Uncertainty and the trade union

105

Economics Letters 9 (1982) l05- I I I North-Holland Publishing Company

UNCERTAINTY Andrew

Uncertainty predictions derived.

UNION

J. OSWALD

OxfordUniversity, Received

AND THE TRADE

Oxford

23 October

OXI

SUL, UK

1981

is introduced into a model of the trade union. The model’s qualitative turn out to be unaltered. Increases in risk are also examined and various results

1. Introduction The economic analysis of trade unions is once more topical. ’ However, one of the most common criticisms of models of trade union behaviour, that the world is too uncertain for unions to behave in the way economists are wont to assume, has not yet been answered. 2 This paper is an attempt to examine just that - the effects of uncertainty on a model of the trade union. The analysis is based on a mixture of Oswald (1981) and McDonald and Solow (1981). The new element, however, is risk: agents are assumed to have subjective estimates of the probability distributions of unknown variables. Assume that the union is utilitarian and has an objective function, under certainty, of U=v(w+s,w)N+u(b)(M-N),

(1)

where v( w + s, w) is the utility of each employed union member, N is the employment of union members, u(b) is the utility of each unemployed member, and M is the total membership of the union. Let v(..) be ’ See, for example, the forthcoming papers of Dertouzos and Pencavel (198 I), MacDonald and Solow (1981) and Oswald (1981). and the work of Pencavel(1981) and Lazear (1981). ’ Ross (1948) gives the classic statement of the argument. Atherton (1973) and Pencavel (I 98 1) are apparently the only studies to touch on this problem.

0165-1765/82/0000-0000/$02.75

0 1982 North-Holland

increasing in its first argument, decreasing in its second, and strictly concave; let u(.) be increasing and strictly concave; define M?as the union wage, b as the level of government unemployment benefit, s as a government income subsidy and w as a ‘comparison wage rate’. Assume that the union faces an industry or firm which solves the conventional problem m;x PF( N ) - wN,

(2)

where p is the price of output and the production function F(N) is increasing and strictly concave. Assume that the union chooses the wage rate and the firm chooses employment. The union’s problem is then to pick an optimal wage rate, w*, subject to the labour demand constraint N = N(w,p) that is defined by the solution to (2). The result is an optimal wage function w* = w*(b,

s, M, 0,~).

(3)

2. The effect of uncertainty We shall consider five risky separately for simplicity. Proposition I. Uncertainty model.

variables

(b, s, M, w,p),

treating

each

does not affect the qualitative predictions

of the

Proof. The main comparative static predictions from this type of framework are derived, for the case without uncertainty, in Oswald (1981). They are that a rise in income subsidy (s) reduces w*, a rise in unemployment benefit (6) increases IV*, a change in membership (M) has no effect on w*, and an increase in product demand (reflected in a higher price, p) can raise or lower w*. The effect of a higher comparison wage, o, has not been studied in this precise framework before; 3 but it is easy to show that if v_ is non-negative a rise in w will increase the union’s optimal wage. 4 These results also hold under uncertainty about ’ Oswald (1979) considers it in a slightly different model. 4 Let consumption c--w+s. Subscripts denote partial derivatives

throughout

the paper

any of b, s, M, w and p. A full set of proofs will not be presented. One example serves to make the point. Say, for example, that s has a probability density function f(s). Then the union solves maxEU= NJ

{u(w+s,u)N+(M-N)u(b)}f(s)

/

subjectto

ds,

(4)

N=N(w,p).

(5)

Hence it follows, using the second-order condition for the existence maximum, that, where E is the expectations operator,

aw*/ah4 = 0,

of a

(6)

sign( aw*/ab)

= sign --N&(b)

sign(aw*/aw)

= sign{NEU,.,

2 0, + N,Eu,}

(7) 20,

(8)

sign(aw*/ap)=sign{N,Eu,.+N,,[E~(w+s,~)-u(b)]}SO

(9)

and these are, in essence, identical conditions to those obtained for the certainty problem. 5 All that the union has to do is to treat the expected values of v,,, u, and so on as though they were the actual ones. The proofs of the proposition for the case of random b, M, w and p are exactly analogous. They are omitted here.

3. Increases in risk A more interesting question is how the union’s optimal wage is affected by changes in the degree of uncertainty about the unknown variables. 6 Proposition 2. An increase in the riskiness creases the union’s optimal wage.

of unemployment

benefit

de-

5 This ignores one very special case, namely that where t;,, changes sign in such a way that ED,, has the opposite sign from O,W. ’ The definition of ‘increasing risk’ follows Rothschild and Stiglitz (1971).

