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Scale efficiency in a dynamic model of Canadian insurance companies.


The purpose of this research is to investigate scale efficiency in the Canadian Insurance Industry. This is an important issue for the industry, itself, and the government agency charged with the responsibilities of monitoring and regulating insurance companies, i.e., the Office of The Superintendent of Financial Institutions. Its obligations include, in addition to setting rules and conventions that cover a company's day to day operations, the responsibility of approving mergers or any other major change in the structure of ownership in the industry. Since social benefits like lower costs and lower policy premiums that could potentially arise from merger activity are determined, largely, by the scale efficiency of the firms involved, the knowledge of firm-specific scale elasticities is central to the successful implementation of policies that affect the industrial organization of the industry.

The Canadian Life Insurance Industry has been the subject of a number of studies. Bernstein and Geehan (1988) provide an overview of the institutional arrangements and a description of how the industry actually functions. Similar material as well as a summary of recent developments including the implications of the 1992 Insurance Companies Act is discussed by Armstrong (1994). Issues of scale and scope have also received considerable attention starting with papers by Halpern and Mathewson (1975), Geehan (1977), Kellner and Mathewson (1983) and most recently by Bernstein (1992). Most of these studies take a similar approach to the problem by assuming that firms are competitive and maximize profits. Details vary depending on the type of data employed but, with the exception of Bernstein, researchers agree that the industry is characterized by constant returns to scale with the possibility of some limited scope economies.(1) Both Kellner and Mathewson and Bernstein utilize data on federally chartered insurance companies. Unlike most studies of the insurance industry, the database employed here allows the researcher to use the number of policies issued by line of insurance as the measure of the line's output.

The measurement of output is, in fact, a contentious issue. A number of output definitions have been proposed. Some of the early literature used premium revenue. In spite of the limitations mentioned by Doherty (1981, 391), this and related measures continue to be used. See, for example, Grace and Timme (1990) and Fields and Murphy (1989) who use "commission dollars". Yuengert (1993), in response to the use of reserves as a measure of output, recommends "additions-to-reserves" as a better representative of the flow of new business. Cummins and Zi (1996), following Doherty, advocate "incurred benefit payments" as the appropriate measure of output along with "additions-to-reserves." All of these measures are "value" measures of output. On the other hand, the measure used in this study is a pure quantity measure. While that may be preferable in principle, there may be problems associated with the heterogeneity generated by differing policy sizes. Consequently, it is not clear at present which measures are most appropriate.

In terms of econometric methodology, however, there are improvements that could be made. Even when there are good measures of output available the estimation procedures currently in use do not treat outputs as endogenous variables. This is a serious deficiency because it leaves the firm's most important observable decision unexplained.(2) Furthermore, assuming that factor prices are the same for all firms regardless of geographical location is an unsatisfactory way of dealing with an unobservable variable in a country like Canada in which large regional disparities exist. Finally, none of the Canadian studies exploit "panel" methods to obtain more efficient parameter estimates or to control for unobservable "fixed effects" even when there are time-series of cross-sections available.(3)

The three deficiencies mentioned above are addressed by developing and estimating an intertemporal product differentiation oligopoly model based on a panel of federally chartered insurance companies for the period 1988 through 1991. Factor prices are treated as unobservable stochastic errors and certain unobservable characteristics of the firm are modeled as "fixed effects."

Significant short run scale economies are found with respect to both the output of new policies and the stock of policies issued in previous periods. For the long run, a new measure of scale efficiency is developed: the stationary elasticity of scale. Given the fitted functional form for the cost function this is the same for all firms and is significantly greater than one. An analysis of the model provides some evidence that certain types of mergers may lead to lower premiums and, consequently, to an increase in the welfare of both producers and consumers of insurance products.

The next section briefly describes the Canadian Insurance Industry and some of the products that are offered. Following this, a model of insurance sales is developed. This model is estimated in the subsequent section. Comments and a discussion of the results end the paper.


