $H( \cdot ) = G( \cdot ) - {G_z}( \cdot )z\ > \ 0$ where $H( \cdot )\ >\ 0$ is due to ${G_z}( \cdot ) \ < \ G( \cdot )/z$ on $z \geq 0$ whenever $G( \cdot )$ is increasing and concave in $z$ with $G(0;v)\ >\ 0$. Using (5.26a, b):

$$ {T_{{Qr}}}( \cdot ) = \frac{{{{\bar{J}}_Q}( \cdot )z}}{{G( \cdot )}} \ < \ 0,\quad {T_{{zr}}}( \cdot ) = \frac{{\bar{J}( \cdot )H( \cdot )}}{{{{[G( \cdot )]}^2}}} \ > \ 0. $$

(5.31)

Differentiate system (5.26a, b) completely with respect to $(Q,z,r)$ and then invert to obtain:

$$ \frac{{{\text{d}}{Q^{*}}}}{{{\text{d}}r}} = \frac{{{T_{{Qz}}}{T_{{zr}}} - {T_{{zz}}}{T_{{Qr}}}}}{\Phi } = \overbrace{{\frac{{{{\bar{J}}_Q}( \cdot )r\bar{J}( \cdot )H( \cdot )}}{{{{[G( \cdot )]}^3}\Phi }}}}^{{ < 0}} + \overbrace{{\frac{{{{\bar{J}}_Q}( \cdot )\bar{B}( \cdot ){G_{{zz}}}( \cdot )z}}{{{{[G( \cdot )]}^3}\Phi }}}}^{{ > 0}}, $$

$$ \frac{{{\text{d}}{Q^{*}}}}{{{\hbox{d}}r}} = \frac{{{T_{{Qz}}}{T_{{zr}}} - {T_{{zz}}}{T_{{Qr}}}}}{\Phi } = \overbrace{{\frac{{{{\bar{J}}_Q}( \cdot )r\bar{J}( \cdot )H( \cdot )}}{{{{[G( \cdot )]}^3}\Phi }}}}^{{ < 0}} + \overbrace{{\frac{{{{\bar{J}}_Q}( \cdot )\bar{B}( \cdot ){G_{{zz}}}( \cdot )z}}{{{{[G( \cdot )]}^3}\Phi }}}}^{{ > 0}}. $$

(5.32)

Both expressions are indeterminate in sign without further information. Plausible functional forms are readily identified such that both derivatives are positive.

For the own-price response, $- {T_{{QQ}}}{T_{{zr}}}/\Phi \ <\ 0$ characterizes the direct effect. The indirect effect, through the effect on $Q$, is ${T_{{Qz}}}{T_{{Qr}}}/\Phi \ >\ 0$. This is because an increase in scale lowers the unit cost of the biosecurity input due to scale economies emphasizing that input’s cost. For the scale response, the direct effect is represented by $- {T_{{zz}}}{T_{{Qr}}}/\Phi \ >\ 0$. The indirect effect on scale when mediated through the biosecurity input is ${T_{{Qz}}}{T_{{zr}}}/\Phi\ < \ 0$.

External effects. Differentiate system (5.26a, b) completely with respect to $(Q,z,v)$ and then use the optimality conditions:

$$ \left( {\begin{array}{*{20}{c}} {\frac{{{{\bar{B}}_{{QQ}}}( \cdot )}}{{G( \cdot )}}} & {\frac{{{{\bar{J}}_Q}( \cdot )r}}{{G( \cdot )}}} \\ {\frac{{{{\bar{J}}_Q}( \cdot )r}}{{G( \cdot )}}} & { - \frac{{{G_{{zz}}}( \cdot )\bar{B}( \cdot )}}{{{{[G( \cdot )]}^2}}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} {\frac{{{\text{d}}{Q^{*}}}}{{{\text{d}}v}}} \\ {\frac{{{\text{d}}{z^{*}}}}{{{\text{d}}v}}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} 0 \\ {\frac{{R( \cdot )\bar{B}( \cdot )}}{{G( \cdot )}}} \\ \end{array} } \right). $$

(5.33)

