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In this paper, we study the interaction of factor-augmenting and factor-eliminating technical progress and their effects on economic growth. Factor-augmenting progress is an old and familiar concept. It appears in economic models as a growing variable that multiplies (and thus "augments") a factor of production. In the CES production function

(1) [Y.sub.t] = A [[[a.sub.t] [([Q.sub.t][K.sub.t]).sup.[gamma]] + (1 - [a.sub.t])[([Z.sub.t][L.sub.t]).sup.[gamma]]].sup.1/[gamma]]

where Y is output; K is physical capital; L is labor; A, Q, and Z are augmentation terms; and a and [gamma] are parameters. Technical progress that increases Q or Z augments their associated factors of production K and L. Increases in Q and Z have the same effects as increases in K and L. Factor-eliminating technical progress is a new concept introduced by Seater (2005), Zuleta (2008b), and Peretto and Seater (2013). Instead of augmenting a factor of production, factor-eliminating progress reduces a factor's influence and perhaps eliminates the factor altogether. In (1), technical progress that raises a reduces labor's influence on the production function and tends to eliminate it as a [right arrow] 1. Factor elimination treats a as an endogenous quantity that can be changed by R&D. Models that allow factor elimination deliver implications that do not arise from factor augmentation and that receive empirical support. To date, factor augmentation and elimination have been considered in isolation. Here, we study a model that includes both.

Factor augmentation and factor elimination each can deliver perpetual economic growth, but the mechanisms are diametrically different. As is well known, if a nonreproducible factor (economic activity cannot create more of it) is essential to production (output is zero if that factor is not used), perpetual economic growth is impossible unless the economy can overcome either that factor's nonreproducibility or its essentiality. Augmentation delivers perpetual growth by effectively making the limiting factor reproducible. For example, suppose that [gamma] [less than or equal to] 0 in (1) so that both K and L are essential, unskilled labor is nonreproducible, and Q, Z, and a are constant. For simplicity, suppose that labor also is constant. Then perpetual growth is impossible. That result is easily seen with the Cobb-Douglas special case of (1) with [gamma] = 0. The marginal return to capital is

[MPK.sub.t] = aA[Q.sup.a] [(ZL).sup.1-a]/[K.sup.1-a.sub.t]

which goes to zero as K grows. Growth stops when MPK falls to the value [MPK.sup.*] equal to the rate of time preference plus capital's depreciation rate. To get perpetual growth, we must either augment labor or eliminate it. Augmentation works through increases in Z. If Z is not constant but instead can be increased by investment, then the term ZL can grow. That raises MPK, which elicits growth in K to maintain MPK = [MPK.sup.*]. The increase in K raises the marginal product of Z, eliciting further increases in Z. Increases in Z and K thus are mutually reinforcing and can continue indefinitely, giving perpetual growth in Y. Without the growth in Z, the nonreproducible factor L acts as a drag on the economy. Growth in Z offsets the drag by effectively making L reproducible. Factor elimination offers a completely different route to perpetual growth. Instead of augmenting a nonreproducible factor, the economy learns to eliminate it. In (1) factor-eliminating technical change appears as an increase in a. Suppose again that Z is constant, but suppose also that a can be increased through expenditure on R&D to learn new production technology. Now it is increases in K and a that are mutually reinforcing. An increase in K raises the marginal value of increasing a, and an increase in a raises the marginal value of increasing K. As a [right arrow] 1, (1) goes to an AK form, in which the nonreproducible factor has been eliminated and so is no longer a drag on MPK or economic growth. Examples of factor elimination include substituting plastic for metal, solar power for fossil fuels, and robots for workers. We might sum up the spirit of the two types of technical change by saying that factor augmentation celebrates the nonreproducible factors of production whereas factor elimination shuns them.

As we can see from Equation (1), it is possible to write a production function that allows both factor augmentation and factor elimination. Nonetheless, the existing literature allows only one or the other. In the augmentation approach to economic growth, some factors of production are restricted to enter only as multipliers (augmenters) of other factors and the parameters of the production function are restricted to be constant:

[Y.sub.t] = f ([Q.sub.t][K.sub.t], [Z.sub.t][L.sub.t]; [OMEGA])

where [OMEGA] is the set of parameters. In the factor elimination approach, reproducible and nonreproducible factors are collected in separate aggregates that enter the production function independently and the production function parameters can change over time:

[Y.sub.t] = f [[R.sub.t], [N.sub.t]; [[OMEGA].sub.t]]

where R = r(K, Q,Z, ...) and N = n(L, ...) are, respectively, aggregates of the reproducible and nonreproducible factors. Because R and N enter separately, this type of model has a structure in which no reproducible factor augments any nonreproducible factor. In other words, the model is restricted to not have augmentation. Economic growth is driven instead by gradual elimination of the nonreproducible factors together with accumulation of the reproducible factors.

When augmentation and elimination are separated into such starkly different competing models, they have corresponding starkly different implications for aspects of the economy's growth path. Most notable is the behavior of factor income shares. Factor augmentation implies that some reproducible factors' shares move in opposite directions from each other because some reproducible factors are bound up by construction with the nonreproducible factors as augmenters of them and others enter independently of the nonreproducibles. In contrast, factor elimination implies that the income shares of all reproducible factors move in the same direction. Similarly, factor augmentation implies that some reproducible factor shares move in the same direction as some nonreproducible factor shares, whereas factor elimination implies that reproducible and nonreproducible factor shares move in opposite directions. Augmentation has no necessary implication for the relation between factor shares on one hand and the level of income on the other, but it does imply that if some reproducible factor shares are positively related to the income level, then other reproducible factor shares will be negatively related to income. Factor elimination implies that reproducible and nonreproducible factor shares are positively and negatively related to income, respectively. These differences flow from the different ways the two approaches group the factors and the different patterns of mutual reinforcement built into the models, explained above.

There is much empirical evidence supporting the predictions of factor elimination over those of factor augmentation. Bound and Johnson (1995) and Krueger (1999) show that unskilled labor's income share in the United States has fallen substantially over time. Zuleta (2008a) shows that the same result holds for a broad array of countries and also shows that the income share of land (a nonreproducible form of capital) has fallen and the income share of physical capital (which is reproducible) has risen for the same countries. Peretto and Seater (2013) provide evidence that the income share of total nonreproducible capital (land plus natural resources) has fallen. Sturgill (2012) and Peretto and Seater (2013) show that the predicted correlations hold for a large sample of countries. Sturgill (2014) shows that crosscountry differences in factor shares explain a significant part of the cross-country difference in income per person. Dawson and Sturgill (2016) show the related result that countries with economic institutions favorable toward investment have higher factor shares for reproducible factors. Sturgill and Zuleta (2017b) show that changes in factor shares explain a significant portion of economic growth for several countries. Factor elimination is a significant phenomenon, both statistically and economically, and is distinctly different from factor augmentation.

What would happen if we were to construct a model that allows both augmentation and elimination? The literature so far has not addressed that possibility, yet even casual consideration suggests likely interesting interactions. For example, in (1) above we see immediately that the ability to accumulate Z leads to a tension in which increases in K and a feed on each other as in Peretto and Seater but increases in Z interact positively with increases in 1 - a and thus negatively with increases in a. Labor augmentation and factor elimination tend to oppose each other, so including both in a model may lead to different conclusions about the economy's growth path than one gets from models restricted to only one type of progress. We develop such a model here. The model is a generalization of the standard factor-augmenting model, obtained by starting with the usual augmentation formulation, keeping the restriction that some factors enter as augmenters of others, but dispensing with the restriction that factor elimination does not occur. Our model is thus a generalization of the usual augmentation model and nests it as a special case.

The main result is quite surprising: when we relax the restriction that excludes factor elimination, the model delivers strongly counterfactual behavior. That is surprising because our model is less restrictive than the standard augmentation model. In contrast, the Peretto-Seater version of factor elimination, which does not impose the augmentation restriction, is consistent with the facts. It thus appears that factor augmentation is an invalid restriction on the form of the production function and that economic growth is not driven by factor augmentation but rather by elimination of the nonreproducible factors and accumulation of the reproducible factors.


We begin with definitions of several central concepts and with some results in the existing literature.

DEFINITION 1. (Reproducibility) A factor of production is reproducible if it is the output of some economic process. Otherwise it is nonreproducible.

