Wednesday, February 17, 2010

The Neoclassical Growth Model and Steady States

THE NEOCLASSICAL GROWTH THEORY: This focuses on capital accumulation, and how it is affected by savings

One important function in the neoclassical growth theory is the AGGREGATE PRODUCTION FUNCTION. This function shows the relationship between total real output and total inputs (sort of like a "macro" version of the production function for individual firms we saw in microeconomics)

REMEMBER from the last leccture? There are three main determinants of economic growth: labour, capital, and technology. Well, with the aggregate production function, we say that output is technology times a function of labour and capital

Y = A x F(N,K) where A = total factor productivity (disembodied technology), N = Labour and Human Capital, and K = capital (both quantity and quality)

Now what happens if we divide through by N?

Well, we get

y = A x F(k) where y is the amount of GDP produced per worker, and k is the amount of physical capital available for each worker

Also, potential output is also representable here

Y* = A x F(Nfe, Kfc) where Nfe is full employment, and Kfc is full capacity. In other words, potential output is technology times a function of labour at its full employment level, and physical capital at its full- capacity level

Some important things to remember:
We assume in the long run that income is at its potential level (that there is no output gap)
L is labour quantity, H is labour quality, and N includes both the quality and quantity of labour.
K includes both the quality and quantity of physical capital
We omit land as a factor input for the sake of simplicity in this model
Technology includes entrepreneurship and savviness


1: In the short run, there are diminishing returns to scale: as more of a variable factor is added to a given amount of fixed factor, the additional output generated by the added factor (the "return") will get increasingly smaller and smaller: they will diminish... ceteris paribus (they will diminish if all other things are held constant) after a certain point (they will not begin to diminish immediately)
But, this is only true of the short run when one factor is increased, and all other factors are held constant!

2: In the long run, there are constant returns to scale: When all factors increase the same amount, output will also increase by that amount (so if I double the amount of workers and also the amount of sewing machines, my sweat shop should double its output of shitty sneakers!)

3: Technology is nuetral: A affects the productivity of K and N equally, so although technology is present, it will not disproportionately impact any one factor.

Image Plz! y = f(k)

4: Steady state equilibrium: Here, the per-capita capital (k) and the per capita output (y) remain constant over time, so /\y = /\k = 0

If the population is growing at n, then income and capital must also grow at the same rate in order to remain in a steady state equilibrium. In other words, in order to be in a stead state equilibrium, the percentage change in output must equal the percentage change in capital, which must = the percentage change in the workforce.

y* and k* are the steady state values (they don't change over time)

Investment required to provide capital for new workers and to replace machines that have worn out (depreciation) is just equal to the national savings in a steady state equilibrium, so New Capital + Replacement Capital = Investment = Savings

If savings is greater than investment, than capital per worker will increase, and thus output per worker will also increase

If savings is just equal to investment, then the capital per worker will be k* and thus output per worker will be y*

When savings is equal to required investment, the economy is in a steady state equilibrium, each worker will have access to k*, and will produce y*


To maintain k at a constant rate, investment depends on both population growth and the depreciation rate. Some of investment will have to go to the new workers

WE ASSUME that the population growth rate is constant: thus, to keep capital per worker constant, you must grow capital by nk (the population growth rate times the amount of capital per worker)
WE ASSUME that the rate of depreciation is constant: thus to keep capital per worker constant, you must grow capital by dk as well (the depreciation rate times the amount of capital per worker)

The level of investment required to fund all of this capital growth to maintain a constant capital-worker ratio can be represented by
I = (n + d)k

Here, we assume that we have a frugal economy (there is no government or international trade)
We also assume that the marginal propensity to save is constant
S/N = sy = sf(k)
in other words, per capita savings are a function of per-capita output, which in turn, is a function of the labour-capital ratio

The net change in the capital-labour ratio is equal to the excess of actual savings over required investment
/\k = to per-capita savings - the capital-labour ratio multiplied by (the population growth rate + the rate of depreciation)

In a steady state, /\k = o, so per-capita savings must be equal to per-capita required investment
sy* = (n + d)k*

Image plz

If we graph the production function, the savings function, and the required investment function with money on the Y axis and the capital labour ratio on the X axis, the savings function and the required investment function will eventually intersect: this point is the steady state equilibrium, E
at E, actual investment is just equal to required investment
the capital-labour ratio k* and standard of living y* are constant
At capital labour ratios lower than k*, savings will be greater than required investment, so the capital labour ratio and the standard of living will both increase.

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