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Systematic risk, wage rates, and factor substitution
NORTH- HOLLAND
Systematic Risk, Wage Rates, and Factor Substitution Cheng-Few Lee, K. C. Chen, and K. Thomas Liaw
The impacts of wage rates, monopoly power, and factor substitution on the systematic risk of a firm are examined. A variable elasticity of substitution production function is employed. Both the long run and the short run are analyzed. In the short run, a higher market power leads to a lower systematic risk, whereas a higher wage rate increases risk. For the long-run analysis, the impact of a wage rate change on systematic risk depends on the degree of input substitutability. In addition, both monopoly power and the degree of input substitutability conditionally reduce the systematic risk of the firm.
I. Introduction The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966) has laid the basis for much of the theoretical and empirical work in the realm of finance. In the context of the CAPM, the relevant risk measure of a security held in a diversified portfolio is its systematic risk or beta. In the literature, a rich body of papers has investigated the firm-specific determinants of systematic risk. Theoretically, systematic risk can be shown as a function of financial leverage [Rubinstein (1973)], growth [Thompson and Senbet (1982)], degrees of operating and financial leverage [Mandelker and Rhee (1984)], and operating leverage [Rubinstein (1973); Lev (1974)]. Similar line of analyses can also be found in Goldenberg and Chiang (1983), Dotan and Ravid (1985), and Narayanaswamy (1988). Subsequently, specific links from the microeconomic variables of the firm to its systematic risk have been developed. Binder (1990) showed that, with competitive product market, beta is negatively related to firm size and concentration due to greater efficiency in production. Hite (1977) and Booth (1981) examined the effects
Department of Finance, Rutgers University, New Brunswick, New Jersey 08903 (CFL); Department of Finance, California State University, Fresno, California 93740 (KCC); Department of Economics and Finance, St. John's University, Jamaica, New York 11439 (KTL). Address reprint request to Professor K. Thomas Liaw, Department of Economics and Finance, St. John's University, College of Business Administration, 8000 Utopia Parkway, Jamaica, New York 11439.
Journal of Economics and Business 1995; 47:267-279 © 1995 Temple University
0148-6195/95/$09.50 SSDI 0148-6195(95)00011-F
268
C.-F. Lee et al. of output and market structure uncertainty on the cost of equity. Thomadakis (1976), Subrahmanyam and Thomadakis (1980), and Chen, Cheng, and Hite (1986) investigated the linkage between market power and systematic risk. Among them, the paper by Subrahmanyam and Thomadakis has attracted significant attention from academics. In their analysis, Subrahmanyam and Thomadakis assumed a fixed-coefficient production technology and developed a theoretical model, linking systematic risk with firm variables such as the labor-capital ratio and monopoly power as measured by the elasticity of demand for the firm's output. Recently, Lee, Liaw, and Rahman (1990) employed a Cobb-Douglas production function to show that a firm's systematic risk is negatively related to its monopoly power and capital-labor ratio. Sun (1993) extended the model to a homogeneous production function case. Sun showed that an increase in one firm's wage rate adversely affects its competitor's systematic risk and that the impact on its own beta coefficient depends on the competitor's reaction function. Wong (1994) simplified Sun's model and improved its tractability, and derived the following observations: (a) a higher wage rate leads to a higher systematic risk and (b) the relationship between one firm's wage rate and its competitor's beta coefficient is negative. Peyser (1994) extended the analysis further to take wage rate uncertainty into consideration. Peyser showed that the relationship between systematic risk and monopoly power depends on the degree of wage rate uncertainty. The purpose of this paper is to extend the analysis with the assumption of a variable elasticity of substitution (VES) production function. The rationale for adopting a VES production function is as follows. First, the VES production function, as shown by Revankar (1967), includes both fixed-coefficient and Cobb-Douglas production functions as special cases. Second, it contains several properties that will facilitate our analysis. Third, as shown by Hicks (1948) and Allen (1956), the elasticity of substitution parameter, in principle, can be a variable depending upon output a n d / o r factor combinations. This is an advantage of a VES production function over the constant elasticity of substitution production function. Based on the above premise, we will investigate the impact of factor substitution on systematic risk. As evidenced by Arrow et al. (1961), for the US economy as a whole, production is neither as flexible as implied by the Cobb-Douglas production function nor as inflexible as the fixed-coefficients case. The production process in the American economy seems to be characterized by some possibilities of substituting one factor for another) Therefore, it is plausible that a firm characterized by inflexible production technique would tend to have higher business risk, ceteris paribus, because major disturbances may arise in the transition of factor substitutions. The rest of this paper is organized as follows. The next section presents model formulation and notation. Using the cash-flow version of the CAPM, the formulation of systematic risk is then derived, with uncertainty in both the output market and the labor market. The comparative statistics in the short run and long run are discussed in the third and fourth sections. The last section concludes this paper.
I Revankar (1967) found the empirical relevance of the VES function in 5 out of 12 industries by using cross sectional data for year 1947.
Systematic Risk, Wage Rates, and Factor Substitution
269
II. The Model There are two sources of uncertainty: price uncertainty and wage rate uncertainty. The first source of uncertainty the firm faces originates from its downward sloping demand curve, ~6 = p(1 + ~),
(1)
where p represents the expected price of output and Y is a random error term with zero mean. Since 6 is an economy wide disturbance, it affects the output prices of all firms in the same way. Following Thomadakis (1976), we introduce a parameter, tz, which is the reciprocal of the price elasticity of demand, to describe the firm's degree of monopoly power in its product market. 2 The firm makes its output decision before the price is known, that is, before 6 is revealed. The marginal revenue (MR) function can now be expressed as MR = (1 + ~)(1 - i.tE)p, where E is an elasticity constant and 0 < ~ < 1. If /z = 0, then the firm is in a perfectly competitive output market. On the other hand, the firm acts as a monopolist i f / z = 1. The firm employs a variable elasticity of substitution (VES) production function [Revankar (1967)] with two factors, capital ( K ) and labor (L): Q = K"(1-*P)[L + ( p - 1 ) K ] "'p,
(2)
w h e r e a , s, and p a r e p a r a m e t e r s w i t h t h e f o l l o w i n g c o n s t r a i n t s : ce>0, 0
O < s p < l,
L / K > (1 - p ) / ( 1 - sp). 3 Equation (2) reduces to the fixed-coefficient production function if p = 0, the Cobb-Douglas case when p = 1, and the linear model provided that p = 1/s. We further assume that the amount of capital employed is purchased with the proceeds of issuing shares to the capital market at the beginning of the period and capital is exhausted in the production process. For notational simplicity, we set the price of capital at unity. The wage rate per unit of labor, which will be paid at the end of the period, is a random variable, = w(1 + t3),
(3)
where w is the expected wage rate and t3 is a random shock in the labor market with zero mean.
