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Dynamic effects of the unemployment insurance tax on temporary layoffs
Journal
of Public
DYNAMIC
Economics
25 (1984) 35-51. North-Holland
EFFECTS OF THE UNEMPLOYMENT TAX ON TEMPORARY LAYOFFS WOLCOWITZ*
Jeffrey Harvard Unioersity, Received
November
INSURANCE
Cambridge, MA 02138, USA
1982, revised version
received July 1983
The reserve ratio method of experience rating the unemployment insurance (UI) tax is an inherently dynamic process. A tax rate is assigned to a firm based on its reserve ratio, the ratio of the balance in its UI account to its taxable payroll. Therefore, in making layoff decisions that will deplete its UI balance, a firm will consider the effect on its future UI tax rate. This study explores the incentive effects of this method of UI financing on temporary layoff behavior of firms in a dynamic context.
1. Introduction The potential role of the financing scheme for unemployment insurance (UI) in encouraging firms to maintain stable work patterns was recognized at the beginning of the UI system in the United States and has been reflected in the experience rating provisions of the legislation.’ Under an experience rated tax scheme, the costs of the UI program are allocated among firms in relation to their history of generating compensable unemployment. Those firms that have had high turnover rates in the past, so that the workers they laid off have received a disproportionately large amount of the benefits disbursed, are assigned a relatively high tax rate. In this way, it is argued, firms internalize the costs of the unemployment they create and see a financial incentive for organizing production in a way that tends to stabilize employment. The most common system of experience rating in the United States is the reserve ratio method.2 Under this system, a bookkeeping account is es*This paper summarizes results from my Ph.D. dissertation. I am grateful to Andrew Abel for his guidance in formulating the problem. Martin Feldstein, Daniel Hamermesh, and James Medoff provided helpful comments and suggestions. ‘For the early history of the UI system in the United States, see Haber and Murray (1966). The most complete review of the experience rating issue is Becker (1972). Experience rating of the UI tax is unique to the United States. In Canada, the Unemployment Insurance Act of 1971 gave the UI Commission the power to establish an experience rated contribution system for large employers, but such a scheme has not been adopted. *As of 1983, the reserve ratio method was used in 31 states and the District of Columbia. The benefit ratio formula was used in 12 states, 4 states used the benefit-wage ratio formula, and 3 states used a payroll variation plan. See U.S. Department of Labor (1982) for descriptions of these methods of experience rating and other aspects of state UI systems, as well as some specific provisions of state programs. 0047~2727/84/$3.00
0
1984, Elsevier Science Publishers
B.V. (North-Holland)
36
J. Wolcowitz, Effects of the unemployment insurance tax
tablished for each firm. This account is credited with the UI tax payments made by the firm, less that portion earmarked for the administrative costs of the system and for a solvency fund that pays benefits that are not charged to a specific firm. Benefits paid to the firm’s ex-employees and laid off employees that the state treats as chargeable to the firm are deducted from the account. The firm’s reserve ratio is defined as the ratio of the balance in its account to its taxable payroll (or the average taxable payroll over several years). The state’s UI tax schedule assigns a tax rate to the firm according to its reserve ratio. This is a negative relation defined by a step function subject to maximum and minimum tax rates. An important characteristic of this system is the dynamic adjustment of the firm’s reserve ratio and tax rate. To the extent that a firm can alter its behavior in response to the UI tax system it will recognize that today’s layoffs will affect the reserve ratio and, thus, the UI tax rate in future periods. This dynamic process has largely been ignored in the existing literature on the effects of the UI tax system on layoffs. Feldstein (1976), Baily (1977) and Topel (1982) have constructed static models of the incentive effects of taxes, simplifying away the full structure of the reserve ratio mechanism; these models are best interpreted as descriptions of steady state behavior. Brechling (1977, 1981) and Topel and Welch (1980) explicitly mode1 the reserve ratio system and consider the adjustment pattern of the tax rate given a change in the layoff rate of a firm, but do not emphasize the incentive effect of the tax on the firm’s behavior, a linkage which runs in the opposite direction. The single exception to this is the final section of Brechling’s (1977) paper where he demonstrates that with a linear tax schedule it is optima1 for the firm to choose a constant layoff rate over time. Below we show that this result is specific to the assumption of linearity of the tax function. The dynamic structure is important in understanding the incentive effects on the layoff behavior of firms. For example, a firm at the maximum tax rate may choose to curtail its layoff rate to accumulate reserves and obtain a lower tax rate. Although it is on a flat segment of the tax schedule, it is incorrect to characterize such a firm as non-experience rated, as past studies have implied. Also, in thinking about empirical analysis of UI tax effects, changes in the firm’s layoff rate caused by changes in the parameters of the UI tax schedule must be disentangled from any changes caused by the dynamic adjustment of the reserve ratio and tax rate. The present study focuses on the dynamic nature of reserve ratio accounting in a mode1 of temporary layoffs. Temporary layoffs are treated as part of the compensation package firms offer workers. These layoffs can be seen as vacations subsidized by the UI program. Despite its importance in determining actual layoff rates, the role of demand variation is omitted to simplify the analysis and focus attention on the role of the UI tax system in determining the rate of temporary layoffs.
J. Wolcowitz,
Effects of the unemployment
insurance
tax
37
Emphasis is placed on the incentive effects of floor and ceiling tax rates. This aspect of the tax schedule has received the most attention in studies of the UI incentive toward layoffs because when the floor and ceiling tax rates are in effect, there appears to be no experience rating. We examine the case of a continuous piecewise linear tax schedule defined by3
z(It) = a -
SC,
a--r,,
rt < r, r_
a--P,
(1.1)
The analysis is carried out in four parts. Section 2 sets up the model and examines the behavior of a firm facing a linear tax schedule, reproducing Brechling’s (1977) result in a continuous time model. Section 3 looks at the dynamic adjustment of the layoff rate when the firm faces a negatively-sloped tax schedule subject to a minimum tax rate. Section 4 explores the kink in the tax schedule formed by the maximum tax rate, raising the possibility of firms adjusting toward different steady states depending on their initial reserve ratios. In section 5 we put these results together for a tax schedule like the one described above. Section 6 suggests some implications for empirical analysis of the effect of UI financing on temporary layoffs.
2. Dynamic
adjustment with a linear tax schedule4
Consider a profit-maximizing firm with a fixed roster of workers, N, and a fixed capital stock, K5 The firm faces an infinite time horizon and is free to adjust its output at each point in time by laying off some fraction u, of its 3The tax schedule analyzed here differs slightly from the form introduced by Brechhng (1977). The form used here corresponds to Brechling’s tax schedule without the step at r=O. Here, the sloped segment of the tax schedule can extend to negative reserve ratios and is only truncated by a single flat segment at the ceiling tax rate. 4The model presented is analytically similar to Abel’s (1982) version of the 4 theory of investment. ‘The assumption of a fixed roster of workers is strong. We would expect a firm to be able to adjust these quantities over a long enough period of time, even if such adjustment is impossible in the short run. Holding the number of workers fixed greatly simplifies the analysis and allows us to concentrate on temporary layoffs because workers are permanently attached to the firm. The analytic simplicity comes from the definition of the reserve ratio as the ratio of the firm’s balance in its LJI account to its taxable payroll. Changes in the firm’s labor force would affect its taxable payroll and, therefore, its reserve ratio and UI tax rate. In terms of temporary layoffs, a spot market for labor in which the number of workers is easily altered from moment to moment would not generate the type of behavior we wish to examine. A more complex labor force adjustment mechanism would be required. The principal issue omitted from the present formulation is consideration of the employment incentive of the reserve ratio UI tax [see Topel and Welch (198O)J. The assumption of a fixed capital stock has little effect on the result because the amount of capital has no direct relation to the reserve ratio system.
38
workers.
