A Multi-Product Cost Study of the U.S. Life Insurance Industry

Transcription

1 A Multi-Product Cost Study of the U.S. Life Insurance Industry By Dan Segal Rotman School of Management University of Toronto 105 St. George St. Toronto, ON M5S-3E6 Canada

2 A Multi-Product Cost Study of the U.S Life Insurance Industry Abstract This paper reports estimates of overall and product specific economies of scale, as well as, economies of scope for the main outputs of the life insurance industry. In addition, the paper presents a number of structural tests of the production technology. The results indicate that the estimated cost function satisfies the regularity conditions, and that the industry exhibits overall and product specific economies of scale. In addition, we document significant economies of scope between the main lines of business.

3 Keywords: Life Insurance, Cost Function, Economies of Scale, Economies of Scope 3

4 A Multi-Product Cost Study of the U.S Life Insurance Industry This paper presents a number of structural tests of the production technology of life insurance companies in the U.S. A full system of translog cost equations is estimated using detailed cross-sectional data on life insurance companies. Using the estimated parameters we test for economies of scale and economies of scope. Economies of scale and scope may affect managerial decision regarding the scale and mix of outputs, and potential cost savings. The potential for cost economies is important for life insurers because of the importance of cost control in the industry. The life insurance industry can be characterized as mature and highly competitive, with fairly homogeneous products and services and comparable providers of insurance. In addition, Few financial inventions can be patented, and most innovations are copied shortly after their introduction. Consequently, success in this industry depends increasingly on the insurer s ability to control costs. Life insurance companies may operate in one or more lines of business, where the main lines of business are life insurance, annuities, and accident & health (A&H) policies. The efficient operation of all three lines of business requires considerable amounts of economies of scale generated by business volume. The life insurance and A&H lines of businesses are forms of insurance, and therefore, they rely on the law of large numbers to insure the collective pooling of insurance risks. Annuities are a form of saving vehicle, and hence, this line of business also requires economies of scale. Since the magnitude of economies of scale may vary by company size, we examine the overall 4

5 scale economies and product specific economies of scale at different points of the data: at the overall mean and at the means of size based quartiles. Most life insurance policies, in particular long duration policies such as whole life and universal-variable polices, provide both risk protection and savings elements. Given this common attribute of long-term life policies and annuities, we expect economies of scope between the life insurance and the annuities lines of business. In addition, since life policies and A&H are forms of insurance, we also hypothesize that there is economies of scope between the life insurance and the A&H lines of business. Economies of scope are said to exist if the cost of producing several products by one firm is smaller than the overall costs of production of several specialty firms. In addition to exploring the economies of scale and scope of the sample firms, this paper conducts a series of tests to determine if the estimated cost function satisfies the regularity conditions: monotonicity in prices and outputs, concavity in prices and positive marginal costs. We also test whether the production technology can be characterized as homothetic, which implies that the production function is separable in prices and outputs. One of the implications of a non-homothetic technology is that economies of scale should be estimated using a cost function rather than a production function because the estimation of economies of scale using a production function assumes that the expansion path is a ray emanating from the origin, which is true only for homothetic technology. Early studies of the production structure of life insurance companies do not incorporate the multi-product nature of the industry and limit their analysis to the estimation of single output measure (see Geehan (1986) for a survey of studies that use single output measure). In addition, these studies report scale economies at the sample 5

6 mean level of output only. Fields (1988) account for the multi-product nature of the industry in the estimation of the cost function, but reports estimates of economies of scale and scope at the sample mean of outputs only. In a more recent study, Grace and Timme (199) provides a more complete analysis of the industry s cost structure and reports estimates of overall and product specific scale economies as well as economies of scope between a variety of products at the sample mean of outputs and at the size based quartiles means of outputs. This study differs from Grace and Timme (199) primarily in the choice of life insurance output and in model specification. The results indicate that the estimated cost function satisfies the regularity conditions at the overall means and at the means of the size based quartiles, and that the production technology is not homothetic. The sample firms exhibit overall and product specific economies of scale. The magnitude of scale economies is decreasing with firm size, although the largest firms still show significantly increasing returns to scale. We also find that the main lines of business exhibit product specific economies of scale for all companies. The results also support the hypothesis of economies of scope between the life insurance and the annuities lines of businesses and between the life insurance and the A&H lines of businesses. The study is organized as follows. An introduction to multiproduct cost function is presented next. Section III describes the model specification and the data. Section IV shows the empirical results, and Section V concludes. II. Multiproduct Cost Functions Duality theory implies that given certain regularity conditions of the cost function (non-negativity, non-decreasing in prices (P) and outputs (Y), concave and continuous in 6