As a good, an insurance policy is rather unusual. First, it can be characterized as an interval-input interval-output commodity rather than the typical textbook point-input point-output good. In the latter case the good is produced at a particular point in time using various stock and flow inputs and the revenue per unit of the good is just its sale price. This is not the case with insurance, which can yield a stream of premiums to the company over the term of the policy. The premiums may vary stochastically or deterministically or be constant depending on the type of policy and the duration of the contract will certainly vary if there are mortality considerations involved. Annuities are a variation on this theme. They can involve a stream of premium payments followed by a stream of payments to the policyholder. Some annuities involve only a single premium.(4)

On the cost side, issuing a policy exposes the company to a stream of policy servicing costs which have to be paid every year in addition to the much larger first year cost.(5) From the point of view of costs, stocks are different from flows and this difference must be explicitly included in any cost function involving the production of insurance policies. In addition to these costs there are the benefits that have to be paid to the beneficiary of the policy at some, possibly uncertain, future date.

A second characteristic of insurance worth noting is that policies are specific to each individual purchaser. The quantity of insurance purchased or the face value of the policy is a choice variable for the consumer along with options chosen. These include premium waivers for health reasons, joint survivorship options and additional coverage privileges, and duration and conversion alternatives. Insurance companies offer quantity discounts on life policies by charging lower premiums per thousand of race value the larger the tace value. These discounts become significant for policies over one hundred thousand dollars.

There are three essential features of the insurance industry, which must be recognized, in any formal model of it. The first is the very large amount of price dispersion in markets for what appear to be almost identical products. For example, a sample of thirty-five federally chartered insurance companies which offered five year renewable and convertible term insurance policies in 1988 starting at the same age with the same face value had premiums ranging from 0.49 to 1.57 times their mean value. However, there is no consensus as to why this should be the case. Brett and Bullock (1993, 69) see this as a consequence of consumer ignorance combined with corporate opportunism.(6) Less cynically, Mathewson (1983) and Dahlby and West (1986) accept that consumers are not fully informed and rely on costly and, therefore, limited consumer search to explain premium dispersion in Canadian life and automobile insurance markets. On the other hand, Winter (1981, 88) suggests that firms may actually be offering different products when they set their premiums to take account of the risk generated by their selection criteria. Product differentiation is, in fact, a very plausible explanation for price dispersion since there are so many ways that firms can achieve this.

Secondly, in spite of a relatively large number of firms, the industry is highly concentrated. In the period under consideration the four largest insurance companies in terms of total assets were in descending order: Manufacturers Life, Sun Life, Great West Life, and Confederation Life. These four companies controlled 56.7 percent of the assets of federally registered Canadian companies. The next three largest companies controlled an additional 24 percent. In terms of total policies underwritten by Canadian companies the four largest sellers of insurance underwrote 45.5 percent of all the policies issued by Canadian companies. There are also foreign companies operating in Canada. Although they are quite numerous, Canadian firms dominate the industry accounting for about 89 percent of policies in force in 1990. The sample used in this study covers about 83 percent of all policies sold in Canada.

Thirdly, the industry is characterized by very large differences in size, range of product, and selling methods. The largest firm in the sample, Manufacturers Life is about one thousand times larger than the smallest, Acadia Life. Larger companies tend to sell more lines of insurance and more variety within each line.(7) Finally, larger firms are more likely to have their own agents operating from a network of branch offices whereas smaller firms tend to utilize independent brokers or market their products directly.


Consider a set of insurance companies that sell a single type of insurance to a homogeneous group of individuals. For firm i let [y.sub.i] (s, t) be the number of policies issued in year s, which are still in force in year t. For notational convenience define [y.sub.i](t,t) as [y.sub.i](t). Policies terminate because of cancellation, the death of the policyholder, or in the case of term or endowment insurance the contract covers the holder for a fixed number of years. Attrition due to mortality is the main focus of attention in the model developed in this section and it is incorporated into the model through the following set of equations:

[E.sub.s] [y.sub.i] (s, t) = [Sigma](s, t) [y.sub.i] (S) s [less than or equal to] t (1)

The symbol [E.sub.s] is the conditional expectation operator given all the information available at time s. The functions [Sigma](s,t) are "survival rates" and are easily computed from a standard life table. As mentioned in the previous section, the operating costs of an insurance company depend, in part, on the number of policies in force. At the beginning of year t the stock of policies in force for company i is defined as

