/69-21_files/69-2100001im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

LESSON 21

ECONOMIC GROWTH (Continued...)

The Golden Rule: introduction

Different values of s lead to different steady states.

How do we know which is the "best" steady state?

Economic well-being depends on consumption, so the "best" steady state has the highest

possible value of consumption per person:

c* = (1s) f(k*)

An increase in s

· leads to higher k and y , which may raise c

· reduces consumption's share of income (1s),

which may lower c*

So, how do we find the s and k* that maximize c*?

The Golden Rule Capital Stock

K*gold = the Golden Rule level of capital, the steady state value of k that maximizes

consumption.

To find it, first express c* in terms of k*:

= y* - i*

- i*

= f (k*)

- δk*

= f (k*)

In general:

i = Δk + δk

In the steady state: i* = δk* because Δk = 0.

Then, graph f(k*) and δk*, and look for the point where the gap between them is biggest.

/69-21_files/69-2100002im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

steady state

δ k*

output and

depreciation

f(k*)

C*gold

i*gold = δk*gold

k*gold

Steady-state

Y*gold = f(k*gold)

capital per

worker, k*

c* = f(k*) - δk*is biggest where the slope of the production function equals the slope of the

depreciation line: MPK = δ

steady state

δ k*

output and

depreciation

f(k*)

C gold

k*gold

Steady-state

capital per

worker, k*

/69-21_files/69-2100003im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

The transition to the Golden Rule Steady State

The economy does NOT have a tendency to move toward the Golden Rule steady state.

Achieving the Golden Rule requires that Policymakers adjust s.

This adjustment leads to a new steady state with higher consumption.

But what happens to consumption during the transition to the Golden Rule?

Starting with too much capital

If k * > k gold

then increasing c* requires a fall in s.

In the transition to the Golden Rule, consumption is higher at all points in time.

Time

/69-21_files/69-2100004im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

Starting with too little capital

If k * < k gold

then increasing c* requires an increase in s.

Future generations enjoy higher consumption, but the current one experiences an initial drop

in consumption.

Time

The basic Solow model cannot explain sustained economic growth. It simply says that high

rates of saving lead to high growth temporarily, but the economy eventually approaches a

steady state.

We need to incorporate two sources of growth to explain sustained economic growth:

population and technological progress.

/69-21_files/69-2100005im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

Population Growth

Assume that the population--and labor force-- grow at rate n. (n is exogenous)

ΔL

= n

EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).

Then ΔL = n L = 0.02 × 1000 = 20,

so L = 1020 in year 2.

Break-even investment

(δ + n)k = break-even investment, the amount of investment necessary to keep k constant.

Break-even investment includes:

δ k to replace capital as it wears out

n k to equip new workers with capital

(otherwise, k would fall as the existing capital stock would be spread more thinly over a

larger population of workers)

The equation of motion for k

With population growth, the equation of motion for k is

Δk = s f(k) - (δ + n) k

Where

S f(k)= actual investment

(δ + n) k = breakeven investment

The impact of population growth

(δ +n2) k

Investment,

(δ +n1) k

break-even

investment

sf(k)

k2*

k1*

Capital per

worker, k

/69-21_files/69-2100006im.jpg" width="695" height="1066" useMap="#Map">

Macroeconomics ECO 403

Prediction:

Higher n ⇒ lower k*.

And since y = f(k) ,

lower k* ⇒ lower y* .

Thus, the Solow model predicts that countries with higher population growth rates will

have lower levels of capital and income per worker in the long run.

The Golden Rule with Population Growth

To find the Golden Rule capital stock, we again express c* in terms of k*:

= y* - i*

= f (k* ) - (δ + n) k*

c* is maximized when

MPK = δ + n

or equivalently,

MPK - δ = n

In the Golden Rule Steady State, the marginal product of capital net of depreciation equals the

population growth rate.