We provide micro foundations for the global production function (GPF) based on the standard microeconomic model, and we develop a parallel probabilistic model with similar properties. The theoretical and probabilistic models of the GPF are integrated in the context of a technology choice problem. We construct a primitive, named the augmented transformation function, to obtain a GPF and its associated joint distribution that includes the output and labor-saving and capital-saving technological innovations. This type of primitive allows us not only to derive the theoretical GPF but also to consistently build a link between the substitutability microparameters (elasticities) and the probabilistic parameters (correlations). We find that the shape of the GPF is determined by all the relations among technological innovations and output and the way they are combined.
Hinweise
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Introduction
Endogenous technology choice models obtain the shape of the GPF from two primitives: (1) the functional form of the local production function (LPF) and (2) the functional form of the technology frontier (TF), which is calculated as a level curve of a predefined joint distribution of labor- and capital-augmenting ideas (Growiec 2008a, b, 2013, 2018; Jones 2005, 2011). Then, the LPF is maximized subject to the TF to find the GPF, microfounding the shape of the GPF by assuming the probabilistic model that gives rise to the TF. Jones (2005) obtains a Cobb‒Douglas GPF from a Leontief LPF and a TF viewed as a level curve of a joint distribution with independently Pareto-distributed ideas. He finds that the shape of the GPF is determined by the distribution of ideas rather than by the shape of the LPF. Growiec (2008a, b) shows that if ideas are independently Weibull-distributed (with the same shape parameter) or Pareto-distributed and dependent according to the Clayton copula, then the GPF can be of the CES class. He shows that the GPF is determined by the distribution of ideas and by the shape of the LPF. Growiec (2013) devises a specification of the R&D sector, microfounding the assumption of Weibull-distributed ideas by using extreme value theory. He finds that the Weibull distribution is an accurate approximation of the true distribution of ideas needed to obtain GPFs of the CES class. Growiec (2018) assumes homothetic primitives and finds that the TF and the GPF are dual objects, confirming that the GPF depends on the shapes of both the LPF and the TF unless one of them has the Cobb‒Douglas form. He proves that if the idea distribution is homothetic and the copula is additively separable, then the TF is of the CES or Cobb‒Douglas form, which translates into a requirement of Pareto or Weibull marginal idea distributions.
In general, the previous literature develops a new method to obtain the GPF and provides relevant insights into the determinants of its shape. However, the functional forms of the primitives impose implicit restrictions on the relations among output and factor-saving ideas and the way they are combined, which have a strong influence on the final GPF. In fact, such assumptions lead us to disregard some of the interdependencies among output and factor-augmenting ideas (Das 2021), to impose the mathematical structure of very specific extreme probability distributions on the TF and the GPF and to envisage the probabilistic model only as a primitive needed to obtain the GPF shape. We hold that these limitations can be avoided by microfounding the shape of the GPF based on the properties of the firm’s production set, which is considered a primitive datum of the theory (Debreu 1959; Diewert 1973; Mas-Colel, 1995).
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Thus, in this paper, we take a new look at the link between the theoretical and probabilistic models of the GPF in the context of the technology choice problem with given input levels. As opposed to the previous literature, we propose to obtain a microfounded theoretical GPF and link it with a probabilistic model by using a unique primitive: a neoclassical augmented transformation function (ATF), which is an equivalent way of describing the properties of the set of feasible production plans augmented by labor- and capital-saving technological innovations (ideas) (Debreu 1959; Diewert 1973; Acemoglu 2003). First, we use the ATF to calculate economically meaningful functional forms for the local production function (LPF) and the technology frontier (TF). Second, we maximize the LPF subject to the TF to obtain the GPF, microfounding the shape of the GPF by using an ATF whose shape is consistent with the assumptions (restrictions) imposed on the original augmented production set (Diewert 1973). In this context, we show that the resulting GPF can also be used to determine the corresponding set of production plans, and thus, we obtain a duality between our ATF, the set of feasible production plans, and the GPF (Diewert 1973). Finally, to integrate the theoretical model with the probabilistic model of the GPF, we propose to construct the proper joint probability distribution associated with our microfounded ATF by using Sklar’s theorem.
By obtaining the theoretical and probabilistic models of the GPF from a unique ATF (convex, quasiconcave, continuous from above, nonincreasing in output, and nondecreasing in the levels of technological innovations and knowledge) that reflects the structure of the augmented production set (Diewert 1993), we add the following findings to the literature: (1) We calculate a general GPF with the following properties: continuous, quasiconcave, and nondecreasing in inputs and in the level of knowledge. More importantly, we show that the shape of the GPF and the elasticity of substitution are determined by all the relations among the technological innovations and output and the way they are combined, described by the firm’s augmented production set rather than by specific functional forms of the LPF and the TF. (2) By mapping the theoretical ATF to a copula, the resulting probabilistic model also reflects the structure of the production set and the ATF (Diewert 1993). Thus, we can consistently relate the substitutability parameter set to the copula’s dependence parameter set through statistical correlation measures such as Kendall’s tau or Spearman’s rho. Specifically, we link the microeconomic parameters (elasticities) to the probabilistic parameters (correlations) without imposing further restrictions on them, and (3) we illustrate our setup by assuming a primitive (ATF) with the CES property that allows us to find closed forms of all the well-known global production functions (linear, Leontief, Cobb‒Douglas, and CES) and build a link between the substitutability microparameters and the statistical parameters (correlations).
This paper is organized as follows. “The Setup” establishes our approach to deliver global production functions based on a transformation function augmented by labor- and capital-saving technological innovations. We also show that the augmented production set, the ATF, and the GPF are dual objects. In addition, we show how our transformation function can be the basis of constructing a joint distribution that makes our setup empirically relevant. In “Illustration of How to Derive a Nested CES GPF and its Distribution Function Based on an Augmented Transformation Function”, we apply our setup to obtain a novel nested CES GPF with two dependence parameters. “Conclusion” concludes and suggests potential directions of future research.
The Setup
Here, we construct our basic object, an augmented transformation function (ATF), which reflects the microeconomic structure of the set of feasible production plans available for firms in a competitive economy. This function is augmented because there are two types of technological innovations available for given levels of labor \(L\) and capital \(K\) in the domain of each production plan. The ATF reflects the relationship among the outputs and the factor-saving innovations, the availability of such innovations, and the substitutability or complementarity between them. We need to introduce this function because it is a primitive that allows us to theoretically derive the local production function (LPF) of a firm and its technology frontier (TF), from which we can derive all the textbook GPFs,1and it is naturally associated with a trivariate joint distribution function by using Sklar’s theorem, which describes all the probabilistic relations among the technological innovations and output.
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Theoretical Setup: A Transformation Function is Used to Obtain a Deterministic Microfounded GPF
We assume that there exists a set of feasible production plans, augmented by capital- and labor-saving technological innovations and a level of knowledge, from which an augmented transformation function (ATF) is realized. The level of knowledge determines the possible pairs of labor- and capital-saving technological innovations. We treat technological innovations as scarce goods that are substitutable or complementary like any other good in the economy. The higher the level of knowledge is, the more possible combinations of technological innovations. We show that the ATF embeds an LPF and a TF that are the basis for deriving a GPF.
Let \(j\in J\subseteq {\mathbb{R}}\) index a discrete set of homogenous firms in a competitive economy. Let \(i\in I\subseteq {\mathbb{R}}\) index a feasible production plan (i.e., a pair of labor-saving and capital-saving technological innovations \({a}_{i}\) and \({b}_{i}\), respectively, an exogenous level of knowledge \({N}_{i}\), the input levels \({L}_{i}\) and \({K}_{i}\), and the output level2\({\widetilde{Y}}_{i}\)), which is available for the j-th firm at any given point in time \(t\in T=\left[0,\tau \right]\). Note that the level of knowledge determines the feasible set of possible pairs of labor- and capital-saving technological innovations. In addition, a multiplicity of pairs \({(a}_{i}, {b}_{i})\) in this set can attain the same level of knowledge \({N}_{i}\).
An augmented production plan\({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{{\widetilde{Y}}_{i},N}_{i})\) is feasible if the combination of technological innovations \(\left({a}_{i},{b}_{i}\right)\) available for the level of knowledge \({N}_{i}\) and the levels of inputs \({L}_{i}\) and \({K}_{i}\) can produce the level of output \({\widetilde{Y}}_{i}\).
Assumption 2.1.1
Let \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{{\widetilde{Y}}_{i},N}_{i})\) represent a single production plan augmented by \({a}_{i},{b}_{i}\), and \({N}_{i}\), and let \({B}_{j}\subseteq {\mathbb{R}}^{6}\) represent the set of augmented feasible production plans for the \(j\)-th firm at time \(t\), where \({\widetilde{Y}}_{i}\) is the output and the available technology pair of innovations \({(a}_{i},{b}_{i})\) is associated with the levels of labor and capital, \({L}_{i}\) and \({K}_{i}\), respectively. Let \({B}_{j}\) be a nonempty, closed, and convex subset of the nonnegative orthant in \({\mathbb{R}}^{6}\) that satisfies the following properties:
a.
