An efficiency-wage model with explicit monitoring

Journal

of International

Economics

32 (1992)

An efficiency-wage monitoring Unemployment Richard Carleton

1799191.

North-Holland

model with explicit

and welfare in an open economy

A. Brecher*

University, Ottawa, Ontario KIS 5B6, Canuda

Received June 1990. revised version

received

November

1990

This paper develops an efficiency-wage model of involuntary unemployment in economy, which uses two factors to produce three goods, one of which is monitoring aimed at reducing shirking by workers in all industries. The analysis tariff on imports may lower employment but, paradoxically, raise social welfare time. As the paper also demonstrates, the optimal policy combination includes not capital and monitoring as well as a subsidy on employment, but also (despite the assumption) a tax on consumption.

a small open (non-traded) shows how a at the same only taxes on small-country

1. Introduction Efficiency-wage models of involuntary unemployment have been receiving much attention in recent years, as documented in surveys by Akerlof and Yellen (1986) Katz (1986) and Stiglitz (1987). The present paper develops an open-economy version of such a model, and uses this version to analyse the employment and welfare effects of various commercial policies. Unemployed labour in our model cannot bid the uniform wage down to the full-employment level, because firms (with costly monitoring) view a higher wage as a device for encouraging efficiency in their employees, who tend to shirk in all sectors of the economy. This situation must be distinguished from the wage-differential models of Bulow and Summers (1986), Copeland (1989), Katz and Summers (1989) and Wilson (1990) who have a shirking-free sector that absorbs all labour not employed in the rest of the economy.’ Whereas *Gratefully acknowledged are helpful comments and suggestions from Jagdish N. Bhagwati, Ehsan U. Choudhri, Richard Clarida. Alan V. Deardorff, Robert C. Feenstra, Ronald Findlay, Christian Gilles, Ngo Van Long, P. Nicholas Rowe, Thomas W. Ross, Lawrence L. Schembri, John D. Wilson, seminar participants at Columbia University, and anonymous referees. ‘Although the Bulow-Summers and Katz-Summers papers briefly consider the possibility of adding ‘wait’ unemployment of the Harris-Todaro (1970) type to their models, this type of unemployment is (arguably) ‘voluntary’, and in any case is not due to efficiency-wage phenomena per se. See also Das (1989) for an analysis of tariff and monetary policy in a shirking model of unemployment, where each country has only a single sector that produces 0022-1996/92/$05.00

c

1992-Elsevier

Science Publishers

B.V. All rights reserved

R.A. Brecher, An efficiency-wage model

180

their wage differential can be neutralized by Bhagwati and Ramaswami’s (1963) tax-cum-subsidy on factor use, our unemployment cannot be fully eliminated. Another distinguishing feature of the present paper is our treatment of monitoring, which we model as a (shirking-detection) good that is domestically produced and endogenously priced in a general-equilibrium framework. This explicit treatment of the resource-using aspects of monitoring crucially alters the effects of commercial policy, as shown below. Extending the Heckscher-Ohlin model of a ‘small’ trading economy that produces two consumer goods using two primary inputs, section 2 lets labour’s efficiency depend on wages, unemployment and monitoring as a non-traded (third) good. Section 3 then shows how a tariff on imports may lower employment but, paradoxically, raise social welfare (by reducing the costs of monitoring) at the same time. As section 4 next demonstrates, the optimal combination of policies includes not only taxes on capital and monitoring as well as a subsidy on employment, but also (despite the smallcountry assumption) a tax on consumption. Concluding remarks are offered in section 5, which highlights some significant departures from Brecher’s (I 974a, b) minimum-wage analysis of unemployment in an open economy. 2. The model Consider a competitive and outputs are described

