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Labor market equilibria under limited liability
Labor MarketEquilibria underLimitedLiability lvan E. Brick and Ephraim F. Sudit
INTRODUCtlON Most treatments of optimal micro-output and -input decision under uncettain demand do not explicitly allow for the probability of default (Sandmo 1971; Leland 1972). It is usually implicitly assumed that in the process of contracting for labor services, neither employers nor employees are significantly a&c&xi in their decisions by the prospect of bankruptcy.’ The fact that such underlying assumptions could be at
times restrictive is aptly underscored by Azariadis (1975):H a rational entrepreneur will not shield the terms of employment of even his most valuable group ofemployees against arbitrary demand fluctuations if doing SCinvolves more thm a token probability that the firm might go bankrupt.“2 Even in nonunionized competitive labor markets labor servicesare supplied and exchanged for some informal set of commitments or implicit labor contracts on the part of firms as to a “reasonable” exmted duration of employment. For a detailed discussion of the nature of such implicit contracts see Azariadis( 1975,t976) and Bailey(1974). It should be interesting to examine the manner in which labor contracting is affected by the prospect of bankruptcy. The institution of limited liability introduces in many instances a si;Gicant positive probability that -Pm ’ Weiss (1976) is the exceptton.For a more detailed referenceto his work, see below. z Azariadis (1975) p. I 184. From the Graduate School of Management. Rutgers University, Newark, New Jersey. Address reprint requeststo Dr. Ivan E..Brick,Assistant Professor 01’Finance and Economics, Graduate School of Management, Rutgers Un.versity, Newark, New Jersey 07102.
Journal of Economicsand Business34.51-58 ( 1982) @ 1982 Temple University
implicit or explicit h&or contracts could be wholly or partially abrogated in the eventuality of default. Without limited liability all cost of default should be borne by the owners of the firm. Labor contracts could be abrogated only under rare circumstances when owners are personally bankrupt. Thus under
limited liability default is much more likely to result in significantcosts for employees than in the absence of limited liability. Such costs include loss of income incurred in the press of search for new jobs (Grossman 1973)as well as loss of pension rights and other seniority privileges. Other things being equal, “conversion” by firms to limited liability status reduces risks for owners and increases risks for employees. Our major objective is to explore the impact of limited liability through the probability of default on decisions made by both firmsand labor in exchanging labor services. Weiss (1976)examines the impact of the probability of default (viathe fmancing decision of the firm) on the production decision. He assumes throughout that in the case of default, workers would not sustain any loss. Assuming that in the absence of limited liabilitythe probability of default does not represent a significant factor in labor contracting decisions,we first explore the impact of the introduction of limited liability on equilibrium employment levels, wage rates, capital inputs, and level of output in perfectly competitive risk-neutral labor, capital, and product markets. We demonstrate that the introduction of limited liability has neutral effects upon production decisions; the optimal quantiries of labor, capital, and the final product do not change. But the nominal wage rate rises to compensate workers for their expected loss 51 0278-2294/82/01/051-08$2.75
1. E. BrickandE.
52
induced by limited liability. We obtain similar results for the quantity-setting monopolist. We conclude by observing that if the owners of firm exhibit risk-averse behavior consistent with Roy’s (1952) safety-first criterion, the neutrality of the production decision in the case of knited liability does not necessarily hold. We demonstrate that limited liability generally decreases the usage of capital. Changes in the quantity of labor depend on the magnitude of complementarity between capita1 and labor and on whether the output elasticity of labor is greater or less than unity. Hence it is possible in both competitive and noncompetitive cases for limited liability to decrease the ux of labor and capital and hence reduce output.
F. Sudit
WU
where fi= random Gutput price Q = quantity of outp\Jt r;i= random wage rate L = quantity OFlabor services K =quantity
of capita1 inputs
6 = price of a unit of capital rJ = risk-free interest rate
PERFECT COMPETITION AND RISK NEliTRAEITY
L’=(present) value of future cash flows
In this section we explore the impact of stockholders’ limited liability upon labor market equilibria when the input and output markets are perfectly competit ive and all economic agents are risk neutral. Assume I) output price uncertainty; 2) all equity firms; 3) single-period framework, with capital and labor inputs determined at the beginning of the period; and 4) no salvage value LJr capital at th:: end of the period. Subject to these assumptions, ‘we show that the introduction of stockholder’s IimiM liability will not akct optinz? usage of labor and capital inputs. Nominal \s,lges. however. will be higher under limited Irability. We assume t:lat capital inputs are paid at the beginning of the period to isolate the effect of limited liability on the labor market. Consequently, labor services are paid at the end of the period on resolution of price uncertainty. The objective function of the firm is to maximize shareholders’ wealth. The firm must therefore determine the optimal amount of capital and labor subject to a production constraint”: M/$X
(l’_cjK)=
L.h
E(jiQ. . .- __.. ix)
____
‘+!I.
