# A stochastic monopsony theory of the business cycle.

Two distinct regimes, contractions and expansions, are generated in a model in which goods markets clear and all individuals are optimizing, strict wage and price takers, have fully rational expectations, and are heterogeneous in both preferences and resource endowments. Involuntary unemployment, asymmetric monetary policy effectiveness, and a changing relationship between real wages and employment over the business cycle are the result of optimizing behavior by monopsonistic, wage-setting, and price-taking firms faced with price uncertainty, an upward-sloped supply of employees, and efficiency wage behavior. Disequilibrium and involuntary unemployment can occur at the level of the individual firm's labor market. (JEL E32, E52, J41, J42)I. INTRODUCTION

For many economists, three stylized facts are central to any plausible theory of the business cycle. (1) Friedman and Schwartz (1963) and Romer and Romer (1994) concluded that monetary shocks have output, employment, and unemployment effects at least some of the time and such effects are asymmetric with respect to the state of the economy. (2) Unemployment is subject to fluctuations associated with recessions and expansions and, in some recessions, has been very high. (1) (3) The relationship between real wages and employment appears to be time period sensitive as reported by Abraham and Haltiwanger (1995).

Although first proposed by Keynes, modern economists such as Benassy (2002) argue that at least some unemployment is best characterized as involuntary and indicates a state of disequilibrium. In this article, we consider a plausible alternative to disequilibrium explanations of these elements of the business cycle based on goods market imperfections, such as those by Benassy (2002) and Dixon and Rankin (1995), who considered monopoly power by firms in the goods market, and Benassy (1995), who assumed monopoly power by unions in the labor market. We assume that the goods market is characterized by competitive market clearing, but firms have some monopsony power as reflected in an upward-sloped or bounded supply of labor to the firm and the ability to set the optimal wage before knowing the price they will receive for their output, which is assumed to be stochastic, and making their employment decision. These modifications are embedded in an otherwise conventional general equilibrium model in which all agents (firms and consumers/workers/capitalists) are fully optimizing and form totally rational expectations and labor may exhibit efficiency wage behavior. (2) This is not a representative agent model and cannot be reduced to such. Individuals are heterogeneous in both their endowments and preferences. Firms may be heterogeneous with respect to their production technology and the interests of firm owners are distinct from those of their employees. The latter is essential to generate the changing regimes that produce predictions that correspond to the stylized facts.

Monetary shocks affect output, prices, employment, and unemployment in an asymmetrical manner consistent with the traditional business cycle. The key to understanding these effects is the labor market. If the expected quantity of money is generated, then the competitive, market-clearing real wage will result. Lower than expected quantities of money generate a deficiency of aggregate demand, lower than expected output prices and real wages above the competitive wage. Real wages above the competitive wage are associated with excess supply of labor and involuntary unemployment (independent of that associated with efficiency wage behavior). This contractionary regime will persist as long as the low quantities of money persist, even though all agents have fully rational expectations, are optimizing, and the goods market clears. In this regime, increases in the quantity of money increase prices, employment, and output, and employment varies negatively with the real wage. Efficiency wage behavior may decrease effort as the real wage rate is decreased.

Higher than expected quantities of money generate high aggregate demand, higher than expected prices, and real wages below the competitive wage. Such wages are associated with excess demand for labor and no involuntary unemployment. The only unemployment may be that associated with efficiency wage behavior. In this expansionary regime, increases in the quantity of money increase prices, lower real wages, and will not affect or will increase employment and output depending on whether the supply of labor is vertical or positively sloped, causing the relationship between real wages and employment to be either zero or positive. Efficiency wage behavior will augment these effects on output as lower real wages may cause more workers to shirk.

The next section presents the basic assumptions and motivations for the model. We then present the formal model of individual behavior of consumers, labor, and firms; aggregate relationships; and graphical illustrations of the labor and goods markets. This is followed by a description of the derivation of the optimal wage and equilibrium values of other endogenous variables using the equilibrium condition that the goods market clears, and the analysis of the effectiveness of monetary policy in recessions and expansions. We end with a summary and conclusions.

