The simplest model of optimal resource depletion is the so-called ‘cake-eating’ problem in which welfare is a discounted integral of B

C utility, utility is a function of consumption, and consumption is equal to the amount of the

(non-renewable) resource extracted. That is:

Ct = Rt and Ft = −Rt

(a) Obtain the Hamiltonian, and the necessary first-order conditions for a welfare maximum.

(b) Interpret the first-order conditions.

(c) What happens to consumption along the optimal path?

(d) What is the effect of an increase in the discount rate?

W UC t t t