e-Brief
Appendix for:
One Percent? For Real? Insights from Modern Growth Theory about
Future Investment Returns
By Steve Ambler and Craig Alexander
For simplicity, we study an economy with a representative household. The household’s planning horizon is
infinite, but its size is not constant. Utility is given by
∞
∞
(
)
U
U == ∑ββiiU
U C
Ctt++ii,,.. N
Ntt++ii,,
ii==11
where 0 < β < 1 is the household’s subjective discount rate. A lower value of β means that the household is
more impatient. The utility of an individual household member at time (t+i) depends on his or her consumption
Ct+i and possibly on other arguments (such as leisure). Nt+i gives the number of household members, which
evolves according to
Nt+i+1 = (1+n) Nt+i,
where n is the (constant) rate of growth of the population (and of the labour force). Households maximize utility
subject to the following budget constraint:
Bt + i +1 =
1
Yt + i + (1 + rt + i ) Bt + i − Ct + i ,
1+ n
(
)
where Bt+i is the value per household member of a risk-free asset held at time (t+i), (1+rt+i) is the gross rate of
return on that asset, and Yt+i is the value of other income. The first-order conditions for the household’s choice
of Ct and Bt+1 are
Ct :
∂U
= λt ,
∂Ct
where λt is the Lagrange multiplier associated with the budget constraint, and
Bt+1 : λt = βλt+1 (1 + rt+1).
The first optimality condition shows that λt is simply equal to the marginal utility of consumption. The second
optimality condition is extremely intuitive. It states that the marginal cost (in terms of utility) of saving one
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Appendix
more (inflation-adjusted) dollar should equal the marginal benefit for the household. The left hand side of the
equation gives the marginal cost. The right hand side gives the marginal benefit, which is just the (certain) gross
real rate of return on the investment, weighted by the marginal utility that it generates and discounted to the
current period using the discount rate β.
If the utility function is isoelastic (a constant elasticity of intertemporal substitution), we have
U (Ct,.) = Ct1–σ + O.T.,
so it is separable between consumption and its other arguments (grouped together as “O.T.” or other terms),
and where σ is the inverse of the elasticity of intertemporal substitution. This immediately gives
∂U
= λt = σ Ct−σ
∂Ct
and
∂U
= λt +1 = σ Ct +1−σ
∂Ct +1
Substituting these expressions into the second condition gives
∂U (Ct , .)
∂Ct
= σ Ct−σ = βσ Ct +1−σ (1 + rt +1 ).
Cancelling σ on both sides and taking logs gives
C

 1
rt + i ≈ log   + σ log  t + i +1 
 β
 Ct + i 
 Ct + i +1 
 Ct + i 
≡ ρ + σ log 
 Ct + i +1 

with log (1⁄β) ≡ ρ. In the steady state along a balanced growth path, log  Ct +i
rate of growth of the economy, and along a steady-state growth path we have
must equal the real per capita
r ≈ ρ + σg,
where
C

g ≈ log  t + i +1 
 Ct + i 
is the average net rate of growth of per capita consumption (and per capita output).
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Appendix
So far, the riskless rate of return does not depend on population growth. Baker, DeLong and Krugman (2002)
suggest modifying the utility function as follows:
∞
∞
(
)
i
1−ϕ
U
U == ∑ββ iU
U C
Ctt++ii,,.. N
Ntt++ii1−ϕ,,
ii==11
where φ captures the degree of deviation from what they call “familial altruism.” With φ = 0, households
love other household members (the ones joining later in time) as much as themselves. This is the classic case
discussed in Ramsey (1928) and developed in detail in Chapter 2 of Romer (2012). They call the case in which
1 > φ > 0 “imperfect familial altruism.” This would lead to following expression for the steady-state real interest
rate:
r ≈ ρ + σg + φn.(1)
With ρ > 0 (households are impatient to some degree), σ > 1 (the elasticity of intertemporal substitution is
less than one according to all empirical estimates), n > 0 (a positive rate of population growth) and φ > 0
(households are less than perfectly altruistic with respect to as-yet unborn members), it must be the case that
r > g.
The riskless rate of return is greater than the growth rate of per capita income.
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