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Relative prices, aggregate prices, and wage indexation
31
J EC0 BUSN 1992; 44:31-45
Relative Prices, Aggregate Prices, and Wage Indexation John A. Carlson and David W. Findlay
When a firm and workers set contingent wages and employment, potential movements in several different prices enter into consideration. The firm’s revenue depends on the selling price of its product. The workers have concerns about the prices of goods they buy, and the firm’s owners may buy a different bundle of goods. Unless owners acquire their own firm’s output or are fairly risk-averse relative to workers, wages will be indexed primarily to the price index of what workers buy and not to variations in the firm’s selling price. This has been reported to be the typical empirical pattern.
I. Introduction Wage indexation is a contract that can protect to varying degrees the real wage received by workers. It has been suggested not only as a means of protecting the real value of nominal variables from fluctuations in prices but also as a policy tool to insulate the real sector of an economy from economic disturbances. In much of the macro literature, the degree of indexation is obtained by optimizing some social objective function on the macro level. The works of Gray (1976), Fischer (1977a), and Cukierman (1980) show, for example, that full indexation will insulate an economy from purely nominal disturbances. When only real shocks occur, full indexation will exacerbate the effects of the disturbances. In the intermediate case in which real and nominal shocks both occur, optimality at the macroeconomic level requires partial indexation. ’ Similar results hold in open-economy macro models, where both wage indexation and exchange rate intervention are used as policy tools. See, for example, Calmfors and Viotti (1983), Aizenman and Frenkel (1985), and Marston and Tumovsky (1985). More recently, Waller and VanHoose (1989) have examined the optimality of wage indexation in an
Address reprint requests to John A. Carlson, Economics Department, Purdue University, West Lafayette, Indiana, 47907 or David W. Findlay, Economics Department, Colby College, Waterville, Maine 04901. ‘Some authors advocate schemes that involve indexing wages to more than just prices. Paxner (1981) argues that it is better to index nominal wages fully to prices and partially to real income. This provides insulation of the real sector from nominal shocks and minimizes the impact of real shocks. Kami (1983) demonstrates that such a policy will dominate one of indexing only to prices. Earro (1977) considers a contract scheme that will bring about an “ex post” equilibrium in the labor market. As Fischer (1977b) and Cukierman (1980) note, however, such schemes are not, as a rule, found in the “real world.”
Journal of Economics and Business
0 1992 Temple University
01&6195/92/$05.00
J. A. Carlsonand D. W. Findlay
32
economy with and without a balanced budget requirement. They also illustrate that complete indexation is optimal when only nominal disturbances occur. In many of the purely aggregative indexation models, the nominal wage is linked to an economywide price index. For a disaggregated economy, in which output and labor are exchanged locally, VanHoose and Waller (1989) note that contracts will specify that the nominal wage be indexed only to local prices. As the authors (p. 707) note, “workers are concerned with the current cost of living in their own location and not the cost of living in some other location. ” Actual indexation, at least in North America, is not imposed by government rules. It is generally negotiated between a firm and its employees (or a union representing the employees). While all workers may be concerned about their real wages (nominal wages deflated by an index of product prices), the firm’s decisions are based on its own product wage (the nominal wage deflated by the firm’s product price). Algoskoufis (1983) is one author who emphasizes the distinction between real wages and product wages in a business-cycle model. Our concern here will be to examine how firms and workers would choose a wage pattern in response to different variations in aggregate and product prices. By doing so, we can address a “fundamental question” posed by Card (1986, pp. 150- 151): “Why not index-link wages directly to market-specific prices? Although there is some historical precedent for indexing wages to industry selling prices, it remains an interesting puzzle as to the nearly universal practice of escalation to the CPI.” Our analysis will also allow us to discuss several aspects of the indexation puzzle recently noted by Gordon (1990). In particular, Gordon (p. 1140) suggests that conflict between the general market basket to which workers would prefer to index, and the firm-specific variables to which firms would prefer to index may prevent the indexation of nominal wages to market-specific prices (and profits). At the micro level the setting of wages contingent on price outcomes is usually studied within the context of implicit contract models. See Azariadis (1975), Baily (1974), and Gordon (1974) for seminal papers in this area. Risk-neutral firms choose state-contingent contracts to maximize expected profits. These same contracts must also maximize the expected utility of the risk-averse workers. The solution suggests that the contract must offer a wage that is independent of the state. It was not initially clear which wage was being fixed. Phelps (1977, p. 152) for one asks, “Is this ‘wage,’ which is state-invariant during the contract, the money wage or the real wage?” Azariadis and Stiglitz (1983) make clear that the real wage is what is fixed in these early implicit contract models. Hart (1983) reviews extensions when there is asymmetric information. The key difference between the analysis that we undertake in this article and previous implicit contract models is that the problem will be set up in terms of different prices that are relevant for different decision makers. There is the selling price for the firm, the price of the bundle of goods bought by workers, and the price of the bundle of goods bought by the owners of the firm with the firm’s profits2 The objectives of this article are similar to the goals of the two-sector, macroeconomic-based, general
‘This is a point suggested implicitly by Hart (1983, p. 15): “So far, we have expressed everything in have assumed that there is a single good in the economy). In reality, of course, firms and workers are interested in the goods money can buy.” While he does not say so explicitly, the bundle of goods bought by the owners of the firm may differ from the bundle of goods bought by the workers.
moneytears (or we
WageIndexation
33
equilibrium explanation provided by Duca and VanHoose (1991). In one sector, wages are specified by contract. In the other sector, wages are determined by Walrasian market clearing. Workers consume output from both sectors; consequently, labor supply is a function of the real wage, which is determined by the sector-specific nominal wage and the economywide consumer price index. Labor demand in each sector is determined by the real (or product) wage, which is a function of the sector-specific price. The authors show that wages will generally be indexed to both economywide and sectoral prices (profits). Duca and VanHoose note that this type of contract “is equivalent to combinations of partial indexation of wages to unanticipated economy-wide inflation and sector-specific profits.” In what follows we consider the possibility of indexing wages not only to the firm’s selling price and to an index of prices of goods purchased by workers but also to an index of prices of goods purchased by the owners of the firm. Section II presents and discusses the first-order conditions of a contract model that can be used to address the following question: If firms and workers were to choose a contingent wage and employment contract, how would the choice be influenced by various relative price outcomes that might occur. The structure of the model is similar to the one presented by Azariadis (1975). Section III then looks at specific cases to help understand when the various prices enter the indexation rule within this setting. Section IV contains concluding comments.
II. A Contract Model with Three Prices Imagine that a group of firms and a group of workers in a labor market are looking ahead to a period in the future, which can be characterized by a set of possible price outcomes. Each such outcome will be denoted as a state s. Let there be S such possible states. Workers and firms have the same perception about the probability that state s will occur. Let this probability be denoted T(S) > 0, and a(l) + . . . + a(S) = 1. Consider the problem of a firm i, which has to announce in advance how many workers it wants to hire and what its wage and employment plans will be once it is known (to both the firm and its employees) that state s has occurred. Assume that each worker is either employed for a fixed number of hours or is unemployed.3 When state s occurs, the firm’s selling price will be Pi
3Hansen (1985, p. 324) argues that “most of the variability in total hours is unambiguously due to fluctuations in the number of employed tither than hours per employed worker.” He then shows how a model that acxumes fixed hours is able to “improve our ability to account for U. S. aggxgate time series data.” This suggests that rigidity in hours per employed worker is not an implausible assumption.
J. A. Carlson and D. W. Findlay
34
directly by the firm. Assume that the utility function U is such that U’ > 0 and U” < 0. The expected utility of a worker who agrees to be available to firm i is
If a worker’s best alternative employment elsewhere provides an expected utility of U* (which of course depends on overall market conditions), the firm must offer a contract such that ECJ is at least as great as U*. We assume that the firm can obtain all the labor it needs by making EU = U*. Since it is costly for the firm to provide any higher utility, it will chose a combination of n(s), w(s), c(s), s = 1, . . . , S, such that EU= U*. The firm’s output is solely a function of labor input. Let f( n( s)) represent the firms output when its labor input is n(s) in state s and let this production function satisfy the following conditions: f’ > 0, f” < 0, f’(0) = 00 and f’(m) = 0. The firm’s nominal profits in state s are represented by the expression
In terms of what the owners wish to buy, these profits in real terms will be denoted by Yts) =
{Pi(s)f[n(s)]
-
Wi(s)n(s)
-
Ci[n
-
n(s)]}lPf(s)*
The utility for the firm in state s will be represented by V( y(s)), V” I 0. If the firm is risk-neutral, then let V( y(s)) = y(s)
(2)
with V’ > 0 and so that V’ = 1
and I’” = 0. The mathematical specification of the first-order conditions is reported in Table 1. The firm chooses a contingent set of wages and employment levels to maximize the expected utility of its real profits subject to the constraints that employment in any state be less than or equal to the firm’s labor force (x(s) is a Kuhn-Tucker Lagrangian multiplier) and that the firm be competitive in the labor market by providing a utility to workers of U* (p is a Lagrangian multiplier). The state S is designated as the one in which none of the firm’s workers are laid off so that n = n(S) is the firm’s desired labor force. The first-order conditions (4a) through (4e) are obtained by taking the partial derivatives of J with respect to the Wi(s), Ci(S), n(s), n(S), and p. From (4f) and (4g), when n(s) < n(S), x(s) = 0; and when n(s) = n(S), x(s) may take on positive values to satisfy (4c) and (4d). Before proceeding to an analysis of the wage indexation results of the model, we need to discuss and interpret some of the implications of the first-order conditions. For each s, condition (4a) can be written in the following form: I++))
= V’(Y@))[ ~&)/~_&)I
[@)/PI
*
(44
Equation (4a’) indicates that the marginal utility of the workers’ real wage is proportional to the marginal utility of the firm’s real profits, adjusted for the ratio of the price index for workers’ purchases to the price index for owners’ purchases, and is equal across all states, since p/n(S) is the same for all s. An increase in Wi(S) lowers the firm’s expected profits and raises the workers’ expected utility. The higher utility of workers in state s allows the firm to offer lower wages, and obtain higher profits, in another state. At the margin no gains to the firm are possible by reallocating wages across states.
