Three-period consumption-savings problem

Consider a three-period decision problem:
max u(c0) + βu(c1) + β
2u(c2)
subject to the flow budget constraints for t = 0, 1, 2:
bt+1 = (1 + r)bt + yt − ct
,
where b0 is the initial wealth of the country.

1. Explain why in this three-period model it cannot be that b3 < 0 and it should not be that b3 > 0.
2. Given this, use the flow budget constraints to derive the intertemporal budget constraint:
   c0 +
   c1
   1 + r
   +
   c2
   (1 + r)
   2
   = (1 + r)b0 + y0 +
   y1
   1 + r
   +
   y2
   (1 + r)
   2
   .
   Interpret this equation.
3. Show that the intertemporal budget constraint is equivalent to
   (1 + r)b0 + nx0 +
   nx1
   1 + r
   +
   nx2
   (1 + r)
   2
   = 0,
   where nxt = yt − ct
   . Explain why it is also equivalent to b0 + ca0 + ca1 + ca2 = 0,
   where cat = rbt + nxt = bt+1 − bt
   .
   When is it possible to have nxt < 0 for every t = 0, 1, 2 and why? Which country
   that we discussed might fit this description? Does it violate the logic that all trade
   deficits must be compensated by trade surpluses?
   1
4. Using your favorite method, derive the intertemporal optimality conditions for t = 0, 1:
   u
   0
   (ct) = β(1 + r)u
   0
   (ct+1).
5. Assume b0 = 0 and β = 1 and r = 0. Solve for consumption c0, net exports nx0,
   and current account ca0, by defining the concept of permanent income ¯y. Interpret
   your results by providing examples for different y0 and ¯y.
   Discuss intuitively how the result change when b0 < 0, β < 1, and r > 0.
   Problem 2: Two-period model with investment
   Consider a two-period economy facing the following budget constraints:
   c1 + k + b ≤ y1,
   c2 ≤ y2 + (1 + r)b,
   where y1 is an exogenous endowment and second-period output
   y2 = Akα
   with 0 < α < 1 and productivity A. Note that k is both first period investment and
   second period capital stock (implicitly assuming full depreciation, δ = 1). Also note that
   initial b0 = 0, and hence b is both first-period current account and second-period net
   foreign assets.
6. Explain how this special environment maps into the general framework of National
   Income Accounts, and in particular why:
   ca1 = y1 − c1 − k = b and ca2 = rb + y2 − c2 = −b.
7. Explain how to derive the intertemporal budget constraint:
   c1 +
   c2
   1 + r
   = y1 − k +
   Akα
   1 + r
   .
8. Given this budget constraint, characterize the optimal capital investment k of the
   country and interpret your results (how does optimal k depend on r and A, and why).
9. Explain why it is possible to determine optimal investment without characterizing
   the optimal consumption-savings decision. In other words, why investment and
   savings decisions separate and when would they not?
10. Why do we expect a country with a high A (relative to y1) to run a current account
    deficit? What may be real-world examples of such countries?