II. ASSUMPTIONS

Although it is not uncommon in macroeconomic models to assume that the labor market always clears at its competitive equilibrium value, whereas the goods market does not, (3) we assume that the goods market is characterized by competitive market clearing to conform with the consensus of economists reflected in many textbooks, as well as the evidence that output prices are flexible presented by Blinder et al. (1998). We assume that firms have monopsony wage-setting power because (1) if one assumes that the wage rate is always set at its competitive market clearing value, involuntary unemployment cannot arise, (2) less than 10% of the labor force are represented by labor unions that may have monopoly power, and (3) the preponderance of the evidence reviewed by Bhaskar et al. (2002) and Boat and Ransom (1997) indicates that firms have some monopsony power in labor markets. Our formulation is consistent with that of a monopsonisticly competitive labor market as advocated by Bhaskar and To (1999).

Although not necessary for the results, we assume that labor may exhibit efficiency wage behavior. Raft and Summers (1987), Krueger and Summers (1986), and Akerlof et al. (1988) have found empirical evidence that efficiency wage payments have productivity-enhancing or cost-reducing benefits to a firm. Solow (1979), Shapiro and Stiglitz (1984), Yellen (1984), and Katz (1986), among others, have argued that efficiency wage behavior can produce involuntary unemployment. On the other hand, the efficiency wage model is insufficient to explain all of the observed data, particularly data associated with recessions. As noted by McCafferty (1990, p. 453), efficiency wage behavior does not generate observations that can be classified as part of the traditional business cycle in any apparent manner because the involuntary unemployment that it generates exists at high prices (during expansions) as well as at low ones (during contractions). Second, Blanchard and Fisher (1989) demonstrated that with efficiency wages short-run monetary changes do not affect output, either with or without product price uncertainty, which is contrary to the empirical evidence on monetary policy effectiveness. Third, the conventional efficiency wage model predicts a correlation between real wages and employment that is always positive, whereas on reviewing a large body of empirical literature, Abraham and Haltiwanger (1995) concluded that the empirical relationship is demonstrably time period sensitive. A subsequent investigation by Holmes and Hutton (1996), updated in Holmes et al. (2004), suggests that the time period sensitivity is related to the business cycle. To generate predictions that are consistent with the empirical evidence, we augment the efficiency wage assumption with two assumptions that are complimentary to but strictly independent of efficiency wage behavior.

(1) As in Jullien and Picard (1998), we assume that the disutility of being employed is separate from that of working instead of shirking. Heterogeneity with respect to the disutility of employment generates a supply of labor to the firm that is an increasing function of the real wage up to some maximum level, (4) rather than a perfectly elastic supply. (2) We assume that the monopsonistic firms set the expected profit-maximizing wage before knowing the price they will receive for their output and choosing the optimal level of employment. (5)

This timing of events is similar to the disequilibrium theory of Benassy (2002, ch. 9) and the real business cycle analysis by Cho and Cooley (1995) in which nominal wages are set before output prices are realized but in a representative agent framework. Why a firm might not contract away the uncertainty that comes with setting wages before output price becomes known may be because supply shocks may preclude full indexing of wage contracts, as suggested by Gray (1976) and Fisher (1977), or because indexation involves a cost to the firm and it is optimal to set totally nonindexed nominal wages as proposed by Ball (1988), or because, as Cooper (1990) and Acemoglu (1995) argue, making nominal wage contracts may be optimal to reduce risk or to hedge an uncertain environment.

We assume that when the wage decision is made, each firm, holding fully rational expectations, has knowledge of the distribution of all exogenous shocks, all behavioral parameters and functions in the model of the whole economy. After the nominal wage is chosen, the output price is learned. Then the firm decides the desired number of employees and output according to its marginal (efficiency) product of labor wage criterion. These desires may not always be fulfilled. In general, the setting of wages before prices and the resulting fluctuations in the real wage paid once the stochastic output price is realized cannot per se explain the stylized facts. It is the combination of this wage setting (which is in the self-interest of the owners of the firms as distinct from that of its employees) with an upward-sloped labor curve that produces these results.