[S.sub.i](t) = [summation over s [less than or equal to] t - 1 [y.sub.i](s, t - 1) (2)

Real costs are assumed to be a function of the stock of policies in force, the flow of new policies, and real factor prices. For reasons to be explained later the cost function is assumed to be multiplicatively separable in quantities and factor prices so that

[C.sub.i] (t) = [C.sup.*][[y.sub.i] (t), [S.sub.i] (t), [c.sub.i] (t)] h([w.sub.i](t)) (3)

where [c.sub.i](t) is a vector of firm characteristics and [w.sub.i](t) is a vector of time varying firm-specific real factor prices. The functional form of [C.sup.*] well as the specific characteristics of each firm will be discussed later.(8)

In what follows future factor prices will be random variables for the firm. It is assumed that decisions are made at the beginning of the period before [w.sub.i](t) is observed. Furthermore, it will be assumed that h([w.sub.i] (t)) is strictly positive and [E.sub.t]ln[[h([w.sub.i] (t))] = 0. Optimal behavior on the part of individual firms requires that they maximize the present value of the stream of expected future profits in a way which takes account of the sequential nature of the decision making process as well as recognizing the presence of other firms with similar but conflicting interests. Given the cost function firm i's objective function may be written as

[Mathematical Expression Omitted]

where [p.sub.i](t) is firm i's premium in period t and [Delta]=1/(1+r), and k([Sigma], r) is the present expected value function of a stream of premiums and depends on the set of survival rates, ([Sigma], and the discount rate, r. A certain proportion, [[Theta].sub.i], of each premium is invested and earns a rate of return, [Rho].sub.i]. Both [[Theta].sub.i] and [[Rho].sub.i] are firm-specific since investment strategies and rates of return vary across firms.(9)

As mentioned earlier insurance companies charge different prices for their products. One model that is consistent with this type of observed behavior is the differentiated product oligopoly model in which each firm faces a demand function [d.sub.i]([p.sub.1](t), . . ., [p.sub.N](t)), where N is the number of firms in the market.(10) The demand for firm i's policies depend not only on its own premium but the premiums of all the other firms offering insurance. In each period firm i picks an intertemporal pricing strategy to maximize the present value of expected future profits subject to the constraint

[y.sub.i](t) = [d.sub.i]([p.sub.1](t), . . ., [p.sub.N] (t)) (5)

given the premiums of all other firms. Thus, observed market premiums are characterized as a Sub-Game Perfect Nash equilibrium in a dynamic game of price or premium competition.

In general, the cost function will be a non-linear function of [S.sub.i](t) making the game described above a truly dynamic one. The reason for this is obvious; policies sold in the current period will be part of the stock for as long as they are in force and will effect the stock costs of policies issued in subsequent periods. In particular, if there are stock increasing returns to scale, firms will have an incentive to sell more policies in the current period than would be justified on the basis of their profitability alone. Ideally, this problem should be solved by backward induction methods; however, such a solution appears to be intractable and an "open loop" solution will be used instead. (11) This is obtained by maximizing the constrained objective function with respect to [p.sub.i](t). The resulting first-order condition is

[Mathematical Expression Omitted]

where [Gamma].sub.i] is equal to (1 + 1/[[Epsilon].sub.ii])(1+[[Theta].sub.i][[Rho.sub.i])] and [[Epsilon].sub.ii] is the "own price elasticity" of the demand for firm i's product and [[Epsilon].sub.ii] [less than] -1. Assuming a steady state, static expectations on mortality rates and factor prices, and that the determinants of mortality and factor prices are statistically independent equation (6) simplifies to

[Mathematical Expression Omitted]

Equation (7) says that discounted marginal revenue is equal to discounted marginal cost. Discounted marginal revenue is the expected present value of premiums and discounted marginal cost is the first year cost together with the present value of expected stock costs that arise through the maintenance of the policy over its lifetime.