2 See Thomadakis (1976) for details. 3 This requirement is to satisfy the assumption that the elasticity of substitution is positive in the empirical relevant range of K/L. See Revankar (1967, 1971) for detailed discussion on the VES production function.
270
C.-F. Lee et al. It is also assumed throughout the paper that the firm is a price taker in both factor and capital markets and that the firm cannot affect the market price of risk. Consequently, there must exist sufficient industries in the economy so that the demand and supply conditions in the capital markets remain unchanged if there are changes from any individual firm. As such, any decisions, either financing or investment, should have negligible impacts upon aggregate output and labor. The finn's uncertain ending cash flow is defined as l ? =/SQ - ~L = (1 +
~)pQ
-
(1 +
~)wL.
(4)
Using the cash flow version of the CAPM, the expected market value of the firm is given by
V = ( E ( g ) - /~Cov(}'~,Rm))(1 q-r) -1,
(5)
where A is the market price of systematic risk, /~m is the uncertain rate of return on market portfolio, and r is the risk-free rate of interest. If we define * = E(1 + ~) - ACov(~,/~m) = 1 - ACov(~,/~m), th = E(1 + t3) - ACov(~3,/~m) = 1 -- ACov(5,/~m), as the certainty equivalents of (1 + ~) and (1 + 5), respectively, the value of the firm simplifies to V = [ptbQ
-
w~bL](1 + r) -1
(5')
Before proceeding further, we summarize the key assumptions as follows:
Assumption 1. A VES production function is employed. Assumption 2. Product market uncertainty is ~ with zero means. Assumption 3. Labor market uncertainty is 5 with zero mean. Assumption 4. Firm is price taker in both factor and capital markets. Assumption 5. The capital asset pricing model holds. Assumption 6. Capital is exhausted in the production process.
Assumption 7. Cov(6, J~rn) and Coy(t3,/~m) are positive. Assumption 8. The firm's beta coefficient is positive. With the basic setting above, we shall carry out the comparative statistics analysis both in the long run and in the short run in the next two sections.
Systematic Risk, Wage Rates, and Factor Substitution
271
III. Comparative Statistics: Short R u n In the short run, the amount of capital is fixed. The goal of the finn is to maximize its net present value (NPV), that is, the difference between its market value ( V ) and its capital expenditure (K); NPV = V - K. The maximization problem can be characterized as follows: Max N P V = V -
K = [ p ~ Q - wq~L](1 + r) -t - K.
(6)
Differentiating (6) and setting the first order condition equal to zero yields ONPV
p ~ ( 1 - tzE) otspQ
OL
L + (1 - p ) K
- w~b = 0.
(7)
From (7) we can solve for the optimal L as follows: pq~(1 - ~ E ) a s p Q L =
- (1 - p ) K .
w4,
(8)
According to the CAPM, systematic risk is measured by the relationship between the rate of return on the firm's stock and the rate of return on the market portfolio. Define the rate of return on the finn as IT-V V where I7 is the ending cash flow to shareholders and V is the expected market value of firm i. Then the systematic risk of the finn, /3, is given by
/3
Var
(9)
,
m
where Cov(Ri, R , , ) is the covariance between the rates of return on the firm and the market portfolio and Var(/~ m) is the variance of return on the market portfolio. Therefore, (9) can be rewritten as (1 nt-
r)(Cov(e,l
m) -
[*(1
-
t E)olsp ) - 1 -
w(pQ)-'(1 - p)K]
/3 = Var(/~m){q~ - [ * ( 1
- txE)asp-
wck(pQ)-l(1 - p)K]}
(9')
272
C.-F. Lee et al. The expression for/3 in (9') shows that systematic risk is dependent on both w and /x, among other variables. We first analyze the impact of a wage rate change on systematic risk by differentiating (9') with respect to the wage rate w:
o/3 Ow
(1 + r)(1 - p)K Cov(O,/l~ra Var(/~){pQ* -pQ[*(1(1 + r)(1 -
IxE)asp- w~b(pQ)-l(1- p)K])
p)qSK(pQ)
-[*(1-
) (10)
I{Cov(Y,/}m)
lzE)olspt~-1- w(pa)-l(1- p)g])
+
Var(/~m){*
-[*(1-
i.zE)otsp-wdp(pQ)-l(1-
P)Kl}
2"
Using Assumptions 7 and 8, it follows that e/3/Ow < O, if p > 1.
(11)
According to Revankar (1967), the elasticity of factor substitution of the VES production function is 4 o-= c r ( K , L ) = 1 + [ ( p -
1)/(1 - s p ) ] [ K / L ] .
(12)
Given any capital-labor ratio, it follows that >
oral,
>
i f p ~ 1.
(13)
Combining expressions (11) and (13) results in 0/3/3w < O, if o- > 1.
(14)
Equation (14) reveals an important finding that the impact of a wage rate change on /3 is dependent on the magnitude of the firm's elasticity of factor substitution. Specifically, the systematic risk of the firm increases with the wage rate if the elasticity of substitution is less than 1. On the other hand, the relationship is negative provided that the elasticity of substitution is greater than 1. This finding is attainable only when the VES production function is employed. As the VES production function subsumes both fixed-coefficient and Cobb-Douglas production functions, our finding here adds more to the literature on the linkage between the firm's systematic risk and its microeconomic variables. Furthermore, this finding cannot be obtained by the constant elasticity of substitution (CES) production function because the elasticity of substitution is constant. 4As discussed earlier, the V E S p r o d u c t i o n function r e d u c e s to the fixed-coefficient m o d e l if p = 0. T h e c o r r e s p o n d i n g elasticity of substitution is tr = 1 + [( p - 1)/(1 - s p ) ] [ K / L ] = 1 + ( p - 1)/[(1 + r X w c ~ ) - l s p - ( p - 1)] = 0 if ~b = 0. O n the o t h e r h a n d , the C o b b - D o u g l a s case ( p = 1) implies o- = 1 + ( p - 1)/[(1 + r X w q b ) - l s p - ( p - 1)] = 1.
