Solow-Swan Model with human capital In class, we laid down the basics of the Solow-Swan Model with human capital. You wi
Posted: Sun May 29, 2022 7:41 pm
Solow-Swan Model with human capital
In class, we laid down the basics of the Solow-Swan Model with
human capital. You will now solve it
completely to determine steady states, growth rates etc that result
from this extension.
Set up:
•The production function is given by: Yt = AKαt (Nht)1−α where N is
a constant population level
and ht is the per-capita level of human capital.
1
•Yt = Ct + St
•Law of motion for per capita physical capital is given by: kt+1 =
(1 −δ)kt + syt
•Law of motion for per capita human capital is given by: ht+1 = (1
−δ)ht + qyt
•rt = ht
ktis the ratio of human to physical capital at time t
Questions:
(a) Divide the laws of motion for per-capita physical capital and
per-capita human capital by kt and
ht respectively. Express the growth rate of per capita physical
capital (gk) and the growth rate
of per capita human capital (gh) solely as a function of parameters
A,δ,α.s,q and rt.
(b) Consider the case in which rt = r = q
s for all t (the economy starts out with the long run ratio
of
human to physical capital). Rewrite the per-capita production
function just in terms of per-capita
capital. Does the production function exhibit diminishing returns
in capital anymore?
(c) Using the result from part b, re-derive the Solow-Swan
equation. That is, find the expressions
that gives you kt+1 as a function of kt and the parameters.
(d) What is different about this version of the Solow-Swan
equation? Is your law of motion for per
capita capital still concave or is it linear? What does this imply
for the evolution of per-capita
capital?
(e) Draw a phase diagram where kt is on the x-axis and kt+1 is on
the y-axis. What can you say
about the steady state in this model? Are there any? What does this
model predict about growth
of GDP per capita? (Hint: You will have 3 cases, 1) where δ =
sAr1−α, 2) where δ < sAr1−α
and 3) δ > sAr1−α. Comment on the existence and stability of
steady states if any?)
(f) Assume that the rate of return for capital is equal to
capital’s marginal product, MPk = ∂Yt
∂Kt.
Does this model help explain low rates-of-return to capital in poor
countries relative to rich
countries?
(g) Revisit the law of motion for human capital, ht+1 = (1 −δh)ht +
qyt. Do you think it’s reasonable
to include depreciation (δh) in the equation? Explain.
(h) Remember that ht stands for per-person human capital. Under
certain conditions, does the law
of motion imply that ht can grow indefinitely? Is that
reasonable?
(i) Bonus question: Suppose we changed the law of motion of human
capital to include an upper
limit. If ht < ̄h then the law of motion is the same as above
until it hits the upper bound ̄h. Why
might we do this? With this new law of motion, repeat part (c).
What are the new predictions
of this model? Is there long-run growth?
In class, we laid down the basics of the Solow-Swan Model with
human capital. You will now solve it
completely to determine steady states, growth rates etc that result
from this extension.
Set up:
•The production function is given by: Yt = AKαt (Nht)1−α where N is
a constant population level
and ht is the per-capita level of human capital.
1
•Yt = Ct + St
•Law of motion for per capita physical capital is given by: kt+1 =
(1 −δ)kt + syt
•Law of motion for per capita human capital is given by: ht+1 = (1
−δ)ht + qyt
•rt = ht
ktis the ratio of human to physical capital at time t
Questions:
(a) Divide the laws of motion for per-capita physical capital and
per-capita human capital by kt and
ht respectively. Express the growth rate of per capita physical
capital (gk) and the growth rate
of per capita human capital (gh) solely as a function of parameters
A,δ,α.s,q and rt.
(b) Consider the case in which rt = r = q
s for all t (the economy starts out with the long run ratio
of
human to physical capital). Rewrite the per-capita production
function just in terms of per-capita
capital. Does the production function exhibit diminishing returns
in capital anymore?
(c) Using the result from part b, re-derive the Solow-Swan
equation. That is, find the expressions
that gives you kt+1 as a function of kt and the parameters.
(d) What is different about this version of the Solow-Swan
equation? Is your law of motion for per
capita capital still concave or is it linear? What does this imply
for the evolution of per-capita
capital?
(e) Draw a phase diagram where kt is on the x-axis and kt+1 is on
the y-axis. What can you say
about the steady state in this model? Are there any? What does this
model predict about growth
of GDP per capita? (Hint: You will have 3 cases, 1) where δ =
sAr1−α, 2) where δ < sAr1−α
and 3) δ > sAr1−α. Comment on the existence and stability of
steady states if any?)
(f) Assume that the rate of return for capital is equal to
capital’s marginal product, MPk = ∂Yt
∂Kt.
Does this model help explain low rates-of-return to capital in poor
countries relative to rich
countries?
(g) Revisit the law of motion for human capital, ht+1 = (1 −δh)ht +
qyt. Do you think it’s reasonable
to include depreciation (δh) in the equation? Explain.
(h) Remember that ht stands for per-person human capital. Under
certain conditions, does the law
of motion imply that ht can grow indefinitely? Is that
reasonable?
(i) Bonus question: Suppose we changed the law of motion of human
capital to include an upper
limit. If ht < ̄h then the law of motion is the same as above
until it hits the upper bound ̄h. Why
might we do this? With this new law of motion, repeat part (c).
What are the new predictions
of this model? Is there long-run growth?