1. Solow (quite hard)

Question 1

Solow model - long run model

What does it depend on (3)

Answer 1

Looks how long run GDP and consumption per capita depend on

savings (investment) rate
growth rate of labour force
Growth of technology

Question 2

What is the main driving force for wealth

Answer 2

Physical Capital accumulation

Question 3

Solow assumptions (3)

Answer 3

Single household/producer owns input and tech to turn input into outputs

Output can be used for consumption or investment

Save for tomorrow

Question 4

Production function: recall output determined by capital labour and tech

Answer 4

Y(t) = F[K(t), A(t)L(t)]

Output is a function of K, A and L
A is tied to L since labour augmenting tech (tech improves labour productivity)

AL is called effective labour

Question 5

Assumptions about the production function (2)

Answer 5

CRS (doubling K and AL doubles output) (also important as allows us to express output in terms of effective labour!)

But if only capital increases, DMR to capital!

Question 6

Find intensive production function (output per unit of effective labour)

Start with F(cK,cAL)

B) what do we find out pure per unit of effective unit of labour is a function of?

Answer 6

Set c=1/AL and sub in. We get:
F(K/AL, 1)
=F(k,1)
=f(k)

y=f(k)

B)
Output per unit of effective labour (y) is a function of capital per effective unit of labour (k)

Question 7

Assumptions for f:
f=F(k,1)
Where k=K/AL

A) what if f(0)

B) what if f’(k)>0
C) what if f’’(k)<0

Answer 7

f(0) means no capital=no output

f’(k)>0: increasing capital increases output

f’’(k)<0: DMR to capital

Question 8

How can we show Y= ALf(k)

Answer 8

As y=f(k) (intensive form of prod function)

y=Y/AL so

Y/AL = f(k)

X by AL
Y = ALf(k)

Question 9

Use this to derive MPK to get = dY/dK = f’(k) : (pg9)

B) when is it high and when is it low (pg 10) and what does this mean,

Answer 9

A) working pg 9

B) High when capital is scarce, MPK low when capital is plentiful. Means a steady state exists

Question 10

Proof for CRS; given CD function is
Y = F(K,AL) = K to the a (AL) to the 1-a

PG 11

Answer 10

Also know CRS as a + (1-a) sums up to 1

Question 11

We also show intensive form of CD production function
Y = K to the a AL to the 1-a

Gets us
y=f(k) = k to the a

Pg12 proof

Question 12

How to draw intensive production function pg 13: axis’ and what shape should it be and why

Answer 12

Y axis f(k)
X axis K

Concave slope, shows as we add more K, f(k) i.e capital per effective unit of labour has diminishing returns

Question 13

Recall we said output Y(t) can be consumed C(t) or invested I(t), to create new physical capital K(t)

So in closed economy what is our output equation

Answer 13

Y = C+I

Question 14

What is savings equation in this close economy

Answer 14

S= sY

s is proportion of output saved (thus savings S)

And also S=I

Question 15

Net Change in physical capital expression (K.)

B) how can we make adjusted expression

Answer 15

K. = I - δK
Gross investment - Depreciation

B) we know I=S and S=sY
So it simplifies to
K. = sY - δK

Question 16

What is the key dynamic of the model

Steps (pg 17, look)

Get to final Solow equation

Answer 16

k = K/AL
(Need this to get our final equation for solow)

k./k = K/K - L./L - A./A
Which is
k.k = K./K - n -g

Recall K = sY-δK
k./k = (sY - δK)/k - n - g
= k./k = sY/K - δ-n-g

Then x by k
sY/K (K/AL) - (δ+n+g)k
= sy - (δ+n+g)k
= sf(k) - (δ+n+g)k

Question 17

So solow equation is
sf(k) - (δ+n+g)k

What is sf(k)

What is (n+g+δ)k

C) whats the equilibrium

D) why is equilibrium useful (imp)

Answer 17

A) actual investment per effective unit of labour

B) break-even investment per unit of effective labour

C) sf(k) = (δ+n+g)k

D) as gives us steady state values of k, which gives us y (since y=f(k) i.e output per effective worker is a function of capital per effective worker! (FC8)

Question 18

Solow steady state diagram pg 19

B) explain it e.g if k<k*

Answer 18

Y axis sf(k) X axis k

Breakeven investment (n+g+δ)k: straight line
Sf(k) concave as MPK positive but DMR to capital

Intersection is where k* steady state

B) Intuition:
If k<k* (klow), sf(k)>(n+g+δ)k i.e actual investment>breakeven investment, so capital per effective worker growth is positive/rising!

Question 19

So A grows at rate g, L grows at rate n.

What rate does K and Y grow at (pg21)

B) What does K/L and Y/L grow at

Answer 19

As k=K/AL, rearrange so K=kAL

So K and Y grows at rate n+g

B) rate g

Question 20

What happens to diagram increases if savings rate (s) increases (pg22)

Answer 20

Shift upwards in sf(k) curve

Savings rate higher so invest more, so steady state capital per effective worker is higher! If k* higher y* (more capital = more output!)