III. THE FORMAL MODEL

In this simple, stochastic, two-period, overlapping generations model, there is no change in population. (6) Agents, indexed with the letter i, are heterogeneous within a generation in both their preferences for work and endowments. K of the young agents are capitalists who inherit their family firms, indexed by j, when young and do not work. Inheritance is assumed to be received by agents when young instead of when old because this avoids the effect of expected income on the savings/consumption choice of the young. The remainder of the young agents, none of whom are capitalists or old and all of whom are wage takers, constitute the potential labor force of N agents. Potential workers are either employed (L), involuntarily unemployed (U) but willing to work at the prevailing or even a lower real wage, or not in the labor force (NL = N- L - U). Employment, unemployment, and those not in the labor force can vary from period to period, depending on the realizations of the variables in the model. The K firms act competitively in the output market and each produces a marketable, composite, perishable consumer good using [L.sub.j] employees (indexed by firm j).

To provide a simple, exogenous source for demand shocks, we assume that money is distributed in period t to the old holders of money balances in the proportion [[mu].sub.t] of the amount of money they saved when young. For an individual i who saved [[M.sup.i.sub.t-1] when young, this is [[M.sup.i.sub.t] = [[mu].sub.t] [[M.sup.i.sub.t-1]. In the most elementary case, where [G.sub.t] is government spending, [T.sub.t] is government tax revenue (or a subsidy if negative), and [M.sub.t] is the quantity of money issued by government and there are no bonds, the government budget constraint is 0 = [G.sub.t] = [T.sub.t] + [M.sub.t] - [M.sub.t-1]. Thus, the total subsidy given to the old is [T.sub.t], = -([summation][[M.sup.i.sub.t] - [summation][[M.sup.i.sub.t-1]). For conceptual clarity and mathematical tractability, we analyze the most elementary case where [M.sub.t] is stationary and distributed i.i.d and has a Bernoulli distribution where the probability of a low quantity, [[M.sup.L.sub.t], is X and the probability of a high quantity, [[M.sup.H.sub.t], is (1 - X).

Demand

To derive aggregate demand, the optimal behavior of all consumers must be derived, aggregated, and related to the behavior of government. Young consumers maximize lifetime utility subject to budget constraints when young and old. Consumer i derives utility from consumption when young, [[C.sup.yi.sub.t] and from consumption when old, [[C.sup.oi.sub.t+1]. The functional form of that portion of the present value of utility derived from [[C.sup.yi.sub.t] and [[C.sup.oi.sub.t+1] is the same for all agents. However, market activities may affect the welfare of each individual differently. A laborer receives disutility of [v.sub.i] from having a job and disutility of [[alpha].sub.i] from working at that job instead of shirking, [v.sub.i] and [[alpha].sub.i] are positive and are different for different individuals. However, if an individual is a capitalist, i [member of] K, is unemployed, i [member of] U, or is not in the labor force, i [member of] NL, then [v.sup.i] and [[alpha].sup.i]= 0.

The utility function for agent i is similar to Jullien and Picard's (1998):

(1) U ([[C.sup.yi.sub.t], [[C.sup.oi.sub.t+ 1]] - [v.sup.i] - [[alpha].sup.i] = [beta]ln ([C.sup.yi.sub.t]) + (1 - [beta]) ln ([[C.sub.oi.sup.t+1]) - [v.sup.i] - [[alpha].sup.i]

where 0 < [beta] < 1, 0 [less than or equal to] [v.sup.i], and [[alpha].sup.i] are constants. Where superscripts or subscripts are obvious, they will be suppressed.

