2. Government spending in the RCK model. Consider the following economy that is very similar to the economy in the RCK model except that we incorporate government spending into the model. Let g₁ denote government spending per worker in period t. The planner can choose the sequence of government spending. The production function takes the form y₁ = f(kt, It), where k is capital per worker and capital depreciates at rate 8. The production function is strictly concave, twice differentiable, and satisfies the usual conditions. In each period, output is divided among consumption, investment and government spending. The planner seeks to maximizes Σ β' u(c) t=0 where u(c) is twice differentiable and strictly concave. Suppose that the initial condition of capital per worker is given, ko > 0 and the transversality condition is satisfied. (a) Set up the maximization problem faced by the planner. (b) Derive the set of equations that characterize an interior solution to the planner's problem. (c) Derive the set of equations that characterize a steady state. (d) Describe how the steady state levels of capital per worker and government spending per worker vary as a function of the discount factor B. Give intuition for your answer. k*. (e) Suppose that y₁ = f(kt, gt) = kg where 0 < a+ < 1. Consider an initial ko With the help of a phase diagram, qualitatively describe the dynamic paths of (kt. 9t, Ct) in the model.
2. Government spending in the RCK model. Consider the following economy that is very similar to the economy in the RCK model except that we incorporate government spending into the model. Let g₁ denote government spending per worker in period t. The planner can choose the sequence of government spending. The production function takes the form y₁ = f(kt, It), where k is capital per worker and capital depreciates at rate 8. The production function is strictly concave, twice differentiable, and satisfies the usual conditions. In each period, output is divided among consumption, investment and government spending. The planner seeks to maximizes Σ β' u(c) t=0 where u(c) is twice differentiable and strictly concave. Suppose that the initial condition of capital per worker is given, ko > 0 and the transversality condition is satisfied. (a) Set up the maximization problem faced by the planner. (b) Derive the set of equations that characterize an interior solution to the planner's problem. (c) Derive the set of equations that characterize a steady state. (d) Describe how the steady state levels of capital per worker and government spending per worker vary as a function of the discount factor B. Give intuition for your answer. k*. (e) Suppose that y₁ = f(kt, gt) = kg where 0 < a+ < 1. Consider an initial ko With the help of a phase diagram, qualitatively describe the dynamic paths of (kt. 9t, Ct) in the model.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:2. Government spending in the RCK model. Consider the following economy that is very
similar to the economy in the RCK model except that we incorporate government spending into
the model. Let g, denote government spending per worker in period t. The planner can choose
the sequence of government spending. The production function takes the form Yt = f(kt, 9t),
where ki is capital per worker and capital depreciates at rate d. The production function is
strictly concave, twice differentiable, and satisfies the usual conditions. In each period, output
is divided among consumption, investment and government spending. The planner seeks to
maximizes
3t u(c)
t=0
where u(c) is twice differentiable and strictly concave. Suppose that the initial condition of
capital per worker is given, ko > 0 and the transversality condition is satisfied.
(a) Set up the maximization problem faced by the planner.
(b) Derive the set of equations that characterize an interior solution to the planner's problem.
(c) Derive the set of equations that characterize a steady state.
(d) Describe how the steady state levels of capital per worker and government spending per
worker vary as a function of the discount factor 3. Give intuition for your answer.
f(kt, 9t) = ke g, where 0 < a + y < 1. Consider an initial ko 7k*.
(e) Suppose that y =
With the help of a phase diagram, qualitatively describe the dynamic paths of (kt, It, Ct)
in the model.
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VIEWStep 3: Derive the set of equations that characterize an interior solution to the planner's problem
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VIEWStep 5: Describe how k* and g* per worker vary as a function of the discount factor B
VIEWStep 6: Describe dynamic paths of Kt, gt, Ct for the following state
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