# Analysis of productivity at the firm level: an application to life insurers.

Analysis of Productivity at the Firm Level: An Application to Life
Insurers: Comment

The non-parametric measurement of total factor productivity (TFP) has become widely accepted and applied to many industries [2], and more recently to life insurers. A limitation of this approach is that differences in TFP may arise due to factors other than production efficiency. Denny, Fuss and Waverman [5], for example, identify two additional attributes to differences in TFP across time: exploitation of economies of scale, and changes in deviation from marginal cost pricing.

To help overcome these limitations, various researchers employed the parametric technique of neoclassical cost estimation [10]. The parametric approach is a more general and detailed specification of the production structure of an industry or a company than the TFP approach. Further, in most recent studies of this type, researchers have utilized flexible functional forms in order to avoid imposing unnecessary restrictions on the production technology (see for example Berndt and Khaled [1] among many others). This approach is appealing but is often expensive to undertake and requires substantial time series observations.

For practitioners, however, TFP analysis has various advantages. TFP is a relative measure showing how the ratio of total output to total input changes from one period to the other. It is relatively inexpensive to perform and since the data are displayed in an index number form, it is easy to identify anomalies in the data. In addition, TFP and the variables used in its construction may reveal valuable information on trends and changes in an industry. Moreover, Diewert and Morrison [7] have added another factor, the terms of trade effect, in explaining productivity.

This Comment extends the study of Weiss [13] in which new techniques for measuring output of life insurers are developed and used in computing divisia and exact indexes of TFP for one stock and one mutual insurer over a five-year interval. Weiss [13] maintains that because the rate of technological change calculated using the Tornqvist approximation (TFP) equals the exact measure of the shift of the variable cost function due to technological change ( t) (see Diewert [6]), the nature of returns to scale is constant. In her words, "Productivity theory suggests that superlative indexes such as the exact index and the Tornqvist-Theil approximation to the divisia index yield similar results if the production function reflects constant returns to scale, regardless of whether the insurer is acting competitively..." [13, p. 74]. Given the result obtained by Weiss, which indicates that the divisia and exact indexes vary directly with each other, it was concluded that "...the sample insurers' production functions exhibited constant returns to scale over the sample periods" [13, p. 74].

While this is true if all inputs are variable, if some of the inputs are fixed (e.g. number of square feet of home office building and constant dollar capital input in Weiss), the statement is in error, primarily because the shift in the variable cost function due to technological change ( t) is greater than the negative value of the overall production relationship's rate of technological change ( t). that is, t at t where at is the mean share of variable relative to total costs (at < 1).

The latter can be proven with the aid of two equations:

t = 1/2 (at + at - 1) t + R ( ) where t is the shift in total cost function due to technological change and R ( ) is a remainder term of first differences which are at least of second order.

t = - t + R* ( ) where t is the rate of technological change corresponding to the joint production function (of all inputs), and R* ( ) is a remainder term of finite differences which are at least of second order.(1)

Now suppose there is a constant mark-up, over marginal cost pricing structure. Such a structure allows for non-constant returns to scale. It follows that with perfectly competitive input markets:

t = -TFP + [1 - 1/2 H/ 1n qi t - 1

+ H/ 1n qi t)] Qc + ( Qp - Qc) where H is the joint total cost function of all outputs, all input prices and t (i.e. 1n TC(t) = H(1n q1(t),..., 1n qN(t), 1n W1(t),..., 1nWM + J(t), t), and Qc and QP are the aggregate output changes using cost elasticity and revenue shares, respectively. A discrete proof of the above, based on the work of Denny, Fuss, and Waverman [5], can be found in Theorem 2 of Chan, Krinsky and Mountain [3].

(1)Detailed proofs of this proposition can be found in Chan, Krinsky and Mountain [3]). Moreover, this is not an exact relationship, even if it is assumed that the variable cost function is translog, because the translog profit, production, variable cost, and total cost functions are not self-dual.

Given that t at t, if the industry is characterized by a uniform markup pricing structure over marginal costs, ( QP = Qc), then

t = TFP - [1 - 1/2 ( H/ 1n qi t - 1 + H/ 1n qit)] QP. Weiss [13] finds that t = - TFP [see 13, pp. 54 and 74]. Thus,

at TFP = TFP - [1 - 1/2 ( H/ 1n qi t-1 + H/ 1nqi t)] QP.

