Introduction and background

Production functions are a staple of economic education and modeling. Yet such functions are problematic. At one level, the conventional forms (e.g., Cobb–Douglas, CES, or Translog) do not describe how firms actually operate. Buried in the mathematics are a number of strong assumptions about efficiency, technology, optimization, and the market environment in which firms operate. After all, the production function is meant to represent the maximum feasible output that can be produced by various combinations of inputs, not what firms really do.

Even if we are willing to abstract from these realities, the production function concept rests on contested theoretical grounds. The measurement and aggregation of capital present unresolved (and unresolvable) difficulties. Although the Cambridge (England) vs. Cambridge (Massachusetts) “capital controversy” has largely been ignored or forgotten, the objections raised by Joan Robinson, Piero Sraffa, and others to the neoclassical treatment of capital have never been laid to rest. The value of a capital asset is equal to the present discounted value of its returns, but in neoclassical theory the rate of return, essential to calculation of the present discounted value, is the value of the marginal product of the capital. Capital has first to be measured to measure itself. The circularity is evident.

There was a great deal more to the Cambridge controversy than this, of course. Still, the validity of key elements of the Cambridge (UK) critique was recognized by major participants from the other side of the pond (Cohen and Harcourt 2003 give a balanced and accessible summary of the state of the debate through 1966; see also the “Comments” on Cohen and Harcourt by Pasinetti (2003), Fisher (2003), and Felipe and McCombie (2003), as well as the 1966 Symposium in the Quarterly Journal of Economics (1966)). Cohen and Harcourt observe that.

With neither side able to deliver a knockout punch, issues of faith and ideology entered the ring with claims about the significance of the results and competing visions of economics. When one-commodity results are not robust in more general models, the lack of definitive evidence leaves room for ideology to play a role in the decision to hang on to a theory or vision. The intensity and passion of the Cambridge controversies were generated not by abstract technical questions about Wicksell effects, but by strong ideological undercurrents like the ethical justification of the return to capital and fundamental methodological questions about comparing deeply differing visions of economics and the extent to which equilibrium is a useful tool of economic analysis (p. 210).

Even entirely within the neoclassical framework, the aggregation required to construct a macroeconomic production function is beset with conceptual problems. Franklin Fisher’s papers establish that measurement of aggregate capital (and hence the existence of a neoclassical aggregate production function) is not feasible except in the most restrictive and unrealistic circumstances (these papers are collected in Fisher 1993; see also Felipe and Fisher 2003). As Fisher puts it, “[t]he existence—and hence the use—of aggregate production functions is very problematic. The conditions for successful aggregation are so stringent that one can hardly believe that actual economies satisfy them” (Fisher 2003, p. 229). An additional devastating critique of the use of conventional neoclassical production functions in empirical work is provided by Felipe and McCombie (2013).

Of course, in principle it would be possible to specify production functions with each different type of capital good listed separately. The “rent” to each of these types of capital would then be the value of its marginal product. However, this approach is not operational from a practical or empirical standpoint. There are simply too many types of fixed inputs that would have to be included for any realistic specification. Some sort of simplification is required for empirical work, and the efficiency-distribution model of production as applied here is facilitates such work without coming up against the insuperable difficulties of capital aggregation.

Houthakker’s (1955–56) combination of elements of neoclassical and activity analysis offers a way of characterizing production and technology other than simply positing the standard neoclassical relationship Q = F(L,K,…and possibly other factors). His approach to specification of production relationships does not suffer from the debilitating flaws exposed by Fisher, Felipe and McCombie, and the Cambridge UK critics. While the approach taken by Houthakker and the papers that have followed him requires its own set of enabling assumptions, these are categorically different from the typical neoclassical setup and offer new possibilities for empirical implementation.

This paper has a twofold purpose: (1) to revisit, using modern symbolic mathematics software, what can be derived from the simplest version of Houthakker’s “efficiency-distribution” model, and (2) to show that the model can offer insights about how the substitutability of human and robotic labor (both taken to be part of the variable input) affects the short-run response of wages to increased automation. The paper will present a series of worked-out examples of the efficiency-distribution model for various specifications of the productivity density function across productive units, followed by a more detailed treatment of the wage/automation relationship for a variety of specific distributions. Throughout, the simplest possible version of the efficiency-distribution model will be employed, to highlight its flexibility and range of potential empirical applications.