This particular functional form has several advantages. Closed-form solutions are easily obtained. Savings are independent of the opportunity cost of current versus future expected consumption and, hence, are independent of expectations. Redistributions of income between agents with the same value of the coefficient [beta] do not affect aggregate demand. Aggregate demand will be equivalent to the simple quantity theory of money.

Budget constraints when young and old are, respectively,

(2) [P.sub.t][[Y.sup.i.sub.t] = [P.sub.t][[C.sub.yi.sup.t] + [[M.sup.i.sub.t] and [P.sub.t+ 1][[C.sup.oi.sub.t+ 1] = [[M.sup.i.sub.t][[mu].sup.t + 1]

where [[Y.sup.i.sub.j] is market income of individual i when young and Pr is the economy wide output price. Because a competitive market has many agents, it is reasonable for the young to assume that their actions will have no effect on output price, and they take their market income and price as given.

The market income of each individual i depends on what he or she does. If i is a capitalist, [[Y.sup.i.sub.t] will be the real profit of firm j, [[pi].sup.j.sub.t] If i is an employee of firm j, [[Y.sup.i.sub.t] equals the real wage, [[w.sup.j.sub.t] = [[W.sup.j.sub.t] / [P.sub.t], paid by the firm. If an individual is unemployed or not currently in the labor force, [[Y.sup.i.sub.t] equals zero. The unemployed and those not in the labor force produce only nonmarket goods (perhaps for sustenance so they do not starve to death) that do not substitute for market goods. We will keep account of them, even when [[Y.sup.i.sub.t] = 0, because their status can vary from period to period.

In period t, a young individual with market income may save, ([[Y.sup.i.sub.t] - [[C.sup.yi.sub.t]), and enjoy the benefit of their frugality when old. A worker or capitalist can only save out of market income by holding money balances equal to [[Y.sup.i.sub.t] - [[C.sup.yi.sub.t] = [[M.sup.i.sub.t]

The young's problem is to choose money balances at time t to maximize

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The young's expectation, [E.sub.t][*], is conditional on the information they have at time t, which includes current output price and the distributions for future output price and money. The first-order condition is--[beta]/[([[Y.sup.i.sub.t] - [[M.sup.i.sub.t]/[P.sub.t]] + [E.sub.t][{(1 - [beta])/([[M.sup.i.sub.t][[mu].sub.t + 1]/[P.sub.t+1])} ([mu].sub.t+1]/[P.sub.t+1] = 0 with respect to [[M.sup.i.sub.t]. Thus,

(4) [[M.sup.i.sub.t]/[P.sub.t] = (1 - [beta])[[Y.sup.i.sub.t] (and [[C.sup.yi.sub.t] = [beta][[Y.sup.i.sub.t]).

Total consumption of market goods by the young at time t is the sum of that for workers, capitalists, the unemployed, and those not in the labor force and equals 13 of their market income.

(5) [c.sup.y.sub.t] = [summation over L] [C.sup.yi.sub.t] + [summation over K] [C.sup.yi.sub.t] + [summation over U] [C.sup.yi.sub.t] + [summation over NL] [C.sup.yi.sub.t] = [summation over K] [L.sub.j][beta][w.sup.j.sub.t] + [summation over K] [beta][[pi].sup.j]

Total consumption of market goods by the N + K old at time t equals the total of their real money balances (augmented by the stochastic, government subsidy or tax paid in money on their savings when young),

(6) [C.sup.o.sub.t] = [summation over N] [m.sup.i.sub.t] + [summation over K] [m.sup.i.sub.t] = [M.sub.t]/[P.sub.t],

where [[m.sup.i.sub.t] = [[M.sup.i.sub.t]/[P.sub.t].

Hence, aggregate income, [Y.sub.t], is

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, for an economy with agents that are heterogeneous in age, endowments of wealth and station, and preferences for employment and work effort, we have derived an aggregate demand function that is similar to the quantity theory of money equation. This result is dependent on both the functional forms assumed and the specification of endowments and station.

