3. The simplest model of optimal resource depletion is the so-called 'cake-eating" problem in which welfare is a discounted integral of utility, utility is a function of consumption, and consumption is equal to the amount of the (non-renewable) resource extracted. (This model is analysed further in Heal (1981); Solow (1974) examines the intuition lying behind the results). The model is: W = [_U(C,)e-Pldt Ct = Rt and S, = -R. So a given constant. St 2 0 for all t e (0, co) We assume that U(C) is strictly concave. (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?2. Using Equation 14.15 in the text (that is, the Hotelling efficiency condition), demonstrate the consequences for the efficient extraction of a non-renewable resource of an increase in the social discount rate, p.3. Recycling of exhaustible resources can relax the constraints imposed by finiteness of non-renewable resources. What determines the efficient amount of recycling for any particular economy? 1. Using the relationship r= p+nc demonstrate that if the utility function is of the special form U(C) = C, the consumption rate of discount (r) and the utility rate of discount are identical.1. Are non-renewable resources becoming more or less substitutable by other productive inputs with the passage of time? What are the possible implications for efficient resource use of the elasticity of substitution between non-renewable resources and other inputs becoming (a) higher, and (b) lower with the passage of time? 2. Discuss the possible effects of technical progress on resource substitutability