## Introduction

## The Setup

^{1}and 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.

### Theoretical Setup: A Transformation Function is Used to Obtain a Deterministic Microfounded GPF

^{2}\({\widetilde{Y}}_{i}\)), 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 \({N}_{i}\).

^{3}

^{4}and \(T{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})=0\) if and only if \({\widetilde{Y}}_{i}=\underset{{\widetilde{Y}}_{i}}{\mathit{max}}\left\{{\widetilde{Y}}_{i}:{(a}_{i}, {L}_{i},{b}_{i},{K}_{i},{\widetilde{Y}}_{i},{N}_{i})\in {B}_{j}\right\}.\)

^{5}

### Probabilistic Setup: Linking the Deterministic Microfounded ATF with its Corresponding Probability Function

^{6}and a given exogenous level of knowledge \(N\), we find that the following monotonically increasing transformation of our ATF

^{7}

^{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 \({\widetilde{Y}}_{i}\),

^{9}assuming 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:

^{10}with respect to \(W\) and \(Y\), respectively.

^{11}assuming continuous and decreasing transformations \({q}_{a}\) and \({q}_{b}\) and an increasing transformation \({q}_{N}\)

^{12}of 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:

^{13}with respect to \(X\) and \({A}_{N}\), respectively, and \({A}_{N}\) is a constant because \(N\) is fixed.

## Illustration of How to Derive a Nested CES GPF and its Distribution Function Based on an Augmented Transformation Function

^{14}:

^{15}we obtain the following CES LPF:

### Relationships Between the Microeconomic and Probabilistic Models

^{16}

Substitutability parameter \({\varvec{\uptheta}}\) | Kendall's Tau | Elasticity of substitution (LPF) | Global production function (GPF) |
---|---|---|---|

\(-1<{\varvec{\theta}}<\boldsymbol{\infty }\), \({\varvec{\theta}}\ne 0\) | \(-1<\tau <1,\) \(\tau \ne 0\) | \(0<\sigma <\infty ,\) \(\sigma \ne 1\) | CES-type function |

\({\varvec{\theta}}=0\) | \(\tau =0\) | \(\sigma =1\) | Cobb‒douglas |

\({\varvec{\theta}}\to \boldsymbol{\infty }\) | \(\tau =1\) | \(\sigma =0\) | Leontief |

\({\varvec{\theta}}=-1\) | \(\tau =-1\) | \(\sigma \to \infty\) | Perfect substitutes |