So

$$ \frac{{{\text{d}}{Q^{*}}}}{{{\hbox{d}}v}} = - \frac{{{{\bar{J}}_Q}( \cdot )r\bar{B}( \cdot )}}{{\Phi {{[G( \cdot )]}^2}}}R( \cdot )\mathop{ = }\limits^{\rm{sign}} R( \cdot ),\quad \frac{{{\text{d}}{z^{*}}}}{{{\hbox{d}}v}} = \frac{{{{\bar{B}}_{{QQ}}}( \cdot )\bar{B}( \cdot )}}{{\Phi {{[G( \cdot )]}^2}}}R( \cdot )\mathop{ = }\limits^{\rm{sign}} R( \cdot ). $$

(5.34)

Explicit solution

Set $C(Q) = {Q^{{{\beta_1} + 1}}},{\beta_1} \ >\ 0$, and $J(Q) = {Q^{{1 - {\beta_2}}}},{\beta_2} \in (0,1)$.¹⁵ Writing $\lambda = {\beta_2}/{\beta_1}$, (5.26a) provides ${Q^{{{\beta_1} - 1}}} = \lambda {Q^{{ - (1 + {\beta_2})}}}rz$ and (5.26a) and (5.26b) solve as:

$$ \frac{{G({z^{*}};v)}}{{{G_z}({z^{*}};v)}} = (1 + \lambda ){z^{*}}, $$

(5.35a)

$$ {Q^{*}}({z^{*}}) = {(\lambda r{z^{*}})^{{1/({\beta_1} + {\beta_2})}}}. $$

(5.35b)

So for these functional forms, and regardless of the choice of some $G(z)$ function that supports an interior solution, the $z$ that minimizes unit cost is independent of unit cost parameter $r$.

Finally, and as a specification distinct from $G( \cdot ) = (1 - v){\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {G}} + v\hat{G}( \cdot )$ previously discussed, let $G(z;v) = z/[\mu + \sigma (v)z]$. This is the logistic function form. With constant $\mu\ > \ 0 \$and $\sigma (v) \ >\ 0$, the external factor influences this distribution through ${\sigma_v}(v)\ <\ 0$ so that ${G_v}( \cdot ) \ >\ 0$ and an increase in the factor reduces the unit cost of output. Then ${G_z}( \cdot ) = \mu /{[\mu + \sigma ( \cdot )z]^2}$, ${G_z}( \cdot )/G( \cdot ) = \mu /\{ z[\mu + \sigma ( \cdot )z]\}$, and $R( \cdot ) = - \mu {\sigma_v}( \cdot )/{[\mu + \sigma ( \cdot )z]^2} \ >\ 0$. An improvement in external biosecurity increases productivity, and also the marginal productivity of the biosecurity input when calculated in percent terms.

System (5.35a, b) then solves as

$$ ({Q^{*}},{z^{*}}) \in \left\{ {(0,0),\left( {{{\left( {\frac{{{\lambda^2}\mu r}}{{\sigma ( \cdot )}}} \right)}^{{1/({\beta_1} + {\beta_2})}}},\frac{{\mu \lambda }}{{\sigma ( \cdot )}}} \right)} \right\}. $$

(5.36)

It will be shown later that $({Q^{*}},{z^{*}}) = (0,0)$ is not optimal, leaving only the interior solution. Notice that the equilibrium value for productivity is $G( \cdot ) = \lambda /[(1 + \lambda )\sigma (v)]$, which is increasing in the beneficial external natural or socioeconomic factor, $v$.

To understand why ${\hbox{d}}{Q^{*}}/{\hbox{d}}r \ > \ 0$ in (5.36), consider the direct and indirect effects of an increase in the price of biosecurity. An increase in $r$ increases the incentive to reduce unit costs by increasing scale. This is the direct effect on scale. The indirect effect, via complementarity relation (5.27), is that a higher value of $r$ reduces the incentive to use the biosecurity input and this reduces the incentive to increase scale. Relation (5.36) shows that the direct effect wins out.

