Solow Model with Technological Progress

2.6.1. Balanced Growth. The models analyzed so far did not feature technological progress. We now introduce changes in A (t) to capture improvements in the technological know-how of the economy.

By balanced growth, we mean a path of the economy consistent with the Kaldor facts (Kaldor, 1963), that is, a path where, while output per capita increases, the capital-output ratio, the interest rate, and the distribution of income between capital and labor remain roughly constant. Figure 2.11, for example, shows the evolution of the shares of capital and labor in the US national income.

Despite fairly large fluctuations, there is no trend in these factors shares. Moreover, a range of evidence suggests that there is no apparent trend in interest rates over long time horizons and even in different societies (see, for example, Homer and Sylla, 1991). These facts and the relative constancy of capital-output ratios until the 1970s have made many economists prefer models with balanced growth to those without. It is not literally true that the share of capital in output and the capital-output ratio are exactly constant. For example, since the 1970s both the capital share and the capital-output ratio may have increased depending on how one measures them.

Also for future reference, note that the capital share in national income is about 1/3, while the labor share is about 2/3. We are ignoring the share of land here as we did in the analysis so far: land is not a major factor of production. This is clearly not the case for the poor countries, where land is a major factor of production. It is useful to think about how incorporating land into this framework will change the implications of our analysis (see Ex­ercise 2.10). For now, it suffices to note that this pattern of the factor distribution of income, combined with economists’ desire to work with simple models, often makes them choose a Cobb-Douglas aggregate production function of the form AK¹Z³L²Z³ as an approximation to reality (especially since it ensures that factor shares are constant by construction).

Figure 2.11. Capital and Labor Share in the U.S. GDP.

For us, the most important reason to start with balanced growth is that it is much easier to handle than non-balanced growth, since the equations describing the law of motion of the economy can be represented by difference or differential equations with well-defined steady states. Put more succinctly, the main advantage from our point of view is that balanced growth is the same as a steady-state in transformed variables—i.e., we will again have k = 0, but the definition of k will change. This will enable us to use the same tools developed so far to analyze economies with sustained growth. It is nevertheless important to bear in mind that in reality, growth has many non-balanced features. For example, the share of different sectors changes systematically over the growth process, with agriculture shrinking, manufacturing first increasing and then shrinking.

FIGURE 2.12. Hicks-neutral, Solow-neutral and Harrod-neutral shifts in isoquants.

2.6.2. Types of Neutral Technological Progress. What are some convenient special for some constant returns to scale function F. This functional form implies that the tech­nology term A (t) is simply a multiplicative constant in front of another (quasi-) production function F and is referred to as Hicks-neutral after the famous British economist John Hicks. Intuitively, consider the isoquants of the function F [K (t),L (t),A (t)] in the L-K space, which plot combinations of labor and capital for a given technology A (t) such that the level of production is constant. This is shown in Figure 2.12. Hicks-neutral technological progress, in the first panel, corresponds to a relabeling of the isoquants (without any change in their shape).

Another alternative is to have capital-augmenting or Solow-neutral technological progress, in the form

This is also referred to as capital-augmenting progress, because a higher A (t) is equivalent to the economy having more capital. This type of technological progress corresponds to the isoquants shifting with technological progress in a way that they have constant slope at a given labor-output ratio and is shown in the second panel of Figure 2.12.

Finally, we can have labor-augmenting or Harrod-neutral technological progress, named after an early influential growth theorist Roy Harrod, who we encountered above in the context of the Harrod-Domar model previously:

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This functional form implies that an increase in technology A (t) increases output as if the economy had more labor.

Of course, in practice technological change can be a mixture of these, so we could have a vector valued index of technology A (t) = (A∏ (t),Ak (t),Al (t)) and a production function that looks like

(2.40)

which nests the constant elasticity of substitution production function introduced in Example 2.3 above. Nevertheless, even (2.40) is a restriction on the form of technological progress, since changes in technology, A (t), could modify the entire production function.

It turns out that, although all of these forms of technological progress look equally plau­sible ex ante, our desire to focus on balanced growth forces us to one of these types of neutral technological progress. In particular, balanced growth necessitates that all technological progress be labor-augmenting or Harrod-neutral. This is a very surprising result and it is also somewhat troubling, since there is no ex ante compelling reason for why technological progress should take this form. We now state and prove the relevant theorem here and return to the discussion of why long-run technological change might be Harrod-neutral in Chapter 15.

2.6.3. The Steady-State Technological Progress Theorem. A version of the fol­lowing theorem was first proved by the early growth economist Hirofumi Uzawa (1961). For simplicity and without loss of any generality, let us focus on continuous time models. The key elements of balanced growth, as suggested by the discussion above, are the constancy of factor shares and the constancy of the capital-output ratio, K (t) / Y (t). Since there is only labor and capital in this model, by factor shares, we mean

By Assumption 1 and Theorem 2.1, we have that (t) + α∣√ (t) = 1.