Xi=Fi(Ki,eiLi),

economy in which the relationships as follows: i=l,2,3,

between

inputs

(1)

where Xi denotes output of good i; Ki and Li, respectively, stand for the inputs of capital and labour used to produce this good; ei represents the level of labour’s efficiency in sector i; while the production function F’ of this sector is increasing in each argument, strictly quasi-concave and homogeneous of degree one. Goods 1 and 2 are the home country’s importable and exportable, respectively, while non-traded good 3 is monitoring,* which attempts to detect shirking by employees in all three industries. In accordance with the efficiency-wage literature, let ei=Ei(wi,Mi/Li,u),

i= 1,2,3,

(2)

varieties of a differentiated product under monopolistic competition. For an open-economy analysis of unemployment arising instead from labour-turnover considerations, see Batra and Naqvi (undated, circa 1989) and Batra and Yamada (undated, circa 1986). ‘This good can be interpreted as. for example, surveillance equipment or personal supervision. The notion of monitoring as a good is an effective analytic device for capturing, in general terms, the important fact that primary factors are absorbed in the detection of shirking. Under the first-degree homogeneity assumed above, none of our analysis will depend on whether the monitoring activity is operated internally by each firm or externally by specific tirms specializing in this activity.

181

R.A. Brecher, An ejjkiency-wage model

where labour’s real wage rate in terms of good 2 is wi in sector i: Mi is this sector’s input of monitoring to detect shirking by workers; u is the economy’s rate of labour unemployment; while the efficiency function, d’, of sector i is increasing in each argument, and is strictly quasi-concave in its first two arguments (given the third). The efficiency of industry i does not depend on the wage rates of the other two industries, because we follow Bowles (1985) in assuming that workers of each firm perceive all firms to be paying the same wage. Profits in sector i are ‘i=piXi--rKi-WiLi-p,Mi,

i=1,2,3,

(3)

where capital’s real rental rate in terms of good 2 is Y throughout the economy, and pi represents the domestic relative price of good i in terms of good 2. Under this representation, the exportable good’s price, pz, is identical given under to 1. The importable good’s price, pl, moreover, is exogenously our small-country assumption. Because monitoring is a non-traded good, however, its price, p3, is endogenously determined in the home market. To obtain the first-order conditions for profit maximization, substitute eqs. (1) and (2) into (3) and then equate to zero the partial derivatives of zi with respect to K,, Li, Mi and wi.3 The resulting conditions are as follows: piFi = r,

i=l,2,3,

(4)

piFL(di-mie”f,,)=wi,

i= 1,2,3,

(5)

piF~E~=p,,

i= 1,2,3,

(6)

i = 1,2,3,

(7)

piF$?i=

1,

where Hi -eiLi and m, = MilLi, while subscripts of functions indicate differentiation (e.g. Fk E aFi/dKi and FkK E aF~/aK,). Dividing eqs. (5) and (6) by (7), we obtain the following conditions:

partial

e”‘= wi;i + mi;;,

i = 1,2,3,

(8)

t?;,lz;=p,,

i= 1,2,3.

(9)

%cluding wi in this list of choice variables is consistent with price-taking behaviour, in sense that the quantity of labour hired by the individual firm does not affect the wage needed achieve each level of efficiency with a given amount of monitoring per worker. As noted Yellen (1984) efficiency-wage models explain why competitive firms refrain from cutting wages the face of involuntary unemployment.

