(a) Formulate the agent's dynamic problem of maximizing her life-time utility with a discounting factor p. (b) Write the Hamiltonian with a co-state variable and the maximum principle conditions. (c) Using the maximum principle conditions from part (b) above, explain the evolution of the co-state variable (i.e.,). 77
(a) Formulate the agent's dynamic problem of maximizing her life-time utility with a discounting factor p. (b) Write the Hamiltonian with a co-state variable and the maximum principle conditions. (c) Using the maximum principle conditions from part (b) above, explain the evolution of the co-state variable (i.e.,). 77
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Question 4.
Suppose that an infinitely lived agent faces the following functions in each period:
Utility function:
u(c) = log(c)
Income (or, output):
y = AkªG1-a
After-tax income allocation:
(1 – t)AKªG²-a = c + k
where c denotes consumption, k denotes capital, i denotes a (constant) proportional
income tax, G denotes government expenditures, and k (= k = 1) denotes investment
in each period. The total population of agents is fixed and normalised to one.
%3D
dt
(a) Formulate the agent's dynamic problem of maximizing her life-time utility with a
discounting factor p.
(b) Write the Hamiltonian with a co-state variable 1 and the maximum principle
conditions.
(c) Using the maximum principle conditions from part (b) above, explain the
evolution of the co-state variable (i.e.,
(d) Using the current valued co-state variable, derive the growth of cons mption
((i.e., 5).
(e) If the government runs a balanced budget (i.e., G = ty), show that the growth
rate of consumption (i.e., )
is a function of t. What is the effect of change in t on
the growth rate of consumption?
(f) Rewrite the expression for the growth rate of consumption derived in part (e) as a
linear first-order autonomous differential equation and find the optimal path of
consumption.
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