The following theorem is a stronger version of a result first stated and proved by Uzawa. Here we will present a proof along the lines of the more recent paper by Schlicht (2006). For this result, let us define an asymptotic path as a path of output, capital, consumption and labor as t → ∞.

Theorem 2.7. (Uzawa) Consider a growth model with a constant returns to scale ag­gregate production function

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with A (t) representing technology at time t and aggregate resource constraint

Suppose that there is a constant growth rate of population, i.e., L (t) = exp (nt) L (0) and that there exists an asymptotic path where output, capital and consumption grow at constant rates, i.e.,Suppose finally that

gκ + δ > 0. Then, along the growth path we have

(1) gγ = gκ = gc; and

(2) asymptotically, the aggregate production function can be represented as:

where

Proof. By hypothesis, as t → ∞, we have Y (t) = exp(gγ (t — τ)) Y (τ), K (t) = exp (gκ (t — τ)) K (τ) and L (t) = exp (n (t — τ)) L (τ) for some τ < ∞. The aggregate re­source constraint at time t implies

Since the left-hand side is positive by hypothesis, we can divide both sides by exp (gκ (t — τ)) and write date t quantities in terms of date τ quantities to obtain

for all t.

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Moreover, this equation is true for t irrespective of the initial τ, thus

establishing the second part of the theorem. ?

A remarkable feature of this result is that it was stated and proved without any reference to equilibrium behavior or market clearing. Also, contrary to Uzawa’s original theorem, it is not stated for a balanced growth path (meaning an equilibrium path with constant factor shares), but only for an asymptotic path with constant rates of output, capital and consump­tion growth. The proposition only exploits the definition of asymptotic paths, the constant returns to scale nature of the aggregate production function and the resource constraint. Consequently, the result is a very powerful one.

Before providing a more economic intuition for this result, let us state an immediate implication of this theorem as a corollary, which will be useful both in the discussions below and for the intuition:

COROLLARY 2.3. Under the assumptions of Theorem 2.7, if an economy has an asymptotic path with constant growth of output, capital and consumption, then asymptotically technolog­ical progress can be represented as Harrod neutral (purely labor-augmenting).

The intuition for Theorem 2.7 and for the corollary is simple. We have assumed that the economy features capital accumulation in the sense that gκ + δ > 0. From the aggregate resource constraint, this is only possible if output and capital grow at the same rate. Either this growth rate is equal to the rate of population growth, n, in which case, there is no technological change (i.e., the proposition applies with g = 0), or the economy exhibits growth of per capita income and capital-labor ratio. The latter case creates an asymmetry between capital and labor, in the sense that capital is accumulating faster than labor. Constancy of growth then requires technological change to make up for this asymmetry—that is, technology to take a labor-augmenting form.

This intuition does not provide a reason for why technology should take this labor­augmenting (Harrod-neutral) form, however. The proposition and its corollary simply state that if technology did not take this form, and asymptotic path with constant growth rates would not be possible. At some level, this is a distressing result, since it implies that balanced growth (in fact something weaker than balanced growth) is only possible under a very strin­gent assumption. It also provides no reason why technological change should take this form. Nevertheless, in Chapter 15, we will see that when technology is endogenous, the intuition in 71

the previous paragraph also works to make technology endogenously more labor-augmenting than capital-augmenting.

Notice also that this proposition does not state that technological change has to be labor-augmenting all the time. Instead, it requires that technological change has to be labor­augmenting asymptotically, i.e., along the balanced growth path. This is exactly the pattern that certain classes of endogenous-technology models will generate.

Finally, it is important to emphasizethat Theorem 2.7 does not require that ically technological change has to be Harrod neutral”. If the aggregate production function is Cobb-Douglas and takes the form

then both Ak (t) and Al (t) could grow asymptotically, while maintaining balanced growth. However, in this Cobb-Douglas example we can defineand the

production function can be represented as

In other words, technological change can be represented as purely labor-augmenting, which is what Theorem 2.7 requires. Intuitively, the differences between labor-augmenting and capital-augmenting (and other forms) of technological progress matter when the elasticity of substitution between capital and labor is not equal to 1. In the Cobb-Douglas case, as we have seen above, this elasticity of substitution is equal to 1, thus different forms of technological progress are simple transforms of each other.

Another important corollary of Theorem 2.7 is obtained when we also assume that factor markets are competitive.

2.7 and that F exhibits constant returns to scale and thus its derivative is homogeneous of degree 0. ?

This corollary, together with Theorem 2.7, implies that any asymptotic path with constant growth rates for output, capital and consumption must be a balanced growth path and can only be generated from an aggregate production function asymptotically featuring Harrod- neutral technological change.