the to by in

I82

R.A. Brecher,

An efficiency-wage

model

To ensure that eqs. (8) and (9) have a unique solution for the wage and monitoring per worker in each industry (given p3 and u), let the function e”’ exhibit increasing or decreasing returns to scale for all values of e, that are respectively less or greater than some borderline level C,, at which returns to scale are (locally) constant. 4 Then, in light of Euler’s Theorem, eqs. (8) immediately imply that ei =C, (for all values of p3 and u). Given this result, eqs. (2) and (9) uniquely determine wi and mi (as illustrated below). Now, assuming that d’ is the same function for all industries, we see that .Ci, e,, wi and m, are each uniform across sectors5 Thus, for each of these five symbols, we can suppress the index i to get C, C, e, w and m. In fig. 1 the iso-efficiency curve EE’ shows combinations of w and m that keep Z(w, m, u) = t? for a particular value of u.~ Firms in all three industries choose the combination corresponding to point A, at which curve EE’ is touched by the line BB’ with slope -p3, as implied by eqs. (9). Given this choice, the length of the line segment OB represents the value of w + p,m = c, which denotes the (wage plus monitoring) cost per worker. As fig. 1 suggests, the proIit-maximizing w, m and c are respectively representable as functions G(p,, u), rG(p,, u) and c’(p,, u) = G(p3, u) + p3ti(pJ, u);~ where KJ~> 0, Kr, 0 (since a rise in p3 would steepen the tangent BB’); 2,<0 (because an increase in u would shift the curve EE’ in toward the origin); while @, and fi, can be positive or negative (depending on how the curve EE’ tilts as it shifts inward). For each combination of p3 and u, ? gives the minimum value of c consistent with 2. Thus, in view of our returns-to-scale restriction on the function P, t/Z gives (for each p3 and u combination) the minimum value of c* z c/e, which is the cost per efficiency 4An analogous restriction yields the S-shaped efficiency-wage curve of Mirrlees (1975, fig. 4.1) and Stiglitz (1976, fig. I). who use this curve to develop Leibenstein’s (1957, ch. 6) model, in which efftciency depends on the wage for nutritional reasons. By allowing for endogenous monitoring as an additional determinant of efftciency, our eqs. (8) provide a straightforward generalization of Stiglitz’s (1.1) which characterizes the proftt-maximizing point on his abovementioned curve. ‘This uniformity allows us to focus sharply on unemployment, without clouding the analysis by introducing additional distortions in the form of wage differentials between industries. 6For the use of a similar diagram, see Bowles (1985). ‘The arguments of these functions need not include e, in view of the above result that profit maximization leads to the same level of efficiency, 2, for all combinations of p3 and u. This result follows from our foregoing assumption that the borderline level of efficiency (between the regions of increasing and decreasing returns to scale) is a constant, 2, and hence is independent of both p3 and u. Such constancy and independence hold if, for example, P(w,m, u) = h[wg(u), mG(u)], where i is a homothetic function, while R and G are increasing functions of u. This particular specification is consistent with the formulation of Bowles (1985), if workers take the unemployment rate to be the probability of not finding employment after dismissal for shirking [as assumed by Bowles and Boyer (1988)] and unemployment insurance is a fixed fraction r of the wage rate (with Osa< I). Under these conditions, the ‘labor extraction function’ derived by Bowles is essentially the same as our P function, with g(u)=u( 1--r) and G(u)- 1. Although he assumes that wages are the only income of employed workers, the level of effort in his model would not be affected by changes in any non-wage income (from capital ownership or other sources) that accrues to employees even if fired.

R.A. Brecher.

An efficiency-wage

model

183

w

c

Monitoring

Per Worker

Fig. 1.

unit of labour. In other words, firms choose the wage and monitoring worker to minimize the unit cost of efficiency. Because the production functions for our three goods are homogeneous degree one, we can rewrite eqs. (4) as

Pif:=‘, and substitute

i= 1,2,3,

of

(10)

eqs. (6) into (5) to yield

pi(fi-_ifi)=c*,

i= 1,2,3,

(11)

where ki = KJH,, fi(ki) -Fi(ki, 1) and fk>O> respectively are the factor reward and marginal ‘Assume f’(cc)=f:(o)=

per

also that co.

the j’

function

is well-behaved,

fi,.” Since c* and fi-kifi product of efficiency units

in the sense that

f’(O)=fi(~;)

=0

of and

R.A. Brecher, An efficiency-wage model

184

labour in sector i, eqs. (10) and (11) are equivalent to the first-order conditions for profit maximization in Komiya’s (1967) model of two factors and three goods, one of which is non-traded. Thus, assuming positive production of both traded goods and ruling out factor-intensity reversals, we can subject these equations to well-known manipulations that yield c*, r, ki and p3 as functions E*(p,), ?(p,), p(p,) and z(pl); where c”p* = k,f’/(k, - k,), fp,= -f’/(k,-k,) and &= -f’f2/pZfif:k(k2-kl), which all accord with the Stolper-Samuelson Theorem; while rrP= f ’ (k2 - k3)/f3 (k, -k,) as shown explicitly by Komiya.’ Since profit-maximizing lirms minimize the unit cost of efficiency and equilibrium in the labour market adopt marginal-productivity pricing, requires that c”C4Pl),

1--Lll~=~*(P,),

(12)

where L denotes total employment, and 1 -L= u because the economy’s fixed endowment of labour is set equal to 1 by choice of units. Eq. (12) can be solved to obtain L as a function f$pl). Since the threat of unemployment is needed to prevent efficiency from falling to zero, z < 1. In other words, equilibrium necessarily involves less than full employment. To clear the market for (non-traded) monitoring, output of good 3 must satisfy the following equation: x3

=~Cn(PI)> 1 -QPdlQPd.

