Endogenous Labor-Augmenting Technological Change

One of the advantages of the models of directed technological change is that they allow us to investigate why technological change might be purely labor-augmenting as required for balanced growth.

In thinking about labor-augmenting technological change, it is useful to consider a two- factor model with H corresponding to capital, that is, H (t) = K (t), in the aggregate pro­duction function (15.3). Correspondingly, let us use Nl and Nk to denote the varieties of machines in the two sectors. Let us also simplify the discussion by assuming that there is no depreciation of capital. Note also that in this case the price of the second factor, K (t), is the same as the interest rate, r (t), since investing in the capital stock of the economy is a way of transferring consumption from one instant to another.

Let us first note that in the context of capital-labor substitution, the empirical evidence suggests that an elasticity of substitution of σ < 1 is much more plausible (whereas in the case of substitution between skilled and unskilled labor, the evidence suggested that σ > 1). An elasticity less than 1 is not only consistent with the available empirical evidence, but it is also economically plausible. For example, with the CES production function an elasticity of substitution between capital and labor greater than 1 would imply that production is possible without labor or without capital, which appears counterintuitive.

Now, recall that when σ < 1, factor-augmenting and factor-biased technologies are re­versed. Therefore, labor-augmenting technological change corresponds to capital-biased tech­nological change. Then, the question becomes: under what circumstances would the economy generate relatively capital-biased technological change? And also, when will the equilibrium technology be sufficiently capital biased that it corresponds to Harrod-neutral technological change? What distinguishes capital from labor is the fact that it accumulates. In other words, most growth models feature some type of capital-deepening, with K (t) /L increasing as the economy grows. This implies that in contrast to the analysis so far, which looked 585

at the effect of one-time changes in relative supplies, the focus now must be on the impli­cations of continuous changes in the relative supply of capital on technological change. In light of this observation, the answer to the first question above is straightforward: capital deepening, combined with Proposition 15.3, implies that technological change should be more labor-augmenting than capital-augmenting.

The next proposition summarizes the main idea of the previous paragraph. For simplic­ity, this proposition treats the increase in K (t) /L as a sequence of one-time increases (full equilibrium dynamics are investigated in the next two propositions).

Proposition 15.12. In the baseline model of directed technological change with H (t) = K (t) as capital, if K (t) /L is increasing over time and σ < 1, then Ng (t) /Nk (t) will also increase over time, i.e., technological change will be “labor-augmenting”.

Proof. Equation (15.27) or eq. (15.37) together with σ < 1 implies that an increase in K (t) /L will raise Ng (t) /Nk (t). ?

This result already gives us important economic insights. The reasoning of directed technological change indicates that there are natural reasons for technology to be more labor­augmenting than capital-augmenting.

Then, the next proposition shows a negative result on the possibility of purely labor­augmenting technological change.

PROPOSITION 15.13. Consider the baseline model of directed technological change with the knowledge spillovers specification and state dependence. Suppose that δ < 1 and capital accumulates according to (15.46). Then, there exists no BGP.

Proof. See Exercise 15.22. ?

Intuitively, even though technological change is more labor-augmenting than capital­augmenting, there is still capital-augmenting technological change in equilibrium. This, com­bined with capital accumulation, is inconsistent with balanced growth. In fact, even a more negative result can be proved (see again Exercise 15.22): in any asymptotic equilibrium, the interest rate cannot be constant, thus consumption and output growth cannot be constant.

In contrast to these negative results, there is a special case that justifies the basic structure of the neoclassical growth model. This takes place when there is extreme state dependence, that is, δ = 1. This case is, in many ways, quite natural and posits that spillovers are limited to the same class of technologies, so that

ÏíSh (t). In this case, it can be verified that technology market equilibrium implies the following relationship in BGP (see Exercise 15.23):

Thus, directed technological change implies that in the long run, the share of capital is con­stant in national income.

In this case, (15.47) combined with (15.46) implies that

Moreover, it can be verified that the equilibrium interest rate is given by (see Exercise 15.24):

Let us now define a constant growth path (CGP) as one in which consumption grows at a constant rate (though, differently from a BGP, the two sectors and the two technology stocks do not need grow at the same rate). From (15.22), this is only possible if r (t) is constant and equal to some r*. Equation (15.48) then implies that (Nl (t) L) / (Nk (t) K (t)) is constant, thus Nk (t) must also be constant. Therefore, eq. (15.48) implies that for the economy to ultimately converge to a CGP, long-run technological change must be purely labor-augmenting. This is summarized in the following proposition:

PROPOSITION 15.14. Consider the baseline model of directed technological change with the two factors corresponding to labor and capital. Suppose that the innovation possibilities frontier is given by the knowledge spillovers specification with extreme state dependence, that is, δ = 1 and that capital accumulates according to (15.46). Then, there exists a CGP allocation in which there is only labor-augmenting technological change, the interest rate is constant and consumption and output grow at constant rates. Moreover, there cannot be any other CGP allocations.

Proof. Part of the proof is provided by the argument preceding the proposition. Exercise 15.25 asks you to complete the proof and show that no other constant CGP allocation can exist. ?

Notice that Proposition 15.14 does not imply that all technological change must be Harrod neutral (purely labor-augmenting). Along the transition path, there can be (and in fact there will be) capital-augmenting technological change. However, in the long run (that is, asymptotically or as t → ∞), all technological change will be labor-augmenting.

It can also be verified that the CGP allocation with purely labor-augmenting technological change is globally stable if σ < 1 (see Exercise 15.26). This is reasonable, especially in view of the results in Proposition 15.7, which indicated that the stability of equilibrium dynamics in the model with the knowledge spillovers requires σ < 1∕δ. Since here there is extreme state dependence (δ = 1) stability requires σ < 1. Intuitively, if capital and labor were gross substitutes (σ > 1), the equilibrium would involve rapid accumulation of capital and capital-augmenting technological change, leading to an asymptotically increasing growth rate of consumption. However, when capital and labor are gross complements (σ < 1), capital accumulation would increase the price of labor more than proportionately and the profits from labor-augmenting technologies increase more than the profits from capital-augmenting ones. This encourages further labor-augmenting technological change. These strong price effects are responsible for the stability of the CGP allocation in Proposition 15.14. Intuitively, an elasticity of substitution between capital and labor that is less than 1 induces the economy to strive towards a balanced allocation of effective capital and labor units (where “effective” here refers to capital and labor units augmented with their complementary technologies). Since capital accumulates at a constant rate, a balanced allocation implies that the productivity of labor should increase faster, in particular, the economy should converge to an equilibrium path with purely labor-augmenting technological progress.

The results in Proposition 15.14 are important, since they provide a justification for the assumption in the Solow and neoclassical growth models that long-run technological change is purely labor-augmenting. Naturally, whether or not this is the case in practice is an empirical matter and is an interesting topic of future empirical research.

15.7.