Consider a worker with a life expectancy T, over which he earns a wage w, receives constant rate interest r on his accumulated savings, or pays his debts at the same interest rate. So when the capital is k, the income will be w + rk. If the consumption is c (t), the accumulation of capital is given by the equation in the image. So k is the state variable and c (t) is the control variable. Suppose the worker has no heirs or received any inheritance, then k (0) = k (T) = 0. Find the optimal consumption path that maximizes the worker's utility function, U (c) = ln (c), if it is constrained by a discount rate .

dk =w+rk - C dt dk =w+rk - C dt