Description of the problem

Ž Fig. 1. Fringe benefits of workers in firms surveyed by the chamber of commerce indices of benefits . as a percent of total compensation, 1951 s100 . there are strong cross-diseconomies of scale, however, employment and hours both fall, but the effects on benefits and wage are ambiguous. It is worth noting that, under reasonable assumptions about cross-economies, an increase in fringe bene- fits can raise equilibrium employment. Note that, at least in the United States, insurance has emerged as not only the fastest growing component of fringe benefits but also the largest. It is possible that increases in insurance and other benefits may have contributed to the growth of U.S. employment during the past several decades.

2. Description of the problem

We conduct the analysis using the social welfare maximization problem, where social welfare refers to the sum of the employer and employee surpluses. Suppose the inverse demand and supply functions of labor are, respectively, W d sW d E sf E;h,G , 1 Ž . Ž . Ž . Ž . W s sW s E sg E;h,G . 2 Ž . Ž . Ž . Ž . These functions are parameterized by standard hours, h, and non-wage benefits, G. Social welfare is given by E d s Z sZ E,h,G s W t yW t dt, 3 Ž . Ž . Ž . Ž . H where E is the equilibrium level of employment. Market forces drive social welfare to its maximum, max Z E, h,G E, h,G G0 , 4 4 Ž . Ž . satisfying the first order conditions: EZ rEEs0, EErEhs0, EErEGs0. 5 Ž . Ž One may analyze the effects of exogenous changes such as an increase in . worker demand for benefits by totally differentiating the equations summarized Ž . by Eq. 5 . In order to conduct such analysis, the second order condition, that the Ž . Hessian matrix of Z E, h,G be negative definite, must hold. Rather than arbitrar- ily imposing this key condition, we derive the market labor demand from the individual firm’s profit-maximization problem in such a way as to be consistent with the second order condition. In fact, our formulation of the firm’s profit-maxi- mization problem leads to the simplest possible functional form for the market labor demand supporting the second order condition. 4 2.1. Firm’s decision Assume that a competitive industry consists of identical firms and that a firm’s labor expenses consist of both wage and non-wage costs as follows: u su n,h,G swhnqC G,n n, 6 Ž . Ž . Ž . where n is employment, h is hours of work per worker, G is the quantity of fringe benefits per worker, and w is the hourly wage rate. 5 We assume the non-wage cost function, C, to satisfy the following conditions: C 0, C 0, C ,0, C s0, C sC ,0. 1 11 2 22 12 21 We call C the marginal non-wage cost of benefits, and C the marginal non-wage 1 2 cost of employment. We specify the marginal non-wage cost of benefits to be Ž . Ž . positive C 0 and rising C 0 . For simplicity, the marginal non-wage cost 1 11 Ž . of employment is assumed be a constant C s0 , which may be any real number 22 Ž . C ,0 . 2 Ž . The expression C sC represents cross-scale effects. If C - 0, an 12 21 12 Ž . Ž . increase in benefits G lowers the marginal non-wage cost of employment C . 2 Ž . Symmetrically, an increase in employment n lowers the marginal non-wage cost 4 Note that the market labor demand and supply functions cannot both be linear. If this were the case, the Hessian matrix for the social welfare function, Z, would not be negative definite. Under the simplification of linear labor supply, the simplest demand form is quadratic. 5 We assume for simplicity that non-wage labor costs are quasi-fixed, or that they are independent of hours of work. Although some non-wage labor costs do depend upon hours of work, approximately Ž . 20 of total labor costs in both the US and OECD countries are quasi-fixed Hart et al., 1988 . Ž . of benefits C . We define these phenomena as cross-economies of scale. In 1 contrast, the case of C 0 represents cross-diseconomies of scale. 12 We argue that cross-economies are empirically plausible, since larger firms are likely to have lower marginal non-wage cost of benefits for the following reasons. First, larger firms can negotiate better deals with benefit providers. Second, larger firms may have more specialized and higher skilled benefit administrators. Third, larger firms may have better technologies for administering benefit programs, e.g., processing claims by using computers rather than manually. 