35
Wage Indexation Similarly,
condition
U’(k + c(s))
(4b) can be rewritten
= q.+))[
as
%(~)Pf(~)l
sothat
the same ratio of marginal unemployed if the firm’s contribution
w
[~WlPl~
utilities holds whether workers are employed or to its unemployed workers is part of the contract.
Table 1. The Langrangian Function and First-Order Conditions S-l
J[ q(s), G(s), n(s)* X(s), P] = s&lW[ 4s) - 441
+ cc ~~lT(S){[no/n(S)l~(~i(s)lP,(s))
[
+[I - n(s)ln(S)]U(k +
ci(s)lpw(s)))
-
1(3)
u*
The first-order conditions (setting equal to zero the partial derivatives of J with respect to the choice variables) are
W,ts): T(s)n(s){ - r(Y(s))lpf(s) +/AV(W(S))/[P,(s)n(S)]} ci(s):
X(s)[ n(s) +pU’(k
n(s):
+ C(S))/[ P,(s)n(S)]}
= 0; s = 1,. . . , s.
?F(S)v(Y(s))[Pi(s)f’(n(s))
- W;:(“) + ci(s)]lpf(s)
- U(k + c(s))]/n(S)
a(S) v’( Y(s)) [ Pi(S)f’(
n(s))
2
0;
s=
l,...,S-
= 0;
- wiCs)] IQ(s)
S-l
n(s) - n(s)
(44
- n(s)] { - y’(Y(s))lpf(s)
+ p*(s)[u(w(s))
n(s):
= 0; s = l,...,S.
1.
WI - x(s)
s = 1,. . . , s - 1.
(44
J. A. Carlson and D. W. Findlay
36 Table 1. (continued)
h(s)[n(S)-n(s)]
=o;
s=l,...,S-1.
@g)
In each state, w(s) = W,(s)/P,,,(s) is the worker’s real wage; c(s) = C,(S)/P,(s)
is the real value of the worker’s unemployment compensationpaid by the
fum; and y(S) = [ Pi(s)f(n(s))
- Wi(s)n(s)
- Ci(s)(n(S)
- n(.V))]/p/(s)
is the
firm’s
real profits.
It can be seen from (4a’) and (4b’) that
U’(k + c(s)) = V(w(s)). In this case k + c(s) = w(s) in each state s and the worker will be indifferent between working and being on layoff. This result implies that the firm, regardless of its degree of risk aversion, will offer workers full insurance within each state. This result also implies that, when unemployment compensation is part of the contract, any influence of particular prices on the nominal wage in the cases to be considered will have a similar influence on the nominal payment by the firm to its unemployed workers. If payments by the firm to the unemployed are determined exogenously and are not part of the contract, equation (4b? can be ignored. If the firm is risk-neutral, so that v’ = 1, tbe indexation rule will not be dependent on any of the first-order conditions other than (4a). In that case, (4a’) becomes
While relative price variations across states may influence employment n(s) and real profits u(s), the assumption of firm risk neutrality will not let those variations influence the real wage solution from equation (4a”). The first three cases analyzed in the next section will assume risk-neutral firms. If the firm is risk-averse, then the firm’s real profits do enter into the indexing rule. To the extent that real profits vary with employment, it also becomes necessary to consider other first-order conditions. Condition (4c) introduces effects of varying employment in each state s # S. If n(s) < n(S), then from (4g) x(s) = 0. Furthermore, when compensation for the unemployed is part of the contract, w(s) = k + c(s) and the last term in (4~) drops out. As a result, condition (4~) reduces to
Pj(S)f’(n(S)) = Wi(S) - C,(S)*
(44
37
Wage Indexation
The left side of (4~‘) is the marginal benefit to the firm from employing an additional worker in state s. The right side is the marginal cost, which is the wage the firm pays minus the cost of compensating an unemployed worker in state s. If C!,.(s)is not part of the contract and is set equal to zero, then condition (4~) can be represented by the following equation:
q(s).r(+))
= wits) - ~,(S)(ccl~(S))[~(W(S)) - ww
v’(.ds)). (44
As in (4c’), the marginal cost of increasing employment in state s is less than the wage the firm pays, because higher employment also reduces the risk to a worker of being unemployed in state s and receiving no compensation from the firm. The shadow value to the firm arises because the higher utility to the workers allows the firm, ceteris paribus, to pay a lower average wage. Equation (4d), the first-order condition with respect to n(S), can be equivalently interpreted as equating the marginal benefit and marginal cost of adding an additional worker to the total number of workers under contract. Given the aggregate solution for n(S) and p, equations (4a’), (2), and (4~“) simultaneously determine variations in w(s), n(s), and u(s) between states in which not all workers under contract will be fully employed.
III. Wage Indexation with Different Relative Prices We shall now concentrate on interpreting the conditions represented by equations (4a’). These can be written as follows:
~wwfw)
= (Pw(s)lPf(s))V’(y(s))(n(S)/~);s =
1, - *a,
s. (5)
To help understand this equation and to answer the question about when firms would want to index to local selling prices, we analyze a series of special cases of equation (5). When the firm is risk-averse, we will have to consider the effects of changes in prices on the firm’s profits.
Case 1 Assume a risk-neutral firm (V’ = 1 and V” = 0) and P,(s) = P,,,(s). If the firm is risk-neutral and if owners of the firm buy the same bundle of goods as workers so that nominal profits are deflated by the same price index as workers, then (5) reduces to
(6) Whatever the solutions for n(S) and p, the marginal utility of the real wage must be the same in all states. Since U’ is strictly decreasing (U” < 0), there is a unique real wage that must hold in each state. Denote this by w. With the real wage constant across all states, the nominal wage is fully indexed to variations in the price of the bundle of goods bought by workers. This is the standard result for risk-neutral firms with risk-averse workers. By reducing the workers’ real-wage variations the firm is able to
J.
38
A. Carlson and D. W. Findlay
decrease the real wage and hence increase its own expected profits. The firm fully insures the workers against fluctuations in the purchasing power of their wages. Note in this case that the nominal wage does not respond to any variations in the price of the firm’s product. To change wages as selling prices change would introduce variations in workers’ real wages. Because of workers’ risk averseness, they would need to be compensated by a higher real wage. This would lower the firm’s expected profits with no offsetting gain to the firm. Recall for this case (and the following two cases) that, because of the optimal risk-sharing properties of the model, differences in employment (and profits) across states do not affect the wage decision of the firm (given by equation (4a’)). Case 2 Assume a risk-neutral firm ( V’ = 1 and I/” = 0) and Pr(s) = 1. Now suppose the risk-neutral firm is concerned with its accumulation of wealth. This would be the case in which owners of the firms wish to save any profits for future consumption, and commodities are not a convenient way of storing wealth. We do not deal with what financial instruments are available but suppose that they are denominated in nominal money units. If the firm’s current objective is in terms of nominal profits to maximize the expected investment in financial markets, then equation (5) becomes V( K(s)lP~(S))
= ‘wtS)( ‘CS)lP) ;
s=
l,...,S.
(7)
A higher P,,,(S) raises the right side of (7) and hence calls for a higher U’, which occurs when w(s) = Wi(s)/P,,,(s) is lower. Therefore when the aggregate price is higher, the real wage is lower. Indexation is less than full, but what indexation there is depends again only on the price of the bundle of goods bought by workers and not on the local product price. Basically what is happening here can be understood by supposing initially that real wages were equal across states. Expected nominal profits could then be increased more by lowering w(s) when PJ S) is high than would be lost by raising w(s) when P,,,(S) is low. The extent of the gain to the firm from variations in real wages between states is limited by the concavity of the workers’ utility function. Assuming that workers’ relative risk aversion is greater than 1 so that nominal wages would be adjusted in the same direction as any variation in the workers’ price index,4
4Consider a utility function with constant relative risk aversion
U(W/P)
= a-
b(W/P)-R+’
so that from (7) b(R
- l)VR
= (n/p)P-R+‘.