7 P, linearly homogeneous in P, and no fixed costs) there exist cost and production functions that are dual to each other. Hence, the structure of the production technology can be analyzed using either a production function or a cost function; the choice should be made based on statistical grounds. Production function estimation is appropriate under the assumptions of profit maximization and endogenous output levels, whereas cost function estimation is preferred under the assumptions of exogenous outputs and input prices. Since the life insurance industry is highly competitive, where each firm tries to maximize its profits, both outputs and input prices are exogenous to the insurers. In general, absent overall risk considerations, life insurance companies meet the demand for their products as determined by the market. The competitiveness of the industry also suggests that firms compete for their inputs (capital and labor), and therefore, input prices are exogenous as well. It follows then that is reasonable to estimate a cost function rather than a production function. In addition, given the multi-product characteristic of the industry, cost function estimation has an empirical advantage over production function estimation becuase cost functions allow incorporating multiple outputs. Direct estimation of a production function for a multi-product industry requires aggregation of the various outputs into one measure of output. Aside from the problematic aggregation and its statistical implications on the properties of the estimated function, production function estimation also limits the researcher ability to investigate certain properties such as product specific economies of scale and economies of scope. 7

8 The economics literature provides numerous cost functions, such as Cobb- Douglas, Leontief, CES, and Linear functions. These functions, however, place a-priori restrictions on either the substitution possibilities among the factors of production or on scale economies. In this study we use the translog cost function, which is a flexible functional form that can be used to approximate any twice-differentiable function without placing a-priori restrictions on the production technology. 1 Furthermore, the function allows testing for the restrictions imposed by the other cost functions. The translog cost function can be written as Ln (C) = α 0 + j α j ln(p j ) + k β k ln(y k ) + ½ j i γ ji ln(p j ) ln(p i ) + ½ k l δ kl ln(y k ) ln(y l ) + j k ρ ik ln(p j ) ln(y k ) + ε (1) Where P i is the price of input i, Y k is output k, C is total costs, and α 0, α j, β k, γ ji, δ kl, ρ ik are the parameters to be estimated. Regularity Conditions To correspond to a well-behaved production structure, the cost function must satisfy the following regularity conditions: continuity, symmetry, linear homogeneity in prices, monotonicity in prices and outputs, and concavity in prices. The continuity of the translog cost function and of its first and second derivatives follow from inspection. Since the function is continues and twice differential, the second cross derivatives are symmetric -- γ ji = γ ij and δ kl = δ lk. Linear homogeneity in prices requires that ln(c(tp,y)) = ln(tc(p,y)) = ln(t) + ln (C) for all P 8

9 Linear homogeneity ensures that the cost-minimizing bundle does not change if all prices are multiplied by the same positive scalar, and therefore, maintains the basic property that only the ratios of the inputs prices affect the allocation of inputs. Linear homogeneity implies the following restrictions on the parameters: j α j = 1, j γ ij = 0, l ρ ik = 0 Monotonicity in outputs requires positive marginal costs (dc/dy>0) and monotonicity in prices (dc/dy>0) requires that total costs increase as input prices increase. For the translog cost function, C/ Y i = (β k + j δ kl ln(y l ) + k ρ ik ln(p j ))*C/Y k (Monotonicity in the k th output ) C/ P i = C/Pi * ln(c)/ ln(p i ) = C*S i /P i (Monotonicity in i th input price) where Si is the cost share of input i. The cost share, which is observable, represents the proportion of total costs that is attributed to input i. The derivation of the monotonicity in prices relies on Sheppard s lemma (Shepard (1970)), which states that the first derivative of the cost function with respect to the price of input i (P i ) is equal to the input itself (X i ) - ln (C)/ ln (P i ) = P i C i /C = P i X i /C = S i where Ci is the first derivative of total costs with respect to input i. It follows that the translog ensures global monotonicity in prices but not in outputs since monotonicity in outputs is a function of the parameters of the cost function. Finally, a cost function must be concave in prices. As the price of an input increases the proportion increase in total costs should be no higher because of the substitution among inputs. One of the main limitations of the translog cost function is that it cannot assure global concavity. Sufficient condition for concavity is that the bordered 9