6 The firm’s production function is given by Q sF n,h , 7 Ž . Ž . where Q denotes output and F , F , F 0 and F , F - 0, implying that the 1 2 12 11 22 marginal products are positive and decreasing and that the two inputs are complements. 7 The firm’s profit function is given by p sp n,h,G spF n,h y whqC G,n n, 8 Ž . Ž . Ž . Ž . where p is the product price and p is assumed to be a concave function. The firm maximizes its profits subject to a fixed reservation level of workers’ utility. Ž . Assume that all workers are identical and have a utility function U sU Y, X, G , for which Y swh denotes a worker’s earnings, X denotes his leisure, and G denotes his benefits. The firm’s decision problem then has the form max p n, h,G spF n,h y whqC G,n n Ž . Ž . Ž . 9 Ž . ½ s.t. n G0, hG0, GG0, and UsU . For any given fixed combination of output price, the wage rate, and reservation Ž . utility p, w, U , the firm selects the optimal combination of employment, hours, Ž . 8 and benefits n, h, G by satisfying the first order conditions associated with Eq. 6 Ž A study by the US Congressional Budget Office Congress of the United States — Congressional . Budget Office, 1992, Table 4, p. 32 reports that, in employment-based health insurance programs, larger firms enjoy greater cost advantages in billing, advertising, sales commissions, claims administra- tion, and general overhead. Thus, total administrative expenses for health insurance as a percent of benefit costs are 40 for firms with 1–4 employees, 18 for firms with 50–99 employees, 8.0 for firms with 2500–9999 employees, and 5.5 for firms with 10,000 or more employees. 7 Ž . Ž . Ž . As in Nadiri and Rosen, 1969 , Brechling, 1977 and Hart, 1984 , capital, which is assumed to be fixed, is ignored in our analysis. Alternatively, one could include G as an argument in the production function, but such an extension is left for future study. 8 Ž . Ž . Ž . Let L l; n, h, G sp n, h, G y l UyU denote the Lagrangian. The first order conditions are: w Ž . x Ž . L s pF y whqC G, n q nC s0, L s pF ywny l dUrdh s0, L syC ny lU s0, L n 1 2 h 2 G 1 G l Ž . sy UyU s0. Ž . Ž 9 . The first order condition with respect to employment i.e., L s0 in Footnote n . 8 generates the firm’s inverse demand for labor, parameterized by h and G as follows: w sw n;h,G s pF yCynC rh. 10 Ž . Ž . Ž . 1 2 The next task is to derive the market demand for labor. 9 2.2. The market demand for labor The simplest model specification satisfying the second order condition is a Ž . negative-definite quadratic form for the welfare function 3 . Assuming a linear labor supply, this functional form is equivalent to having the following market labor demand function: w d sf E;h,G st h,G qb h,G E Ž . Ž . Ž . U 2 2 sa q a hqa h q a Gqa G qb h,G E, 1 Ž . Ž . Ž . Ž . 1 2 3 4 Ž . Ž . where the intercept t h, G is quadratic and the slope b h, G is linear. Note that Ž . U Ž . 10 Eq. 1 is linear in E and quadratic in h, G . Ž . The derivative Eb rEG is critical to our analyses. Eq. A7 shows that EbrEGs Ž . y2C r Kh , indicating that the sign of EbrEG depends solely on the sign of 12 C . As noted earlier, the term C defines cross-scale effects. Therefore, Eb rEG 12 12 Ž . G0 represents cross-economies of scale C F0 while EbrEG-0 represents 12 Ž . cross-diseconomies of scale C 0 . 12 Ž . U Ž . Note that Eq. 1 is derived from the p-max problem 9 , and it is the simplest form yielding the second order condition for welfare maximization. Appendix A provides a detailed derivation 11 of the market demand for labor from the firm’s Ž . Ž . U demand for labor 10 . The demand function 1 is a linear approximation of the Ž Ž . inverse market demand with a quadratic intercept and a linear slope see Eqs. A4 9 Ž . The firm’s demand for labor is a function of the full labor cost s wageqbenefit cost . The inverse demand functions plotted against the wage, therefore, generate a family of curves, where each curve is parameterized by h and G. 10 Ž . U One could further simplify Eq. 1 by removing the first order terms h and G, but such a change does not affect the analysis. 