Expression ${z^{*}}$ in (5.36) also indicates that the direct and indirect effects of an increase in the value of $r$ exactly offset. Although (5.36) does not show ${\hbox{d}}{z^{*}}/{\hbox{d}}r\ > \ 0$, this is possible, i.e., there could conceivably be a positive own-price effect in long-run equilibrium. The possibility arises because the endogenous scale choice alters the effective unit cost of the input in a manner broadly similar to how the income effect mediates price response in demand theory. As with Giffen goods, the indirect effect can overwhelm the direct effect.

It follows from ${\sigma_v}( \cdot )\ < \ 0$ that ${\hbox{d}}{z^{*}}/{\hbox{d}}v\ > \ 0$ and ${\hbox{d}}{Q^{*}}/{\hbox{d}}v \ >\ 0$, so that more external biosecurity coaxes out more internal biosecurity and increases scale. The example supports the hypothesis that more public health inputs do not crowd out (i.e., do complement) private activities to safeguard animal health but do promote large-scale production. In the example, strengthening the external biosecurity environment complements incentives for internal biosecurity and also encourages a scaling up of production activities. With better external biosecurity, the animal production format is more likely to become industrial than back-yard.

Finding the optimum in the explicit solution: Pose the problem as having two stages:

$$ \mathop{{\min }}\limits_z \mathop{{\min }}\limits_{{Q|z}} T( \cdot ). $$

(5.37)

Upon inserting (5.35b) into the given $C( \cdot )$ and $J( \cdot )$ functions, cost function (5.25) becomes:

$$ \begin{array}{*{20}{c}} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T}}( \cdot ) = \frac{{{{({Q^{*}})}^{{{\beta_1}}}} + {{({Q^{*}})}^{{ - {\beta_2}}}}r{z^{*}}}}{{{z^{*}}/[\mu + \sigma ( \cdot ){z^{*}}]}} = \mu \frac{{{{({Q^{*}})}^{{{\beta_1}}}}}}{{{z^{*}}}} + \sigma ( \cdot ){{({Q^{*}})}^{{{\beta_1}}}} + {{({Q^{*}})}^{{ - {\beta_2}}}}r[\mu + \sigma ( \cdot ){z^{*}}]} \\ { = \Gamma \times [\mu {{({z^{*}})}^{{ - {\beta_2}/({\beta_1} + {\beta_2})}}} + \sigma ( \cdot ){{({z^{*}})}^{{{\beta_1}/({\beta_1} + {\beta_2})}}}],\quad \Gamma \equiv [{\lambda^{{{\beta_1}/({\beta_1} + {\beta_2})}}} + {\lambda^{{ - {\beta_2}/({\beta_1} + {\beta_2})}}}]{r^{{{\beta_1}/({\beta_1} + {\beta_2})}}} > 0,} \\ \end{array} $$

(5.38)

which is the solution to the inner conditional optimization problem in (5.37). When ${z^{*}} = 0$ then ${({z^{*}})^{{{\beta_1}/({\beta_1} + {\beta_2})}}} = 0$ but ${({z^{*}})^{{ - {\beta_2}/({\beta_1} + {\beta_2})}}} \to \infty$ so that $({Q^{*}},{z^{*}}) = (0,0)$ maximizes unit cost and cannot be optimal. To establish the problem’s overall convexity, differentiate the final expression for ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {T}}( \cdot )$ in (5.38) twice with respect to the remaining endogenous variable, $z$, and insert ${z^{*}} = \mu \lambda /\sigma (v)$ to obtain

$$ \frac{{\Gamma {\beta_2}[\mu ({\beta_1} + 2{\beta_2}) - \sigma ( \cdot ){\beta_1}{z^{*}}]{{({z^{*}})}^{{ - (2{\beta_1} + 3{\beta_2})/({\beta_1} + {\beta_2})}}}}}{{{{({\beta_1} + {\beta_2})}^2}}} = \frac{{\Gamma {\beta_2}\mu {{({z^{*}})}^{{ - (2{\beta_1} + 3{\beta_2})/({\beta_1} + {\beta_2})}}}}}{{{\beta_1} + {\beta_2}}} \ >\ 0. $$

(5.39)

Thus the problem is indeed convex and the interior solution is optimal.