-bK
wage
riW
Equation (1) is the net present value of the firm and equation (1A) is the production constraint of the firm. Equation (1B) describes the nature of the randomness of the wage rate 6. If the fiml can meet all its labor costs, it will pay the contractual wage G. if the firm cannot completely satisfy labor claims, the entire proceeds of the firm is then devoted to satisfying labor claims and the randomness of the wage rate is attributable in its entirety to the randomness in the output price and limited liability in case of detaultW4 The loss incurred by workers as a result of default in the context of limited liability needs further elaboration. Suppose the period under consideration in equation (1) is one year. If the fum defaults, workers exercising their seniority rights in bankruptcy proceedings can claim compensation only for the services rendered. If their labor contract exa%ds one year, workers are likely to incur losses associated with possible periods of unemployment, search costs for a new job, and loss of nonvested pension rights. Therefore equation (1B) is a proxy for labor losses upon the firm’s default. .
(1)
sub&t to G(l_.. K)=Q
i? =contractual
04
’ Although the randomnessof H’is attributable to the randomwss ofp. owners or workers in a competitivesettingare neither able or ncx-essarily willing to tie the contractual wage rate w to the price of the product. Nonetheless,sina owners and WOIkers share total revenue net of capital costs, they may incorporate p&t sharing schemesinto their wage contracts. This would make w indinzctiya function of p. The analysisof roles of profit sharing in the context of changmg liabilit: forms is beyond the scopeof our presentanalIsis but should be il~terestingto pursue in further aresearch.(We are indebted to a re.‘&e for alerting u to this possibility.)
LabarKarketEquiliblia Diffkrentiating equation(1)partiallywith respectto t iind K and rearmnag, we obtain the following first-order conditions:
53 riskless industry:
where w. is the riskss wage rate. Thus the change in the wage rate induced by a change in labor input
&C/at of equation (l), can be expressed as
Iu@;~LGJf(p)& e
dG
(1 -F(u))-’
-=
ix
WkR
aQ-w(-J
GL= -
c
c?L
G&= dQ>O
ix ix
U
=----
Q'
Analogous to the certainty case, the optimal amounts of labor and capital are determined so as to equate the respective value of their marginal products to their marginal factor costs. But owing to limited liability the vAxs of the marginal products in equations (2)and (3)are not simply the products of the expected price and marginal productivity; rather, the marginal productivities of labor and capital will reflect the limitied (truncated) loss faced by the employers due to limited liability. Becauseof Iimited liabilityenjoyed by st&cholders and the randomness of output price, a positive probability of the firm’s not being able to completely satisfy the claims of labor is likely. Rational behavior implies that the limited liability industry would have to pay higher wages to attract labor. Consequently, the marginal factor cost functions for labor and capital in equations (2)and (3)appear similar to their cotlnterparts in the imperfectly competitive inputmarket cases, where the firm‘sdemand for factors of production affectsthe prices of those inputs. We have not departed from the perfect competition case. Rather, in the case of uncertainty, perfect competition allo\Ang for probability of default, implies the existence of a schedule of prices for the different level of uncertainty. No single firm can affect this schedule by its factor demand. But as the firm employs more labor and/or more capital, it is affecting in the process labor’s eipected loss and thereby the wage rate changes. Risk neutrality implies that the expected wage rate of the risky industry should equal the wage rate ofthe
(5)
0
where F(*l ) is the cumulative density function off@). Similarly,the change in the wage rate induced by a change in the capital input X$X of equation (3)can be expressed as
is
ix=-
sp
!!f(p)djj
o
(1 -F(u))-
L
l.
(6)
From equation (5) we can infer that
srs,<
--
i?L
>
0
. LGL
’
(7)
If
Q
Note that LGLjQ is the output elasticity of labor. Multiplying both sides of equation (2) by L/(E(j 1jj > a)Q) we obtain
LGL -=
Q
WIPNQ
'
(8)
For the firm to plan to stay in business, ii~Lshould be less than E@ 1j > a)(), the expect@ revenue for the limited liability firm. Consequently, %@L must be positive so that the conditions of equation (7)are not violated. It follows that an optimiz%g firm would produce in a range where the outf ut elasticityof labor is less than unity and tke labor supply function is positively sloped. From equation (6), because GK> 0, it is obvious that dG/aK < 0. In view of the signs of a6,1LVL and &G/Z& it would appear that the introduction of limited liability will lead to a more capital-intensive production (i.e.,a higher K/L). Substituting the values of &+X and X$X from equations (5) and (6) into the first-order conditions of equations (2) and (3), respectively, and collecting terms, we have: G,E(i) = E(6) = w.
(9)
I. E. BrickandE. F. Sudit G KE(j5)=
(10)
b(1+ ‘,I.
Equations (9) and (10) are the first-order conditions of L and K for the no limited liability case. Thus the limited itability does not alter the optimul leue1oJ‘L trnd K. Our results in equations (9) and (10) are attributable to the following economic phenomena. After the introduction of limited liability, employees are corn: pensated by higher wages to keep their expected wealth intact ‘@e shareholders can “afford” to pay the higher wages without erosion in their expected wealth owing to the limited (truncated) loss they face from limited liability. But the second-order conditions for optimality in the iimited liability case are more complicated than the no-limited htibility and/or perfect certainty case. In particular. concavity of the production functions in K and E do not guarantee sufficiency of the first-order conditions. For a more detailed discussion, see Appendix A. A transition from a perfectly competitive to an imperfectly competitive market structure is expected, in the framework of c!assica.l static price theory, to a&ct optimal production decisions. It seems interesting to explore whether changing liability forms could have an intervening impact on changes in production decisions for the quantity setting monopolist. In Appendix B we demonstrate the neutral impact of limited liability on the quantity setting monopolist assuming risk-neutral preferences.