To find the effect on W of an arbitrarily small change in P in going from one state toward another, differentiate this expression and collect terms with the result that dW/ W = [(R - 1)/R]
dP/P.
Thus, with R > 1, W moves with P but in less than full proportion.
39
Wage Indexation
the contract could be specified in terms of guaranteeing the nominal wage that gives the highest real wage (for the lowest possible value of P,,,(s)) and then specifying a partial indexing rule in accordance with equation (7). Case 3 Assume a risk-neutral firm (I/’ = 1 and V” = 0) and Pf(s) = Pi(s). Now suppose that the firm wishes to acquire its own goods. It is not clear that many firms would want to do this. If firms are so diversified as to produce a representative bundle of goods or there is a one-commodity model, there would be little point in making the distinction between the product price and an aggregate price. One possible situation in which this case could be relevant is a foreign-owned firm that is in business to extract a product from its operations in the domestic economy. Whatever the rationale, this case does result in some indexing to product prices with risk-neutral firms. With V’ = 1 and P,(S) = Pi(S), equation (5) becomes U’(y(S)/Pw(s))
= (P,(s)/Pi(s))(n(S)/~);
s=
l,***,s*
(8)
As in case 2 there is less than full indexation to variations in the aggregate price, P,,,(s), ceteris paribus. What is different about this case is that there is now some indexation to the local price to the extent that it does not move with the aggregate price. Ceteris paribus, at higher Pi(s), U’ must be lower, which calls for a higher w(s). Hence, if the firm is trying to maximiz e its product profits, then real wages will be higher when local prices are higher relative to the aggregate price. In this case, where the firm’s objective is intluenced by the local price, the firm has something to gain by adjusting wages to local prices and will accept a somewhat higher real wage to take advantage of variations in relative prices. Case 4 Assume a risk-averse firm ( V’ > 0 and V” < 0), Pf(s) = P,,,(s), and no variations in employment across states. Now consider the situation when the same deflator is used for wages and for profits, but the firm is risk-averse. The firm’s real price, a(s) = Pi(S)/P,(S), in this specification will play a role similar to a productivity shift term in a model specified by Simonsen (1983). Equation (5) now takes the form
(9) With a risk-averse firm, variations in real profits y(s) influence the real wage, which in turn affects the firm’s real profits. This feedback needs to be explored. Suppose states are ordered from the lowest to the highest relative price p(s) = Pi(S)/P,(S). Then, using (9) and comparing adjacent states (for s = 1, . . . , S - 1, we have
[qw+dw) -
v(w)]/u’(w) = [ V(Y + dY) -
qY)]/v'(Y),
where dw = w(s + 1) - w(s) and dy = y(s + 1) - y(s). Define r,,. = -[V(w
+ dw) - V(w)]/[U’(w)dw]
(10)
J. A. Carlson and D. W. Findlay
40
and ‘f = - [
V’(Y+ dY)-
qY>]l[
fqY)dY]
as the absolute risk aversion of workers and the firm, respectively. These are discrete state counterparts to - LJ”/ II’ and - V”/ V’, the usual measures of absolute risk aversion at particular values of w and y. It then follows from these definitions and from (10) that
01)
dw = (ff/fw)dY.
Equation (11) indicates that when the firm is risk-averse, it chooses to engage in some profit sharing. When real profits are high, the firm will also raise the real wage of its workers. The analysis, however, is incomplete at this point, because there is a feedback from higher wages to the firm’s real profits. Under the assumption that Pr(s) = P,(s), equation (2) can be rewritten: Y(S)
=p(s)f(n(s)) - J+)n(s) - cW[nW -
n(s)].
(12)
For a given total employment n(S), variations in y(s) across states depend on variations in the firm’s real price p(s), real wage w(s), real compensation of its unemployed workers c(s), and its employment n(s). If we subtract y(s) from y(s + 1) and collect terms, the firm’s real profits can be seen to vary between states according to the following formula: dy = [p(s
+ l)(df/dn)
- n(s)(dw
- w(s + 1) + c(s + l)]dn
- dc) - n(S)dc,
+f(n(s))dp (13)
where dp = p(s + 1) - p(s), dc = c(s + 1) - c(s), dn = n(s + 1) - n(s), and df = f( n( s + 1)) - f( n( s)). Note that df /dn, like f’, is a marginal increase in output per unit increase in labor input, except that f’ is the effect of marginal adjustments within a particular state, whereas df /dn is the increase in output relative to the increase in employment in going from one state to the next better state in terms of the firm’s real price. We need to distinguish two situations. In this case, we consider the nature of the feedback when there are no variations in employment across states. In the next case, we will analyze the effects of variations in employment. MacDonald (1986) has argued within a two-state version of this model that relative price variations would have to be extremely large in order for the firm to plan any variations in employment across states. If price variations are not sufficiently large, then dn = 0. That will be assumed throughout this case. When dn = 0, equation (13) reduces to essentially the same formula whether or not unemployment compensation is part of the contract. The formula can be written dy = fdp - ndw,
(13’)
41
Wage Indexation
where f = f(n(s)). If unemployment compensation is not part of the contract, dc = 0 and n = n(s) in (13’). If unemployment compensation is part of the contract, then dc = dw and n = n(S) in (13). This shows the nature of the feedback from wages to profits when there are no variations in employment. If the firm’s real price increases, its real profits rise. When it chooses to engage in profit sharing as in equation (1 l), the rise in real wages reduces the firm’s real profits. This real-wage feedback, however, moderates rather than reverses the tendency for the firm to index wages to its product price. Formally, if dy from (13’) is substituted into (1 l), the resulting formula is dwldp
= (r,lr,)fl(l
+ (+w)n)
> 0.
(14)
The fact that dw / dp is positive means that the firm increases the real wage that it pays when its real product price is higher. Case 5 Assume a risk-averse firm (V’ > 0 and V” < 0), Pr(s) = P,,,(s), and employment varies across states. When the contract calls for variations in employment, the dn tern in (13) can no longer be ignored. Assume for now that unemployment compensation is part of the contract, In this case, equation (13) becomes dy = [ p(s + l)(df/dn) +f(n(s))dp
- w(s + 1) + c(s + l)]dn
- n(S)dw.
(13”)
When employment varies between states, we need to consider the equation corresponding to the optimal choice for contingent employment in each state. Equation (4~) discussed above comes from those first-order conditions. With n(s) c n(S), we have x(s) = 0. When unemployment compensation is part of the contract, U( w(s)) = U( k + c(s)) and the last term in (4~‘) drops out. Dividing by P,(s) yields &)Y(+))
= w(s) - c(s).
(15)
Note that if there were constant marginal product of labor between states s and s + 1, df /dn would equal f’( n(s)), the bracketed term in (13’9 would equal zero, and case 5 would reduce to case 4. With diminishing marginal product of labor, df /dn is larger than f’(n(s + l)), and the bracketed term in (13”) is positive. The expansion of employment adds to the firm’s real profits. Since w(s) - c(s) = k for all states when the firm’s payment of unemployment compensation is part of the contract, it must be true from equation (15) that a higher real price calls for higher employment. Suppose p were higher. If pf’ is constant, then $ must be lower and with diminishing product of labor this occurs with higher employment. Thus, we can write dn = b(s)dp,
(16)
where b is a positive number that depends on the curvature of the production function between states s and s + 1.
42
J. A. Carlson and D. W. Findlay
We can use equations (ll), (139, and (16) to eliminate dn and dy to get a relationship (i.e., indexation rule) between dw and dp:
dwldp =
<+w>[f+ b(p(dfldn) -
w+
c)]/(l + (Q/+)
’
0.