The classic factors of production are physical capital, land, and labor (i.e., workers). Physical capital is unambiguously reproducible, arising from investment by economic agents. Land and labor are not so obvious. Dredging can add to the stock of land, and the growth rate of the population is at least partly an economic choice. However, additions to land are minuscule compared to the total stock of land, so we can treat land as nonreproducible. Labor is more problematic. On one hand, it is true that family size is influenced by people's choices, and choices take into account economic incentives. On the other hand, families do not generally come into existence for the purpose of creating children in order to take advantage of an economic opportunity. In the subsequent analysis, we treat population as exogenous and therefore nonreproducible in the sense of Definition 1. We also expand the set of reproducible factors to include the qualities of capital and labor. We thus have the general production function

(2) [Y.sub.t] = f ([K.sub.t], [Z.sub.t], [Q.sub.t], [L.sub.t]; [[OMEGA].sub.t])

where [K.sub.t] is physical capital as before, [L.sub.t] is unskilled labor, [Q.sub.t] is physical capital's quality, [Z.sub.t] is labor's quality, and [[OMEGA].sub.t] is the vector of function's parameters. The most frequent interpretation of Z is human capital, but it also may be technical progress embodied in physical capital that augments labor.

The growth literature virtually always imposes two important restrictions on the production function: (1) constant parameters and (2) factor augmentation. The first restriction means that all elements of [OMEGA] are constant. The second restriction means that some factors do not enter as independent factors but instead multiply other factors.

DEFINITION 2. (Factor Augmentation) A factor of production A augments another factor B if A multiplies B, e.g., Y =f(A, B, *) = g(AB, *).

In our case, machine and labor quality usually are assumed to augment physical capital and unskilled labor:

(3) [Y.sub.t] = f ([Q.sub.t][K.sub.t], [Z.sub.t][L.sub.t]; [OMEGA]).

The factors Q and K enter (3) in a completely symmetric way, and similarly for Z and L. Mathematically one can just as well say that K and L augment Q and Z as the other way around. There is a slight economic difference in that Q and K are both reproducible and so are symmetric in every sense, whereas Z is reproducible but L is not. We follow convention and say that Q and Z augment K and L, respectively.

To explain the relation between augmentation and economic growth, we need the concept of essentiality in production.

DEFINITION 3. (Essentiality) A factor of production X is essential for the Y if Y = f(X, *) = 0 whenever X=0.

We have an important result originally due to Phelps (1965).

THEOREM 1. (Augmentation and Economic Growth) If technical progress is restricted to factor augmentation, perpetual economic growth requires augmentation of all essential nonreproducible factors of production.

The economic intuition behind this theorem is straightforward. Suppose that Q and Z are constant in (3) and that K and L are essential. As K accumulates, the nonreproducible factor L drags down the marginal product of K, which eventually falls to a point where further investment in K is unprofitable. Capital accumulation stops, and economic growth peters out. Labor-augmenting progress, represented by the growth of Z, prevents that outcome. Growth in Z raises the marginal product of K, and K can grow forever without its marginal product falling.

There are two points to note here. First, when the nonreproducible factor is nonessential, perpetual growth can occur without labor-augmenting progress. Indeed, growth can occur with no technical progress at all. For example, if [gamma] > 0 in (1), then neither K nor L alone is essential. Either can be zero and output still is positive as long as the other factor is positive. If y is sufficiently large, capital accumulation is self-sustaining and delivers perpetual growth of output even if Q and Z are constant (Barro and Sala-i-Martin 2004). We ignore models with [gamma] > 0 because production functions that allow positive output with no labor input are unrealistic. Furthermore, growth by simply building more of the same type of capital also is unrealistic. Humanity did not reach its current state of economic development by making ever more stone-age tools but rather by building ever better tools and learning to use them effectively. We therefore limit attention to models in which growth is driven by technical progress. Second, augmentation of K through increases in capital quality Q does not deliver perpetual growth. It drives the marginal product of augmented capital QK down to the point where further investment in either Q or K is unprofitable just as increases in K do. In light of that result, capital quality adds nothing important to what follows, so we dispense with it to simplify the presentation. Equation (3) reduces to

(4) [Y.sub.t] = f ([K.sub.t], [Z.sub.t][L.sub.t]; [OMEGA]).

Factor elimination imposes different restrictions on the production function. It offers an alternative route to perpetual economic growth.

DEFINITION 4. (Factor-eliminating Technical Progress) Technical progress that changes the production function's parameters such that the marginal product of a factor goes to zero for all values of that factor is said to eliminate that factor.

For example, in the generic production function (2), technical progress eliminates L if technical progress drives the parameter vector [[OMEGA].sub.t] to some critical set [bar.[OMEGA]] for which FL = 0 for all values of L, that is, [F.sub.L] [right arrow] 0 as [[OMEGA].sub.t] [right arrow] [bar.[OMEGA]]. More specifically, in the CES production function (1), [partial derivative]Y/[partial derivative]L [right arrow] 0 as the parameter vector (A, [alpha], [gamma]) [right arrow] (A, 1, [gamma]).

Factor elimination is not the same as inessentiality. Factor elimination renders a factor inessential, of course, but inessentiality does not guarantee factor elimination. Inessentiality thus is necessary but not sufficient for factor elimination. For example, in (1) the factor L is eliminated and thus inessential if [alpha] = 1 irrespective of the value of [gamma], but L has a positive marginal product and thus is not eliminated if [gamma] > 0 (guaranteeing inessentiality) but [alpha] < 1.

Peretto and Seater (2013) study factor elimination and its effects on economic growth in a model where factor elimination is the only type of technical progress. Their production function has the form

(5) [Y.sub.t] = f [[R.sub.t], [N.sub.t]; [[OMEGA].sub.t]]

where R is a vector of all reproducible factors including physical capital, human capital, and factor quality levels, and A? is a vector of all nonreproducible factors including unskilled labor, land, and natural resources. Firms' R&D changes the parameter vector [OMEGA],. A special case restricts R and N to aggregates of all types of reproducible and nonreproducible factors, respectively, rather than having each factor enter the function separately:

[Y.sub.t] = f [r([K.sub.t][Z.sub.t]), n([L.sub.t][A.sub.t][M.sub.t])]

where r and n are the aggregator functions for reproducible and nonreproducible factors, A (for "acreage") is land, and M (for "minerals") is "natural capital" (natural resources). Peretto and Seater show that optimal R&D moves [[OMEGA].sub.t] to the critical set [bar.[OMEGA]] at least asymptotically and possibly in finite time. (1) Restricting attention to a Cobb-Douglas production function, they obtain the following important result:

THEOREM 2. (Factor Elimination and Economic Growth) Factor elimination can generate perpetual economic growth.

The Peretto-Seater model yields testable implications for the behavior of factor shares that do not emerge from factor augmentation models. One of the main implications of the Peretto-Seater model is that all reproducible factors' income shares should rise over time at the expense of the nonreproducible factors' shares. As explained in the Introduction, models of factor augmentation by construction cannot deliver that implication. Instead they imply that income shares either do not change or they rise for some reproducible factors and fall for others. From (4), the factor shares for K and ZL are Kf, If and ZL[f.sub.2]/f. An increase in one share (say that for K) necessarily reduces the other share (that for Z). The evidence, however, is that all reproducible factor shares, that is, those for both K and Z, have risen over time. A related difference between factor augmentation and elimination is that augmentation implies that some reproducible factor shares move in the same direction as some nonreproducible factor shares (e.g., those for Z and L), whereas factor elimination implies that reproducible and nonreproducible factor shares move in opposite directions (those for R vs. those for N). Augmentation and elimination also have different implications for the relation between factor shares and the level of income. Under augmentation, there is no necessary relation between factor shares and the level of income, but augmentation does imply that if some reproducible factor shares are positively related to the income level, then other reproducible factor shares will be negatively related to income because of the way augmentation separates the factors in the production function. In contrast, factor elimination implies that reproducible and nonreproducible factor shares are positively and negatively related to income, respectively.

We thus have two models of perpetual growth through technical progress. Each imposes restrictions on the general production function (2). To the best of our knowledge, all growth models based on factor augmentation impose two restrictions: Z enters only by augmenting labor and the parameter set [OMEGA] so that factor elimination is excluded. The factor elimination models considered in the literature to date have only one general restriction, which is that no factor of production augments any other. There is no model in the literature that allows both kinds of technical progress. We present such a model here by generalizing the standard factor augmentation model to allow factor elimination. We follow the most obvious path and impose the functional form of Equation (4), thus maintaining the restriction that Z enters only as an augmenter of labor, but we relax the restriction that the parameter set [[OMEGA].sub.t] is constant, allowing it to change endogenously through R&D. Our model thus nests the standard augmentation approach. It does not nest the Peretto-Seater version of factor elimination because it changes Z from an independent factor of production to one that augments labor. The relations among the three approaches are important in interpreting our results.


Our model comprises three groups of economic agents: a representative household, competitive producers of final goods, and monopolistic producers of intermediate goods. The model admits both factor-augmenting and factor-eliminating progress, but otherwise its structure is generally the same as in Peretto and Seater (2013).