Since 1/2 ( H/ 1nqi t - 1 + H/ 1n qi t) represents a weighted cost elasticity with respect to output, it also equals the reciprocal of returns to scale 0 (see Denny, Fuss, and Waverman [5]). Thus, 0 = 1/(1 + (TFP/ QP) (at - 1)). For 0 > 1 (i.e. increasing returns to scale), and since at < 1, TFP must be of the same sign as QP. For 0 < 1 (i.e., decreasing returns to scale) TFP and QP must be of opposite signs.

Using data found in Weiss [13] and background data provided by Weiss, we were able to calculate mean estimates of a, TFP and QP for a mutual insurer for the period 1975 through 1979 and for a stock insurer for the period 1976 through 1980. These mean values along with the associated computations of 0 are presented in Table 1. Weiss' data set indicates increasing returns to scale for the stock insurer 0 = 1.0893 (dividend growth model) and 1.0984 (CAPM) and decreasing returns to scale (0 = 0.8093) for the mutual insurer.

These findings, although specifically associated with this data set, should augment previous studies which sought evidence of economies of scale in the life insurance industry (see for example, Praetz [12], Geehan [9], Kellner and Matthewson [10], and Fields [8]). Geehan [9] found statistically (not economically) significant returns to scale in the Canadian life insurance industry but his evidence is inconclusive. Praetz [12] concludes his study on a sample of 90 insurers conducting business in New York by stating that "the evidence of economies of size is remarkably strong" (p. 532). Praetz's results were later criticized by Kott [11] who claimed that they are based on the misuse of regression analysis. Daly, Geehan and Rao [4] found that the operations of relatively small insurers are characterized by significant scale economies, while those of the relatively large one exhibit significant scale diseconomies. More recently, Fields [8] using a Translog cost function reports significant diseconomies of scale and significant economies of scope, at the mean output, for the life insurers in his sample. "...The economies of scale finding is not consistent with the findings of Kellner and Matthewson for Canadian data. The scope finding is also contradictory to their results..." [8, p. 120].

Weiss' [13] research "...represents a first attempt to meaningfully measure productivity for life insurers and because it is a first attempt, a number of extensions are possible" ([13, p. 79]. The purpose of this discussion is to do just that. Economies of scale can be identified when TFP and exact measure are computed.

Table : Calculations of Returns to Scale

The non-parametric measurement of total factor productivity (TFP) has become widely accepted and applied to many industries [2], and more recently to life insurers. A limitation of this approach is that differences in TFP may arise due to factors other than production efficiency. Denny, Fuss and Waverman [5], for example, identify two additional attributes to differences in TFP across time: exploitation of economies of scale, and changes in deviation from marginal cost pricing.

To help overcome these limitations, various researchers employed the parametric technique of neoclassical cost estimation [10]. The parametric approach is a more general and detailed specification of the production structure of an industry or a company than the TFP approach. Further, in most recent studies of this type, researchers have utilized flexible functional forms in order to avoid imposing unnecessary restrictions on the production technology (see for example Berndt and Khaled [1] among many others). This approach is appealing but is often expensive to undertake and requires substantial time series observations.

For practitioners, however, TFP analysis has various advantages. TFP is a relative measure showing how the ratio of total output to total input changes from one period to the other. It is relatively inexpensive to perform and since the data are displayed in an index number form, it is easy to identify anomalies in the data. In addition, TFP and the variables used in its construction may reveal valuable information on trends and changes in an industry. Moreover, Diewert and Morrison [7] have added another factor, the terms of trade effect, in explaining productivity.

This Comment extends the study of Weiss [13] in which new techniques for measuring output of life insurers are developed and used in computing divisia and exact indexes of TFP for one stock and one mutual insurer over a five-year interval. Weiss [13] maintains that because the rate of technological change calculated using the Tornqvist approximation (TFP) equals the exact measure of the shift of the variable cost function due to technological change ( t) (see Diewert [6]), the nature of returns to scale is constant. In her words, "Productivity theory suggests that superlative indexes such as the exact index and the Tornqvist-Theil approximation to the divisia index yield similar results if the production function reflects constant returns to scale, regardless of whether the insurer is acting competitively..." [13, p. 74]. Given the result obtained by Weiss, which indicates that the divisia and exact indexes vary directly with each other, it was concluded that "...the sample insurers' production functions exhibited constant returns to scale over the sample periods" [13, p. 74].

While this is true if all inputs are variable, if some of the inputs are fixed (e.g. number of square feet of home office building and constant dollar capital input in Weiss), the statement is in error, primarily because the shift in the variable cost function due to technological change ( t) is greater than the negative value of the overall production relationship's rate of technological change ( t). that is, t at t where at is the mean share of variable relative to total costs (at < 1).