If \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{{\widetilde{Y}}_{i},N}_{i})\in {B}_{j}\) and \({(a}_{l}, {L}_{l},{b}_{l},{K}_{l},{N}_{l})\ge {(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{N}_{i}),\) then \({(a}_{l}, {L}_{l},{b}_{l},{K}_{l},{{\widetilde{Y}}_{i},N}_{l})\in {B}_{j}\),3
b.
\(0=\left(0,0,\text{0,0},\text{0,0}\right)\in {B}_{j}\), and if for an augmented production plan \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{{\widetilde{Y}}_{i},N}_{i})\in {B}_{j}\), \({a}_{i}=0\), \({L}_{i}=0\), \({b}_{i}=0\), \({K}_{i}=0\) or \({N}_{i}=0\), then \({\widetilde{Y}}_{i}=0\) (possibility of inaction and no free lunch),
c.
If \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{{\widetilde{Y}}_{i},N}_{i})\in {B}_{j}\) and \(0\le \widetilde{{Y}_{\zeta }}\le {\widetilde{Y}}_{i},\) then \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},\widetilde{{Y}_{\zeta }},{N}_{i})\in {B}_{j}\), and
d.
For each vector\({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{N}_{i})\ge 0,\), the set \(\left\{{{\widetilde{Y}}_{i} :(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\in {B}_{j}\right\}\) of feasible outputs is bounded from above.
Assumption 2.1.1 not only states the substitutability or complementarity between \({L}_{i}\) and \({K}_{i}\) and \({a}_{i}\) and \({b}_{i}\), respectively, but also describes the relationship of such inputs and the level of the available factor-saving technological innovations and output. Properties (i) and (iii) indicate that if we can produce a certain quantity of output with a given amount of inputs and technological innovations, then higher quantities can produce at least the same level of output. Property (ii) indicates that firms can decide not to produce and not incur costs for production and that all inputs and technological innovations are necessary for production. Property (iv) indicates that with finite levels of inputs and technological innovations, we can produce only finite amounts of output.
Note that we might use convex analysis to define and solve the firm’s technology choice problem, but it could be unmanageable. To have a functional representation of \({B}_{j}\), we define an augmented transformation function that indicates if a feasible augmented production plan \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\) of a is technologically efficient for a profit-maximizing firm.
Definition 2.1.1
Let \({B}_{j}\) satisfy Assumption 2.1.1; then, \(T:{\mathbb{R}}_{+}^{6}\to {\mathbb{R}}\) defined on the vector \({{\varvec{z}}}_{{\varvec{i}}}={(a}_{i}, {L}_{i},{b}_{i},{K}_{i}, {\widetilde{Y}}_{i},{N}_{i})\) as
is an augmented transformation function (ATF). Note that \({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\ge 0\)4 and \(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})=0\) if and only if \({\widetilde{Y}}_{i}=\underset{{\widetilde{Y}}_{i}}{\mathit{max}}\left\{{\widetilde{Y}}_{i}:{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\in {B}_{j}\right\}.\)
Then, an augmented transformation function inherits the mathematical properties of the set \({B}_{j}\), such as the shape of its level surfaces and its quasiconcavity and continuity. To analyze the firm’s technology choice problem, we are interested in choosing the optimal levels of feasible technological innovations \({a}_{i}\) and \({b}_{i}\) that allow the firm to obtain the maximum amount of production for given input levels.
Proposition 2.1.1
If the transformation function\(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\)satisfies Definition 2.1.1, then it is monotone, concave, quasiconcave, continuous from above, nonincreasing in\({\widetilde{Y}}_{i}\), and nondecreasing in\({a}_{i},{L}_{i},{b}_{i},{K}_{i}\), and\({N}_{i}\).
Proof
See Appendix A.1.
Definition 2.1.2
Given the augmented transformation function (ATF) \(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\), we can define the following set:
If\(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\)satisfies the conditions given in Proposition 2.1.1, the set \({B}_{j}^{^{\prime}}\), defined in 2.1.2, satisfies the conditions of a proper set of augmented feasible production plans.
Then, we have proven that there exists a duality between the set of augmented feasible production plans, \({B}_{j}\), and the augmented transformation function, \(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\), because we can define each object from the other one, maintaining their properties.
Now, we suppose that \(L\) and \(K\) are fixed for a given time \(t\) and a representative firm. As a result, subscripts are not provided for these variables in the remainder of the derivations. Thus, we can write the transformation function as \(T\left({a}_{i},{b}_{i},{\widetilde{Y}}_{i},{N}_{i}\right):=T\left({a}_{i},{L,b}_{i},K,{\widetilde{Y}}_{i},{N}_{i}\right)\). Figure 1 shows an augmented transformation function for a given level of knowledge \(N\).
Fig. 1
Surface level of the set of augmented production plans \({B}_{j}\) and the augmented transformation function \(T\left({a}_{i}, {b}_{i},{\widetilde{Y}}_{i},N\right)\) for given levels of \(L,K,\) and \(N\)
×
Then, Assumption 2.1.1 and Definition 2.1.1 imply that a representative firm will choose to produce at the maximum of \(\left\{{\widetilde{Y}}_{i} :T\left({a}_{i},{b}_{i},{\widetilde{Y}}_{i},{N}_{i}\right)\ge 0\right\}\) for each pair \({(a}_{i}, {b}_{i})\ge 0\) at given levels of \(L\) and\(K\). Then, an LPF can be obtained from the transformation function. Note that the level of knowledge \(N\) at time \(t\) is a given exogenous variable because it is unrelated to the level of production in the short run. Consequently, the LPF can be considered a function that does not depend on the level of knowledge\(N\). In addition, the maximum of \(T\) always exists due to Weierstrass’s theorem, as it is a continuous function over the range of \({\widetilde{Y}}_{i}\) and the set \(\left\{{\widetilde{Y}}_{i} :T\left({a}_{i},{b}_{i},{\widetilde{Y}}_{i}\right)\ge 0\right\}\) of feasible outputs is a compact set.5
Definition 2.1.3
The LPF gives the maximum amount of output \({\widetilde{Y}}_{i}\) that can be produced in the set \({B}_{j}\) for given levels of \(L\) and \(K\), i.e.,
For a certain time \(t\), the level of \({\widetilde{Y}}_{i}\) does not influence the exogenous level of knowledge; then, we define the technology frontier (TF) as the set of combinations of technology pairs \(\left({a}_{i},{b}_{i}\right)\) and the level of knowledge \(N\) such that
Definition 2.1.1 and Proposition 2.1.1 imply that for given levels of \(L\) and \(K\), the ATF has embedded in it the following microeconomic information: (1) the availability of labor- and capital-saving technological innovations \({a}_{i}\) and \({b}_{i}\); (2) the existing substitutability or complementarity between the technological innovations and their relationships; and (3) the relationship among technological innovations and output through the LPF for given inputs in the economy. Thus, we can consistently obtain an LPF and a TF from our ATF for the representative firm that chooses an optimal technology pair \(\left({a}_{i},{b}_{i}\right)\) from \({B}_{j}\). As opposed to the previous literature, we derive the LPF and the TF from a unique primitive, the ATF. In contrast, in the literature, it has been assumed that the LPF and TF already exist. Our model has a different nature because we do not need to use extreme value theory (probability theory) to provide a microfounded model. Then, the GPF is obtained by maximizing the former subject to the latter.
Now, for a given level of knowledge N and levels of \(L\) and \(K\), we can also define a global production function (GPF) [see, e.g., (Growiec 2008a; Jones 2005)] for the representative firm as the maximum amount of output \({Y}_{i}^{*}\) that results from solving the problem of maximizing the LPF subject to the TF (as in Fig. 2). That is,
Fig. 2
Derivation of the LPF, TF, and GPF from the augmented transformation function
×
\({Y}_{i}^{*}(L,K,N):=\underset{{a}_{i},{b}_{i}}{\mathit{max}}f\left({a}_{i},{b}_{i};L,K\right)\) subject to \({H}_{N}\left({a}_{i},{b}_{i}\right)\).
Proposition 2.1.3
The GPF \({Y}_{i}^{*}(L,K,N)\) satisfies the following properties: monotone, nondecreasing in inputs and level of knowledge, quasiconcave, and satisfying\({Y}_{i}^{*}\left(\text{0,0},0\right)=0.\)
Proof
See Appendix A.1.
Definition 2.1.5
Given the GPF, \({Y}_{i}^{*}(L,K,N)\), that satisfies the properties of Proposition 2.1.2, we can define the set \({B}_{j}^{{^{\prime}}{^{\prime}}}\) as follows:
If \({Y}_{i}^{*}(L,K,N)\) satisfies the conditions in Proposition 2.1.4, the set \({B}_{j}^{{^{\prime}}{^{\prime}}}\) defined in 2.1.5 satisfies the properties of a proper augmented feasible production set in Assumption 2.1.1.
Proof
See Appendix A.1.
Note that there exists a duality defined among \({B}_{j}\), the ATF and the GPF.