A. Households

A representative household has a fixed endowment L of a nonreproducible factor that it supplies inelastically to the firms. In general the nonreproducible factor can be unskilled labor or a natural resource, but we refer to it as labor for ease of discussion. The household owns the firms. The household saves a fixed fraction .v of its income, eliminating consumption-saving choice from the model. (2)

B. Final Good Producers

The final good Y is produced by competitive firms using the technology

(6) [Y.sub.t] = [[[[integral].sup.1.sub.0] [X.sup.[epsilon]] di].sup.[epsilon]/[epsilon]-1], [epsilon] > 1

where the [X.sub.i] are intermediate goods described below and [epsilon] is the elasticity of substitution among them. The mass of varieties of intermediate good X is fixed at 1. Final good producers maximize their profit

[[pi].sub.Y] = [Y.sub.t] - [[integral].sup.1.sub.0] [][]di Jo

subject to (6) where [P.sub.Yt] and [] are the price of [Y.sub.t] and [], respectively. We take the final good Y as the numeraire and normalize [P.sub.Yt] to 1. The resulting demand for [] is given by the well-known formula

(7) [] = [Y.sub.t][[P.sup.-[epsilon]]/[P.sup.1-[epsilon].sub.Xt]


[P.sub.Xt] = [([[integral].sup.1.sub.0] [P.sup.1-[epsilon]]di).sup.1/1-[epsilon]]

is the Dixit-Stiglitz price index for Y, a composite of the prices for the intermediate goods []. The firm's expenditure can be written

[[integral].sup.1.sub.0] [][]di = [P.sub.Xt][Y.sub.t]. Jo

Because final producers are competitive, their profit is zero and their revenue equals their expenditure, leading to

[P.sub.Xt] = [P.sub.Yt] = 1,

allowing us to drop [P.sub.Xt] from the remainder of the analysis. In particular, the demand for [] simplifies to

[] = [Y.sub.t][P.sup.-[epsilon]].

C. Intermediate Good Producers

Intermediate goods firms are the source of all the important action in the model. We start with a standard model of labor augmentation, but we generalize it by relaxing the universal implicit restriction that there is no factor-eliminating technical change. Thus, our model is a generalization of the standard model of perpetual growth.

Each intermediate firm i has a monopoly in the production of good X,. It invests in the accumulation of physical capital, R&D to augment its workers, and R&D to develop new production technologies. The intermediate firm's production process combines the outcomes of those activities:

[] = f ([], [][]; [[OMEGA]])

where as before [K.sub.i] is physical capital, [L.sub.i] is unskilled labor, and [Z.sub.i] is a state variable that augments labor. We want to allow both factor augmentation and factor elimination. To that end, we keep K and Z separate, in contrast to Peretto and Seater (2013), who use an aggregate of all reproducible factors in the theoretical part of their article. We also allow the parameter set [OMEGA] to vary over time in response to the firm's R&D. Doing that relaxes the usual restriction in the standard growth model that there is no factor elimination. The standard factor-augmenting model is thus nested in our model. Our model is not a special case of Peretto and Seater (2013), nor vice versa. We restrict Z to be an augmenter of L, whereas Peretto and Seater restrict it to be an independent factor similar to K. Our model of factor elimination is thus an alternative to Peretto and Seater's version.

To make further progress, we must choose a specific form for the production function. All previous work on factor elimination has used the Cobb-Douglas function. Our purpose here is to study the interaction of factor elimination and factor augmentation, not to study the implications of factor elimination with other production functions. We therefore follow the literature and use a Cobb-Douglas production function for intermediate goods:

(8) [mathematical expression not reproducible].

With this function, the parameter set [[OMEGA]] = {A, []}.

Physical capital is accumulated through investment:

(9) [] = [I.sub.Kit] - [[delta].sub.K][]

where [I.sub.K] is gross investment in K and [[delta].sub.K] is the depreciation rate of physical capital. As in Peretto and Seater (2013) the firm has available a set [0, [[alpha].sub.i]] from which it chooses the capital elasticity [a.sub.i] and the firm engages in R&D to increase the upper bound a, of the choice set. The equation for the evolution of a is

(10) [mathematical expression not reproducible]

where [I.sub.[alpha]i] is R&D expenditure on technical progress in units of final good, g([[alpha].sub.i]) is the productivity of R&D, and improvements in [[alpha].sub.i] are forms of knowledge that are fully excludable from other firms. The function g([alpha]) is defined on the closed interval [0,1], and some of its properties are g([alpha]) [greater than or equal to] 0 for all [alpha], g(0) > 0, g'([alpha]) < 0 for [alpha] [member of] [0, 1), and g'(1) [less than or equal to] 0. We discuss the function g(<x) in more detail below. (3)

It is simplest to treat Z as determined by the firm rather than by the household. That makes the firm responsible for all critical decisions in the model. Firms, of course, are owned by the households and can be regarded as the households' agents, so we could just as well have the household determine the path of Z directly, but the results would be identical to those here. We can think of Z as training, which is a form of human capital embodied in the worker, or labor-augmenting technical progress embodied in physical capital. The accumulation equation for Z is

(11) [] = b[I.sub.Zit]

where [I.sub.Zit] is investment in Z and b > 0 is the productivity of investment in Z. We have assumed that Z does not depreciate, being a form of knowledge. Reasonable arguments can be made for assuming otherwise, but the analysis is simpler when Z does not depreciate. Nothing important changes if Z does depreciate. We discuss the case where Z depreciates briefly after we present the main analysis.

The intermediate firm's maximization problem is

(12) [mathematical expression not reproducible]

subject to (7), (8), (9), (10), and (11), and where [[bar.r].sub.t] = [1/t] [[integral].sup.t.sub.0] [r.sub.s]ds is the average interest rate between time 0 and t. It is convenient to separate the firm into two divisions, production and investment, and rewrite (12) as

(13) [mathematical expression not reproducible]

where the first and second integrals are the present values of production and investment division profits, respectively. The investment division "rents" K and Z to the production division and receives implicit payments to increases in a through induced changes in [p.sub.Kit] and [p.sub.Zit] because increases in a allow more efficient allocation of the firm's resources and therefore increase output. Separating the two divisions of the firm is convenient because it separates the firm's intratemporal decisions (production) from its intertemporal decisions (investment).

The term inside the first integral in (13) is the dividend [] paid to the household:

(14) [] [equivalent to] [][] - [p.sub.Kit][] - [w.sub.t][] - [p.sub.Zit]Zit.

The first half of the integral in (13) thus is the value [] of the firm, that is, the present discounted value of the firm's dividend payments:

[mathematical expression not reproducible].

Differentiating [] with respect to time gives

[[??]] = [] + [r.sub.t][]

which can be written as the familiar

[[??]] = [[[??]]/[r.sub.t]] + [[]/[r.sub.t]].

For simplicity, we assume that entry costs are zero. At every moment, firms enter or leave to make V equal to entry costs, which means that [] = 0 at all times. That in turn implies that [] = 0 at all times, too, which finally implies that [] = 0 at all times, so the firm pays no dividends but instead retains all its earnings for its various types of investment. The household's benefit from owning the firm is increasing consumption.

The firm's optimization problem can be solved in two steps. The production division chooses P, L, K, and Z, taking [p.sub.K], [p.sub.Z], w, and a as given. Then the investment division chooses [I.sub.K], [I.sub.Z], and [I.sub.[alpha]]. We consider those steps in turn.

Production Division. The production division solves

[mathematical expression not reproducible]

subject to (7) and (8). It has available the range of technologies [0, [[alpha]]]. As in Peretto and Seater (2013), it uses only the two extremes:

THEOREM 3. Choice of Technology: A firm that has a positive capital stock and has available Cobb-Douglas technologies with constant returns to scale and capital elasticities in the range a [member of] [0, [alpha]] uses the lowest and the highest capital elasticities.

Proof. See the Appendix. a

As we show below, the firm may use only one of the two technologies, or it may use both by running two plants, one with the primitive technology and the other with the advanced technology. Formally, using the two technologies rather than just one convexities the firm's decision set by allowing the firm to produce with a weighted combination of the two extremes. (4) More intuitively, the firm can create two capital/labor ratios, one larger and one smaller than 1. The larger ratio is used in the advanced plant, where the relatively high capital elasticity "rewards" a high capital/labor ratio, and conversely for the primitive plant. (5) The firm uses only one technology in certain corner cases, explained below.