J. Wolcowitz,
Thus, its output
Eficts
of the unemployment
insurance tax
at time t is given by
It is assumed that fi >O and f’ii ~0. By treating the output of the firm as numeraire, this also represents the total revenue of the firm at time t. The firm faces direct input costs for capital and labor. The rental price of capital is assumed to be fixed over time, so the total capital costs are fixed and can be ignored in determining the profit-maximizing path of layoffs. The firm’s wage bill is the wage rate at time t, w,, times the number of employees who are working, wt( 1 - u,)N. To focus attention on the reserve ratio system, we make very simple assumptions about the benefit rate paid to workers on layoff and the taxable wage base on which the firm must pay the UI tax rate. During periods of layoff, the worker receives benefits at the rate b which is assumed to be unrelated to the wage rate the worker would receive if working and the worker’s previous wages.‘j It is assumed, further, that the wage rate at all times is greater than the benefit rate. The taxable wage base is similarly simplified to be a fixed amount, G, independent of the worker’s stream of wages7 When the firm has reserve ratio rl, it faces the tax rate (a-v,) and pays a total UI tax bill of (a-sr,)E,N. The essence of reserve ratio experience rating is expressed by the assumption s > 0: the higher the reserve ratio, reflecting a better record in terms of keeping layoffs at a low level, the lower the firm’s tax rate. If s=O, the UI tax system is non-experience rated and the firm pays a fixed annual contribution per worker unrelated to its layoff history. The firm’s layoff decision at one moment affects its ability to earn profits at subsequent points in time through the UI tax rate. At time t, the firm contributes taxes to its UI account at the rate (a-sr,)GN. At the same time, its account is being depleted because of benefit payments to its workers who are on layoff at rate u&N. Thus, at time t, the balance in the firm’s reserve account is changing at the rate (a -sr,)iGN -u,bN. The firm’s reserve ratio is the ratio of its reserve balance to its taxable payroll, GN. Therefore, the dynamics of the firm’s reserve ratio can be described by8 ‘In reality, benefits are a fraction of the worker’s wage over a specified period, subject to minimum and maximum amounts. Thus, the current model is consistent with wages being low enough to entitle workers to the minimum benefit or so high that they receive the maximum benefit. The limited duration of benefits is ignored because it is unlikely to be a binding constraint for temporary layoffs. ‘The taxable wage base is usually the first $X a worker earns in a calendar year. The treatment in this model is consistent with the worker earning more than $X over the year despite periods of layoff. ‘The continuous adjustment of the reserve ratio is a simplification of the actual process which sets the tax rate for an entire year based on the reserve ratio on a fixed computation date. While this captures the essence of reserve ratio accounting, it may not be a good approximation of reality in a model of cyclical fluctuations. The true lags in adjusting the tax rate may lead the pattern of tax rates to bear little resemblance to the pattern of layoffs and demand.
J. Wolcowitz, Effects of the unemployment insurance tax
f,=a--ssr,-t4-
39
b
(2.1)
_.
W
The choice variables open to the firm at each point in time are the wage rate and the layoff rate. Workers are assumed to be identical and we assume that at each point in time the firm must offer workers a combination of wage rate and layoff rate yielding an expected utility level of V. Assuming no risk aversion, the representative worker’s utility level will be a function of the expected income level and the expected leisure resulting from time on layoff.’ Each of the identical workers has an equal probability of being on layoff each period. The expected income at time t is (1 - u,)w, + u,b.” The expected fraction of time on layoff is u,. Therefore, the firm faces a market utility constraint at time t requiring
(2.4
V[(l-u,)w,+utb,u,]=P It is further assumed that V, > 0, V, > 0, V, I < 0, V,, < 0, and V, 2 > 0. Given the discount rate, p, the firm’s problem is to choose time paths and w that maximize the present value of the firm’s profits:
of u
(2.3) subject market
to the dynamic adjustment of the reserve constraint (2.2). The appropriate Lagrangian
ratio (2.1) and the labor for maximization is:
&=f[(l-u,)N,K]-w,(l-u,)N-(a-sr,)GN+&
a-ssr,-uu,; c
:1
where 2, is the marginal value of the reserve ratio or the shadow price of a unit of the reserve ratio and qt is the marginal value to the firm of providing its workers with a lower level of utility. The optimal behavior of the firm ‘The assumption that instantaneous utility depends only on the current expected income and layoff rate is made to keep the problem tractable. One might expect workers who are permanently attached to a firm to have a utility constraint based on the entire pattern of their wages and layoffs. However, in this case the difficulty of the mathematics increases substantially because the maximization of the value of the firm does not reduce to a pair of autonomous equations that can be studied with a phase diagram. “We ignore the role of the income tax and the fact that wage income and benefits get taxed at different rates. This has been emphasized by Feldstein (1976, 1978) as an important part of the UI system’s incentive toward layoffs. However, he indicates that experience rating does not interact with the provisions that benefits and wages are taxed at different rates in terms of their layoff incentives.
40
J. Woicowitz,
Efficts
of the unemployment insurance tax
must satisfy the dynamic reserve constraint (2.2), and three additional dL >= &A,
-NJ’,
2-h --(1
aw*
ratio constraint (2.1), the labor first-order conditions:
+w,N-
market
(2.4)
-z~~)N+QI/,(~
-u,)=O,
(2.5)
(2.6) To examine the role of i. in determining the firm’s optimal behavior at a point in time, we focus on (2.2), (2.4) and (2.5). Using (2.5) to solve for Q, substituting into (2.4), and rearranging terms results in
f
1
+A
Lh+5.
'fiN
(2.7)
v,
The left-hand side of this equation is the marginal benefit to the firm of reducing layoffs. It is the sum of the marginal revenue product of labor and the value of the increase in the reserve ratio that results from a reduction in layoffs, where h/tiN is the amount that the reserve ratio increases from a marginal reduction in layoffs. The right-hand side is the marginal cost to the firm of reducing layoffs: the wage increase equivalent to the worker’s lost benefits and leisure. Thus, (2.7) says the firm should equate its marginal benefits and marginal costs in choosing its layoff rate.” Totally differentiating eq. (2.7) and the market utility constraint (2.2), we obtain: du, d3.,=
(1 -u,)V,(h/EN) A
1
where d is the determinant of the coefficients of these two value of d is negative by the assumptions of the model and second-order condition for profit maximization subject to a numerator is positive, so du,/di., is unambiguously negative.
equations. The as a necessary constraint. The An increase in
“Solving (2.7) for j’l puts the equation in the form of the first-order condition presented by Feldstein (1976). Feldstein’s J, the net unemployment insurance subsidy, is defined by (li.,,EN)h. the excess of the benefits to which the workers are entitled when on layoff over the U1 tax cost of these benefits to the lirm. Feldstein models the UI tax bill of the firm to consist of a lump-sum levy plus an amount equal to the fraction e of the benefits paid to the firm’s workers. The analogue to e in the current model is I,/GiV.
