Measuring the effect of management on production: a two-tier stochastic frontier approach

Abstract

We revisit the production frontier of a firm, and we examine the effects that the firm’s management has on output. In order to estimate these effects using a cross-sectional sample while avoiding the costly requirement of obtaining data on management as a production factor, we develop a two-tier stochastic frontier model where management is treated as a latent variable. The model is consistent with the microeconomic theory of the firm, and it can estimate the effect of management on the output of a firm in monetary terms from different angles, separately from inefficiency. The approach can thus contribute to the cost–benefit analysis related to the management system of a company, and it can facilitate research related to management pay and be used in studies of the determinants of management performance. We also present an empirical application, where we find that the estimates from our latent-variable model align with the results obtained when we use the World Management Survey scores that provide a measure of management.

Appendices

Appendices

1.1 A The distribution of management

Introduced by Stacy (1962), the Generalized Gamma distribution has density

$$\begin{aligned} f_m(m) = \frac{p}{\alpha ^d \mathrm{\Gamma }(d/p)} m^{d-1}\exp \{-m^p/\alpha ^p\},\;\;\; \alpha ,p,d>0. \end{aligned}$$

Assume a subfamily of this distribution with $p=d=\gamma <1,\; \alpha >0$. Then the density becomes

$$\begin{aligned} f_m(m) = \frac{\gamma }{\alpha ^\gamma } m^{\gamma -1}\exp \{-m^\gamma /\alpha ^\gamma \},\;\;\; \alpha >0, 0< \gamma <1. \end{aligned}$$

Consider the random variable $w=m^\gamma$. We have

$$\begin{aligned} w=m^\gamma \implies m = w^{1/\gamma } \implies \frac{\partial m}{\partial w} = \frac{1}{\gamma } w^{(1/\gamma ) -1}. \end{aligned}$$

Applying a change of variables, the density of w is

$$\begin{aligned} f_w(w)= & {} \left| \frac{\partial m}{\partial w}\right| \cdot f_m(w^{1/\gamma }) = \frac{1}{\gamma } w^{(1/\gamma ) -1} \cdot \frac{\gamma }{\alpha ^\gamma } \left[ w^{1/\gamma }\right] ^{\gamma -1}\exp \{-[w^{1/\gamma }]^\gamma /\alpha ^\gamma \}\\&\quad \implies f_w(w) = \frac{1}{\alpha ^\gamma }\exp \{-w/\alpha ^\gamma \}. \end{aligned}$$

But this is the density of an Exponential random variable with scale parameter $\sigma _w =\alpha ^\gamma$.

B Distribution and moments of management metrics

We derive below the distribution of the various metrics we use in the main text, under the assumption that the variables w, u follow Exponential distributions and are independent.

1.1 B.1 Management contribution to output

Applying the change-of-variables technique, we have

$$\begin{aligned} {{\mathrm{Mc}}} = 1- e^{-w} \implies w = -\ln (1- {{\mathrm{Mc}}}) \implies \frac{\partial w}{\partial {{\mathrm{Mc}}}} = \frac{1}{1-{{\mathrm{Mc}}}}. \end{aligned}$$

Then

$$\begin{aligned} f_{{{\mathrm{Mc}}}}({{\mathrm{Mc}}}) = \frac{1}{1-{{\mathrm{Mc}}}} \frac{1}{\sigma _w}\exp \left\{ \frac{-1}{\sigma _w} \big [-\ln (1- {{\mathrm{Mc}}})\big ]\right\} = \frac{1}{\sigma _w} \left( 1-{{\mathrm{Mc}}}\right) ^{1/\sigma _w -1}. \end{aligned}$$

This is the density of a Beta distribution, with parameters $\alpha =1, \beta =1/\sigma _w$, and applying the moment expressions for the specific parameters we obtain

$$\begin{aligned} E({{\mathrm{Mc}}})= & {} \frac{\alpha }{\alpha +\beta }=\frac{1}{1+1/\sigma _w}=\frac{\sigma _w}{1+\sigma _w},\\ \mathrm{Var(Mc)}= & {} \frac{\alpha \beta }{(\alpha + \beta )^2(\alpha +\beta +1)} = \frac{1/\sigma _w}{(1 + 1/\sigma _w)^2(2+1/\sigma _w)} \\= & {} \frac{1/\sigma _w}{(1+\sigma _w)^2(1+2\sigma _w)}\cdot \frac{1}{1/\sigma _w^3}\\= & {} \frac{\sigma _w^2}{(1+\sigma _w)^2(1+2\sigma _w)} \implies \mathrm{SD(Mc)}=\frac{E({{\mathrm{Mc}}})}{\sqrt{1+2\sigma _w}} \end{aligned}$$