With the technologies limited to the extremes, the production function for X becomes

(15) X = A [[Z(L - [L.sub.[alpha]]) + [K.sup.[alpha]] (Z[L.sub.[alpha]]).sup.1-[alpha]]]

where [L.sub.[alpha]] [less than or equal to] L denotes labor employed at the plant with advanced technology [alpha]. The production division's profit is

[mathematical expression not reproducible]

by using (7) and (15). The first-order conditions for profit maximization, shown in the Appendix, give the solutions for [L.sub.[alpha]], w, [p.sub.K], and [p.sub.Z]. The solution for [L.sub.[alpha]] is

(16) [L.sub.[alpha]] = [K/Z] [(1 - [alpha]).sup.1/[alpha]]

which is unattainable whenever [L.sub.[alpha]] > L, that is, whenever the firm wants to assign to the advanced plant more workers than the total number it has hired. We thus have unconstrained and constrained alternatives for [L.sub.[alpha]]:

(17) [mathematical expression not reproducible].

In the unconstrained case, the firm runs two plants, each using one of the extreme technologies. In the constrained case, the firm runs only the advanced plant.

The two solutions for [L.sub.[alpha]] induce similar unconstrained and constrained solutions in the firm's other demand functions. The Appendix shows that the inverse demands for L, K, and Z are

(18) [mathematical expression not reproducible],

(19) [mathematical expression not reproducible]

(20) [mathematical expression not reproducible].

We can rewrite the production function in a way that gives a different perspective on the solution to the production division's problem and that also simplifies the subsequent analysis. Define

k [equivalent to] K/Z[L.sub.[alpha]]

and write the unconstrained part of (17) as

k = [(1/1 - [alpha]).sup.1/[alpha]] > 1.

The production division's allocation of labor to the advanced plant is independent of factor prices and makes the capital/labor ratio in the advanced plant greater than 1 and increasing in the capital elasticity a. We can write the production function as

X = A [L + K ([k.sup.[alpha]] - 1)/k].

For any chosen values of L and K, the production division maximizes X by making an appropriate allocation of labor across the two plants, that is, by choosing k as arg [max.sub.k][([k.sup.[alpha]] - l)/k] = [(1 - [alpha]).sup.-1/[alpha]]. As long as k < K/(ZL), we have an unconstrained solution with both types of plant operating. Otherwise only the advanced plant runs. Note that increases in k and a feed on each other. A higher value of a increases the desired value of k (desired ratio of physical capital to augmented labor). Conversely, a higher value of k increases the marginal product of [alpha]. Increasing one raises the value of the other, leading to a self-reinforcing interaction between K and [alpha] and a "self-defeating" interaction between Z and [alpha].

For the maximizing value of k, we have

(21) [mathematical expression not reproducible].

Peretto and Seater (2013) show that m' > 0, m" > 0, m(0) = 0, m(1) = 1, m'(0) = [e.sup.-1], and m'(1) = [infinity].

Investment Division. The investment division's optimization problem is

[mathematical expression not reproducible]

subject to (9), (II), (10), [K.sub.0], [Z.sub.0], and [[alpha].sub.0]. The current-value Hamiltonian is

[mathematical expression not reproducible]

where [PSI], [PHI], and [mu] are the costate variables and the Lagrange multipliers [[omega].sub.K], [[omega].sub.[alpha]], and [[omega].sub.Z] satisfy the Kuhn-Tucker conditions

[mathematical expression not reproducible].

The Hamiltonian is linear in the control variables, so we have a bang-bang control problem with investment being only one type at any moment, except for some knife-edge cases explained below. The relative rates of return to the three types of investment determine which type is active. The first-order conditions (see the Appendix) give the following solutions for the rates of return:

(22) [mathematical expression not reproducible]

(23) [mathematical expression not reproducible]

(24) [mathematical expression not reproducible].

We derive the time paths of each kind of investment below.

The rate of return to R&D [r.sub.[alpha]] is the same as in Peretto and Seater except for the presence of the term

[2 - [alpha] - [m ([alpha])/[alpha]m' ([alpha])]

which arises from the explicit payment to Z. The Appendix shows:

[mathematical expression not reproducible]

so m([alpha])/[alpha]m' ([alpha]) declines monotonically as [alpha] goes from zero to one.


We now discuss the general equilibrium dynamics of the economy. Our model is one of monopolistic competition among firms with identical production functions. That model yields a symmetric equilibrium.

In symmetric equilibrium Y = X, so the expressions for wage, prices, and the rates of return simplify:

(25) [mathematical expression not reproducible]

(26) [mathematical expression not reproducible]

(27) [mathematical expression not reproducible]

(28) [mathematical expression not reproducible]

(29) [mathematical expression not reproducible]

(30) [mathematical expression not reproducible].

In the unconstrained case, the return [r.sub.Z] to investment in Z is a constant. We assume the parameter values involved are such that [r.sub.Z] > 0 so that investment in Z may be desirable.

A stable symmetric equilibrium exists only if no firm can take over its whole market. That requires that the firm's return to innovation [r.sub.[alpha]] decreases in [alpha]. The investment division in our model faces strong increasing returns due to the production division's ability to allocate resources between two plants/technologies. To control that destabilizing force and ensure that [r.sub.[alpha]] decreases in [alpha], we must impose some restrictions on the function g([alpha]) that we explain briefly here and in more detail below. (6) The critical term in the expression (24) for [r.sub.[alpha]] is g([alpha])m'(a)[2 - [alpha] - m([alpha])/[alpha]m'([alpha])], which is the product of the two functions g([alpha]) and

[mathematical expression not reproducible]

where the second line follows from the definitions of m([alpha]) and m'([alpha]) and some straightforward algebra. The function n(a) is positive, convex, and increasing for all (a) and goes to infinity as [alpha] goes to 1 (see Figure 1). For g([alpha])n([alpha]) to be decreasing for all a, we must have g'([alpha]) < 0 and g([alpha]) go to zero faster than n(a) goes to infinity as a goes to 1, so we impose those conditions. The discussion from here is cumbersome unless we work with a specific functional form for g([alpha]). We impose the convenient form g([alpha]) = [(1 - [alpha]).sup.2] . The main conclusions are the same for a broad class of functional forms, as we explain later when discussing the g-function and its foundations in more detail. The plot of the product g([alpha])n([alpha]) when g([alpha]) = [(1 - [alpha]).sup.2] is shown in Figure 2. The function is indeed decreasing in a as required for symmetric equilibrium.

A. Equilibrium Loci

We explain the economy's dynamic behavior with phase diagrams. To construct those, we first derive several equilibrium loci: (1) three arbitrage loci pertaining to rates of return, (2) the labor constraint boundary dividing the phase space into the regions in which the labor constraint does or does not bind, and (3) a stationarity locus showing where net investment is zero. We then assemble the equilibrium loci into the phase diagrams. As in Peretto and Seater (2013), the constrained case is somewhat more tedious and gives results whose main features are the same as for the unconstrained case, so henceforth we restrict attention to the unconstrained case, that is, when firms have more labor than they want to allocate to the advanced plant.

Arbitrage Locus [r.sub.K] = [r.sub.Z]. Using (28) and (29), solving [r.sub.K] = [r.sub.Z] yields

(31) m([alpha]) = bL + [epsilon][[delta].sub.K]/([epsilon] - 1)A.

The right side is constant, so there is a unique value [??] of [alpha], independent of both K and Z, at which [r.sub.K] = [r.sub.Z].

[mathematical expression not reproducible].

Arbitrage Locus [r.sub.[alpha]] = [r.sub.Z]. Using (29) and (30), solving [r.sub.[alpha]] = [r.sub.Z] yields

(32) K = (1 - [1/[epsilon])AbL/(1 - [1/[epsilon]]) Ag([alpha])n([alpha])

which is independent of Z. Substituting for g([alpha]) and using the definition and properties of m([alpha]) given in (21) and immediately following it, we can write

(33) K = (1 - [1/[epsilon])AbL/(1 - [1/[epsilon]]) A[(1 - [alpha]).sup.2]n([alpha])

which is convex and upward sloping for [alpha][epsilon][0,1]. We have

[mathematical expression not reproducible]

and from the properties of m'

[mathematical expression not reproducible]

which imply that the vertical intercept of the [r.sub.K] = [r.sub.Z] locus is

[mathematical expression not reproducible].

Arbitrage Locus [r.sub.K] = [r.sub.[alpha]]. Using (28) and (30), solving [r.sub.K] = [r.sub.[alpha]] yields

(34) K = (1 - [1/[epsilon]]) Am([alpha] - [[delta].sub.K]/(1 - [1/[epsilon]) Ag([alpha])n([alpha])

which is independent of Z. Substituting for g([alpha]) and using the definition and properties of m([alpha]), we have

K = (1 - 1/[epsilon])Am([alpha]) - [[delta].sub.K]/(1 - 1/[epsilon])A[(1 - [alpha]).sup.2]n([alpha])

which is convex and upward sloping for [alpha][epsilon][0, 1]. The denominator at a = 0 is the same as for the [r.sub.Z] = [r.sub.[alpha]] locus, and the limit of the numerator is -[[delta].sub.K], giving

[mathematical expression not reproducible].