SP KI-Y-FlRST
PREFERENCES
Qume that the objective function of the owners of tt~ firm is to maxim& expected profits such that the pr&ability of future loss is less than B (i.e., Prob df 4)s R). This risk-averse behavior is consistent with Roy’s (1952) safety-first criterion (see also Arzac ( 1976)).Assume further that workers are risk-neutral. Introduction of limited liability guarantees that Prob ( t’;~0) = 0. Hence if the safety constrtint is binding in a no limited liability environment. the optimal production decision will drrfer from the limited liability case. We can compare the aptimal production decision of both the limited liabilitjf case and thz no limited liability case where the safety constraint is binding by wns of a diagram in the K, E plane. The first-order conditions for the perfectly competitive firm under limited liability. ;malogous to equations (9) and (lo), are
VK=!GKE(fi)(l +‘/)-
’ -6=O.
WA)
The slope of the curve V, =0 dL _GIC;K _dK IJi=O = G-’ KL The slope of the curve VL= 0 dL iii-
_GKL c’L=o
=
(12)
.
-- GLL
Assuming the production function is concave in K and L, the slopes of both curves ae both positive if capital and labor are complements (G xL > 0) and are negative if capital and labor are competitive (G kL < 0). To determine which curve is the more steep at !he point of optimum, take the difference between equations ( 11) and (12), or
------- --(Gd21 _I_ =- tGKKGLL GLLGKL
Hence if G KL> ( < )O,the curve vK = 0 is more (less) steep than VL=O, at the optimum. The optimal level of inputs for the perfectly competitive firm L*, KS is depicted in Figure 1 where GKL>O and in Figure 2 where GKL ~0. The objective function for the perfectly competitive firm in a no limited liability environment with safetyfirst preferences is
Et&Q wd+ - -- ----
MAX
h,L
1+
--
sI(
- A(F(Q) - B).
I.,f
The first-order conditions are Z, = C’, - if(a)(Qw,> -
y,LGL)/Q2 =0
(13)
By equation (Id), at the optimum b’K~0, since KKK~0. The curve ZK =0 must clearly be everywhere to the right of the curve b’K= 0. The relationship between ZL and V, depends on the magnitude of the output elasticity of laborLG,lQ. IfLGL/Q c 1 then by equation (13). VL~0. Because VLt ~0, ZL=O is everywhere below V, = 0. If GLLc 0, then as shown by Figure 4, limited liability increases the amount of labor employed and decreases capital usage. If
0
: I
K*
K
K*
K**
K
FIGURE 1
FIGURE3
GKL~0, then it is possiblefor total output to decline (seeFigure 3).But if&/Q > 1,then by equation(131, V, 1 implies that E@)Q
imply that limited liability can reduce total output even for Gk,
FIGURE 2
FIGURE 4
L L.
I.*
I!.”
I
K*
I
K””
K
I. E. Brickand E. F. Wit
56 L
LIMFI*ED LIABILITY AND TRANSACTION CO!STS ASSOCIATED WITH KBRTFOLIO DIVERSIFKATION
I
I I __.-,i-
Kf
#t*
n ?’
FIGURE 5
probabilistic lower limit. can be viewed as a reasonably conservative approach to decision making. somewharr in line with the reasoning behind the maximin criterion commonly used in game situations. Third, the safety-first criterion by its nature allows for the possibility of firms being risk-neutral under limited liability and becoming risk-averse under full hability as a result of being exposed to a more risky environment.’ It would be interestins to extend our results to tlhe case where owners exhibit continuous risk-averse utility functions. but the mathematics _sofar have proven intractable.
FIGURE6
Without limited liability, transaction costs associated with portfolio diversification would be prohibitive. The costs of keeping track of the legal and fmancial obligations of all shareholders and enforcing those obligations in the eventuality of default will be very high. Such transaction costs are likely to result in minimum limits on percentage of ownership and thereby place restraints on equity diversifications. These barriers to diversification could affect the personal wealth composition of most individuals, owners, as well as workers (via pension plans). A general equilibrium analysis of the impact of transaction costs alleviated by limited liability would constitute an interesting phase in the future research in this area, but it is beyond the scope of this paper. Transaction costs associated with diversification: have a direct bearing on our analysis. Suppose thaz utility functions of owners, as well as workers, are subject to change as a result of a learning process and exposure to events. In that case, owners may turn from risk neutral to risk averse, following a transition from limited to full liability, if such transition subjects them to greater exposure to pains of absorbing losses. Workers may turn from risk neutral to risk averse in the wake of a transition from full to limited liability, if they become frequently exposed to layoffs.’ Both groups can mitigate such effects via diversification. Yet under full liability, diversification could be extremely costly for owners for the reasons noted above. Under limited liability, workers whose skills constitute the bulk of their human capital can diversify only by learning alternate skills, a costly process of diversification.
