OG Model of Optimal Saving
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OG Model of Optimal Saving (II)
Is the market equilibrium optimal? Or can we improve on it?
We have found equilibrium savings and rate of interest, the
quantity and price generated by a decentralized market. The chosen
combination of consumption when young and old, is constrained by a
given budget, subject to the clearing of the market.
We do not know, however, whether this equilibrium level of
saving leads to the best possible, or optimal, combination of
consumption; is there a way to redistribute resources unconstrained by
individual budgets that does better than the market?
To answer, imagine a central planner with total control over
the economy. He wants to allocate the goods available among
the young and the old.
OG Model of Optimal Saving
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I. The Constraint Facing the Central Planner
At any date he cannot allocate more goods than are available
in the economy: the feasible set
Nt w1 + N t−1w2 ≥ Nt c 1,t + Nt− 1 c 2,t
Dividing through by Nt, and noting that Nt = nNt–1, the
feasible set is
w1 
c
w2
 c1,t  2,t
n
n
A stationary allocation is one that gives the members of every
generation the same lifetime consumption pattern; i.e., in a
stationary allocation, c1,t = c1 and c2,t = c2 for every period t.
w1 
w2
c
 c1  2
n
n
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II. The Golden Rule (Phelps, AER 1965)
— Maximize steady-state lifetime utility
Of all study states, choose the one that maximizes utility of
future generations, ignoring initial old.
MaxU c1 ,c2 
subject to the feasible set for stationary allocations
w1 

w2
c
 c1  2
n
n
U 1 c1 
n
U 2 c2 
The slope of the feasible set implies that when we take a unit
of consumption from each young person, we can give exactly n
more units of consumption to each old.
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Therefore, the marginal cost and benefit of taking 1 extra good
from each young person in order to give n goods to each of the
old:
Marginal cost
Marginal benefit
in goods
1
n
in utility
u1
nu2
Question: is the golden rule satisfied by the competitive equilibrium?
U1 c1 
r
U 2 c2 
U 1 c1 
n
U 2 c2 
Only if r = n. But r is determined by the endowment pattern;
nothing constrains it to be exactly equal to the economy’s
growth rate.
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III. Pareto Optimality
The golden rule cares only about the future generations. It
ignores the initial old.
To apply a welfare criterion that considers both types of
people, we turn to the concept of Pareto optimality:
Allocation A is Pareto superior to allocation B if and only if in
allocation A everyone is at least as well off as in allocation B
and at least one individual is better off in A.
An allocation is Pareto optimal (or Pareto efficient) if and only if
no one can be made better off without making someone else
worse off.
Note that an allocation is Pareto optimal if and only if no
feasible allocation is Pareto superior to it.
OG Model of Optimal Saving
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In the below graph,
 which 2 allocations U, V, W, and X are Pareto optimal?
 which allocations are Pareto superior to another allocation?
 Explain with an example:
An allocation that is not
Pareto optimal can nevertheless
be Pareto superior to another
allocation.
An allocation that is Pareto
optimal does not have to be
Pareto superior to all other
allocations.
OG Model of Optimal Saving
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1. A brief review of the Kuhn-Tucker conditions
What if x is constrained to be non-negative?
2 possibilities:
(1) w'(x) = 0, x > 0, as in
the graph above.
(2) Constraint is binding.
Unconstrained max would be
negative, as in left graph. Here x
= 0 is optimal if w'(x)|x = 0 < 0.
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2. Formal conditions defining Pareto optima
Maximize utility of the future generations subject to
— a constraint that the utility of the initial old V(c2) fall
below a given level V. This defines a minimum level of
consumption for the old.
— the feasible set for stationary allocations
max U(c1,c2) + θ[V(c2) − V] + λ[w1 + w2/n− c1 − c2 /n]
FOCs:
U1(c1,c2) − λ = 0
U2(c1,c2) +θV’(c2) − λ/n = 0
w1 
w2
c
 c1  2
n
n
θ[V(c2) − V] = 0 (Kuhn-Tucker condition)
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• If the constraint is non-binding, V(c2) –V > 0, θ = 0
At θ = 0, we have the golden rule:
U 1 cˆ1 , cˆ2 
n
ˆ
ˆ
U 2 c1 , c2 
• If the constraint is binding, V(c2) = V, 0 ≤ θ and
c2  cˆ2 ,
U 1 c1 , c2 
n
U 2 c1 , c2 
Nothing pins down the value of V, the minimum utility that
we reserve for the initial old. Therefore, the set of Pareto
Optima consists of the allocations satisfying these conditions for
all feasible values of V.