Identification of a Production Function with Age Limit for Production Capacities

Abstract

The microlevel dynamics of the age-limited production capacities differentiated by the moments of creation set the macrolevel production function. The microdescription is based on the hypothesis of the capacity loss at a constant rate and constant number of jobs from the moment of the creation of the production unit to its closure, when the age limit is exceeded. An analytical expression for the endogenous production function with the given maximum age of the capacities is obtained in characteristic exponential growth modes with the constant share of new capacities. A transient growth mode with varying incremental capital intensity of new capacities is considered. The production function’s parameters can be determined even with significant variations of the new capacity’s share in the total capacity, which occurred in the Russian economy. For this purpose, the initial microeconomic model of the production capacity’s dynamics is used in the numerical calculations of the production function. The parameters are estimated indirectly based on a comparison of the results of the calculations by the model with the statistical data over 1970–2017. The obtained value of the average age limit of capacities A = 25 for the Russian economy explains the vanishing of cost inflation in 2017. The identification of the endogenous production function parameters also show that the value of the average incremental capital intensity for the entire Russian economy decreased significantly from 1970 to 2017. The decrease is explained by the increase in the share of the primary industry in the output.

APPENDIX A

Proof of Theorem 1. Statement 1 follows directly from (9). In order to prove Statement 2 (another formulation of Lemma 1), it is sufficient to divide (5) by $M(t)$, to find the derivative of $M(t)$ in the obtained equation according to (9), and to use Statement 1. The expression of production function (10) in Statement 3 directly follows from the substitution of the constant share of the new capacity $\sigma$ into relations (6). Under balanced growth (9), the values $M(t)$ and $Y(t)$ grow at a constant rate; therefore $f(t,x) = {\text{const}}$ and Statement 4 follows from (10), which completes the proof of Theorem 1.

Proof of Theorem 2. According to Lemma 1, the growth rate $\gamma$ (7) is determined by relation (8). Equation (8) has a unique positive solution if the derivative of the left-hand side of this equation is greater than the derivative of the right-hand side at $\varphi = 0$, which satisfies the condition $A > {1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-0em} \sigma }$ of Lemma 1. Statement 1 is proved by substituting (7) and (8) into the parametric expression for the production function (6). Finally, we obtain relation (10). Statement 2 follows from the definition of the production function $Y(t) = M(t)f(t,x)$ and equalities (10) and (12). Statement 3 follows from Statement 2, condition (13), and Eq. (2) describing the dynamics of the lowest labor intensity. In order to prove Statement 4, we take into consideration that it follows from condition (13) that the growth rate of the new capacities $J(t)$ coincides with the growth rate of the aggregate capacity $M(t)$, and then, according to (11) and (12), the share of savings in relation to the output decreases, ${{b(t)J(t)} \mathord{\left/ {\vphantom {{b(t)J(t)} {Y(t)}}} \right. \kern-0em} {Y(t)}} = ({{{{b}_{0}}{{J}_{0}}} \mathord{\left/ {\vphantom {{{{b}_{0}}{{J}_{0}}} {{{Y}_{0}}}}} \right. \kern-0em} {{{Y}_{0}}}})\exp ( - \beta \sigma t)$, and consequently, the share of consumption ${{C(t)} \mathord{\left/ {\vphantom {{C(t)} {Y(t)}}} \right. \kern-0em} {Y(t)}} = 1 - {{b(t)J(t)} \mathord{\left/ {\vphantom {{b(t)J(t)} {Y(t)}}} \right. \kern-0em} {Y(t)}}$ increases. According to Statement 2, the average consumption in the transient mode is determined by the relation

$$c(t) = \left( {f({x \mathord{\left/ {\vphantom {x {\nu (t)}}} \right. \kern-0em} {\nu (t)}}) - \sigma {{b}_{0}}\exp \left( { - \beta \sigma t} \right)} \right){{\nu (t)} \mathord{\left/ {\vphantom {{\nu (t)} {\left( {{{\nu }_{0}}x} \right)}}} \right. \kern-0em} {\left( {{{\nu }_{0}}x} \right)}}\exp \left( {\varepsilon \sigma t} \right),$$

i.e., the average consumption in the transition mode (at ${x \mathord{\left/ {\vphantom {x {\nu (t)}}} \right. \kern-0em} {\nu (t)}} = {\text{const}}$) grows faster than in the balanced growth mode (9). This completes the proof of Theorem 2.