CONCLLSIONS This paper explores t hc impact of limlted liability (via the probability of default) on labor market equilibria in i_x‘rfectlycompetitive labor, capital, and product markets, as well as in imperfect product markets. For perfectly competitive and risk-netural economic agents, the effmts of limited liability on production decisions are shown to be neutral: the optimal quantities of labor, capital, and final product do not change. This neutrality is also shown to hokl for the quantity-setting monopolist. Finally, it is observed that neutrality of the production decision will not hold ;lnder safety-first risk aversion. In particular, we ’
We are Indebted to a referee for pointing to this possibility.
demonstrate that limited liability generally decreases the usage of capital. The changes in the quantity of labor depend on the magnitud: of complementarity between capital and labor and on whether the output elasticity of labor is greater 0:’&s than unity. Hence i: is possible in both compet&e and noncompetitive cases for limited liability to decrease the usage of labor and capital and hence reduce output APPENDIX A
Examination of equations (A-2) through (A-4) reveals that the concavity of the production function G(K, L) is not sufficient to satisfy our secondorder conditions. In particular, it can be shown that the second term of the right-hahd expression in equations (A-2)an d(A-3) is positive. This contrasts with the optimal solution for the competitive labor markets without limited liability where the concavity of the production function guarantees the prevalence of the second-order maximization conditions.
!SeWItd-Order Conditions For the perfect competitive case, necessary and sufficient second-order conditions for value maximization by the firm require that dZ(V-dK) <~ (7L’
P(V-SK)
.
<*
VW
SK2
and
t?L2
The Quantity-Setting Monopolist
QRQ)E(&-W)L -SK
U
jjGLL_
= uo
$ -2;;)
subject to G(L,
x_f(/-wp( I+ q)- ’ _
t
Q((‘i+/t?L)+ +(Q - LG L) ----
Q2
~(caG~-L(~~ll~L)-~U‘(u~l
(B1)
l+rl
L.K x
a
Consider a risk-neutral quantity+etting monopolist who faces perfectly competitive factor markets and a random demand function j=&Q)t, where z is a multiplicat;ve random disturbance term. Analogous to equation (I). the firm’s objective function is MAX (V-SK)=
where
S2(P-SK)
APPENDIX B
V=Q it
69=
> +rJ-’
PQ ,L
(A-2)
(B-IA) i+L
for &
-
for k
6?L __er(Q,
W(Q)
(B-1B)
The first-order conditions imply
>s
C’D G
7
G,Q+G,&Q)
z
1
=S(l +r,)+L x_f(j)dp( I+ I$- ’ Q(i,i$L)
+ k(Q -LGL)
-(
x (aG, -(Xl/WY,) xJ(a)(l + r$
’
(A-4)
;;
s”ll(;)di
l-7
J1
k(;)dc
(B-3)
where rx=(ti?L/(QD(Q)) and h(s.) is the probability density function of r. Equations (B-2) and (B-3) are analogous to equations (2) and (3) except for their allowance for monopoly power as captured by (2D/i3Q)GLQ and (dD/2Q)GKQ. Replacing a with CY in equation (4) and differentiating with respect to L
I.E.BrickandE.F.Sutiit and K, we have t-,;. t-L
1
;h&i< L ; Gl_Q+ C,D[Q) - QD(Q, 0 )S ( = _ _ ..- .--. --I------ ---- --- I_---..---. *
L'
strictly comparable to the perfectly competitive firm and is not duplicated here,’ but it is possible that for a monopolist LGL iQ > l?
I
1f
k3,1; W-4)
_ Caution should be exercised since G’lif>Q
does not
’ Unlike in perfect comiletition. LGL,Q> t does not imply E(p@ < H’L.
wbstltutmg and (‘i
iK
equations (B-4) and (B-5) for iGs,iL m equations [B-2) and (B-3), we have
(B-Q
(B-7)
REFERENCES Arzac, E. R. Feb. 1976. Profits and safety in the theory of the firm under price uncertainty. International Economic Review 17( 1). Azariadis, C. Dec. 1975. Implicit contracts and underemployment equilibria. Journai o.f Political Economy 83(6). Azariadis, C. Feb. 1976. On the incidence of unemyloyment. Review of Economic Studies 43. Bailey, M. N. Jan. 1974. Wages and employment
der uncertain Studies 4 1.
demand.
Review of
unEconomic
Grossman, H. Dec. 1973. Aggregate demand, job search and employment. Journal 0,’ Political Economy 81(6). Leland, H. E. June 1972. Theory of the fiim facing uncertain demand. American Economic Review 62. SC>Y, A. July 1952. Safety-first assets. Econometrica.
and the holding
of
Sandmo, A. March 1971. CZI he theory of the competitive firm under prke unctxtainty. American Economic Review 6 1.
r-i, (I =
4? (I ,*(I
Weiss, N. 1976. A Simuitaneous Solution to the Real and Financial Decisions of the Firm Under Uncertainty: An Integrotiors of the Akoclassical Theory of the Firm and Finance TheorJp, Columbia Uniwrsity, Ph.D. Thesis.