(13)

Thus, in equilibrium, H3=rii[n(p,), l-~(p,)]Z(p,)/f3[K3(p1)]-A3(pI) and K, = k”3(pI)fi3(pI) = R3(pI). By focusing on the case of incomplete specialization in the production of traded goods, we are implicitly assuming that (K-Z?3)/(FZ-fi3) lies strictly between c’ and E2 at the given pl, where Z? is the economy’s fixed endowment of capital. In other words, once industry 3 has absorbed enough capital and labour to satisfy the demand for monitoring, the ratio of residual factor totals must lie within the cone of diversilication for goods 1 and 2. These totals then determine X, and X, in the usual way.

3. Tariff policy Under domestic

our small-country price of good

‘For derivations appears to require

assumption, an import tariff always increases the tariffs impact 1. Thus, to determine

of all four of these formulae, see Kemp (1969, pp. 17 and 139), although replacement by /, on the right-hand side of the first line in his eqs. (6.8).

the on

JO

R.A. Brecher, An efficiency-wage model

employment, differentiate eq. (12) totally forward manipulation shows that

with respect

18.5

to pi. Then,

straight-

(14) when we recall the formulae for c”p*and rcP, while observing that E,,=m (in view of Shephard’s Lemma). The expression in square brackets in eq. (14) must be negative, since m/t?f 3 =(M3/L3)/(EF3/ifL3) = M,/X, < 1 for the monitoring industry to yield a positive net output. Consequently, with E,
s=W,,D,,L),

(15)

where s is the level of social welfare; D, and D, respectively are the aggregate consumption levels of goods 1 and 2, which are both assumed to be normal; while the social-welfare function g is increasing in each of its tirst two arguments, and is strictly quasi-concave in these two given the third.” The employment level enters S not only indirectly by affecting national income and hence consumption levels, but also directly because an employment expansion means an equal decrease in off-the-job leisure and (given 2~0) a smaller increase in on-the-job leisure from shirking. Assuming that this net reduction in leisure tends to diminish welfare (given D, and D2), we have s,
(16)

Pl~l+~,=P,x,+x,,

where the exogenously given p1 is the international relative price of good in terms of good 2. Since firms break even, we also know that PIXI +X,+4pJX3

=~(P,)~+~*(P,)aP,),

1

(17)

“‘Under the following four assumptions, S can be derived directly from the Bergson socialwelfare function /?(s,,s,,...,s,), the arguments of which are the utility levels of the economy’s t individuals: (i) sj= u(D, j, Dzj) - 6,+(e) for person j( = 1,2,. , t), where D, is this person’s demand for good i( = 1, 2), while aj equals 1 if individual j is employed but equals 0 otherwise; (ii) the function o is homogeneous of degree one; (iii) 4 is a linear function of e; and (iv) p(sr . sa, , s,)= s1+ s2+. + s,. Assumptions (i) - (iii) provide a straightforward generalization of the utility function adopted by Shapiro and Stiglitz (1984). Under this generalization, assumption (iv) ensures that social welfare is independent of the income distribution among individuals, as implied by eq. (15). This independence is a commonly adopted simplification in the theory of international trade.

186

R.A. Brecher, An efficiency- wage model

in view of eqs. (3). We now differentiate eqs. (15)-(17) thereby establishing that dsldp, = [(Kzw + gJ + d,p, L&l

and

dL/dp,

(13) totally

with

- 3,p, Lrii,n,,

respect

to pl,

(18)

because we restrict attention to the case of a small tariff in the neighbourhood of free-trade equilibrium at which p1 =pl.” Although g,_,< 0, we assume that S2w +gL>O, because the involuntary nature of unemployment in our mode1 means that the marginal utility of wage income plus the marginal utility of on-the-job leisure exceeds the marginal utility of off-the-job leisure for the individual worker. Inspection of eq. (18) then shows that a tariff may improve social welfare, even when employment falls as a result of protection. For the sake of concreteness, consider the case where n,>O>dL/dpr (as when k, < k, and k,ck3) and &,0.14 This paradoxical possibility of welfare gain despite employment loss highlights the crucial difference made by our explicit recognition of monitoring costs. If these costs were ignored, the right-hand side of eq. (18) would reduce to ($,w+3,)dL/dp,, thereby giving rise to the misleading conclusion that a tariff must change employment and welfare in the same direction.