Systematic Risk, Wage Rates, and Factor Substitution
273
The relationship between the systematic risk of the firm and its market power in the product market can be examined by differentiating expression (9') with respect to /z: (1 + r)dpEaspdp-' Coy(g,
at~ 3/x
Rm)
I x E ) a s p - w~b(pQ)-'(1- p)K]}
Var(/~m)(q~ - [ q ~ ( 1 -
(1 + r)~Easp Var(/~m){* - [ * ( 1 -
l~E)asp-wd~(pQ)-t(1-
p)KI) 2
<0. (15)
The result indicates that a higher degree of monopoly power in the product market will unambiguously lower the systematic risk of a firm, ceteris paribus. Sullivan (1978), Subrahmanyam and Thomadakis (1980), Chen, Cheng, and Hite (1986), Lee, Liaw, and Rahman (1990), and Sun (1993) had reached a similar conclusion. Based on the CAPM, the firm with a higher market power in its product market can raise capital at a lower cost (via a lower required rate of return).
IV. C o m p a r a t i v e Statistics: L o n g R u n Similar to the setup in the short run, the long-run goal of the firm is also to maximize its net present value by choosing both K and L to achieve the objective: Max NPV = V - K =
[p~pQ- w~bL](1 + r) -1 - K .
(16)
Substituting (2) into (16) and differentiating it with respect to K and L, the first order conditions are p@(1 - u E )
a(1 - sp)Q asp(p- 1)Q/(1 -l ~+ L+(p~-I-)-K) +r) -1=0
(17)
and pqb(1 -- IzE)aspQ L + (1 - p ) K
- w4, = 0.
(18)
Equations (17) and (18) can be used to solve for K and L as functions of Q 5: apdp(1 - ~E)(1 - sp)Q
K=
(1 + r ) -
(p-
1)w~b
and L
(19) [(1 + r)(w(9)-'sp - (p - 1)] a p ~ ( 1 - tzE)Q (1 + r ) -
(p-
1)w~b
5All variables K, L, and Q are evaluated at their optima in the followinganalysis.
274
C.-F. l e e et al.
The optimal levels of capital and labor can then be substituted into (4) and (5'). Using (4) and (5'), we can express the systematic risk of the firm in (9') as follows: Cov(/SQ - ffL,/~m)
(1 + r)
p f ~ a -- wc~Z
War(/~m)
3=
(Cov(Y,/~m)[1 + r -
(1 +
r)
(p-
1)wtb]
- [ ( 1 + r)sp~b -1 - w ( p -
1)]~(1
{4~[1 + r - ( p - 1)w~b] -[(1 + r)4o-lsp- (p-
Var(/~,,)
- tzE)Cov(~,
Rm)~ J,/
1)w]adP(1 - /xE)th}
(20) Simply put, the equilibrium relationship in equation (20) integrates the systematic risk of the firm with its monopoly power in the product market (/z), the relative factor price (w), and the degree of factor substitution (tr), among other variables. First, differentiating (20) with respect to w, we obtain 0[3
- ( 1 + r) Cov(~,/~m)( P -- 1)~b - ( p - 1 ) a ~ ( 1 - / z E ) C o v ( ~ , / ~ m )
Ow
Var(/~m)
¢[1 + r - ( p - 1)w~b] -[(1 + r)¢-lso
+
- ( p -
1)wladP(1 - / x E ) ~ b
(1 + r) Var(/~,.)
{Coy(e,/~m)[1 +
r- (p-
-[(1 + r)~-~sp x
- w(p
1)w4,] -
1)]aq~(1 - ~ E ) C o v ( ~ , / ~ ) } ~
{¢P[1 + r - ( p - 1)w~b] - [ ( 1 + r ) 4 o - l s p - ( p - 1)w]aq~(1 -/zE)4~} 2
where ~O= q~4~(p - 1) - ( p - 1)a¢~b(1 - ~E). The expression above can be simplified to to - ( 1 + r ) ( p - 1)(1 - s p ) [ a ( 1 - r)q~(1 - /zE)]
0[3 -
-
Ow
[cov( ,
cov( , m)l (21)
=
Var(/~m){q~[1 + r - ( p - 1)w~b] -[(1 + r)4~-Isp - ( p - 1)w]a¢(1 - ~E)4~} 2
The sign of the above expression depends on ( p - 1) and [ C o v ( ~ , / ~ , , ) Cov(~,/~m)]. If [Cov(~,/~m) - Cov(~,/~m)] > 0, i.e., the covariance of the firm's product market uncertainty with the market portfolio returns is greater than that of the labor market uncertainty with the market portfolio returns, then O[3/Ow~.O,
>
ifp~.l.
(22)
Systematic Risk, Wage Rates, and Factor Substitution
275
Combining expressions (13) and (22) yields
0,
if
1.
This shows that systematic risk and the wage rate are positively related if the elasticity of substitution is less than 1. On the other hand, the systematic risk decreases with the wage rate if the elasticity of substitution is greater than 1. When the elasticity of substitution is at unity, the firm's systematic risk is not affected by the changes in the wage rate. Overall, the results here are similar to those obtained in the short-run analysis. Again, this result could have been obtained only by the VES production function, and not by other production functions, e.g., the CES production. The results are reversed if [Cov(£ Rm) - Cov(z3,/},,)] < 0. In addition, any changes in the wage rates will not affect the firm's systematic risk if [Cov(g, Rm) - Cov(t3, R m ) ] = O. Second, the impact of monopoly power in the product market on the firm's systematic risk can be examined by differentiating (20) with respect to /z, the degree of monopoly power. Then we get c~/3 31t
(1 + r )
[(1 +r)qb l s p - w ( p -
Var(/~m) qb[1 + r -
(p-
1)](aqbE)Cov(tT,/~m)
1)w4,]
-[(1 + r)~b-lsp- (p-
1)wlaqb(1 - /zE)~b
(1 + r) Var(/~,n) {Cov(g,/~m)[1 + r -
(p-
1)w~b]
- [ ( 1 + r)~b-lsp - w ( p - 1)]aO(1 - ttE)Cov(5,/~m)}qJ X
{qb[1 + r -- ( p - 1)w~b] -[(1 + r)~b-lsp- (p-
1 ) w ] a ~ ( 1 -/zE)~b} 2
where ¢J = [(1 + r)ck-lsp - ( p - 1 ) w ] a ~ b E . The expression above can be simplified (see Appendix for derivation) to -aopqbE(1 + r ) [ w 2 ( p - 1)2~b - (1 + r ) ( p - 1)(1 + s p ) w +(i
3tt
+
r)2s
1][Cov( ,
-
cov( ,
Var(/~m){qb[1 + r - ( p - 1)w~b]
(23)
- [ ( 1 + r)c~-lsp - ( p - 1 ) w ] a ~ ( 1 - ttE)~b} ~ It follows, if [w2(p - 1)2~b - (1 + r ) ( p - 1)(1 + sp)w + (1 + r)2sp~b-1] 6 is positive, that c)[3/3tz<0,
if[Cov(J,/~m)-Cov(~,/~m) ] >0.
