Reference no: EM133589636

Mathematical Economics

Problem 1:

In an economy, let S(t) denote the stock of the fuel and E(t) the rate of extraction of fuel (and energy use) at any time t, with S(0) = S₀. Then, we have S?≡ dS/dt = -E. Energy use, E, makes possible the production of goods and services for consumption, C, which creates utility but generates a flow of pollution, and P, which creates disutility. The consumption and pollution functions are C ≡ C(E) = E + 1 and P ≡ P (E) = √E. The social utility function depends on consumption and pollution: U ≡ U (C, P ) = ln (C) - P² = ln(E + 1) - 1E². An Energy Board is appointed to plan and chart the optimal time path of the energy-use variable E to maximize the social utility over a specified period [0, 1].

1: Write the Board's dynamic optimization problem.
2: Write the Hamiltonian function of the Board's problem and check whether the conditions of the maximum principle are sufficient for the social utility maximization problem.
3: Solve the Board's problem and discuss how the optimal control, state, and costate vari- ables evolve with time. [20 points]

Problem 2:

A firm's net total payoff is given by P (K, I) ≡ π(K) - C(I) = K + 1 - 1/2 I², where Π is the profit, K is the capital stock, C is the adjustment cost, and I is the net investment. The firm's goal is to choose an optimal investment path I to maximize the total net payoff in a finite horizon [0, 1], discounted at a positive rate ρ. The initial capital stock is given by K(0) = K₀.

We recall that the marginal change in the capital with time is the investment: K? = I.

1: Write the firm's dynamic optimization problem.
2: Write the current-value Hamiltonian functions of the firm's problem and the maximum- principle conditions.
3: Using the current-value Hamiltonian, solve the firm's problem.
4: Using the current-value Hamiltonian functions, check whether the Arrow conditions for the maximum principles hold.