J. Wolcowitz,
Effects of the unemployment
insurance
tax
41
the marginal value of the reserve ratio at a point in time will decrease the fraction of the workforce on layoff at that time. This allows the firm to build its reserve ratio more rapidly. Solving for the effect of a change in the marginal value of the reserve ratio on the wage rate yields: dw, PC= dL,
(-b/GN)[I/,(b-w,)+V’J A
The sign of this expression will be the same as the sign of the bracketed term, the marginal utility of layoffs, taking account of both the increased leisure and reduced income that results from additional layoffs. If the extra utility from additional leisure outweighs the loss in utility from substituting UI benefits for wages, the wage rate must fall to compensate for an increase in layoffs. Therefore, a decrease in I, that increases u, must also decrease w,, so dw,/dL, is positive. Similarly, if the marginal utility of lost income outweighs the marginal utility from additional leisure, dw,/d& will be negative. To describe the behavior of 2 and r over time we must examine the laws of motion of 2 and r. The equation of motion for the reserve ratio is given by (2.1). Because of the negative relation between the layoff rate and A, the rate of adjustment of the reserve ratio can be defined by the current level of the reserve ratio and its current marginal value, ;1. The dynamic behavior of the marginal value of the reserve ratio is defined by (2.6). This equation can be interpreted as saying that the value to the firm of having the marginal unit of its reserve ratio at time t, the sum of the change in the firm’s tax bill s@tN and any increase in the value of holding reserves A,, must equal the required return on the reserve ratio due to depreciation and the interest rate [s +p]&. The reserve ratio depreciates at the rate s because an additional unit of the reserve ratio at a point in time reduces the tax rate the firm faces and leaves an increment of less than a unit for the firm’s reserve ratio in the subsequent period. Observe from (2.1) that ai/ar = - s. The laws of motion for I and J. form a pair of autonomous equations, so phase diagram analysis can be used to show the adjustment of 1 and I over time, as shown in fig. 1. The i = 0 locus slopes upward because an increase in r reduces the tax rate and must be offset by a decrease in u. A higher level of J. generates this lower layoff rate. At a given level of r, raising IV above the i=O locus will cause u to fall and the reserve ratio to grow, so i is positive. Similarly, below this locus, the reserve ratio will be falling. The x=0 locus is given by 2 =sGN/(s +p). The locus shows the present value of the stream of tax savings from the marginal unit of the reserve ratio and is represented by a horizontal line. At points above the A= 0 locus, the
42
J. Wolcowitz, Effects of the unemployment insurance tax
x
I I I
r
r* Fig. 1
required return on the reserve ratio to cover depreciation and interest, [s + p]A, exceeds that period’s tax savings from a unit of the reserve ratio, stiN, so x must be positive. Below this locus, x will be negative. The steady-state equilibrium is the intersection of the two loci at point E. The steady state is a saddlepoint equilibrium and the optimal path is along the /i=O locus. Thus, at all times 2 is sGN/(s+p), implying that the optimal combination of layoffs and wages that the firm provides its workers is the same in all periods. The marginal tax rate is constant from the linearity assumption, so the firm always faces the same incentives with respect to layoffs and does not adjust the layoff rate.r’ In the case of a non-experience rated UI tax system (i.e. s=O), the A=0 locus is defined by I =O. The equation of motion for rt is independent of the current reserve ratio. Because the tax rate is fixed, there is a single layoff rate that will generate a reserve balance at all levels of the reserve ratio. Therefore, the i =0 locus will be a horizontal line. The solution implies that 2 is constant at zero. If the firm’s profit-maximizing layoff rate is greater than the layoff rate consistent with the reserve balance, the reserve ratio will fall by a fixed amount each period and will approach negative infinity. If the profit-maximizing layoff rate is less than the reserve balance layoff rate, the “That the optimal level of the control variable is fixed over time comes from the fact that the functional to be maximized is linear in the state variable. Corresponding results have been established for a model of advertising expenditures by Gould (1970) and for models of investment in physical capital by Gould (1968) and Abel (1982). See Wolcowitz (1982) for a demonstration that when the tax function is twice continuously differentiable, but the second derivative is not identically zero, a constant layoff rate over time is not implied.
J. Wolcowitz, Effects of the unemployment insurance
tax
43
reserve ratio grows without bound. Should the firm’s optimal layoff rate coincide with the layoff rate that generates the reserve balance, the firm will remain at whatever reserve ratio it faced at the outset.
3. Dynamic
adjustment with a floor tax rate
Now assume that the UI tax schedule is composed of two linear portions. At reserve ratios less than f, the UI tax rate is a--r,. At reserve ratios above ?. the tax rate is fixed at the floor tax rate, a-s?. For r1 ?, the laws of motion are the same as when the entire tax schedule is non-experience rated.i3 For r < f, the i=O locus is upward sloping. When I> f, this locus is flat at ;5 defined by u(x) =(a-s?)G)lb. Because the tax schedule is continuous, the i= 0 locus is continuous. When I< ?, the value of 3, that satisfies A= 0 is sGN/(s + p). The 2 =0 locus is defined by 1,=0 for I > ?. These two flat segments of the A=0 locus are joined by a vertical segment at r =i;.14 The key to understanding the dynamic adjustment in this case is recognizing that when the firm is on the same segment of the tax schedule as the steady state it is approaching, the marginal tax rate for the firm will be constant so 2 will have the value that satisfies x=0; in this case, A, the layoff rate, and the wage rate will be constant during the approach to the steady state. However, when the firm is on the other segment of the tax schedule (or on either segment if the steady state is at the kink), it does not face a constant marginal tax rate because it will cross the kink during its adjustment to the steady state. While in this situation, the firm will not be along the horizontal i= 0 locus and will not choose a constant layoff rate while on this linear segment of the tax schedule. This adjustment process can be seen more clearly by examining the three possible configurations of the ?= 0 and i =0 loci shown in fig. 2. In fig. 2(a) the i=O locus intersects the i =0 locus at a reserve ratio along the sloped portion of the UI tax schedule. At reserve ratios less than F; the only part of the tax schedule relevant to the firm’s adjustment to the steady state is the 13The necessary conditions for optimization can be derived by treating the tax function as [a-~(t(r,r,+a~,!)], where aI, and a,, are control variables open to the firm in pursuit of profits. To ensure that (a,,, c+) be (1,O) when rl
1,
~11% = 0, a*&-F))LO.
r4This comes from the convexity of the problem implied by the increasing marginal costs of steady-state layoff rates. The Kuhn-Tucker restrictions on the multipliers associated with the constraints in footnote 13 do not rule out the possibility that i=O if I is between 0 and sGN/ (s+p) at r=F, so the x=0 locus will be continuous with a vertical portion from 1=0 to 1=&N/(s+p).
44
J. Wolcowitz,
Effects of the unemployment
insurance tax
(‘4
Fig. 2
sloped segment. The value of 2 will be .GN/(s+p) during the adjustment to the steady state over this range of reserve ratios, so the optimal combination of layoffs and wages will be constant. However, when the firm has a reserve ratio greater than r, the marginal tax rate is increasing during the adjustment to the steady state because the firm crosses the kink in the tax schedule. From the requirement that the path of adjustment of r and 2. lead to
J. Wolcowitz,
Effects of the unemployment
insurance tax
45
(I, sGN/(s + p)),” the laws of motion imply that 1, rises over time while the reserve ratio is greater than ?. As the lirm approaches ?, it sees a higher present value of taxes associated with layoffs, and reduces its layoff rate. The adjustment process in fig. 2(c) is completely analogous. In fig. 2(b), where the i = 0 and 3:= 0 loci intersect at r = P, the firm never acts as if the tax schedule is linear. Along each segment of the tax schedule the firm sees a non-constant marginal tax rate because its adjustment process takes it to the kink. Despite the linearity of each segment of the tax schedule, the firm does not necessarily choose a constant layoff rate-wage rate package along a segment.
4. Dynamic
adjustment with a ceiling tax rate
Consider once again a firm that faces a continuous piecewise linear tax schedule. However, now the UI tax rate is U-V, for Y,& and a-q for r1r, the optimal path to this steady state is along the x=0 locus. It is also possible to construct a path for at least some values of r
46
J. Wolcowitz,
Effects
ofthe unemployment
Fig. 3
insurance tax
J. Wolcowitz,
Effects of the unemployment
insurance tax
41
path for some values of r>r that join it at @,0).17 The problem here is to determine over which ranges of reserve ratios the firm will choose each strategy. Consider a firm with initial reserve ratio !: choosing between the II =sGNl (s +p) and ll=O paths. Define the constant layoff rate on the 1 =sGN/(s +p) path as ui and the constant layoff rate on the il=O path as u2. The increment in profits from choosing the first path over the second has two components: (1) the reduction in the pretax profits from choosing a lower layoff rate; and (2) the extra profits generated by the lower UI tax bill at each subsequent point in time. The expression for the first component is
(4.1) (b/G)(u-u,) is the change in the reserve ratio from the disbursement of benefits and ;1 is the present value of the extra profits from an extra unit of the reserve ratio. The present value of a continuous flow of $1 is l/p. At the margin, the extra profits from an increment in the layoff rate must equal the valuation of the forgone increment to the reserve ratio. Expression (4.1) sums these marginal valuations over the range of layoff rates from u1 to u2 as a measure of the lost profits from choosing layoff rate ul. The extra profits generated by the lower tax bill is 1 s --bN(g-u,), Ps+P
where u is the layoff rate that generates the reserve balance at the ceiling tax rate. The reduction in the benefits disbursed to the firm’s workers under the chosen strategy is bN(u, -uJ. For the increment bN(u-u,) the marginal cost is s/(s+p),l’ whereas the marginal cost of an additional dollar of benefits disbursed is 0 for the increment bN(u, -u). Combining (4.1) and (4.2), the extra profits from choosing the 2 = sGN/(s + p) path over the ;1=0 path for a firm with initial reserve ratio r are sivN/(s
+p)
1
ud,? .