All three arbitrage loci are independent of Z, so their graph is the same in any [alpha] - K plane in ([alpha], K, Z)-space. Figure 3 shows the three loci for in the [alpha] - K plane at an arbitrary value of Z. The third-state variable Z would be plotted along a third axis orthogonal to the [alpha] - K plane.

The three loci divide the phase space into six regions labeled accordingly in the diagram. Table 1 shows the relations among the three rates of return in each region. The bang-bang nature of the firm's optimization problem means that only the asset with the highest rate of return in a region is accumulated. Thus at any time only one type of investment takes place, except when the economy is on one of the boundaries between regions. The arbitrage loci are thus also the switching loci for the controls.

The information in Table 1 is not quite sufficient to determine the economy's dynamics. We also must know whether total asset accumulation is positive or negative, which is determined by the stationarity equation and its corresponding equilibrium locus, discussed next.

Stationarity Locus. The stationarity locus is the set of points in the phase space where total net investment is zero:

[mathematical expression not reproducible].

To derive its properties, begin with the equality between gross investment and saving:

[mathematical expression not reproducible].

Net investment is zero on the stationarity locus, so the equation simplifies to

[[delta].sub.K]K = sA [ZL + m([alpha]) K].

After some algebra, we obtain the expression for the stationarity locus:

(35) [mathematical expression not reproducible]

The numerator is positive if sAL is positive, which must hold for accumulation of Z to be possible. (7)

The form of the stationarity locus depends on whether [mathematical expression not reproducible]:

(1) If [s.sub.A]/[[delta].sub.K] > 1, then from (35) there is a value of [alpha], [??] [member of] [0,1], such that in m ([??]) = [delta]/sA. The locus has two branches. For the region 0 [less than or equal to] [alpha] < [??], the K-intercept is

K = sAbL[bar.Z]/b[[delta].sub.K].

As [alpha] [up arrow] [bar.[alpha]] on this branch, K [right arrow] [infinity]. For the region [??] < [alpha] [less than or equal to] 1, the intercept at a = 1 is

K = sAbLZ/b [[[delta].sub.K] - [s.sub.A] < 0.

The locus drops to-[infinity] as [alpha] [down arrow] [??].

(2) If sA/[[delta].sub.K] < 1, the denominator of (35) is positive because m([alpha]) [less than or equal to] 1 [for all] [alpha] and is monotonically decreasing in a because m([alpha]) is monotonically increasing in [alpha]. The K-intercept is

K = Z[sAL/[[delta].sub.K]]

because m(0) = 0, and the intercept at [alpha] = 1 is

K = sAbLZ/b [[[delta].sub.K] - sA].

The two possibilities for the stationarity locus are shown in Figures 4 and 5 as dotted curves, labeled with their defining equation.

Labor Constraint Boundary. Finally, we must know whether we are operating in the part of the phase space that is constrained or unconstrained by labor supply. The boundary between those regions is given by

[K/Z] [(1 - [alpha]).sup.1/[alpha]] = L

which we can rearrange as

(36) K = ZL[(1 /1 - [alpha]).sup.1/[alpha]].

The constraint does not bind if the left side is less than the right. The boundary's slope is

[mathematical expression not reproducible]

which is increasing in [alpha]. Application of L'Hopital's Rule shows that K = eZL for [alpha] = 0, K [right arrow] [infinity] as [alpha] [right arrow] 1, and dK/d[alpha] = eZL/2. The term in brackets in the expression for the slope is positive for [alpha][epsilon](0,1). The boundary therefore is increasing and asymptotic to the vertical line [alpha] = 1 as [alpha] [right arrow] 1. It is shown in Figures 4 and 5 as the dot-dashed line and labeled LCB.

In the discussion that follows, we restrict attention to the unconstrained region, which is the part of the phase space below the stationarity locus. The unconstrained case is simple, and furthermore the economy always moves into the unconstrained region if it does not start there, as the laws of motion in the phase diagrams will show. Thus, the unconstrained case may prevail at all times and always will prevail at least eventually, so restricting attention to it is not much of a limitation. Behavior in the constrained region offers only minor additional insight, so we ignore it here. See Peretto and Seater (2013) for discussion of the constrained case in a simpler model but whose main results carry over here.

B. Phase Diagrams

There are two versions of the phase diagram corresponding to the two shapes of the stationarity locus. One obtains in the "high saving" case, that is, when the saving rate s is sufficiently high that sA/[[delta].sub.K] > 1. The other obtains in the "low saving" case when s is low enough that sA/[[delta].sub.K] < 1. The high saving case is simpler, so we start with that.

High Savings. Figure 4 shows the phase diagram for the case of high savings. It has two more loci than Figure 3: the stationarity locus and the labor constraint boundary. Recall that the three arbitrage loci are independent of Z, so their positions are the same for all values of Z. In contrast, the foregoing discussion shows that the slopes and intercepts of the stationarity locus and the labor constraint boundary depend positively on Z. Thus, both loci are higher and steeper for higher values of Z. However, the slopes and intercepts all are linear in Z, so the relative positions of the two loci never change. Furthermore, as noted above, the Cobb-Douglas form of the intermediate goods production function implies that Z is necessary for production for positive a and thus must have a positive value for the advanced production function to be useful. With those properties in mind, consider the economy's behavior in the six regions of the phase plane.

In regions I and II, [r.sub.K] exceeds the other two rates of return, and the economy invests only in K. Both [alpha] and Z are constant. The economy follows a path such as [p.sub.1]. Irrespective of whether the economy started in region I or II, it eventually arrives at the [r.sub.K] = [r.sub.[alpha]] locus and then moves along that locus, following path [p.sup.*] forever. The economy grows by accumulating physical capital and increasing capital's exponent in the production function. As time passes, K [right arrow] [infinity] and [alpha] [right arrow] 1. (8) This behavior is the same as in Peretto and Seater (2013) which omits Z. In regions I and II, labor-augmenting Z is irrelevant. (9)

In regions III and IV, [r.sub.[alpha]] exceeds [r.sub.K] and [r.sub.Z], so the economy invests only in a. From a starting point anywhere in region III and from the part of region IV on or above the path [p.sub.3] the economy eventually reaches the [r.sub.k] = [r.sub.[alpha]] locus, following a path such as [p.sub.2]. The economy then moves along that locus forever, following path [p.sup.*]. Once again, the result is the same as in Peretto and Seater, and once again labor-augmenting Z is irrelevant. Behavior is different for a starting point in the part of region IV below path [p.sub.3]. The economy follows a path such as [p.sub.4], reaching the [r.sub.Z] = [r.sub.[alpha]] locus and passing into region V, where behavior is very different from anything in Peretto and Seater, as explained next.

In regions V and VI, the rate of return [r.sub.Z] exceeds [r.sub.[alpha]] and [r.sub.K]. so only Z is accumulated. In the [alpha] - K plane, the economy follows path [p.sub.5] in Figure 4, with K declining through depreciation and [alpha] constant. However, because the economy is accumulating Z in regions V and VI, it moves continuously along the third dimension, that is, the Z-axis, not shown in the phase diagrams and orthogonal to the [alpha] - K plane, coming out of the page toward us. The economy not only moves down the path [p.sub.5] but also out parallel to the Z-axis. Thus in regions V and VI, a does not change, K [right arrow] 0, and Z [right arrow] [infinity]. Further more, all of regions V and VI are below the stationarity locus where total asset accumulation is positive, so Z increases by more than K falls. This behavior is completely different from that in Peretto and Seater, where the economy always accumulates [alpha], K, or both and ultimately moves up path [p.sup.*] with K and possibly [alpha] growing forever.

We discuss these results below after we examine the low saving case and look briefly at some possible variations of the model.

Low Savings. We define low saving as a saving rate s sufficiently small that sA/[[delta].sub.K] < 1. When the saving rate is that low, the stationarity locus has a single branch and lies everywhere below the labor constraint boundary. The corresponding phase diagram is shown in Figure 5. The main difference from the high saving case is that regions I and II now are divided into two parts. In sub-regions la and Ha, gross investment in K exceeds depreciation, net investment therefore is positive, and K grows, just as in the high saving case. In subregions lb and lib, however, gross investment in K is less than depreciation, net investment is negative, and K falls. The heavy part of the stationarity locus to the upper right of point E thus is a basin of attraction for the economy. For any starting point in regions I, II, and III or in the part of region IV above path [p.sub.5], the economy eventually arrives at the stationarity locus and then stops moving. The point E on the stationarity locus is all by itself the basin of attraction for a substantial part of the phase space. All starting points in regions la and IIa on or to the left of path [p.sub.1] and in regions III and IV between paths [p.sub.4] and [p.sub.5] (including those paths themselves) go to point E. These results are the same as in Peretto and Seater. The stationarity locus on and to upper right of point E is also a stagnation locus for the economy: for starting points in the appropriate part of the phase space, the economy will arrive at the stationarity locus and then remain there forever with no growth in [alpha], K, or Z, and thus no growth in X or Y, either.