The efficiency-distribution model

The most basic version of the Houthakker model is given by Solow (1967) and Levhari (1968), and their notation and outline will be followed here. Although both Johansen (1972) and Sato (1975) examined multi-factor specifications, the Solow–Levhari exposition is sufficient to highlight the main advantages of the efficiency-distribution approach. Let each productive unit (or “cell” in Houthakker’s terminology) produce output of an additively homogeneous good using a variable factor X and an unmeasured fixed factor according to a Leontief fixed-proportions relationship. Normalize the price of output to be unity, and let the price of the variable factor be p, determined exogenously by market forces. Each cell has a requirement t of the variable factor and produces one unit of output. A cell will be in operation if it creates a non-negative quasi-rent, that is, if t p ≤ 1 (or t ≤ 1/p). If there are a large number of cells with varying requirements, their productivities follow (to a close approximation) a continuous distribution g(t). The function g(t) can be thought of as the number (or density) of firms that use t units of the variable factor to produce one unit of the good. With suitable choice of units, g will be an ordinary probability density function. The two basic equations of the model then are, with p, t, and g(t) all positive:

$$Q = \mathop \int \limits_{0}^{1/p} g\left( t \right)dt$$
(1)

for total output, and

$$X = \mathop \int \limits_{0}^{1/p} t g\left( t \right)dt$$
(2)

for the total amount of the variable factor used in production.

These two simple equations embody a considerable amount of information about aggregate production relationships. Letting u = 1/p, the fundamental theorem of calculus implies that

$$\frac{dQ}{{dX}} = \frac{dQ/du}{{ dX/du}} = \frac{g\left( u \right)}{{u g\left( u \right)}} = \frac{1}{u} = p$$
(3)

which is just the value of the marginal productivity of the variable input that would hold under competitive conditions for a well-behaved neoclassical production function. In addition

$$\frac{{d^{2} Q}}{{dX^{2} }} = \frac{d}{dX}\left[ {\frac{dQ}{{dX}}} \right] = \frac{{\frac{d}{du}\left[ {\frac{dQ}{{dX}}} \right]}}{{\frac{dX}{{du}}}} = \frac{{ - \frac{1}{{u^{2} }}}}{u g\left( u \right)} = \frac{{ - p^{3} }}{{g\left( {1/p} \right)}}.$$
(4)

This necessarily is negative, because p and g are positive. Therefore, the production relationship shows diminishing returns in the variable input X. The variables Q and X are both parametric functions of p, with the exact form of these functions depending on the efficiency-distribution g.

From Eq. (1) we see that for a suitable choice of units 0 ≤ Q ≤ 1, because Q is just the expression for the cumulative density of g; Q → 1 as u → ∞. Similarly from (2), X → 0 as u → 0 and X → Mean[g(t)] as u → ∞ (if the mean of g exists). The relationship between Q and X meets three of the four Inada conditions (Uzawa 1963; Inada 1963): Q → 0 as X → 0; d2Q/dX2 < 0; and dQ/dX → ∞ as X → 0. The slope dQ/dX is always positive and decreasing as X increases, but cannot reach 0, because X has a maximum value in the ordinary cases, where the mean of g exists. The share of the variable factor in total income, s = pX/Q, is also a function of p and the parameters of g. Use of the chain rule and the substitution u = 1/p yields the following:

$$\frac{ds}{dp} = - \frac{{g( {1/p} )}}{{p^{2} Q}}( {1 - s} ) + \frac{s}{p}.$$
(5)

The first term on the right of this equation is always negative (because 0 < s < 1) and the second term is always positive if the equations are to be economically meaningful, so the sign of ds/dp is ambiguous. The return to the fixed factor(s) is simply the residual left over after remuneration of the variable factor(s). Thus the functional distribution of income does not depend on any measurement of “capital” or attribution of its “marginal product.” For example (as in Solow 1967), if g is the “Pareto-like” density Ath−1 with constants A and h, 0 < h < 1, it is easy to show that the share of the variable factor will be h/(1 + h) just as if the production function were Cobb–Douglas. This will be a valid probability density function with finite support [0, B] provided (1/h)ABh = 1.

Unlike the traditional formulation of the production function that requires explicit measurement of capital in Q = F(X, K), the efficiency-distribution model is grounded in two more easily measured quantities: s, the share of the variable factor in total output, and g, the probability distribution of the productivities of the producing cells. Determining s empirically is relatively easy; knowing g depends on how closely the observed distribution of the productivities of individual producing units resembles an underlying hypothesized productivity distribution of cells.