The first-order condition that this equation represents is based on a combination of "typical" term and whole life policies each starting at age thirty. k([Sigma], r) is, therefore, a weighted average of present value functions for each type of policy. Under 1985-87 mortality rates and a discount rate of 5 percent the present discounted expected value of a stream of one-dollar premiums is 18.13 dollars for a whole life policy and 12.51 dollars for a twenty-year term policy.(12)

It would also have been desirable to have had a first order condition for annuities as well. The reason why this was not done as well as a discussion of other aggregation problems is deferred to the next section.


The functional form for the deterministic part of the cost function is given by

[Mathematical Expression Omitted]

The ([Alpha], [Beta]) parameters of the model, which are the same for all firms, will be estimated using a panel of data on federally chartered insurance companies for the years 1988 to 1991 which has been made available by The Office of The Superintendent of Financial Institutions. While this database is extremely rich and detailed with respect to the insurance aspects of these companies there is no information on input quantities or prices. For this reason the four equations of (8), one for each year, are estimated in logarithms and ln(h([w.sub.i](t))) being a firm-specific time-varying unobservable, serves as the error term.(13) This solution to the problem of missing factor price data was made possible by the assumption that the cost function was multiplicatively separable. The same procedure is applied to the four equations of (7) which are then differenced to remove ln([[Gamma].sub.i]), which is treated as a "fixed effect."(14)

Because of the relatively small sample sizes, it was necessary to aggregate over type of policy in order to obtain reliable parameter estimates. There are a large number of ways of classifying different types of policy. For the data base used in this study, disregarding accident and sickness coverage, insurance is classified as either life insurance or annuity. Within each of these classes there are both participating/non-participating and individual/group sub-classifications. Furthermore, within the area of life insurance, policies are categorized as whole life, term, or endowment. Consequently, any econometric study of the Canadian Insurance industry will be forced to add various types of policy together to form an aggregate simply because there are too many different types given the number of companies for which there is data available. In this study all types are aggregated into one representative policy, although the date of issue is partially considered by distinguishing between new policies and those already in existence. Although this solution is not ideal, it appears to be the best one available. Attempts to further disaggregate the stock-flow model by adding either the individual/group or the life insurance/annuity classification produced coefficients with large standard errors. The stock-flow classification was assumed to be the more fundamental and was used as the base-line classification to which others were added since it is a property that all policies have whether they are individual or group or participating or non-participating etc.

The functional form for the cost function depends on six parameters. It is multiplicatively separable in (y, S) and -additive in the firm's business characteristics. This latter property prevents the cost function from taking a zero value when a firm has no annuity or group business and there are some firms in the sample which fall into this category. A number of other functional forms were tried, including some, which were additively separable in (y,S). This property is desirable since it allows a firm's cost function to be positive even when it has sold no policies in the current period, provided that it sold in previous periods. Unfortunately, functional forms with this property like the quadratic, for example, produced negative marginal products for some of the firms and, therefore, could not be used.

The endogenous variables are (ln([C.sub.i] (t)),ln([y.sub.i](t)),[c.sub.i](t)) and (ln([p.sub.i](t))-ln([p.sub.i](t-1)) and the exogenous variables are ln([S.sub.i](t)) and ln([a.sub.i](t)), the total assets of firm i. [a.sub.i](t) do not appear in the model except as instrumental variables. Total costs, [C.sub.i](t), cover all running expenses like rental, interest, labor costs, agents' commissions and investment expenses as well as all of the costs that arise as obligations from policies like death claims, annuity payments, and dividends. The variables [y.sub.i](t) and [S.sub.i](t) are the total number of policies sold and in force in year t, respectively. Both of these are aggregates over the eight categories generated by the three classifications: insurance/annuity, individual/group, and participating/non-participating. The variable, [p.sub.i](t), is the average premium on a representative insurance policy. This is constructed as a weighted average of term and whole life premiums; the weights being the respective shares of these two types of insurance in total insurance sales. Costs, premiums, and assets are valued in 1988 Canadian dollars. The vector [c.sub.i](t) has three elements representing the characteristics of firm i's business; they are the shares of annuities and group policies in total sales in each year and a dummy variable which takes on the value one if a firm has branches in addition to its head office. In order to capture some of the potential differences in cost over line of business, these variables were included as explanatory variables in the cost function.(15)

Means and standard deviations for these variables are displayed in the third column of Table 1. Notice that since cs is a dummy variable its mean is the proportion of companies with multiple branches. The sample consists of forty-two companies. There are fifty-two listed as federally chartered companies of which forty-seven show positive values for all of these variables for all four years. Five companies were dropped from the sample due to data errors.

