Lecture 12 - Using The Solow Growth Model (Golden Rule)

Question 1

What is the golden rule savings rate?

Answer 1

Finding the savings rate that maximises consumption (known as benevolent)

Question 2

How to find the golden rule savings rate

(Learn beginning and final formula)

Answer 2

Use basic Solow model. (Add * as maximising)
Recall national income identity y=c+i
c* = y* - i* = f(k) - δk

Differentiate with respect to k*, set = 0 to find min/max (in this case we want max consumption)

C* = f’(k) - δ = 0
C = MPK - δ = 0
MPK=δ is the k* associated with the golden saving rate. (k*gold)

To then find the saving rate that kgold is achieved.
add s to the solow model. ‘\;\
sf(kgold) -δkgold = 0
s = δkgold / f(k*gold)

Question 3

Golden rule equation and diagram (LOOK AT PG4, ANNOTATIONS ARE USEFUL)

2. What if MPK>δ or <δ

Answer 3

Golden rule requires MPK =δ , steady state is at golden rule level of capital k*gold

If MPK >δ capital stock (K) is below k*gold, so an increase in capital will increase consumption.

If MPK<δ capital stock is above k*gold, and so an increase in capital will decrease consmption.

Question 4

Intuition of the golden rule: why does consumption increase when below steady state vs decrease when above.

Answer 4

Diminishing returns to k

y (income) grows at a faster rate than i when below the golden rule level, and so consumption increases. (Using c=y-i we can see this)

Above the golden rule level, y is growing slower than i so consumption falls.

Question 5

5/12

Assume savings rate is currently higher than the golden rule. What happens to output consumption and investment if saving rates fall down to the golden rule level. E.g through an increase in tax.

Answer 5

Consumption increases - saving less means spending more

Investment (per worker) - falls as saving is now less than breakeven investment

Output - Less savings per worker to keep capital per worker (k) constant (savings<breakeven investment) so output falls as workers have less capital to use.

So when capital stock falls to k*gold, economy settles at the golden rule where consumption is maximised!!!

Question 6

What then happens to consumption overtime

Answer 6

Increases as mentioned in last slide.

But as output per worker is now lower, so consumption falls. (Less output means less income)

But still higher than originally since savings rate is lower

Question 7

So why was savings rate high in the first place at a high steady state, if consumers want to maximise consumption c*???

Answer 7

Households are influenced by other factors and constraints and so are forced to save more than optimal for them, so cannot maximise consumption.

Question 8

So if the savings rate is far away from the golden rule, what does this imply the need for?

Answer 8

Policy intervention to lower savings rates to allow consumption to increase.

e.g tax cuts reduce income thus increasing MP (saving is a luxury - keynes consumtion)

Question 9

Now consider opposite situation: savings rate is below the golden rule.

What would the effect on consumption, investment and output be if the saving rate increased to the golden rule level? E.g through a cut in tax

Answer 9

Consumption - an immediate fall (as we save more)

Investment increases (savings>breakeven inv) and so we are able to increase capital per worker, and as a result of that….

Output - with more capital pw, can increase output. Output increases income and so savings and investment increases further, and so consumption also recovers.

Till we get to kgold, where consumption is maximised

Question 10

So why does consumption initially fall then recover, for an increase in savings rate up to the golden rule level.

Answer 10

The increased investment+output (from saving>BIE increasing investment and capital pw) means income rises for further saving/investment and also consumption.

Question 11

1. Golden rule formula
2. Modified golden rule with population growth n
3. Modified golden rule with population growth and technical progress

Answer 11

1. c* = f(k*) - δk differentiate to get MPK = δ
2. c* = f(k) - (δ+n)k differentiate to get MPK = δ + n
3. c* = f(k) - (δ+n+g)k differentiate to get
   MPK = δ + n + g

Question 12

Is the golden rule a good choice?

Answer 12

Yes if household utility is only derived from consumption.
(If they only focus on consumption U(C) and if U’(C)> 0 (Marginal utility positive))

But households may care about other things like the environment/ethics.

Question 13

How to capture households consideration for other factors?

Assume the factor to be environment

Answer 13

U(C-Z)

Z=environmental damage by consumption
Z=F(C)

Differentiate to find maximum utility… we find consumption depends on the sign of (1-F’(C))

If an increase in C causes a greater increase in Z, i.e if 1-F’(C)) is negative, then C is utility reducing!!!

Question 14

3 different scenarios on how to use Solow model

Answer 14

1. Income per worker rises causing less children (remember quantity quality model!!)
2. Saving rate rises with wealth (capital per worker, since more capital to use increase output/income per worker)
3. Solow model with human capital

Question 15

Scenario 1:
Increase in income per worker causing less children

Answer 15

So we nee to use Solow with population growth

Initally at k₀ , an increase in income per worker y₀>y₁ means s>i so steady state increases k₀>k₁.

Now at higher y, less children (n). Shift down in breakeven point leads to a further higher steady state (k₁ to k₂, y₁ to y₂)

Question 16

Scenario 2 : Saving rate rises with wealth (measured by capital per worker k)

Assume economy starts at k₀

Answer 16

We dont have to go beyond basic Solow model
sf(k) = i = δk (since no mention of tech progress or labour force)

s is MPS

So if s increases with wealth (measured by k), as s increases, k increases.

Visualised by shifts up in saving.

Question 17

Scenario 3: Solow model with human capital

Answer 17

Rewrite production function as Y= F(K, HN) where H is human capital. (Similar to technical progress where it was Y= F(K,EN) where E was efficiency of workers.

Then divide prod function by n to get pw terms
y = F(k,H)

Use basic solow model as no mention of n or g.

Graphically shown by the pw prod function with higher H₁ above H₀. (High human capital=high capital per worker k)

Question 18

Solow model predicts that countries with population growth (n) have

Answer 18

Low income per worker y.

(As lower k, we also have lower y using the balanced growth path - i.e both grow at rate n)

Question 19

So Malthus is population growth,
Kremer is technical progress built upon the fact of population growth causing innovation and diversity (homogeneity) of humans