(17)
In comparing (17) with (14), when employment did not vary between states, we can see that they differ only in that (17) has the term b( p(df /dn) - w + c) in the numerator. With the value of the incremental output exceeding the net cost of employing the extra labor, the firm’s real profits tend to increase between states more than if employment does not change, and this increment in real profits is shared with workers when the firm is risk-averse (r, > 0). If unemployment compensation is not part of the contract, the last term in (4c”) plays precisely the same role as c(s) does when unemployment compensation is part of the contract. It represents an implicit cost of an unemployed worker, and that cost is not incurred at the margin when employment is higher. The algebra, however, becomes much messier and is not worth reproducing here. Case 6 Assume a risk-averse firm ( v’ > 0, V” < 0), and that P’(S) is not always equal to PJs). We can rewrite (5) once more as
we))
= (n(s)/r)r(S)~‘“[f(S)X(S)l)
(18)
where r(s) = P,,,(s)/PJs) is the ratio of the price index for the bundle of goods bought by the workers to the price index for the bundle of goods bought by the owners, and X(S) is nominal profits deflated by the price index for the bundle of goods bought by the workers. In the prior two cases we have assumed that r(s) equaled unity in every state. Now consider what equation (18) says about indexing wages to variations in r(s) for given values of x(s). The derivative of the right side of (19) with respect to f(s) will be positive if
r-XV”(m) + V’(m) > 0, and this holds if the firm’s coefficient of relative risk aversion is less than unity, that is, R= -yV”( y)/ V’(y) < 1. In other words, with R < 1, when a state occurs in which P/(s) falls relative to P,,,(S) so that r(s) is higher, the right side of (18) is increased. Equation (18) then calls for U’ to be higher, which means, with diminishing marginal utility, that real wages should be lower. Thus, if it could be part of the contract, the firm would want to some extent to index wages to the price index of the goods that the profits will be used to buy. If the firm has a relative risk averseness that is greater than unity, the argument is reversed. In states in which P,(S) falls relative to P,,,(s), the firm will want to increase the real wage. A peculiar possibility arises in this formulation if the firm’s owners have a risk averseness greater than unity and they want to take their profits in terms of the firm’s product so that Pr(s) equals the product price Pi(s). Then, ceteris paribus, the rule calls for a decrease in wages relative to the aggregate price index for workers
43
Wage Indexation
when the firm’s product price is higher relative to the aggregate price. Try selling that in wage negotiations! Whatever the degree of firm risk averseness, the difficulty in explaining the rationale for an indexation rule that includes the specification of a price index for a bundle of goods other than goods that workers buy would probably outweigh any gain in utility to the owners from including that in the contract. Thus, while variations in Pr(s) call for some optimal variations in real wages when firms are risk-averse, practical considerations may rule them out in actual contracts. Because of the peculiar implications of this example and the possible ambiguity given certain values of R, we will not proceed further in this case to examine the feedback effects of the above indexation rule.
IV. Concluding Comments Equation (11) shows the extent to which a risk-averse firm’s real wages move with its real profits. Shave11 (1976) has developed a similar formula. State-contingent real incomes are given exogenously and one party, say V, chooses a payment to a second party U to maximize V’s utility subject to holding U’s utility constant. In the case in which both parties have the same perceptions about the probability of each state occurring and U’s outside income does not vary, Shavell’s result can be written
w’(s) = y’(s)[ q/c’, +
L>l .
Our equation (11) would look the same if Y’ = ~JJ + dw denoted the total revenue that the firm had to divide between itself and the workers in going from one state to another and dw = w’ were the change in the total wage bill. Blinder (1977, p. 76) utilizes Shavell’s formula to make the following suggestion: “Firms which tend to do well in inflation [ Y’(s) > 0] will be eager to give generous escalators as a form of insurance to workers-in return for lower mean wages. Conversely, and for the same reasons, firms which tend to fare poorly in inflation [ Y’(s) c 0] will be reluctant to escalate wages very much, even though that implies higher mean wage costs.” Blinder is implicitly assuming that what indexation there is will be to changes in the aggregate price. If a firm’s price moves with the aggregate price so that its real price does not change, then such firms even if they are highly risk-averse will keep real wages constant. Our analysis introduces different prices explicitly. The results indicate that risk-averse firms will tend to increase real wages when their own real prices (and hence real profits) are higher. The extent of such wage adjustments, as we noted above, depends on risk aversion of the firm relative to that of workers. Weitzman (1985) has stressed the desirable implications of greater profit sharing. The fact that it is not more widespread suggests, within the structure of our model, that firms are not very risk-averse relative to workers. Our analysis complements that of Duca and VanHoose (1991). In a multisector, risk-neutral framework, they show that the degrees of wage indexation to an economywide price index and to profits will depend on the labor demand and supply elasticities. Our analysis indicates that owners’ objectives and particularly their degree of risk averseness can also influence the degrees of indexation to an aggregate price index and to firm’s profits. Now consider Card’s “puzzle as to the nearly universal practice of escalation to the CPI. ” If workers are very risk-averse relative to firms (r, / rw close to zero) and if
44
J. A. Carlsoa and D. W. Findlay
firms are not concerned with acquiring their own product, then our analysis provides a ready answer to the puzzle. The only gain to the firm from reducing workers’ risk is in moderating fluctuations in the workers’ real wage and hence indexing primarily to the price of the bundle of goods that workers buy. Note also that our result does not require asymmetric information. It may be that firms have better information than workers about product prices and that this could contribute to indexation to a well-publicized aggregate price index, such as the CPI, but this is not a necessary rationale. Even with full symmetric information about all prices, the optimal contract in cases 1 and 2 calIs for indexation only to the aggregate price index for what workers buy. There is therefore no incentive for workers to develop ways of getting full information about their firm’s selling prices.
We are grateful to David D.
VanHoose
for helpful suggestions on au earlier draft.
References Aizenman, J., and Frenkel, J. A. (June 1985) Gptirnal wage indexatioa, foreign exchange intervention, and monetary policy. American Economic Review, 75: 402-423. Algoskoufis, G. S. (1983) The labor market in an equilibrium business cycle model. Journal of Monetary Economics 11: 117-128. Azariadis, C. (December 1975) Implicit contracts and underemployment equilibria. Political Economy 83: 1183-1202.
Journal of
Azariadis, C., and Stiglitz, J. E. (1983) Implicit contracts aad fixed price equilibria. Quarterly Journal of Economics 98(Suppl.): l-22. Baily, M. N. (January 1974) Wages and employment under uncertain demand. Review of Economic Studies 41: 37-50. Barro, R. J. (July 1977) Long-term contracting, sticky prices, and monetary policy. Journal of Monetary Economics 3: 305-316. Blinder, A. S. (1977) Indexing the economy through financial intermediation. Journal of Monetary Economics S(Supp1.): 69-105. Cahnfors, L., and Viotti, S. (November 1983) Wage indexation, the Scandinavian model and macroeconomic stability in the open economy. Oxford Economic Papers 34: 547-566. Card, D. (June 1986) An empirical model of wage indexation provisions in union contracts. Journal of Political Economy 94: 144-175. Cukierman, A. (1980) The effects of wage indexation on macroeconomic fluctuations. Journal of Monetary Economics 6: 147- 175. Duca, J. V., and VanHoose, D. D. (November 1991) Optimal wage indexatioa in a multisector economy. International Economic Review 32(4):859-868. Fischer, S. (1977a) Wage indexation and macroeconomic stability. Journal of Monetary Economics S(Supp1.): 107-147. Fischer, S. (February 1977b) Long-term contracts, rational expectations, and the optimal money supply rule. Journal of Political Economy 85: 191-205. Gordon, D. F. (1974) A neoclassical theory of Keynesian unemployment. Economic Inquiry 12: 431-459. Gordon, R. J. (September 1990) What is new-Keynesian economics? Journal of Economic Literature 28: 1115-1171. Gray, J. A. (April 1976) Wage indexation: A macroeconomic approach. Journal of Monetary Economics 2: 22 l-235. Hart, 0. D. (January 1983) Gptimal labour contracts under asymmetric information: An introduction. Review of Economic Studies 50:3-35. Hansen, G. D. (November 1985) Indivisible labor and the business cycle. Journal of Monetary Economics 16: 309-327.
Wage Indexation
45
Karni, E. (April 1983) On optimal wage indexation. Journal of Political Economy 91: 282-292. MacDonald, I. M. (January 1986) Can the risk-shifting employment model explain fluctuating employment. Economic Inquiry 24: 25-41. Marston, R. C., and Tumovsky, S. J. (November 1985) Macroeconomic stabilization through taxation and indexation: The use of firm-specific information. Journal of Monetary Economics 16: 375-395. Paxner, E. A. (1981) On indexation and macroeconomic stability. In Development in an Inflationary World (M. J. Flanders and A. Raxin, eds.). New York: Academic Press. Phelps, E. S. (1977) Indexation issues: A comment on the Blinder and Fischer papers. Journal of Monetary Economics S(Supp1.): 149-159. Shave& S. (February 1976) Sharing risks of deferred payment. Journal of Political Economy 84: 161-168. Simonsen, M. H. (1983) Indexation: Current theory and the Brazilian experience. In Inflation, Debt and Zndexation (R. Dombusch and M. H. Simonsen, eds.). Cambridge, MA: MIT Press, pp. 99-132. VanHoose, D. D., and Waller, C. J. (October 1989) Islands, indexation and monetary policy. Economic Inquiry 27: 705-717. Waller, C. J., and VanHoose, D. D. (February 1989) Endogenous wage indexation and optimal monetary policy with and without a balanced budget. Journal of Economics and Business 41: 21-31. Weitxman, M. L. @ecember 1985) The simple macroeconomics of profit sharing. American Economic Review 75: 937-953.