Moreover, when one of the parameters of the Beta distribution is equal to unity, the distribution becomes identical to the Kumaraswamy distribution (see Jones 2009), which gives us a simple closed-form quantile function,

$$\begin{aligned} Q(p) = 1-(1-p)^{1/\beta } = 1-(1-p)^{\sigma _w},\;\;\;\ p \in (0,1). \end{aligned}$$

From this, we obtain

$$\begin{aligned} \mathrm{med(Mc)} = 1- 2^{-\sigma _w}. \end{aligned}$$

1.2 B.2 Management as an output shifter.

We have

$$\begin{aligned} {{\mathrm{Ms}}} = \exp \{w\} \implies \ln ({{\mathrm{Ms}}}) = w \implies \frac{\partial w}{\partial {{\mathrm{Ms}}}} = \frac{1}{{{\mathrm{Ms}}}}. \end{aligned}$$

Then

$$\begin{aligned} f_{{{\mathrm{Ms}}}}({{\mathrm{Ms}}}) = \frac{1}{{{\mathrm{Ms}}}} \frac{1}{\sigma _w}\exp \left\{ \frac{-1}{\sigma _w} \ln ({{\mathrm{Ms}}})\right\} = \frac{1/\sigma _w}{{{\mathrm{Ms}}}^{(1+1/\sigma _w)}}\,. \end{aligned}$$

This is the density of a Pareto distribution with minimum value 1 and shape parameter $\alpha = 1/\sigma _w$.

If they exist, the moments are given by

$$\begin{aligned} E({{\mathrm{Ms}}})= & {} \frac{\alpha }{\alpha -1}=\frac{1/\sigma _w}{1/\sigma _w-1}=\frac{1}{1-\sigma _w},\\ \mathrm{Var(Ms)}= & {} \frac{\alpha }{(\alpha -1 )^2(\alpha -2)} = \frac{1/\sigma _w}{(1/\sigma _w -1)^2(1/\sigma _w-2)} \\= & {} \frac{1/\sigma _w}{(1-\sigma _w)^2(1-2\sigma _w)}\cdot \frac{1}{1/\sigma _w^3}\\= & {} \frac{\sigma _w^2}{(1-\sigma _w)^2(1-2\sigma _w)} \implies \mathrm{SD(Ms)}=\frac{\sigma _wE({{\mathrm{Ms}}})}{\sqrt{1-2\sigma _w}}. \end{aligned}$$

For the median of this Pareto distribution, we have the quantile function

$$\begin{aligned} Q({p}) = \frac{1}{(1-p)^{1/\alpha }} = \frac{1}{(1-p)^{\sigma _w}},\;\;\;\ p \in (0,1) \implies \mathrm{med(Ms)} = 2^{\sigma _w}. \end{aligned}$$

1.3 B.3 Technical efficiency

Here we have

$$\begin{aligned} \mathrm{TE} = e^{-u} \implies u = -\ln (\mathrm{TE}) \implies \frac{\partial u}{\partial \mathrm{TE}} = -\frac{1}{\mathrm{TE}}. \end{aligned}$$

Then

$$\begin{aligned} f_{\mathrm{TE}}(\mathrm{TE}) = \frac{1}{\mathrm{TE}} \frac{1}{\sigma _u}\exp \left\{ \frac{-1}{\sigma _u} \left[ -\ln (\mathrm{TE})\right] \right\} = \frac{1}{\sigma _u} \left( \mathrm{TE}\right) ^{1/\sigma _u -1}. \end{aligned}$$

This is the density of a Beta distribution, with parameters $\alpha =1/\sigma _u, \beta =1$, and applying the moment expression for the specific parameters we obtain

$$\begin{aligned} E(\mathrm{TE})= & {} \frac{\alpha }{\alpha +\beta }=\frac{1/\sigma _u}{1+1/\sigma _u}=\frac{1}{1+\sigma _u},\\ \mathrm{Var(TE)}= & {} \frac{\alpha \beta }{(\alpha + \beta )^2(\alpha +\beta +1)} = \frac{1/\sigma _u}{(1 + 1/\sigma _u)^2(2+1/\sigma _u)} \\= & {} \frac{1/\sigma _u}{(1+\sigma _u)^2(1+2\sigma _u)}\cdot \frac{1}{1/\sigma _u^3}\\= & {} \frac{\sigma _u^2}{(1+\sigma _u)^2(1+2\sigma _u)} \implies \mathrm{SD(TE)}=\frac{\sigma _uE(\mathrm{TE})}{\sqrt{1+2\sigma _u}}. \end{aligned}$$