(;,Q + G,D( Q,
h ,, - 1(; 1 A)’ > 0. The analysis.
henwf’rt
h, is
Accepted 1, May 1981
INTRODUCtlON Most treatments of optimal micro-output and -input decision under uncettain demand do not explicitly allow for the probability of default (Sandmo 1971; Leland 1972). It is usually implicitly assumed that in the process of contracting for labor services, neither employers nor employees are significantly a&c&xi in their decisions by the prospect of bankruptcy.’ The fact that such underlying assumptions could be at
times restrictive is aptly underscored by Azariadis (1975):H a rational entrepreneur will not shield the terms of employment of even his most valuable group ofemployees against arbitrary demand fluctuations if doing SCinvolves more thm a token probability that the firm might go bankrupt.“2 Even in nonunionized competitive labor markets labor servicesare supplied and exchanged for some informal set of commitments or implicit labor contracts on the part of firms as to a “reasonable” exmted duration of employment. For a detailed discussion of the nature of such implicit contracts see Azariadis( 1975,t976) and Bailey(1974). It should be interesting to examine the manner in which labor contracting is affected by the prospect of bankruptcy. The institution of limited liability introduces in many instances a si;Gicant positive probability that -Pm ’ Weiss (1976) is the exceptton.For a more detailed referenceto his work, see below. z Azariadis (1975) p. I 184. From the Graduate School of Management. Rutgers University, Newark, New Jersey. Address reprint requeststo Dr. Ivan E..Brick,Assistant Professor 01’Finance and Economics, Graduate School of Management, Rutgers Un.versity, Newark, New Jersey 07102.
Journal of Economicsand Business34.51-58 ( 1982) @ 1982 Temple University
implicit or explicit h&or contracts could be wholly or partially abrogated in the eventuality of default. Without limited liability all cost of default should be borne by the owners of the firm. Labor contracts could be abrogated only under rare circumstances when owners are personally bankrupt. Thus under
limited liability default is much more likely to result in significantcosts for employees than in the absence of limited liability. Such costs include loss of income incurred in the press of search for new jobs (Grossman 1973)as well as loss of pension rights and other seniority privileges. Other things being equal, “conversion” by firms to limited liability status reduces risks for owners and increases risks for employees. Our major objective is to explore the impact of limited liability through the probability of default on decisions made by both firmsand labor in exchanging labor services. Weiss (1976)examines the impact of the probability of default (viathe fmancing decision of the firm) on the production decision. He assumes throughout that in the case of default, workers would not sustain any loss. Assuming that in the absence of limited liabilitythe probability of default does not represent a significant factor in labor contracting decisions,we first explore the impact of the introduction of limited liability on equilibrium employment levels, wage rates, capital inputs, and level of output in perfectly competitive risk-neutral labor, capital, and product markets. We demonstrate that the introduction of limited liability has neutral effects upon production decisions; the optimal quantiries of labor, capital, and the final product do not change. But the nominal wage rate rises to compensate workers for their expected loss 51 0278-2294/82/01/051-08$2.75
1. E. BrickandE.
52
induced by limited liability. We obtain similar results for the quantity-setting monopolist. We conclude by observing that if the owners of firm exhibit risk-averse behavior consistent with Roy’s (1952) safety-first criterion, the neutrality of the production decision in the case of knited liability does not necessarily hold. We demonstrate that limited liability generally decreases the usage of capital. Changes in the quantity of labor depend on the magnitude of complementarity between capita1 and labor and on whether the output elasticity of labor is greater or less than unity. Hence it is possible in both competitive and noncompetitive cases for limited liability to decrease the ux of labor and capital and hence reduce output.
F. Sudit
WU
where fi= random Gutput price Q = quantity of outp\Jt r;i= random wage rate L = quantity OFlabor services K =quantity
of capita1 inputs
6 = price of a unit of capital rJ = risk-free interest rate
PERFECT COMPETITION AND RISK NEliTRAEITY
L’=(present) value of future cash flows
In this section we explore the impact of stockholders’ limited liability upon labor market equilibria when the input and output markets are perfectly competit ive and all economic agents are risk neutral. Assume I) output price uncertainty; 2) all equity firms; 3) single-period framework, with capital and labor inputs determined at the beginning of the period; and 4) no salvage value LJr capital at th:: end of the period. Subject to these assumptions, ‘we show that the introduction of stockholder’s IimiM liability will not akct optinz? usage of labor and capital inputs. Nominal \s,lges. however. will be higher under limited Irability. We assume t:lat capital inputs are paid at the beginning of the period to isolate the effect of limited liability on the labor market. Consequently, labor services are paid at the end of the period on resolution of price uncertainty. The objective function of the firm is to maximize shareholders’ wealth. The firm must therefore determine the optimal amount of capital and labor subject to a production constraint”: M/$X
(l’_cjK)=
L.h
E(jiQ. . .- __.. ix)
____
‘+!I.