6 This is related to the constraint on the wage rate. It can be shown that [w2(p - 1)2~b- (1 + rX p - 1)(1 +sp)w+(1 +r)2spdp l ] > 0 i f w> a~E(1 +rX(1 +sp)+ 1/(1 +sp) 2 - 4 s p ) / ( 2 ( p - 1)).
276
C.-F. Lee et al. As shown, to support the conclusion by Subrahmanyam and Thomadakis (1980) and others that market power reduces systematic risk, the covariance of the firm's product market uncertainty must be greater than that of the uncertainty in the labor market. Peyser (1994) employed Tabin's q as a proxy of monopoly power and reached a similar conclusion. The outcome is plausible through various contractual mechanisms that reduce wage rate uncertainty. Finally, we turn to the degree of factor substitution. Differentiating (20) with respect to p yields 813
- (1 + r)
3p
Var(/~m) ×
Coy(&/~m)w~b + [(1 + r ) q ) - l s - w]ot~(1 - ~E)Cov(t3,/~m) • [1 + r - ( p -
1)w~b]
-[(1 + r)ck-lsp+
( p - 1)w]aqb(1 -- /xE)~b
(1 + r) Var(/~m) {Cov(~,/~m)[1 + r - ( p -
1)~b]
- [ ( 1 + r)4~-lsp -
w(o
-
1)]~(1
- tzE)Cov(&/~m)}O
X
{~[1 + r - ( p - 1)w~b] -[(1 + r)qb-lsp-
(p-
1)w]adO(1 - /~E)q~} 2
where ® = w~b + [(1 + r ) 4 ) - l s - w ] a ~ b ( 1 - / z E ) . The expression above can be simplified to a ~ ( 1 + r)2(1 - /~E){(1 + r ) s 4 ) -1 + ws - w}
ol3 3p
[cov( , &)- cov(o,&)] Var(/~m){~[1
+ r -
( P -
- [ ( 1 + r)49 1 s o -
(24)
1)w~b]
(p-
1 ) w ] a ~ ( 1 -txE)4~} 2
It can be shown that if [(1 + r)sck -1 + ws - w] 7 is positive and the covariance of the product market uncertainty is greater than that of the labor market uncertainty, the greater (less) the degree of input substitutability between capital and labor, and the lower (higher) the systematic risk of the firm. Put differently, if it is relatively easy for a firm to substitute capital for labor, the process of technical transformation can be smoothly accomplished and the firm is exposed to a lower business risk. On the other hand, if inflexible technology requires major disturbances in order to absorb the additional capital, the business risk of the firm will substantially increase. 7 If [(1 + r)sqb -1 + WS -- W] > 0, it impliesthat w < ((1 + r)s~k-1)/(1 - s). Note also that 0 < s < 1, by assumption in equation (2).
Systematic Risk, Wage Rates, and Factor Substitution
277
V. Conclusions This paper has attempted to integrate the relationship between operating leverage, monopoly power, and factor substitution with the systematic risk of a firm. A VES production function is employed. Also--an important improvement over most of other papers in this area--both price and wage rate uncertainty are take into consideration as in Peyser (1994). Several significant results are derived. First, the impact of a wage rate change on systematic risk depends on the degree of input substitutability both in the short run and in the long run. Second, the firm's systematic risk is negatively correlated with market power in its product market and with the degree of factor substitution in the long run.
Appendix D e r i v a t i o n o f E x p r e s s i o n (23)
0/3 c~/z
(1 + r)
[(1 +
r)¢o-'sp
-
p
w(
-
1)](a~E)Cov(tS,/~m)
Var(Rm ) qb[1 + r - ( p - 1)wqS] -[(1 + r)c~-lsp- (p-
1)w]aqb(1 --/xE)q5
{Cov(~,/~m)[1 + r -- ( p -- 1)w~b] - [ ( 1 + r ) q S - l s p - w ( p - 1)] (1 + r)
c~*(1
Var(km) {~[1 + r -
(p-
~E)Cov(?:,km)}~0
-
1)w4~]
- [ ( 1 + r)49 ~sp - ( p - 1)w]aqb(l - /~E)&} 2
[(1 + r ) a - ~ s p
-
q~[l+r(1 + r)
w(p
(p-
-
1)](~,E)Cov(~, kin)
1)w4~]
- [ ( 1 + r ) 4 ~ - l s p - ( p - 1)w]aq~(1 - /~E)~b
Var(/}m) {qb[l + r - - ( p - - 1)w&] -[(1 +r)¢
~sp-- ( p - -
1)w]otqb(1 --/zE)~b} 2
{Cov(~, kin)[1 + r - ( p - 1)w~] - [ ( 1 + r ) g J - l s p - w ( p - 1)]
-
(1 + r) Var(R~) {q~[1 + r -
(p-
£)),
1)w&]
-[(1 + r)~b-lsp-
(p-
1)w]a~(1 - /~E)t~} 2
278
C.-F. Lee et al. -(1
+ r)tp[(1 + r) - ( p -
1)w4)][Cov(g,l~ m]] - Coy(t3,/~,,]] \ sj I. \
Var(/~m)((I)[1 + r - ( p - 1)w4)] -[(1 + r)4)-isp - (p - 1)w]a~(1
-/xE)4)) 2
w h e r e q, = [(1 + r ) 4 ) - l s p - ( p - 1)w]o~(I)4)E. T h e expression above can then be simplified to - a ~ 4 ) E ( 1 + r) [ w 2 ( p - 1)24) - (1 + r ) ( p - 1)(1 + s p ) w + (1 + r)Zsp4) -1]
3/3 0/x
[Cov(e, g m ) -
Cov(u, gm) ]
Var(/}m){(I)[1 + r - ( p - 1)w4)] - [ ( 1 + r ) 4 ) - l s p - ( p - 1 ) w ] a ( I ) ( 1 - /xE)4)}:
We thank seminar participants at Rutgers University, New Brunswick, for helpful comments.
