So far, our presentation of the Solow model has assumed an unchanging relationship between the inputs of capital and labor and the output of goods and services. Yet the model can be modified to include exogenous technological progress, which over time expands society’s production capabilities.

To incorporate technological progress, we must return to the production function that relates total capital *K* and total labor *L* to total output *Y*. Thus far, the production function has been

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

We now write the production function as

*Y* = *F*(*K*, *L* × *E*),

where *E* is a new (and somewhat abstract) variable called the efficiency of labor. The efficiency of labor is meant to reflect society’s knowledge about production methods: as the available technology improves, the efficiency of labor rises, and each hour of work contributes more to the production of goods and services. For instance, the efficiency of labor rose when assembly-line production transformed manufacturing in the early twentieth century, and it rose again when computerization was introduced in the late twentieth century. The efficiency of labor also rises when there are improvements in the health, education, or skills of the labor force.

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The term *L* × *E* can be interpreted as measuring the *effective number of workers*. It takes into account the number of actual workers *L* and the efficiency of each worker *E*. In other words, *L* measures the number of workers in the labor force, whereas *L* × *E* measures both the workers and the technology with which the typical worker comes equipped. This new production function states that total output *Y* depends on the inputs of capital *K* and effective workers *L* × *E*.

The essence of this approach to modeling technological progress is that increases in the efficiency of labor *E* are analogous to increases in the labor force *L*. Suppose, for example, that an advance in production methods makes the efficiency of labor *E* double between 1980 and 2015. This means that a single worker in 2015 is, *in effect*, as productive as two workers were in 1980. That is, even if the actual number of workers (*L*) stays the same from 1980 to 2015, the effective number of workers (*L* × *E*) doubles, and the economy benefits from the increased production of goods and services.

The simplest assumption about technological progress is that it causes the efficiency of labor *E* to grow at some constant rate *g*. For example, if *g* = 0.02, then each unit of labor becomes 2 percent more efficient each year: output increases as if the labor force had increased by 2 percent more than it really did. This form of technological progress is called *labor augmenting*, and *g* is called the rate of labor-augmenting technological progress. Because the labor force *L* is growing at rate *n*, and the efficiency of each unit of labor *E* is growing at rate *g*, the effective number of workers *L* × *E* is growing at rate *n* + *g*.

Because technological progress is modeled here as labor 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, the analytic tools we used in **Chapter 8** to study the Solow model with population growth are easily adapted to studying the Solow model with labor-augmenting technological progress.

We begin by reconsidering our notation. Previously, before we added technological progress, we analyzed the economy in terms of quantities per worker; now we can generalize that approach by analyzing the economy in terms of quantities per effective worker. We now let *k* = *K*/(*L* × *E*) stand for capital per effective worker and *y* = *Y*/(*L* × *E*) stand for output per effective worker. With these definitions, we can again write *y* = *f*(*k*).

Our analysis of the economy proceeds just as it did when we examined population growth. The equation showing the evolution of *k* over time becomes

Δ*k* = *sf*(*k*) − (*δ* + *n* + *g*)*k*.

As before, the change in the capital stock Δ*k* equals investment *sf*(*k*) minus break-even investment (*δ* + *n* + *g*)*k*. Now, however, because *k* = *K*/(*L* × *E*), break-even investment includes three terms: to keep *k* constant, *δk* is needed to replace depreciating capital, *nk* is needed to provide capital for new workers, and *gk* is needed to provide capital for the new “effective workers” created by technological progress.1

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As shown in Figure 9-1, the inclusion of technological progress does not substantially alter our analysis of the steady state. There is one level of *k*, denoted *k**, at which capital per effective worker and output per effective worker are constant. As before, this steady state represents the long-run equilibrium of the economy.

Figure 9.1: FIGURE 9-1: **Technological Progress and the Solow Growth Model** Labor-augmenting technological progress at rate *g* enters our analysis of the Solow growth model in much the same way as did population growth at rate *n*. Now that *k* is defined as the amount of capital per effective worker, increases in the effective number of workers because of technological progress tend to decrease *k*. In the steady state, investment *sf*(*k*) exactly offsets the reductions in *k* attributable to depreciation, population growth, and technological progress.

Table 9-1 shows how four key variables behave in the steady state with technological progress. As we have just seen, capital per effective worker *k* is constant in the steady state. Because *y* = *f*(*k*), output per effective worker is also constant. It is these quantities per effective worker that are steady in the steady state.

Figure 9.2: TABLE 9-1: **Steady-State Growth Rates in the Solow Model With Technological Progress**

From this information, we can also infer what is happening to variables that are not expressed in units per effective worker. For instance, consider output per actual worker *Y*/*L* = *y* × *E*. Because *y* is constant in the steady state and *E* is growing at rate *g*, output per worker must also be growing at rate *g* in the steady state. Similarly, the economy’s total output is *Y* = *y* × (*E* × *L*). Because *y* is constant in the steady state, *E* is growing at rate *g*, and *L* is growing at rate *n*, total output grows at rate *n* + *g* in the steady state.

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With the addition of technological progress, our model can finally explain the sustained increases in standards of living that we observe. That is, we have shown that technological progress can lead to sustained growth in output per worker. By contrast, a high rate of saving leads to a high rate of growth only until the steady state is reached. Once the economy is in steady state, the rate of growth of output per worker depends only on the rate of technological progress. *According to the Solow model, only technological progress can explain sustained growth and persistently rising living standards*.

The introduction of technological progress also modifies the criterion for the Golden Rule. The Golden Rule level of capital is now defined as the steady state that maximizes consumption per effective worker. Following the same arguments that we have used before, we can show that steady-state consumption per effective worker is

*c** = *f*(*k**) − (*δ* + *n* + *g*)*k**.

Steady-state consumption is maximized if

*MPK* = δ + *n* + *g*,

or

*MPK* − *δ* = *n* + *g*.

That is, at the Golden Rule level of capital, the net marginal product of capital, *MPK* − *δ*, equals the rate of growth of total output, *n* + *g*. Because actual economies experience both population growth and technological progress, we must use this criterion to evaluate whether they have more or less capital than they would at the Golden Rule steady state.