10 hessian matrix of C(Y, P) is negative semi-definite. The bordered hessian of the translog cost function is computed as, B H = S S S / P1 / P / P 3 1/ P 1 S ( γ / P1P( γ 1/ P1P3( γ / P 1 S1( S S1S S1S 1)) 3 ) ) 1/ P1P( γ 1/ P 1/ P ( γ S + P3( γ / P 1 + S + 3 S1S ( S S ) 1)) S 3 ) 1/ P1P3( γ 1/ P 1/ P 3 P3( γ ( γ S / P S1S S S ( S 3 S 3 ) 3 ) 1)) Functional Form The translog cost function is general enough to encompass homothetic and homogeneous production structures. A cost function is homothetic if and only if it is separable in input prices and outputs, that is, it can be written as C(p, y) = h(p)φ(y), for any p>0 and y>0. The translog cost function corresponds to a homothetic production function if ρ ik = 0 () Linear homogeneous production function is a special class of homothetic production function. A production function is homogeneous of degree i if it can be expressed as C (p, y)=y i h(p). Linear homogeneous production function (i=1) implies constant average costs and, therefore, constant returns to scale. 3 Linear homogeneous production function requires in addition to () that δ kl = 0 4 (3) Unitary elasticity of substitution implies that the percentage change in the i th input equals to the percentage change in the j th input price. Unitary elasticity of substitution of the production function requires the second order terms of the prices in the cost function to be zero. That is, 10

11 γ ji = 0 (4) Elasticity of Substitution, and Economies of Scale and Scope Elasticity of substitution Further insights into the reasonableness of the cost function are provided by examining the Allen-Uzawa own-price elasticities and partial elasticities of substitution. Economics theory requires that the own-price elasticities be negative and that the substitution matrix be negative semi-definite. The Allen-Uzawa elasticities of substitution can be computed from the cost function as σ ij = CC ij /C i C j For the translog function, the partial elasticities of substitution are computed as σ ij = (γ ji + S i S j )/S i S j (5) and the own price elasticities are calculated as σ ii = (γ ii + S i (S i 1))/S i (6) Economies of Scale Elasticity of scale (ESC) measures how the output responds as inputs are increased proportionally. Constant return to scale implies that as inputs are increased by λ so does output. If economies (diseconomies) to scale exist, the output increases by a proportion greater (smaller) than λ. However, since ESC is computed at points on a ray emanating from the origin, this measure of scale economies would be biased for any nonhomogeneous production function. 11

12 Elasticity of size (ESZ) measures the percentage increase in total cost as output increases along the expansion path. 5. ESZ is computed as θ (y) = ln(c)/ ln(y) = ( C/ Y)/(Y/C) = MC/AC (7) IRS/CRS/DRS (increasing/constant/decreasing) return to scale prevails if θ<1/θ =1/ θ>1. For a multi-product cost function, overall scale economies may be computed as, SE (Y) = ( k ln(c)/ ln(y k )) (Panzar & Willig 1977) Thus, for the translog cost function, SE (Y) = [ k β k + k l δ kl ln(y l ) + k j ρ ik ln(p j )] (8) and product specific economies of scale is computed as θ k (Y) = ln(c)/ ln(y k ) = β k + l δ kl ln(y l ) + j ρ ik ln(p j ) (9) Economies of scope Economies of scope are said to exist if the cost of producing n goods by one firm is less costly than the combined costs of production of m specialty firms. Cost complementarities measure how the costs of producing particular output for a multiproduct firm vary with the levels of output of other commodities. Panzar and Willig (1981) argue that economies of scope and cost complementarities arise from sharable, quasi-public input -- an input that may be shared by two or more product lines without complete congestion. Total economies of scope may be computed as, SC = [ C(Y 1,0) + C(0,Y ) C(Y 1,Y )]/ C(Y 1,Y ) SC measures the percentage reduction in costs due to joint production. A sufficient condition for overall economies of scope is that the cost function exhibits weak cost complementarities between each pair of outputs. That is, 1

13 C(Y)/ Y k Y l <=0, k l. This is equivalent to testing whether δ kl + β k β l <=0 (10) for any pair k, l outputs. III. Data and Model Specification Data The insurance financial data were obtained from the regulatory annual statement filed by insurers as reported by the National Association of Insurance Commissioners (NAIC) life insurance data tapes for 1995 through1998. Because the NAIC tapes do not include the number of employees and agents whom insurers employ--information required to adequately estimate labor and its price we collected these data from two sources: responses to a survey that requested the number of companies employees and agents, and the Life Office Management Association s (LOMA s) Expense Management Program (EMaP). 6 The initial sample consisted of 733 observations (company-years). We excluded from the sample companies that had fewer than 10 employees and agents (78 observations), firms that did not sell either term or whole life policies (46 observations), and those for which the data show negative direct premiums, revenues, benefits, commissions, amount of insurance, labor-related expenses, or capital expenses (10 observations). The final sample consists of 489 observations: 11 firms in 1995, 16 firms in 1996, and 11 firms in each of 1997 and

