Growth (Solow model; Endogenous growth) Flashcards
How is the production function written?
Y = f(K,L) → e.g. Y = KαL1-α
Can add in exogenous technology:
- Y = Af(K,L) → e.g. Y = AKαL1-α
- Y = f(K, AL) → e.g. Y = Kα(AL)1-α
How do you get output in per-worker terms (intensive form)?
Divide through by L…
Y = Kα(AL)1-α
y = Y/L = Kα(AL)1-α / L
y = A1-αKα (L1-α/L)
y = A1-α (K/L)α
y = A1-α kα
How do you get output in effective worker terms?
Divide through by AL…
Y = Kα(AL)1-α
- y *= Y/AL = Kα(AL)1-α / AL
- y* = Kα(AL)-α
- y* = (K/AL)α
- y* = kα
How do you include a) labour force growth, and b) increase in rate of growth of technology, in the model?
a) Need intensive form (y = Y/L), then labour force growth, n, is added to depreciation line
b) Need effective worker form (y = Y/AL), then depreciation line is ∂ + n + x
Why is the production function concave?
Because of diminishing marginal returns (to capital)
What is capital accumulation?
k• = dk/dt = sy - δk
Growth in capital stock over time equals savings (a proportion of output) minus depreciation on existing capital stock.
The distance between the savings and depreciation lines on the diagram.
What is the (∂ + n + x)k line a depictor of?
Line showing the rate at which capital is used up of its own accord.
What happens if people save a greater proportion of their income?
The bigger ‘s’ is, the closer the investment line is to the production function.
The rate of capital accumulation increases, causing a movement along the depreciation line to new steady state equilibrium.
What is the Golden Rule?
The optimal savings rate to maximise consumption.
max consumption s.t. being at the steady state
max (1-s) f(k*) subject to sf(k*) = (δ+n)k*
L = (1-s) f(k*) + λ[sf(k*) - (δ+n)k*]
Partial derivatives wrt s and k*
FOCs: λ = 1 and f’(k*) = δ+n → MPK = δ+n
How do you find the optimum level of savings?
Differentiate the production function to get the equation for its gradient (MPK), and then set that equal to δ+n (gradient of the capital decumulation line). Work through to find the level of k* at the golden level.
Since sy = (δ+n)k* at this point, you can then work out s, the optimal savings rate.
In the Cobb-Douglas case:
y = kα, y’ = αkα-1
y’ = MPK = δ+n = αkα-1
sy = (δ+n)k = (αkα-1)k
sy = αkα = αy
s = α
Why do we have problems with the Golden Savings rule (with a Cobb-Douglas function)?
It suggests that the optimal rate of saving ‘s’ is equal to the share of output paid to capital ‘α’, which in the UK is typically 30-35%…but our savings rate is not that high.
Why might the Golden Rule of savings be wrong? [4]
- If the economy is not at the steady state - might be able to gain more by saving more (increasing savings will increase your growth rate); calculation involves assuming steady state equilibrium
- Impatience - the Golden Rule gives a savings rate such that the same amount is saved each year, which will not be the case if people are impatient (lower ‘s’) - Ramsey
- If savings affects technology, then the production function will be shifting up each period, so the ‘s’ to maximise consumption will be higher - Aghion & Howitt (2007)
- Capital flows - if some savings go abroad, then the savings curve will be lower than you think it is
Why do we need endogenous growth models?
To explain why technology (A) grows - why diminishing returns to capital can be removed from the production function (and the production / savings lines can be made straight).
What are the three endogenous growth models?
- Arrow, learning by doing
- Romer, R&D model
- Lucas, human capital model
Explain Arrow’s “learning by doing” model
Learning by doing removes diminishing returns to capital by assuming knowledge spillovers, so that technology is increasing in k. This effect offsets the diminishing returns to capital.
Firm has yi = Akiα
Technology = A = Bkβ, where β = knowledge spillovers
So, y = Bkβkα = Bkα+β
If α+β<1, we have diminishing MPK. But, the more β increases, the less MPK diminishes, increasing the chance that the economy will grow for a long time.
Eventually get y = Bk, with constant returns to capital, and all lines straight (no crossovers, indefinite growth).
Explain Romer’s R&D model
R&D models assume no diminishing returns to technology accumulation. This means that technology continually grows, so the marginal product of capital increases period by period.
Labour works LY in production and LR in R&D, so technology accumulates by A•=LRA.
A•/A = gA = LRφ, where φ<1 represents diminishing returns to workers
y = Akα → lny = lnA + αlnk → gy = gA + αgk
So gy = LRφ + αgk, and increasing the number of people in R&D permanently raises the growth of output.
sy = (δ+n)k → y/k = (n+δ)/s → y ∝ k, gy= gk
So gy = LRφ / (1-α)
So the rate of growth in output is constant, rather than diminishing?
Explain Lucas’s human capital model
Human capital models supplement regular capital with another type of capital (human capital) which is assumed not to have diminishing returns.
LY in production and LH studying
Human capital accumulates by A•=LHA
A•/A = gA = LH → gY = LH - αgk
If you get people to study more, you permanently increase the growth rate
What are Jones’s critiques of the endogenous models?
- Is knowledge/human capital development infinite?
- Assumption that the exponent of A is 1 so that gy increases permanently and constantly
Other problems:
- Free-rider problem, investment under-provided - solved with patents
- Education is a merit good, underconsumed - solved with grants/subsidies
Explain Jones’s semi-endogenous model
A• = LRφ Aη → gA = LRφ / A1-η
If η<1, diminishing returns to R&D or human capital - innovation slows down
If η>1, increasing returns - growth rate explodes