As in the high saving case, behavior differs from Peretto and Seater is the part of the phase space that starts in or leads into regions V and VI. There, behavior is the same as in the high saving case: ultimately only Z is accumulated, K dissipates through depreciation, and a stagnates at a value below 1. Output does grow forever through accumulation of labor-augmenting Z.

C. Amplifications and Extensions

g Function. Here, we present some economic intuition for the function g. Recall that g([alpha]) is the productivity of [I.sub.[alpha]] in the R&D technology (10). We suppose that the firm's R&D activity accumulates knowledge S and that knowledge transforms into values of [alpha] with diminishing returns. We assume the usual linear investment relation

[??] = [I.sub.[alpha]]

and denote the transformation from knowledge to technology by the "learning function"

[alpha] = h (S)

with h'(S) < 0. We then have

[mathematical expression not reproducible].


g([alpha]) [equivalent to] h' [[h.sup.-1] ([alpha])]

we obtain (10). For [??] > 0, we need the marginal product of knowledge positive for [alpha] < 1, so we impose h'(S) > 0 for all S<S, and h'(S) = 0 for S [greater than or equal to] [S.sub.1], where [S.sub.1] [equivalent to] [h.sup.-1](1) is the value of S, possibly infinite, at which a first equals 1.

As explained earlier, stability of symmetric equilibrium requires that a firm's incentive to pursue further innovations must decrease in a, which requires g'([alpha]) < 0. We then can conclude that h" < 0:

[mathematical expression not reproducible]

because h' > 0 [??] d[h.sup.-1]([alpha])/d[alpha] > 0. Thus the transformation of knowledge S into the capital elasticity of output [alpha] is a concave function. Finally, to guarantee that g([alpha]) goes to zero fast enough to dominate n([alpha]) going to infinity as a goes to 1, we must impose the condition that h(a) is sufficiently concave for g([alpha]) to have the requisite behavior. (10) Two functional forms that satisfy the foregoing requirements are g = 1 - [alpha] and g = [(1 - [alpha]).sup.2], which arise from the respective learning functions h(S) = 1 - exp (-S) and h(S) = S/(1 + S), both of which are positive and concave, equal 0 at S = 0, and go asymptotically to 1 as S [right arrow] [infinity].

Symmetric equilibrium requires no further restrictions on the functions h(S) and g([alpha]).

Costs of Adjustment. The Hamiltonian for our model is linear in the control variables, leading to the bang-bang nature of the solution. That aspect of the solution is very convenient because it means we have only one control active at any time, except on the boundaries of regions of the phase space. Introducing costs of adjustment would eliminate the Hamiltonian's linearity and thus the bang-bang property, making the model analytically intractable. Although we cannot solve the model with adjustment cost, we strongly suspect that none of our major conclusions would change. In models of exogenous growth, adjustment costs warp the phase space, leading to slight changes in the steady state and the dynamic adjustment paths but leaving all the important properties of the model unchanged (Abel and Blanchard 1983). As far as we have been able to tell from tentative explorations of the model, the same would be true here. Adjustment costs would alter positions of the equilibrium loci, the adjustment paths, and the critical points in the phase diagrams and would make all controls active. Nonetheless, the main conclusions would remain: for low values of K and a the economy would converge to a solution in which only investment in Z is positive, and for high values of K and a would converge to a path on which investment in Z disappears. Investment in Z still would be dominant near the origin, and a and K investment still would be dominant elsewhere, leading asymptotically to the same results as the bang-bang version. It is not the bang-bang nature of our solution that delivers our strong results but rather the AZ or AK structure of the production function that emerges as investment in K, Z, and a proceeds, a property of the solution that is not changed by introducing costs of adjustment.

Depreciation of Z. Adding depreciation of Z would change the results only slightly. With depreciation of Z, investment in physical capital K is positive on the stationarity locus in regions I and II. In those regions, investment in both a and Z is zero, and the depreciation of Z makes total net investment negative unless offset by positive net investment in physical capital K. Thus in regions I and II, we would have growth of K. That would complicate the dynamics slightly because now the economy is not moving in a fixed K-a plane when it is in regions I and II but instead would also be moving along the third dimension with Z declining. Otherwise the results are the same as discussed above.


Our results divide into two parts. In a neighborhood of the origin, labor augmentation has a higher rate of return than either factor elimination or physical capital investment, so factor elimination and physical capital investment never occur. Investment in Z is the only investment that happens. The economy grows forever purely by perpetually augmenting labor with investment in K and a playing no role. Outside that neighborhood, the opposite holds. There, labor augmentation has a lower rate of return than other kinds of investments and so never happens. As a result, Z stays constant at its initial endowment, and the economy grows (in the high saving case) through accumulation of [alpha] and K or (in the low saving case) settles in a stagnant steady state with no growth. Labor augmentation plays no role.

We can see the economic intuition for these results by reconsidering the production function given by (15):

X = A [Z (L - [L.sub.[alpha]]) + [K.sup.[alpha]] [(Z[L.sub.[alpha]]).sup.1-[alpha]].

In the Peretto-Seater model, Z [equivalent to] 1 always. In that case there is a "virtuous cycle" operating between [alpha] and K: an increase in a raises the marginal value of K, so more K is accumulated, but then an increase in K raises the marginal value of a and more R&D is done to increase [alpha]. The same forces operate here if [alpha] and K are sufficiently large. In that case, the economy looks just like Peretto-Seater, which explains the second set of results. However, for [alpha] = 0, the production function takes on an AK form:

[mathematical expression not reproducible]

where B [equivalent to] AL is constant because both A and L are constant. In that case, it is optimal simply to invest in Z and not bother with investing in either a or K. The economy then grows forever as in any AA'-type of model (here, a BZ model). The same results hold in the neighborhood of the origin comprising all of regions V and VI. The economic intuition is the same as at the origin. Furthermore, if the economy starts in the low saving case but also starts in either region V or VI, it avoids the stagnant part of the phase space and grows forever even with a low saving rate. That is because in an AK model growth is positive for any positive rate of saving.

The problem with these results is that they are inconsistent with economic history, in contrast to the Peretto-Seater model. The result here says that if the economy starts near the origin it grows entirely by labor augmentation and never accumulates physical capital and never eliminates anything. If the economy starts anywhere else it never invests in in any kind of progress that augments labor--no schooling, no worker training, no labor-augmenting technical progress. It just builds more capital and progressively eliminates labor. Neither possibility resembles the path of human progress.

These results thus raise a deep question about the nature of the production function and the place of labor augmentation in it. In Peretto and Seater's model, "capital" is a generic term for all reproducible factors of production. When they get perpetual growth arising from perpetually rising K, the K in question includes not only physical capital but also labor-augmenting technical progress and conventional human capital (e.g., education, training, and experience). They cite much evidence that factor shares (represented by a in our model) have shifted over time from nonreproducible to reproducible factors of production. Land, natural resources, and unskilled ("raw") labor all have experienced declines in their shares of national income, whereas physical and human capital have experienced increases in their income shares, as documented in the Introduction. Our variation of the factor elimination model follows a vast literature in separating a subset of technical knowledge from all other reproducible factors and writing it as a labor-augmenting term. Doing that, however, converts the model from one that can explain several phenomena that previously were unexplained into one that is wildly inconsistent with the historical data. We thus have two contrasting results. On one hand, the original Peretto-Seater specification with reproducible and nonreproducible factors entering independently is consistent with a wide variety of time series and cross-section evidence on the behavior of economic growth, the evolution of reproducible and nonreproducible types of capital, and the evolution of the factor shares of those types of capital. On the other hand, changing the specification to one in which some knowledge is separated from other reproducible inputs and is written as a labor-augmenting factor leads to model behavior that is strongly at variance with the historical record. In short, the data fail to reject the Peretto-Seater specification but do reject our version. The implication is as straightforward as it is striking: the usual specification that imposes the labor augmentation restriction is incorrect.

One may ask how such a result can possibly be true. Did not Phelps (1965) show that long-term economic growth requires technical change that augments the essential nonreproducible factors, such as labor? We obviously have seen economic growth, so must not there be factor-eliminating technical progress? The answer is that Phelps's result holds only in a model that admits only factor augmentation. Once the model is expanded to admit factor elimination, Phelps's condition for perpetual growth no longer is necessary, as Peretto and Seater (2013) show. Our result here goes farther: not only is factor augmentation not necessary for growth when there also is factor elimination, it is inconsistent with the historical evidence.