Question 21

Level effects vs growth effects of the increase in savings rate s

Answer 21

Savings rate effects on growth is temporary, capital accumulates faster only temporarily (until new k* is reached). Increases level of output/capital, but not the growth rate!

In long run, K/L and Y/L only depend on g (as mentioned)

Question 22

What is consumption expression

B) expand to get the steady state (balanced growth path) pg24

Answer 22

c = (1-s)f(k)

Expand to get

c* = f(k) - sf(k)
B)
And then as steady state sf(k) = (n+g+δ)k
= f(k) - (n+g+δ)k*

Question 23

Then FOC with respect to saving (see how consumption changes in response to savings rate

Pg 24

IMPORTANT TO LEARN FINAL EQUATION (HINT CHAIN RULE SO JS RMB FINAL)

Answer 23

Chain rule
dc/ds = [f’(k) - (n+g+δ)] dk/ds

Question 24

Consumption relationship with saving using the equation

Answer 24

Consumption c* rises WITH saving if f’(k) > (n+g+δ) (this will be if k small, since MPK f’(k*) high)

Falls WITH saving if < (this will be if k* large = low MPK f’(k*)

Unchanged if =

Question 25

The level of k* where f’(k*) = (n+g+δ) is known as what B) draw diagram to show this and the consumption/saving relationship pg25

Answer 25

Golden rule level of capital stock - consumption is maximised

Question 26

What happens if savings rate is above the gold rule savings rate (pg26 my own drawing) B) what if savings rate < golden rule savings rate

Answer 26

Dynamically inefficient - economy is saving too much. They can enjoy higher consumption (red line is larger than green line) but do not. B) consumption not as high as at golden rule . When savings rate increases, consumption increases.

Question 27

Economic interpretation of the golden rule, f’(k) = n+g+δ needs rearranging B) what is interpretation

Answer 27

Just move δ to other side f’(k*) - δ = n+g B) Net rate of return on capital = growth rate of economy

Question 28

So overall effect of savings (an increase) on A) k B) y C) c

Answer 28

Increase in s increases steady state k and y (since more investment, higher k capital pew and thus y output pew C) Consumption ambiguous (rises with saving if f’(k) > etc… falls with saving if <

Question 29

What does steady state income per capita depend on (2) and pos or neg rel B) what does steady state GROWTH of income per capita depend on C) in steady state, what does K grow at same rate as

Answer 29

Savings rate (pos) Population growth n (negative - see by diag, it creates steeper slope of breakeven inv) B) rate of tech progress g (Recall level vs growth effects, only depend on g in long run!) C) output!!! Constant capital-output ratio (K and Y grow at n+g!!)

Question 30

Now consider long run effects of s on output Assume f(k) = k to the a, then find steady state output y* Pg30 final equation (the one to see rich vs poor!)

Answer 30

Final y* = k to the a = (s/n+g+δ) to the a/1-a

Question 31

Now use this steady state output equation for country analysis. Assume savings rate in rich country (H) is 4x larger than poor. sH = 4sP also assume a (capital share) =1/3 Find y*H in terms of y*P (pg31) B) intuition of result

Answer 31

y*H = (sH/n+g+δ) to the a/1-a sub in sH = 4sP = (4sP/n+g+δ) to the a/1-a take 4 outside the bracket 4 (a/1-a) x (sP/n+g+δ) sub in values of a=1/3 to get final y*H = 2y*P b) savings rate 4x larger than poor, means its output/income is 2x as large.

Question 32

So not much change in income differences. Solow model underestimates cross-country income differences what about for population growth effects

Answer 32

Also population growth rate effects are too small

Question 33

So savings rate and population effects in the model are understimated, what about rates of return (MPK?)

Answer 33

Overestimated

Question 34

To show, recall f(k) = k to the a What should MPK be (simple deriv) pg 32 b) simplify further to express in terms of y not k

Answer 34

MPK = f’(k) = ak to the a-1 simplify further as k to the a-1 = y to the a-1/a so MPK = ay to the a-1/a (in human capital model MPK = a(n+g)/sK MPH = β(n+g)/sH

Question 35

So MPK = ay to the a-1/a Suppose income in rich country is 10x poor. yH = 10yP What is MPK of rich country (MPKh) pg 32 b) intuition

Answer 35

sub yH = 10yP MPKh = a(10yP) to the a-1/a MPKh = MPKp/100 b) if income in rich country is 10x the poor, the poor country’s MPK is 100x larger! Way too large, solow model overestimates rates of return!!!

Question 36

So rate of return differences are overestimated what IRL makes the high size of difference we get unlikely

Answer 36

international capital markets

Question 37

So savings rate effects underestimated, rate of return overestimated, what about speed of (conditional) convergence?

Answer 37

How quick they reach their balanced growth path steady state = Convergence is predicted too fast!