Labor

The labor/leisure substitution by any individual is discrete. He or she chooses to seek a job or not. Similar to Jullien and Picard (1998), we assume that an employed worker will supply one unit of labor, for example, 40 hours a week, which is the same for all laborers. According to equation (1), showing up for work or being employed imposes a fixed disutility of [v.sub.i] on agent i. A laborer takes the nominal wage, [[W.sup.j.sub.t] from firm j and output price, [P.sub.t], as exogenous and, if employed in period t, earns [[Y.sup.i.sub.t] = [[w.sup.j.sub.t] = [[W.sup.j.sub.t]/[P.sub.t] . The choice to be a labor market participant must maximize utility (discontinuous in vi) given in equation (3). If [[w.sup.j.sub.t] [greater than or equal to] [v.sup.i], then utility is maximized if an individual i participates. Otherwise it is optimal to choose not to be in the labor force. Assuming that [v.sup.i] is different for each laborer and varies continuously in the labor force from 0 to [[v.sup.[psi], the participation rate or supply of labor is an increasing function of the real wage up to the level determined by [[v.sup.[psi]. (7) Denote as [[w.sup.k.sub.t] the real wage equal to [[v.sup.[psi]. For [w.sub.t][greater than or equal to] [[w.sup.k.sub.t], all potential workers are in the labor force. This makes the market supply of labor kinked at [[w.sup.k.sub.t], where it becomes vertical and equal to [N.sub.t].

To simply further analysis, we now assume that K is a finite number of representative firms, such that the labor supply function to a firm, [S.sub.j] ([[w.sup.k.sub.t]), is merely its proportionate share of the market supply of labor (8) and the wage will be the same for all firms, [[w.sup.j.sub.t] = [w.sub.t]. Assuming an exponential function,

(8) [S.sub.j]([w.sub.t]) = ([w.sub.t])sup.[psi] = [[L.sup.j.sub.t], if [w.sub.t] [member of] (0, [[w.sup.k.sub.t]),

and [S.sub.j]([w.sub.t]) = [[N.sup.j.sub.t], if [w.sub.t] [greater than or equal to][[w.sup.k.sub.t],

0 < [psi] < [[psi].sup.*]. x [[psi].sup.*] is an upper bound determined by the sufficient condition for a solution to this model, specified later. [[N.sup.j.sub.t] is the maximum potential supply of labor to firm j. The aggregate supply of labor is S([w.sub.t]) = K x [S.sub.j]([w.sub.t]) = K x ([[w.sub.t)sup.[psi] if [w.sub.t] [member of](0, [[w.sup.k.sub.t]), and if [w.sub.t][greater than or equal to][[W.sup.k.sub.t] , S([w.sub.t]) = K x [[N.sup.j.sub.t] = [N.sub.t].

An employed laborer can choose to shirk or be productive. Productive effort by agent i associated with firm j, [[e.sup.ji.sub.t] = 1, involves a disutility of [[alpha].sup.i]. The choice to be productive must also maximize utility (discontinuous in [[alpha].sup.i) given in equation (3), subject to the budget constraints in equation (2). If [w.sup.t] [greater than or equal to] [[alpha].sup.i], then the optimal choice of employee i is to be productive. Otherwise it is to shirk.

[[alpha].sup.i] is assumed to vary continuously among the set of employees from a low value of 0 to a high value of [[alpha].sup.[phi]]. In this event, the supply of effort is an increasing function (e.g., exponential) of the real wage up to a real wage equal to [[alpha].sup.[phi]]. We term the real wage at which the most effort-adverse employee is not shirking as [[w.sup.*.sub.t] = [[alpha].sup.[phi]]. For the finite set of representative firms, K, it follows that as the real wage increases so will average effort, [summation][[e.sup.ji.sub.t]/[L.sup.j.sub.t] = e([w.sup.t])(where the summation is over [L.sup.J]) for firm j up to a maximum value of 1, which occurs at [[w.sup.*.sub.t]. Hence, assume

(9) e([w.sub.t]) = ([[w.sup.t]/[w.sup.*.sub.t]).sup.[phi]], if [w.sub.t] [member of] (0,[w.sup.*.sub.t]) and l, if [w.sub.t] [greater than or equal to] [w.sup.*.sub.t].