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Mossin, J. 1966. Equilibrium in a capital asset market. Econometrica 34:768-784. Narayanaswamy, C. R. 1988. A mean-variance synthesis of corporate financial theory: A note. Journal of Finance 43:529-530. Peyser, P. S. 1994. Beta, market power and wage rate uncertainty. Journal of Industrial Economics 46:217-226. Revankar, N. S. 2967. Production functions with variable elasticity of substitution and variable returns to scale. Doctoral dissertation, University of Wisconsin. Revankar, N. S. 1971: A Coolies of variable elasticity of substitution production functions. Econometrica 39:60-71. Rubinstein, M. E. 1973. A mean-variance synthesis of corporate financial theory. Journal of Finance 28:167-182. Sharpe, W. 1964. Capital asset price: A theory of market equilibrium under conditions of risk. Journal of Finance 19:425-442. Subrahmanyam, M., and Thomadakis, S. 1980. Systematic risk and the theory of the firm, Quarterly Journal of Economics 95:437-451. Sullivan, T. G. 1978. The cost of capital and the market power of firms. Ret,iew of Economics andStatistics 60:209-217. Sun, L. 1993. Market power, wage rate, and systematic risk: A homogeneous production approach. Journal of Economics and Business 45:99-108. Thomadakis, S. 1976. A model of market power, valuation and the firm's return. Bell Journal of Economics 7:150-162. Thompson, H. E., and Senbet, L. W. 1982. Growth and risk. Journal of Financial and Quantitati~,e Analysis 17:331-340. Wong, K. P. 1994. Effects of wage rate on systematic risk in a Cournot duopoly. Journal of Economics and Business 46:227-233.
Systematic Risk, Wage Rates, and Factor Substitution Cheng-Few Lee, K. C. Chen, and K. Thomas Liaw
The impacts of wage rates, monopoly power, and factor substitution on the systematic risk of a firm are examined. A variable elasticity of substitution production function is employed. Both the long run and the short run are analyzed. In the short run, a higher market power leads to a lower systematic risk, whereas a higher wage rate increases risk. For the long-run analysis, the impact of a wage rate change on systematic risk depends on the degree of input substitutability. In addition, both monopoly power and the degree of input substitutability conditionally reduce the systematic risk of the firm.
I. Introduction The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966) has laid the basis for much of the theoretical and empirical work in the realm of finance. In the context of the CAPM, the relevant risk measure of a security held in a diversified portfolio is its systematic risk or beta. In the literature, a rich body of papers has investigated the firm-specific determinants of systematic risk. Theoretically, systematic risk can be shown as a function of financial leverage [Rubinstein (1973)], growth [Thompson and Senbet (1982)], degrees of operating and financial leverage [Mandelker and Rhee (1984)], and operating leverage [Rubinstein (1973); Lev (1974)]. Similar line of analyses can also be found in Goldenberg and Chiang (1983), Dotan and Ravid (1985), and Narayanaswamy (1988). Subsequently, specific links from the microeconomic variables of the firm to its systematic risk have been developed. Binder (1990) showed that, with competitive product market, beta is negatively related to firm size and concentration due to greater efficiency in production. Hite (1977) and Booth (1981) examined the effects
Department of Finance, Rutgers University, New Brunswick, New Jersey 08903 (CFL); Department of Finance, California State University, Fresno, California 93740 (KCC); Department of Economics and Finance, St. John's University, Jamaica, New York 11439 (KTL). Address reprint request to Professor K. Thomas Liaw, Department of Economics and Finance, St. John's University, College of Business Administration, 8000 Utopia Parkway, Jamaica, New York 11439.
Journal of Economics and Business 1995; 47:267-279 © 1995 Temple University
0148-6195/95/$09.50 SSDI 0148-6195(95)00011-F
268
C.-F. Lee et al. of output and market structure uncertainty on the cost of equity. Thomadakis (1976), Subrahmanyam and Thomadakis (1980), and Chen, Cheng, and Hite (1986) investigated the linkage between market power and systematic risk. Among them, the paper by Subrahmanyam and Thomadakis has attracted significant attention from academics. In their analysis, Subrahmanyam and Thomadakis assumed a fixed-coefficient production technology and developed a theoretical model, linking systematic risk with firm variables such as the labor-capital ratio and monopoly power as measured by the elasticity of demand for the firm's output. Recently, Lee, Liaw, and Rahman (1990) employed a Cobb-Douglas production function to show that a firm's systematic risk is negatively related to its monopoly power and capital-labor ratio. Sun (1993) extended the model to a homogeneous production function case. Sun showed that an increase in one firm's wage rate adversely affects its competitor's systematic risk and that the impact on its own beta coefficient depends on the competitor's reaction function. Wong (1994) simplified Sun's model and improved its tractability, and derived the following observations: (a) a higher wage rate leads to a higher systematic risk and (b) the relationship between one firm's wage rate and its competitor's beta coefficient is negative. Peyser (1994) extended the analysis further to take wage rate uncertainty into consideration. Peyser showed that the relationship between systematic risk and monopoly power depends on the degree of wage rate uncertainty. The purpose of this paper is to extend the analysis with the assumption of a variable elasticity of substitution (VES) production function. The rationale for adopting a VES production function is as follows. First, the VES production function, as shown by Revankar (1967), includes both fixed-coefficient and Cobb-Douglas production functions as special cases. Second, it contains several properties that will facilitate our analysis. Third, as shown by Hicks (1948) and Allen (1956), the elasticity of substitution parameter, in principle, can be a variable depending upon output a n d / o r factor combinations. This is an advantage of a VES production function over the constant elasticity of substitution production function. Based on the above premise, we will investigate the impact of factor substitution on systematic risk. As evidenced by Arrow et al. (1961), for the US economy as a whole, production is neither as flexible as implied by the Cobb-Douglas production function nor as inflexible as the fixed-coefficients case. The production process in the American economy seems to be characterized by some possibilities of substituting one factor for another) Therefore, it is plausible that a firm characterized by inflexible production technique would tend to have higher business risk, ceteris paribus, because major disturbances may arise in the transition of factor substitutions. The rest of this paper is organized as follows. The next section presents model formulation and notation. Using the cash-flow version of the CAPM, the formulation of systematic risk is then derived, with uncertainty in both the output market and the labor market. The comparative statistics in the short run and long run are discussed in the third and fourth sections. The last section concludes this paper.
I Revankar (1967) found the empirical relevance of the VES function in 5 out of 12 industries by using cross sectional data for year 1947.
Systematic Risk, Wage Rates, and Factor Substitution
269
II. The Model There are two sources of uncertainty: price uncertainty and wage rate uncertainty. The first source of uncertainty the firm faces originates from its downward sloping demand curve, ~6 = p(1 + ~),
(1)
where p represents the expected price of output and Y is a random error term with zero mean. Since 6 is an economy wide disturbance, it affects the output prices of all firms in the same way. Following Thomadakis (1976), we introduce a parameter, tz, which is the reciprocal of the price elasticity of demand, to describe the firm's degree of monopoly power in its product market. 2 The firm makes its output decision before the price is known, that is, before 6 is revealed. The marginal revenue (MR) function can now be expressed as MR = (1 + ~)(1 - i.tE)p, where E is an elasticity constant and 0 < ~ < 1. If /z = 0, then the firm is in a perfectly competitive output market. On the other hand, the firm acts as a monopolist i f / z = 1. The firm employs a variable elasticity of substitution (VES) production function [Revankar (1967)] with two factors, capital ( K ) and labor (L): Q = K"(1-*P)[L + ( p - 1 ) K ] "'p,
(2)
w h e r e a , s, and p a r e p a r a m e t e r s w i t h t h e f o l l o w i n g c o n s t r a i n t s : ce>0, 0
O < s p < l,
L / K > (1 - p ) / ( 1 - sp). 3 Equation (2) reduces to the fixed-coefficient production function if p = 0, the Cobb-Douglas case when p = 1, and the linear model provided that p = 1/s. We further assume that the amount of capital employed is purchased with the proceeds of issuing shares to the capital market at the beginning of the period and capital is exhausted in the production process. For notational simplicity, we set the price of capital at unity. The wage rate per unit of labor, which will be paid at the end of the period, is a random variable, = w(1 + t3),
(3)
where w is the expected wage rate and t3 is a random shock in the labor market with zero mean.
