Baseline Model of Directed Technological Change

In this section, I present the baseline model of directed technological change, which uses the expanding varieties model of endogenous technological change and the lab-equipment specification of the innovation possibilities frontier.

where Yl (t) and Yh (t) denote the outputs of two intermediate goods. As the indices indicate, the first is L-intensive, while the second is H -intensive. The parameter ε ∈ [0, ∞) is the elasticity of substitution between these two intermediate goods. The assumption that (15.3) features a constant elasticity of substitution simplifies the analysis but is not crucial for the results. How relaxing this assumption affects the results is discussed at the end of this chapter.

The resource constraint of the economy at time t takes the form

where, as before, X (t) denotes total spending on machines and Z (t) is aggregate R&D spending.

The two intermediate goods are produced competitively with the following production functions:

be equal to this relative price.

Using the latter equation, the relative factor prices in this economy are obtained as

The first line defines ω (t) as the relative wage of factor H compared to factor L. The second line uses the definition of marginal product combined with (15.16) and (15.17), and the third line, (15.19), uses (15.18). I refer to σ as the (derived) elasticity of substitution between the two factors, since it is exactly equal to

To complete the description of equilibrium in the technology side, we need to impose the following free-entry conditions:

which uses the fact that Nl (t) Vl (t) + Nh (t) Vh (t) is the total value of corporate assets in this economy.

Let us next define the BGP equilibrium as an equilibrium path where consumption grows at the constant rate, g*, and the relative price p (t) is constant. From (15.10) this implies that pl (t) and p∣∕ (t) are also constant.

Let Vl and Vh be the BGP net present discounted values of new innovations in the two sectors. Then, (15.9) implies that

where r* is the BGP interest rate, while pl and p∏ are the BGP prices of the two intermediate goods. The comparison of these two values is of crucial importance. As discussed intuitively 570

(1) The price effect manifests itself because Vh/Vl is increasing in đí/pl. The greater is this relative price, the greater are the incentives to invent technologies complement­ing the H factor. Since goods produced by relatively scarce factors will be relatively more expensive, the price effect tends to favor technologies complementing scarce factors.

(2) The market size effect is a consequence of the fact that Vh/Vl is increasing in H∕L. The market for a technology is the workers (or the other factors) that will be using and working with this technology. Consequently, an increase in the supply of a factor translates into a greater market for the technology complementing that factor. The market size effect encourages innovation for the more abundant factor.

The above discussion is incomplete, however, since prices are endogenous. Combining (15.24) together with (15.18), relative prices can be eliminated and the relative profitability of the technologies becomes

So the two factors will be gross substitutes when the two intermediate goods are gross sub­stitutes in the production of the final good.

Next, using the two free-entry conditions (15.20) and (15.21), and assuming that both of them hold as equalities, we obtain the following BGP technology market clearing condition:

technology. Equation (15.27) contains most of the economics of directed technology. However, before discussing this, it is useful to characterize the BGP growth rate of the economy. This is done in the next proposition:

PROPOSITION 15.1. Consider the directed technological change model described above. Suppose that

Then, there exists a unique BGP equilibrium in which the relative technologies are given by (15.27), and consumption and output grow at the rate

Proof. The derivation of (15.29) is provided by the argument preceding the proposition. Exercise 15.2 asks you to check that condition (15.28) ensures that free-entry conditions (15.20) and (15.21) must hold, to verify that this is the unique relative equilibrium technology, to calculate the BGP equilibrium growth rate and to verify that the transversality condition is satisfied.

It can also be verified that there are simple transitional dynamics in this economy whereby starting with technology levels N∣∣ (0) and Np (0), there always exists a unique equilibrium path and it involves the economy monotonically converging to the BGP equilibrium of Propo­sition 15.1. This is stated in the next proposition:

Proof. See Exercise 15.3. ?

More interesting than the aggregate growth rate and the transitional dynamics behavior of the economy are the results concerning the direction of technological change and its effects on relative factor prices. These are studied in the next subsection.

15.3.2. Directed Technological Change and Factor Prices. Let us start by study­ing (15.27). This equation implies that, in BGP, there is a positive relationship between the relative supply of the H factor, H/L, and the relative factor-augmenting technologies, only when σ > 1. In contrast, if the derived elasticity of substitution, σ, is less than 1, the relationship is reversed. This might suggest that, depending on the elasticity of substitution between factors (or between the intermediate goods), changes in factor supplies may induce technological changes that are biased in favor or against the factor that is becoming more

H/L raises the relative marginal product and the relative wage of the factor H compared to factor L.