Expression (4.3) moves continuously and monotonically negative values as u decreases from u2 to ur. Therefore,
(4.3) from positive to there must exist a
“The ‘intersection’ of the i=O and x=0 loci at r= r can be viewed as the unstable state that must lie between two stable steady states in sitiations of multiple equilibria. “This term corresponds to Feldstein’s (1976) e. See footnote 11.
steady
48
J. Wolcowitz, Effects of the unemployment insurance tax
value A* with a corresponding layoff rate u * for which (4.3) equals zero and the firm is indifferent between the two paths. If g>g*, a firm with reserve ratio r will prefer the path toward the steady state along the sloped portion of the tax schedule. If ug*, at any reserve ratio greater than or equal to r, the path toward r* will be preferred. Also, for at least some reserve ratios less than I_‘,the firm will find the path toward r* optimal. It can be shown that for A>0 there exists a tinite reserve ratio i
r,=I-~e-“‘j(U-l(,)esxdx. f If lim,,_, r,= + CO, the reserve ratio declines monotonically as the system adjusts toward (c,O). The optimal behavior of the firm is to choose the path leading to A=0 for all initial reserve ratios. However, it is also possible that “See Wolcowitz (1982) for derivations of these results.
J. Wolcowitz, Effects of the unemployment
insurunce
tax
49
if the laws of motion appropriate to the sloped segment of the tax schedule applied at all reserve ratios, lim,, _ m r, = - co. In this situation there exists a reserve ratio between r and the reserve ratio where the path crosses the i=O locus at which the firm is indifferent between the two adjustment strategies; at higher reserve ratios the firm will prefer to approach the steady state at r* and at lower reserve ratios the firm prefers the path that leads to 2 =O. Finally, if lim,, _ o. rr = r*, r* becomes the cutoff reserve ratio.20 5. Combining
the floor and ceiling tax rates
If a firm faces a continuous piecewise linear tax schedule characterized by both a floor tax rate and a ceiling tax rate, as defined by (l.l), the dynamic behavior of the firm again depends on the positions of the x=0 and ;=O loci. As the vectors of motion for r and 2 depend on the segment of the tax schedule as in the previous sections, the above analysis carries over to this more realistic situation. There are, however, two results worth noting. First, in the case of two potential steady states, if one of the steady states is at f, the range of reserve ratios over which the path toward a negatively infinite reserve ratio dominates extends to higher reserve ratios than if the sloped portion of the tax schedule were truncated by a lower tax rate corresponding to a higher reserve ratio. The floor tax rate reduces the gain from curtailing layoffs. Second, ?Land the layoff rate will not necessarily move monotonically over time. If the firm is on one flat segment and is adjusting toward a steady state on the other, layoffs will decrease while moving toward the sloped segment and then increase while on the sloped segment.
6. Conclusion The reserve ratio method of experience rating the unemployment insurance (UI) tax is an inherently dynamic process. A tax rate is assigned to a firm on the basis of its reserve ratio, calculated as the ratio of the balance in the firm’s UI account to its taxable payroll. Therefore, in making layoff decisions that will deplete its UI balance, we expect a firm to consider the effect on its future UI tax rate. This paper developed a model of temporary layoffs as part of the compensation package a firm provides its workers. We considered a tax function that is continuous but piecewise linear with three segments corresponding to a ceiling tax rate, a floor tax rate, and a negatively sloped segment between them. Because the entire tax schedule is not linear, the optimal behavior of a firm is not to offer the same package of layoffs and wages at each reserve ratio. Moreover, despite the linearity of each segment “‘See Wolcowitz
(1982) for derivations
of these results.
50
J. Wolcowitz,
Effects of the unemployment
insurnnce tnx
of the tax schedule, maximizing the value of the firm does not imply in general a strategy of offering a constant compensation package while it remains on a single segment of the tax schedule. Only if the firm is on the same segment of the tax schedule as the steady state it is approaching will it choose a constant layoff rate over time. If the lit-m crosses both kinks in the tax schedule during its adjustment toward the steady state, layoffs will decrease and then increase over time. Not only is it optimal for the firm to adjust its layoff rate over time, but this adjustment may not be monotonic toward the steady state layoff rate. These results have several implications. First, it is incorrect to conclude that firms currently at the floor or ceiling tax rates are completely nonexperience rated. The dynamic adjustment toward the sloped segment of the tax schedule may provide an incentive to curtail layoffs. Similarly, firms along the sloped portion of the tax schedule may increase their layoffs as they move toward the floor or ceiling tax rate. Thus, simple tabulations based on current reserve ratios do not fully indicate the extent of experience rating. Another implication is related to the estimation of the effect of the UI tax schedule on the rate of temporary layoffs. One approach is a cross-section over states or a time series within a state in order to get variation in the parameters of the tax system. While it might appear that such a sample is necessary because all firms in a state face an identical tax schedule, there is within state variation in the tax effects based on differences in current reserve ratios. However, different reserve ratios reflect different past behavior and may be correlated with other determinants of the layoff rate, such as vintage of the capital stock. It is also important to note that between two points in time the change in the layoff rate will be the result of two factors: changes in the UI tax schedule and changes in the reserve ratio along the path to the steady rate. These components could be offsetting. Careful modeling of the dynamic adjustment of the firm’s layoff decision is needed to disentangle the response of the firm to a change in the UI tax schedule. Finally, even in situations where the firm’s steady-state layoff behavior does not change when the UI tax schedule changes, the layoff rate may change at some reserve ratios. If policy is aimed at changing steady-state behavior, it is important to recognize that the behavior at a fixed reserve ratio may overstate the long-run responsiveness of the layoff rate. Again, careful modeling of the firm’s entire decision process is needed to disentangle these effects. An important extension of this work would be a more realistic treatment of temporary layoff behavior. Layoffs are largely a response to variations in demand. Therefore, it would be useful to incorporate demand fluctuations into the model to see how the UI tax system affects the layoff response to
J. Wolcowitz, EfSects of the unemployment
insurance tax
51
these fluctuations. Extensions of the range of behavioral response to the UI tax system open to the firm, such as changes in the labor force and inventory behavior, also need to be explored.
References Abel, A.B., 1982, Dynamic effects of permanent and temporary tax policies in a q model of investment, Journal of Monetary Economics 9. Baily, M.N., 1977, On the theory of layoffs and unemployment, Econometrica 45. Becker, J.M., 1972, Experience rating in unemployment insurance: An experiment in competitive socialism (The Johns Hopkins University Press, Baltimore). Brechling, F., 1977, The incentive effects of the U.S. unemployment insurance tax, in: R.G. Ehrenberg, ed., Research in labor economics, vol. 1 (JAI Press, Greenwich, Connecticut). Brechling, F., 1981, Layoffs and unemployment insurance, in: S. Rosen, ed., Studies in labor markets (The University of Chicago Press). Feldstein, M., 1976, Temporary layoffs in the theory of unemployment, Journal of Political Economy 84. Feldstein, M., 1978, The effect of unemployment insurance on temporary layoff unemployment. American Economic Review 68. Gould, J.P., 1968, Adjustment costs in the theory of investment of the firm, Review of Economic Studies 35. Gould, J.P., 1970, Diffusion processes and optimal advertising policies, in: E.S. Phelps, ed., Microeconomic foundations of employment and inflation theory (W.W. Norton & Company, New York). Haber, W. and M.G. Murray, 1966, Unemployment insurance in the American economy (Richard D. Irwin, Inc., Homewood, Illinois). Topel, R.H., 1982, Unemployment insurance, experience rating, and the occurrence of unemployment, Center for the Study of the Economy and the State Working Paper 021, University of Chicago. Topel, R.H. and F. Welch, 1980, Unemployment insurance: Survey and extensions, Economica 47. U.S. Department of Labor, Unemployment Insurance Service, 1982, Comparison of state unemployment insurance laws (U.S. Government Printing Offlice, Washington, D.C.). Wolcowitz, J., 1982, Dynamic effects of the unemployment insurance tax on temporary layoffs, Ph.D. dissertation, Harvard University.
of Public
DYNAMIC
Economics
25 (1984) 35-51. North-Holland
EFFECTS OF THE UNEMPLOYMENT TAX ON TEMPORARY LAYOFFS WOLCOWITZ*
Jeffrey Harvard Unioersity, Received
November
INSURANCE
Cambridge, MA 02138, USA
1982, revised version
received July 1983
The reserve ratio method of experience rating the unemployment insurance (UI) tax is an inherently dynamic process. A tax rate is assigned to a firm based on its reserve ratio, the ratio of the balance in its UI account to its taxable payroll. Therefore, in making layoff decisions that will deplete its UI balance, a firm will consider the effect on its future UI tax rate. This study explores the incentive effects of this method of UI financing on temporary layoff behavior of firms in a dynamic context.
