Week 6 - Amendment
November 13, 2015
Question 1 - Derivation of
k ∗ /y ∗
We want to nd an expression for the steady state capital to income ratio. It is
equivalent to look for the ratio in levels or in terms of eective workers
K∗
=
Y∗
K∗
EN
Y∗
EN
=
k∗
y∗
First, we derive the steady state value of capital per eective worker, k∗ . Note
that the 0 ∗0 denotes steady state values. Start from the steady state condition
for the capital stock in the long run. The value of capital per eective worker
which solves this condition is k∗ .
MPK
= r̄ + δ
1+µ
Find an expression for M P K using the aggregate production function Y =
K α (EN )1−α .
MPK =
∂Y
∂K
=
αK α−1 (EN )1−α
=
α
=
K α−1
(EN )−(1−α)
α−1
K
α
= αk α−1
EN
Plug the expression for M P K back into optimality condition for capital and
solve for k.
αk α−1
1+µ
k α−1
= r̄ + δ
=
(r̄ + δ)(1 + µ)
α
1
k
k
(r̄ + δ)(1 + µ)
α
α
(r̄ + δ)(1 + µ)
=
∗
=
1
α−1
1
1−α
Second, we derive the steady state value for output per eective worker, y ∗ .
Start from the aggregate production function and divide it by EN to get output
per eective worker.
K α (EN )1−α
= K α (EN )−α
EN
α
K
=
= kα
EN
Y
≡y
EN
=
Steady state output is the aggregate production function evaluated at steady
state capital. Thus, plug in k∗ to get steady state output per worker.
y∗
Third, compute the ratio
Ramsey model is
k∗
=
y∗
K ∗/EN
Y ∗/EN
K∗
= ∗
Y
= k ∗α
α
1−α
α
=
(r̄ + δ)(1 + µ)
k∗/y ∗
. The long run capital to income ratio in the
=
α
(r̄ + δ)(1 + µ)
=
=
α
(r̄ + δ)(1 + µ)
α
(r̄ + δ)(1 + µ)
Question 3 - Increase in
1
1−α
α
(r̄ + δ)(1 + µ)
−α
1−α
1−α
1−α
µ
In class we discussed changes in the depreciation rate and the rate of impatience.
Now, we want to know how the steady state value of capital per eective worker
changes for an increase in the price mark-up µ. The graph depicts the following
condition
αk α−1
− δ = r̄
1+µ
The downward sloping curve denoted LHS depits the left-hand side of this condition. This LHS can also be interpreted as the net gain from investing in the
capital stock. The y-axis plots this gain in terms of r as a function of k.
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First, consider an increase in µ for a given value of k. For a given k, a higher
µ decreases the value of the LHS:
αk α−1
↓ −δ
(1 + µ) ↑
Thus, the curve depicting the LHS shifts downwards, i.e. for a given level of k,
the net gain from investing in the capital stock is lower.
Second, consider the slope of the LHS curve:
Slope =
∂LHS
α (α − 1) α−2
α (1 − α) α−2
=
k
=−
k
∂k
1+µ
1+µ
where the last equality is just to point out that the slope is negative because
0 < α < 1. Now, consider an increase in µ:
−
α (1 − α) α−2
k
↓
(1 + µ) ↑
Hence, a higher price mark-up decreases the slope of the LHS, the LHS curve
becomes atter.
There are thus two changes of the LHS curve in response to a higher markup: a downward shift and a anti-clockwise rotation making the curve atter.
Both act to decrease the steady state capital stock per eective worker for a
given long-term real interest rate: The LHS decreases for a given k, the RHS
with r̄ stays unchanged, so the LHS has to increase to equal r̄ again. This is
achieved at a lower capital stock per eective worker, where the M P K and thus
the return on investment are higher.
How can we explain this result intuitively? First, why does the LHS decrease
and become atter, for a given k, for a higher price mark-up? A higher mark-up
increases the costs of investing in the capital stock and decreases the marginal
value product of capital (recall MPK*MR was shown to equal M P K/(1 + µ) in
the week 4 seminar). It increases the costs because the price for capital goods is
higher (we assume there is only one type of good, which can either be consumed
or invested to be turned into capital, thus the price for both is the same, i.e.
P = (1 + µ)M C ), and it decreases the marginal value product of capital, for a
given k, because the higher mark-up indicates the demand curve, and thus the
marginal revenue curve, is steeper, thus rms nd it optimal to choose a lower
level of production (recall the rst seminar in week 3). Both act to decrease and
tilt the LHS curve, which is depicting the net gain from investing in the capital
stock.
Second, why does this lead to a lower steady state capital stock per eective
worker? For a given k the net gain from investing in the capital stock will be
lower. Firms nd it optimal to invest in the capital stock until this net gain
equals the costs of borrowing as given by r̄ (recall that the model assumes rms
borrow money to invest in one unit of capital and repay the loan next period).
At the initial k, the cost of borrowing are higher than the net gain of investing
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in capital. Thus, rms will acquire a lower capital stock, at a higher marginal
product of capital and thus a higher net gain from investing, such that the net
gain is again equal to the costs of investing in the capital stock.
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