Figure 15.3 illustrates the results of Propositions 15.3 and 15.4, referring to H as skilled labor and L as unskilled labor as in the first application discussed in Section 15.1.

Figure 15.3. The relationship between the relative supply of skills and the skill premium in the model of directed technical change.

The curve marked with CT corresponds to the constant-technology relative demand from eq. (15.19). It is always downward-sloping because it holds the relative technologies, Nh/Nl, constant, and thus only features the usual substitution effect. The fact that this curve is downward-sloping follows from basic producer theory. The curve marked as ETi applies when technology is endogenous, but the condition in Proposition 15.4, that σ > 2, is not satisfied. Proposition 15.3 states that even in this case an increase in H/L will induce skill- biased (H-biased) technological change. This implies that when H/L is higher than its initial level, the induced-technology effect will shift the constant-technology demand curve CT to the right (i.e., as technology changes, it will be another CT curve, above the original one, that will apply). When it is below, the same effect will shift CT to the left. Consequently, the locus of points that the endogenous-technology demand, ETi, is shallower than CT. (This can be also verified comparing the relative demand functions with constant and endogenous technology, (15.19) and (15.30), and noting that σ — 2 is never less than —1/u). There is an obvious analogy between this result and Samuelson’s LeChatelier principle, which states that long-run demand curves, which apply when all factors can adjust, must be more elastic than the short-run demand curves which hold some factors constant. We can think of the endogenous-technology demand curve as adjusting the “factors of production” corresponding to technology. However, the analogy is imperfect because the effects here are caused by general equilibrium changes, while the LeChatelier principle and the basic producer theory focus on partial equilibrium effects.

A complementary intuition for this result can be obtained by going back to the importance of non-rivalry of ideas as discussed in Chapter 12. Here, as in the basic endogenous technology models of the last two chapters, the non-rivalry of ideas leads to an aggregate production function that exhibits increasing returns to scale (in all factors including technologies). It is the increasing returns to scale in the production possibilities set of the economy that leads to potentially upward-sloping relative demand curves. Put differently, the market size effect, which results from the non-rivalry of ideas and is at the root of aggregate increasing returns, can create sufficiently strong induced technological change to increase the relative marginal product and the relative price of the factor that has become more abundant.

15.3.3. Implications. The results of Propositions 15.3 and 15.4 are not only of theo­retical interest, but also shed light on a range of important empirical patterns. As already discussed above, one of the most interesting applications is to changes in the skill premium. For this application, suppose that H stands for college-educated workers. In the US labor market, the skill premium has shown no tendency to decline despite a very large increase in the supply of college educated workers. On the contrary, following a brief period of decline during the 1970s in the face of the very large increase in the supply of college-educated work­ers, the skill (college) premium has increased very sharply throughout the 1980s and the 1990s, to reach a level not experienced in the postwar era. Figure 15.1 above showed these general patterns.

In the labor economics and parts of the macroeconomics literature, the most popular explanation for these patterns is skill-biased technological change. For example, computers or new IT technologies are argued to favor skilled workers relative to unskilled workers. But why should the economy adopt and develop more skill-biased technologies throughout the past 30 years, or more generally throughout the entire 20th century? This question becomes more relevant once we remember that during the 19th century many of the technologies that were fueling economic growth, such as the factory system and the major spinning and weaving innovations, were unskill-biased rather than skill-biased.

Thus, in summary, the following stylized facts are relevant:

• Secular skill-biased technological change increasing the demand for skills throughout the 20th century.

• Possible acceleration in skill-biased technological change over the past 25 years.

• A range of important technologies biased against skill workers during the 19th cen­tury.

Propositions 15.3 and 15.4 provide us with a framework for thinking about these issues. Proposition 15.3 implies the increase in the relative supply of skills will always induce skill- biased technological change. In addition, Proposition 15.4 implies that, when σ > 2, the long-run (endogenous-technology) relationship between the relative supply of skills and the skill premium is positive. More specifically:

1. According to Propositions 15.3 and 15.4, the increase in the number of skilled workers that has taken place throughout the 20th century should cause steady skill-biased technolog­ical change. Therefore, models of directed technological change offer a natural explanation for the secular skill-biased technological developments of the past century.

2. The more rapid increase in the number of skilled workers over the past 25 years, shown in Figure 15.1, should also induce an acceleration in skill-biased technological change. How this class of models might account for the dynamics of factor prices in the face of endogenously changing technologies is discussed below.

