a. Formulate the planning problem for this economy in the space of sequences and form the pertinent Lagrangian. Find a formula for the optimal steady state level of capital. How does a permanent increase in ψ affect the steady values of k, c and x? b. Formulate the planning problem for this economy recursively (i.e., compose a Bellman equation for the planner). Be careful to give a complete description of the state vector and its law of motion. (‘Finding the state is an art.’) c. Formulate an (Arrow-Debreu) competitive equilibrium with time 0 trades, assuming the following decentralization. Let the household own the stocks of capital and labor and in each period let the household rent them to the firm. Let the household choose the investment rate each period. Define an appropriate price system and compute the first-order necessary conditions for the household and for the firm. d. What is the connection between a solution of the planning problem and the competitive equilibrium in part (c)? Please link the prices in part (c) to corresponding objects in the planning problem. e. Assume that k0 is given by the steady state value that corresponds to the assumption that ψt had been equal to 1 forever, and had been expected to remain equal to 1 forever. Qualitatively describe the evolution of the economy from time 0 on. Does the jump in ψ at t = 4 have any effects that precede it?
Anticipated productivity shift An infinitely lived representative household has preferences over a stream of consumption of a single good that are ordered by
Where u is a strictly concave, twice continuously differentiable one period utility function, β is a discount factor, and ct is time t consumption. The technology is:
Here f(kt)nt is output, where f’ > 0, f” > 0, f t is capital per unit of labor input, and nt is labor input. The household supplies one unit of labor inelastically. The initial capital stock k0 is given and is owned by the representative household. In particular, assume that k0 is at the optimal steady value for k presuming that ψt had been equal to 1 forever. There is no uncertainty. There is no government.
a. Formulate the planning problem for this economy in the space of sequences and form the pertinent Lagrangian. Find a formula for the optimal steady state level of capital. How does a permanent increase in ψ affect the steady values of k, c and x?
b. Formulate the planning problem for this economy recursively (i.e., compose a Bellman equation for the planner). Be careful to give a complete description of the state vector and its law of motion. (‘Finding the state is an art.’)
c. Formulate an (Arrow-Debreu) competitive equilibrium with time 0 trades, assuming the following decentralization. Let the household own the stocks of capital and labor and in each period let the household rent them to the firm. Let the household choose the investment rate each period. Define an appropriate price system and compute the first-order necessary conditions for the household and for the firm.
d. What is the connection between a solution of the planning problem and the competitive equilibrium in part (c)? Please link the prices in part (c) to corresponding objects in the planning problem.
e. Assume that k0 is given by the steady state value that corresponds to the assumption that ψt had been equal to 1 forever, and had been expected to remain equal to 1 forever. Qualitatively describe the evolution of the economy from time 0 on. Does the jump in ψ at t = 4 have any effects that precede it?
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