Probabilistic Setup: Linking the Deterministic Microfounded ATF with its Corresponding Probability Function
Now, we construct a probabilistic version of our augmented transformation function, which can be achieved using Sklar’s theorem. Specifically, we construct a joint distribution function of \({a}_{i},{b}_{i}\),\({\widetilde{Y}}_{i}\), and \({N}_{i}\) for given levels of \(L\) and \(K\) by using a multivariate copula, which allows us to link the availability and substitutability microparameters to their probabilistic counterparts (location and correlation probabilistic parameters).
Assumption 2.2.1
There exists a random vector \(\left({g}_{a}\left({a}_{i}\right),{g}_{b}\left({b}_{i}\right),{g}_{ \widetilde{Y}}\left({\widetilde{Y}}_{i}\right),{g}_{N}\left({N}_{i}\right)\right)\) into a probability space \(\left({\Omega }_{i}, {\mathcal{F}}_{i}, {P}_{i}\right)\) (Spanos, 1986) that includes monotone transformations of both labor- and capital-saving technological innovations \(\left({a}_{i}, {b}_{i}\right)\), output \({\widetilde{Y}}_{i}\), and the level of knowledge \({N}_{i}\), which are available to the \(\text{j}\)-th firm at a given point in time for given levels of \(L\) and \(K\).
Let \(\left({g}_{a}\left(a\right),{g}_{b}\left(b\right),{g}_{ \widetilde{Y}}\left(\widetilde{Y}\right),{g}_{N}\left(N\right)\right)\) be a realization of this vector. Then, this assumption implies that there exist marginal distribution functions \(U,V\),\(W\), and \(X\) that describe its probabilistic behavior. It also implies that there exists a joint distribution of the random vector \(\left({g}_{a}\left({a}_{i}\right),{g}_{b}\left({b}_{i}\right), {g}_{\widetilde{Y}}\left(\widetilde{{Y}_{i}}\right),{g}_{N}\left({N}_{i}\right)\right),\) which is determined by a copula \(C\left(U,V,W,X\right)\)\((\text{Nelsen}, 2006)\). Notably, our setup is general enough that it allows for more than one substitutability parameter that might determine the shape of the GPF. First, we define a 4-dimensional copula as follows:
Definition 2.2.1
A \(4\)-dimensional copula of the random vector \(\left({g}_{a}\left(a\right),{g}_{b}\left(b\right),{g}_{\widetilde{Y}}\left(\widetilde{Y}\right),{g}_{N}\left(N\right)\right)\) is a continuous and nondecreasing real-valued function \(C\), parametrized with the set of parameters \(\Theta\), from \({\mathcal{l}}^{4}=\left[\text{0,1}\right]\times \left[\text{0,1}\right]\times \left[\text{0,1}\right]\times \left[\text{0,1}\right]\) to \(\mathcal{l}=\left[\text{0,1}\right]\) that links the marginal distributions \(U=U\left({g}_{a}\left(a\right);{\Theta }_{a}\right), V=V\left({g}_{b}\left(b\right);{\Theta }_{b}\right)\),\(W=W\left({g}_{\widetilde{Y}}\left(\widetilde{Y}\right);{\Theta }_{\widetilde{Y}}\right)\), and \(X=X\left({g}_{N}\left(N\right);{\Theta }_{N}\right)\) together by using Sklar’s theorem to obtain the joint distribution.
where \({\Theta }_{a}\),\({\Theta }_{b}\),\({\Theta }_{\widetilde{Y}}\), and \({\Theta }_{N}\) are the margins’ parameter sets and where \(\Theta\) is the copula’s parameter set that describes the probabilistic relationship, i.e., the probabilistic dependence structure, between the involved random variables.
Theorem 2.2.1
Using Assumptions2.1.1and 2.2.1 and assuming a nonempty set\({B}_{j}\ne \left\{0\right\}\), for a particular realization\(\left({g}_{a}\left(a\right),{g}_{b}\left(b\right),{g}_{\widetilde{Y}}\left(\widetilde{Y}\right),{g}_{N}\left(N\right)\right)\)of the random vector\(\left({g}_{a}\left({a}_{i}\right),{g}_{b}\left({b}_{i}\right),{g}_{ \widetilde{Y}}\left({\widetilde{Y}}_{i}\right),{g}_{N}\left({N}_{i}\right)\right)\)6and a given exogenous level of knowledge\(N\), we find that the following monotonically increasing transformation of our ATF7
where\(U\left({g}_{a}\left(a\right);{\Theta }_{a}\right), V\left({g}_{b}\left(b\right);{\Theta }_{b}\right),\)and\(X\left({g}_{N}\left(N\right);{\Theta }_{N}\right)\)are distributions of the monotonically increasing transformations of a, b, and N, respectively, and \(W\left({g}_{\widetilde{Y}}\left(\widetilde{Y}\right);{\Theta }_{\widetilde{Y}}\right)\)is the distribution of a monotonically decreasing transformation of \(\widetilde{Y}\). In addition, by using Sklar’s theorem, there exists a 4-dimensional distribution function with margins \(U, V, W,\)and\(X\)given by
which is a probabilistic version of our deterministic augmented transformation function\({T}_{C}\left({g}_{a}\left(a\right),{g}_{b}\left(b\right),{g}_{\widetilde{Y}}\left(\widetilde{Y}\right),{g}_{N}\left(N\right)\right)\)and has the properties defined in Assumption 2.1.1.
Proof
See Appendix A.1.
Thus, as shown in Fig. 1, we have constructed a probabilistic version of our transformation function not only by building a particular copula with margins for \({g}_{a}\left(a\right),{g}_{b}\left(b\right),{g}_{\widetilde{Y}}\left(\widetilde{Y}\right),\) and \({g}_{N}\left(N\right)\) but also by selecting specific values for the copula’s parameter set. These parameters must be consistent with the theoretical parameters of the transformation function. Once we have a probabilistic version of the transformation function, we can derive the LPF and the TF from it by following the procedure indicated in the next corollary.
Corollary 2.2.1
The LPF and the TF can be obtained by using Definition2.2.1and Eq. (1), as follows:
(a)
The functional form of a deterministic LPF can be obtained as follows. By marginalizing\(C\left(a,b, \widetilde{Y},N\right)\)with respect to the random variable\({N}_{i}\)\((X=1\))8, using the continuous and nondecreasing transformations\({h}_{a}\)and\({h}_{b}\)of\({a}_{i}\), and\({b}_{i}\), respectively, and a decreasing transformation\({h}_{\widetilde{Y}}\)of\({\widetilde{Y}}_{i}\),9assuming specific values for the copula dependence parameter set\(\Theta\),fixing the probability\({F}_{abY}\left({h}_{a}\left(a\right),{h}_{b}\left(b\right),{h}_{\widetilde{Y}}\left(\widetilde{Y}\right);\Theta \right)={C}_{ab\widetilde{Y}}\left(a,b, \widetilde{Y}\right)\)at the level\({P}_{0}\)in Eq. (1), and rearranging the resulting equation, the following result is obtained:
The deterministic TF can be obtained by marginalizing\(C\left(a,b, \widetilde{Y},N\right)\)with respect to the random variable\({\widetilde{Y}}_{i}\)\(\left(W=1\right)\),11assuming continuous and decreasing transformations\({q}_{a}\)and\({q}_{b}\)and an increasing transformation\({q}_{N}\)12of the variables\({a}_{i}\), \({b}_{i}\), and\({N}_{i}\), respectively, taking specific values for the copula parameter set\(\Theta\), fixing the probability\({F}_{abN}\left({q}_{a}\left(a\right),{q}_{b}\left(b\right),{q}_{N}\left(N\right);\theta \right)={C}_{abN}\left(a,b,N\right)\)at a constant level\({P}_{1}\), and rearranging the related equation in terms of\(a\)and\(b\)as follows:
A relevant case is one in which we assume that joint distribution is given by a generalized Archimedean copula family, with a probabilistic dependence parameter set \(\Theta\), as shown below.
Proposition 2.2.1
Based on Theorem 2.1.1 and Corollary 2.1.1, the shape of the LPF and its elasticity of substitution\(\sigma\)are determined not only by the substitutability parameter set \(\Theta\), i.e., the copula’s dependence parameter set in the probabilistic space but also by the parameters that reflect theavailability of technological innovations in the economy \({\Theta }_{a}\)and\({\Theta }_{b}\), i.e., the margins’ parameters in the probabilistic space. In addition, if the marginalized copula\({C}_{abN}\)that represents the TF belongs to the family of generalized Archimedean copulas, the substitutability parameter set\(\Theta\)can be mapped to the dependence parameter set through a statistical correlation measure\(\mu =\mu \left(\Theta \right)\)such as either Kendall’s tau or Spearman’s rho.
Proof
See Appendix A.1.