2 See Thomadakis (1976) for details. 3 This requirement is to satisfy the assumption that the elasticity of substitution is positive in the empirical relevant range of K/L. See Revankar (1967, 1971) for detailed discussion on the VES production function.
270
C.-F. Lee et al. It is also assumed throughout the paper that the firm is a price taker in both factor and capital markets and that the firm cannot affect the market price of risk. Consequently, there must exist sufficient industries in the economy so that the demand and supply conditions in the capital markets remain unchanged if there are changes from any individual firm. As such, any decisions, either financing or investment, should have negligible impacts upon aggregate output and labor. The finn's uncertain ending cash flow is defined as l ? =/SQ - ~L = (1 +
~)pQ
-
(1 +
~)wL.
(4)
Using the cash flow version of the CAPM, the expected market value of the firm is given by
V = ( E ( g ) - /~Cov(}'~,Rm))(1 q-r) -1,
(5)
where A is the market price of systematic risk, /~m is the uncertain rate of return on market portfolio, and r is the risk-free rate of interest. If we define * = E(1 + ~) - ACov(~,/~m) = 1 - ACov(~,/~m), th = E(1 + t3) - ACov(~3,/~m) = 1 -- ACov(5,/~m), as the certainty equivalents of (1 + ~) and (1 + 5), respectively, the value of the firm simplifies to V = [ptbQ
-
w~bL](1 + r) -1
(5')
Before proceeding further, we summarize the key assumptions as follows:
Assumption 1. A VES production function is employed. Assumption 2. Product market uncertainty is ~ with zero means. Assumption 3. Labor market uncertainty is 5 with zero mean. Assumption 4. Firm is price taker in both factor and capital markets. Assumption 5. The capital asset pricing model holds. Assumption 6. Capital is exhausted in the production process.
Assumption 7. Cov(6, J~rn) and Coy(t3,/~m) are positive. Assumption 8. The firm's beta coefficient is positive. With the basic setting above, we shall carry out the comparative statistics analysis both in the long run and in the short run in the next two sections.
Systematic Risk, Wage Rates, and Factor Substitution
271
III. Comparative Statistics: Short R u n In the short run, the amount of capital is fixed. The goal of the finn is to maximize its net present value (NPV), that is, the difference between its market value ( V ) and its capital expenditure (K); NPV = V - K. The maximization problem can be characterized as follows: Max N P V = V -
K = [ p ~ Q - wq~L](1 + r) -t - K.
(6)
Differentiating (6) and setting the first order condition equal to zero yields ONPV
p ~ ( 1 - tzE) otspQ
OL
L + (1 - p ) K
- w~b = 0.
(7)
From (7) we can solve for the optimal L as follows: pq~(1 - ~ E ) a s p Q L =
- (1 - p ) K .
w4,
(8)
According to the CAPM, systematic risk is measured by the relationship between the rate of return on the firm's stock and the rate of return on the market portfolio. Define the rate of return on the finn as IT-V V where I7 is the ending cash flow to shareholders and V is the expected market value of firm i. Then the systematic risk of the finn, /3, is given by
/3
Var
(9)
,
m
where Cov(Ri, R , , ) is the covariance between the rates of return on the firm and the market portfolio and Var(/~ m) is the variance of return on the market portfolio. Therefore, (9) can be rewritten as (1 nt-
r)(Cov(e,l
m) -
[*(1
-
t E)olsp ) - 1 -
w(pQ)-'(1 - p)K]
/3 = Var(/~m){q~ - [ * ( 1
- txE)asp-
wck(pQ)-l(1 - p)K]}
(9')
272
C.-F. Lee et al. The expression for/3 in (9') shows that systematic risk is dependent on both w and /x, among other variables. We first analyze the impact of a wage rate change on systematic risk by differentiating (9') with respect to the wage rate w:
o/3 Ow
(1 + r)(1 - p)K Cov(O,/l~ra Var(/~){pQ* -pQ[*(1(1 + r)(1 -
IxE)asp- w~b(pQ)-l(1- p)K])
p)qSK(pQ)
-[*(1-
) (10)
I{Cov(Y,/}m)
lzE)olspt~-1- w(pa)-l(1- p)g])
+
Var(/~m){*
-[*(1-
i.zE)otsp-wdp(pQ)-l(1-
P)Kl}
2"
Using Assumptions 7 and 8, it follows that e/3/Ow < O, if p > 1.
(11)
According to Revankar (1967), the elasticity of factor substitution of the VES production function is 4 o-= c r ( K , L ) = 1 + [ ( p -
1)/(1 - s p ) ] [ K / L ] .
(12)
Given any capital-labor ratio, it follows that >
oral,
>
i f p ~ 1.
(13)
Combining expressions (11) and (13) results in 0/3/3w < O, if o- > 1.
(14)
Equation (14) reveals an important finding that the impact of a wage rate change on /3 is dependent on the magnitude of the firm's elasticity of factor substitution. Specifically, the systematic risk of the firm increases with the wage rate if the elasticity of substitution is less than 1. On the other hand, the relationship is negative provided that the elasticity of substitution is greater than 1. This finding is attainable only when the VES production function is employed. As the VES production function subsumes both fixed-coefficient and Cobb-Douglas production functions, our finding here adds more to the literature on the linkage between the firm's systematic risk and its microeconomic variables. Furthermore, this finding cannot be obtained by the constant elasticity of substitution (CES) production function because the elasticity of substitution is constant. 4As discussed earlier, the V E S p r o d u c t i o n function r e d u c e s to the fixed-coefficient m o d e l if p = 0. T h e c o r r e s p o n d i n g elasticity of substitution is tr = 1 + [( p - 1)/(1 - s p ) ] [ K / L ] = 1 + ( p - 1)/[(1 + r X w c ~ ) - l s p - ( p - 1)] = 0 if ~b = 0. O n the o t h e r h a n d , the C o b b - D o u g l a s case ( p = 1) implies o- = 1 + ( p - 1)/[(1 + r X w q b ) - l s p - ( p - 1)] = 1.