14 Model Specification Outputs Like all service sectors, the life insurance industry presents difficulties of output definitions and measurement. Most studies identify outputs with lines of business - that is, life policies, annuities, and accident and health (A&H) - whereas some add investment income as an additional output. The major differences among studies of the cost structure of the industry are in output measurement. Geehan (1986) provides a useful discussion of the issues involved, and compares the output measures of major studies. Grace and Timme (199), Gardner and Grace (1993), and Fecher et al. (1993) measure outputs as the dollar value of premiums and annuity considerations. Premiums are, however, a questionable measure of life policies. They represent not physical output but rather revenues (price times number of policies). Furthermore, for whole life insurance policies, only a portion of the premium covers the risk-bearing that life insurance companies provide to the insured. The remaining portion covers the savings element of the policy; it therefore actually belongs to the insured and cannot be considered as revenue of the insurer. Yungert (1993) measures outputs by additions to reserves. The major problem with this measure is that reserves change when policies age, regardless of whether new policies are sold. In addition, the change in reserves measures the change in liabilities, rather than the output of the selling effort. In a more recent study, Cummins and Zi (1998) distinguish between the two principal services provided by life insurance companies: risk bearing/pooling, and intermediation services. As a measure of the former, they use incurred benefits by line of business, whereas for the latter they use 14

15 additions to reserves. Here again the output measure is disputable. Benefits represent obligations that were incurred in the past; hence they measure past cumulative output, not current output. Following Cummins and Zi (1998), we characterize the outputs by the service provided. Life policies give either pure risk protection (through term life policies) or a mix of risk protection and intermediation services (through whole life policies). Annuities can be viewed as saving vehicles and, therefore, the service they provide can be characterized as intermediation. A&H policies, on the other hand, provide risk protection service alone. The risk bearing/pooling services that companies provide on new life insurance policies can be approximated by the total amount of insurance sold during the year 7. That amount measures the outcome of the selling effort and the additional risk that the company bears and, therefore, can represent the output of the life insurance line of business. 8 Furthermore, this output measure may be appropriate for all types of life policies, both term life and whole life. Profits and losses from annuities arise from the difference between the actual return on investments and the return credited to the contracts. Assuming a positive spread, the larger the annuity considerations (premiums) the larger is the expected profit. Hence, a plausible proxy for this output is annuity considerations, which represent the increase in the earning base of this line of business. A&H policies primarily provide risk protection. Since one cannot quantify the amount of risk associated with each new policy, we use A&H premiums as a proxy for 15

16 these policies output. In equilibrium, where the risk associated with A&H policies is priced correctly, premiums serve as a good proxy for risk. To sum up, we use three outputs: amount of life insurance, total annuity considerations and total A&H premiums. 9 Inputs and Inputs Prices For this study we employ three inputs: labor, capital, and other. Labor is defined as the number of employee-days. The price of labor is computed as the total cost of employees and agents divided by their total number. Capital comprises two components: financial capital, defined as book value of equity plus the asset valuation reserve (AVR) 10 and physical capital, defined as the sum of capital expenses - rent, rental of equipment, and depreciation 11. We define the price of capital as the opportunity cost of holding the financial capital and measure it as the difference between the ratio of five years total net income to total financial capital (return on equity) and the ratio of total investment income to total assets (return on investments) over the same period 1,13. Our third input ( other ) consists of all operating expenses other than labor and capital expenses. Most of these expenses are related directly to selling and servicing policies. We use the number of policies sold and terminated during the year as a proxy for the number of policies sold and serviced during the year. And we quantify the price of this third input as the related expenses divided by the total number of policies sold or terminated 14. Table 1 shows descriptive statistics of the variables used in the estimation of the cost function. The table indicates that a great amount of variation is present across insurers, and that the sample consists of relatively small and relatively large companies. 16

