4. Consider an infinite horizon representative agent with the utility function u(c(t)), where u' > 0 and u"

4. Consider an infinite horizon representative agent with the utility function u(c(t)), where u' > 0 and u" < 0. Suppose that c(t), w(t), and x(t) are the consumption level, wage rate, and assets of the agent at time t. The interest rate, r(t), and the wage rate are exogenously- determined in competitive markets. Let p and n be the discount and population growth rates, respectively. The agent's inter-temporal optimisation problem can be written as: max dt s. t. = w(t) + (r(t) n)x(t) c(t) and x(0) is given c(t) 1 Write down the Hamiltonian equation. Derive the first-order conditions. Find the exact Euler equation. What 'is the complete system of first-order equations?