Lecture 2 - Solow Model

Question 1

Endogenous variable

Answer 1

A variable that’s changes or determined by its relationship with other variables in the model

Question 2

Exogenous variable

Answer 2

A variable whose measure is determined outside the model and is imposed on the model

Question 3

What is the Solow Model built around

Answer 3

1. a production function and 2. a capital accumulation equation, with endogenous (output Y= Y(t), capital K = K(t)) and exogenous (population L=L(t)) variables changing over time

Question 4

What is the production function, and what are the two assumptions

Answer 4

Y= F(K,L) or y= f(k), constant returns to scale and diminishing marginal returns

cobb- douglas production function: F(K,L) = K ^α L ^{1- α}

Question 5

What are the proofs for constant returns to scale and diminshing marginal returns

Answer 5

constant returns to scale: LF (K, L) / L = LF(K/L, L/L) = LF(k, 1) = LF(k) which is y=f(k) (with f(k) = F(k, 1))

Question 6

Proposition

In the basic Solow model what is growth in income per worker

Answer 6

growth in income per worker is directly proportional to growth in capital per worker
g_y = αg_k

Question 7

What is the capital accumulation equation?

Answer 7

K˙ = sY −δK or ˙k = sy −(δ +n)k or ˙k = sk^α - (δ +n)k

change in capital = total savings - depreciation of capital

Question 8

Proof

What is the equation for the expression ˙k

assuming that population grows exogenously at a rate g _L = n (fertility) with g_k = g_K - g_L ≡ K˙ /K − L˙/L gives

Answer 8

˙k = sy - (δ + n)k

Question 9

Definition

What is the balanced growth path (steady-state)

Answer 9

long run trajectory where endogenous variables grow at constant rates, given the parameters of the model

Question 10

Proposition

What does the concavity of f(k) ensure

Answer 10

a unique balanced-growth path (or steady state) where k˙ = 0 hence g_k = 0 and g _y = 0

since capital and output are directly proportional, we are looking for k^* such that g_k = 0 ,k˙ = 0

Question 11

What is the Solow diagram for the evolution of physical capital per worker

Question 12

Proposition

What are the equations in the steady-state for capital and output per worker

Answer 12

k^* = (s/ δ + n) ^{1 / (1-α)} and y^* = (s/ δ + n) ^{α / (1-α)}

Question 13

Proof

What is the proof of the steady state equations using the capital accumulation equation

Question 14

Definition

Absolute Convergence

Answer 14

Irrespective of structural characteristics, economic growth will be larger for poorer countries

Question 15

Definition

Conditional convergence

Answer 15

economic growth will be faster for poorer countries then they will converge to richer countries only if they have indentical structural characteristics (s, δ, n)

Question 16

What is equation for the rate of growth

Answer 16

g_k = sk^α-1 - (δ +n)

Since g_k = ˙k/k to find the rate of growth, divide the capital accumulation equation through by k

Question 17

What type of convergene does the Solow Model predict

Answer 17

Conditional Convergence because a wealthy country could still have a higher growth rate if it has a higher savings rate, lower fertility or less depreciation

rate of growth derivatives proof

Question 18

What is the rate of growth dependent on

Answer 18

structural characteristics

Question 19

What are the derivatives of rate of growth

Answer 19

strictly decreasing and concave

∂g_k / ∂k = (α - 1)sk^α-2 <0 and ∂²g_k / ∂k² = (α-1)(α-2)sk^α-3 > 0

Question 20

In the long run there is no economic growth, what diagram represents this

Question 21

Proposition

In the Solow model with technological progress what is growth in income per worker

Answer 21

growth in income per worker is a liner, convex combination of the rate of growth in capital per worker and rate of technological progress
g_y = αg_k + (1-α)g_A

Question 22

Proof

What is the proof for the growth in income per worker in a Solow model with technological progress

Answer 22

Take logs and then derivatives using y = A^{1- α} + k^α

log(y) = log(A^{1- α}) + log(k^α)
log(y) = 1- αlog(A) + αlog(k)
g_y = (1-α)g_A + αg_k

Question 23

Proposition

What is economic growth driven by in the Solow Model with technological progress

Answer 23

exogenous technological progress in the balanced growth path: g_y = g_k = g_A = g

Constant growth: all varables grow at the same rate

Question 24

Proof

Prove that in the balanced growth path, k grows at some constant rate g

Solow Model with technological progress

Answer 24

From the capital accumulation equation dividing by k:
k˙/k = sy/k - (δ+n) means g = sy/k - (δ+n) .
sy/k = δ+n+g
y/k = (δ+n+g) / s
Hence y and k are directly proportional we can deduce that they grow at the same rate: g_y = g_k.
Since g_y = αg_k + (1-α)g_A we deduce that g_y = g_k = g_A = g

Question 25

Proposition

What are capital and output per worker directly proportional to in the balanced growth path

Solow Model with technological progress

Answer 25

to total factor productivity

Question 26

What are the equations for capital and output per worker in the balanced growth path

Answer 26

k^* = A(s/(g+δ+n))^1/(1-α) and y^* = A(s/(g+δ+n))^α/(1-α)