Download Command optimum, the equivalence theorem, and the turnpike

Advanced Macroeconomics.
16.09.2003.
Note 5
Christian Groth
Command optimum, the equivalence theorem, and the
turnpike property
This note is a follow-up on Proposition 1 in Note 4, saying that as long as the bequest
motive is operative (i.e., bt+1 > 0), the market economy brings about the same allocation as
would a command economy ruled by a social planner with an ”effective” intergenerational rate
of discount equal to the private one. A particular aspect of this equivalence proposition is that
the same modified golden rule appears in both regimes. This is the theme of Section 2, while
Section 3 touches upon the fact that the modified golden rule has a so-called turnpike property.
1
The command optimum
As we have seen, according to the Diamond OLG model the allocation in the economy may
suffer from dynamic inefficiency, hence absence of Pareto optimality. But even if the allocation
is Pareto optimal, it may not be satisfactory from a societal point of view.
In fact, Pareto optimality is a very weak optimality criterion. For example, if current
generations have a very high rate of time preference (high impatience), they save very little,
and the economy may gradually shrink to the detriment of future generations. At the same
time, this can easily be a Pareto optimal allocation.1
A more interesting criterion requires construction of a social welfare function. How should
we do that? No definite, objective answer can be given. In the last instance a social welfare
function relies on political and ethical choices.
We will consider a hypothetical centrally planned economy with a benevolent and omniscient
social planner who can dictate every aspect of production and distribution within the constraints
given by technology and initial resources. The demography, technology and private preferences
are as in the standard Diamond model. We assume the social planner
1
Similarly, if the Danish queen received almost all consumption goods (and saturation were impossible), while
the rest of the population received only just what is needed for subsistence, that could be a Pareto optimum.
1
• respects the private preferences as to the division of own consumption across time;
• discounts the utility of future generations at rate R.
If R = 0, all generations have the same weight in the social welfare function, if R > 0, future
generations have less weight than current generations, and if R < 0, the future generations have
more weight.2
1.1
Finite planning horizon
To begin, assume the planner has a finite planning horizon, i.e., the planner cares only about
utility in the T + 1 current and future periods. The social welfare function is:
W0 = (1 + ρ)−1 u(c20 ) +
T −1
[
(1 + R)−(t+1) [u(c1t ) + (1 + ρ)−1 u(c2t+1 )]
t=0
+(1 + R)
−(T +1)
u(c1T ).
(1)
Notation is as usual, ρ is the private rate of time preference, and u > 0, u < 0. Until further
notice, the social planner’s intergenerational rate of discount, R, may be positive or negative,
but we require, of course, R > −1. The number of young is Lt = L0 (1 + n)t , where L0 > 0,
and n > −1.
The technology is given by the aggregate production function
Yt = F (Kt , Nt ) ≡ Nt f (kt ),
(2)
where K is capital, N is employment, and F is a neoclassical production function, homogeneous
of degree one, hence, f > 0, f < 0.3 Only the young can work, and they all supply inelastically
one unit of labour per period. The social planner ensures full employment, so that Nt = Lt .
The dynamic resource constraint is
Kt+1 = Kt + Yt − Ct − δKt ,
K0 > 0,
(3)
where δ is the capital depreciation rate, 0 ≤ δ ≤ 1. Aggregate consumption, Ct , satisfies
Ct = Lt c1t + Lt (1 + n)−1 c2t .
2
(4)
Note that if all human beings (present as well as future) should have the same weight, then the required
discount factor would be
1
= 1 + n,
1+R
implying R = (1 + n)−1 − 1 < 0 for n > 0, i.e., R ≈ −n (since R ≈ log(1 + R) = − log(1 + n) ≈ −n for n
"small").
3
For simplicity, we ignore technical progress.
2
Combining (3) and (4) and isolating c1t gives
c1t = (1 − δ)kt + f (kt ) − (1 + n)kt+1 − (1 + n)−1 c2t ,
k0 > 0 given.
(5)
This is just a book-keeping relation saying that consumption as young equals what is available
per young minus what is used for other purposes (investment and consumption of the old).
The planner may give some weight to what happens after period T. Therefore, we introduce
the terminal condition
kT +1 ≥ k̄T +1 ,
(6)
where k̄T +1 ≥ 0.