Corollary 2.2.2
Based on Proposition 2.\(2.1\), let \(C\) be the copula that represents our transformation function. If the marginalized copula function \({C}_{ab}\left(U,V;{\Theta }_{abN}\right)={C}_{abN}\left(U,V,1;{\Theta }_{abN}\right)\), which determines the joint distribution of \(a\) and \(b\), belongs to a generalized Archimedean family (with generator \(\psi\)); then, we can write the transformation function as
Additionally, if the derivatives of the functions \(U\) and \(V\) are separable as
$${U}^{{^{\prime}}}\left(a,L;{\Theta }_{a}\right)={U}_{1}\left(a;{\Theta }_{a}\right){U}_{2}\left(L;\Theta \right)\text{ and }{V}^{{^{\prime}}}\left(b,K;{\Theta }_{b}\right)={V}_{1}\left(b;{\Theta }_{b}\right){V}_{2}\left(K;\Theta \right),$$
then the elasticity of substitution \(\sigma\) depends only on \(\Theta\) through a dependence measure \(\mu\); i.e., \(\sigma =r\left(\mu \left(\Theta \right)\right)\), where \(r\) is a real function.
Additionally, if the marginalized copula \({C}_{ab}\left(U,V;{\Theta }_{abN}\right)\) is described by a single dependence parameter \(\theta\)\(\left({\Theta }_{abN}=\left\{\theta \right\}\right)\), then \(\theta\) comes out as the substitutability parameter of the derived LPF and constitutes the unique link between the elasticity of substitution \(\sigma =\sigma \left(\theta \right)\) and the probabilistic correlation measure \(\mu =\mu \left(\theta \right)\), as follows:
(a)
\(\sigma =\sigma \left({m}^{-1}\left(\mu \right)\right),\) where \(\mu =m\left(\theta \right)\);
(b)
\(\mu =\mu \left({n}^{-1}\left(\sigma \right)\right),\) where \(\sigma =n\left(\theta \right)\).
Proof
See Appendix A.1.
The construction of the transformation function in Corollary 2.2.1 is a relevant case in production theory.
Next, as we have deterministic versions of the LPF and the TF, we can obtain the GPF from the firm’s technology choice problem (Jones and Manuelli, 1990):
such that \(H\left(a,b\right)={X}_{N}^{-1}\left({C}_{X}^{-1}\left(U\left({q}_{a}\left(a\right);{\Theta }_{a}\right), V\left({q}_{b}\left(b\right);{\Theta }_{b}\right)\right),{ P}_{1}\right)={A}_{N}\), where \(N\) represents the given level of knowledge at time t.
The solution to (2) yields different closed functional forms of the textbook GPFs, depending on the following values contained in our suitable monotonic transformations of the margins \(U,V,W,\) and \(X\): the substitutability parameters, the associated copula parameter set \(\Theta\); the availability parameters \({\Theta }_{a}, {\Theta }_{b}\), and \({\Theta }_{N}\); the output parameters \({\Theta }_{\widetilde{Y}}\); and the associated tail-thickness parameters.
Proposition 2.2.2
Given Assumption 2.1.1 and the resulting microfounded LPF and TF, the firm’s technology choice problem (2) has a unique solution if and only if the LPF and the TF are tangential at the point \({[a}^{*}\left(L,K;{\Theta }_{a},{\Theta }_{b},{\Theta }_{\widetilde{Y}},{\Theta }_{N},\Theta \right)\), \({b}^{*}(L,K;{\Theta }_{a},{\Theta }_{b},{\Theta }_{\widetilde{Y}},{\Theta }_{N},\Theta )]\), which can occur only if the LPF is convex and the TF is concave or if both are convex but the LPF is more sharply curved than the TF in the feasible augmented technology set for a given \(L\) and \(K\). We obtain the following general GPF \({Y}^{*}={Y}^{*}\left(L, K, N;{\Theta }_{a},{\Theta }_{b},{\Theta }_{\widetilde{Y}},{\Theta }_{N},\Theta \right)\) by substituting the optimal values of \({a}^{*}\) and \({b}^{*}\) into the LPF.
Proof
See Appendix A.1.
Note that as the GPF inherits the parameters of the LPF and the TF, the elasticity of substitution \({\sigma }_{GPF}\) in the GPF is determined by the substitutability parameter set \(\Theta\), i.e., the copula’s dependence parameter set in the probabilistic space and by the availability parameter set \({\Theta }_{a}\), \({\Theta }_{b}\), \({\Theta }_{\widetilde{Y}}\), and \({\Theta }_{N}\), i.e., the margins’ parameters set in the probabilistic space. In other words,
This setup is microfounded because if we wish to obtain a GPF, we assume a general ATF by describing the substitutability or complementarity between the technological innovations and their relationship with the output level. Next, we can associate this microeconomic structure with a particular probability distribution (i.e., a copula) with its margins and connect it to the data with observables (i.e., patents and output), without having data on \(L\) and \(K\), to validate the empirical predictions of the model. Specifically, from Corollary 2.2.2, we can derive formulas that relate the substitutability between the innovations and the relationship between the factors and output to a particular probabilistic dependence measure, such as Kendall’s tau correlation coefficient, and the availability parameters to tail-thickness parameters.
Illustration of How to Derive a Nested CES GPF and its Distribution Function Based on an Augmented Transformation Function
Next, we use our setup to derive a novel nested CES GPF that inherits its properties from a well-behaved ATF, augmented by labor- and capital-saving technological innovations. The derived nested CES GPF is determined both theoretically and probabilistically by the parameters \(\theta\) and \(\delta\), which are defined in the parameter space \(\Theta\). Here, \(\theta\) describes the substitutability (dependence) between the two types of technological innovations, whereas \(\delta\) characterizes the combined effect of the pair of technological innovations on the output. This GPF is general enough to derive textbook GPFs (Leontief, Cobb–Douglas, CES, and linear). We build the probabilistic version of the transformation function (ATF), \(T\), which is useful to link the microparameters of our nested CES GPF with its probabilistic counterparts and approximate its empirical shape. Finally, such a link allows us to describe a method to compute the microparameters of the GPF by simply estimating two Kendall’s tau correlation coefficients: (1) between the two types of technological innovations and (2) between the combined pair of technological innovations and output.
that satisfies Definition 2.1.1 and Proposition 2.1.1, where \(\left(a,L,b,K,\widetilde{Y}, N\right)\gg 0\), which is described by the substitutability parameters \(\theta\) and \(\delta\) and the availability parameters \(\alpha\) and \(\beta\) for output \(\widetilde{Y}\) and fixed levels of \(L\), \(K\), and \(N\). In addition, based on Theorem 2.2.1, we can construct a probabilistic version of this transformation function T (which is the basis to build a nested CES function) by using the following nested \(4\)-variate copula14:
This copula describes the joint behavior of the random variables \({a}_{i}{, b}_{i}, {\widetilde{Y}}_{i}\), and \({N}_{i}\) through the dependence parameters \(\delta\) and \(\theta\), whose domains are defined on \(\left[-1,\infty \right]\). If we marginalize with respect to \({N}_{i}\)\(\left(X=1\right)\) in Eq. (3), we obtain the trivariate copula that describes the joint behavior of the random variables \({a}_{i}\), \({b}_{i}\), and \({\widetilde{Y}}_{i}\). Similarly, if we marginalize with respect to \({Y}_{i} (W=1)\) in Eq. (3), we obtain the following trivariate copula that describes the joint behavior of the random variables \({a}_{i}\), \({b}_{i}\), and \({N}_{i}\).
Next, let us assume that the labor- and capital-saving technological innovations, knowledge level, and output are random draws from the marginal Pareto distributions (Growiec 2008a; Jones 2005); then, the functional form is given by
Using Corollary 2.2.1 and Assumption 3.1 and assuming consistent transformations \({h}_{a}\left({a}_{i}\right)\),\({{h}_{b}(b}_{i})\), and \({h}_{\widetilde{Y}}\left({Y}_{i}\right)\) of the random variables \({a}_{i},\)\({b}_{i}\), and \({Y}_{i}\), respectively, and replacing \({\widetilde{Y}}_{i}\) with \({Y}_{i}\), we obtain a trivariate probabilistic version of the LPF
Corollary 2.2.1 guarantees that by using transformations \({q}_{a}\left({a}_{i}\right)\),\({{q}_{b}(b}_{i})\), and \({q}_{N}\left({N}_{i}\right)\) of the random variables \({a}_{i},\)\({b}_{i}\), and \({N}_{i}\), respectively, we can obtain a probabilistic version of the TF as
Note that random variables \({a}_{i}\) and \({b}_{i}\) are strongly correlated through the copula parameter \(\theta\) in the innovative process. Then, they are mutually correlated with \({Y}_{i}\) when we consider the full production process through the parameter \(\delta\) and are independent of \(N\). These relations are statistically and microeconomically meaningful.