J EC0 BUSN 1992; 44:31-45
Relative Prices, Aggregate Prices, and Wage Indexation John A. Carlson and David W. Findlay
When a firm and workers set contingent wages and employment, potential movements in several different prices enter into consideration. The firm’s revenue depends on the selling price of its product. The workers have concerns about the prices of goods they buy, and the firm’s owners may buy a different bundle of goods. Unless owners acquire their own firm’s output or are fairly risk-averse relative to workers, wages will be indexed primarily to the price index of what workers buy and not to variations in the firm’s selling price. This has been reported to be the typical empirical pattern.
I. Introduction Wage indexation is a contract that can protect to varying degrees the real wage received by workers. It has been suggested not only as a means of protecting the real value of nominal variables from fluctuations in prices but also as a policy tool to insulate the real sector of an economy from economic disturbances. In much of the macro literature, the degree of indexation is obtained by optimizing some social objective function on the macro level. The works of Gray (1976), Fischer (1977a), and Cukierman (1980) show, for example, that full indexation will insulate an economy from purely nominal disturbances. When only real shocks occur, full indexation will exacerbate the effects of the disturbances. In the intermediate case in which real and nominal shocks both occur, optimality at the macroeconomic level requires partial indexation. ’ Similar results hold in open-economy macro models, where both wage indexation and exchange rate intervention are used as policy tools. See, for example, Calmfors and Viotti (1983), Aizenman and Frenkel (1985), and Marston and Tumovsky (1985). More recently, Waller and VanHoose (1989) have examined the optimality of wage indexation in an
Address reprint requests to John A. Carlson, Economics Department, Purdue University, West Lafayette, Indiana, 47907 or David W. Findlay, Economics Department, Colby College, Waterville, Maine 04901. ‘Some authors advocate schemes that involve indexing wages to more than just prices. Paxner (1981) argues that it is better to index nominal wages fully to prices and partially to real income. This provides insulation of the real sector from nominal shocks and minimizes the impact of real shocks. Kami (1983) demonstrates that such a policy will dominate one of indexing only to prices. Earro (1977) considers a contract scheme that will bring about an “ex post” equilibrium in the labor market. As Fischer (1977b) and Cukierman (1980) note, however, such schemes are not, as a rule, found in the “real world.”
Journal of Economics and Business
0 1992 Temple University
01&6195/92/$05.00
J. A. Carlsonand D. W. Findlay
32
economy with and without a balanced budget requirement. They also illustrate that complete indexation is optimal when only nominal disturbances occur. In many of the purely aggregative indexation models, the nominal wage is linked to an economywide price index. For a disaggregated economy, in which output and labor are exchanged locally, VanHoose and Waller (1989) note that contracts will specify that the nominal wage be indexed only to local prices. As the authors (p. 707) note, “workers are concerned with the current cost of living in their own location and not the cost of living in some other location. ” Actual indexation, at least in North America, is not imposed by government rules. It is generally negotiated between a firm and its employees (or a union representing the employees). While all workers may be concerned about their real wages (nominal wages deflated by an index of product prices), the firm’s decisions are based on its own product wage (the nominal wage deflated by the firm’s product price). Algoskoufis (1983) is one author who emphasizes the distinction between real wages and product wages in a business-cycle model. Our concern here will be to examine how firms and workers would choose a wage pattern in response to different variations in aggregate and product prices. By doing so, we can address a “fundamental question” posed by Card (1986, pp. 150- 151): “Why not index-link wages directly to market-specific prices? Although there is some historical precedent for indexing wages to industry selling prices, it remains an interesting puzzle as to the nearly universal practice of escalation to the CPI.” Our analysis will also allow us to discuss several aspects of the indexation puzzle recently noted by Gordon (1990). In particular, Gordon (p. 1140) suggests that conflict between the general market basket to which workers would prefer to index, and the firm-specific variables to which firms would prefer to index may prevent the indexation of nominal wages to market-specific prices (and profits). At the micro level the setting of wages contingent on price outcomes is usually studied within the context of implicit contract models. See Azariadis (1975), Baily (1974), and Gordon (1974) for seminal papers in this area. Risk-neutral firms choose state-contingent contracts to maximize expected profits. These same contracts must also maximize the expected utility of the risk-averse workers. The solution suggests that the contract must offer a wage that is independent of the state. It was not initially clear which wage was being fixed. Phelps (1977, p. 152) for one asks, “Is this ‘wage,’ which is state-invariant during the contract, the money wage or the real wage?” Azariadis and Stiglitz (1983) make clear that the real wage is what is fixed in these early implicit contract models. Hart (1983) reviews extensions when there is asymmetric information. The key difference between the analysis that we undertake in this article and previous implicit contract models is that the problem will be set up in terms of different prices that are relevant for different decision makers. There is the selling price for the firm, the price of the bundle of goods bought by workers, and the price of the bundle of goods bought by the owners of the firm with the firm’s profits2 The objectives of this article are similar to the goals of the two-sector, macroeconomic-based, general
‘This is a point suggested implicitly by Hart (1983, p. 15): “So far, we have expressed everything in have assumed that there is a single good in the economy). In reality, of course, firms and workers are interested in the goods money can buy.” While he does not say so explicitly, the bundle of goods bought by the owners of the firm may differ from the bundle of goods bought by the workers.
moneytears (or we
WageIndexation
33
equilibrium explanation provided by Duca and VanHoose (1991). In one sector, wages are specified by contract. In the other sector, wages are determined by Walrasian market clearing. Workers consume output from both sectors; consequently, labor supply is a function of the real wage, which is determined by the sector-specific nominal wage and the economywide consumer price index. Labor demand in each sector is determined by the real (or product) wage, which is a function of the sector-specific price. The authors show that wages will generally be indexed to both economywide and sectoral prices (profits). Duca and VanHoose note that this type of contract “is equivalent to combinations of partial indexation of wages to unanticipated economy-wide inflation and sector-specific profits.” In what follows we consider the possibility of indexing wages not only to the firm’s selling price and to an index of prices of goods purchased by workers but also to an index of prices of goods purchased by the owners of the firm. Section II presents and discusses the first-order conditions of a contract model that can be used to address the following question: If firms and workers were to choose a contingent wage and employment contract, how would the choice be influenced by various relative price outcomes that might occur. The structure of the model is similar to the one presented by Azariadis (1975). Section III then looks at specific cases to help understand when the various prices enter the indexation rule within this setting. Section IV contains concluding comments.
II. A Contract Model with Three Prices Imagine that a group of firms and a group of workers in a labor market are looking ahead to a period in the future, which can be characterized by a set of possible price outcomes. Each such outcome will be denoted as a state s. Let there be S such possible states. Workers and firms have the same perception about the probability that state s will occur. Let this probability be denoted T(S) > 0, and a(l) + . . . + a(S) = 1. Consider the problem of a firm i, which has to announce in advance how many workers it wants to hire and what its wage and employment plans will be once it is known (to both the firm and its employees) that state s has occurred. Assume that each worker is either employed for a fixed number of hours or is unemployed.3 When state s occurs, the firm’s selling price will be Pi
3Hansen (1985, p. 324) argues that “most of the variability in total hours is unambiguously due to fluctuations in the number of employed tither than hours per employed worker.” He then shows how a model that acxumes fixed hours is able to “improve our ability to account for U. S. aggxgate time series data.” This suggests that rigidity in hours per employed worker is not an implausible assumption.
J. A. Carlson and D. W. Findlay
34
directly by the firm. Assume that the utility function U is such that U’ > 0 and U” < 0. The expected utility of a worker who agrees to be available to firm i is
If a worker’s best alternative employment elsewhere provides an expected utility of U* (which of course depends on overall market conditions), the firm must offer a contract such that ECJ is at least as great as U*. We assume that the firm can obtain all the labor it needs by making EU = U*. Since it is costly for the firm to provide any higher utility, it will chose a combination of n(s), w(s), c(s), s = 1, . . . , S, such that EU= U*. The firm’s output is solely a function of labor input. Let f( n( s)) represent the firms output when its labor input is n(s) in state s and let this production function satisfy the following conditions: f’ > 0, f” < 0, f’(0) = 00 and f’(m) = 0. The firm’s nominal profits in state s are represented by the expression
In terms of what the owners wish to buy, these profits in real terms will be denoted by Yts) =
{Pi(s)f[n(s)]
-
Wi(s)n(s)
-
Ci
-
n(s)]}lPf(s)*
The utility for the firm in state s will be represented by V( y(s)), V” I 0. If the firm is risk-neutral, then let V( y(s)) = y(s)
(2)
with V’ > 0 and so that V’ = 1
and I’” = 0. The mathematical specification of the first-order conditions is reported in Table 1. The firm chooses a contingent set of wages and employment levels to maximize the expected utility of its real profits subject to the constraints that employment in any state be less than or equal to the firm’s labor force (x(s) is a Kuhn-Tucker Lagrangian multiplier) and that the firm be competitive in the labor market by providing a utility to workers of U* (p is a Lagrangian multiplier). The state S is designated as the one in which none of the firm’s workers are laid off so that n = n(S) is the firm’s desired labor force. The first-order conditions (4a) through (4e) are obtained by taking the partial derivatives of J with respect to the Wi(s), Ci(S), n(s), n(S), and p. From (4f) and (4g), when n(s) < n(S), x(s) = 0; and when n(s) = n(S), x(s) may take on positive values to satisfy (4c) and (4d). Before proceeding to an analysis of the wage indexation results of the model, we need to discuss and interpret some of the implications of the first-order conditions. For each s, condition (4a) can be written in the following form: I++))
= V’(Y@))[ ~&)/~_&)I
[@)/PI
*
(44
Equation (4a’) indicates that the marginal utility of the workers’ real wage is proportional to the marginal utility of the firm’s real profits, adjusted for the ratio of the price index for workers’ purchases to the price index for owners’ purchases, and is equal across all states, since p/n(S) is the same for all s. An increase in Wi(S) lowers the firm’s expected profits and raises the workers’ expected utility. The higher utility of workers in state s allows the firm to offer lower wages, and obtain higher profits, in another state. At the margin no gains to the firm are possible by reallocating wages across states.