One then may ask how it can be that the more restrictive standard factor augmentation model is consistent with the facts but the less restrictive model, which nests the standard model, is not. The answer is in two parts. First, the standard model is not consistent with all the facts. For example, it does not generate either of the two facts mentioned in the Introduction, the evolution of the relative income shares of reproducible and nonreproducible factors of production and the correlation between the reproducible factors' income share and the level of per capita income. Second, the standard augmentation model prohibits factor elimination by construction (through constancy of the production function's parameters) and so excludes the only alternative source of endogenous perpetual growth in an economy with essential nonreproducible factors of production. Unsurprisingly, indeed inevitably, the model then ascribes all explanatory power to the single remaining growth mechanism, and the data, being allowed no alternative, are forced to agree with that conclusion. When the restriction on factor elimination is relaxed and the model is allowed full scope, it produces counterfactual behavior. In contrast, an alternative model, that of Peretto and Seater (2013) in which the reproducible factors enter independently of the nonreproducible factors, obtains results consistent with the data. All that information together suggests that the factor augmentation restriction is a misspecification.

The foregoing conclusion must be treated as tentative because we have shown it only for Cobb-Douglas production. That is the only functional form that has been used so far in the factor elimination literature, so we limited our attention to it. No one has explored the properties of factor elimination with other production functions, so at this point we do not know how factor elimination and factor augmentation behave with other production functions. Perhaps the most obvious function to try next is the CES, which includes the Cobb-Douglas as a special case. It may be that the results on factor elimination and factor augmentation are sensitive to the magnitude of the elasticity of substitution, though it is not obvious that they will be. (11)


We have presented a model of economic growth that allows both factor-augmenting and factor-eliminating technical progress. The model is a generalization of the standard factor augmentation model, relaxing the standard restriction that the production function's parameters are constant and instead allowing them to change endogenously in a way that permits factor elimination. The model thus nests the standard factor augmentation model as a special case. We expected to see an interesting interplay between factor augmentation and factor elimination, but we found no such thing. Instead, we found behavior that is strongly counterfactual. In contrast, the factor elimination model presented by Peretto and Seater (2013) is not rejected by the data. That model has a different restriction, not permitting augmentation but instead requiring that the reproducible and nonreproducible factors enter the production function independently of each other. The results imply that factor augmentation is an invalid restriction. Consequently, the results also imply that economic growth is not driven by factor augmentation, contrary to the usual approach, but rather is driven by factor elimination coupled with accumulation of the reproducible factors of production, which include human as well as physical capital.

Many important questions arise from the foregoing conclusions. We mention a few here. First, there are different types of reproducible assets. What is the right specification for distinguishing among them and capturing their interaction and effect on economic growth? Peretto and Seater's (2013) approach was to add them together into one reproducible asset. Perhaps we would obtain further insight by treating them as separate factors in the production function, such as

[mathematical expression not reproducible]

or by specifying an aggregator other than simple summation for combining them into a single aggregate reproducible factor, such as

[mathematical expression not reproducible].

Second, we have restricted attention in our analysis to Cobb-Douglas forms of the production function because to date that is the only framework that has been used for studying factor-eliminating technical progress. Would any conclusions change for other production functions? Third, how does our main result affect the analysis of the current hot topic of the change in the income distribution over the last few decades, with a smaller fraction of the population commanding a larger fraction of the income? If human capital is not treated in the traditional way as an augmenter of labor but rather as interacting with physical capital in a way not previously considered, perhaps new insights into the relative returns to investing in human and physical capital will emerge. Fourth, if reproducible assets cannot be treated as factor augmenters, then the traditional way of decomposing economic growth needs reconsideration. Growth decompositions often use a Cobb-Douglas specification with fixed exponents and allocate growth among accumulation of capital, growth of labor, and increases in total factor productivity (TFP). In such a framework, TFP is mathematically equivalent to factor augmentation, and TFP growth can be regarded as an amalgam of growth of the factor-augmenting processes. With endogenous exponents, that equivalence no longer holds, and it is not clear what the source of any growth in TFP is. Endogenous exponents introduce an index number problem that must be addressed before any statistical analysis can be done. The same issues affect cross-country comparisons of income and economic growth, which traditionally have taken the Cobb-Douglas exponents as fixed when in fact they vary over time in a systematic and endogenous way. Sturgill (2014) and Sturgill and Zuleta (2017a, 2017b) have begun examining those issues with interesting results.



The Lagrangian for the firm's problem of technology choice with the production function defined in (8) is

(A1) [mathematical expression not reproducible]

(A2) [mathematical expression not reproducible]

(A3) [mathematical expression not reproducible].

From (A2) it follows that

[mathematical expression not reproducible].

Similarly, from (A3) we can write

[PSI] = Aa [([K.sub.a]).sup.a-1] [Z.sup.1-a] [([L.sub.a]).sup.1-a]

Substituting for [L.sub.a] gives

[PSI] = a [(1 - a).sup.1-a/a] A[(AZ/[lambda]).sup.1-a/a].

From here the argument is the same as in Peretto and Seater (2013), establishing that the two and only the two extreme technologies are used in the unconstrained case and only the advanced technology is used in the constrained case (i.e., [L.sub.a] [greater than or equal to] L).


Then the first-order conditions for profit maximization are:

(A4) [mathematical expression not reproducible]

(A5) [mathematical expression not reproducible]

(A6) [mathematical expression not reproducible]

(A7) [mathematical expression not reproducible].


The current value Hamiltonian is

(A8) [mathematical expression not reproducible].

The necessary conditions are:

(A9) [partial derivative]H/[partial derivative][PSI] = [I.sub.K] - [[delta].sub.K]K

(A10) [partial derivative]H/[partial derivative][PHI] = g([alpha])[I.sub.[alpha]]

(A11) [partial derivative]H/[partial derivative][mu] = b[I.sub.Z] - [[delta].sub.Z]Z

(A12) [??] =-[[partial derivative]H/[partial derivative]K] + r[PSI] =-[p.sub.K] + [PSI][[delta].sub.K] + r[PSI]

(A13) [mathematical expression not reproducible]

(A14) [??] =-[[partial derivative]H/[partial derivative]Z] + r[mu] =-[p.sub.Z] + [mu][[delta].sub.Z] + r[mu]

(A15) [partial derivative]H/[partial derivative][I.sub.K] =-1 + [PSI] + [[omega].sub.K] = 0

(A16) [partial derivative]H/[partial derivative][R.sub.[alpha]] =-1 + [PHI]g([alpha]) + [w.sub.[alpha]] = 0

(A17) [partial derivative]H/[partial derivative][I.sub.Z] =-1 + b[mu] + [[omega].sub.Z] = 0

(A18) [mathematical expression not reproducible]

(A19) [mathematical expression not reproducible]

(A20) [mathematical expression not reproducible].

If [I.sub.K] > 0, then [[omega].sub.K] = 0 then from (A15) it follows that [PSI] = 1 and thus [??] = 0. Substituting this result into (A12) implies

(A21) r = [p.sub.K] - [[delta].sub.K].

If [R.sub.[alpha]] > 0, then [[omega].sub.[alpha]] = 0 then from (A16) it follows that -1 + [PHI])g([alpha]) = 0 and thus [PHI] = 1/g([alpha]), implying

ln[PHI] =-lng ([alpha]).

Take the derivative to obtain

(A22) [??]/[PHI] =-[1/g([alpha])]g'([alpha])[d[alpha]/dt].

On the other hand, if we rewrite (A13) as [PHI]/[PHI], we get

[mathematical expression not reproducible].

Substituting [PHI] = 1/g([alpha]) and (A22) for [PHI]/[PHI] gives

[mathematical expression not reproducible].

Finally, substituting (10) gives

[mathematical expression not reproducible]

which after simplification yields

(A23) [mathematical expression not reproducible].

Finally, if [I.sub.Z] > 0, then [[omega].sub.Z] = 0. From (A17) it follows -1 + d[mu] = 0, which implies [mu] = 1/d and that [??] = 0. Using this result in (A 14), we get

(A24) [mathematical expression not reproducible].

Proceed by calculating and [partial derivative][p.sub.K]/[partial derivative][alpha] and [partial derivative][p.sub.Z]/[partial derivative][alpha] for (A23). From (19) we have:

[mathematical expression not reproducible].

Before calculating the derivatives of [p.sub.K] with respect to [alpha], we introduce m([alpha]) function. Following Peretto and Seater (2013), we can write

m([alpha]) = [alpha][k.sup.[alpha]-1]

where k = K/Z[L.sub.[alpha]]. Substituting m([alpha]) into the expression for [p.sub.K], we get

(A25) [mathematical expression not reproducible].

Then, [partial derivative][p.sub.K]/[partial derivative][alpha] for unconstrained and constrained cases, respectively, will be:

[mathematical expression not reproducible]

which we can rewrite as:

(A26) [mathematical expression not reproducible],

Next, from (20), we have the expression for [p.sub.Z] equal to:

[mathematical expression not reproducible].