The efficiency-distribution idea has been applied, expanded, and elaborated, notably by Cornwall (1971), Johansen (1972), Sato (1975), Rosen (1978), Jones (2005), Growiec (2013), Mangin (2015) and DeCanio (2016), although it has not received the attention Solow may have anticipated when he wrote in his 1967 survey article that “I cannot resist mentioning a result due to Houthakker [1955–56], in the hope that someone will take it up and push it further….Houthakker himself treats only the Pareto distribution, which gives rise to the Cobb–Douglas. The calculations can also be carried out with exponential or gamma-type distributions; they lead to a legitimate but not especially convenient aggregate production function. Can anyone think of other interesting cases? Even some numerical calculations would be worth having” (op. cit, pp. 47–48). The present paper continues in the spirit of Solow’s hoped-for development.

Examples of one-parameter distribution functions

Integration of Eqs. (1) and (2) may be simple or complicated, but powerful algorithms for finding closed-form antiderivatives are built into modern symbol manipulation software, such as Mathematica and MATLAB. Neither Mathematica nor MATLAB was available when the initial work on the Houthakker model was carried out. Mathematica was introduced in 1988 (Wolfram 2018). PC–MATLAB was commercially introduced in 1984 (Moler 2020). Once the parametric (in p) versions of Q and X are known, it is often possible to write down an implicit relationship between Q and X by eliminating the parameter p. In other cases, such elimination is not possible but the relationship between Q and X can still be evaluated, because both are functions of p.

In empirical applications, it cannot be known in advance which distribution of g would best fit the data. For this reason, and to show the remarkable diversity of functional forms that can result from application of the efficiency-distribution model of production, a variety of distributions have been used in the examples given below. All the calculations performed using Mathematica 12 (2020).


Example 1—uniform distribution: \({{g}}\left( {{t}} \right)\, = \,{1}/{{b}}, \, 0\, \le \,{{t}}\, \le \,{{b}}\) This is the simplest of all continuous probability density functions, and the corresponding formulas for Q and X are also quite simple:

$$Q = \frac{1}{b p}\, \,{\text{and}}\,\, X = \frac{1}{{2 b p^{2} }}$$
(6)

from which it follows immediately that

$$Q = \sqrt{\frac{2}{b}} X^{1/2} .$$
(7)

This is just a Cobb–Douglas function, with the share of the variable factor equal to 1/2 independent of its price. The Uniform is the only example in this paper (other than Solow’s example above) of a distribution with a finite support; Q = 1 if p = 1/b, and the maximum value of X is b/2.


Example 2—exponential distribution: \(g(t) = \lambda {e}^{-t\lambda }\,\,\mathrm{for}\,\,t > 0\,\,\mathrm{ and }\,\,\lambda > 0\) The parametric forms of Q and X are

$$Q = 1 - {\text{e}}^{ - ( \lambda /p)} \,{\text{and}}\, X = \frac{1}{\lambda } - \frac{{{\text{e}}^{ - (\lambda /p)} \left( {p + \lambda } \right)}}{\lambda p}.$$
(8)

Elimination of p between these two equations gives an implicit functional relationship between Q and X:

$$\lambda X = Q - {\text{Log}}\left[ {\frac{1}{1 - Q}} \right] + Q{\text{Log}}\left[ {\frac{1}{1 - Q}} \right].$$
(9)

The derivative of Q with respect to X is easily calculated from this equation by total differentiation and collecting terms,

$$\frac{dQ}{{dX}} = \frac{\lambda }{{{\text{Log}}\left[ {\frac{1}{1 - Q}} \right]}} = p.$$
(10)

The second equality in (10) follows from the expression for Q in terms of p in (8), and (10) is just a restatement of (3). It also must be the case that 0 < Q < 1, because the natural logarithm is not defined for non-positive arguments. Here and elsewhere, what matters most for the efficiency distribution model of production is the shape of the distribution g. It can also be seen that the upper bound of X is (1/λ) from the second equation of (8). Of course, 1/λ is the mean of the Exponential distribution.

Plotting Q against X, with the parameter p ranging from 0.0001 to 1 in increments of 0.01, shows that the relationship between output and the variable input exhibits the shape of an ordinary neoclassical production function. This is displayed in Fig. 1.

Fig. 1
figure 1

Q as a function of X. exponential distribution with λ = 1/3

Full size image

The variable factor’s share s of the value of output is given by

$$s = \frac{p X}{Q} = \frac{{\left[ {p - e^{{ - \left( {\lambda /p} \right)}} \left( {p + \lambda } \right)} \right]}}{{\left[ {1 - e^{{ - \left( {\lambda /p} \right)}} } \right]\lambda }}.$$
(11)

The share s ranges from 0 (as p → 0) to 1/2 (as p → ∞), which puts an upper limit on the variable factor’s share if the efficiency distribution g is exponential.