35
Wage Indexation Similarly,
condition
U’(k + c(s))
(4b) can be rewritten
= q.+))[
as
%(~)Pf(~)l
sothat
the same ratio of marginal unemployed if the firm’s contribution
w
[~WlPl~
utilities holds whether workers are employed or to its unemployed workers is part of the contract.
Table 1. The Langrangian Function and First-Order Conditions S-l
J[ q(s), G(s), n(s)* X(s), P] = s&lW[ 4s) - 441
+ cc ~~lT(S){[no/n(S)l~(~i(s)lP,(s))
[
+[I - n(s)ln(S)]U(k +
ci(s)lpw(s)))
-
1(3)
u*
The first-order conditions (setting equal to zero the partial derivatives of J with respect to the choice variables) are
W,ts): T(s)n(s){ - r(Y(s))lpf(s) +/AV(W(S))/[P,(s)n(S)]} ci(s):
X(s)[ n(s) +pU’(k
n(s):
+ C(S))/[ P,(s)n(S)]}
= 0; s = 1,. . . , s.
?F(S)v(Y(s))[Pi(s)f’(n(s))
- W;:(“) + ci(s)]lpf(s)
- U(k + c(s))]/n(S)
a(S) v’( Y(s)) [ Pi(S)f’(
n(s))
2
0;
s=
l,...,S-
= 0;
- wiCs)] IQ(s)
S-l
n(s) - n(s)
(44
- n(s)] { - y’(Y(s))lpf(s)
+ p*(s)[u(w(s))
n(s):
= 0; s = l,...,S.
1.
WI - x(s)
s = 1,. . . , s - 1.
(44
J. A. Carlson and D. W. Findlay
36 Table 1. (continued)
h(s)[n(S)-n(s)]
=o;
s=l,...,S-1.
@g)
In each state, w(s) = W,(s)/P,,,(s) is the worker’s real wage; c(s) = C,(S)/P,(s)
is the real value of the worker’s unemployment compensationpaid by the
fum; and y(S) = [ Pi(s)f(n(s))
- Wi(s)n(s)
- Ci(s)(n(S)
- n(.V))]/p/(s)
is the
firm’s
real profits.
It can be seen from (4a’) and (4b’) that
U’(k + c(s)) = V(w(s)). In this case k + c(s) = w(s) in each state s and the worker will be indifferent between working and being on layoff. This result implies that the firm, regardless of its degree of risk aversion, will offer workers full insurance within each state. This result also implies that, when unemployment compensation is part of the contract, any influence of particular prices on the nominal wage in the cases to be considered will have a similar influence on the nominal payment by the firm to its unemployed workers. If payments by the firm to the unemployed are determined exogenously and are not part of the contract, equation (4b? can be ignored. If the firm is risk-neutral, so that v’ = 1, tbe indexation rule will not be dependent on any of the first-order conditions other than (4a). In that case, (4a’) becomes
While relative price variations across states may influence employment n(s) and real profits u(s), the assumption of firm risk neutrality will not let those variations influence the real wage solution from equation (4a”). The first three cases analyzed in the next section will assume risk-neutral firms. If the firm is risk-averse, then the firm’s real profits do enter into the indexing rule. To the extent that real profits vary with employment, it also becomes necessary to consider other first-order conditions. Condition (4c) introduces effects of varying employment in each state s # S. If n(s) < n(S), then from (4g) x(s) = 0. Furthermore, when compensation for the unemployed is part of the contract, w(s) = k + c(s) and the last term in (4~) drops out. As a result, condition (4~) reduces to
Pj(S)f’(n(S)) = Wi(S) - C,(S)*
(44
37
Wage Indexation
The left side of (4~‘) is the marginal benefit to the firm from employing an additional worker in state s. The right side is the marginal cost, which is the wage the firm pays minus the cost of compensating an unemployed worker in state s. If C!,.(s)is not part of the contract and is set equal to zero, then condition (4~) can be represented by the following equation:
q(s).r(+))
= wits) - ~,(S)(ccl~(S))[~(W(S)) - ww
v’(.ds)). (44
As in (4c’), the marginal cost of increasing employment in state s is less than the wage the firm pays, because higher employment also reduces the risk to a worker of being unemployed in state s and receiving no compensation from the firm. The shadow value to the firm arises because the higher utility to the workers allows the firm, ceteris paribus, to pay a lower average wage. Equation (4d), the first-order condition with respect to n(S), can be equivalently interpreted as equating the marginal benefit and marginal cost of adding an additional worker to the total number of workers under contract. Given the aggregate solution for n(S) and p, equations (4a’), (2), and (4~“) simultaneously determine variations in w(s), n(s), and u(s) between states in which not all workers under contract will be fully employed.
III. Wage Indexation with Different Relative Prices We shall now concentrate on interpreting the conditions represented by equations (4a’). These can be written as follows:
~wwfw)
= (Pw(s)lPf(s))V’(y(s))(n(S)/~);s =
1, - *a,
s. (5)
To help understand this equation and to answer the question about when firms would want to index to local selling prices, we analyze a series of special cases of equation (5). When the firm is risk-averse, we will have to consider the effects of changes in prices on the firm’s profits.
Case 1 Assume a risk-neutral firm (V’ = 1 and V” = 0) and P,(s) = P,,,(s). If the firm is risk-neutral and if owners of the firm buy the same bundle of goods as workers so that nominal profits are deflated by the same price index as workers, then (5) reduces to
(6) Whatever the solutions for n(S) and p, the marginal utility of the real wage must be the same in all states. Since U’ is strictly decreasing (U” < 0), there is a unique real wage that must hold in each state. Denote this by w. With the real wage constant across all states, the nominal wage is fully indexed to variations in the price of the bundle of goods bought by workers. This is the standard result for risk-neutral firms with risk-averse workers. By reducing the workers’ real-wage variations the firm is able to
J.
38
A. Carlson and D. W. Findlay
decrease the real wage and hence increase its own expected profits. The firm fully insures the workers against fluctuations in the purchasing power of their wages. Note in this case that the nominal wage does not respond to any variations in the price of the firm’s product. To change wages as selling prices change would introduce variations in workers’ real wages. Because of workers’ risk averseness, they would need to be compensated by a higher real wage. This would lower the firm’s expected profits with no offsetting gain to the firm. Recall for this case (and the following two cases) that, because of the optimal risk-sharing properties of the model, differences in employment (and profits) across states do not affect the wage decision of the firm (given by equation (4a’)). Case 2 Assume a risk-neutral firm ( V’ = 1 and I/” = 0) and Pr(s) = 1. Now suppose the risk-neutral firm is concerned with its accumulation of wealth. This would be the case in which owners of the firms wish to save any profits for future consumption, and commodities are not a convenient way of storing wealth. We do not deal with what financial instruments are available but suppose that they are denominated in nominal money units. If the firm’s current objective is in terms of nominal profits to maximize the expected investment in financial markets, then equation (5) becomes V( K(s)lP~(S))
= ‘wtS)( ‘CS)lP) ;
s=
l,...,S.
(7)
A higher P,,,(S) raises the right side of (7) and hence calls for a higher U’, which occurs when w(s) = Wi(s)/P,,,(s) is lower. Therefore when the aggregate price is higher, the real wage is lower. Indexation is less than full, but what indexation there is depends again only on the price of the bundle of goods bought by workers and not on the local product price. Basically what is happening here can be understood by supposing initially that real wages were equal across states. Expected nominal profits could then be increased more by lowering w(s) when PJ S) is high than would be lost by raising w(s) when P,,,(S) is low. The extent of the gain to the firm from variations in real wages between states is limited by the concavity of the workers’ utility function. Assuming that workers’ relative risk aversion is greater than 1 so that nominal wages would be adjusted in the same direction as any variation in the workers’ price index,4
4Consider a utility function with constant relative risk aversion
U(W/P)
= a-
b(W/P)-R+’
so that from (7) b(R
- l)VR
= (n/p)P-R+‘.
