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ECON10003 THE SOLOW-SWAN MODEL OF ECONOMIC GROWTH

创建日期：2024-05-27 17:21:08

所属分类：经济Economics

标签： ECON10003

THE SOLOW-SWAN MODEL OF ECONOMIC GROWTH

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ECON10003

LECTURE 15: THE SOLOW-SWAN MODEL OF ECONOMIC GROWTH I
THIS LECTURE
• Solow-Swan growth model, part one
– savings and capital accumulation
– implications of diminishing returns to capital
– transitional dynamics
• BOFAH chapter 15
SOLOW-SWAN MODEL OF GROWTH
• Based on two important long-run relationships:
(i) amount of capital determines output produced (per period)
(ii) amount of output determines new investment (per period)
• Together, these two relationships determines the amount of capital accumulation
over time.
• Question: Can capital accumulation sustain long-run growth?
SOLOW-SWAN MODEL OF GROWTH
• We start with the following aggregate production function with the standard properties that we
discussed in the previous lecture ! = !,
• For now, to keep things simple, we will assume and to be constant. We will have them vary
over time in the next lecture.
• National income accounting ! = ! + !
• Note here that we are assuming a closed economy and no government purchases.
CAPITAL ACCUMULATION, SAVINGS AND
INVESTMENT
• Recall that the change in the capital stock is investment less depreciation (i.e., a constant fraction
of the capital stock depreciates each period)!"# − ! = ! − !, 0 < < 1
• A constant fraction of output is saved each period ! = !, 0 < < 1
• Since the economy is closed, investment per period is equal to savings! = ! = !
• Consumption per period is therefore ! = (1 − )!
CAPITAL ACCUMULATION
• Since investment ( = saving) is proportional to output the change in the
capital stock is therefore, !"# − ! = ! − ! = ! − !
• As ! = ! , , we may write!"# − ! = ! , − !
• This tells us how much capital there will be next period !"# given how much capital there is in this period !
SOLOW-SWAN DIAGRAM
If investment > depreciation,
capital is increasing, !"# > !
If investment < depreciation,
capital is decreasing, !"# < !
If investment = depreciation,
capital is unchanged, !"# = !
STEADY-STATE CAPITAL ∗
• If investment = depreciation, capital is does not change, !"# = !, when this occurs, we are said to be in a steady state.
• This steady state capital stock which we shall denote as ∗, satisfies the
equation ∗, = ∗
• Therefore, for a given production function ∗, we can solve for ∗ in
terms of , , and the parameters of the production function.
• Graphically, this is the point where the investment curve ∗,
intersects the depreciation line ∗
STEADY-STATE OUTPUT AND CONSUMPTION
• Once we have determined the steady-state capital stock ∗ we can
then easily obtain the steady state output ∗ by simply plugging ∗
into the production function ∗ = (∗, )
• Steady-state consumption is then given by ∗ = (1 − )∗
STEADY-STATE OUTPUT AND CONSUMPTION
DIMINISHING RETURNS TO CAPITAL
• Question: Can capital accumulation sustain long-run growth?
• No.
• In the short run, if ! < ∗, then the marginal product of capital is relatively high and the
economy grows
• But there are diminishing returns to capital. As ! increases, the marginal product of
capital falls and growth slows down!
• In the long run, capital ! → ∗ and growth slows down to zero.
INCREASE IN THE SAVINGS RATE
• To see that capital accumulation cannot sustain long-run growth, consider a permanent increase in the
savings rate from to ′ >
• Shifts up investment curve along an unchanged depreciation line
– steady state capital increases from ∗to ∗′
– steady state output increases from ∗ to ∗′
• On impact, investment exceed depreciation, capital accumulates and economy grows
• But over time, diminishing returns set in and, in the long run, growth slows back down to zero.
INCREASE IN THE SAVINGS RATE
INCREASE IN SAVINGS RATE
LEVEL EFFECTS AND GROWTH EFFECTS
• Increase in the savings rate from to ′ >
– increases the level of capital and output in the long run
– but has no long run effect on the growth of capital and output
• Increase in savings rate moves us along production function, increasing
capital per worker for a given level of productivity A.
LEVEL EFFECT
BUT NO LONG RUN GROWTH EFFECT
COBB-DOUGLAS EXAMPLE
• Now let’s work with the Cobb-Douglas production function
, = !"#!
• In this case, the steady state capital stock ∗ solves
∗!"#! = ∗
• To derive the solution, we first divide both sides of the above equation by ∗! to obtain,
∗"#! = "#!
• Then we raise both sides to the power of 1/(1 − ) which yields
∗ = ""#!
IMPLICATIONS OF THE MODEL
• The steady state capital ∗ is:
- Increasing in the savings rate , productivity , and labour
- Decreasing in depreciation rate
• Question: So what can generate sustained growth?
• Answer: Productivity growth.
• Increases in A shift up the production function, allowing more output at any given level of capital
per worker.
NEXT LECTURE
• Solow-Swan growth model, part two
– productivity growth, employment growth, factor shares
– the convergence hypothesis
– growth accounting

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