It is reasonable to expect for each company that the errors in the cost equations will be correlated over time. This may also be true for the errors in the equations for the first-order conditions. On the other hand, firms are assumed to act independently and no inter-firm correlation is expected. An estimation method which permits this type of error structure and also allows for the endogeneity of (ln([y.sub.i](t)), [c.sub.i](t)) is non-linear three stage least squares.(16) As a minimum distance estimator it requires consistent starting values.(17) These were obtained by applying non-linear least squares to the seven equation system in which (ln([y.sub.i](t), [c.sub.i](t)) were replaced by their fitted values from a seemingly unrelated set of regressions. Non-linear three stage least squares estimates of the six parameters are displayed in the fifth column of Table 1. Parameter values of 1 result from the required normalization and no parameter are associated with ln([a.sub.i](t)) since it is used only as an instrument.

As is clear from the standard deviations of the parameter estimates reported in Table 1, both [[Alpha].sub.y] and [[Alpha].sub.s] are highly significant and significantly less than 1, indicating the presence of both stock and flow increasing returns to scale. But stocks and flows are related; in particular, they are proportional in a stationary state. This fact can be used to define the stationary or long run elasticity of scale(18)

[Eta] = 1/([Alpha].sub.y] + [[Alpha].sub.s]) (9)

Naturally, values of [Eta] greater than one indicate increasing returns to scale. Given the particular form of the cost function this is the same for all firms and should, therefore, be seen as an industry scale elasticity. Some sort of intertemporal notion of scale efficiency is needed here since policies sold in the present period commit the firm to a stream of future costs required to service these policies. For this sample, the estimated value of [Eta] is 1.45 and its standard deviation is 0.165 making it significantly greater than one.

As one would expect, the type of business in which the firm engages affects the structure of it costs. Firms with a higher proportion of policies issued through group plans have lower costs. On the other hand, firms, which specialize in annuities, do not appear to derive any significant advantage or suffer any extra cost from this. The distribution method employed by the firm also affects its costs; those firms, which market their products through their own system of branch offices, have significantly higher costs than similar sized single branch firms.

Two other variables were considered as potential explanatory variables for costs. In Section 2 it was noted that some of the variation in premiums was due to differences in the face values of the policies offered. This suggests the possibility that costs could also be affected by average policy size. However, this variable(19) was not significant when it was included as a regressor in equations (7) and (8), nor were the parameter estimates changed very much by its inclusion. This is not surprising even if costs do depend on policy size since there is no evidence that firms specialize by size of policy; average policy size is not correlated with firm size. Another type of specialization could also influence the firm's cost structure. Winter (1981), in the paper cited in Section 2, mentions that some American insurance companies have selected their clientele on the basis of their expected mortality risks. If this is true and mortality rates are correlated with firm size then, as one of the referees suggested, the parameter estimates would be biased because of an omitted variable. However, this bias is likely to be small since most of the variation in the logarithm of claims is explained by the variables already present in the model. Furthermore, when the model was re-estimated using costs less claims as the dependent variables in equation (3), [Eta] actually rose slightly indicating that scale economies are the explanation of the parameter values in Table 1 and they are not simply an artifact of a selection process involving mortality.

As a check on the ability of the model to explain all firms in the sample the model was run on two sub-samples: the first excludes the five smallest firms and the second excludes the five largest firms. The results in each case were similar to those reported in Table 1 in the sense that there were no significant changes in the values of the estimated coefficients.

For the average firm in the sample the cost of issuing a policy which includes agents commissions and evaluation expenses is 1121.53 dollars, a reasonable figure if one half to three quarters of the first premium is paid as a commission to the seller of the policy. After the first year the average company pays an additional 499.63 dollars per year to maintain the policy. Given an average premium of 827.16, these are not unreasonable values for issue and maintenance costs. The average firm lies between the ninth and tenth largest firms and is, therefore, also large in terms of total assets. Consequently these numbers are more representative of the larger firms in the sample.