To find the effect on W of an arbitrarily small change in P in going from one state toward another, differentiate this expression and collect terms with the result that dW/ W = [(R - 1)/R]
dP/P.
Thus, with R > 1, W moves with P but in less than full proportion.
39
Wage Indexation
the contract could be specified in terms of guaranteeing the nominal wage that gives the highest real wage (for the lowest possible value of P,,,(s)) and then specifying a partial indexing rule in accordance with equation (7). Case 3 Assume a risk-neutral firm (I/’ = 1 and V” = 0) and Pf(s) = Pi(s). Now suppose that the firm wishes to acquire its own goods. It is not clear that many firms would want to do this. If firms are so diversified as to produce a representative bundle of goods or there is a one-commodity model, there would be little point in making the distinction between the product price and an aggregate price. One possible situation in which this case could be relevant is a foreign-owned firm that is in business to extract a product from its operations in the domestic economy. Whatever the rationale, this case does result in some indexing to product prices with risk-neutral firms. With V’ = 1 and P,(S) = Pi(S), equation (5) becomes U’(y(S)/Pw(s))
= (P,(s)/Pi(s))(n(S)/~);
s=
l,***,s*
(8)
As in case 2 there is less than full indexation to variations in the aggregate price, P,,,(s), ceteris paribus. What is different about this case is that there is now some indexation to the local price to the extent that it does not move with the aggregate price. Ceteris paribus, at higher Pi(s), U’ must be lower, which calls for a higher w(s). Hence, if the firm is trying to maximiz e its product profits, then real wages will be higher when local prices are higher relative to the aggregate price. In this case, where the firm’s objective is intluenced by the local price, the firm has something to gain by adjusting wages to local prices and will accept a somewhat higher real wage to take advantage of variations in relative prices. Case 4 Assume a risk-averse firm ( V’ > 0 and V” < 0), Pf(s) = P,,,(s), and no variations in employment across states. Now consider the situation when the same deflator is used for wages and for profits, but the firm is risk-averse. The firm’s real price, a(s) = Pi(S)/P,(S), in this specification will play a role similar to a productivity shift term in a model specified by Simonsen (1983). Equation (5) now takes the form
(9) With a risk-averse firm, variations in real profits y(s) influence the real wage, which in turn affects the firm’s real profits. This feedback needs to be explored. Suppose states are ordered from the lowest to the highest relative price p(s) = Pi(S)/P,(S). Then, using (9) and comparing adjacent states (for s = 1, . . . , S - 1, we have
[qw+dw) -
v(w)]/u’(w) = [ V(Y + dY) -
qY)]/v'(Y),
where dw = w(s + 1) - w(s) and dy = y(s + 1) - y(s). Define r,,. = -[V(w
+ dw) - V(w)]/[U’(w)dw]
(10)
J. A. Carlson and D. W. Findlay
40
and ‘f = - [
V’(Y+ dY)-
qY>]l[
fqY)dY]
as the absolute risk aversion of workers and the firm, respectively. These are discrete state counterparts to - LJ”/ II’ and - V”/ V’, the usual measures of absolute risk aversion at particular values of w and y. It then follows from these definitions and from (10) that
01)
dw = (ff/fw)dY.
Equation (11) indicates that when the firm is risk-averse, it chooses to engage in some profit sharing. When real profits are high, the firm will also raise the real wage of its workers. The analysis, however, is incomplete at this point, because there is a feedback from higher wages to the firm’s real profits. Under the assumption that Pr(s) = P,(s), equation (2) can be rewritten: Y(S)
=p(s)f(n(s)) - J+)n(s) - cW[nW -
n(s)].
(12)
For a given total employment n(S), variations in y(s) across states depend on variations in the firm’s real price p(s), real wage w(s), real compensation of its unemployed workers c(s), and its employment n(s). If we subtract y(s) from y(s + 1) and collect terms, the firm’s real profits can be seen to vary between states according to the following formula: dy = [p(s
+ l)(df/dn)
- n(s)(dw
- w(s + 1) + c(s + l)]dn
- dc) - n(S)dc,
+f(n(s))dp (13)
where dp = p(s + 1) - p(s), dc = c(s + 1) - c(s), dn = n(s + 1) - n(s), and df = f( n( s + 1)) - f( n( s)). Note that df /dn, like f’, is a marginal increase in output per unit increase in labor input, except that f’ is the effect of marginal adjustments within a particular state, whereas df /dn is the increase in output relative to the increase in employment in going from one state to the next better state in terms of the firm’s real price. We need to distinguish two situations. In this case, we consider the nature of the feedback when there are no variations in employment across states. In the next case, we will analyze the effects of variations in employment. MacDonald (1986) has argued within a two-state version of this model that relative price variations would have to be extremely large in order for the firm to plan any variations in employment across states. If price variations are not sufficiently large, then dn = 0. That will be assumed throughout this case. When dn = 0, equation (13) reduces to essentially the same formula whether or not unemployment compensation is part of the contract. The formula can be written dy = fdp - ndw,
(13’)
41
Wage Indexation
where f = f(n(s)). If unemployment compensation is not part of the contract, dc = 0 and n = n(s) in (13’). If unemployment compensation is part of the contract, then dc = dw and n = n(S) in (13). This shows the nature of the feedback from wages to profits when there are no variations in employment. If the firm’s real price increases, its real profits rise. When it chooses to engage in profit sharing as in equation (1 l), the rise in real wages reduces the firm’s real profits. This real-wage feedback, however, moderates rather than reverses the tendency for the firm to index wages to its product price. Formally, if dy from (13’) is substituted into (1 l), the resulting formula is dwldp
= (r,lr,)fl(l
+ (+w)n)
> 0.
(14)
The fact that dw / dp is positive means that the firm increases the real wage that it pays when its real product price is higher. Case 5 Assume a risk-averse firm (V’ > 0 and V” < 0), Pr(s) = P,,,(s), and employment varies across states. When the contract calls for variations in employment, the dn tern in (13) can no longer be ignored. Assume for now that unemployment compensation is part of the contract, In this case, equation (13) becomes dy = [ p(s + l)(df/dn) +f(n(s))dp
- w(s + 1) + c(s + l)]dn
- n(S)dw.
(13”)
When employment varies between states, we need to consider the equation corresponding to the optimal choice for contingent employment in each state. Equation (4~) discussed above comes from those first-order conditions. With n(s) c n(S), we have x(s) = 0. When unemployment compensation is part of the contract, U( w(s)) = U( k + c(s)) and the last term in (4~‘) drops out. Dividing by P,(s) yields &)Y(+))
= w(s) - c(s).
(15)
Note that if there were constant marginal product of labor between states s and s + 1, df /dn would equal f’( n(s)), the bracketed term in (13’9 would equal zero, and case 5 would reduce to case 4. With diminishing marginal product of labor, df /dn is larger than f’(n(s + l)), and the bracketed term in (13”) is positive. The expansion of employment adds to the firm’s real profits. Since w(s) - c(s) = k for all states when the firm’s payment of unemployment compensation is part of the contract, it must be true from equation (15) that a higher real price calls for higher employment. Suppose p were higher. If pf’ is constant, then $ must be lower and with diminishing product of labor this occurs with higher employment. Thus, we can write dn = b(s)dp,
(16)
where b is a positive number that depends on the curvature of the production function between states s and s + 1.
42
J. A. Carlson and D. W. Findlay
We can use equations (ll), (139, and (16) to eliminate dn and dy to get a relationship (i.e., indexation rule) between dw and dp:
dwldp =
<+w>[f+ b(p(dfldn) -
w+
c)]/(l + (Q/+)
’
0.