1. Introduction The potential role of the financing scheme for unemployment insurance (UI) in encouraging firms to maintain stable work patterns was recognized at the beginning of the UI system in the United States and has been reflected in the experience rating provisions of the legislation.’ Under an experience rated tax scheme, the costs of the UI program are allocated among firms in relation to their history of generating compensable unemployment. Those firms that have had high turnover rates in the past, so that the workers they laid off have received a disproportionately large amount of the benefits disbursed, are assigned a relatively high tax rate. In this way, it is argued, firms internalize the costs of the unemployment they create and see a financial incentive for organizing production in a way that tends to stabilize employment. The most common system of experience rating in the United States is the reserve ratio method.2 Under this system, a bookkeeping account is es*This paper summarizes results from my Ph.D. dissertation. I am grateful to Andrew Abel for his guidance in formulating the problem. Martin Feldstein, Daniel Hamermesh, and James Medoff provided helpful comments and suggestions. ‘For the early history of the UI system in the United States, see Haber and Murray (1966). The most complete review of the experience rating issue is Becker (1972). Experience rating of the UI tax is unique to the United States. In Canada, the Unemployment Insurance Act of 1971 gave the UI Commission the power to establish an experience rated contribution system for large employers, but such a scheme has not been adopted. *As of 1983, the reserve ratio method was used in 31 states and the District of Columbia. The benefit ratio formula was used in 12 states, 4 states used the benefit-wage ratio formula, and 3 states used a payroll variation plan. See U.S. Department of Labor (1982) for descriptions of these methods of experience rating and other aspects of state UI systems, as well as some specific provisions of state programs. 0047~2727/84/$3.00
0
1984, Elsevier Science Publishers
B.V. (North-Holland)
36
J. Wolcowitz, Effects of the unemployment insurance tax
tablished for each firm. This account is credited with the UI tax payments made by the firm, less that portion earmarked for the administrative costs of the system and for a solvency fund that pays benefits that are not charged to a specific firm. Benefits paid to the firm’s ex-employees and laid off employees that the state treats as chargeable to the firm are deducted from the account. The firm’s reserve ratio is defined as the ratio of the balance in its account to its taxable payroll (or the average taxable payroll over several years). The state’s UI tax schedule assigns a tax rate to the firm according to its reserve ratio. This is a negative relation defined by a step function subject to maximum and minimum tax rates. An important characteristic of this system is the dynamic adjustment of the firm’s reserve ratio and tax rate. To the extent that a firm can alter its behavior in response to the UI tax system it will recognize that today’s layoffs will affect the reserve ratio and, thus, the UI tax rate in future periods. This dynamic process has largely been ignored in the existing literature on the effects of the UI tax system on layoffs. Feldstein (1976), Baily (1977) and Topel (1982) have constructed static models of the incentive effects of taxes, simplifying away the full structure of the reserve ratio mechanism; these models are best interpreted as descriptions of steady state behavior. Brechling (1977, 1981) and Topel and Welch (1980) explicitly mode1 the reserve ratio system and consider the adjustment pattern of the tax rate given a change in the layoff rate of a firm, but do not emphasize the incentive effect of the tax on the firm’s behavior, a linkage which runs in the opposite direction. The single exception to this is the final section of Brechling’s (1977) paper where he demonstrates that with a linear tax schedule it is optima1 for the firm to choose a constant layoff rate over time. Below we show that this result is specific to the assumption of linearity of the tax function. The dynamic structure is important in understanding the incentive effects on the layoff behavior of firms. For example, a firm at the maximum tax rate may choose to curtail its layoff rate to accumulate reserves and obtain a lower tax rate. Although it is on a flat segment of the tax schedule, it is incorrect to characterize such a firm as non-experience rated, as past studies have implied. Also, in thinking about empirical analysis of UI tax effects, changes in the firm’s layoff rate caused by changes in the parameters of the UI tax schedule must be disentangled from any changes caused by the dynamic adjustment of the reserve ratio and tax rate. The present study focuses on the dynamic nature of reserve ratio accounting in a mode1 of temporary layoffs. Temporary layoffs are treated as part of the compensation package firms offer workers. These layoffs can be seen as vacations subsidized by the UI program. Despite its importance in determining actual layoff rates, the role of demand variation is omitted to simplify the analysis and focus attention on the role of the UI tax system in determining the rate of temporary layoffs.
J. Wolcowitz,
Effects of the unemployment
insurance
tax
37
Emphasis is placed on the incentive effects of floor and ceiling tax rates. This aspect of the tax schedule has received the most attention in studies of the UI incentive toward layoffs because when the floor and ceiling tax rates are in effect, there appears to be no experience rating. We examine the case of a continuous piecewise linear tax schedule defined by3
z(It) = a -
SC,
a--r,,
rt < r, r_
a--P,
(1.1)
The analysis is carried out in four parts. Section 2 sets up the model and examines the behavior of a firm facing a linear tax schedule, reproducing Brechling’s (1977) result in a continuous time model. Section 3 looks at the dynamic adjustment of the layoff rate when the firm faces a negatively-sloped tax schedule subject to a minimum tax rate. Section 4 explores the kink in the tax schedule formed by the maximum tax rate, raising the possibility of firms adjusting toward different steady states depending on their initial reserve ratios. In section 5 we put these results together for a tax schedule like the one described above. Section 6 suggests some implications for empirical analysis of the effect of UI financing on temporary layoffs.
2. Dynamic
adjustment with a linear tax schedule4
Consider a profit-maximizing firm with a fixed roster of workers, N, and a fixed capital stock, K5 The firm faces an infinite time horizon and is free to adjust its output at each point in time by laying off some fraction u, of its 3The tax schedule analyzed here differs slightly from the form introduced by Brechhng (1977). The form used here corresponds to Brechling’s tax schedule without the step at r=O. Here, the sloped segment of the tax schedule can extend to negative reserve ratios and is only truncated by a single flat segment at the ceiling tax rate. 4The model presented is analytically similar to Abel’s (1982) version of the 4 theory of investment. ‘The assumption of a fixed roster of workers is strong. We would expect a firm to be able to adjust these quantities over a long enough period of time, even if such adjustment is impossible in the short run. Holding the number of workers fixed greatly simplifies the analysis and allows us to concentrate on temporary layoffs because workers are permanently attached to the firm. The analytic simplicity comes from the definition of the reserve ratio as the ratio of the firm’s balance in its LJI account to its taxable payroll. Changes in the firm’s labor force would affect its taxable payroll and, therefore, its reserve ratio and UI tax rate. In terms of temporary layoffs, a spot market for labor in which the number of workers is easily altered from moment to moment would not generate the type of behavior we wish to examine. A more complex labor force adjustment mechanism would be required. The principal issue omitted from the present formulation is consideration of the employment incentive of the reserve ratio UI tax [see Topel and Welch (198O)J. The assumption of a fixed capital stock has little effect on the result because the amount of capital has no direct relation to the reserve ratio system.
38
workers.
