Problem 2. A consumer has initial wealth yo and discount rate o. Her flow utility function is u(c) = log(c). The consumer's problem is: max cy. [ e-of log c(t )at s.t. dy dt = ry(t) - c(t) y(0) = yo y(T) 20 Find the optimal paths of consumption, c(t), and wealth, y(t). Hint-suggested steps: (1) Setup the Lagrangian or Hamiltonian and write down the first order conditions. (2) The first order condition for A is of the form -dA = AA(t). The solution to this differential equation is A(t) = A(0)e-At. (3) Solve for c(t) up to the unknown constant A(0). (4) Argue that y(T) = 0. (5) Substitute your equation for c(t) into dy = ry(t) - c(t). (6) A differential equation of the form dy = ry(t) - AeBt dt has solution y(t) = Kert _ _A Bt B-r for some constant K. (7) Use the initial condition, y(0) = yo, and terminal condition, y(T) = 0, to solve for K and A(0)