Lemma 3.1
Deterministic versions of the LPF and TF can be derived by using Assumption 3.1 and Corollary 2.2.1 for given levels of \(L\) and \(K\). First, by fixing probability level \({P}_{0}\) in Eq. (4), defining \({\gamma }_{a}^{-\alpha \theta }\equiv {\gamma }_{L}^{-\alpha \theta }\left(1+{L}^{-\theta }\right) and {\gamma }_{b}^{-\beta \theta }\equiv {\gamma }_{K}^{-\beta \theta }\left(1+{K}^{-\theta }\right)\) and choosing \({\left({a\gamma }_{L}\right)}^{-\alpha \theta }\)+\({\left({b\gamma }_{K}\right)}^{-\beta \theta }=1,\)15 we obtain the following CES LPF:
where \(A=1+{P}_{0}^{-\delta }>1,\)\({A}_{N}=1+{P}_{1}^{-\theta }{\left({\gamma }_{N}N\right)}^{\varepsilon \theta }\), and \(-1\le \delta ,\theta \le \infty .\)
Proof
See Appendix A.2.2.
Next, the firm’s technology choice problem consists of maximizing (6) given (7); then, its solution is given in the next lemma.
Lemma 3.2
Based on Proposition 2.2.1and Eqs. (6) and (7), the producer’s optimization problem has a unique solution\({(a}^{*},{b}^{*})\)that is substituted into the LPF to obtain the following optimal CES GPF:
Relationships Between the Microeconomic and Probabilistic Models
Note that the copula dependence parameters \(\delta\) and \(\theta\) come out directly as the microeconomic parameters that determine the shape of the nested CES GPF in Eq. (7). These two parameters also describe the dependence between the output and the combined pair of innovations \(\delta\) and between the innovations themselves \(\theta\). Thus, the set of microparameters \((\theta\),\(\delta )\) constitutes a link between their corresponding copulas, which are defined on the probability space \(\left(\Omega ,\mathcal{F},P\right)\), and the deterministic TF and the LPF, which are defined on the augmented feasible plans set \({B}_{j}\). To obtain a simpler interpretation of the LPF and TF, from a probabilistic point of view, the dependence parameters \(\delta\) and \(\theta\) can be mapped to the interval \(\left[-\text{1,1}\right]\) by using two dependence measures: the dependence between \({a}_{i}\) and \({b}_{\text{I}}\) and the dependence between \(\left({a}_{i}{,b}_{\text{i}}\right)\) and \({Y}_{i}\).
As we use a nested Clayton copula with generator \(\psi \left(t\right)\) for the production model, we represent the copula as \(C\left(U,V,W\right)={\psi }_{2}^{-1}[{\psi }_{2}\left(W\right)+{\psi }_{2}({\psi }_{1}^{-1}\left[{\psi }_{1}\left(U\right)+{\psi }_{1}\left(V\right)\right])]\), with correlation parameters \(\theta\) and \(\delta\). Then, the dependence measure Kendall’s tau can be obtained by using the following formula
Based on Assumption 3.1, Lemma 3.1, and Eqs. (4), (5), and (6), we can link the parameter \({\sigma }_{Yab}\), which describes the relationship between the output (\(Y\)) and the effect of both technological innovations (\({\left[{U}^{-\theta }{+V}^{-\theta }-1\right]}^{-\frac{1}{\theta }}\)), with the parameter \(\delta\) through Kendall’s tau correlation coefficient -1 \(\le {\tau }_{wuv}\le 1\), and the elasticity of substitution \({\sigma }_{ab}\) between \({a}_{i}\) and \({b}_{I}\) in the LPF with the substitutability parameter \(\theta\) through Kendall’s tau correlation coefficient \(-1\le {\tau }_{uv}\le 1\). Thus, we can obtain empirical estimates of the correlations \({\tau }_{uv}\) and \({\tau }_{wuv}\) by using data on labor- and capital-saving technological innovations (i.e., patents) and output data to compute the numerical values of the parameters \(\delta\) and \(\theta\) and the elasticities \({\sigma }_{ab}=\frac{1}{1+\theta }\) and \({\sigma }_{Yab}=\frac{1}{1-\eta \delta }\). Conversely, if we assume values of \({\sigma }_{ab}\) and \({\sigma }_{Yab}\), we can obtain estimates of the correlation between the two types of innovations \({ \tau }_{uv}\) and between output and innovations\({\tau }_{wuv}\).
Then, we have the following relationships:
(a)
Elasticity of substitution \({\sigma }_{ab}=\frac{1-{\tau }_{uv}}{1+{\tau }_{uv}},\) where \(\theta =\frac{2{\tau }_{uv}}{1-{\tau }_{uv}}\).
Elasticity of substitution \({\sigma }_{Yab}=\frac{1-{\tau }_{wuv}}{1-\left(1+2\eta \right){\tau }_{wuv}},\) where \(\delta =\frac{2{\tau }_{wuv}}{1-{\tau }_{wuv}}\).
(d)
Dependence measure between output and innovations (Kendall’s tau)\({\tau }_{wuv}=\frac{\delta }{2+\delta }.\)
Proof
See Appendix A.2.2.
Note that as we know the joint distribution of \({a}_{i},\)\({b}_{i}\) and \({Y}_{i}\) defined in Eq. (4), we can also recover the values of \({\tau }_{uv}\), \({\tau }_{wuv}\), \({\sigma }_{ab}\), and \({\sigma }_{Yab}\) by using the formulas in (a), (b), (c), and (d). The link between \({\sigma }_{ab}\) and \({\tau }_{uv}\), i.e., between the probabilistic space (\(\Omega ,\mathcal{F},P\)) and the set of augmented production plans \({B}_{j}\), is the substitutability parameter \(\theta\), with \(\left[-1,\infty \right]\) as its range of values. Thus, we can vary the values of the parameter \(\theta\) to obtain all the nested classic textbook production functions. In addition, the shape of the GPF and its elasticity of substitution depend not only on the substitutability parameters, i.e., the copula’s dependence parameters and statistically approximated by Kendall’s tau correlation coefficient but also on the availability parameters, i.e., functional forms of the margins and the values of their location parameters.
Note that the link between \({\sigma }_{KL}\) and \({\tau }_{c}\) is the substitutability parameter \(\theta\). The range of values of the substitutability parameter in our transformation function is \([-1,\infty ]\); thus, we can vary the values of the parameter \(\theta\) to obtain all the classic production functions. We confirm that the shape of the GPF and its elasticity of substitution depend not only on the substitutability parameter, which is reflected by the copula dependence parameter but also on the availability parameters, i.e., the functional forms of the margins and the values of their parameters. Table 1 shows a summary of the derived GPFs when the LPF and the TF share the same set of parameter values \(-1\le \theta \le \infty\). Note that specific values of \(\theta\) are associated with different values of the dependence measure \({\tau }_{c}\), the elasticity of substitution \({\sigma }_{KL}\), and the shape of the GPF.16
Table 1
GPFs derived from a transformation function \((-1\le \theta \le \infty\)); Source: Author calculations
The shape of the GPF is obtained from a unique ATF that reflects the structure of the set of production plans augmented by labor- and capital-saving technological innovations, which is the primitive datum in the theory. We show that the shape of the GPF is determined by all the relations among technological innovations and output and the way they are combined, rather than by the shape of two primitives, the LPF and the TF. By mapping the ATF to its corresponding copula, using Sklar’s theorem, we ensure that the probabilistic model of the GPF has the same structure as the production set and, consequently, we can consistently link the elasticities and the correlations. Thus, we provide micro foundations for the GPF exclusively based on the microeconomic model, and we develop a parallel probabilistic model with similar properties. This approach allows us to relax the restrictive assumptions on the functional forms of the primitives, including the probability models often used to determine the shape of the TF, which impose strong restrictions on the shape of the calculated GPFs. We could also propose an ATF and its corresponding production set that gives rise to GPFs with different substitutability properties (i.e., an ATF with Leontief or Cobb‒Douglas properties).
Acknowledgements
The article is a collateral result of the Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica, Project PAPIIT IN301421, and the Programa de Apoyo a Proyectos para Innovar y Mejorar la Educación, Project PAPIME PE301422, both of the Dirección General de Asuntos del Personal Académico (DGAPA) of the National Autonomous University of Mexico, UNAM. The authors thank Débora Martínez Ventura, Lizeth Guerrero González, Ruth Martínez Ventura, Gonzalo Curiel Vázquez, and Raúl Orozco Aguilar for their technical assistance.