J. Wolcowitz,
Thus, its output
Eficts
of the unemployment
insurance tax
at time t is given by
It is assumed that fi >O and f’ii ~0. By treating the output of the firm as numeraire, this also represents the total revenue of the firm at time t. The firm faces direct input costs for capital and labor. The rental price of capital is assumed to be fixed over time, so the total capital costs are fixed and can be ignored in determining the profit-maximizing path of layoffs. The firm’s wage bill is the wage rate at time t, w,, times the number of employees who are working, wt( 1 - u,)N. To focus attention on the reserve ratio system, we make very simple assumptions about the benefit rate paid to workers on layoff and the taxable wage base on which the firm must pay the UI tax rate. During periods of layoff, the worker receives benefits at the rate b which is assumed to be unrelated to the wage rate the worker would receive if working and the worker’s previous wages.‘j It is assumed, further, that the wage rate at all times is greater than the benefit rate. The taxable wage base is similarly simplified to be a fixed amount, G, independent of the worker’s stream of wages7 When the firm has reserve ratio rl, it faces the tax rate (a-v,) and pays a total UI tax bill of (a-sr,)E,N. The essence of reserve ratio experience rating is expressed by the assumption s > 0: the higher the reserve ratio, reflecting a better record in terms of keeping layoffs at a low level, the lower the firm’s tax rate. If s=O, the UI tax system is non-experience rated and the firm pays a fixed annual contribution per worker unrelated to its layoff history. The firm’s layoff decision at one moment affects its ability to earn profits at subsequent points in time through the UI tax rate. At time t, the firm contributes taxes to its UI account at the rate (a-sr,)GN. At the same time, its account is being depleted because of benefit payments to its workers who are on layoff at rate u&N. Thus, at time t, the balance in the firm’s reserve account is changing at the rate (a -sr,)iGN -u,bN. The firm’s reserve ratio is the ratio of its reserve balance to its taxable payroll, GN. Therefore, the dynamics of the firm’s reserve ratio can be described by8 ‘In reality, benefits are a fraction of the worker’s wage over a specified period, subject to minimum and maximum amounts. Thus, the current model is consistent with wages being low enough to entitle workers to the minimum benefit or so high that they receive the maximum benefit. The limited duration of benefits is ignored because it is unlikely to be a binding constraint for temporary layoffs. ‘The taxable wage base is usually the first $X a worker earns in a calendar year. The treatment in this model is consistent with the worker earning more than $X over the year despite periods of layoff. ‘The continuous adjustment of the reserve ratio is a simplification of the actual process which sets the tax rate for an entire year based on the reserve ratio on a fixed computation date. While this captures the essence of reserve ratio accounting, it may not be a good approximation of reality in a model of cyclical fluctuations. The true lags in adjusting the tax rate may lead the pattern of tax rates to bear little resemblance to the pattern of layoffs and demand.
J. Wolcowitz, Effects of the unemployment insurance tax
f,=a--ssr,-t4-
39
b
(2.1)
_.
W
The choice variables open to the firm at each point in time are the wage rate and the layoff rate. Workers are assumed to be identical and we assume that at each point in time the firm must offer workers a combination of wage rate and layoff rate yielding an expected utility level of V. Assuming no risk aversion, the representative worker’s utility level will be a function of the expected income level and the expected leisure resulting from time on layoff.’ Each of the identical workers has an equal probability of being on layoff each period. The expected income at time t is (1 - u,)w, + u,b.” The expected fraction of time on layoff is u,. Therefore, the firm faces a market utility constraint at time t requiring
(2.4
V[(l-u,)w,+utb,u,]=P It is further assumed that V, > 0, V, > 0, V, I < 0, V,, < 0, and V, 2 > 0. Given the discount rate, p, the firm’s problem is to choose time paths and w that maximize the present value of the firm’s profits:
of u
(2.3) subject market
to the dynamic adjustment of the reserve constraint (2.2). The appropriate Lagrangian
ratio (2.1) and the labor for maximization is:
&=f[(l-u,)N,K]-w,(l-u,)N-(a-sr,)GN+&
a-ssr,-uu,; c
:1
where 2, is the marginal value of the reserve ratio or the shadow price of a unit of the reserve ratio and qt is the marginal value to the firm of providing its workers with a lower level of utility. The optimal behavior of the firm ‘The assumption that instantaneous utility depends only on the current expected income and layoff rate is made to keep the problem tractable. One might expect workers who are permanently attached to a firm to have a utility constraint based on the entire pattern of their wages and layoffs. However, in this case the difficulty of the mathematics increases substantially because the maximization of the value of the firm does not reduce to a pair of autonomous equations that can be studied with a phase diagram. “We ignore the role of the income tax and the fact that wage income and benefits get taxed at different rates. This has been emphasized by Feldstein (1976, 1978) as an important part of the UI system’s incentive toward layoffs. However, he indicates that experience rating does not interact with the provisions that benefits and wages are taxed at different rates in terms of their layoff incentives.
40
J. Woicowitz,
Efficts
of the unemployment insurance tax
must satisfy the dynamic reserve constraint (2.2), and three additional dL >= &A,
-NJ’,
2-h --(1
aw*
ratio constraint (2.1), the labor first-order conditions:
+w,N-
market
(2.4)
-z~~)N+QI/,(~
-u,)=O,
(2.5)
(2.6) To examine the role of i. in determining the firm’s optimal behavior at a point in time, we focus on (2.2), (2.4) and (2.5). Using (2.5) to solve for Q, substituting into (2.4), and rearranging terms results in
f
1
+A
Lh+5.
'fiN
(2.7)
v,
The left-hand side of this equation is the marginal benefit to the firm of reducing layoffs. It is the sum of the marginal revenue product of labor and the value of the increase in the reserve ratio that results from a reduction in layoffs, where h/tiN is the amount that the reserve ratio increases from a marginal reduction in layoffs. The right-hand side is the marginal cost to the firm of reducing layoffs: the wage increase equivalent to the worker’s lost benefits and leisure. Thus, (2.7) says the firm should equate its marginal benefits and marginal costs in choosing its layoff rate.” Totally differentiating eq. (2.7) and the market utility constraint (2.2), we obtain: du, d3.,=
(1 -u,)V,(h/EN) A
1
where d is the determinant of the coefficients of these two value of d is negative by the assumptions of the model and second-order condition for profit maximization subject to a numerator is positive, so du,/di., is unambiguously negative.
equations. The as a necessary constraint. The An increase in
“Solving (2.7) for j’l puts the equation in the form of the first-order condition presented by Feldstein (1976). Feldstein’s J, the net unemployment insurance subsidy, is defined by (li.,,EN)h. the excess of the benefits to which the workers are entitled when on layoff over the U1 tax cost of these benefits to the lirm. Feldstein models the UI tax bill of the firm to consist of a lump-sum levy plus an amount equal to the fraction e of the benefits paid to the firm’s workers. The analogue to e in the current model is I,/GiV.