1.2
Solution of the social planner’s problem
−1
The social planner’s problem is: Choose a plan [c20 , (c1t , c2t+1 )Tt=0
, c1T ] to maximize W0 subject
to the constraints (5) and (6). In order to solve the problem we insert (5), both as it looks and
shifted one period backward, into (1) and maximize w.r.t. c2t and kt . Setting ∂W0 /∂c2t = 0
and ∂W0 /∂kt = 0 we get the FOCs:
u (c1t )
,
t = 0, 1, ..., T,
1+n
1 − δ + f (kt )
u (c1t−1 ) = (1 + R)−1
u (c1t ),
t = 1, ..., T.
1+n
(1 + ρ)−1 u (c2t ) = (1 + R)−1
(7)
(8)
Condition (7) is a M C = M B condition referring to the division of consumption across generations in the same period. It states that, from the point of view of the social planner, the
utility loss by decreasing consumption per old by one unit and transfer to the young in the
same period must be equal to the utility gain by the young, discounted by the intergenerational
discount rate R. The rate of transformation is 1/(1 + n), since, for every old there are 1 + n
young.
Condition (8) is a M C = M B condition referring to the division of consumption across
time and generations. It states that the utility loss by decreasing the consumption of the
young in period t − 1 by one unit must be equal to the utility gain by the young in the
next period discounted by the intergenerational discount rate R. The rate of transformation is
(1 − δ + f (kt ))/(1 + n), since the saved unit is invested and gives a gross return of 1 − δ + f (kt )
in period t, but at the same time, for every young in period t − 1 there are 1 + n young in
period t.
3
Replacing u (c1t ) in (7) by its value from (8) and ordering gives u (c1t−1 ) = (1+ρ)−1 u (c2t )(1+
f (kt ) − δ). Replacing t by t + 1, we end up with
u (c1t ) = (1 + ρ)−1 u (c2t+1 )(1 + f (kt+1 ) − δ).
(9)
This relation is identical to the familiar condition for individual intertemporal utility optimization in the Diamond model, if we insert the equilibrium relation rt+1 = f (kt+1 ) − δ. This
shows that the central planner’s first-order conditions respect the first-order condition that the
individual chooses for herself in the market economy.
To ensure that not only the "direction", but also the "level" of the dynamic path is correct,
yet another condition is needed for optimality. This is the transversality condition
(10)
kT +1 = k̄T +1 .
Given the terminal condition (6), the alternative to (10) is kT +1 > k̄T +1 , but that would imply
overaccumulation.
Now, the allocation over time is determined by (5), (7), (9), and (10). The first three of
these equations are exactly the same as those describing the market economy according to the
Barro model (when the bequest motive is operative), namely equation (9), (6) (replacing t by
t − 1), and (7) in Note 4. This gives a hint that the Equivalence Proposition (Proposition 1 in
Note 4) holds. But the Barro model has an infinite horizon, and a condition like (10), which
comes from a finite horizon, does not enter. To get a comparable situation we therefore now
assume an infinite planning horizon in the social planner’s problem.
1.3
Infinite planning horizon
In the social welfare function (1) we let T → ∞. In order to be sure that the sum of discounted
utilities converges, we assume R > 0. The natural terminal condition will no longer be (6), but
just
lim kt ≥ 0,
(11)
t→∞
since, when T → ∞ in the criterion function (1), the planner already gives weight to what
happens in every future period.4 The necessary transversality condition can now be shown to
be
lim
1→∞
kt
1−δ+f (kt )
t
Πτ =1 1+n
4
= 0.
(12)
Hence, there is no need to introduce a particular minimum terminal capital stock apart from the conceptual
non-negativity requirement (11).
4
This precludes overaccumulation. With rt = f (kt )−δ exactly the same transversality condition
can be shown to hold in the market equilibrium of the Barro model (because it is a representative
agent model).
The conclusion is that the equations describing the social planner’s solution are identical
to those of the market equilibrium as long as the intergenerational rate of discount is the same
and the bequest motive is operational. This proves the Equivalence Proposition (Proposition
1 in Note 4), and this proposition can be considered a generalization of the corresponding
proposition for the Ramsey model (see Romer, 2001, Chapter 2).
2
The modified golden rule
In view of the result above, it can be no surprise that also the social planner’s solution satisfies
the modified golden rule. This is the rule that the net marginal product of capital5 must obey
in a steady state in order that this steady state can be optimal, given the social planner’s
intergenerational rate of discount, R. The rule is
1 + f (k ∗ ) − δ = (1 + R)(1 + n) = 1 + n + R + Rn ≈ 1 + n + R,
(13)
f (k ∗ ) − δ ≈ n + R,
(14)
that is,
where k ∗ is the steady state capital intensity, k ∗ > 0.