-bK
wage
riW
Equation (1) is the net present value of the firm and equation (1A) is the production constraint of the firm. Equation (1B) describes the nature of the randomness of the wage rate 6. If the fiml can meet all its labor costs, it will pay the contractual wage G. if the firm cannot completely satisfy labor claims, the entire proceeds of the firm is then devoted to satisfying labor claims and the randomness of the wage rate is attributable in its entirety to the randomness in the output price and limited liability in case of detaultW4 The loss incurred by workers as a result of default in the context of limited liability needs further elaboration. Suppose the period under consideration in equation (1) is one year. If the fum defaults, workers exercising their seniority rights in bankruptcy proceedings can claim compensation only for the services rendered. If their labor contract exa%ds one year, workers are likely to incur losses associated with possible periods of unemployment, search costs for a new job, and loss of nonvested pension rights. Therefore equation (1B) is a proxy for labor losses upon the firm’s default. .
(1)
sub&t to G(l_.. K)=Q
i? =contractual
04
’ Although the randomnessof H’is attributable to the randomwss ofp. owners or workers in a competitivesettingare neither able or ncx-essarily willing to tie the contractual wage rate w to the price of the product. Nonetheless,sina owners and WOIkers share total revenue net of capital costs, they may incorporate p&t sharing schemesinto their wage contracts. This would make w indinzctiya function of p. The analysisof roles of profit sharing in the context of changmg liabilit: forms is beyond the scopeof our presentanalIsis but should be il~terestingto pursue in further aresearch.(We are indebted to a re.‘&e for alerting u to this possibility.)
LabarKarketEquiliblia Diffkrentiating equation(1)partiallywith respectto t iind K and rearmnag, we obtain the following first-order conditions:
53 riskless industry:
where w. is the riskss wage rate. Thus the change in the wage rate induced by a change in labor input
&C/at of equation (l), can be expressed as
Iu@;~LGJf(p)& e
dG
(1 -F(u))-’
-=
ix
WkR
aQ-w(-J
GL= -
c
c?L
G&= dQ>O
ix ix
U
=----
Q'
Analogous to the certainty case, the optimal amounts of labor and capital are determined so as to equate the respective value of their marginal products to their marginal factor costs. But owing to limited liability the vAxs of the marginal products in equations (2)and (3)are not simply the products of the expected price and marginal productivity; rather, the marginal productivities of labor and capital will reflect the limitied (truncated) loss faced by the employers due to limited liability. Becauseof Iimited liabilityenjoyed by st&cholders and the randomness of output price, a positive probability of the firm’s not being able to completely satisfy the claims of labor is likely. Rational behavior implies that the limited liability industry would have to pay higher wages to attract labor. Consequently, the marginal factor cost functions for labor and capital in equations (2)and (3)appear similar to their cotlnterparts in the imperfectly competitive inputmarket cases, where the firm‘sdemand for factors of production affectsthe prices of those inputs. We have not departed from the perfect competition case. Rather, in the case of uncertainty, perfect competition allo\Ang for probability of default, implies the existence of a schedule of prices for the different level of uncertainty. No single firm can affect this schedule by its factor demand. But as the firm employs more labor and/or more capital, it is affecting in the process labor’s eipected loss and thereby the wage rate changes. Risk neutrality implies that the expected wage rate of the risky industry should equal the wage rate ofthe
(5)
0
where F(*l ) is the cumulative density function off@). Similarly,the change in the wage rate induced by a change in the capital input X$X of equation (3)can be expressed as
is
ix=-
sp
!!f(p)djj
o
(1 -F(u))-
L
l.
(6)
From equation (5) we can infer that
srs,<
--
i?L
>
0
. LGL
’
(7)
If
Q
Note that LGLjQ is the output elasticity of labor. Multiplying both sides of equation (2) by L/(E(j 1jj > a)Q) we obtain
LGL -=
Q
WIPNQ
'
(8)
For the firm to plan to stay in business, ii~Lshould be less than E@ 1j > a)(), the expect@ revenue for the limited liability firm. Consequently, %@L must be positive so that the conditions of equation (7)are not violated. It follows that an optimiz%g firm would produce in a range where the outf ut elasticityof labor is less than unity and tke labor supply function is positively sloped. From equation (6), because GK> 0, it is obvious that dG/aK < 0. In view of the signs of a6,1LVL and &G/Z& it would appear that the introduction of limited liability will lead to a more capital-intensive production (i.e.,a higher K/L). Substituting the values of &+X and X$X from equations (5) and (6) into the first-order conditions of equations (2) and (3), respectively, and collecting terms, we have: G,E(i) = E(6) = w.
(9)
I. E. BrickandE. F. Sudit G KE(j5)=
(10)
b(1+ ‘,I.
Equations (9) and (10) are the first-order conditions of L and K for the no limited liability case. Thus the limited itability does not alter the optimul leue1oJ‘L trnd K. Our results in equations (9) and (10) are attributable to the following economic phenomena. After the introduction of limited liability, employees are corn: pensated by higher wages to keep their expected wealth intact ‘@e shareholders can “afford” to pay the higher wages without erosion in their expected wealth owing to the limited (truncated) loss they face from limited liability. But the second-order conditions for optimality in the iimited liability case are more complicated than the no-limited htibility and/or perfect certainty case. In particular. concavity of the production functions in K and E do not guarantee sufficiency of the first-order conditions. For a more detailed discussion, see Appendix A. A transition from a perfectly competitive to an imperfectly competitive market structure is expected, in the framework of c!assica.l static price theory, to a&ct optimal production decisions. It seems interesting to explore whether changing liability forms could have an intervening impact on changes in production decisions for the quantity setting monopolist. In Appendix B we demonstrate the neutral impact of limited liability on the quantity setting monopolist assuming risk-neutral preferences.