J. Wolcowitz,
Effects of the unemployment
insurance
tax
41
the marginal value of the reserve ratio at a point in time will decrease the fraction of the workforce on layoff at that time. This allows the firm to build its reserve ratio more rapidly. Solving for the effect of a change in the marginal value of the reserve ratio on the wage rate yields: dw, PC= dL,
(-b/GN)[I/,(b-w,)+V’J A
The sign of this expression will be the same as the sign of the bracketed term, the marginal utility of layoffs, taking account of both the increased leisure and reduced income that results from additional layoffs. If the extra utility from additional leisure outweighs the loss in utility from substituting UI benefits for wages, the wage rate must fall to compensate for an increase in layoffs. Therefore, a decrease in I, that increases u, must also decrease w,, so dw,/dL, is positive. Similarly, if the marginal utility of lost income outweighs the marginal utility from additional leisure, dw,/d& will be negative. To describe the behavior of 2 and r over time we must examine the laws of motion of 2 and r. The equation of motion for the reserve ratio is given by (2.1). Because of the negative relation between the layoff rate and A, the rate of adjustment of the reserve ratio can be defined by the current level of the reserve ratio and its current marginal value, ;1. The dynamic behavior of the marginal value of the reserve ratio is defined by (2.6). This equation can be interpreted as saying that the value to the firm of having the marginal unit of its reserve ratio at time t, the sum of the change in the firm’s tax bill s@tN and any increase in the value of holding reserves A,, must equal the required return on the reserve ratio due to depreciation and the interest rate [s +p]&. The reserve ratio depreciates at the rate s because an additional unit of the reserve ratio at a point in time reduces the tax rate the firm faces and leaves an increment of less than a unit for the firm’s reserve ratio in the subsequent period. Observe from (2.1) that ai/ar = - s. The laws of motion for I and J. form a pair of autonomous equations, so phase diagram analysis can be used to show the adjustment of 1 and I over time, as shown in fig. 1. The i = 0 locus slopes upward because an increase in r reduces the tax rate and must be offset by a decrease in u. A higher level of J. generates this lower layoff rate. At a given level of r, raising IV above the i=O locus will cause u to fall and the reserve ratio to grow, so i is positive. Similarly, below this locus, the reserve ratio will be falling. The x=0 locus is given by 2 =sGN/(s +p). The locus shows the present value of the stream of tax savings from the marginal unit of the reserve ratio and is represented by a horizontal line. At points above the A= 0 locus, the
42
J. Wolcowitz, Effects of the unemployment insurance tax
x
I I I
r
r* Fig. 1
required return on the reserve ratio to cover depreciation and interest, [s + p]A, exceeds that period’s tax savings from a unit of the reserve ratio, stiN, so x must be positive. Below this locus, x will be negative. The steady-state equilibrium is the intersection of the two loci at point E. The steady state is a saddlepoint equilibrium and the optimal path is along the /i=O locus. Thus, at all times 2 is sGN/(s+p), implying that the optimal combination of layoffs and wages that the firm provides its workers is the same in all periods. The marginal tax rate is constant from the linearity assumption, so the firm always faces the same incentives with respect to layoffs and does not adjust the layoff rate.r’ In the case of a non-experience rated UI tax system (i.e. s=O), the A=0 locus is defined by I =O. The equation of motion for rt is independent of the current reserve ratio. Because the tax rate is fixed, there is a single layoff rate that will generate a reserve balance at all levels of the reserve ratio. Therefore, the i =0 locus will be a horizontal line. The solution implies that 2 is constant at zero. If the firm’s profit-maximizing layoff rate is greater than the layoff rate consistent with the reserve balance, the reserve ratio will fall by a fixed amount each period and will approach negative infinity. If the profit-maximizing layoff rate is less than the reserve balance layoff rate, the “That the optimal level of the control variable is fixed over time comes from the fact that the functional to be maximized is linear in the state variable. Corresponding results have been established for a model of advertising expenditures by Gould (1970) and for models of investment in physical capital by Gould (1968) and Abel (1982). See Wolcowitz (1982) for a demonstration that when the tax function is twice continuously differentiable, but the second derivative is not identically zero, a constant layoff rate over time is not implied.
J. Wolcowitz, Effects of the unemployment insurance
tax
43
reserve ratio grows without bound. Should the firm’s optimal layoff rate coincide with the layoff rate that generates the reserve balance, the firm will remain at whatever reserve ratio it faced at the outset.
3. Dynamic
adjustment with a floor tax rate
Now assume that the UI tax schedule is composed of two linear portions. At reserve ratios less than f, the UI tax rate is a--r,. At reserve ratios above ?. the tax rate is fixed at the floor tax rate, a-s?. For r1 ?, the laws of motion are the same as when the entire tax schedule is non-experience rated.i3 For r < f, the i=O locus is upward sloping. When I> f, this locus is flat at ;5 defined by u(x) =(a-s?)G)lb. Because the tax schedule is continuous, the i= 0 locus is continuous. When I< ?, the value of 3, that satisfies A= 0 is sGN/(s + p). The 2 =0 locus is defined by 1,=0 for I > ?. These two flat segments of the A=0 locus are joined by a vertical segment at r =i;.14 The key to understanding the dynamic adjustment in this case is recognizing that when the firm is on the same segment of the tax schedule as the steady state it is approaching, the marginal tax rate for the firm will be constant so 2 will have the value that satisfies x=0; in this case, A, the layoff rate, and the wage rate will be constant during the approach to the steady state. However, when the firm is on the other segment of the tax schedule (or on either segment if the steady state is at the kink), it does not face a constant marginal tax rate because it will cross the kink during its adjustment to the steady state. While in this situation, the firm will not be along the horizontal i= 0 locus and will not choose a constant layoff rate while on this linear segment of the tax schedule. This adjustment process can be seen more clearly by examining the three possible configurations of the ?= 0 and i =0 loci shown in fig. 2. In fig. 2(a) the i=O locus intersects the i =0 locus at a reserve ratio along the sloped portion of the UI tax schedule. At reserve ratios less than F; the only part of the tax schedule relevant to the firm’s adjustment to the steady state is the 13The necessary conditions for optimization can be derived by treating the tax function as [a-~(t(r,r,+a~,!)], where aI, and a,, are control variables open to the firm in pursuit of profits. To ensure that (a,,, c+) be (1,O) when rl
1,
~11% = 0, a*&-F))LO.
r4This comes from the convexity of the problem implied by the increasing marginal costs of steady-state layoff rates. The Kuhn-Tucker restrictions on the multipliers associated with the constraints in footnote 13 do not rule out the possibility that i=O if I is between 0 and sGN/ (s+p) at r=F, so the x=0 locus will be continuous with a vertical portion from 1=0 to 1=&N/(s+p).
44
J. Wolcowitz,
Effects of the unemployment
insurance tax
(‘4
Fig. 2
sloped segment. The value of 2 will be .GN/(s+p) during the adjustment to the steady state over this range of reserve ratios, so the optimal combination of layoffs and wages will be constant. However, when the firm has a reserve ratio greater than r, the marginal tax rate is increasing during the adjustment to the steady state because the firm crosses the kink in the tax schedule. From the requirement that the path of adjustment of r and 2. lead to
J. Wolcowitz,
Effects of the unemployment
insurance tax
45
(I, sGN/(s + p)),” the laws of motion imply that 1, rises over time while the reserve ratio is greater than ?. As the lirm approaches ?, it sees a higher present value of taxes associated with layoffs, and reduces its layoff rate. The adjustment process in fig. 2(c) is completely analogous. In fig. 2(b), where the i = 0 and 3:= 0 loci intersect at r = P, the firm never acts as if the tax schedule is linear. Along each segment of the tax schedule the firm sees a non-constant marginal tax rate because its adjustment process takes it to the kink. Despite the linearity of each segment of the tax schedule, the firm does not necessarily choose a constant layoff rate-wage rate package along a segment.
4. Dynamic
adjustment with a ceiling tax rate
Consider once again a firm that faces a continuous piecewise linear tax schedule. However, now the UI tax rate is U-V, for Y,& and a-q for r1
46
J. Wolcowitz,
Effects
ofthe unemployment
Fig. 3
insurance tax
J. Wolcowitz,
Effects of the unemployment
insurance tax
41
path for some values of r>r that join it at @,0).17 The problem here is to determine over which ranges of reserve ratios the firm will choose each strategy. Consider a firm with initial reserve ratio !: choosing between the II =sGNl (s +p) and ll=O paths. Define the constant layoff rate on the 1 =sGN/(s +p) path as ui and the constant layoff rate on the il=O path as u2. The increment in profits from choosing the first path over the second has two components: (1) the reduction in the pretax profits from choosing a lower layoff rate; and (2) the extra profits generated by the lower UI tax bill at each subsequent point in time. The expression for the first component is
(4.1) (b/G)(u-u,) is the change in the reserve ratio from the disbursement of benefits and ;1 is the present value of the extra profits from an extra unit of the reserve ratio. The present value of a continuous flow of $1 is l/p. At the margin, the extra profits from an increment in the layoff rate must equal the valuation of the forgone increment to the reserve ratio. Expression (4.1) sums these marginal valuations over the range of layoff rates from u1 to u2 as a measure of the lost profits from choosing layoff rate ul. The extra profits generated by the lower tax bill is 1 s --bN(g-u,), Ps+P
where u is the layoff rate that generates the reserve balance at the ceiling tax rate. The reduction in the benefits disbursed to the firm’s workers under the chosen strategy is bN(u, -uJ. For the increment bN(u-u,) the marginal cost is s/(s+p),l’ whereas the marginal cost of an additional dollar of benefits disbursed is 0 for the increment bN(u, -u). Combining (4.1) and (4.2), the extra profits from choosing the 2 = sGN/(s + p) path over the ;1=0 path for a firm with initial reserve ratio r are sivN/(s
+p)
1
ud,? .