17 Total costs range from $173,000 million to $13.5 billion, the amount of term life insurance sold ranges from 30,000 to $81 billion. The measures of input prices also vary substantially: Pk, the price of capital, ranges from 0.07 percent to 65 percent, whereas Pm, the price of materials, ranges from $0. to $7,557. The mean labor share is 60 percent, while the means of capital share and materials share are 5 percent and 10 percent respectively. In 1998, the aggregate total assets of the sample firms were about $657 billion, approximately one-third of the total assets in the industry. Thus, our sample covers a material portion of the total assets of the industry. [Insert Table 1] IV. Estimation and Empirical Results Section describes, in general terms, the translog cost function and its properties. We use the following model to estimate the cost function of the life insurance industry: Ln (C/P M ) = α 0 + ϕlnpol + j α j ln(p j /P M ) + i β i ln(y i ) + ½ j i γ ji ln(p j /P M ) ln(p i /P M ) + ½ i l δ kl ln(y i ) ln(y l ) + j i ρ ji ln(p j /P M ) ln(y i ) + ξ (11) S L = α L + γ LL ln (P L /P M ) + γ LK ln (P K /P M ) + i ρ Li ln(p L /P M ) ln(y i ) + ε L (1) S K = α K + γ KK ln (P K /P M ) + γ LK ln (P L /PM) + i ρ Ki ln(p L /P M ) ln(y i ) + ε K (13) where C is total cost, 15 LNPOL is the natural log of the average size of life policies (in terms of the amount of insurance), i indexes the outputs (Y) (amount of insurance, annuity considerations, and A&H premiums), and j indexes the input prices (P) (labor, capital and materials). P L is the price of labor, P K is the price of capital, P M is the price of materials, and S L and S K are the share equations of labor and capital, respectively. We 17

18 divide total cost and all prices by P M to impose the regularity condition of linear homogeneity in prices 16. We use maximum likelihood estimation of constrained system of regression equations (seemingly unrelated regression equations (SURE)) to jointly estimate equations 11 through 13. The input share equations are derived as ln (C)/ ln (P i ) = P i C i /C = P i X i /C = S i The inclusion of the share equations in the estimation increases the efficiency of the estimated parameters since many additional observations are added without adding any parameters. In addition, the joint estimation assures that the estimated shares would be as close as possible to the observed shares. The model includes only two share equations because the sum of the shares is one and, therefore, only two shares are independent. Also, the disturbances of the share equations must sum to zero, hence their covariance matrix is singular. This precludes the estimation unless one share equation is omitted. 17 The estimated cost function appears in table. The parameters that are not estimated directly are inferred from the linear homogeneity restrictions as follows: α M = 1 - α L - α K γ KM = - γ KL - γ KK γ LM = - γ KL - γ LL γ MM = γ LL + γ KK + *γ KL ρ Mi = - ρ Li - ρ Ki (i indexes the outputs AMT, ANN, AH) [Insert Table ] 18

19 Functional form: Table 3 presents the test statistics for the homotheticity, the homogeneity and the unitary elasticity of substitution hypotheses. We test the hypotheses using likelihood ratio tests. The table shows that we reject all the hypotheses at less than 1% confidence level. [Insert Table 3] Concavity: Table 4 presents the bordered hessian (evaluated at the means of the estimated factor shares) of the cost function with respect to the factor prices. The bordered hessian matrix of a concave, twice differential, and continuous function is semidefinite. A sufficient condition for a matrix to be negative (positive) semi-definite is nonnegative (non-positive) eigenvalues. The eigenvalues of the matrix presented in Table 4 are less or equal to zero, indicating that the estimated cost function is concave. The bordered hessians evaluated at the medians of the factor shares and at the mean factor shares of each size quartile are also negative semi-definite. [Insert Table 4] Elasticities of substitution: We use the Allen-Uzawa own-price elasticities and partial elasticities of substitution to estimate the elasticities of substitution among the demand factors. The partial and own price elasticities are computed by equations (5) and (6), respectively. Inputs are substitutes if σ ij >0, and complements if σ ij <0. The estimated own price elasticities, using the average estimated shares of the overall sample, are: σ LL =-0.34, σ KK = -0.56, and σ MM = The own price elasticities are significantly 18 less than zero at the 1% level. The partial elasticities are: σ LK =0.75, 19

20 σ LM = 0.86, and σ KM =0.8 which are also significantly different from zero at the 1% level, indicating that the factors are substitutes. Economies of scale: Overall scale economies is a measure of total cost elasticity as the firm expands its outputs, holding the output mix constant. We measure overall economies of scale using the parameter estimates given in Table at the overall mean and at the mean of each quartile of size (measured as total assets) vectors of outputs. Product specific economies of scale provide insight into the source of the overall scale economies. It can be measured by the curvature of the marginal cost of the product; if the marginal cost is decreasing (increasing) than the product exhibits increasing (decreasing) returns to scale. We compute products specific economies of scale as ω (Yi) = (δ ii θ i (Y i ))*C/Y i where θ i (Y i ) is the first derivative of the natural log of total costs with respect to the natural log of output i (see equation (9)), C is total costs, Y i is output i, and δ ii is the estimated parameter from the translog cost function. Table 5 reports the estimates of the ray and of the products specific scale economies for the overall sample and for each size quartile. The results in Table 5 indicate that the industry exhibits both overall and product specific economies of scale. The overall elasticity of scale is 0.6 and it is significantly different from one at the 1% level, and the curvature of the marginal costs is significantly negative for all outputs. The table also shows that the ray scale economies measure increases monotonically in moving from the first size quartile to the fourth quartile, and that for all quartiles the scale measure differs significantly from one. The curvatures of 0