Example 3—Lindley distribution: \(g(t) = \frac{{e^{ - t\delta } \left( {1 + t} \right)\delta^{2} }}{1 + \delta },t \ge 0, \delta > 0.\) The Lindley Distribution is one of a family of one-parameter distributions related to the exponential but with different parameter configurations (see Sharma et al. 2019 for a list of eleven such distributions). For the Lindley distribution integrating (1) and (2) yields

$$Q = 1 - \frac{{{\text{e}}^{ - (\delta /p)} \left( {p + \delta + p\delta } \right)}}{{p\left( {1 + \delta } \right)}} {\text{and }}X = \frac{{2 + \delta - {\text{e}}^{ - (\delta /p)} \left( {2 + \delta + \frac{{\delta \left( {\delta + p\left( {2 + \delta } \right)} \right)}}{{p^{2} }}} \right)}}{{\delta \left( {1 + \delta } \right)}} .$$
(12)

The first equation of (12) confirms that 0 ≤ Q ≤ 1. The maximum value of X is the mean of the distribution, (2 + δ)/δ(1 + δ). Furthermore, the limit of s as p → 0 is 1/2. Not much more can be said about the production function based on the Lindley distribution. Mathematica’s symbolic manipulation algorithms are not able to find an expression for Q in terms of X by elimination of p from the two equations in (12). Of course, in an empirical implementation the restrictions on the distribution would necessarily be met, assuming the underlying productivity distribution followed the Lindley.


Example 4—the Half-Normal distribution: \(g\left( t \right) = \frac{2 \theta }{\pi }e^{{ - \frac{{t^{2} \theta^{2} }}{\pi }}} , t \ge 0.\) Because the Half-Normal is essentially a Normal that is truncated at zero, it has only the single shape parameter θ. For this distribution, we have the following expressions for Q and X in terms of p:

$$Q = {\text{Erf}}\left[ {\frac{\theta }{p \sqrt \pi }} \right] {\text{\,and\,}} X = \frac{1}{\theta } \left[ {1 - {\text{e}}^{{ - \frac{{\theta^{2} }}{{p^{2} \pi }}}} } \right],$$
(13)

where Erf[] is the Gaussian error function, defined by \(\mathrm{Erf}\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_{0}^{z}{e}^{-{t}^{2}}dt.\) Both Q and X are positive for p > 0, but the maximum attainable value for Q is 1 and for X is 1/θ as p approaches 0 from above. Eliminating p we obtain

$$Q = {\text{Erf}}\left( {\sqrt {{\text{Log}}\left[ {\frac{1}{1 - X\theta }} \right] } } \right), 0 \le X < \frac{1}{\theta }.$$
(14)

The variable factor’s share pX/Q for the Half-Normal ranges between 0 and 1/2, the upper bound being reached as p → ∞.


Example 5—Half-Cauchy distribution: \(g\left( t \right) = \frac{2}{{{\pi }\left( {1 + \frac{{{\varvec{t}}^{2} }}{{{\varvec{\nu}}^{2} }}} \right){ \nu }}},t \ge {\mathbf{0}}.\) Only the spread parameter ν appears here, because the distribution is truncated at 0. It might be thought that this distribution is an unlikely candidate for the efficiency-distribution representation. The Cauchy is notoriously ill-behaved; neither its mean nor its variance exists because of its “fat tails.” Nevertheless, the Half-Cauchy does yield a workable production function.

The parametric representations are given by

$$Q = \frac{2}{\pi }{\text{ArcTan}}\left[ {\frac{1}{p\nu }} \right] {\text{\,and\,}} X = \frac{\nu }{\pi } \left( { - 2 {\text{Log}}\left[ \nu \right] + {\text{Log}}\left[ {\frac{1}{{p^{2} }} + \nu^{2} } \right]} \right) .$$
(15)

In Mathematica, the output of the ArcTan function is given in radians, so as p → 0 ArcTan[1/pν] → π/2, and therefore, Q → 1. As p → ∞ ArcTan[1/pν] → 0. Thus the range of Q is 0 ≤ Q ≤ 1 as it must be. As p → 0, X becomes indefinitely large, reflecting the fact that the mean of the Cauchy is undefined. As p → ∞ the two Logarithmic terms in X add to zero so X → 0.

