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Econ 714: Deriving the PIH Formula
Sang Yoon (Tim) Lee
March 10, 2007
Solve the deterministic individual problem
max {ct,at+ 1 }
∞
t= 0
β tu(ct)
s.t. ct + at+ 1 = yt + ( 1 + r)at ∀t
F.O.C.’s are
ct : u′(ct) = λ t at+ 1 : λ t = β ( 1 + r) λ t+ 1
Assuming β ( 1 + r) = 1 , we obtain
u′(ct) = u′(ct+ 1 ).
Hence if u(·) is strictly concave, ct+j = ct for all j. Then writing out the budget constraints,
ct + at+ 1 = yt + ( 1 + r)at ct + at+ 2 = yt+ 1 + ( 1 + r)at+ 1 ct + at+ 3 = yt+ 2 + ( 1 + r)at+ 2 ct + at+ 4 = yt+ 3 + ( 1 + r)at+ 3 · · ·
Multiply iteratively by (^11) +r to get
ct + at+ 1 = yt + ( 1 + r)at 1 1 + r ct +
1 + r at+ 2 =
1 + r yt+ 1 + at+ 1
(
1 + r )^2 ct + (
1 + r )^2 at+ 3 = (
1 + r )^2 yt+ 2 + (
1 + r )at+ 2
(
1 + r )^3 ct + (
1 + r )^3 at+ 4 = (
1 + r )^3 yt+ 3 + (
1 + r )^2 at+ 3 · · ·
1
Then adding up LHS’s and RHS’s, notice that the a’s cancel out, so we obtain
∞ ∑ j= 0
1 + r )jct + lim J→∞
1 + r )J^ at+J+ 1 =
∞ ∑ j= 0
1 + r )jyt+j + ( 1 + r)at
By the TVC or borrowing constraint, the last term of LHS is 0. Hence
1 + r r ct =
∞ ∑ j= 0
1 + r )jyt+j + ( 1 + r)at
So
ct = r 1 + r
yt +
∞ ∑ j= 1
1 + r )jyt+j + ( 1 + r)at
Friedman’s conjecture is that this holds even in the stochastic case:
ct =
r 1 + r
yt +
∞ ∑ j= 1
1 + r )j E tyt+j + ( 1 + r)at
This is called ”certainty equivalence,” a principle that is exploited in many other appli- cations as well to show that the solution to a stochastic problem coincides with its deter- ministic counterpart. Of course, it is something that has to be shown case by case, not a
universal property.