Expression (4.3) moves continuously and monotonically negative values as u decreases from u2 to ur. Therefore,
(4.3) from positive to there must exist a
“The ‘intersection’ of the i=O and x=0 loci at r= r can be viewed as the unstable state that must lie between two stable steady states in sitiations of multiple equilibria. “This term corresponds to Feldstein’s (1976) e. See footnote 11.
steady
48
J. Wolcowitz, Effects of the unemployment insurance tax
value A* with a corresponding layoff rate u * for which (4.3) equals zero and the firm is indifferent between the two paths. If g>g*, a firm with reserve ratio r will prefer the path toward the steady state along the sloped portion of the tax schedule. If u
r,=I-~e-“‘j(U-l(,)esxdx. f If lim,,_, r,= + CO, the reserve ratio declines monotonically as the system adjusts toward (c,O). The optimal behavior of the firm is to choose the path leading to A=0 for all initial reserve ratios. However, it is also possible that “See Wolcowitz (1982) for derivations of these results.
J. Wolcowitz, Effects of the unemployment
insurunce
tax
49
if the laws of motion appropriate to the sloped segment of the tax schedule applied at all reserve ratios, lim,, _ m r, = - co. In this situation there exists a reserve ratio between r and the reserve ratio where the path crosses the i=O locus at which the firm is indifferent between the two adjustment strategies; at higher reserve ratios the firm will prefer to approach the steady state at r* and at lower reserve ratios the firm prefers the path that leads to 2 =O. Finally, if lim,, _ o. rr = r*, r* becomes the cutoff reserve ratio.20 5. Combining
the floor and ceiling tax rates
If a firm faces a continuous piecewise linear tax schedule characterized by both a floor tax rate and a ceiling tax rate, as defined by (l.l), the dynamic behavior of the firm again depends on the positions of the x=0 and ;=O loci. As the vectors of motion for r and 2 depend on the segment of the tax schedule as in the previous sections, the above analysis carries over to this more realistic situation. There are, however, two results worth noting. First, in the case of two potential steady states, if one of the steady states is at f, the range of reserve ratios over which the path toward a negatively infinite reserve ratio dominates extends to higher reserve ratios than if the sloped portion of the tax schedule were truncated by a lower tax rate corresponding to a higher reserve ratio. The floor tax rate reduces the gain from curtailing layoffs. Second, ?Land the layoff rate will not necessarily move monotonically over time. If the firm is on one flat segment and is adjusting toward a steady state on the other, layoffs will decrease while moving toward the sloped segment and then increase while on the sloped segment.
6. Conclusion The reserve ratio method of experience rating the unemployment insurance (UI) tax is an inherently dynamic process. A tax rate is assigned to a firm on the basis of its reserve ratio, calculated as the ratio of the balance in the firm’s UI account to its taxable payroll. Therefore, in making layoff decisions that will deplete its UI balance, we expect a firm to consider the effect on its future UI tax rate. This paper developed a model of temporary layoffs as part of the compensation package a firm provides its workers. We considered a tax function that is continuous but piecewise linear with three segments corresponding to a ceiling tax rate, a floor tax rate, and a negatively sloped segment between them. Because the entire tax schedule is not linear, the optimal behavior of a firm is not to offer the same package of layoffs and wages at each reserve ratio. Moreover, despite the linearity of each segment “‘See Wolcowitz
(1982) for derivations
of these results.
50
J. Wolcowitz,
Effects of the unemployment
insurnnce tnx
of the tax schedule, maximizing the value of the firm does not imply in general a strategy of offering a constant compensation package while it remains on a single segment of the tax schedule. Only if the firm is on the same segment of the tax schedule as the steady state it is approaching will it choose a constant layoff rate over time. If the lit-m crosses both kinks in the tax schedule during its adjustment toward the steady state, layoffs will decrease and then increase over time. Not only is it optimal for the firm to adjust its layoff rate over time, but this adjustment may not be monotonic toward the steady state layoff rate. These results have several implications. First, it is incorrect to conclude that firms currently at the floor or ceiling tax rates are completely nonexperience rated. The dynamic adjustment toward the sloped segment of the tax schedule may provide an incentive to curtail layoffs. Similarly, firms along the sloped portion of the tax schedule may increase their layoffs as they move toward the floor or ceiling tax rate. Thus, simple tabulations based on current reserve ratios do not fully indicate the extent of experience rating. Another implication is related to the estimation of the effect of the UI tax schedule on the rate of temporary layoffs. One approach is a cross-section over states or a time series within a state in order to get variation in the parameters of the tax system. While it might appear that such a sample is necessary because all firms in a state face an identical tax schedule, there is within state variation in the tax effects based on differences in current reserve ratios. However, different reserve ratios reflect different past behavior and may be correlated with other determinants of the layoff rate, such as vintage of the capital stock. It is also important to note that between two points in time the change in the layoff rate will be the result of two factors: changes in the UI tax schedule and changes in the reserve ratio along the path to the steady rate. These components could be offsetting. Careful modeling of the dynamic adjustment of the firm’s layoff decision is needed to disentangle the response of the firm to a change in the UI tax schedule. Finally, even in situations where the firm’s steady-state layoff behavior does not change when the UI tax schedule changes, the layoff rate may change at some reserve ratios. If policy is aimed at changing steady-state behavior, it is important to recognize that the behavior at a fixed reserve ratio may overstate the long-run responsiveness of the layoff rate. Again, careful modeling of the firm’s entire decision process is needed to disentangle these effects. An important extension of this work would be a more realistic treatment of temporary layoff behavior. Layoffs are largely a response to variations in demand. Therefore, it would be useful to incorporate demand fluctuations into the model to see how the UI tax system affects the layoff response to
J. Wolcowitz, EfSects of the unemployment
insurance tax
51
these fluctuations. Extensions of the range of behavioral response to the UI tax system open to the firm, such as changes in the labor force and inventory behavior, also need to be explored.
References Abel, A.B., 1982, Dynamic effects of permanent and temporary tax policies in a q model of investment, Journal of Monetary Economics 9. Baily, M.N., 1977, On the theory of layoffs and unemployment, Econometrica 45. Becker, J.M., 1972, Experience rating in unemployment insurance: An experiment in competitive socialism (The Johns Hopkins University Press, Baltimore). Brechling, F., 1977, The incentive effects of the U.S. unemployment insurance tax, in: R.G. Ehrenberg, ed., Research in labor economics, vol. 1 (JAI Press, Greenwich, Connecticut). Brechling, F., 1981, Layoffs and unemployment insurance, in: S. Rosen, ed., Studies in labor markets (The University of Chicago Press). Feldstein, M., 1976, Temporary layoffs in the theory of unemployment, Journal of Political Economy 84. Feldstein, M., 1978, The effect of unemployment insurance on temporary layoff unemployment. American Economic Review 68. Gould, J.P., 1968, Adjustment costs in the theory of investment of the firm, Review of Economic Studies 35. Gould, J.P., 1970, Diffusion processes and optimal advertising policies, in: E.S. Phelps, ed., Microeconomic foundations of employment and inflation theory (W.W. Norton & Company, New York). Haber, W. and M.G. Murray, 1966, Unemployment insurance in the American economy (Richard D. Irwin, Inc., Homewood, Illinois). Topel, R.H., 1982, Unemployment insurance, experience rating, and the occurrence of unemployment, Center for the Study of the Economy and the State Working Paper 021, University of Chicago. Topel, R.H. and F. Welch, 1980, Unemployment insurance: Survey and extensions, Economica 47. U.S. Department of Labor, Unemployment Insurance Service, 1982, Comparison of state unemployment insurance laws (U.S. Government Printing Offlice, Washington, D.C.). Wolcowitz, J., 1982, Dynamic effects of the unemployment insurance tax on temporary layoffs, Ph.D. dissertation, Harvard University.