0 [less than or equal to] [phi] < [[phi]sup.*] x [[phi].sup.*] is an upper bound that can be 1 and is determined along with [[phi]sup.*] by the sufficient condition for a solution to the model. For [w.sub.t], < [w.sup.*.sub.t], e([w.sub.t)' = [phi][([w.sub.t]/[w.sup.*.sub.t]).sup.[phi]][([w.sub.t).sup.-1]] [greater than or equal to] 0 as [phi] [greater than or equal to]0 and e([w.sub.t]) < 1, and for [w.sub.t] [greater than or equal to] [w.sup.*.sub.t], e([w.sub.t]) = 1 and e([w.sub.t])' = 0. The latter occurs if [phi]) = 0.

Firms

Representative firm j produces output, [[Q.sup.j.sub.t], using a Cobb-Douglas production function.

(10) [[Q.sup.j.sub.t] = F([a.sup.J.sub.t], e([w.sup.t]) x [[L.sup.j.sub.t]) = [[a.sup.j.sub.t][(e([w.sub.t]) x [L.sup.j.sub.t].sup.[theta]], for 0 < [theta] < 1.

The firm will hire [[L.sup.sub.j] employees who put. forth average effort, e([w.sub.t]). The variable [[a.sup.j.sub.t] incorporates both capital and technology. For simplicity, assume that [[a.sub.j.sub.t] = 1.

A monopsonistic firm chooses [W.sub.t] prior to period t. Because it is fully rational, the firm knows the probability distribution of the quantities of money that may be supplied and all of the behavioral parameters and functions in the model. After it has chosen [W.sub.t], the firm learns Pt. Then, knowing the real wage, the firm decides on the optimal number of its employees. Its optimal employment plans satisfies the marginal (efficiency) product of labor wage criterion,

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and aggregate desired employment by all firms is [L.sub.t] = [[summation].sub.K] [[L.sup.j.sub.t] = [[summation].sub.k] [D.sup.j]([w.sub.t]). It is straightforward to demonstrate that as 0 < [theta] < 1, d[D.sup.j]/ d[w.sub.t] < 0, if 0 [less than or equal to] [theta] [less than or equal to] 1, and if [theta] > 1, for [theta] < 1/[theta]. (9)

As Benassy (2002, pp. 8-9) points out, assuming strictly voluntary exchange the firm will not desire to hire more than the number defined by equation (11) and more employees than the number defined by the supply function (equation [8]) will not want to be hired. Hence, the number of employees actually hired by a firm must be consistent with the lesser of these two equations,

(12) [L.sup.j.sub.t]([w.sub.t]) = Min[[S.sup.j]([w.sub.t]), [D.sup.J]([w.sub.t])].

Market employment is the aggregate employment of all firms, [L.sub.t] = [[summation].sub.k] [L.sup.j.sub.t] = [[summation of].sub.k] Min[[S.sup.j]([w.sub.t]), [D.sup.j]([w.sub.j]], and aggregate supply is given by equations (10) evaluated at (9) and (12), aggregated over all firms,

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Equilibrium occurs when the goods market clears, [summation of] AS([S.sup.j][[w.sub.t]]) = [Y.sub.t]. Given the optimal choice of the nominal wage, the equilibrium price and output are then simultaneously determined (along with the real wage, employment, and unemployment for each firm) by the equality of aggregate demand and aggregate supply.

