Week 3 - Solow-Swan & Romers AK model Flashcards
What production function do we use in the SS model?
A cobb-douglas:
Yt = Kt^a*Lt^1-a, 0<=a<=1
How do you rewrite a CD production function in terms of output and capital per worker?
Divide both sides by L to get:
yt=kt^a
Draw the cobb-douglas PF
Look at slides
True or false: The marginal product of capital is fixed in this model
False. This is true in the HD model.
What is the marginal product of capital?
dyt/dkt = akt^a-1
What is the equation for investment?
Closed economy, so investment comes from saving a fixed fraction of output:
It = St = sYt
What is the equation for the evolution of capital stock?
K˙t = It - δK
K˙ = change in K
δ = depreciation rate
What assumption do we make about the population and the labour force?
Everyone works, so population = labour force
What is the equation for labour force Lt?
Lt = L0e^nt
What is the population growth rate?
(Take logs of both sides then differentiate wrt t)
n
How do you calculate the growth rate of capital per worker? What is it?
- Start with Kt/Lt = kt and take logs from both sides n differentiate wrt t
k˙/k = K˙/K - n
How do you get capital accumulation in per worker terms? What is it?
- Divide the capital accumulation equation by Kt
- sub into growth rate of capital per worker equation (remembering gK = K˙t/Kt)
k˙t = syt - (n + δ)kt
What is syt in the capital per worker equation
Investment per worker
What is (n + δ)kt in the capital per worker equation
Amount of investment per person needed to keep kt constant
Draw the basic Solow diagram
Look at lecture slides for answer
Draw the solow diagram and the production function on the same graph. Label consumption
Look at lecture slides
What happens when you increase the investment rate in the SS model? Show on a diagram
- The steady state of capital per worker increases
- Also increases the level of income per capita in the long run
- Temporary effect on the growth rate of income per capita
What happens when you increase population growth in the SS model? Show on a diagram
The steady state of capital per worker decreases.
How do you find the steady state k*? What is it?
- Where capital per worker is constant, so k˙=0
- solve for k* in the capital accumulation per worker equation
k* = (s/n+δ)^1/1-a
What is the steady state y*?
y* =k*^a = (s/n+δ)^a/1-a
Do changes in s and n have a permanent effect on the growth of output per worker
No, they permanently affect the level of output per worker by temporarily increasing the growth rate whilst on the transition path to the new steady state.
How do you show in the algebra that the growth rate in the steady state is 0?
- Divide capital evolution equation by kt to get:
k˙t/kt = s(yt/kt) - (n+δ)
as a-1<0, the growth rate of kt (and therefore yt) gradually declines as kt gets larger
Show using a diagram that the growth rate in the steady is 0
Look at lecture slides for answer.
What must we introduce to the SS production function to generate sustained growth in income per capita? What does the new PF look like?
Technological progress
Yt = (Kt^a)(AtLt)^1-a
Is technological progress endogenous or exogenous to our model?
Exogenous
What is the growth rate of technology?
At=A0e^gt
gA = g
When can we say the SS model is in equilibrium?
Where capital, output, population and consumption all grow at a constant rate.
Also called the balanced growth path
What 2 things have to be the same on the balanced growth path in the SS model?
Output per worker and capital per worker have the same growth rate.
(Derived from the capital accumulation equation)
How do you calculate the growth rate of output per worker in the SS model with technological progress? What is it?
- Start with the PF with technological progress
- Divide by L to get per worker terms
- Take logs n differentiate wrt time to get:
gy = agk + (1-a)gA - On the balanced growth path, gy = gk
- Plugging in, we get gy=gk=ga
On the balanced growth path, what do the growth rate of output per worker and capital per worker equal?
Growth rate of exogenous technological growth gA
True or false, technological progress is the source of sustained per capita growth?
True
In the long run of the SS model with technological progress, kt is no longer constant so we have to define a new steady state variable. What is it?
ekt = kt/At = Kt/AtLt
(The capital technology ratio)
e = tilde
Rewrite the production function in terms of ekt
eyt = ekt^a
(The output technology ratio)
How do you calculate ek˙? What is it?
- Write definition of ek
- Take logs n differentiate wrt time
- Sub into the capital accumulation equation
ek˙= seyt - (n + g + δ)ekt
How do you solve for the balanced growth path? What is it?
set ek˙ = 0
ey=ek^a = (s/n+g+δ)^a/1-a
Using the previous equation, how do you get an equation for y in the steady state? What is it?
multiply both sides by A:
y*t = A (s/n+g+δ)^a/1-a
What would happen if g = 0 and A0 = 1?
There would be no technological progress, bringing us back to the basic SS model
What was a problem with the SS Model/why did economists look to expand on it
Because it says technological progress is the key source of sustained income per capita in the long run, but it does not explain how this happens
What is the basic concept for Romers AK model of growth
Romers exploits the notion of learning by doing to build an endogenous growth model
What are the 2 key features of knowledge?
- Knowledge is created as a side product of investing: Leaning by investing
- It has public good characteristics:
- non-rival
- often non-excludable
What is the consequence of investment creating knowledge which has public good characteristics?
Investment creates a positive externality: A firms investment raises the stock of knowledge, which can be used by all firms in the economy
What is the production function in the AK model
Yit = At(Kit^a)Lit^1-a
What it the total level of knowledge in the economy (At) a positive function of?
The capital labour ratio
What does At =
At = Abar(Kt/Lt)^β
Would investment from an individual firm have an impact aggregate knowledge in the economy? Why?
No. Each firm is so small that, by investing, they are not making much of an impact on A. This is why we get diminishing marginal returns to capital at the firm level.
What happens when loads of firms invest?
This increases K/L, increasing At
Since firms experience diminishing marginal returns to investing in capital, why would they invest?
The positive externality to investment can - at the aggregate level - offset the diminishing MPK at the firm level
What does the new production function for Romers AK model look like when we incorporate the equations for At and assume a + β = 1
Yt = Abar(Kt^a+β)Lt^1-a-β
Assuming a + β = 1:
Yt = AbarKt
What does a + β = 1 mean?
Where the positive externality to investment exactly offsets the diminishing MPk at the firm level
what is gy?
gy = gk
Incorporating the capital accumulation equation, what is gy?
gy = sAbar - (n+δ)
What is the interpretation of this equation?
permanent changes in s and n have long-run growth effects.
What does a + β > 1 imply? What about a + β < 1
- Accelerating growth in output per worker, which is implausible
- Means we still have diminishing returns to capital, so we’re back in Solow-Swan territory
