Growth (Solow model; Endogenous growth)

Question 1

How is the production function written?

Answer 1

Y = f(K,L) → e.g. Y = K^αL^1-α

Can add in exogenous technology:

• Y = Af(K,L) → e.g. Y = AK^αL^1-α
• Y = f(K, AL) → e.g. Y = K^α(AL)^1-α

Question 2

How do you get output in per-worker terms (intensive form)?

Answer 2

Divide through by L…

Y = K^α(AL)^1-α

y = Y/L = K^α(AL)^1-α / L

y = A^1-αK^α (L^1-α/L)

y = A^1-α (K/L)^α

y = A^1-α k^α

Question 3

How do you get output in effective worker terms?

Answer 3

Divide through by AL…

Y = K^α(AL)^1-α

• y *= Y/AL = K^α(AL)^1-α / AL
• y* = K^α(AL)^-α
• y* = (K/AL)^α
• y* = k^α

Question 4

How do you include a) labour force growth, and b) increase in rate of growth of technology, in the model?

Answer 4

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

Question 5

Why is the production function concave?

Answer 5

Because of diminishing marginal returns (to capital)

Question 6

What is capital accumulation?

Answer 6

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.

Question 7

What is the (∂ + n + x)k line a depictor of?

Answer 7

Line showing the rate at which capital is used up of its own accord.

Question 8

What happens if people save a greater proportion of their income?

Answer 8

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.

Question 9

What is the Golden Rule?

Answer 9

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

Question 10

How do you find the optimum level of savings?

Answer 10

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 = k^α, y’ = αk^α-1
y’ = MPK = δ+n = αk^α-1
sy = (δ+n)k = (αk^α-1)k
sy = αk^α = αy
s = α

Question 11

Why do we have problems with the Golden Savings rule (with a Cobb-Douglas function)?

Answer 11

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.

Question 12

Why might the Golden Rule of savings be wrong? [4]

Answer 12

1. 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
2. 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
3. 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)
4. Capital flows - if some savings go abroad, then the savings curve will be lower than you think it is

Question 13

Why do we need endogenous growth models?

Answer 13

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).

Question 14

What are the three endogenous growth models?

Answer 14

1. Arrow, learning by doing
2. Romer, R&D model
3. Lucas, human capital model

Question 15

Explain Arrow’s “learning by doing” model

Answer 15

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 y_i = Ak_i^α
Technology = A = Bk^β, where β = knowledge spillovers
So, y = Bk^βk^α = Bk^α+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).

Question 16

Explain Romer’s R&D model

Answer 16

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 L_Y in production and L_R in R&D, so technology accumulates by A^•=L_RA.

A^•/A = g_A = L_R^φ, where φ<1 represents diminishing returns to workers

y = Ak^α → lny = lnA + αlnk → g_y = g_A + αg_k
So g_y = L_R^φ + αg_k, 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, g_y= g_k
So g_y = L_R^φ / (1-α)

So the rate of growth in output is constant, rather than diminishing?

Question 17

Explain Lucas’s human capital model

Answer 17

Human capital models supplement regular capital with another type of capital (human capital) which is assumed not to have diminishing returns.

L_Y in production and L_H studying

Human capital accumulates by A^•=L_HA

A^•/A = g_A = L_H → g_Y = L_H - αg_k

If you get people to study more, you permanently increase the growth rate

Question 18

What are Jones’s critiques of the endogenous models?

Answer 18

1. Is knowledge/human capital development infinite?
2. Assumption that the exponent of A is 1 so that g_y 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

Question 19

Explain Jones’s semi-endogenous model

Answer 19

A^• = L_R^φ A^η → g_A = L_R^φ / A^1-η

If η<1, diminishing returns to R&D or human capital - innovation slows down

If η>1, increasing returns - growth rate explodes