In the conventional efficiency wage model, the optimal nominal wage is given by [W.sub.t] = [P.sub.t]e/ e', when [w.sub.t] [member of] (0, [[w.sup.*.sub.t]). However, as in Solow (1979), the firm faces a perfectly elastic supply of labor [S.sub.j]([w.sub.t]) (i.e., [v.sup.i] identical across agents) and knows both e([w.sub.t]) and the competitive market clearing output price, [P.sub.t]. In this model the firm must choose the nominal wage it pays ignorant of the price it will receive for its output (or the realized quantity of money) and subject to a supply of labor that is either upward-sloped or vertical instead of perfectly elastic. Subject to equations (9), (10), and (12), the firm chooses the optimal wage, [W.sub.t], to maximize expected profit,

(14) E[[PI].sup.j] = E[F(e[[w.sub.t]] x [[L.sup.j.sub.t][[w.sub.t]]) - ([w.sub.t) x [[L.sub.j.sup.t] ([w.sub.t])].

The optimal nominal wage for the firm to set is given by the solution to the implicit equation.

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Given this wage, the firm subsequently learns the realized values of [P.sub.t] and [w.sub.t] and hires its employees.

Intuition into the complexity of the monopsonistic firm's choice of wage in an uncertain environment is gained with the help of Figure 1. In Figure 1. the firm's marginal product of or demand for labor when there is no shirking, e(w) = 1, is labeled D([w.sub.t], e[[w.sub.t]] = 1). Alternatively, if shirking begins for real wages below [[w.sup.*.sub.t], then the demand for labor is the dashed line labeled D([w.sub.t] e[[w.sub.t]] < 1). The supply of labor, labeled S([w.sub.t]), is vertical (and equal to [N.sub.t]) at [[w.sup.k.sub.t]). The real wage at which the marginal product of labor intersects the supply of labor is the value that would obtain in a competitive market. It is labeled [[w.sup.C.sub.t] for e(w) = 1, and it is labeled [[w.sup.C*.sub.t] for e([w.sub.t])< 1. We shall focus on the case of no shirking, D([w.sub.t], e[w] = 1), for the discussion that follows. (10)

[FIGURE 1 OMITTED]

The Bernoulli distribution generates high and low quantities of money, [[M.sup.H.sub.t] and [[M.sup.L.sub.t], prices, [[P.sup.H.sub.t] and [[P.sup.L.sub.t], and real wages, [W.sub.t]/[P.sup.L.sub.t]) and [W.sub.t]/[[P.sup.H.sub.t]. For any wage above [w.sup.C.sub.t], the number of employees a firm desires and can hire is determined by D([w.sub.t], e[w] = 1). For example, at the high real wage, [W.sub.t]/[P.sup.L.sub.t], firms will hire [L.sup.L.sub.t], and this will produce involuntary unemployment of [U.sub.t]. For real wages below [w.sup.C.sub.t], the supply of labor determines the number of employees who desire employment and the firm can hire. For example, for the low real wage, [W.sub.t]/[P.sup.H.sub.t], the firm can only hire [L.sup.H.sub.t] employees. For any real wages below [w.sup.C.sub.t], involuntary unemployment will be zero.

Whenever a real wage below [w.sup.C.sub.t] is realized, the firm will make an extra monopolistic real profit per employee in excess of the return to capital. At [W.sub.t]/[P.sup.H.sub.t], the firm makes the extra profit measured approximately by the rectangle [D.sup.-1]([L.sup.H.sub.t]) - [W.sub.t]/[P.sup.H.sub.t] x [[L.sup.H.sub.t]. Whenever a real wage above [w.sup.C.sub.t] is realized, the firm will make the same profit as it would if it were a competitive firm hiring the same number of employees. The existence of extra profit whenever the realized real wage is below [w.sup.C.sub.t], which occurs with probability (1 - X), explains why the firm owner might never choose to set the competitive market clearing wage. This can occur because the owners of firms are distinct from their employees. (11) If they were the same, as in a representative agent model, it would always be in their self-interest to behave as a competitive firm.

It should be obvious that it will never be optimal for the firm to choose a wage such that the resulting lower real wage, [W.sub.t]/[P.sup.H.sub.t], will not be on the upward-sloped part of the labor supply curve. One can clearly see from Figure 1 that there are three possibilities for the higher real wage, [W.sub.t]/[P.sup.H.sub.t]. (1) It is on the upward-sloped part of the labor supply curve, (2) on the vertical part of the labor supply curve, or (3) on the demand for labor.