21 the marginal costs of all outputs across quartiles are negative and significantly different from zero, indicating that product specific economies exist even for the largest firms. [Insert Table 5] A potential reason for the increase in the scale measure with the companies size is increasing marginal costs. Table 6 provides estimates of the marginal costs by quartile of size. The table shows that the marginal cost of amount of insurance is almost constant in the first three quartiles. However, in the fourth quartile the marginal cost is twice as much as in the third quarter (0.6 versus 0.13). The marginal cost of annuities declines monotonically with firm size from 0.3 in the first quartile to 0.1 in the fourth quartile. The marginal cost of accident and health is the same for the first and second quartile (0.6); it decreases in the third quartile to 0.18, but increases to 0.3 in the fourth quartile. Overall, we find that the marginal cost of AMT increases with size, the marginal cost of annuities decreases with size, and the marginal cost of AH does not change monotonically with size. Given that the overall scale economies increases with size, and that the life insurance line of business is the largest among the three lines of business, the reason for the increase in the scale measure with size is probably the increase in the marginal cost of AMT. The results in Tables 5 and 6 are presented graphically in Figures 1 through 4. [Insert Table 6] 1

22 Economies of scope: we test for economies of scope between each pair of outputs by analyzing the pair-wise cost complementaries among the three outputs. The pair-wise cost complementaries are computed as CC kl = C(Y)/ Y k Y l =δ kl + β k β l for any pair k, l outputs. Cost complementary between two outputs exists if CC kl is negative. Table 7 shows the cost complementaries between the three outputs. The results indicate that the cost complementaries statistic is negative and significant (at the 5% level) for the life insurance and annuities lines of business and for the life insurance and A&H lines of business (at 10% level). Since negative cost complimentary is sufficient for economies of scope we conclude that there exist significant economies of scope between the life insurance and the annuities and the accident and health lines of business. Thus, a joint production of all three lines of business by one firm would be cheaper than the overall cost of producing these separate products separately. V. Summary This study investigates the cost structure of the life insurance industry. A translog cost function is estimated to avoid imposing any restrictions on the cost structure. The estimated function satisfies the regularity conditions, and hence is assumed to properly represent the production technology. We test for homotheticity, homogeneity, unitary elasticity of substitution, economies of scale and economies of scope. Our results reject the hypotheses of homothetic production function (and therefore of homogeneity) and of unitary elasticity of substitution among the inputs. The inputs are found to be substitutes. The sample firms exhibit significant increasing returns to scale

23 and product specific returns to scale. The magnitude of scale economies increases with size; nonetheless, the largest firms still exhibit significant increasing returns to scale. A potential reason for the increasing scale economies measure is the increase in the marginal costs of the life insurance line of business output (amount of insurance) with firms size. We also find economies of scope between the life insurance line of business and annuities and accident and health. Economies of scope arise from sharable input, which is defined as an input that may be shared by two or more product lines without complete congestion. Life insurance and accident and health policies offer some form of insurance, whereas most life policies and annuities provide a form of savings. Thus, the similarity in the operations of these lines of business allows for the economies of scope. 3

24 References Barten, A., Maximum Likelihood Estimation of a Complete System of Demand equations, European Economic Review, 1969, 1, 7-73 Chambers, Robert G., Applied Production Analysis, 1988 Cummins, David J. and Hongmin Zi, Comparison of Frontier Efficiency Models: An Application to the U.S Life Insurance Industry, Journal of Productivity Analysis, 1998, 10, Fields, Joseph A., Expense Preference Behavior in Mutual Life Insurers, Journal of Financial Services Research, 1988, 1, Gardner, L., and M. F. Grace, X-Efficiency in the U.S Life Insurance Industry, Journal of Banking and Finance, 1993, 17, Geehan, Randall, Economies of Scale in Insurance: Implications for Regulation, The Insurance Industry in Economic Development, 1986, Grace, Martin F. and Stephen G. Timme, An Examination of Cost Economics in the United States Life Insurance Industry, Journal of Risk and Insurance, 199, 59, Life Office Management Association (LOMA), Inc., Expense Management Program (Emap) Manual, Expense Year 1997 February 1998, 1998 Panzar, John C. and Robert D. Willig. "Economies Of Scale In Multi-Output Production," Quarterly Journal of Economics, 1977, v91(3), Panzar, John C., and Robert D. Willig, Economies of Scope, AEA Papers and Proceedings, Sustainability Analysis, 1981, 68-7 Yungert, A. M., The Measurement of Efficiency in Life Insurance: Estimates of a Mixed Normal-Gamma Error Model, Journal of Banking and Finance, 1993, 17,