Eliminating p from the two equations in (15) gives the following relationship between Q and X (after setting an arbitrary constant in the solution to zero):

$$Q = \frac{2}{\pi } \left( {{\text{ArcTan}}\left[ {\sqrt { - 1 + {\text{e}}^{{\frac{\pi X}{\nu }}} } } \right]} \right).$$
(16)

It is straightforward to calculate that s, the share of the variable factor in total output, falls between 0 and 1/2 for any value of ν. Even for the badly behaved Cauchy distribution, the efficiency-distribution approach to the production function yields economically reasonable results. Despite this, the production function arising from the Half-Cauchy surely looks strange to anyone brought up with conventional Cobb–Douglas, CES, or TransLog functional forms.

Distributions with more than one parameter

Other examples of one-parameter distributions could be explored in the same way as the examples given in the previous section. However, the efficiency-distribution representation of production relationships is not restricted to one-parameter cases. It should be emphasized that the same model can be implemented for any probability density function whatsoever.


Example 6—the LogNormal distribution \(g\left( t \right) = \frac{1}{{t \nu \sqrt {2\pi } }} e^{{ - \frac{{\left( { - \mu + Log\left[ t \right]} \right)^{2} }}{{2\nu^{2} }}}} , \, t > 0.\) Integrating Eqs. (1) and (2) yields

$$Q = \frac{1}{2}{\text{Erfc}}\left[ {\frac{{\mu + {\text{Log}}\left[ p \right]}}{\sqrt 2 \nu }} \right]{\text{\,and\,}} X = \frac{1}{2}{\text{e}}^{{\mu + \frac{{\nu^{2} }}{2}}} {\text{Erfc}}\left[ {\frac{{\mu + \nu^{2} + {\text{Log}}\left[ p \right]}}{\sqrt 2 \nu }} \right],$$
(17)

where Erfc[z] is the complementary error function, Erfc[z] = 1 – Erf[z]. For the LogNormal it is possible to eliminate p from the two equations in (1) to obtain the production function:

$$Q = \frac{1}{2}{\text{Erfc}}\left[ {\frac{1}{2}\left( { - \sqrt 2 \nu + 2 {\text{InverseErfc}}\left[ {2{\text{e}}^{{ - \mu - \frac{{\nu^{2} }}{2}}} X} \right]} \right)} \right],$$
(18)

where InverseErfc is the inverse complementary error function obtained as the solution for z in y = Erfc(z). The Erfc function ranges between 0 and 2, so the range of Q is 0 to 1. The domain of the InverseErfc function is (0,2) and its range is (− ∞, + ∞). The maximum value that X can attain is \({e}^{\mu +\frac{{\nu }^{2}}{2}}\), the mean of the LogNormal, so this maximum can become indefinitely large as μ and/or ν become large. As p → 0, both Q and X approach their maximum values, because Erfc[− ∞] = 2.

The variable factor’s share s is given by

$$s = \frac{{{\text{e}}^{{\mu + \frac{{\nu^{2} }}{2}}} p {\text{Erfc}}\left[ {\frac{{\mu + \nu^{2} + {\text{Log}}\left[ p \right]}}{\sqrt 2 \nu }} \right]}}{{{\text{Erfc}}\left[ {\frac{{\mu + {\text{Log}}\left[ p \right]}}{\sqrt 2 \nu }} \right]}}.$$
(19)

The limit of s as p → 0 is zero, and the limit as p → ∞ is 1, so an underlying LogNormal g is consistent with the full range of the possible variable factor’s share.


Example 7—the gamma distribution: \(g\left( t \right) = \frac{{e^{ - t/\beta } t^{ - 1 + \alpha } \beta^{ - \alpha } }}{\Gamma \left[ \alpha \right]}, t > 0.\) Here, Γ[z] is the familiar Gamma function, satisfying \(\Gamma (z)={\int }_{0}^{\infty }{t}^{z-1}{e}^{-t}dt\) provided the real part of z is positive, and Γ[n] = (n – 1)! for any positive integer n. Integrating (1) and (2) yields:

$$Q{ } = { }1 - \frac{{{\Gamma }\left[ {\alpha ,\frac{1}{p\beta }} \right]}}{{{\Gamma }\left[ \alpha \right]}} {\text{\,and\,}} X = \beta \left( {\alpha - \frac{{{\Gamma }\left[ {1 + \alpha , \frac{1}{p\beta }} \right]}}{{{\Gamma }\left[ \alpha \right]}}} \right).$$
(20)

In this expression, the incomplete gamma function Γ[a,z] satisfies \(\Gamma [a,z]={\int }_{z}^{\infty }{t}^{a-1}{e}^{-t}dt\). The software of Mathematica is not able to eliminate p between these two equations. The two equations of (20) constitute an acceptable production function nevertheless. As in the general case, 0 ≤ Q ≤ 1. The mean of the Gamma distribution is α β, so the maximum value of X is α β. The factor share s ranges from 0 to α/(1 + α), with an upper limit of one as α increases.


