So far we have assumed that technology is fixed: i.e. that there is an unchanging relationship between the inputs of capital and labour and output. We now include technological progress, which increases production for any given level of inputs.

To incorporate technological progress, modify the production function. So far we have used the following production function to relate total capital KK:

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

We now write the production function as:

Y=F(K,L×E)Y = F(K, L \times E)

where EE is a new variable called the efficiency of labour. The efficiency of labour reflects worker's knowledge about how to produce goods and services: as technologies improve, workers become more efficient and each hour of work contributes more output. For example, the efficiency of labour rose when assembly line production transformed manufacturing in the early twentieth century. The efficiency of labour also rises with improvements in the education or skills of workers.

The term L×EL \times E measures the effective number of workers. It takes into account the number of workers and how efficient each worker is. In this new production function, total output depends on both the inputs of capital and on a measure of both the number of workers and their skills (or technical know-how) with which they come equipped.

This approach to modeling technological progress is that increases in the efficiency of labour EE have a similar effect to increases in the labour force LL. For example, suppose that over time an advance in production methods makes a single worker as productive as two workers. As the efficiency of labour EE doubles, so does the effective number of workers L×EL \times E, which leads to an increase in output.

We assume that technological progress causes the efficiency of labour EE to grow at some constant rate gg. For example, if g=0.02g = 0.02 then each unit of labour becomes 2 percent more efficient every year. The effect on total output is identical to that of a 2 percent marginal increase in the number of workers. Hence, we call this approach to technological progress labour augmenting, and gg is called the rate of labor-augmenting technological progress.

Because the labour force LL is growing at rate nn and the efficiency of each unit of labour EE is growing at rate gg, the effective number of workers is growing at rate n+gn + g.

Because technological progress is modeled as labour augmenting, it fits into the model in much the same way as population growth. Technological progress does not cause the actual number of workers to increase, but because each worker in effect comes with more units of labor over time, technological progress causes the effective number of workers to increase. Thus, we can easily adapt the model with population growth to a model with labor-augmenting technological progress.

Previously, when there was no technological progress, we analyzed the economy in terms of quantities per worker. We now generalize that approach by analyzing the economy in terms of quantities per effective worker. We now have k=K/(L×E)k = K/(L \times E) on the X axis stand for capital per effective worker, and y=Y/(L×E)y = Y/(L \times E) on the Y axis stand for output per effective worker. With these definitions, we can again write y=f(k)y = f(k).

We agument our model just as we did when we introduced population growth. The equation for the evolution of kk over time becomes:

Δk=sf(k)(δ+n+g)k\Delta k = s f(k) - (\delta + n + g) k

The change in the capital stock per effective worker equals investment per effective worker minus break even investment per effective worker. Now, break even investment includes three terms: to keep kk constant, we must replace depreciating capital through δk\delta k, provide capital for new workers through nkn k and gkg k is needed to provide capital to the new 'effective workers' created by technological progress.


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