(17)
In comparing (17) with (14), when employment did not vary between states, we can see that they differ only in that (17) has the term b( p(df /dn) - w + c) in the numerator. With the value of the incremental output exceeding the net cost of employing the extra labor, the firm’s real profits tend to increase between states more than if employment does not change, and this increment in real profits is shared with workers when the firm is risk-averse (r, > 0). If unemployment compensation is not part of the contract, the last term in (4c”) plays precisely the same role as c(s) does when unemployment compensation is part of the contract. It represents an implicit cost of an unemployed worker, and that cost is not incurred at the margin when employment is higher. The algebra, however, becomes much messier and is not worth reproducing here. Case 6 Assume a risk-averse firm ( v’ > 0, V” < 0), and that P’(S) is not always equal to PJs). We can rewrite (5) once more as
we))
= (n(s)/r)r(S)~‘“[f(S)X(S)l)
(18)
where r(s) = P,,,(s)/PJs) is the ratio of the price index for the bundle of goods bought by the workers to the price index for the bundle of goods bought by the owners, and X(S) is nominal profits deflated by the price index for the bundle of goods bought by the workers. In the prior two cases we have assumed that r(s) equaled unity in every state. Now consider what equation (18) says about indexing wages to variations in r(s) for given values of x(s). The derivative of the right side of (19) with respect to f(s) will be positive if
r-XV”(m) + V’(m) > 0, and this holds if the firm’s coefficient of relative risk aversion is less than unity, that is, R= -yV”( y)/ V’(y) < 1. In other words, with R < 1, when a state occurs in which P/(s) falls relative to P,,,(S) so that r(s) is higher, the right side of (18) is increased. Equation (18) then calls for U’ to be higher, which means, with diminishing marginal utility, that real wages should be lower. Thus, if it could be part of the contract, the firm would want to some extent to index wages to the price index of the goods that the profits will be used to buy. If the firm has a relative risk averseness that is greater than unity, the argument is reversed. In states in which P,(S) falls relative to P,,,(s), the firm will want to increase the real wage. A peculiar possibility arises in this formulation if the firm’s owners have a risk averseness greater than unity and they want to take their profits in terms of the firm’s product so that Pr(s) equals the product price Pi(s). Then, ceteris paribus, the rule calls for a decrease in wages relative to the aggregate price index for workers
43
Wage Indexation
when the firm’s product price is higher relative to the aggregate price. Try selling that in wage negotiations! Whatever the degree of firm risk averseness, the difficulty in explaining the rationale for an indexation rule that includes the specification of a price index for a bundle of goods other than goods that workers buy would probably outweigh any gain in utility to the owners from including that in the contract. Thus, while variations in Pr(s) call for some optimal variations in real wages when firms are risk-averse, practical considerations may rule them out in actual contracts. Because of the peculiar implications of this example and the possible ambiguity given certain values of R, we will not proceed further in this case to examine the feedback effects of the above indexation rule.
IV. Concluding Comments Equation (11) shows the extent to which a risk-averse firm’s real wages move with its real profits. Shave11 (1976) has developed a similar formula. State-contingent real incomes are given exogenously and one party, say V, chooses a payment to a second party U to maximize V’s utility subject to holding U’s utility constant. In the case in which both parties have the same perceptions about the probability of each state occurring and U’s outside income does not vary, Shavell’s result can be written
w’(s) = y’(s)[ q/c’, +
L>l .
Our equation (11) would look the same if Y’ = ~JJ + dw denoted the total revenue that the firm had to divide between itself and the workers in going from one state to another and dw = w’ were the change in the total wage bill. Blinder (1977, p. 76) utilizes Shavell’s formula to make the following suggestion: “Firms which tend to do well in inflation [ Y’(s) > 0] will be eager to give generous escalators as a form of insurance to workers-in return for lower mean wages. Conversely, and for the same reasons, firms which tend to fare poorly in inflation [ Y’(s) c 0] will be reluctant to escalate wages very much, even though that implies higher mean wage costs.” Blinder is implicitly assuming that what indexation there is will be to changes in the aggregate price. If a firm’s price moves with the aggregate price so that its real price does not change, then such firms even if they are highly risk-averse will keep real wages constant. Our analysis introduces different prices explicitly. The results indicate that risk-averse firms will tend to increase real wages when their own real prices (and hence real profits) are higher. The extent of such wage adjustments, as we noted above, depends on risk aversion of the firm relative to that of workers. Weitzman (1985) has stressed the desirable implications of greater profit sharing. The fact that it is not more widespread suggests, within the structure of our model, that firms are not very risk-averse relative to workers. Our analysis complements that of Duca and VanHoose (1991). In a multisector, risk-neutral framework, they show that the degrees of wage indexation to an economywide price index and to profits will depend on the labor demand and supply elasticities. Our analysis indicates that owners’ objectives and particularly their degree of risk averseness can also influence the degrees of indexation to an aggregate price index and to firm’s profits. Now consider Card’s “puzzle as to the nearly universal practice of escalation to the CPI. ” If workers are very risk-averse relative to firms (r, / rw close to zero) and if
44
J. A. Carlsoa and D. W. Findlay
firms are not concerned with acquiring their own product, then our analysis provides a ready answer to the puzzle. The only gain to the firm from reducing workers’ risk is in moderating fluctuations in the workers’ real wage and hence indexing primarily to the price of the bundle of goods that workers buy. Note also that our result does not require asymmetric information. It may be that firms have better information than workers about product prices and that this could contribute to indexation to a well-publicized aggregate price index, such as the CPI, but this is not a necessary rationale. Even with full symmetric information about all prices, the optimal contract in cases 1 and 2 calIs for indexation only to the aggregate price index for what workers buy. There is therefore no incentive for workers to develop ways of getting full information about their firm’s selling prices.
We are grateful to David D.
VanHoose
for helpful suggestions on au earlier draft.
References Aizenman, J., and Frenkel, J. A. (June 1985) Gptirnal wage indexatioa, foreign exchange intervention, and monetary policy. American Economic Review, 75: 402-423. Algoskoufis, G. S. (1983) The labor market in an equilibrium business cycle model. Journal of Monetary Economics 11: 117-128. Azariadis, C. (December 1975) Implicit contracts and underemployment equilibria. Political Economy 83: 1183-1202.
Journal of
Azariadis, C., and Stiglitz, J. E. (1983) Implicit contracts aad fixed price equilibria. Quarterly Journal of Economics 98(Suppl.): l-22. Baily, M. N. (January 1974) Wages and employment under uncertain demand. Review of Economic Studies 41: 37-50. Barro, R. J. (July 1977) Long-term contracting, sticky prices, and monetary policy. Journal of Monetary Economics 3: 305-316. Blinder, A. S. (1977) Indexing the economy through financial intermediation. Journal of Monetary Economics S(Supp1.): 69-105. Cahnfors, L., and Viotti, S. (November 1983) Wage indexation, the Scandinavian model and macroeconomic stability in the open economy. Oxford Economic Papers 34: 547-566. Card, D. (June 1986) An empirical model of wage indexation provisions in union contracts. Journal of Political Economy 94: 144-175. Cukierman, A. (1980) The effects of wage indexation on macroeconomic fluctuations. Journal of Monetary Economics 6: 147- 175. Duca, J. V., and VanHoose, D. D. (November 1991) Optimal wage indexatioa in a multisector economy. International Economic Review 32(4):859-868. Fischer, S. (1977a) Wage indexation and macroeconomic stability. Journal of Monetary Economics S(Supp1.): 107-147. Fischer, S. (February 1977b) Long-term contracts, rational expectations, and the optimal money supply rule. Journal of Political Economy 85: 191-205. Gordon, D. F. (1974) A neoclassical theory of Keynesian unemployment. Economic Inquiry 12: 431-459. Gordon, R. J. (September 1990) What is new-Keynesian economics? Journal of Economic Literature 28: 1115-1171. Gray, J. A. (April 1976) Wage indexation: A macroeconomic approach. Journal of Monetary Economics 2: 22 l-235. Hart, 0. D. (January 1983) Gptimal labour contracts under asymmetric information: An introduction. Review of Economic Studies 50:3-35. Hansen, G. D. (November 1985) Indivisible labor and the business cycle. Journal of Monetary Economics 16: 309-327.
Wage Indexation
45
Karni, E. (April 1983) On optimal wage indexation. Journal of Political Economy 91: 282-292. MacDonald, I. M. (January 1986) Can the risk-shifting employment model explain fluctuating employment. Economic Inquiry 24: 25-41. Marston, R. C., and Tumovsky, S. J. (November 1985) Macroeconomic stabilization through taxation and indexation: The use of firm-specific information. Journal of Monetary Economics 16: 375-395. Paxner, E. A. (1981) On indexation and macroeconomic stability. In Development in an Inflationary World (M. J. Flanders and A. Raxin, eds.). New York: Academic Press. Phelps, E. S. (1977) Indexation issues: A comment on the Blinder and Fischer papers. Journal of Monetary Economics S(Supp1.): 149-159. Shave& S. (February 1976) Sharing risks of deferred payment. Journal of Political Economy 84: 161-168. Simonsen, M. H. (1983) Indexation: Current theory and the Brazilian experience. In Inflation, Debt and Zndexation (R. Dombusch and M. H. Simonsen, eds.). Cambridge, MA: MIT Press, pp. 99-132. VanHoose, D. D., and Waller, C. J. (October 1989) Islands, indexation and monetary policy. Economic Inquiry 27: 705-717. Waller, C. J., and VanHoose, D. D. (February 1989) Endogenous wage indexation and optimal monetary policy with and without a balanced budget. Journal of Economics and Business 41: 21-31. Weitxman, M. L. @ecember 1985) The simple macroeconomics of profit sharing. American Economic Review 75: 937-953.