Consumer “Habit Persistence” problem (35%). Consider the following infinite horizon utility maximization
problem. Notice that utility depends on both current consumption and previous period
consumption. The habit persistence parameter shows how much of current consumption is influenced
by previous consumption patterns.
max[
Σ︁∞
=0
(ln + ln −1)]
s.t. + +1 ≤
,
> 0,
0 < < 1,
0 < < 1,
> 0,
0 > 0, and −1 given.
(a) Write down the Lagrangian for this optimization problem. (You do not have to solve it (yet), just
write down the Lagrangian equation.) (5 marks)
(b) State the control and state variables. (5 marks)
(c) Write down the Bellman equation. Follow the convention to drop the time subscripts - that is
express = , +1 = ′, −1 = −1 and so on. (5 marks)
(d) Derive the Euler equation. (8 marks)
(e) Show that the solution of the Bellman equation has the form + ln + ln −1 where , and
are constants. Express these constants in terms of parameters , , and . (This is the method
of undetermined coefficients or the so-called “guess and verify method”.) (6 marks)
(f) What is the role of habit persistence in the solution to this model? Does current consumption
depend on −1, based on consumer habit, given this setup? What is the intuition behind this result?
Hint: Show that the optimal policy is of the form = where is a constant. (6 marks)
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