Continuous time Ramsey: Let the utility and production functions of the representative household and firm, in the continuous time Ramsey model, are given by, respectively,

u(C(t)) = C(t)^1/ 1 .... (1)

Y (t) = K(t) ^ L(t)^ 1 .... (2)

where c (t) represents the consumption of a household member; Y (t), K (t) and L(t) denote output, capital and labor, respectively. Each household member owns a unit of labor that supplies to firms and earn a wage w (t). The household also owns an asset A(t) at time t, and receives interest income for exchange of its service. The size of the household grows at n rate.

(a) Define the household problem, derive the optimal conditions and the Euler equation and interpret.