Are distance measures effective at measuring efficiency? DEA meets the vintage model

Abstract

In this paper I develop a model of capacity expansion that accounts for differences in the productivity of the installed capital due to technical progress exhibited by the ex ante production function. A putty-clay set-up is assumed, meaning flexible input coefficients and substitution possibilities ex ante, but fixed input coefficients ex post. Based on the model, I generate a capacity distribution of DMUs (vintages) describing an industry with a homogeneous output and perform an efficiency analysis employing data envelopment analysis, a popular non-parametric method for estimating efficiency. The results show that in some circumstances older vintages might appear on the efficiency frontier, unlike some newer vintages that are found to be inefficient, despite benefiting from the advancement of the technology.

Appendices

Appendix 1

When firms expect exponential wage growth, with growth parameter a, then:

$$C_{t} = w_{K} \left( t \right) \cdot K_{t} + \sum\limits_{s = t}^{\infty } {w_{L} \left( s \right)L_{t} \cdot e^{ - rt} } .$$

(8)

However, now w(t) = w ₀ · e ^at, a ≥ 0, where a is the rate of wage growth.

Then the cost the firms face is:

$$C_{t} = w_{K} (t) \cdot K_{t} + L_{t} \sum\limits_{s = t}^{\infty } {w_{0} \cdot e^{(a - r)t} } = w_{K} (t) \cdot K_{t} + w_{0} \cdot L_{t} \frac{{e^{(a - r)t} }}{{1 - e^{(a - r)} }},$$

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where a − r < 0 <=> a < r.

Hence, for the relation above to exist, it is required that the rate of wage growth is always smaller than the discount rate.

Appendix 2

The optimisation programme associated with the investment decision at time t is:

$$\begin{aligned} & \mathop {\hbox{min} }\limits_{{K_{t} ,L_{t} }} C_{t} = w_{K} (t) \cdot K_{t} + \sum\limits_{s = t}^{\infty } {w_{L} (s) \cdot L_{t} \cdot e^{ - rt} } \\ & such \, as \\ & A_{0} \cdot e^{\delta \cdot t} \cdot K_{t}^{\alpha } \cdot L_{t}^{\beta } = Q_{0} \cdot e^{gt} (1 - e^{ - g} ). \\ \end{aligned}$$

(10)

Since

$$\sum\limits_{s = t}^{\infty } {(e^{ - r} )^{s} = \frac{{e^{ - rt} }}{{1 - e^{ - r} }}}$$

(11)

and

$$w_{L} (s) = w_{L} (t),\quad {\text{for}}{\kern 1pt} \, s > t,$$

(12)

the optimisation programme becomes:

$$\begin{aligned} & \mathop {\hbox{min} }\limits_{{K_{t} ,L_{t} }} C_{t} = w_{K} (t) \cdot K_{t} + w_{L} (t) \cdot L_{t} \cdot \frac{{e^{ - rt} }}{{1 - e^{ - r} }} \\ & such \, as \\ & A_{0} \cdot e^{\delta \cdot t} \cdot K_{t}^{\alpha } \cdot L_{t}^{\beta } = Q_{0} \cdot e^{gt} (1 - e^{ - g} ) \\ \end{aligned}$$

(13)

First-order conditions for optimality imply⁹:

$$w_{K} \left( t \right) = \alpha \cdot A_{0} \cdot e^{\delta \cdot t} \cdot K_{t}^{\alpha - 1} \cdot L_{t}^{\beta }$$

(14)

$$w_{L} (t) \cdot \frac{{e^{ - rt} }}{{1 - e^{ - r} }} = \beta \cdot A_{0} \cdot e^{\delta \cdot t} \cdot K_{t}^{\alpha } \cdot L_{t}^{\beta - 1}$$

(15)

$$A_{0} \cdot e^{\delta \cdot t} \cdot K_{t}^{\alpha } \cdot L_{t}^{\beta } = Q_{0} \cdot e^{gt} \left( { 1 - e^{ - g} } \right).$$

(16)

Following this we can write:

$$\frac{{w_{K} (t)}}{{w_{L} (t)}} \cdot \frac{{1 - e^{ - r} }}{{e^{ - rt} }} = \frac{\alpha }{\beta } \cdot \frac{{L_{t} }}{{K_{t} }}.$$

(17)

Substituting L _t with $\frac{{w_{K} (t)}}{{w_{L} (t)}} \cdot \frac{{1 - e^{ - r} }}{{e^{ - rt} }} \cdot \frac{\beta }{\alpha } \cdot K_{t}$, it follows that:

$$K_{t}^{\alpha + \beta } = \frac{{Q_{0} }}{{A_{0} }} \cdot e^{(g - \delta - r\beta ) \cdot t} \cdot \left( {\frac{\alpha }{\beta } \cdot \frac{{w_{L} (t)}}{{w_{K} (t)}}} \right)^{\beta } \cdot \frac{{(1 - e^{ - g} )}}{{(1 - e^{ - r} )^{\beta } }}.$$

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Hence, the optimal instalment will be:

$$K_{t}^{*} = \left( {\frac{{Q_{0} }}{{A_{0} }}} \right)^{1/\varepsilon} \cdot e^{{\left( {\frac{g - \delta - r\beta }{\varepsilon }} \right) \cdot t}} \cdot \left( {\frac{\alpha }{\beta } \cdot \frac{{w_{L} (t)}}{{w_{K} (t)}}} \right)^{\beta/\varepsilon} \cdot \frac{{(1 - e^{ - g} )^{1/\varepsilon}}}{{(1 - e^{ - r} )^{\beta/\varepsilon}}},\quad {\text{where}}\,\,\varepsilon { = }\alpha { + }\beta.$$

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Similarly:

$$L_{t}^{*} = \left( {\frac{{Q_{0} }}{{A_{0} }}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \varepsilon }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\varepsilon $}}}} \cdot e^{{\left( {\frac{g - \delta - r\alpha }{\varepsilon }} \right) \cdot t}} \cdot \left( {\frac{\beta }{\alpha } \cdot \frac{{w_{K} (t)}}{{w_{L} (t)}}} \right)^{{{\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha \varepsilon }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\varepsilon $}}}} \cdot \frac{{(1 - e^{ - g} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \varepsilon }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\varepsilon $}}}} }}{{(1 - e^{ - r} )^{{{\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha \varepsilon }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\varepsilon $}}}} }}.$$

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