25 Table 1 Descriptive Statistics Variable Mean Minimum Maximum Total Cost ($000) 14, ,459,700 Labor Share 60% 9% 96% Capital Share 5% % 89% Materials Share 15% 0.03% 60% AMT ($000) 4,13, ,071,900 ANN ($000) 445, ,44,900 AH ($000) 10, ,56,190 Price of labor 33,055 15,000 10,000 Price of materials ,557 Price of capital AVGPOL 76,43 1,8 86,6 Total assets ($000) 5,159,50 1,86 18,035,000 Notes: 1. Labor Share, Capital Share and Material Share are the proportion of labor, capital and material expenses from total costs.. AMT is the total amount of insurance sold. 3. ANN (AH) is total annuity (accident and health) premiums. Zero values were replaced by AVGPOL is the average policy size in terms of amount of insurance. 5. Total assets are from the annual statement submitted to the National Association of Insurance Commissioners (NAIC). 5

26 Table Translog Cost Function Estimates (Standard errors in parentheses) Parameter Coefficient Parameter Coefficient INTERCEPT (1.73) LNPOL (0.031) LABOR 0.14 (0.094) CAPITAL (0.088) MATERIALS (0.045) AMT (0.196) ANN 0.13 (0.055) AH (0.045) LABOR (0.008) LABOR*CAPITAL (0.007) LABOR*MATERIALS (0.004) CAPITAL (0.007) CAPITAL*MATERIALS (0.003) MATERIALS 0.01 (0.003) AMT (0.013) AMT*ANN (0.004) AMT*AH (0.004) ANN (0.00) ANN*AH (0.00) AH 0.0 (0.003) LABOR*AMT (0.004) LABOR*ANN (0.00) LABOR*AH 0.00 (0.00) CAPITAL*AMT (0.004) CAPITAL*ANN (0.00) CAPITAL*AH (0.00) MATERIALS*AMT (0.003) MATERIALS*ANN (0.001) MATERIALS*AH (0.001) 6

27 Table 3 Test Statistics for Homotheticity, Homogeneity and Unitary Elasticity of Substitution Homotheticity Homogeneity Unitary Elasticity Homotheticity and Unitary Elasticity Homogeneity and Unitary Elasticity Number of restrictions Test Statistic P-value Notes: 1. Homotheticity implies that ρ ji =0.. Homogeneity implies that ρ ji =0 and δ kl =0. 3. Unitary elasticity of substitution implies that γ ji =0. 4. Test Statistic is the likelihood ratio test statistic, which is assumed to have Chi-Square distribution. Table 4 The Bordered Hessian Of The Cost Function, Evaluated At The Means Of The Estimated Factor Shares 0.00E E-05.61E E E E E E-08.61E E E E E E E E-06 Table 5 Overall and product specific Economies of Scale (Standard Errors in Parentheses) Output Sample Mean 1 st Quartile nd Quartile 3 rd Quartile 4 th Quartile All 0.6** (0.0) AMT -0.34E-11** (0.54E-1) ANN -0.13E-09** (0.17E-10) AH -0.19E-08** (0.E-09) 0.38** (0.0) -0.71E-10** (0.14E-10) -0.1E-06** (0.6E-07) -0.88E-07** (0.9E-08) 0.57** (0.0) -0.19E-10** (0.3E-11) -0.11E-07** (0.95E-09) -0.15E-07** (0.99E-09) 0.68** (0.03) -0.64E-11** (0.57E-1) -0.77E-09** (0.57E-10) -0.36E-08** (0.8E-09) 0.84** (0.5) -0.15E-11** (0.79E-13) -0.55E-10** (0.47E-11) -0.63E-09** (0.6E-10) Notes: 1. AMT is the total amount of insurance sold.. ANN (AH) is total annuity (accident and health) premiums. 3. The quartiles are based on size. 4. The overall economies of scale ( All ) are significantly different from 1 at 1% level. 5. All estimates of product specific economies of scale are significantly different from zero at 1% level. 7