Example 8—the Beta Prime distribution: \(g\left( t \right) = \frac{{t^{ - 1 + \alpha } \left( {1 + t} \right)^{ - \alpha - \beta } }}{{Beta\left[ {\alpha , \beta } \right]}},t > 0.\) Here, the beta function is given by Beta[α,β] = Γ[α] Γ[β] / Γ[α + β]. For this distribution, integration of (1) and (2) yields:

$$Q = \frac{{p^{ - \alpha } \Gamma \left[ \alpha \right]{\text{Hypergeometric\,2F1\,Regularized}}\left[ {\alpha ,\alpha + \beta ,1 + \alpha , - \frac{1}{p}} \right]}}{{{\text{Beta}}\left[ {\alpha ,\beta } \right]}},$$
(21)

and

$$X = \frac{{p^{ - 1 - \alpha } \Gamma \left[ {1 + \alpha } \right] {\text{Hypergeometric}}2{\text{F}}1{\text{Regularized}}\left[ {1 + \alpha ,\alpha + \beta ,2 + \alpha , - \frac{1}{p}} \right]}}{{{\text{Beta}}\left[ {\alpha ,\beta } \right]}},$$

where the Hypergeometric2F1Regularized function is given by \({ }_{2}{F}_{1}(a,b;c;z)/\Gamma \left(c\right), \mathrm{with} { }_{2}{F}_{1}\left(a,b;c;z\right)=\sum_{k=0}^{\infty }{\left(a\right)}_{k}{\left(b\right)}_{k}/{\left(c\right)}_{k}{z}^{k}/k!.\) The functions Beta, Gamma, and Hypergeometric2F1Regularized are functions that can be manipulated symbolically and evaluated numerically. In the efficiency-distribution model, 0 ≤ Q ≤ 1 as in the other examples, and X has a maximum value of α/(β – 1) provided β > 1. (If β ≤ 1 the distribution has no mean.) The share of X in total output is between 0 and α/(1 + α). Both Q and X exhibit the behavior of ordinary production relationships when analyzed as parametric functions of p.

These examples could be multiplied for as many probability distributions as can be written down. At the time of this writing, Wikipedia lists over 100 different continuous probability distributions, and this does not include mixed distributions, the “lifetime distributions” reported by Sharma et al. (2019), and empirical distributions that do not have a closed-form representation. If the integrals of Eqs. (1) and (2) can be expressed in closed form, (parameterized) production functions are implied. If closed-form integration is not possible, the production functions can still be approximated by numerical methods.

An application to Robotics

The great advantage of estimating production relationships based on the efficiency-distribution model is that it is not necessary to measure capital or the other fixed factors. An interesting application is to calculate, for several different efficiency distributions, how the wage paid to human workers would change in the short run in response to an increase in robotic labor. An empirical example of this calculation was carried out in DeCanio (2016), but only for a few distributions fitted to limited data sets. Other theoretical possibilities (such as the overlapping generations model of the sensitivity of wages to automation by Gasteiger and Prettner (2020). and the studies cited by them) are not explored here.

First, let human and robotic labor be combined into the input X in such a way as to highlight the degree of substitutability between them. The simplest and most natural way to do this is to specify a constant elasticity of substitution (CES) form:

$$X = A \left[ {\theta L^{{\frac{\sigma - 1}{\sigma }}} + \left( {1 - \theta } \right)M^{{\frac{\sigma - 1}{\sigma }}} } \right]^{{\frac{\sigma }{\sigma - 1}}} ,$$
(22)

where σ is the elasticity of substitution between human labor L and robot labor M, and θ is a distribution parameter that can range from 0 to 1. This functional form for X allows the full range of possibilities, from human and robotic labor being perfect substitutes (as σ → ∞) to their being completely complementary (as σ → 0). With this specification, the elasticity of the human wage w with respect to an increase in M can be derived (following the steps laid out in DeCanio (2016)):

