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Investment and Capital Accumulation: The evolution of the capital stock is given by
Kt+1=(1δ)Kt+It

change in capital per worker over time
 equal to saving per worker minus depreciation
 K_{t+1}/NK_{t}/N=sY_{t}/NδK_{t}/N

if there is increase in capital per worker
investment>depreciation

effect of lower saving rate
 lower output per worker
 hence temporary lower(negative) growth
 no effect on long term growth

for the developing countries, using PPP method results in … standard of living; exchange rate method … the standard of living because
higher
underestimate
the lower a country’s output per capita, the lower the prices of food and basic services in that country.

The Convergence of Output per Person since 1950(Asia and OECD countries)
if a country show lower GDP per capita(horizonal), it shows higher annual GDP growth rate(vertical)

CONVERGENCE: Countries with small capital stocks should grow rapidly because of
Diminishing returns of capital

Constant returns to scale
if the scale of operation is doubled—that is, if the quantities of capital and labor are doubled—then output will also double.

Decreasing returns to capital
 increases in capital, given labor, lead to smaller and smaller increases in output as the level of capital
 increases.

Capital accumulation by itself cannot sustain growth, why?
 Due Decreasing returns to capital, maintaining a steady increase in output per worker would require larger and larger increase in capital per worker.
 At some point the economy will be unable to save and invest enough to further increase capital

Sustained growth requires
technological progress

what increases output level in long run
 increase in tech state
 increase in saving rate
 decrease in dep'n rate

INTERACTIONS BETWEEN OUTPUT AND CAPITAL
The amount of capital determines the amount of output being produced
The amount of output determines the amount of saving and, in turn, the amount of capital accumulated over time

the relation between output and investment

Derive steadystate consumption per worker
 c = y – i
 c = f(k) – sf(k)
 since sf(K*)=δK*
 c = f(k*) – δK*

saving rate determines the allocation of ... between ... and ...
 output
 consumption and investment

golden rule level of consumption can be achieved by
 a particular level of saving rate such that where sf(K*)=δK* AND
 marginal product of capital=δ(+gn+ga)

testing constant return of scale
the powers add up to one

initial saving rate always equal to certain value signals
initially at steady state

Derive the steady state level of investment with population growth and tech progress
 In SS, K/AN is contant
 K must grow at the same rate as A&N; gk=gn+ga
 ΔK=(gn+ga)*K
 I must be sufficient (1) to replace depreciated capital and (2) to allow the capital stock to increase by gn*K(providing new efficient units with SS capital)
 I=(δ+gn+ga)*K

economies with higher rates of population growth will have
lower level of capital thus lower income

growth rate approximation
z=x^{a
}g_{z}=ag_{x}

what is growing at (ga+gn) in steady state
 effective labour
 capital
 output

Why would reduction in budget deficit lead to increase in output and investment in the long run?
Reduction in budget deficit is equivalent to increase in public saving
More saving, Shift up the long run investment curve; gives higher output per capital

suppose Y=Bsq(K)sq(AN), find k and y in terms of s, delta, ga, gn
k=(B*s/(delta+ga+gn))^{2
}y=B*sq(K)

An increase in the rate of technological progress will ....the steadystate levels of capital and output per effective worker
decrease