28 Table 6 Marginal Costs by Size (Standard Errors in Parentheses) Output Overall 1 st Quartile nd Quartile 3 rd Quartile 4 th Quartile AMT 0.017** (0.001) 0.011** (0.001) 0.014** (0.001) 0.013** (0.0008) ANN 0.07** 0.3** 0.** 0.09** (0.005) (0.04) (0.015) (0.07) AH 0.6** 0.6** 0.7** 0.18** (0.018) (0.0) (0.018) (0.014) Notes: 1. AMT is the total amount of insurance sold.. ANN (AH) is total annuity (accident and health) premiums. 3. The quartiles are based on size. 4. The estimated marginal costs are significantly different from 0 at 1%. 0.06** (0.0) 0.1** (0.009) 0.3** (0.09) Table 7 Economies of Scope (Standard Errors in Parentheses) ANN AH AMT -0.11** (0.067) * (0.046) ANN 0.04 (0.0085) 1. AMT is the total amount of insurance sold.. ANN (AH) is total annuity (accident and health) premiums. 8

29 Figure 1 Economies of Scale Elasticity of Scale Quartile Figure Marginal Cost of Amount of Insurance Cost per Thousand Quartile 9

30 Figure 3 Marginal Cost of Annuities Quartile Figure 4 Marginal cost of A&H Quartile 30

31 ENDNOTES 1 Although the translog cost function represents a very general production structure, it cannot be derived from explicit production function. Proof (Chambers): one of the variables in the right hand-side of the function is γ jj ln(p j ). For a concave function the first (second) derivative is positive (non-positive) over the relevant range. Direct computation of the second derivative gives γ jj (1- ln(p j ))/ P j. Thus, if γ jj >=0, ln(pj)<=1 to ensure concavity. Hence, no parametric restriction on γ ji guarantees global concavity. 3 If the production function is homogeneous of degree greater (less) than one then the returns to scale are constantly increasing (decreasing). 4 Note that if we reject the homotheticity hypothesis the homogeneity hypothesis is also rejected. 5 At cost minimizing points, the two measures coincide, as ESC = ESZ EMaP is a detailed expense study of life insurance companies that chose to participate in the program. LOMA agreed to provide the data as part of a study of the cost structure of the life insurance industry. 7 By using this measure we implicitly ignore the intermediary output associated with whole life policies. In this type of policies, insurance companies make a profit both on the insurance and on the investments of the savings portion of the policy. However, we believe that the main output of the life insurance line of business is the insurance risk assumed by the insurer. Second, given the data limitations, it is impossible to separate the premiums on whole life policies into their insurance and savings components. 31

32 8 Another potential proxy is the change in the amount of insurance in force during the year. It would measure the net additional amount of risk that the company assumes during the year. However, this measure could take on negative values in cases of reinsurance or when the amount of insurance paid is greater than the amount of insurance sold in any given year. 9 Cummins and Zi (1998) and Grace and Timme (199) control also for group and individual policies in the cost function. Given our sample size, we do not control for group and individual policies because of lack of degrees of freedom. Another important aspect that might affect the results is the marketing distribution system of the firm. Insurers use various marketing distribution systems such as branch offices, agencies and direct marketing. The results reported here are possibly associated with the distribution system. Most insurers, however, employ more than one distribution system and hence one cannot determine the unique distribution system of each firm. 10 The AVR does not reflect future obligations (as do other reserves) but is set aside to protect against an extreme decline in the value of the assets that back up liabilities. 11 We are aware that the financial capital is a stock variable while physical capital is a flow variable. We assume that flow is a fixed proportion of the stock. 1 We measure these ratios over five years, rather than averaging the yearly ratios, in order to mitigate the influence of extreme fluctuations in the returns ratios on the price of capital. If the price of capital in a particular case is negative--that is, if the five-year 3

33 investment return was greater than the return on equity--we compute the price of capital as the average price of capital of the sample for that year. 13 We do not account for the price of the physical capital in the aggregate price of capital since the related expenses are rather negligible compared to the magnitude of the financial capital. 14 The data do not contain information as to the number of insureds under A&H group master policies. Therefore, we used the number of master policies in the computation. 15 Since we include financial capital as input, total costs consists of both actual costs and the opportunity cost of holding financial capital times the amount of financial capital. 16 The results of the estimation of the cost function are invariant to the price we use as a scaler. 17 The choice of shares to be included in the analysis does not affect the maximum likelihood estimates (Barten 1969). 18 The standard errors for the own price elasticities take into account the variability of the estimated shares. 33