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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First, monotonicity holds because T is nondecreasing in \({a}_{i},{b}_{i},\) and \({N}_{i}\) and nonincreasing in \({\widetilde{Y}}_{i}\). Next, to prove concavity with respect to \({a}_{i}\), \({b}_{i}\), and \({N}_{i}\), we let \({{\varvec{z}}}_{0}=\left({a}_{i0}, {L}_{i0},{b}_{i0}, {K}_{i0},{\widetilde{Y}}_{i},{N}_{i0}\right)\) and \({{\varvec{z}}}_{1}=\left({a}_{i1},{L}_{i1},{b}_{i1},{K}_{i1},{\widetilde{Y}}_{i},{N}_{i1}\right)\) be two augmented production plans in \({B}_{j}\). By Assumption 1, \(\omega {{\varvec{z}}}_{0}+\left(1-\omega \right){{\varvec{z}}}_{1}\) is also in \({B}_{j}\) for all \(\uplambda \in \left[\text{0,1}\right]\), given that \({B}_{\text{j}}\) is a convex set. Thus,
To prove continuity from above, let us fix \({\widetilde{Y}}_{i}\) and a given decreasing sequence \(\{({{a}_{l},{L}_{l},{b}_{l},{K}_{l},{N}_{l})\}}_{l\in {\mathbb{N}}}\) that goes to \(\left({a}_{i},{L}_{i},{b}_{i},{K}_{i}{N}_{i}\right)\) such that \(\left({a}_{l},{L}_{l},{b}_{l},{K}_{l}{N}_{l}\right)\ge \left({a}_{i},{L}_{i},{b}_{i},{K}_{i}{N}_{i}\right)\) for all \(l\in {\mathbb{N}}\). From (i) in Assumption 2.1.1, let \({z}_{i}=\left({a}_{i},{L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i}\right)\) such that \({\widetilde{Y}}_{i}=\underset{{\widetilde{Y}}_{i}}{\mathit{max}}\left\{{\widetilde{Y}}_{i}:{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\in {B}_{j}\right\}.\) Then, \(\left({a}_{l},{L}_{l},{b}_{l},{K}_{l},{N}_{l}\right)\ge \left({a}_{i},{L}_{i},{b}_{i},{K}_{i}{N}_{i}\right)\) and \({z}_{l}=\left({a}_{l},{L}_{l},{b}_{l},{K}_{l},{\widetilde{Y}}_{i},{N}_{l}\right)\in {B}_{j}\), which implies that \(T\left({z}_{l}\right)\ge T\left({z}_{i}\right)=0\). Since \(T\) is a monotone function, \(\{T\left({a}_{l},{L}_{l},{b}_{l},{K}_{l},{\widetilde{Y}}_{i},{N}_{l}\right){\}}_{l\in {\mathbb{N}}}\) converges to \(T\left({(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i}\right)\).
Finally, let \({z}_{0}=\left({a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i0},{N}_{i}\right)\) and \({z}_{1}=\left({a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i1},{N}_{i}\right)\) be two augmented production plans in \({B}_{j}\). By Assumption 2.1.1, \(\omega {z}_{0}+\left(1-\omega \right){z}_{1}\) is also in \({B}_{j}\) for all \(\omega \in \left[\text{0,1}\right]\). Then,
which implies that \(T\) is quasiconcave in \({\widetilde{Y}}_{i}.\) If \({\widetilde{Y}}_{i0}\le {\widetilde{Y}}_{i}\), then from (iii) in Assumption 2.1.1, \(\text{T}\left({a}_{i},{L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i0},{N}_{i}\right)\ge T\left({a}_{i},{L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i1},{N}_{i}\right),\) which implies that \(T\) is nonincreasing in \({\widetilde{Y}}_{i}\). \(\square\).
Proof of Proposition 2.1.3
First, the monotonicity and nondecreasing characteristics in inputs and the level of knowledge properties are inherited by the ATF \(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\). Then,
because by definition of the ATF, \(f\left({a}_{i},0,{b}_{i}, 0\right)=\underset{{\widetilde{Y}}_{i}\in {B}_{j}}{\mathit{max}}\{{\widetilde{Y}}_{i} : T\left({a}_{i},0,{b}_{i},0,{\widetilde{Y}}_{i}\right)\ge 0\}=0\) (property (ii) in Assumption 2.1.1). Finally, to prove cuasiconcavity, let \({z}_{0}=\left({a}_{i}, {L}_{i0},{b}_{i},{K}_{i0},{N}_{i0}\right)\) and\({z}_{1}=\left({a}_{i}, {L}_{i1},{b}_{i},{K}_{i1},{N}_{i1}\right)\), such that \({z}_{0\widetilde{Y}}=\left({a}_{i}, {L}_{i0},{b}_{i},{K}_{i0},{\widetilde{Y}}_{i},{N}_{i0}\right)\) and \({z}_{1\widetilde{Y}}=\left({a}_{i}, {L}_{i1},{b}_{i},{K}_{i1},{\widetilde{Y}}_{i},{N}_{i1}\right)\) are in \({B}_{j}\) for all\(\omega \in \left[\text{0,1}\right]\). Then, \(\omega {z}_{0\widetilde{Y}}+\left(1-\omega \right){z}_{1\widetilde{Y}}\) is also in \({B}_{j}\) for all\(\omega \in \left[\text{0,1}\right]\). Then, by quasiconcavity of\(f\left({a}_{i},{b}_{i};\text{0,0}\right)\), we have that
(i) As \(0\in {B}_{j}^{{^{\prime}}{^{\prime}}}\), \({B}_{j}^{{^{\prime}}{^{\prime}}}\ne \varnothing\), and is closed by definition (due to \(\left({a}_{i},{L,b}_{i},K,{\widetilde{Y}}_{i},{N}_{i}\right)\ge 0\), \(\left({a}_{i},{L,b}_{i},K,{\widetilde{Y}}_{i},{N}_{i}\right)\ge 0,\) and \(T\left({a}_{i},{L,b}_{i},K,{\widetilde{Y}}_{i},{N}_{i}\right)\ge 0\)). (ii) Convexity fits because \(T\) and \({Y}_{i}\) are concave. (iii) Fits because \(T\) is monotone. Property (iv) holds because \(0\le {\widetilde{Y}}_{i}^{^{\prime}}\le {\widetilde{Y}}_{i}\) implies \(0\le {\widetilde{Y}}_{i}^{^{\prime}}\le {\widetilde{Y}}_{i}\le {Y}^{*}\left(L,K,N\right)\), and by monotonicity of \(T\). (v) Given \({a}_{i},{L}_{i},{b}_{i},{K}_{i},\) and \({N}_{i}\), the set \(\{{\widetilde{Y}}_{i}|({a}_{i},{L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\in {B}_{j}^{{^{\prime}}{^{\prime}}}\}\) is bounded above by \({Y}_{i}^{*}\left(L,K,N\right)\). \(\square\)
Proof of Theorem 2.2.1
\({T}_{C}\left(a,b,\widetilde{Y},N\right)\) is a well-defined function since \({B}_{j}\) has at least an interior point z. Then, \(T\left(z\right)>0\), which implies that \(\mathit{max}\ T\left({a}_{i}, {b}_{i}, {\widetilde{Y}}_{i},N\right)>0\). In addition, by definition, \({0\le T}_{C}\left(a,b,\widetilde{Y},N\right)\le 1\), which is the range of a proper copula function. Next, we show that \({T}_{C}\left(a,b,\widetilde{Y},N\right)\) satisfies the conditions to be represented as a copula function (see Nelsen (2006)). From (ii) in Assumption 1, \({T}_{C}\left(0,b,\widetilde{Y},N\right)={T}_{C}\left(a,0,\widetilde{Y},N\right)=0\), \({T}_{C}\left(a,b,\mathit{max}\ \widetilde{Y},N\right)=0\) holds because \(\left(a,L,b,K,\mathit{max}\ \widetilde{Y},N\right)\) lies on the boundary of \({B}_{j}\). These two properties are equivalent to the fact that the proposed transformation function is grounded like a copula function. \({T}_{C}\left(a,\mathit{max}\ b,0,N\right), { T}_{C}\left(\mathit{max}\ a,b,0,N\right)\), \({T}_{C}\left(a,b,0,\mathit{max}\ N\right)\), and \({T}_{C}\left(\mathit{max}\ a,\mathit{max}\ b,\widetilde{Y},\mathit{max}\ N\right)\) have the properties of the marginal distribution of \({a}_{i},{b}_{i},{\widetilde{Y}}_{i}\), and \({N}_{i}\). Additionally, \({T}_{C}\left(0,\mathit{max}\ b,\text{0,0}\right)={T}_{C}\left(\mathit{max}\ a,\text{0,0},0\right)={T}_{C}\left(\text{0,0},0,\mathit{max}\ N\right)= {T}_{C}\left(\mathit{max}\ a,\mathit{max}\ b,\mathit{max}\ \widetilde{Y},\mathit{max}\ N\right)=0\) and \({T}_{C}\left(\mathit{max}\ a,\mathit{max}\ b,0,\mathit{max}\ N\right)=1\), which ensures that the cumulative probability over the entire domain is equal to 1 as in a well-defined distribution function. \({T}_{c}\) inherits the properties of being continuous from above and nondecreasing for \({g}_{a}\left(a\right)\),\({g}_{b}\left(b\right)\), and \({g}_{\widetilde{Y}}\left(\widetilde{Y}\right)\) from T. We show that the \({T}_{C}\)-volume is nonnegative for the 3-variated case, which naturally extends to the 4-variated copula. Then, let us assume \({a}_{0}\le {a}_{1}, {b}_{0}\le {b}_{1}\), and \({\widetilde{Y}}_{0}\le {\widetilde{Y}}_{1}\). We define the 3-box
Given that \({g}_{a}\left(a\right)\) and \({g}_{b}\left(b\right)\) are nondecreasing and \({g}_{\widetilde{Y}}\left(\widetilde{Y}\right)\) is nonincreasing, the \({T}_{C}\)-volume of the 4-box \(B\) is given by
In particular, when \(\mu =\frac{1}{2}\), we can define \({T}_\frac{l}{2}={\frac{1}{2}[T}_{C}\left({a}_{1}, {b}_{0},{\widetilde{Y}}_{l}\right)+{T}_{C}\left({a}_{0}, {b}_{1},{\widetilde{Y}}_{l}\right)]\).