Because of the difficulty in dealing with more than one type of output traditional methods for measuring scope economies were not available. However, the number of lines of business offered by Canadian insurance companies is highly correlated with the size of the firm. Since there are increasing returns to scale there are advantages to diversification if this leads to increased sales even if there are no intrinsic benefits to product diversification per se. Thus, the fact that large firms have a more diversified range of products can not be used as an argument to justify the presence of scope economies in large firms although it does suggest that they are not present in small companies.

Although the model is highly aggregated the first order condition is specific to the life insurance side of the business. As was pointed out earlier it would have been beneficial to have had a first order condition for annuity contracts as well since the average company does seventeen percent of its business in this area. Two difficulties prevented this. First, the features of annuities are not easily captured by a representative policy for reasons mentioned in footnote 4. More importantly, however, the information on annuity premiums is not sufficient to distinguish between new and existing policies because of the presence of what the industry calls "flexible premium contracts".

Before turning to the policy implications of this study it is worth mentioning two additional points about the estimation procedure. The parameter estimates are not very sensitive to the choice of interest rate used for discounting future costs and revenues; using 3 percent or 7 percent instead of 5 percent as a discount rate has no effect on any of the conclusions. Secondly, one of the implicit assumptions underlying the model was that there was no heteroscedasticity present. This assumption appears to be satisfied since White's robust standard errors are very similar to those reported in Table 1.


When there are increasing returns to scale mergers and higher levels of concentration may serve the public interest. Farrell and Shapiro (1990) give conditions in the standard homogeneous product oligopoly model under which mergers increase net welfare. For the model employed in this paper, however, there are no general results. When average cost declines with the size of the firm mergers permit the product to be produced at a lower cost. This benefits the firm and also produces a social benefit because resources are used more efficiently. In certain circumstances the consumer may also be rewarded by lower premiums. While it is impossible to give a precise set of conditions which guarantee lower premiums the model suggests that certain types of merger may benefit consumers as well as producers. The proper way to see what the effects of mergers are on premiums is to compare various simulated merger solutions. Unfortunately, this requires the computation of the Nash equilibrium for every simulation, an extremely burdensome task given the complexity of the model. What can be done is to compute the effect on the firm's premium of an increase in its stock of policies in force that would occur if it did merge with another company and took over servicing the other company's stock of policies. This is obtained by differentiating equation (7) to get d[p.sub.i](t)/d[S.sub.i](t).

While this term involves [[Gamma].sub.i] which can be computed for each firm given the data it also involves the unobservables, [[Epsilon].sub.ii] and [[Theta].sub.i][[Rho].sub.i]. Given [[Gamma].sub.i], d[p.sub.i](t)/d[S.sub.i](t) can be computed conditional on [[Theta].sub.i] and [[Rho].sub.i]. For plausible values of these two parameters ie those satisfying the inequality, 0.005 [less than or equal to] [[Theta].sub.i][[Rho].sub.i] [less than or equal to] 0.1, the sign of this derivative is negative for all firms. It is, therefore, reasonable to expect that mergers would lead to lower premiums and, consequently, an increase in consumer surplus.

Finally, one would expect that an industry characterized by such significant scale efficiencies would also exhibit considerable merger activity. This is, in fact, what has happened in the Canadian life insurance industry. By the end of 1994, ten of the smaller firms in the sample were no longer listed as federally chartered companies. Some left the industry while others combined to form new larger firms. There have also been some new entrants, including three of the Chartered Banks who see long run opportunities in expanding their client base. Apparently, there are more mergers on the way. Executives in the larger firms are already aware that there are advantages to size.(20) It is their perception that mergers, acquisitions and strategic alliances will be the outcome of a major restructuring that will see a much smaller number of firms in the industry.

The author wishes to express his gratitude to the Editor and two anonymous referees for helpful comments. Thanks also go to Greg Placidi, formerly of the Canadian Life and Health insurance Association, for helpful discussion. A number of industry executives consented to interviews on the condition that their anonymity was guaranteed. Their help is appreciated as well as that of D. S. Rudd, John Leroux, and the SSHRC for financial support.