Systematic Risk, Wage Rates, and Factor Substitution
273
The relationship between the systematic risk of the firm and its market power in the product market can be examined by differentiating expression (9') with respect to /z: (1 + r)dpEaspdp-' Coy(g,
at~ 3/x
Rm)
I x E ) a s p - w~b(pQ)-'(1- p)K]}
Var(/~m)(q~ - [ q ~ ( 1 -
(1 + r)~Easp Var(/~m){* - [ * ( 1 -
l~E)asp-wd~(pQ)-t(1-
p)KI) 2
<0. (15)
The result indicates that a higher degree of monopoly power in the product market will unambiguously lower the systematic risk of a firm, ceteris paribus. Sullivan (1978), Subrahmanyam and Thomadakis (1980), Chen, Cheng, and Hite (1986), Lee, Liaw, and Rahman (1990), and Sun (1993) had reached a similar conclusion. Based on the CAPM, the firm with a higher market power in its product market can raise capital at a lower cost (via a lower required rate of return).
IV. C o m p a r a t i v e Statistics: L o n g R u n Similar to the setup in the short run, the long-run goal of the firm is also to maximize its net present value by choosing both K and L to achieve the objective: Max NPV = V - K =
[p~pQ- w~bL](1 + r) -1 - K .
(16)
Substituting (2) into (16) and differentiating it with respect to K and L, the first order conditions are p@(1 - u E )
a(1 - sp)Q asp(p- 1)Q/(1 -l ~+ L+(p~-I-)-K) +r) -1=0
(17)
and pqb(1 -- IzE)aspQ L + (1 - p ) K
- w4, = 0.
(18)
Equations (17) and (18) can be used to solve for K and L as functions of Q 5: apdp(1 - ~E)(1 - sp)Q
K=
(1 + r ) -
(p-
1)w~b
and L
(19) [(1 + r)(w(9)-'sp - (p - 1)] a p ~ ( 1 - tzE)Q (1 + r ) -
(p-
1)w~b
5All variables K, L, and Q are evaluated at their optima in the followinganalysis.
274
C.-F. l e e et al.
The optimal levels of capital and labor can then be substituted into (4) and (5'). Using (4) and (5'), we can express the systematic risk of the firm in (9') as follows: Cov(/SQ - ffL,/~m)
(1 + r)
p f ~ a -- wc~Z
War(/~m)
3=
(Cov(Y,/~m)[1 + r -
(1 +
r)
(p-
1)wtb]
- [ ( 1 + r)sp~b -1 - w ( p -
1)]~(1
{4~[1 + r - ( p - 1)w~b] -[(1 + r)4o-lsp- (p-
Var(/~,,)
- tzE)Cov(~,
Rm)~ J,/
1)w]adP(1 - /xE)th}
(20) Simply put, the equilibrium relationship in equation (20) integrates the systematic risk of the firm with its monopoly power in the product market (/z), the relative factor price (w), and the degree of factor substitution (tr), among other variables. First, differentiating (20) with respect to w, we obtain 0[3
- ( 1 + r) Cov(~,/~m)( P -- 1)~b - ( p - 1 ) a ~ ( 1 - / z E ) C o v ( ~ , / ~ m )
Ow
Var(/~m)
¢[1 + r - ( p - 1)w~b] -[(1 + r)¢-lso
+
- ( p -
1)wladP(1 - / x E ) ~ b
(1 + r) Var(/~,.)
{Coy(e,/~m)[1 +
r- (p-
-[(1 + r)~-~sp x
- w(p
1)w4,] -
1)]aq~(1 - ~ E ) C o v ( ~ , / ~ ) } ~
{¢P[1 + r - ( p - 1)w~b] - [ ( 1 + r ) 4 o - l s p - ( p - 1)w]aq~(1 -/zE)4~} 2
where ~O= q~4~(p - 1) - ( p - 1)a¢~b(1 - ~E). The expression above can be simplified to to - ( 1 + r ) ( p - 1)(1 - s p ) [ a ( 1 - r)q~(1 - /zE)]
0[3 -
-
Ow
[cov( ,
cov( , m)l (21)
=
Var(/~m){q~[1 + r - ( p - 1)w~b] -[(1 + r)4~-Isp - ( p - 1)w]a¢(1 - ~E)4~} 2
The sign of the above expression depends on ( p - 1) and [ C o v ( ~ , / ~ , , ) Cov(~,/~m)]. If [Cov(~,/~m) - Cov(~,/~m)] > 0, i.e., the covariance of the firm's product market uncertainty with the market portfolio returns is greater than that of the labor market uncertainty with the market portfolio returns, then O[3/Ow~.O,
>
ifp~.l.
(22)
Systematic Risk, Wage Rates, and Factor Substitution
275
Combining expressions (13) and (22) yields
0,
if
1.
This shows that systematic risk and the wage rate are positively related if the elasticity of substitution is less than 1. On the other hand, the systematic risk decreases with the wage rate if the elasticity of substitution is greater than 1. When the elasticity of substitution is at unity, the firm's systematic risk is not affected by the changes in the wage rate. Overall, the results here are similar to those obtained in the short-run analysis. Again, this result could have been obtained only by the VES production function, and not by other production functions, e.g., the CES production. The results are reversed if [Cov(£ Rm) - Cov(z3,/},,)] < 0. In addition, any changes in the wage rates will not affect the firm's systematic risk if [Cov(g, Rm) - Cov(t3, R m ) ] = O. Second, the impact of monopoly power in the product market on the firm's systematic risk can be examined by differentiating (20) with respect to /z, the degree of monopoly power. Then we get c~/3 31t
(1 + r )
[(1 +r)qb l s p - w ( p -
Var(/~m) qb[1 + r -
(p-
1)](aqbE)Cov(tT,/~m)
1)w4,]
-[(1 + r)~b-lsp- (p-
1)wlaqb(1 - /zE)~b
(1 + r) Var(/~,n) {Cov(g,/~m)[1 + r -
(p-
1)w~b]
- [ ( 1 + r)~b-lsp - w ( p - 1)]aO(1 - ttE)Cov(5,/~m)}qJ X
{qb[1 + r -- ( p - 1)w~b] -[(1 + r)~b-lsp- (p-
1 ) w ] a ~ ( 1 -/zE)~b} 2
where ¢J = [(1 + r)ck-lsp - ( p - 1 ) w ] a ~ b E . The expression above can be simplified (see Appendix for derivation) to -aopqbE(1 + r ) [ w 2 ( p - 1)2~b - (1 + r ) ( p - 1)(1 + s p ) w +(i
3tt
+
r)2s
1][Cov( ,
-
cov( ,
Var(/~m){qb[1 + r - ( p - 1)w~b]
(23)
- [ ( 1 + r)c~-lsp - ( p - 1 ) w ] a ~ ( 1 - ttE)~b} ~ It follows, if [w2(p - 1)2~b - (1 + r ) ( p - 1)(1 + sp)w + (1 + r)2sp~b-1] 6 is positive, that c)[3/3tz<0,
if[Cov(J,/~m)-Cov(~,/~m) ] >0.