4. Optimal policy To determine the optimal combination of ad valorem taxes (subsidies) within our model, we first introduce taxes of ~~ on employment and rM on monitoring. (Negative values of these taxes indicate positive subsidies.) Then, “The following information has been used to derive eq. (18): S,/Sz=p,, since consumers set their marginal rate of substitution equal to their (domestic) price ratio; and F# + F:CL - n#X, X, =O, when we recall the formulae for i,, Cz and nP while noting that K/t?=k,L, +k&,+k,L, and L-L,+L,+L,. 121f Z(w,m,u)~2[wu( 1 -x),m! along the lines of footnote 7, Ci
R.A. Brecher, An efficiency-wage model

w and p3 in eqs. (3) must be multiplied by 1 +z, = TL( >O) and T,( >O), respectively. Thus, conditions (9) and (12) are replaced by

e”,/e”,= P, T&l,

187

1 +r,

E

(9’)

1 -Q/Z=

2*(&Q),

(12’)

where p- T,n(p,)/T,; while by redefinition, c= w+pm and c* = T,c/e. In fig. 1, the slope of the tangent to the iso-efliciency curve at the profit-maximizing point now equals -p, as implied by eq. (9’). The vertical intecept of this tangent correspondingly becomes G(p, U) + pfi(p, U) = c”(p,u). Thus, the firm’s cost per efficiency unit of labour under profit maximization is now measured by the left-hand side of eq. (12’). The optimal-policy problem for the home government is to maximize $D,, D,, L) subject to Di=Bi(y,4), y=G(p,

i-1,2,

1 -L)L+R,

P*~l+~*=~l~1(Pl,,,L)+~2(Pl,p,L),

(19) (20)

(21)

where y denotes national expenditure by home consumers, whose relative price of good 1 in terms of good 2 is represented by 4; R stands for after-tax returns to capital;” 6’ is the Marshallian demand function for consumer good i; while the profit-maximizing output of this good is given by the function 2’. To achieve a better understanding of this function, note that Cz’(p,,p,L) =P{pl, K-z?[El(p, 1 -L)L, P(pJ,ZL-zP[fi(p, 1 -L)L,/qp,)]} for i= 1, 2, where fi3=kL/f 3(k”3)=H3 and c3=L3fi3= K,. Thus, for well-known reasons, the output levels of goods 1 and 2 depend only on their relative price, pl, and their combined factor-use totals, K-K, and CL -H,. Before solving the maximization problem, let us first show that R =0 at the optimum. If R were positive, we could lower it and simultaneously raise p to keep y constant (since fiO>O) in eq. (20). This action would reduce CiL (since tip ~0) and hence free some of both primary factors for use in the production of traded goods, thereby raising the right-hand side of eq. (21). Then the left-hand side of this equation can be raised equivalently (e.g. by a reduction in 4) for an improvement in welfare. Once R has been lowered to zero, further lowering is impossible, because capital’s disposable income clearly cannot be negative. ‘5Eq. (20) implicitly assumes no unemployment insurance. Nevertheless, for the case (in footnote 7) where the efficiency function is d[wu( 1 -x),m], we could expand the definition of R to include such insurance, without changing the results derived below.

188

R.A. Brecher, An ejkiency-wage

With R set equal to zero, the Lagrangian problem is as follows: I/(&J,94,P>L, 4

model

function

for our maximization

= qal CNp,1- L)L, 41,d2 C%P,1-W,ql,L) +;l{pJqbqp, 1 -L)L,q]

where i is the Lagrange multiplier. Setting VP, V,, V,,, I/, and I/, equal zero, we derive the following first-order conditions for a maximum: PI-P1=0,

to

(22) (23) (24)

