1. An economy produces manna according to the production function y = a), when y is the rate of production and K is the stock of capital. Manna can be consumed or invested: y = c + i where c is the rate of consumption and i. is the rate of investment Consumption generates utility according to the function u(c). Both utility and production functions are twice-continuously differentiable; the production function is strictly increasing and concave, while utility is concave with a global maximum at some c*. The capital stock evolves according K = i - δK.

There is no discounting, and the economy ends at time T (which is known). The goal of the economy's planner is to maximize the flow of utility over the planning horizon. I want you to solve this dynamic programming problem using the calculus of variations approach.

a) Write out the optimization problem, including any relevant constraints.

b) Produce the Euler equation for this problem

c) Give as much of the solution as you can obtain, and provide an intuitive explanation of your results.

2. Robinson Crusoe is stranded on a very small, unpleasant island. With the wrecking of his ship upon the rocks, his entire ouzo has been lost, except for several bunny rabbits that have managed to swim to shore.

Now all good biologists (like Crusoe, who got an A in biology) know that, if leR to itself, the growth of an animal population will follow a logistic process. Letting Crusoe's consumption at time t be C(t) and the bunny population at time t be S(t), we may therefore describe the evolution of the bunny population by

S(t) = g(S(t)- c(t))
       = S(t)(0[1-(S(t)/K]- C(t)

where K is a known constant Crusoe sees no hope of ever being rescued and assumes he will be forced to sustain himself forever on bunny stew; he derives utility U(C) from consuming C units of bunnies. Crusoe's time rate of preference is Ρ.

a) Formally mite out Crusoe's optimal control problem, including all relevant information.

b) Find the maximum principle, give an economically intuitive interpretation; and provide any relevant trinsvasa. conditions.

c) Discuss the paths for Crusoe's optimal consumption and the bunny population.

d) Assuming that U(C) = ln(C), solve for the steady state values of S and C, and then construct a phase diagram for the problem.

3. A firm has an initial endowment of yo units of an asset, which it can ex-tract and sell. At every point in time, the asset sells for the price p. At any point in time where extraction takes place, total costs are C_i, if the firm does not extract, costs are 0. Let the firm's extraction rate at time t be u(t) and let the remaining amount of the asset possessed by the firm be y(t).

The firm has access to two extraction technologies, call them type 1 and type 2. The firm starts off using type 1, and can switch to type 2 at any time it wishes; to do so entails a one-time expense of A. At any point where the firm extracts using the type 1 technology, its maximum extraction rate is Ω_i(t) = ω₁ + δ₁y(t). Likewise, its maximum rate of extraction using the type 2 technology is Ω₂(t) = ω2 + δ₁y(t). The parameters satisfy, ω₂ > ω₁ oil and δ₁ > δ₁. At every point in time, Y. = -u. Using the optimal control framework, find the firm's optimal extraction path, the optimal time (if any) at which it switches technologies, and optimal time at which it ceases extraction altogether.