$$\begin{gathered} \frac{\partial w}{{\partial M}} = \frac{\partial }{\partial M}\left[ {\frac{\partial Q}{{\partial L}}} \right] = \frac{\partial }{\partial M}\left[ {\frac{dQ}{{dX}} \frac{\partial X}{{\partial L}}} \right] \hfill \\ \quad \;\;\; = \frac{\partial X}{{\partial L}} \frac{\partial }{\partial M}\left( {\frac{dQ}{{dX}}} \right) + \frac{dQ}{{dX}} \frac{{\partial^{2} X}}{\partial L \partial M} \hfill \\ \quad \;\;\; = \frac{\partial X}{{\partial L}} \frac{d}{dX} \left( {\frac{dQ}{{dX}}} \right)\frac{\partial X}{{\partial M}} + \frac{dQ}{{dX}} \frac{{\partial^{2} X}}{\partial L \partial M} \hfill \\ \quad \;\;\; = \frac{\partial X}{{\partial L}}\frac{\partial X}{{\partial M}} \frac{{d^{2} Q}}{{dX^{2} }} + \frac{dQ}{{dX}} \frac{{\partial^{2} X}}{\partial L \partial M}. \hfill \\ \end{gathered}$$
(23)

Substituting from Eqs. (4) and (3), we have

$$= \frac{\partial X}{{\partial L}} \frac{\partial X}{{\partial M}} \left[ {\frac{{ - p^{3} }}{{g\left( {1/p} \right)}}} \right] + p \frac{{\partial^{2} X}}{\partial L \partial M}$$
(24)

and converting to an elasticity yields

$$\epsilon_{wM} = \frac{M}{w} \frac{\partial w}{{\partial M }} = \frac{M}{w}\left[ {\frac{\partial X}{{\partial L}} \frac{\partial X}{{\partial M}} \left( {\frac{{ - p^{3} }}{{g\left( {1/p} \right)}}} \right) + p \frac{{\partial^{2} X}}{\partial L\partial M}} \right].$$
(25)

This expression can be simplified considerably by an appropriate choice of units. In particular, L can be measured so that at the point at which the calculations are made, L = 1, and similarly M can be measured in units such that M = 1. Even though there are difficulties in aggregating “labor” and “robots” into variables, such as L and M, these factors do have natural units that are not subject to all the drawbacks to the aggregation of capital. The CES form is used here as a simple way to focus on the degree of substitutability between human and robot labor, without denying that there may be aggregation problems with both of them. With this simplification

$$\frac{\partial X}{{\partial L}} = A\theta , \frac{\partial X}{{\partial M}} = A\left( {1 - \theta } \right), {\text{\,and\,}} \frac{{\partial^{2} X}}{\partial L\partial M} = - \frac{{A\left( { - 1 + \theta } \right)\theta }}{\sigma },$$
(26)

so that after some further simplification

$$\epsilon_{wM} = \frac{{p\left( {1 - \theta } \right)\left[ {g(1/p) - Ap^{2} \sigma } \right]}}{{g\left( {1/p} \right)\sigma }}.$$
(27)

The first term in the brackets of (27) is always positive and the second term in the brackets is negative, because 0 < σ < ∞, so the sign of \({\epsilon }_{wM}\) in (27) is ambiguous. What can pin this sign down further is the requirement that p and A be such that the share of X in the value of Q is equal to s. Suppose, for example, that the efficiency distribution g is LogNormal. Then p must be the solution of Eq. (19) above for the empirically determined value of the X-share s. Call this value p*. Setting p = p* also fixes Q and X to be Q* and X* from (17), and this in turn determines A from p X/Q = s. Let A* be this value of A and recall that the units were chosen so that L and M are equal to 1 in (26). Therefore, p* A*/Q* = s, or. A* = s Q*/p*. Therefore, the value of the elasticity of w with respect to M is given by Eq. (27) after substituting p* for p and A* for A.

This value of this units-free measure of the sensitivity of wages to the proliferation of robots depends only on the underlying productivity distribution g, the empirically determined X-share s, and σ, the elasticity of substitution between L and M. The point at which \({\epsilon }_{wM}\) = 0 is the point at which its value switches from positive to negative as σ increases (that is, as robots become better substitutes for humans). This value will not depend on θ, because (1 − θ) is always positive in Eq. (27). Thus the critical point at which \({\epsilon }_{wM}\) becomes negative depends only on the empirically determined s, the underlying productivity distribution, and σ.

