The Rest of the Growth Model:

The rest of the growth model is straightforward. First comes the need to keep track of the quantities of the model over time. Do so by attaching to each variable--like the capital stock or the efficiency of labor or output per worker or the labor force--a little subscript telling what year it applies to. Thus K1999 will be the capital stock in year 1999. If we want to refer to the efficiency of labor in the current year (but don't much care what the current year is), we will use a t (for "time) as a placeholder to stand in for the numerical value of the current year. Thus we write: E_t. And if we want to refer to the efficiency of labor in the year after the current year, we will write: E_t+1.

We assume—once again making a simplifying leap of abstraction--that the labor force L of the economy is growing at a constant proportional rate given by the value of a parameter n. Note that n does not have to be the same across countries, and can shift over time in any one country). Thus between this year and the next the labor force will grow so that:

L_t+1 = (1+n) x L_t   

Assume, also, that the efficiency of labor E is growing at a constant proportional rate given by a parameter g. (Note that g does not have to be the same across countries, and can shift over time in any one country.) Thus between this year and the next year:

E_t+1 = (1+g) x E_t

Last, assume that a constant proportional share, equal to a parameter s, of real GDP is saved each year and invested. These gross investments add to the capital stock, so a higher amount of savings and investment means faster growth for the capital stock.

But the capital stock does not grow by the full amount of gross investment. A fraction δ (the Greek letter lower-case delta, for depreciation) of the capital stock wears out or is scrapped each period. Thus the actual relationship between the capital stock now and the capital stock next year is:

K_t+1 = K_t + (s x Y_t) - (δ x K_t)

The level of the capital stock next year will be equal to the capital stock this year, plus the savings rate s times this year's level of real GDP, minus the depreciation rate δ times this year's capital stock.

That is all there is to the growth model: three assumptions about rates of population growth, increases in the efficiency of labor, and investment, plus one additional equation to describe how the capital stock grows over time. Those plus the production function make up the growth model. It is simple. But understanding the processes of economic growth that the model generates is more complicated.

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