16. Consider an economy with a continuum of agents, with $i \in [0, 1]$. To formalize the consequences of
decentralized conflict over economic distribution as simply as possible, suppose that each agent competes
against the economy's average. Suppose agent $i$ allocates an amount of time $l_i$ to labor, an amount $x_i$ to
protection and an amount $z_i$ to predation to solve the following problem:
$$ \max_{(l_i, x_i, z_i)} \{p(x_i, z) f(l_i) + (1 - p(x, z_i)) f(l) \} $$
subject to $l_i + x_i + z_i \leq 1$, where $(f)$ denotes the production technology and $p$ denotes the conflict technology.
(16.a) What are the implications of the above problem for the optimal choices of $l_i, x_i$ and $z_i$? Explain.
(16.b) Define a symmetric equilibrium of the model.
(16.c) Now suppose that the production technology is given by $f(l_i) = Al_i$ and the conflict technology is
given by
$$ p(x_i, z) = \frac{\pi x_i^m}{\pi x_i^m + z^m} \text{ and } p(x, z_i) = \frac{\pi x^m}{\pi x^m + z_i^m} $$
if $x + z > 0$, with $m \in (0, 1)$ and $\pi > 0$, and with $p(0, 0) = p_0 \in [0, 1]$.
(16.d) If $\pi = \infty$, what is the optimal allocation $(l_i, x_i, z_i)$?
(16.e) Suppose $\pi < \infty$, and suppose that the choices of protection and predation in a symmetric equilib-
rium are such that $x^* = z^*$. What does this imply for the security of property? Explain.