Taking partial derivative of [p.sub.Z] with respect to a, we will get the following expressions for unconstrained and constrained cases, respectively:


[mathematical expression not reproducible]

which we can rewrite as:

[mathematical expression not reproducible].


[mathematical expression not reproducible]

which we can rewrite as:

[mathematical expression not reproducible].

Rewriting the expression for the partial derivative of [p.sub.Z] with respect to [alpha] for both constrained and unconstrained cases together we get

(A27) [mathematical expression not reproducible].

Substituting (A26) and (A27) into (A23) gives the expressions for the rate of return on [alpha], [r.sub.[alpha]], for both unconstrained and constrained cases.


(A28) [mathematical expression not reproducible].


(A29) [mathematical expression not reproducible].

Substituting (A25) into (A21), we will get the expression for the rate of return on capital, [r.sub.K], such that

(A30) [mathematical expression not reproducible].

Finally, substituting (20) into (A24), we will get the expression for the rate of return on Z:

(A31) [mathematical expression not reproducible].

Next, we rewrite (A28) and (A31) using definition of m([alpha]). Start with (A31). We will work with unconstrained case only. Rewrite making the following substitution k = K/Z[L.sub.[alpha]].

[r.sub.Z] = (1 - [1/[epsilon]]) [Y.sup.1/[epsilon]] [X.sup.-1/[epsilon]] Ab [(L - [L.sub.[alpha]]) + (1 - [alpha]) [k.sup.[alpha]][L.sub.[alpha]]] - [[delta].sub.Z].

Substitute k = [(1 - [alpha]).sup.-1/[alpha]] into the above expression.

[mathematical expression not reproducible].

Now work with the unconstrained part of (A28). Rewrite (A28) as

[mathematical expression not reproducible].

Substitute k = K/Z[L.sub.[alpha]] and multiply and divide the second part of the expression by K.

[mathematical expression not reproducible].

Again, using k = K/Z[L.sub.[alpha]], we can rewrite the above expression as

[mathematical expression not reproducible].

Now substitute k = [(1 - [alpha]).sup.-1/[alpha]] into the above expression and do some algebra to get

[mathematical expression not reproducible].

Multiply and divide the second large term by [alpha]:

[mathematical expression not reproducible].

From page 2 of the appendix (Peretto and Seater 2013), it follows that

m([alpha]) = [alpha][(1 - [alpha]).sup.1-[alpha]/[alpha]].

Using this definition, we can rewrite the above expression as

[mathematical expression not reproducible]

which can be written as

[mathematical expression not reproducible].

Using the definition of m'([alpha]) from the appendix (Peretto and Seater 2013), we can rewrite the above expression as:

[mathematical expression not reproducible]

and do some algebra to obtain

[mathematical expression not reproducible]

Referring again to the appendix (Peretto and Seater 2013), we can rewrite the above expression making the substitution for the last term as

[mathematical expression not reproducible].

PROPERTIES OF M([alpha])/[alpha]M'([alpha])


[mathematical expression not reproducible].

The limit of the numerator is indeterminate, while the limit of the denominator approaches [e.sup.-1] (see Peretto and Seater 2013 [Appendix, page 2]). We apply L'Hopital's rule to determine the limiting behavior of m([alpha])/[alpha]

[mathematical expression not reproducible].

Thus, we can write:

[mathematical expression not reproducible].

Similarly, for [alpha] [right arrow] 1, we can write:

[mathematical expression not reproducible].


h'([alpha]) = [alpha][(m').sup.2] - [m' + [alpha]m"] m/[([alpha]m').sup.2].

Determine sign of the numerator [alpha][(m').sup.2] - [m' + [alpha]m"]m using the expressions for m and m derived in Peretto and Seater (2013)

[mathematical expression not reproducible]

where (*) > 0, and [[1/[alpha]] ln [1/1-[alpha]] - 1 - [1[alpha]] ln [1/1-[alpha]] - [[alpha]/1-[alpha]]] = [-1 - [[alpha]/1-[alpha]]] < 0, making overall sign of the slope of m([alpha])/[alpha]m'([alpha]) negative.


CES: Constant Elasticity of Substitution

TFP: Total Factor Productivity

doi: 10.1111/ecin.12711


Abel, A. B., and O. J. Blanchard. "An Intertemporal Model of Saving and Investment." Econometricci, 51(3), 1983, 675-92.

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Sturgill, B., and H. Zuleta. "Variable Factor Shares and the Index Number Problem: A Generalization." Economics Bulletin, 37, 2017a, 30-37.

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* This project was supported by a grant from the Corporate Fund "Fund of Social Development," Nazarbayev University.

Seater: Visiting Scholar, Economics Department, Boston College, Chestnut Hill, MA 02467-3806. Phone 617-552-4550, Fax 617-552-2308, E-mail

Yenokyan: Biostatistician, Department of Epidemiology, Bloomberg School of Public Health, Johns Hopkins University. Baltimore, MD 21205. Phone 410-614-5038, Fax 410-955-7587, E-mail

(1.) Movement of [OMEGA] toward [bar.[OMEGA]] means that the Euclidean distance between [OMEGA] and [bar.[OMEGA]] decreases.

(2.) Endogenizing the savings rate would introduce another dimension in the state-costate space and render the model intractable. Sturgill (2010) extends the Peretto-Seater model, which does not have a state variable representing factor-augmenting technology, to include endogenous savings and shows that the main conclusions do not change. The nature of his argument suggests that the same conclusions would apply to the model of the present paper.

(3.) See also Peretto and Seater (2013).

(4.) Just as in chattering control in optimal control problems and in mixed strategies in game theory.

(5.) See Peretto and Seater (2013) for more discussion.

(6.) See Peretto (1996, 1998, 1999a, 1999b) for a thorough discussion of the assumptions that support symmetric equilibria in growth models based on cost-reducing innovation.

(7.) The first-term sAL is what saving would be if all labor were allocated to the primitive plant, which is the initial state of the economy. The second term is the depreciation rate of Z relative to the productivity b of investment [I.sub.Z] in Z. If the former does not exceed the latter, Z will not be accumulated. We are interested only in the case where Z is accumulated, so we impose sAL - ([[delta].sub.Z]/b) > 0.

(8.) If the [r.sub.K] = [r.sub.[alpha]], locus intersects the vertical line [alpha] = 1, the locus becomes vertical, growth in a stops, and K [right arrow] [infinity] as t [right arrow] [infinity]. As in Peretto and Seater (2013), the advanced production process has become AK, and all labor is allocated to the primitive production process. Perpetual economic growth is driven by perpetual capital accumulation.

(9.) In Peretto and Seater (2013), [alpha] may reach 1 in finite time, when it stops changing. Under suitable parameter restrictions, that does not happen and [alpha] grows forever, going asymptotically to 1. For simplicity, we have ignored the possibility a [alpha] reaching 1 in finite time.

(10.) A similar assumption of "sufficient concavity" is routine in quality-ladder models, where the cost of innovation is assumed to be increasing and convex in quality to offset the fact that the profit from monopolizing the market also is increasing and convex in quality. The offset usually is assumed to be exact so that the return to innovation is independent of quality, which simplifies the analysis.

(11.) We also have a constant saving rate, but as mentioned previously Sturgill (2010) finds that extending Peretto and Seater (2013) to an endogenous saving rate changes no important implications of that model. We do not see any reason to think that conclusion would be different here, but of course the only way to be sure would be to repeat our model with an endogenous saving rate. Such a model would have four state variables (K, Z, [alpha], s) and almost surely would be analytically intractable, requiring numerical analysis.

Caption: FIGURE 1 The Function n([alpha])

Caption: FIGURE 2 The Composite Function g([alpha])n([alpha])

Caption: FIGURE 3 Arbitrage Loci

Caption: FIGURE 4 High Savings

Caption: FIGURE 5 Low Savings
Rates of Return Relations

Region        Rates of Return

Region I      [r.sub.K] > [r.sub.z] > [r.sub.[alpha]]
Region II     [r.sub.K]  > [r.sub.[alpha]] > [r.sub.z]
Region III    [r.sub.[alpha]] > [r.sub.K] > [r.sub.z]
Region IV     [r.sub.[alpha]] > [r.sub.z] > [r.sub.K]
Region V      [r.sub.z] > [r.sub.[alpha]] > [r.sub.K]
Region VI     [r.sub.z] > [r.sub.K] > [r.sub.[alpha]]

Region        Asset Accumulated

Region I              K
Region II             K
Region III         [alpha]
Region IV          [alpha]
Region V              z
Region VI             z
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Author:Seater, John; Yenokyan, Karine
Publication:Economic Inquiry
Article Type:Report
Date:Jan 1, 2019

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