11 Ž . X Ž The derivation assumes a product-price feedback given by p s p E , p -0 i.e., an increase in E . raises aggregate output and therefore reduces the product price , and proceeds in two steps. We first obtain the second order Taylor approximations for the firm’s production and cost functions. This is equivalent to assuming that both the cost and the production functions are quadratic. Using more general functional forms would make the model too complex to handle. We then linearly approximate Ž . the market demand function in terms of E assuming that h, G affect the intercept quadratically and the slope linearly. Ž . Ž . Ž . Fig. 2. a Linear labor demand W s f E; h, G for a fixed G ; b If h is also fixed at h , the labor Ž . demand curve W s f E; h , G obtains. As G increases from G to G , the intercept drops. When 1 Ž . d b rdG0, the curve becomes flatter. c When dbrdG-0, the curve becomes steeper. Ž .. and A5 . As noted earlier, these approximations yield the simplest social welfare function needed for comparative statics. 12 Ž . Fig. 2a portrays the labor demand W sf E; h, G with the value of G fixed at G , and Fig. 2b portrays a special case of a two-dimensional demand curve Ž . W sf E obtained by fixing h at h as well. Note that an increase in G always 13 Ž reduces the intercept, because Et rEG-0. With cross-economies of scale i.e., . Ž . Eb rEG0 , an increase in G makes the demand curve flatter Fig. 2b and with Ž . cross-diseconomies, steeper Fig. 2c . 12 The negative definiteness of the welfare function is usually assumed. In our model, this condition almost follows from the standard conditions on the cost and production functions. Since our derivation shows that a 0, b - 0, and a - 0, we only need to assume that a - 0. See Footnote 16 for a 4 2 related discussion. 13 Ž . This follows from Eq. A4 and C 0. 1 Ž . Ž . Fig. 3. a Linear labor supply W s g E; h, G for a fixed G . The surface shifts down when G Ž . Ž . increases. b If h is also fixed h , the horizontal supply curve W s g E; h , G obtains. When G increases from G to G , the curve shifts downward. 1 2.3. Market supply of labor We abstract from the complexities of the worker’s labor participation decision by specifying the inverse market labor supply function to be of the following form, Ž . which is constant with respect to E and parameterized by h, G : U s w sg E;h,G sa qa hya G, 2 Ž . Ž . 1 2 where w s is the workers’ asking wage and a , a , a 0 are constants. The 1 2 simplifying assumption that the supply curve is horizontal, 14 shifting downward as either G increases or h decreases, is consistent with the worker’s utility function given earlier in the context of the firm’s problem. For example, when Ž . U Y, X, G is linear and a worker takes w, h, and G as fixed, U leads to the max Ž . U same relationship among w, h, and G as in Eq. 2 . Ž . Fig. 3a shows the linear labor supply function W sg E; h, G with G fixed at G . If one also fixes h at h , one obtains a horizontal supply curve. This supply Ž . curve shifts down when G increases from G to G Fig. 3b . 1 14 Ž . Our main predictions i.e., effects in the presence of strong cross-scale effects continue to hold with upward sloping curves. This result can be demonstrated as follows. Suppose the horizontal supply curves in Fig. 6 are replaced by upward sloping curves. By assuming that an increase in benefits has no effect on the slope of the supply curve and that it only shifts down the curve, one derives the same predictions about employment because of the large changes in the slope of demand curves caused by the strong cross-scale effects. Ž . Fig. 4. The equilibrium locus map, where each locus L records the equilibria W, E for a fixed h and all G, h and h are the upper and lower bounds for hour, and h U is the equilibrium hour. 1 3 2.4. Market equilibrium The competitive market equilibrium is the solution that maximizes the sum total of the surpluses of employers and employees. The process of reaching this equilibrium is illustrated in three steps. First, the market optimizes on E for each Ž . fixed h, G by setting EZ rEEs0. In the second step, the market chooses the optimum quantity of G by setting EZ rEGs0. In the third step, the market chooses the optimum value of h by setting EZ rEhs0. The first step generates a map of locus points for all potential equilibrium Ž . combinations W, E as a function of G and h. The result is shown in Fig. 4, Ž . where each curve traces W, E for all G and a fixed h. As one moves down on the curve, G increases and w decreases. Fig. 5 depicts a special case of Fig. 4 Ž . Ž . obtained by fixing h and G at h , G ; it shows the equilibrium point w , E Ž . Ž . Fig. 5. Equilibrium W , E for a fixed h , G . where the demand and supply curves intersect each other. The second step chooses the optimal point on each curve by equating the marginal social cost of G with its marginal social value. 