Презентация на тему Growth theory: the economy in the very long run

Rule Level of Capital
8-3 Population Growth

growth,
technological progress
Level & Growth of output



A

2010
Indonesia
Uruguay
Poland
Senegal
Kyrgyz Republic
Nigeria
Zambia
Panama
Mexico
Georgia
Peru

for Goods
Growth in the Capital Stock and the Steady

The Supply in the Solow model is based on the PF:
Y = F(K, L).

Assumption:
the PF has constant returns to scale:
zY = F(zK, zL), for any positive number z.

If z = 1/L →
Y/L = F(K/L, 1).

= K/L is capital per worker
f(k) = F(k,

8-1 The Accumulation of Capital

The Supply and Demand for Goods
Growth in the Capital Stock and the Steady State
Approaching the Steady State: A Numerical Example
How Saving Affects Growth

Y/L = F(K/L, 1)

of capital per worker k determines the amount of

for Goods
Growth in the Capital Stock and the Steady

Output per worker y is divided between consumption per worker c and investment per worker i:
y = c + i.
G - we can ignore here and NX – we assumed a closed economy.

The Solow model assumes that people
save a fraction s of their income
consume a fraction (1 − s).
We can express this idea with the following CF:
c = (1 − s)y,
0 < s (the saving rate) < 1

Gnt. policies can influence a nation’s s
What s is desirable ?

for Goods
Growth in the Capital Stock and the Steady

Assamption:
We take the saving rate s as given.

To see what this CF implies for I,
we substitute (1 − s)y for c
in the national income accounts identity:
y = (1 − s)y + i =>
i = sy
s is the fraction of y devoted to i.

for Goods
Growth in the Capital Stock and the Steady

The 2 main ingredients of the Solow model—
the PF and the CF.
For any given capital stock k,
y = f(k)
determines how much Y the economy produces, and
s (i = sy)
determines the allocation of that Y between C & I.

for Goods
Growth in the Capital Stock and the Steady

The capital stock (CS) is a key determinant of output,
its changes can lead to economic growth.

2 forces influence the CS.
Investment is expenditure on new plant and equipment, and it causes the CS to rise.
Depreciation is the wearing out of old capital, and it causes the CS to fall.

Investment per worker i = sy
We can express i as a function of the CS per worker:
i = sf(k).
This equation relates the existing CS k to the
accumulation of new capital i.

the allocation of output between C & I.
For

wears out every year. Depreciation is therefore proportional to

δ = the rate of depreciation
= the fraction of the capital stock that wears out each period

= i – δk
Since

Δk = s f(k) – δk

The basic idea: Investment increases the capital stock, depreciation reduces it.

central equation
Determines behavior of capital over time…
…which, in turn,

Δk = s f(k) – δk

cover depreciation [sf(k) = δk ],
then capital per

Δk = s f(k) – δk

δk
k2

δk
k2

δk
k2
k3

δk
k3
Summary: As long as k < k*, investment will exceed

the steady state k*.
On the horizontal axis, pick

production function, divide through by L:
Then substitute y =

of k = 4.0

k y c i k

4 4.584 2.141 1.499 0.642 0.458 0.184
…
10 5.602 2.367 1.657 0.710 0.560 0.150
…
25 7.351 2.706 1.894 0.812 0.732 0.080
…
100 8.962 2.994 2.096 0.898 0.896 0.002
…
 9.000 3.000 2.100 0.900 0.900 0.000

s = 0.3, δ = 0.1, and y

Use the equation of motion Δk = s f(k) − δk to solve for the steady-state values of k, y, and c.

the saving rate raises investment…
…causing k to grow toward

y = f(k) , higher k* ⇒ higher y*

person



































































































100
1,000
10,000
100,000
0
5
10
15
20
25
30
35
Investment as percentage of output
(average 1960-2000)
Income per
person

2000

(log scale)

to different steady states. How do we know which

of capital, the steady state value of k that

To find it, first express c* in terms of k*:
c* = y* − i*
= f (k*) − i*
= f (k*) − δk*

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

point where the gap between them is biggest.
The

δk* is biggest where the slope of the production function

steady-state capital per worker, k*

MPK = δ

economy does NOT have a tendency to move toward

a fall in s.
In the transition to the

t0

c

i

y

an increase in s.
Future generations enjoy higher consumption,

time

t0

c

i

y

grow at rate n. (n is exogenous.)

EX: Suppose

amount of investment necessary to keep k constant.
Break-even

the equation of motion for k is
Δk = s

(δ +n)k

worker, k
(δ +n1) k
k1*
An increase in n

leading to a lower steady-state level of k.

y = f(k) , lower k* ⇒ lower y*.

person



































































































100
1,000
10,000
100,000
0
1
2
3
4
5
Population Growth
(percent per year; average 1960-2000)
Income
per Person

in 2000

(log scale)

Golden Rule capital stock, express c* in terms of

In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate.

population growth will outstrip the Earth’s ability to produce

that population growth contributes to economic growth.
More people

the long run, a country’s standard of living depends
positively

CHAPTER 7 Economic Growth I

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