1 Bernstein is the first to impose a market structure consistent with increasing returns to scale. Although his model is somewhat vague on this point he assumes that." an insurer has some price setting ability over policy coverage and can charge a price that is different from other insurers" (p. S96). Given that different insurance companies charge different prices for the same generic product, adopting an implicitly imperfectly competitive framework is a major step forward regardless of the scale issue.

2 Modern texts on microeconomics, such as Mas-Colell (1995, p. 136), treat output as endogenous and represent it as the firm's "supply correspondence" which arises when profit is maximized. For a more detailed discussion of the econometric methodology employed in the study of scope and scale economies in financial institutions see Breslaw and McIntosh (1996)

3 Mairesse (1992) noted the first and third comments in his discussion of the Bernstein paper.

4 An example of this, a "term annuity", is designed to compete with guaranteed income certificates that banks and trust companies offer. These pay the annuitant or his/her estate a guaranteed sum at the end of the term. There is no insurance component in the product.

5 For a discussion of this point see Pedoe and Rudd (1993, Ch. 12). They suggest that first year costs are between ten and twelve times annual costs for subsequent years.

6 They write in their book Insure Sensibly: A Guide to Life and Disability Insurance "There isn't a consumer product on the market for which there are such vast differences in price for essentially the same thing. ... The adage, 'You get what you pay for,' is not necessarily true for life insurance."

7 Some, but not all, large companies have substantial foreign operations.

8 A better representation of costs could be obtained by using the distribution of policies in force, rather than the sum as defined by equation (2). Unfortunately, there is not sufficient data available to estimate such a model. On the other hand, distributional effects are not captured by the present formulation only when a firm's policy distribution deviates from its steady state value. Consequently, the present formulation can be seen as steady state approximation to the true cost function.

9 Investment activities of firms are not explained in this model. While these are central to the operations of an insurance company the assumption here is that reasonable models of output and premium determination can be constructed without the added complexity of dealing with the firm's portfolio decisions.

10 See Shapiro (1989, p. 346-8) for a description.

11 For a review of the literature on dynamic games see Fudenberg and Tirole (1991, Ch. 13). They discuss Markov Perfect Equilibria and Eaton and Engers (1990) deal with the two-firm product differentiation oligopoly model in a Markovian framework by assuming that firms set prices in alternating periods. Unfortunately, the problem under consideration here is not Markovian since insurance policies have a finite lifetime of more than one period.

12 Equation (7) is forward looking. However, mortality expectations assume that existing rates will continue. Rates for the 1985-87 period from Statics Canada (1990, p. 16) were the most recent ones available for use in the sample period.

13 This is not the only way to deal with factor prices. Cummins and Weiss (1993, section 3), for example, combine firm data with industry level index numbers to get estimates of firm specific factor prices.

14 The simultaneous equations approach to panel data methods is outlined in Chamberlin (1984).

15 Weiss (1986 p. 58) emphasizes the importance of variation in "life insurer production cost by line of business" and cites a large number of studies to support this point.

16 See Amemiya (1985, pages 257-8) for details. As a minimum distance estimator it requires consistent starting values.

17 In this particular case the parameter estimates are not sensitive to the choice of starting values. The same values of the parameter estimates are obtained from a wide variety of starting values. This is reassuring for a number of reasons, including confirmation of the local identifiability of the parameters.

18 When S and y are proportional the cost function can be written in terms of y alone and the usual definition of the elasticity of scale as the ratio of average cost to marginal cost applies.

19 This was calculated as a simple average over all insurance policies. There was not sufficient data available to construct a weighted average reflecting the shares of term and life policies.

20 This is clear from what "Industry elder statesmen and current leaders" told the authors of a Report on Insurance, which appeared in the Globe and Mail May 10, 1994.


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James McIntosh is at the Department of Economics, Concordia University, Montreal, Quebec, Canada.
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Author:McIntosh, James
Publication:Journal of Risk and Insurance
Date:Jun 1, 1998
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