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Effect of infrastructure design on commons dilemmas

Model Type
SES Library XPP model

 

The authors address the question of how infrastructure design affects SES sustainability in two stages. First, they explore the effects of design variations in shared infrastructure on long-term system behavior in a model system. They examine two types of distribution infrastructure, one with and one without upstream−downstream asymmetry, and different threshold characteristics of infrastructure maintenance. Second, they evaluate how these design variations influence the robustness of system function to an economic shock.  

The model assumes that there are N farming households spread across two villages (village 1 and village 2) that manage a shared irrigation infrastructure. Each farmer is endowed with the same amount of available labor (l) each year and the same acreage (a). A farmer appropriates q units of water from the system and allocates labor among three activities: farming, maintaining infrastructure and outside employment at a wage rate w. Governance is represented in the model by the following rules. The expected maintenance labor contribution is proportional to the farmer's acreage (assumed to be same for all farmers in this model). Water allocations are also proportional to acreage, but only among the water rights holders. Only farmers who contributed labor to the infrastructure maintenance before the planting season obtain water rights. Farmers choose between two strategies: group conformist (G) and opportunist (O). Gs follow and enforce rules, and strive to maximize the total welfare of the two villages. Os break the rules and attempt to maximize the individual net income. The model tracks the fraction of Gs in the village i and the resulting performance of the irrigation system.

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Name Initial Value Description
$X_1$ - Fraction of group conformists (G) in village 1
$X_2$ - Fraction of group conformists (G) in village 2
Name Default Value Description
$j$ - output elasticity of farming labor for agricultural yield
$k$ - output elasticity of irrigated water for agricultural yield
$p$ - price per unit of agricultural yield
$b$ - productivity coefficient for the inputs of production
$N_1$ - number of farmers in village 1
$N_2$ - number of farmers in village 2
$l$ - available labor per farmer
$a_1$ - acreage of farmer in village 1
$a_2$ - acreage of farmer in village 2
$I_{max}$ - maximum infrastructure efficiency
$\psi$ - half-saturation point of $L_m$ yielding $I_{max} / 2$ infrastructure efficiency
$\epsilon$ - half-width of the threshold slope for infrastructure maintenance
$S$ - volume of river discharge
$w$ - wage for outside employment
$R$ - total amount of freely available alternative water
$\gamma_s$ - maximum enforcement costs for monitoring opportunists in the same village
$\gamma_o$ - maximum enforcement costs for monitoring opportunists in the other village
$\delta$ - maximum penalty cost imposed on $O$s
$\sigma$ - tolerance for water theft shown by $G$s when water is abundant in the system (\leq 1.0)
$$\begin{equation}\r \frac{dX_i}{dt} = X_i(\pi_i^G - \overline{\pi_i}) \r \end{equation}$$
$$\begin{equation}\r Q = I(L_m)S(t)\r \end{equation}$$
$$I(L_m) = \r \begin{cases}\r 0 & 0 \leq L_m < \psi - \epsilon \\\r \frac{I_{max}}{2\epsilon} (L_m - \psi + \epsilon) & \psi - \epsilon \leq L_m \leq \psi + \epsilon\\\r I_{max} & \psi + \epsilon \leq L_m \leq L\r \end{cases}$$
$$\begin{equation}\r l = l_f + l_m + l_e\r \end{equation}$$
$$\begin{equation}\r \pi_i = pb(l_f)^j(q_i + r_i)^ka_i^(1-j-k) + wl_e\r \end{equation}$$
$$\begin{equation}\r \begin{aligned}\r & \underset{L_f,L_m}{ ext{max}}\r {\Pi = pbL_f^j[I(L_m)S + R]^k A^(1-j-k) + w L_e} \\\r & ext{s.t.} L_f + L_m + L_e = L \\\r \end{aligned}\r \end{equation}$$
$$\begin{equation}\r \pi_1^G = pb(l_f^G)^j(q_1^G + r_1)^ka_1^(1-j-k) + w(l_e^G) - [\gamma_s(1 - X_1) + \gamma_o(1 - X_2)] \\\r \pi_2^G = pb(l_f^G)^j(q_2^G + r_2)^ka_2^(1-j-k) + w(l_e^G) - [\gamma_s(1 - X_2) + \gamma_o(1 - X_1)]\r \end{equation}$$
$$\begin{equation}\r \pi_1^O = pb(l_f^O)^j(q_1^O + r_1)^ka_1^(1-j-k) + w(l_e^O) - \delta\bigg(1 - \sigma \frac{Q(L_m)}{Q(L_m*)}\bigg)q_1^O\frac{(X_1 + X_2)}{2}\\\r \pi_2^O = pb(l_f^O)^j(q_2^O + r_2)^ka_2^(1-j-k) + w(l_e^O) - \delta\bigg(1 - \sigma \frac{Q(L_m)}{Q(L_m*)}\bigg)q_2^O\frac{(X_1 + X_2)}{2}\r \end{equation}$$
$$\begin{equation}\r Q(L_m) = I(L_m)S\r Q(L_m^*) = I(L_m^*)S\r Abdundance = \frac{Q(L_m)}{Q(L_m^*)}\r \end{equation}$$
$$\begin{equation}\r q_1^O \leq \frac{Q}{N_1} \\\r q_1^G = min[\frac{Q}{N^G} , (Q - Q_1^O) / N_1^G]\r \end{equation}$$
$$\begin{equation}\r q_2^O \leq \frac{Q - Q_1^O - Q_1^G)}{N_2}\\\r q_2^G = \frac{[Q - Q_1^O - Q_1^G - Q_2^O]}{N_2^G}\r \end{equation}$$
2014_may17_asymmetry.ode
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