From the concavity of \({T}_{C},\) we have \({T}_\frac{l}{2}-{T}_{C}\left({a}_{0}, {b}_{0},{\widetilde{Y}}_{l}\right)\le {T}_{C}\left({a}_{1}, {b}_{1},{\widetilde{Y}}_{l}\right)-{T}_\frac{l}{2}\), which implies that
\({T}_{C}\) is nondecreasing in \({\widetilde{Y}}_{\text{l}}\), \({T}_{C}^{0}\le {T}_{C}^{1}\), and \({{T}_{C}\left(M\right)=T}_{C}^{1}-{T}_{C}^{0}\ge 0\), which implies that the transformation function \({T}_{C}\) behaves as a 3-increasing function and is similar to a 3-copula (and then the 4-copula) function. \(\square\)
Proof of Corollary 2.2.1
The proof easily follows by applying the steps described in this corollary.
Proof of Proposition 2.2.1
Since the LPF and the TF depend on the substitutability parameter set \(\Theta\); the LPF depends on the availability parameters in the economy, \({\Theta }_{a}\),\({\Theta }_{b}\), and \({\Theta }_{N}\); and the TF depends on \({\Theta }_{a}\),\({\Theta }_{b}\), and \({\Theta }_{\widetilde{Y}}\), the shape of the resulting GPF depends on these \(5\) sets of parameters. The marginal technical rate of substitution (MRTS) between \(L\) and \(K\)\(MRTS\left(U\left({ a}^{*},L;{\Theta }_{a}\right), V\left({ b}^{*},K;{\Theta }_{b}\right),{\Theta }_{\widetilde{Y}},\Theta \right)\) also depends on these parameter sets; then, the elasticity of substitution of the GPF is defined by
In addition, based on Ida et al. (2014), any measure of dependence \(\mu\) based on the copula \(C\) of a generalized Archimedean copula should be of the form \(\mu = \iint_{{I^{2} }} {f(u,v,C(u,v;\Theta ))dC(u,v;\Theta )}\), where \(f=f\left(u,v,C\right)\) is an appropriate smooth positive function. Then, the dependence measure only depends on the copula’s parameter set, i.e., \(\mu =\mu \left(\Theta \right)\). \(\square\)
Proof of Corollary 2.2.2
Based on Proposition 2.1.2 and the definition of a generalized Archimedean copula, we can denote the transformation function as
As \(u\) and \(v\) are simple variables, \({\left[{\psi }^{-1}\right]}^{^{\prime}}\left(u;\Theta \right)={\left[{\psi }^{-1}\right]}^{^{\prime}}\left(v;\Theta \right)\), which reduces to
Finally, suppose that \({\Theta }_{abN}=\left\{\theta \right\}\). If \(\mu =m\left(\theta \right)\) has an inverse (\({m}^{-1}\)), then we obtain \(\theta ={m}^{-1}\left(\mu \right)\) and \(\sigma =\sigma \left({m}^{-1}\left(\mu \right)\right)\). Similarly, if \(\sigma =n\left(\theta \right)\) has an inverse, then \(\mu =\mu \left(\theta \right)=\mu \left({n}^{-1}\left(\sigma \right)\right)\). \(\square\)
Proof of Proposition 2.2.2
Let \(Y=f\left(a,b;L, K\right)\) be the LPF and \(H\left(a,b\right)={A}_{N}\) be the TF. Both are defined in the \(\left(a,b\right)\) space for given values of\(L\),\(K\), and\(N\). A competitive firm maximizes \(\widetilde{Y}\) over the set of feasible technology pairs\(D=\left\{\left(a,b\right): {T}_{C}\left(a,b,\widetilde{Y},N\right)\ge 0\right\}\). Since we have shown that the transformation function \({T}_{C}\) is represented by a copula in the probability space with two increasing and nondecreasing margins of \(a\) and\(b\), respectively, the derived deterministic LPF \(f\left(a,b;L, K\right)\) also increases over the compact set \(D\) and attains its maximum at the TF by (i) and (iii) of Assumption 1. Then, the maximization of the LPF\({Y}_{i}=f\left({a}_{i},{b}_{i};L,K\right)=\underset{{B}_{j}}{\mathit{max}}\{{\widetilde{Y}}_{i}:{T}_{C}\left({a}_{i},{b}_{i},{\widetilde{Y}}_{i}\right)\ge 0\}\), which is subject to the TF \(H\left({a}_{i},{b}_{i}\right)=\left\{\left({a}_{i},{b}_{i}\right) :{T}_{C}\left({a}_{i},{b}_{i},{N}_{i}\right)=0\right\}\) occurs when they coincide over the TF, with the unique solution\({(a}^{*}\),\({b}^{*})\) when these curves are tangent at this point, which can occur only if the LPF is convex and the TF is concave or the LPF is sharper than the TF in the feasible augmented technology set.
In addition, given that the LPF and TF are represented by the distribution functions \(F\left({h}_{a}\left(a\right),{h}_{b}\left(b\right),{h}_{\widetilde{Y}}\left(Y\right);{\Theta }_{a},{\Theta }_{b},{\Theta }_{\widetilde{Y}},\Theta \right)\) and \(F\left({q}_{a}\left(a\right),{q}_{b}\left(b\right),{q}_{N}\left(N\right);{\Theta }_{a},{\Theta }_{b},{\Theta }_{N},\Theta \right)\), respectively, they are determined not only by the substitutability parameter set \(\Theta\) but also by the parameters \({\Theta }_{a}\), \({\Theta }_{b}\), \({\Theta }_{\widetilde{Y}}\), and \({\Theta }_{N}\). Then, once we maximize the LPF, which is subject to the TF, the optimal values \({a}^{*}\) and \({b}^{*}\) also depend on \(\Theta\), \({\Theta }_{a},{\Theta }_{b}\), \({\Theta }_{\widetilde{Y}}\), and \({\Theta }_{N}\).
Since we obtain the GPF by substituting \({a}^{*}\) and \({b}^{*}\) into the LPF, we have
Appendix A.2: Proofs for the Microfounded CES Model
A.2.1. Proofs of Random Variable Transformations
Here, we show how to obtain the margins of the transformed random variables used in our examples. For a Pareto distributed random variable \({\text{c}}_{\text{I}}\), its density function is given by
We use the results from Appendix A.2.1 for the transformed Pareto random variables \({h}_{a}({a}_{i})=\frac{1}{{a}_{i}},{{h}_{b}(b}_{i})=\frac{1}{{b}_{i}}\) and \({h}_{\widetilde{Y}}\left({Y}_{i}\right)=-{Y}_{i}\) to obtain (4) by substituting their marginal distributions into the nested Clayton copula (3). Similarly, we use the following Pareto random variables \({q}_{a}\left({a}_{i}\right)=-{a}_{i}\), \({q}_{b}\left({b}_{i}\right)=-{b}_{i}\), and \({q}_{N}\left({N}_{i}\right)=\frac{1}{{N}_{i}}\) to obtain (5).
A.2.2. Model Proofs
Proof of Lemma 3.1
Fixing the probability \({P}_{0}=P\left({a}_{i}>\frac{1}{a},{b}_{i}>\frac{1}{b},{Y}_{i}>Y\right)\) in Eq. (4) and solving for \({\left(\frac{Y}{{\gamma }_{Y}}\right)}^{\eta },\) we have
Now, we define \({\gamma }_{a}={\gamma }_{L}{\left(1+{L}^{-\theta }\right)}^{-\frac{1}{\alpha \theta }}\) and \({\gamma }_{b}={\gamma }_{K}{\left(1+{K}^{-\theta }\right)}^{-\frac{1}{\beta \theta }}\) and substitute them into the previous equation to obtain
We change the scale to obtain \({\left(a{\gamma }_{L}\right)}^{-\alpha \theta }+{\left(b{\gamma }_{K}\right)}^{-\beta \theta }=1\), and finally, we obtain
Then, we combine the first-order conditions to obtain \({b}^{2\beta \theta }=\frac{1+{L}^{\theta }}{1+{K}^{\theta }}{a}^{2\alpha \theta }\). Substituting the previous equation into the constraint and solving for \({a}^{\alpha \theta }\), we have
Consider the copulas \(C\left(W,Z\right)=max\left\{{\left({W}^{-\delta }+{Z}^{-\delta }-1\right)}^{-\frac{1}{\delta }}, 0\right\}\), where \(Z=max\left\{{\left[{U}^{-\theta }{+V}^{-\theta }-1\right]}^{-\frac{1}{\theta }},0\right\}\) and \(C\left(U,V\right)=max\left\{{\left[{U}^{-\theta }{+V}^{-\theta }-1\right]}^{-\frac{1}{\theta }},0\right\}\). Then, by using the generators \(\psi \left(t\right)=\frac{1}{\delta }\left({t}^{-\delta }-1\right)\) and \(\psi \left(t\right)=\frac{1}{\theta }\left({t}^{-\theta }-1\right)\) in the next formula \({\tau }_{c}=1+4{\int }_{0}^{1}\frac{\psi \left(t\right)}{{\psi }^{^{\prime}}\left(t\right)}dt\) and applying simple algebra, we obtain \({\tau }_{wuv}=\frac{\delta }{2+\delta }\) and \({\tau }_{uv}=\frac{\theta }{\theta +2}\), respectively. However, using the formula \(\sigma =\left(\frac{dA}{dB}\right)\left(\frac{B}{A}\right)\), we calculate the elasticity of substitution of transformation and local production functions.