The latter can be proven with the aid of two equations:

t = 1/2 (at + at - 1) t + R ( ) where t is the shift in total cost function due to technological change and R ( ) is a remainder term of first differences which are at least of second order.

t = - t + R* ( ) where t is the rate of technological change corresponding to the joint production function (of all inputs), and R* ( ) is a remainder term of finite differences which are at least of second order.(1)

Now suppose there is a constant mark-up, over marginal cost pricing structure. Such a structure allows for non-constant returns to scale. It follows that with perfectly competitive input markets:

t = -TFP + [1 - 1/2 H/ 1n qi t - 1

+ H/ 1n qi t)] Qc + ( Qp - Qc) where H is the joint total cost function of all outputs, all input prices and t (i.e. 1n TC(t) = H(1n q1(t),..., 1n qN(t), 1n W1(t),..., 1nWM + J(t), t), and Qc and QP are the aggregate output changes using cost elasticity and revenue shares, respectively. A discrete proof of the above, based on the work of Denny, Fuss, and Waverman [5], can be found in Theorem 2 of Chan, Krinsky and Mountain [3].

(1)Detailed proofs of this proposition can be found in Chan, Krinsky and Mountain [3]). Moreover, this is not an exact relationship, even if it is assumed that the variable cost function is translog, because the translog profit, production, variable cost, and total cost functions are not self-dual.

Given that t at t, if the industry is characterized by a uniform markup pricing structure over marginal costs, ( QP = Qc), then

t = TFP - [1 - 1/2 ( H/ 1n qi t - 1 + H/ 1n qit)] QP. Weiss [13] finds that t = - TFP [see 13, pp. 54 and 74]. Thus,

at TFP = TFP - [1 - 1/2 ( H/ 1n qi t-1 + H/ 1nqi t)] QP.

Since 1/2 ( H/ 1nqi t - 1 + H/ 1n qi t) represents a weighted cost elasticity with respect to output, it also equals the reciprocal of returns to scale 0 (see Denny, Fuss, and Waverman [5]). Thus, 0 = 1/(1 + (TFP/ QP) (at - 1)). For 0 > 1 (i.e. increasing returns to scale), and since at < 1, TFP must be of the same sign as QP. For 0 < 1 (i.e., decreasing returns to scale) TFP and QP must be of opposite signs.

Using data found in Weiss [13] and background data provided by Weiss, we were able to calculate mean estimates of a, TFP and QP for a mutual insurer for the period 1975 through 1979 and for a stock insurer for the period 1976 through 1980. These mean values along with the associated computations of 0 are presented in Table 1. Weiss' data set indicates increasing returns to scale for the stock insurer 0 = 1.0893 (dividend growth model) and 1.0984 (CAPM) and decreasing returns to scale (0 = 0.8093) for the mutual insurer.

These findings, although specifically associated with this data set, should augment previous studies which sought evidence of economies of scale in the life insurance industry (see for example, Praetz [12], Geehan [9], Kellner and Matthewson [10], and Fields [8]). Geehan [9] found statistically (not economically) significant returns to scale in the Canadian life insurance industry but his evidence is inconclusive. Praetz [12] concludes his study on a sample of 90 insurers conducting business in New York by stating that "the evidence of economies of size is remarkably strong" (p. 532). Praetz's results were later criticized by Kott [11] who claimed that they are based on the misuse of regression analysis. Daly, Geehan and Rao [4] found that the operations of relatively small insurers are characterized by significant scale economies, while those of the relatively large one exhibit significant scale diseconomies. More recently, Fields [8] using a Translog cost function reports significant diseconomies of scale and significant economies of scope, at the mean output, for the life insurers in his sample. "...The economies of scale finding is not consistent with the findings of Kellner and Matthewson for Canadian data. The scope finding is also contradictory to their results..." [8, p. 120].

Weiss' [13] research "...represents a first attempt to meaningfully measure productivity for life insurers and because it is a first attempt, a number of extensions are possible" ([13, p. 79]. The purpose of this discussion is to do just that. Economies of scale can be identified when TFP and exact measure are computed.

Table : Calculations of Returns to Scale

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Title Annotation: | comment and author's reply |
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Author: | Chan, M.W. Luke; Krinsky, Itzhak; Mountain, Dean C.; Weiss, Mary A. |

Publication: | Journal of Risk and Insurance |

Date: | Jun 1, 1989 |

Words: | 1285 |

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