ECON 4350: Growth and Investment
Excercises for seminar 3
Spring 2007
Discussion topic 1
There is not a clear-cut answer to the following questions. You should,
however, think about them and increase your awareness to these issues.
1. Why is it more plausible that we can experience increasing returns to
scale when we adopt a broader concept of capital that also incorporates
human capital. Try finding some examples.
2. Assume both consumption, C, physical capital K and human capital
H are produced from the inputs K, H and raw-labor L. Suppose that
the production function for human-capital was
H = Kαh
Hηh
L1−αh−ηh
differed from that characterizing production of physical capital (out of
Y )
Y = Kαy
Hηy
L1−αy−ηy
What do you think are reasonable assumptions concerning the factorintensities
in the two production functions (i.e. αh, ηh, αy and ηy)?
Convergence
The following exercise takes you through various manipulations of the Solowmodel
in order to derive expressions that we will need when we turn to
empirical studies of convergence.
1. Prove the following rule: If GDP per capita grows at g per cent per
year, then it will double in approximately 70/g years.
2. Show that if
˙z(t) ≡
dz(t)
dt
= β(z∗
− z(t)) (1)
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then
z(t) − z(0) = (1 − e−βt
)z∗
+ (1 − e−βt
)z(0) (2)
Explain the role of β.
3. Assume instead that
˙z(t) ≡
dz(t)
dt
= H(z(t))
where H(z∗) = 0. Under what conditions does the following approximation
make sense?
˙z(t) ≡
dz(t)
dt
−H (z∗
)(z∗
− z(t)) (3)
4. In the text-book Solow model, Show that
d ln(ˆk)
dt
=
sf(eln ˆk)
eln ˆk
− (n + x + δ) ≡ H(ln ˆk)
Where the last equality defines the function H(ln ˆk).
5. Why is H(ln ˆk∗) = 0? Show that
H (ln ˆk∗
) = (1 − α∗
)(n + x + δ) ≡ β
where α∗ = f (ˆk∗)ˆk∗/f(ˆk∗), i.e. the capital elasticity at ˆk∗, and the
last equality defines the constant β.
6. Discuss the roles of the different parameters in β.
7. Find an estimate of the time it takes for ln(ˆk) to go halfway to ln(ˆk∗).
8. Show that
d ln(ˆy∗
dt
= α∗ d ln(ˆk∗)
dt
and that around the steady state we have approximately
ln(ˆy) − ln(ˆy∗
) α∗
(ln(ˆk) − ln(ˆk∗
))
9. Use these result to conclude that we use the approximation/linearization
d ln(ˆy(t))
dt
= β(ln(ˆy∗
) − ln(ˆy(t))) (4)
10. (This question is tedious, but straightforward. Only do it if you
find the time) Show that in the augmented Solow-model with a CDproduction
function
β = (1 − α − η)(n + x + δ) (5)
(Hint: Define the function d ln(ˆy(t))
dt = H(ln ˆk, ln ˆh) and linearize around
the steady-state).
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11. Use these results to derive equation (16) in MRW (Mind the differences
in notation).
12. Why was it convenient to choose z = ln k, instead of z = k, when
doing the approximation (3)?
13. If we are to use MRW (16) as a framework for regression, is it legitimate
to set β equal across countries?
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