15 The third step selects the optimum hours h U , or the optimum curve L in Fig. 4. 2 To understand the curvature in Fig. 4, let us start at a very small value of G Ž . i.e., at the top end point of L . As G increases, more workers decide to work 2 and they are willing to work at lower wages than before. As a result, the equilibrium employment increases and the wage decreases, and the curve moves Ž . down and to the right. After a certain level of G is reached i.e., beyond point A , benefits become less attractive than before because of the rising marginal non-wage cost of benefits, so a further increase in benefits reduces both the equilibrium employment and the wage, and the curve moves down and to the left. The above process is equivalent to solving the social welfare maximization Ž . Ž . problem summarized by Eq. 4 , with the welfare function 3 specified as 2 Z E, h,G s a ya q a ya hqa h Ž . Ž . 1 1 2 2 2 q a qa Gqa G E qb h,G E r2qconstant. Ž . Ž . 4 3 2 4 U 3 Ž . Ž . Ž . Ž . Ž . Ž . The optimality conditions appear in Eqs. B1.1 , B1.2 , B1.3 , B2.1 , B2.2 and Ž . 16 B2.3 . As discussed earlier, the derivative Eb rEG represents cross-scale effects. The effect of Eb rEG on equilibrium is illustrated in Fig. 4, where h h sh U h 1 2 3 denote, respectively, the upper bound, the equilibrium level, and the lower bound of hours. The three corresponding L curves are L , L , and L . Moving down the 1 2 3 L curve increases G and decreases w. For convenience, Fig. 4 depicts three potential equilibrium points on the curve L . Readers are advised, however, that a 2 unique map of locus points is associated with each value of Eb rEG, so that only one equilibrium point is relevant on a particular curve. Note also that Fig. 4 Ž . indicates the maximum employment for each fixed h as being attained at Ž equilibrium only when Eb rEGs0 i.e., when the slope of the labor demand curve . is constant with respect to G ; otherwise, equilibrium employment is less than the maximum feasible level. These properties are summarized in the following proposition. 15 Ž . Ž . By Eqs. B1.2 and B1.3 , the conditions, EZ rEhs0 and, EZrEGs0 can be rewritten as Ž . Ž 2 .Ž . Ž . Ž 2 .Ž . a q2 a h Eq E r2 EbrEh s a E, a q2 a G Eq E r2 EbrEG sy a E. The left-hand 1 2 1 3 4 2 Ž . right-hand sides in both expressions represent the effects of hours and benefits on labor demand Ž . supply . Because the surplus integrates the difference between the demand and supply curves, the Ž . Ž . left-hand right-hand side is interpreted as the marginal social value marginal social cost . 16 The second order condition is consistent with our assumptions on the cost and production functions. When, Eb rEhsEbrEGs0, the second order condition is equivalent to b -0, a -0, and 2 a - 0, where b - 0 and a - 0 follow directly from the firm’s decision problem. 4 4 Ž . Ž . Proposition 1. a In the absence of cross-scale effects i.e., Eb rEGs0 , competi- tive equilibrium employment is at its maximum feasible level for a given h; however, in the presence of cross-scale effects, equilibrium employment is less Ž . than its maximum feasible level; b In the presence of cross-economies of scale Ž . i.e., Eb rEG0 , the equilibrium wage lies above that wage associated with Ž . Ž . maximum employment point B in Fig. 4 ; c In the presence of cross-disecono- Ž . mies of scale i.e., Eb rEG-0 , the equilibrium wage lies below that wage Ž . associated with maximum employment point C in Fig. 4 . Under competition, the market maximizes the sum of the employer and employee Ž . surpluses, not employment. In a there are no cross-scale effects and maximizing Ž . Ž . surpluses happens to maximize employment. In b and c , however, there are cross-scale effects and maximizing surpluses does not maximize employment. We proceed to evaluate the effects of an exogenous increase in non-wage benefits on employment, hours, and wages. Increases in G are traced to an Ž Ž . U . increase in the demand for G i.e., an increase in a in Eq. 2 , to a decrease in 2 Ž Ž . U . 17 the cost of providing G i.e., an increase in a in Eq. 1 , and an increase in 3 the mandated benefits.

3. Effects of increased demand for G