6 This is related to the constraint on the wage rate. It can be shown that [w2(p - 1)2~b- (1 + rX p - 1)(1 +sp)w+(1 +r)2spdp l ] > 0 i f w> a~E(1 +rX(1 +sp)+ 1/(1 +sp) 2 - 4 s p ) / ( 2 ( p - 1)).
276
C.-F. Lee et al. As shown, to support the conclusion by Subrahmanyam and Thomadakis (1980) and others that market power reduces systematic risk, the covariance of the firm's product market uncertainty must be greater than that of the uncertainty in the labor market. Peyser (1994) employed Tabin's q as a proxy of monopoly power and reached a similar conclusion. The outcome is plausible through various contractual mechanisms that reduce wage rate uncertainty. Finally, we turn to the degree of factor substitution. Differentiating (20) with respect to p yields 813
- (1 + r)
3p
Var(/~m) ×
Coy(&/~m)w~b + [(1 + r ) q ) - l s - w]ot~(1 - ~E)Cov(t3,/~m) • [1 + r - ( p -
1)w~b]
-[(1 + r)ck-lsp+
( p - 1)w]aqb(1 -- /xE)~b
(1 + r) Var(/~m) {Cov(~,/~m)[1 + r - ( p -
1)~b]
- [ ( 1 + r)4~-lsp -
w(o
-
1)]~(1
- tzE)Cov(&/~m)}O
X
{~[1 + r - ( p - 1)w~b] -[(1 + r)qb-lsp-
(p-
1)w]adO(1 - /~E)q~} 2
where ® = w~b + [(1 + r ) 4 ) - l s - w ] a ~ b ( 1 - / z E ) . The expression above can be simplified to a ~ ( 1 + r)2(1 - /~E){(1 + r ) s 4 ) -1 + ws - w}
ol3 3p
[cov( , &)- cov(o,&)] Var(/~m){~[1
+ r -
( P -
- [ ( 1 + r)49 1 s o -
(24)
1)w~b]
(p-
1 ) w ] a ~ ( 1 -txE)4~} 2
It can be shown that if [(1 + r)sck -1 + ws - w] 7 is positive and the covariance of the product market uncertainty is greater than that of the labor market uncertainty, the greater (less) the degree of input substitutability between capital and labor, and the lower (higher) the systematic risk of the firm. Put differently, if it is relatively easy for a firm to substitute capital for labor, the process of technical transformation can be smoothly accomplished and the firm is exposed to a lower business risk. On the other hand, if inflexible technology requires major disturbances in order to absorb the additional capital, the business risk of the firm will substantially increase. 7 If [(1 + r)sqb -1 + WS -- W] > 0, it impliesthat w < ((1 + r)s~k-1)/(1 - s). Note also that 0 < s < 1, by assumption in equation (2).
Systematic Risk, Wage Rates, and Factor Substitution
277
V. Conclusions This paper has attempted to integrate the relationship between operating leverage, monopoly power, and factor substitution with the systematic risk of a firm. A VES production function is employed. Also--an important improvement over most of other papers in this area--both price and wage rate uncertainty are take into consideration as in Peyser (1994). Several significant results are derived. First, the impact of a wage rate change on systematic risk depends on the degree of input substitutability both in the short run and in the long run. Second, the firm's systematic risk is negatively correlated with market power in its product market and with the degree of factor substitution in the long run.
Appendix D e r i v a t i o n o f E x p r e s s i o n (23)
0/3 c~/z
(1 + r)
[(1 +
r)¢o-'sp
-
p
w(
-
1)](a~E)Cov(tS,/~m)
Var(Rm ) qb[1 + r - ( p - 1)wqS] -[(1 + r)c~-lsp- (p-
1)w]aqb(1 --/xE)q5
{Cov(~,/~m)[1 + r -- ( p -- 1)w~b] - [ ( 1 + r ) q S - l s p - w ( p - 1)] (1 + r)
c~*(1
Var(km) {~[1 + r -
(p-
~E)Cov(?:,km)}~0
-
1)w4~]
- [ ( 1 + r)49 ~sp - ( p - 1)w]aqb(l - /~E)&} 2
[(1 + r ) a - ~ s p
-
q~[l+r(1 + r)
w(p
(p-
-
1)](~,E)Cov(~, kin)
1)w4~]
- [ ( 1 + r ) 4 ~ - l s p - ( p - 1)w]aq~(1 - /~E)~b
Var(/}m) {qb[l + r - - ( p - - 1)w&] -[(1 +r)¢
~sp-- ( p - -
1)w]otqb(1 --/zE)~b} 2
{Cov(~, kin)[1 + r - ( p - 1)w~] - [ ( 1 + r ) g J - l s p - w ( p - 1)]
-
(1 + r) Var(R~) {q~[1 + r -
(p-
£)),
1)w&]
-[(1 + r)~b-lsp-
(p-
1)w]a~(1 - /~E)t~} 2
278
C.-F. Lee et al. -(1
+ r)tp[(1 + r) - ( p -
1)w4)][Cov(g,l~ m]] - Coy(t3,/~,,]] \ sj I. \
Var(/~m)((I)[1 + r - ( p - 1)w4)] -[(1 + r)4)-isp - (p - 1)w]a~(1
-/xE)4)) 2
w h e r e q, = [(1 + r ) 4 ) - l s p - ( p - 1)w]o~(I)4)E. T h e expression above can then be simplified to - a ~ 4 ) E ( 1 + r) [ w 2 ( p - 1)24) - (1 + r ) ( p - 1)(1 + s p ) w + (1 + r)Zsp4) -1]
3/3 0/x
[Cov(e, g m ) -
Cov(u, gm) ]
Var(/}m){(I)[1 + r - ( p - 1)w4)] - [ ( 1 + r ) 4 ) - l s p - ( p - 1 ) w ] a ( I ) ( 1 - /xE)4)}:
We thank seminar participants at Rutgers University, New Brunswick, for helpful comments.
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