First, we define \({Z=\left[{\left(a{\gamma }_{L}\right)}^{-\alpha \theta }{L}^{-\theta }+{\left(b{\gamma }_{K}\right)}^{-\beta \theta }{K}^{-\theta }\right]}^{\frac{1}{\eta \theta }}\) from the transformation function and obtain the rate of substitution (RS) as \(\text{RS}=\varphi {\left(\frac{Z}{Y}\right)}^{\eta \delta -1}\); then, we obtain the derivative
Thus, we obtain the result \({\sigma }_{Yab}={\left(\frac{dLn\left|RS\right|}{dLn\left(\frac{Y}{Z}\right)}\right)}^{-1}=\frac{1}{1-\eta \delta }\). Second, the technical rate of substitution (TRS) is given by \(\text{TRS}=\frac{\varphi }{1-\varphi }{\left(\frac{K}{L}\right)}^{\theta +1}\); then, we obtain the derivative
Thus, we obtain the result \({\sigma }_{KL}={\left(\frac{dLn\left|TRS\right|}{dLn\left(\frac{K}{L}\right)}\right)}^{-1}=\frac{1}{1+\theta }\).
Finally, to obtain the identities, we rewrite \(\delta\) and \(\theta\) as functions of \({\sigma }_{Yab}\), \({\sigma }_{KL}\),\({\tau }_{wuv}\) and \({\tau }_{uv}\) to obtain \(\delta =\frac{2{\tau }_{wuv}}{1-{\tau }_{wuv}}\) and \(\theta =\frac{2{\tau }_{uv}}{1-{\tau }_{uv}}\). \(\square\)
Appendix B: General Definitions and Theorems
Definition B.1
A 4-dimensional copula is a function \(C\) from \({I}^{4}=\left[\text{0,1}\right]\times \left[\text{0,1}\right]\times \left[\text{0,1}\right]\times \left[\text{0,1}\right]\) to \(I=\left[\text{0,1}\right]\) that satisfies the following properties:
(1)
For every \(\left(U, V,W.X\right)\in {I}^{4}\), if at least one of such variables is zero, then \(C\left(U,V,W,X\right)=0\);
(2)
\(C\left(\text{1,1},1,N\right)=N;\)\(C\left(\text{1,1},W,1\right)=W\); \(C\left(1,V,\text{1,1}\right)=V\), and \(C\left(U,\text{1,1},1\right)=U\);
(3)
For every \({\varvec{r}},\boldsymbol{ }{\varvec{s}}\in {I}^{4}\) such that \({\varvec{r}}\le {\varvec{s}}\),\({V}_{C}(M)=\sum sgn({\varvec{c}})F({\varvec{c}})\ge 0\);
where \(M=\left[{r}_{1},{s}_{1}\right]\times \left[{r}_{2},{s}_{2}\right]\times \left[{r}_{3},{s}_{3}\right]\times \left[{r}_{4},{s}_{4}\right]\) and the sum is over all the vertices \(c=\left({c}_{1}, {c}_{2},{c}_{3},{c}_{4}\right)\) of \(M\) (each \({c}_{k}\) is equal to either \({r}_{k}\) or \({s}_{k}\)) and \(sgn\left({\varvec{c}}\right)\) is given by
$$sgn\left({\varvec{c}}\right)=\left\{\begin{array}{ll} 1,&\quad if\ {c}_{k}={r}_{k} \text{ for an even number }of\ {k}^{^{\prime}}s,\\ -1,&\quad if\ {c}_{k}={r}_{k} \text{ for an odd number }of\ {k}^{^{\prime}}s.\end{array}\right.$$
Theorem
(Sklar’s Theorem) Let \(C\) be a 4-copula and \(U, V,W,\) and \(X\) be the marginal distributions of the random variables \({a}_{i},{b}_{i}\)\({\widetilde{Y}}_{i}\), and \({N}_{i}\) respectively. Then, there exists a 4-dimensional distribution function \(F\) with margins \(U, V,\) and \(W\) defined by
Conversely, let \(F\) be a 4-dimensional joint distribution function with margins \(U, V,\text{W},\) and \(X\). Then, for all \(\left(a,b,\widetilde{Y},N\right)\) in \({\overline{\mathbb{R}} }^{4}\) there exists a 4-dimensional copula \(C\) defined by (B.1). If \(U, V,W\), and \(X\) are all continuous, then \(C\) is unique; otherwise, \(C\) is uniquely determined on its range \(Ran\left(U\right)\times Ran\left(V\right)\times Ran\left(W\right)\times Ran\left(X\right).\)
In this paper, we analyze a technology choice problem, which is different from the classic production problem of choosing the inputs and the output level by maximizing profits.
We say that vector \({\varvec{x}}=({x}_{1},{x}_{2},\dots ,{x}_{n})\) is either greater than or equal to \({\varvec{y}}=({y}_{1},{y}_{2},\dots ,{y}_{n})\) if \({x}_{i}{\ge y}_{i}\) for all \(i=\text{1,2},\dots ,n.\)
The expression \({\varvec{x}}=\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\ge \left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)={\varvec{y}}\) means that \({x}_{i}{\ge y}_{i}\) for all \(i=\text{1,2},\dots ,n.\)
$$\left\{{{\widetilde{Y}}_{i} : T(a}_{i}, {b}_{i},{\widetilde{Y}}_{i})\ge 0\right\}$$ is a compact set, as it is a closed interval and bounded from above and below ($${\widetilde{Y}}_{i}\ge 0$$, and (iv) of Assumption 2.1.1).
Notice that this is the general case derived from our setup. A particular and relevant case is the one in which \(\text{Y}\) is deterministic, assuming that we have a realization of the random variables \({\text{a}}_{\text{i}},{\text{ b}}_{\text{i}}\) and \({\text{N}}_{\text{i}}\), for a given \(\text{L}\) and \(\text{K}\). Therefore, in this particular case, a trivariate copula on the transformation of \({\text{a}}_{\text{i}}\), \({\text{b}}_{\text{i}}\) and \({\text{N}}_{\text{i}}\) can be used. We thank an anonymous referee for useful comments on this issue.
The maximum in \(T\left({a}_{i}, {b}_{i}, {\widetilde{Y}}_{i},N\right)\) exists because, as \(N\) is a fixed exogenous variable, there is a limit for the size of knowledge at time \(t\). Then, as \(L\) and \(K\) are fixed, there exists a bounded set of feasible outputs that can be attained by this firm.
In the LPF, \({g}_{a}\left(a\right),{g}_{b}\left(b\right),\) and \({g}_{\widetilde{Y}}\left(\widetilde{Y}\right)\) are denoted as \({h}_{a}(a)\), \({h}_{b}(b)\), and \({h}_{\widetilde{Y}}\left(\widetilde{Y}\right)\), respectively.
In the TF, \({g}_{a}\left(a\right),{g}_{b}\left(b\right),\) and \({g}_{\widetilde{Y}}\left(\widetilde{Y}\right)\) are denoted as \({q}_{a}(a)\), \({q}_{b}(b)\), and \({q}_{N}\left(N\right)\), respectively.
In the technology choice literature, no attempt has been made to derive a setup that leads to an estimable empirical model by constructing a link between the microeconomic parameters and the statistical parameters, as we propose here. This is a straightforward method to estimate the GPF for any sector of the economy by using data on the output and technological innovations (i.e., patents) at the firm level.
Expressions \(\varphi ={\left({a\gamma }_{L}\right)}^{-\alpha \theta }>0\) and \(1-\varphi ={\left({b\gamma }_{K}\right)}^{-\beta \theta }>0\), which are known as the technological change indices of a CES function, are shaped by the availability parameters (tail indices α, β) and the substitutability parameters (copula´s dependence parameters θ and δ). In addition, Eq. (6) describes a general nested CES production function, as it might consider nonlinear and linear relationships between technological innovations and the factors of production L and K.
The cases presented in this table are the ones in which the LPF and GPF have the same shape. However, there are cases where the LPF and GPFs are not of the same class, which includes the cases developed by (Acemoglu 2003; Growiec 2008a; Jones 2005).