Proof.

Let b have density

maxEU= W

J

has first-order

function

{v(w+s,w)N+

g(b). Then the solution

(M-N)u(b)}g(b)

to

db

(10)

condition

v,.N+N,[v(w+s,w)-Eu(b)]

=O.

(11)

By ignoring the expectations operator E and applying the main result of Rothschild and Stiglitz (1971) we need only show that XJ/aw = 0 is concave in b. But

&cs,= -u’(b)N,

$(Uw) =-u”(b)N, so the proposition

(12)

20,

GO,

(13)

is established

Proposition 3. If employed men have non-increasing absolute risk aversion a rise in the riskiness of the income subsidy increases the union’s optimal wage. Proof. Again we are interested first-order condition:

&J

=v,,N+v,.N,

$(u.,=v‘Jv+vJv,

in the convexity

or concavity

GO,

(14)

20.

Let A z -v,,/v,. Then, because tion follows immediately.

of the

(15) &4/ac

-=c0 implies vCC,> 0, the proposi-

Proposition 4. A change in the riskiness of union membership on the optimal union wage.

has no effect

Prooj: The proposition follows automatically from the fact that M does not appear in the first-order condition for a maximum.

A.J. O.w~~ld / Uncertainty

and the trude union

109

Proposition 5. The effect on the union’s optimal wage of an increase in the riskiness of product demand is ambiguous. If F(N) = (I /(I - y))N’-Y then, local&, the introduction of this type of uncertainty has no effect. Proof. The effects of product demand uncertainty are complicated: in general the results depend on the fourth derivative of the production function F(N). To prove the second part of the proposition we need to note first of all that $(UW)=v,N,+[v(w+s.w)-u(b)]N&O,

(16)

(17) Substitution

from F(N)

=01

= (l/( 1 - y))N’ -Y eventually gives

at certainty

equilibrium’

(19)

Proposition 6. The effect of the union’s optimal wage of an increase in the riskiness of the comparison wage rate is ambiguous. If the marginal utility of consumption is a convex function of the comparison wage then an increase in the riskiness of the comparison wage raises the union’s optimal wage. Proof

(21) Hence v,,, b 0 ensures the necessary convexity. Because the general case is so complicated it is illuminating to consider some special cases of the

utility

function

U( w + s, w). Three possible

Example I.

o(..) =p(w+s)

Example 2.

v(..)=u(w+s-a).

Example 3.

++(a),

cases are (22) (23) (24)

For the first of these a rise in the riskiness of the comparison wage increases the union’s optimal wage. In the second example the same is true under declining absolute risk aversion. The third is too complicated to give a clear result.

4. Conclusion The purpose of this paper has been to show that uncertainty can be incorporated into a microeconomic model of the trade union. The model’s qualitative predictions are unaltered by this change. It therefore seems hard to justify the argument that the existence of imperfect information destroys the case for an economic theory of the union. Increases in risk have also been studied. One result, Proposition 2, is particularly striking: if uncertainty about the level of unemployment benefit increases (because of unsystematic indexing under inflation, say), the optimal union wage is unambiguously reduced. Some other results are also derived.

References Atherton, W., 1973, The theory of union bargaining goals (Princeton University Press. Princeton. NJ). Dertouzos, J.N. and J.H. Pcncavel. 19XI, Wage and employment determination under trade unionism: The international typographical union. Journal of Political Economy, forthcoming. Lazear, E., 19XI, A competitive theory of monopoly unionism, Mimeo. (Chicago, IL). MacDonald. I.M. and R.M. Solow. 1981. Wage bargaining and employment, American Economic Review, forthcoming. Oswald. A.J., 1979, Wage determination in an economy with many trade unions, Oxford Economic Papers 3 I, 369-3X5. Oswald, A.J.. 19X1, The microeconomic theor)r of the trade union, Economic Journal. forthcoming.

Pencavel, J.H., 1981, The empirical performance of a model of trade union behaviour. Mimeo. (Stanford, CA). Ross, A.M., 1948, Trade union wage policy (University of California Press, Berkeley and Los Angeles, CA). Rothschild, Journal

M. and J.E. Stiglitz. 197 I, Increasing of Economic Theory 3. 66-84.

risk

II: Its economic

consequences,