The quantile function here is

$$\begin{aligned} Q(p) = p^{1/\alpha } =p^{\sigma _u} ,\;\;\;\ p \in (0,1). \end{aligned}$$

From this we obtain

$$\begin{aligned} \mathrm{med(TE)} = 2^{-\sigma _u}. \end{aligned}$$

1.4 B.4 The management net multiplier

We are examining the random variable $\exp \{z\},\;z = w-u$, with w, u independent. When they exist, we can obtain the mean and standard deviation of ${{\mathrm{Mm}}} = \exp \{w-u\}$ using the moment generating function of the Exponential distribution,

$$\begin{aligned} E[e^{w-u}]= & {} E[\exp \{w\}] \cdot E[\exp \{-u\}]= \frac{1/\sigma _w}{(1/\sigma _w-1)}\frac{1/\sigma _u}{(1/\sigma _u+1)} \\= & {} [(1-\sigma _w)(1+\sigma _u)]^{-1}. \end{aligned}$$

For the variance, we need

$$\begin{aligned} E[\exp \{z\}^2]= & {} E[\exp \{2w\}\exp \{-2u\}]=\frac{1/\sigma _w}{(1/\sigma _w-2)}\frac{1/\sigma _u}{(1/\sigma _u+2)}\\= & {} [(1-2\sigma _w)(1+2\sigma _u)]^{-1}. \end{aligned}$$

So

$$\begin{aligned} \mathrm{Var(Mm)} = [(1-2\sigma _w)(1+2\sigma _u)]^{-1}- [(1-\sigma _w)(1+\sigma _u)]^{-2}. \end{aligned}$$

Regarding the median of this distribution, it is a well-known result that the difference of two independent Exponential random variables $z = w-u$ has density

$$\begin{aligned} {{f}_{z}}\left( z \right)= & {} \frac{1}{{{\sigma }_{w}}+{{\sigma }_{u}}} {\left\{ \begin{array}{ll} \text {exp}\left\{ {z}/{{{\sigma }_{u}}}\; \right\} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z\le 0 \\ {} \\ \text {exp}\left\{ -{z}/{{{\sigma }_{w}}}\; \right\} \,\,\,\,\,\,\,\,\,\,\,\,z>0\,. \\ \end{array}\right. } \end{aligned}$$

This leads to the distribution function

$$\begin{aligned} F_Z(z)= {\left\{ \begin{array}{ll} \frac{\sigma _u}{\sigma _w+\sigma _u}\exp \{z/\sigma _u\} &{} z \le 0 \\ \\ 1-\frac{\sigma _w}{\sigma _w+\sigma _u}\exp \{-z/\sigma _w\} &{} z > 0\,. \\ \end{array}\right. } \end{aligned}$$

For the ${{\mathrm{Mm}}}$ variable, we have

$$\begin{aligned} \Pr \left( e^z \le {{\mathrm{Mm}}}\right) = \Pr \left( z \le \ln {{\mathrm{Mm}}} \right) = F_{Z}\left( \ln {{\mathrm{Mm}}}\right) , \end{aligned}$$

and so

$$\begin{aligned} F_{{\mathrm{Mm}}}({{\mathrm{Mm}}})= {\left\{ \begin{array}{ll} \frac{\sigma _u}{\sigma _w+\sigma _u}{{\mathrm{Mm}}}^{1/\sigma _u} &{} {{\mathrm{Mm}}} \le 1 \\ \\ 1-\frac{\sigma _w}{\sigma _w+\sigma _u}{{\mathrm{Mm}}}^{-1/\sigma _w} &{} {{\mathrm{Mm}}} > 1\,. \\ \end{array}\right. } \end{aligned}$$

The corresponding quantile function is

$$\begin{aligned} Q_{{\mathrm{Mm}}}({p})= {\left\{ \begin{array}{ll} \left( \frac{\sigma _w+\sigma _u}{\sigma _u}\cdot p \right) ^{\sigma _u} &{} 0< {p} \le \frac{\sigma _u}{\sigma _w+\sigma _u} \\ \\ \left( \frac{\sigma _w+\sigma _u}{\sigma _w}\cdot (1-p) \right) ^{-\sigma _w} &{} \frac{\sigma _u}{\sigma _w+\sigma _u}< {p} < 1 \,. \\ \end{array}\right. } \end{aligned}$$

From this, we can obtain the expression for the median shown in the main text.