SP KI-Y-FlRST
PREFERENCES
Qume that the objective function of the owners of tt~ firm is to maxim& expected profits such that the pr&ability of future loss is less than B (i.e., Prob df 4)s R). This risk-averse behavior is consistent with Roy’s (1952) safety-first criterion (see also Arzac ( 1976)).Assume further that workers are risk-neutral. Introduction of limited liability guarantees that Prob ( t’;~0) = 0. Hence if the safety constrtint is binding in a no limited liability environment. the optimal production decision will drrfer from the limited liability case. We can compare the aptimal production decision of both the limited liabilitjf case and thz no limited liability case where the safety constraint is binding by wns of a diagram in the K, E plane. The first-order conditions for the perfectly competitive firm under limited liability. ;malogous to equations (9) and (lo), are
VK=!GKE(fi)(l +‘/)-
’ -6=O.
WA)
The slope of the curve V, =0 dL _GIC;K _dK IJi=O = G-’ KL The slope of the curve VL= 0 dL iii-
_GKL c’L=o
=
(12)
.
-- GLL
Assuming the production function is concave in K and L, the slopes of both curves ae both positive if capital and labor are complements (G xL > 0) and are negative if capital and labor are competitive (G kL < 0). To determine which curve is the more steep at !he point of optimum, take the difference between equations ( 11) and (12), or
------- --(Gd21 _I_ =- tGKKGLL GLLGKL
Hence if G KL> ( < )O,the curve vK = 0 is more (less) steep than VL=O, at the optimum. The optimal level of inputs for the perfectly competitive firm L*, KS is depicted in Figure 1 where GKL>O and in Figure 2 where GKL ~0. The objective function for the perfectly competitive firm in a no limited liability environment with safetyfirst preferences is
Et&Q wd+ - -- ----
MAX
h,L
1+
--
sI(
- A(F(Q) - B).
I.,f
The first-order conditions are Z, = C’, - if(a)(Qw,> -
y,LGL)/Q2 =0
(13)
By equation (Id), at the optimum b’K~0, since KKK~0. The curve ZK =0 must clearly be everywhere to the right of the curve b’K= 0. The relationship between ZL and V, depends on the magnitude of the output elasticity of laborLG,lQ. IfLGL/Q c 1 then by equation (13). VL~0. Because VLt ~0, ZL=O is everywhere below V, = 0. If GLLc 0, then as shown by Figure 4, limited liability increases the amount of labor employed and decreases capital usage. If
0
: I
K*
K
K*
K**
K
FIGURE 1
FIGURE3
GKL~0, then it is possiblefor total output to decline (seeFigure 3).But if&/Q > 1,then by equation(131, V,
imply that limited liability can reduce total output even for Gk,
FIGURE 2
FIGURE 4
L L.
I.*
I!.”
I
K*
I
K””
K
I. E. Brickand E. F. Wit
56 L
LIMFI*ED LIABILITY AND TRANSACTION CO!STS ASSOCIATED WITH KBRTFOLIO DIVERSIFKATION
I
I I __.-,i-
Kf
#t*
n ?’
FIGURE 5
probabilistic lower limit. can be viewed as a reasonably conservative approach to decision making. somewharr in line with the reasoning behind the maximin criterion commonly used in game situations. Third, the safety-first criterion by its nature allows for the possibility of firms being risk-neutral under limited liability and becoming risk-averse under full hability as a result of being exposed to a more risky environment.’ It would be interestins to extend our results to tlhe case where owners exhibit continuous risk-averse utility functions. but the mathematics _sofar have proven intractable.
FIGURE6
Without limited liability, transaction costs associated with portfolio diversification would be prohibitive. The costs of keeping track of the legal and fmancial obligations of all shareholders and enforcing those obligations in the eventuality of default will be very high. Such transaction costs are likely to result in minimum limits on percentage of ownership and thereby place restraints on equity diversifications. These barriers to diversification could affect the personal wealth composition of most individuals, owners, as well as workers (via pension plans). A general equilibrium analysis of the impact of transaction costs alleviated by limited liability would constitute an interesting phase in the future research in this area, but it is beyond the scope of this paper. Transaction costs associated with diversification: have a direct bearing on our analysis. Suppose thaz utility functions of owners, as well as workers, are subject to change as a result of a learning process and exposure to events. In that case, owners may turn from risk neutral to risk averse, following a transition from limited to full liability, if such transition subjects them to greater exposure to pains of absorbing losses. Workers may turn from risk neutral to risk averse in the wake of a transition from full to limited liability, if they become frequently exposed to layoffs.’ Both groups can mitigate such effects via diversification. Yet under full liability, diversification could be extremely costly for owners for the reasons noted above. Under limited liability, workers whose skills constitute the bulk of their human capital can diversify only by learning alternate skills, a costly process of diversification.