To show this result, notice that steady state requires kt = k ∗ , c1t = c∗1 , and c2t = c∗2 , for all
t = 0, 1, 2, ..., where k ∗ , c∗1 , and c∗2 are positive constants. Hence, in (8) we can replace both
u (c1t−1 ) and u (c1t ) with u (c∗1 ), which can thereby be eliminated so that we end up with (13).
To be sure that a positive k ∗ satisfying the modified golden rule exists, we need the combined
parameter and technology condition
lim f (k) > (1 + R)(1 + n) + δ − 1 > lim f (k).
k→0
k→∞
This inequality is certainly satisfied, if (1 + R)(1 + n) > 1 − δ, and the production function
satisfies the Inada conditions.
We see that with R = 0 (all generations have equal weight) the rule (13) is the well-known
simple golden rule saying that the maximum sustainable consumption output per unit of labour
5
Or in the market economy with perfect competition, the rate of interest.
5
is obtained in a steady state where the net marginal product of capital is equal to the rate of
growth of the labour force, n.6 In fact, though with R = 0 maximization of W0 does not
make sense (when T → ∞), one can in such a case use the so-called overtaking criterion or
the catching-up criterion as optimality criteria.7 Thus it can be shown that even if R = 0,
the social planner’s problem is well-defined, using the overtaking criterion or the catching-up
criterion, and the conditions (5), (7), and (9) are still necessary for an optimal solution. Hence,
in steady state with R = 0, the social planner’s solution satisfies the usual golden rule condition,
f (k ∗ ) − δ = n. Therefore, R = 0 implies k ∗ = kg , where kg is the golden rule capital intensity.
With the golden rule as the benchmark case, the term modified golden rule refers to the fact
that if R > 0 (current generations have more weight than future generations), then the social
planner prefers a permanent capital intensity lower than kg . Though society could consume
more in the long run, if kg were strived for, this would not compensate for the cost in terms of
the lower current consumption than would otherwise be possible. This cost is higher relative
to the long-run benefit, the higher is R.
What about stability? It can be shown that whether R > 0 or R = 0, the optimal path
converges over time from arbitrary k0 > 0 to the unique steady state k ∗ , the modified golden
rule. If the social planner cares equally about all generations, the economy tends to the golden
rule, and consumption per unit of labour is maximized.8
Interestingly, a related stability property holds when the planning horizon is finite, that is,
when T < ∞. We conclude this note by commenting on this so-called turnpike property.
3
The turnpike property
Return to the problem considered in Section 1.1 above. Given a finite planning horizon T, the
turnpike proposition is the following statement:
If the planning horizon T is large, then the best (i.e., optimal) way to go from
any initial k0 > 0 to a specified terminal capital stock k̄T +1 ≥ 0 is to stay close to
the golden rule capital stock k ∗ , defined in (13), for a long time.
6
Or, more generally, equal to the natural rate of growth of GDP, n + g, where g is the (constant) rate of
Harrod-neutral technical progress.
7
See Seierstad & Sydsaeter (1987).
8
It might seem unjust and unethical towards future generations to use a positive intergenerational discount
rate R. There would be less to this point of view, if technical progress were added to the analysis. Then, in
view of the future generations being favoured by better technology, they would, in spite of R > 0, tend to end
up with higher life-time utility than current generations.
6
k
k*
kT +1
ko
0
T T +1
1
t
Figure 1:
Fig. 1 illustrates this turnpike property of the optimal path. The intuition is that k ∗ gives
the best trade off between consumption now and consumption later, given the growth rate of
the labour force, n, and the intergenerational rate of discount, R.9 Further, the initial cost
(in terms of consumption) of moving up to k ∗ is compensated when, finally, we move down
to k̄T +1 . It is like a motorway (am.: turnpike): Though the shortest way to go from Hellerup
to Helsinore may be to take the road along the coast, "Strandvejen", it is faster to use the
motorway.
4
Literature
Azariadis, C., 1993, Intertemporal Macroeconomics, Oxford.
Blanchard, O., & S. Fischer, 1989, Lectures on Macroeconomics, Cambridge (Mass.).
de la Croix, D., & P. Michel, 2002, A Theory of Economic Growth, Cambridge.
Romer, D., 2001, Advanced Macroeconomics, 2. ed., New York.
Seierstad, A., & K. Sydsaeter, 1987, Optimal Control Theory with Economic Applications,
Amsterdam.
9
In fact, the turnpike proposition holds, whether R > 0, R = 0, or R < 0.
7