3. Can the framework also explain the prevalence of skill-replacing/labor-biased techno­logical change in the late 18th and 19th centuries? While we know less about both changes in relative supplies and technological developments during these historical periods, available evidence suggests that there were large increases in the number of unskilled workers available to be employed in the factories during this time periods. Bairoch (1988, p. 245), for example, describes this rapid expansion of the supply of unskilled labor as follows:

“... between 1740 and 1840 the population of England... went up from 6 million to 15.7 million.... while the agricultural labor force represented 60-70% of the total work force in 1740, by 1840 it represented only 22%.”

Habakkuk’s well-known account of 19th-century technological development (pp. 136-137) also emphasizes the increase in the supply of unskilled labor in English cities, and attributes it to five sources. First, “technical changes in agriculture increased the supply of labor available to industry” (p. 137). Second, “population was increasing very rapidly” (p. 136). Third, labor reserves of rural industry came to the cities. Fourth, “there was a large influx of labor from Ireland” (p. 137). Finally, enclosures released substantial labor from agriculture.² According to our model of directed technological change, this increase in the relative supply of unskilled labor should have encouraged unskill-biased technological change, and this is consistent with the patterns discussed above.

In addition to accounting for the recent skill-biased technological developments and for the historical technologies that appear to have been biased towards unskilled workers, this

₂

²Although there is general agreement that the supply of unskilled labor in British cities increased between the 18th and 19th centuries, there is disagreement among economic historians about the importance of various factors. For example, Habakkuk’s emphasis on the importance of enclosures is challenged by Allen (1992, 2004). The exact contribution of the enclosures movement to the increase in the supply of unskilled labor is secondary for the argument here.

Figure 15.4. Dynamics of the skill premium in response to an exogenous increase in the relative supply of skills, with an upward-sloping endogenous- technology relative demand curve.

framework also gives a potential interpretation for the dynamics of the college premium during the 1970s and 1980s. It is reasonable to presume that the equilibrium skill bias of technologies, Nh/Nl, is a sluggish variable determined by the slow buildup and development of new technologies (as the analysis of transitional dynamics in Proposition 15.2 shows). In this case, a rapid increase in the supply of skills would first reduce the skill premium as the economy would be moving along a constant technology (constant Nh/Nl) curve as shown in Figure 15.4. After a while technology would start adjusting, and the economy would move back to the upward sloping relative demand curve, with a relatively sharp increase in the college premium. This approach can therefore explain both the decline in the college premium during the 1970s and its subsequent large surge, and relates both of these phenomena to the large increase in the supply of skilled workers.

If, on the other hand, σ < 2, then the long-run relative demand curve will be downward sloping, though again it will be shallower than the short-run relative demand curve. Following the increase in the relative supply of skills there will again be an initial decline in the college premium, and as technology starts adjusting the skill premium will increase. But it will end up below its initial level. To explain the larger increase in the college premium in the 1980s, in this case we would need some exogenous skill-biased technological change. Figure 15.5 draws this case.

Figure 15.5. Dynamics of the skill premium in response to an increase in the relative supply of skills, with a downward-sloping endogenous-technology relative demand curve.

Consequently, a model of directed technological change can shed light both on the sec­ular skill bias of technology and on the relatively short-run changes in technology-induced factor prices. Other implications of these results will be discussed below. However, before doing this, a couple of further issues are worth noting. First, Proposition 15.4 shows that upward-sloping relative demand curves arise only when σ > 2. In the context of substitution between skilled and unskilled workers, an elasticity of substitution much higher than 2 is un­likely. Most estimates put the elasticity of substitution between 1.4 and 2. One would like to understand whether σ > 2 is a feature of the specific model discussed here and how different assumptions about the technology of production or the innovation possibilities frontier affect this result. This issue will be discussed in Section 15.4. Second, we would like to understand the relationship between the market size effect and the scale effects, in particular, whether the results on induced technological change are an artifact of the scale effect (which many economists do not view as an attractive feature of endogenous technological change models). Section 15.5 shows that this is not the case and exactly the same results apply when scale effects are removed. Third, we would like to apply these ideas to investigate whether there are reasons for technological change to be endogenously labor-augmenting in the neoclassical growth model. This will be investigated in Section 15.6. Finally, it is also useful to con­trast equilibrium allocation to the Pareto optimal allocation. We will start with this latter comparison in the next subsection.

Proof. See Exercise 15.7.

15.4.