CONCLLSIONS This paper explores t hc impact of limlted liability (via the probability of default) on labor market equilibria in i_x‘rfectlycompetitive labor, capital, and product markets, as well as in imperfect product markets. For perfectly competitive and risk-netural economic agents, the effmts of limited liability on production decisions are shown to be neutral: the optimal quantities of labor, capital, and final product do not change. This neutrality is also shown to hokl for the quantity-setting monopolist. Finally, it is observed that neutrality of the production decision will not hold ;lnder safety-first risk aversion. In particular, we ’
We are Indebted to a referee for pointing to this possibility.
demonstrate that limited liability generally decreases the usage of capital. The changes in the quantity of labor depend on the magnitud: of complementarity between capital and labor and on whether the output elasticity of labor is greater 0:’&s than unity. Hence i: is possible in both compet&e and noncompetitive cases for limited liability to decrease the usage of labor and capital and hence reduce output APPENDIX A
Examination of equations (A-2) through (A-4) reveals that the concavity of the production function G(K, L) is not sufficient to satisfy our secondorder conditions. In particular, it can be shown that the second term of the right-hahd expression in equations (A-2)an d(A-3) is positive. This contrasts with the optimal solution for the competitive labor markets without limited liability where the concavity of the production function guarantees the prevalence of the second-order maximization conditions.
!SeWItd-Order Conditions For the perfect competitive case, necessary and sufficient second-order conditions for value maximization by the firm require that dZ(V-dK) <~ (7L’
P(V-SK)
.
<*
VW
SK2
and
t?L2
The Quantity-Setting Monopolist
QRQ)E(&-W)L -SK
U
jjGLL_
= uo
$ -2;;)
subject to G(L,
x_f(/-wp( I+ q)- ’ _
t
Q((‘i+/t?L)+ +(Q - LG L) ----
Q2
~(caG~-L(~~ll~L)-~U‘(u~l
(B1)
l+rl
L.K x
a
Consider a risk-neutral quantity+etting monopolist who faces perfectly competitive factor markets and a random demand function j=&Q)t, where z is a multiplicat;ve random disturbance term. Analogous to equation (I). the firm’s objective function is MAX (V-SK)=
where
S2(P-SK)
APPENDIX B
V=Q it
69=
> +rJ-’
PQ ,L
(A-2)
(B-IA) i+L
for &
-
for k
6?L __er(Q,
W(Q)
(B-1B)
The first-order conditions imply
>s
C’D G
7
G,Q+G,&Q)
z
1
=S(l +r,)+L x_f(j)dp( I+ I$- ’ Q(i,i$L)
+ k(Q -LGL)
-(
x (aG, -(Xl/WY,) xJ(a)(l + r$
’
(A-4)
;;
s”ll(;)di
l-7
J1
k(;)dc
(B-3)
where rx=(ti?L/(QD(Q)) and h(s.) is the probability density function of r. Equations (B-2) and (B-3) are analogous to equations (2) and (3) except for their allowance for monopoly power as captured by (2D/i3Q)GLQ and (dD/2Q)GKQ. Replacing a with CY in equation (4) and differentiating with respect to L
I.E.BrickandE.F.Sutiit and K, we have t-,;. t-L
1
;h&i< L ; Gl_Q+ C,D[Q) - QD(Q, 0 )S ( = _ _ ..- .--. --I------ ---- --- I_---..---. *
L'
strictly comparable to the perfectly competitive firm and is not duplicated here,’ but it is possible that for a monopolist LGL iQ > l?
I
1f
k3,1; W-4)
_ Caution should be exercised since G’lif>Q
does not
’ Unlike in perfect comiletition. LGL,Q> t does not imply E(p@ < H’L.
wbstltutmg and (‘i
iK
equations (B-4) and (B-5) for iGs,iL m equations [B-2) and (B-3), we have
(B-Q
(B-7)
REFERENCES Arzac, E. R. Feb. 1976. Profits and safety in the theory of the firm under price uncertainty. International Economic Review 17( 1). Azariadis, C. Dec. 1975. Implicit contracts and underemployment equilibria. Journai o.f Political Economy 83(6). Azariadis, C. Feb. 1976. On the incidence of unemyloyment. Review of Economic Studies 43. Bailey, M. N. Jan. 1974. Wages and employment
der uncertain Studies 4 1.
demand.
Review of
unEconomic
Grossman, H. Dec. 1973. Aggregate demand, job search and employment. Journal 0,’ Political Economy 81(6). Leland, H. E. June 1972. Theory of the fiim facing uncertain demand. American Economic Review 62. SC>Y, A. July 1952. Safety-first assets. Econometrica.
and the holding
of
Sandmo, A. March 1971. CZI he theory of the competitive firm under prke unctxtainty. American Economic Review 6 1.
r-i, (I =
4? (I ,*(I
Weiss, N. 1976. A Simuitaneous Solution to the Real and Financial Decisions of the Firm Under Uncertainty: An Integrotiors of the Akoclassical Theory of the Firm and Finance TheorJp, Columbia Uniwrsity, Ph.D. Thesis.
(;,Q + G,D( Q,
h ,, - 1(; 1 A)’ > 0. The analysis.
henwf’rt
h, is
Accepted 1, May 1981