the agent discounts future utility at rate , with and . Here, is the state variable which evolves according to the above equation, and is the control variable. The Hamiltonian becomes The optimality conditions are in addition to the transversality condition . If we let , then log-differentiating the first optimality condition with respect to yields Inserting this equation into the second optimality condition yields which is known as the Keynes–Ramsey rule, which gives a condition for consumption in every period which, if followed, ensures maximum lifetime utility. References Further reading Optimal control