In light of this corollary, we can provide further intuition for Theorem 2.7. Suppose the production function takes the special formThe corollary

implies that factor shares will be constant. Given constant returns to scale, this can only be the case when total capital inputs, Ak (t) K (t), and total labor inputs, Al (t) L (t), grow at the same rate; otherwise, the share of either capital or labor will be increasing over time. The fact that the capital-output ratio is constant in steady state (or the fact that capital accumulates) implies that K (t) must grow at the same rate as Al (t) L (t). Thus balanced growth can only be possible if Ak (t) is asymptotically constant.

2.6.4. The Solow Growth Model with Technological Progress: Continuous Time. Now we are ready to analyze the Solow growth model with technological progress. We will only present the analysis for continuous time. The discrete time case can be analyzed analogously and we omit this to avoid repetition. From Theorem 2.7, we know that the production function must a representation of the form

Y (t) = F [K (t),A (t) L (t)],

with purely labor-augmenting technological progress asymptotically. For simplicity, let us assume that it takes this form throughout. Moreover, suppose that there is technological progress at the rate g, i.e.,

The simplest way of analyzing this economy is to express everything in terms of a nor­malized variable. Since “effective” or efficiency units of labor are given by A (t) L (t), and F exhibits constant returns to scale in its two arguments, we now define k (t) as the effective capital-labor ratio, i.e., capital divided by efficiency units of labor,

(2.43)

Differentiating this expression with respect to time, we obtain

It should be clear that if y (t) is constant, income per capita, y (t), will grow over time, since A (t) is growing. This highlights that in this model, and more generally in models with technological progress, we should not look for “steady states” where income per capita is constant, but for balanced growth paths, where income per capita grows at a constant rate, while some transformed variables such as y (t) or k (t) in (2.44) remain constant. Since these transformed variables remain constant, balanced growth paths can be thought of as steady states of a transformed model. Motivated by this, in models with technological change throughout we will use the terms “steady state” and balanced growth path interchangeably.

Substituting forfrom (2.42) into (2.44), we obtain:

which is very similar to the law of motion of the capital-labor ratio in the continuous time model, (2.32). The only difference is the presence of 9, which reflects the fact that now k is no longer the capital-labor ratio but the effective capital-labor ratio. Precisely because it is the effective capital-labor ratio, k will remain constant in the balanced growth path of this economy.

An equilibrium in this model is defined similarly to before. A steady state or a balanced growth path is, in turn, defined as an equilibrium in which k (t) is constant. Consequently, we have (proof omitted):

PROPOSITION 2.11. Consider the basic Solow growth model in continuous time, with Harrod-neutral technological progress at the rate g and population growth at the rate n. Sup­pose that Assumptions 1 and 2 hold, and define the effective capital-labor ratio as in (2.43). Then there exists a unique steady state (balanced growth path) equilibrium where the effective capital-labor ratio is equal to k* ∈ (0, ∞) and is given by

Per capita output and consumption grow at the rate g.

Equation (2.47), which determines the steady-state level of effective capital-labor ratio, emphasizes that now total savings, sf (k), are used for replenishing the capital stock for three distinct reasons. The first is again the depreciation at the rate δ. The second is population growth at the rate n, which reduces capital per worker. The third is Harrod- neutral technological progress at the rate g. Recall that we now need to keep the effective capital-labor ratio, k, given by (2.43) constant. Even if K/L is constant, k will now decline because of the growth of A. Thus the replenishment of the effective capital-labor ratio requires investments to be equal to (δ + g + n) k, which is the intuitive explanation for equation (2.47).

The comparative static results are also similar to before, with the additional comparative static with respect to the initial level of the labor-augmenting technology, A (0) (since the level of technology at all points in time, A (t), is completely determined by A (0) given the assumption in (2.41)). We therefore have:

Proposition 2.12. Suppose Assumptions 1 and 2 hold and let A (0) be the initial level of technology. Denote the balanced growth path level of effective capital-labor ratio by

for each t.

Proof. See Exercise 2.18.

Finally, we also have very similar transitional dynamics.

PROPOSITION 2.13. Suppose that Assumptions 1 and 2 hold, then the Solow growth model with Harrod-neutral technological progress and population growth in continuous time is as­ymptotically stable, i.e., starting from any k (0) > 0, the effective capital-labor ratio converges to a steady-state value

Proof. See Exercise 2.19. ?

Therefore, the comparative statics and dynamics are very similar to the model without technological progress. The major difference is that now the model generates growth in output per capita, so can be mapped to the data much better. However, the disadvantage is that growth is driven entirely exogenously. The growth rate of the economy is exactly the same as the exogenous growth rate of the technology stock. The model specifies neither where this technology stock comes from nor how fast it grows.

2.7.