TM- 1 =[~~Lb,‘/(s,+~)(a,l+D,6:)-LE,]/c,

(25)

and eq. (21), respectively.r6 We know that the optimal value of 1”is negative, since it equals the change in maximum welfare for a small tightening of the budget constraint (21). Thus, from eqs. (23) and (24) it follows that 5, +l=np3b,‘/p(d~ + DID:) ~0, because 6: ~0 under our normality assumption, while the expression in parentheses is negative by the Slutsky decomposition. Therefore eqs. (23) and (25) imply that q-p1 >O and TM- 1 >O, respectively. In other words, at the optimum there must be positive taxes on both consumption and monitoring. As indicated by eq. (22), however, production should be neither taxed nor subsidized. Since capital, consumption and monitoring are all being taxed at positive rates, employment must be subject to a positive subsidy (i.e. TLO, which all hold for well-known reasons; K,/!?: = - c*/r, because the marginal rate of factor substitution equals the factor-price ratio; K:>O>fi:, given downwardsloping isoquants; and hence (X:I?: + XhI?:)Ez ~0, because Samuelson’s (1953) ‘reciprocity’ relations imply that the Rybczynski terms Xi and X?:, equal the Stapler-Samuelson terms i, and c:~ respectively. The derivation of eq. (23) makes use of the property that 46: +D,i = -D,, since qD’(y,q)+I?(y,q)zy. To obtain eq. (24), note that: qb:+D:= 1, given the identity in the immediately preceding sentence; 3, +& =0 by well-known reasoning; and, with A4 mmL, rl?&+c*I?$=p,, because a unit increase in X, (=M) at constant prices will raise cost and revenue by the same amount. Eqs. (23) and (24) are used in the derivation of (25).

R.A. Brecher, An eficiency-wage

model

189

consumption would have a negligible impact on social welfare if the resulting revenues were redistributed to consumers in a non-distortive fashion, which here means raising R above its optimal level of zero. Then, as shown above, welfare could be increased by reducing R. The optimality of tax-free production, moreover, is consistent with the prescription of Diamond and Mirrlees (1971). Since the subsidy on employment tends to raise L while the tax on may monitoring has the opposite effect,” the optimal level of employment be either more or less than the laissez-faire level.” This ambiguity arises because our optimal combination of policies is, in fact, only a ‘second-best’ solution. A first-best optimum would require a fine for shirking, in consonance with Bhagwati and Ramaswami’s (1963) prescription for a policy that goes directly to the source of distortion. If set optimally, this line would achieve full employment, by replacing unemployment as an efficiencypreserving device. Imposing such a line, however, would be analytically the same as requiring workers to post performance bonds of the type discussed in the efficiency-wage literature. Thus, the well-known problems with these bonds apply equally to the shirking tine, thereby making the first-best policy infeasible.

5. Conclusion This paper has developed an efficiency-wage model of involuntary unemployment in a small open economy, which uses two factors to produce three goods, one of which is (non-traded) monitoring aimed at reducing shirking by employees. The unemployment persists because jobless workers are unable to bid the economy’s uniform wage down to the full-employment level. In this respect, the present efficiency-wage model bears some resemblance to Brecher’s (1974a, b) minimum-wage model of a trading economy. Despite this limited resemblance, the two models differ in at least the following three important respects. First, the real wage remains fixed at an exogenously imposed floor in the minimum-wage case, but varies endogenously in the efficiency-wage case. Second, when there is no monopoly power in trade, an import tariff can raise the minimum-wage country’s welfare only if employment increases at the same time, although a tariff in the efficiency-wage model may (by reducing the costs of monitoring) paradoxically lead to a simultaneous rise in welfare “After setting p= T,n(p,)/T, and p1 =p,, totally differentiate equation dL/dT,=w/FUT,
(12’) to find that (given a constant model

of a closed

190

R.A. Brecher, An efficiency-wage

model

and unemployment. I9 Third a labour subsidy is a first-best policy for a small minimum-wage econoky, but is only part of a second-best policy combination (which includes also taxes on capital, monitoring and consumption) for a small efficiency-wage economy. As these differences suggest, the present model has distinctive implications for comparative statics and optimal policy in an open economy with involuntary unemployment. Although we have focused attention on the small-country case, our analysis could be extended readily to cover the case of a large (or even closed) economy, by using the above results to derive and apply the offer curve of an efficiency-wage country with explicit monitoring. “Another tariff-related difference would arise if w in our 0 function were replaced by u(w,q)= wu( l,q), where u is the indirect utility function corresponding to 0 in footnote 10. With this replacement, the right-hand side of eq. (14) would gain the expression ~wu,/uF,
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