Recent macroeconomic data suggest that the share of labor compensation in total output has been somewhere between 0.4 and 0.6 in recent years, depending on how the share is measured. (U.S. Bureau of Economic Analysis, 2020; Giandrea and Sprague 2017). Note that the share of labor compensation is not exactly the same as the share of the variable factor(s) X in the model. The current value of “robotic services” is not known, but is surely small relative to the share of human labor. For present illustrative purposes, the labor share is a good proxy for the share of X in total compensation. In addition, it may be pointed out that BLS estimates of the labor share are higher than the BEA estimates, because the BEA estimates include only wages and salaries, while the BLS estimates include proprietors’ imputed income as well as all other forms of compensation, such as commissions, tips, bonuses, severance payments and early retirement buyout payments, regular supplementary allowances, such as housing allowances, the exercising of nonqualified stock options, in-kind earnings such as transit subsidies, meals, and lodging, employer contributions to employee pension and insurance funds and to government social insurance. Giandrea and Sprague discuss details about the issues and methodologies involved.

One more element is needed to make the results for different distributions comparable: The parameter(s) of the distributions need to be chosen to make the means of the distributions equal. A mean of 3.0 was picked to make the results roughly comparable to the empirical examples reported in DeCanio (2016), where the estimated mean of the best-fitting LogNormal distribution was 2.984. Results are shown in Table 1.

Table 1 Critical values of σ* such that \(\epsilon_{wM} < {\text{if}}\sigma > \sigma^{*} ,\) different efficiency distributions, all having mean = 3.0
Full size table

There is some variation in the value of the critical σ across distributions, but all values of σ* calculated for these distributions are below 2. This means that for a not-implausible degree of substitutability between human and robotic labor, the introduction of robots will depress the human wage. One other result stands out in Table 1. The critical value of σ increases with the X-share, for these particular distributions having a mean of approximately 3. This means that, at least for the distributions listed in Table 1, a fall in the X-share makes it likelier that wage rate will decline with the introduction of robotic labor.

It should be noted that several efficiency distributions imply limits on the share of output going to the variable input(s). In part this is a reflection of the original specification of equations of the model—Eqs. (1) and (2). Total output Q is restricted to the interval [0,1], which is not a problem for the model, because units of Q can always be chosen to conform to this range. On the other hand, the maximum value attainable by X is the mean of the distribution. In applications, the efficiency distribution would be fitted to actual data, so if, for example, the empirical value of s were larger than the theoretical maximum allowable for a distribution, that distribution would not be an appropriate one to try to estimate.

It also should also be kept in mind that the share of output s received as compensation by the variable factor is exogenous to the model. The efficiency distribution approach is agnostic about the functional distribution of income, because it makes no attempt to assign “marginal products” to capital or the other fixed factors. The individual productive “cells” that make up the distribution will operate as long as they produce a positive return, but p, the price of the variable factor, is determined by market forces outside the model. It is not necessary to hold the Marxian notion that “the share of labour in output depends on bargaining power” (Robinson 1963, p. 80) to treat s as determined exogenously. Indeed, there are many reasons to question the neoclassical theory of distribution in a modern economy—in large bureaucracies the marginal contribution of any individual is impossible to determine, rent-seeking behavior at all levels can influence compensation, and network positioning can affect compensation of otherwise identical individuals (DeCanio et al., 2000), and the economics of stock prices is plagued by the yet-unresolved equity premium puzzle first examined by Mehra and Prescott (1985).

It is natural to ask if there is any relationship between the parameters of the underlying efficiency distribution and other features of the production relationship. For example, with g LogNormal, σ* is increasing in both s and ν. In general, the relationship is an empirical question, with the answer depending on the distribution that best fits the data. Numerical experimentation with different underlying distributions does not disclose any obvious patterns. In most cases I have examined, σ* increases with the share of the value of output going to the variable factor, but this does not appear to be a universal rule.

Conclusions

The efficiency-distribution model of production introduced by Houthakker in the 1950s is worthy of more attention than it has received. It gives the analysis of production relationships an empirical foundation different from what is ordinarily taken for granted in constructing macroeconomic production functions. The efficiency-distribution approach is quite flexible, allowing for almost any shape of the distribution function. Modern symbolic mathematical software enables calculation of parameterized production functions starting from almost any underlying efficiency distribution. Avoidance of the need to measure capital sidesteps the aggregation problems that have dogged production theory from the time of the Cambridge controversies.

The efficiency-distribution model also offers the possibility of new insights into how the substitution of robot labor for human labor might affect wages. The critical value of the elasticity of substitution between human and robotic labor that marks the point at which more robotic labor lowers the wage is not unrealistically large, and is even less than one for some of the distributions examined. The advance of AI technologies will surely increase overall productivity, but we should not lose sight of the possibility that it may negatively affect some workers. Designing measures to mitigate such negative potential impacts ought to be a priority for policy research.