[FIGURE 1 OMITTED]

Figure 2 portrays the aggregate supply AS([W.sub.t], e[[w.sub.t]] - 1) that is derivable from Figure 1. Although not realized, the real wage [W.sub.t]/ [P.sup.C.sub.t] = [w.sup.C.sub.t] and the associated price [P.sup.C.sub.t] are critical to the analysis. Lower prices and higher real wages than these are associated with reduced demand for labor by firms and hence output, so that aggregate supply is positively sloped. Between [P.sup.C.sub.t] and [P.sup.k.sub.t] the supplies of labor and output are vertical. Above [P.sup.k.sub.t] the aggregate supply curve is negatively sloped because increases in output price decrease the real wage and reduce the participation rate of labor.

[FIGURE 2 OMITTED]

We have portrayed three aggregate demands. AD([M.sup.H.sub.t]) is associated with the high quantity of money, and AD([M.sup.L.sub.t]) is associated with the low quantity of money. The intersection of these aggregate demand curves and the aggregate supply curve produce the goods market-clearing equilibrium values of [P.sup.H.sub .t], [[Y.sup.H.sub.t], and [P.sup.L.sub.t], [Y.sup.L.sub.t], respectively, portrayed in Figure 2. A third aggregate demand, AD([M.sup.C.sub.t]), portrayed by a dashed line and associated with a quantity of money [M.sup.C.sub.t], larger than [M.sup.L.sub.t], intersects the aggregate supply curve AS([W.sub.t], e[w.sub.t] = 1) at the transitional price level [P.sup.C.sub.t] and full employment output [Y.sup.C.sub.t]. Clearly, a recession is associated with quantities of money less than [M.sup.C.sub.t], and in this regime fluctuations in the quantity of money will affect output, generate a negative relationship between real wages and employment, and can produce involuntary unemployment.

Now consider the effect of efficiency wage behavior that begins for real wages below [[w.sup.*.sub.t], as portrayed by the dashed line in Figure 1. The aggregate supply curve derived from this labor market, AS([W.sub.t], e[w.sub.t] < 1), is portrayed in Figure 3. Maximum output at [P.sup.C*.sub.t] in Figure 3 corresponds to maximum employment, [L.sub.t] = [N.sub.t], at [W.sub.t]/[P.sup.C*.sub.t] = [[w.sup.C*.sub.t], in Figure 1. Below [P.sup.*.sub.t], an increase in output price leads firms to hire more employees along the demand for labor curve D([w.sub.t], e[[w.sub.t]] = 1). Between [P.sup.*.sub.t] and [P.sup.C*.sub.t], an increase in price decreases the real wage and both the number of workers hired and their productivity due to increased shirking along the curve labled D([w.sub.t], e[[w.sub.t]] < 1). Consequently, AS([W.sub.t], e[w.sub.t]] < 1) is steeper for prices above [P.sup.*.sub.t] than for prices below [P.sup.*.sub.t], when efficiency wage behavior is inoperative. To illustrate this, the relevant portion of AS([W.sub.t], e[[w.sub.t]] = 1) is portrayed as a dashed line. The output price [P.sup.*.sub.t]. that results in a real wage at which the labor market with efficiency wages clears is higher than the output price [P.sup.C.sub.t] that results in a real wage at which the labor market clears without efficiency wages.

[FIGURE 3 OMITTED]

Between [P.sup.C*.sub.t]. and [P.sup.k*.sub.t] output is negatively related to price increases because decreases in the real wage decrease average effort, although employment [N.sub.t] in Figure 1 remains the same. If the price increases above [P.sup.k*.sub.t] and the real wage decreases below [w.sup.k.sub.t], further increases in price will decrease both the supply of employees as well as their effort. For this reason, the aggregate supply curve is negatively sloped above [P.sup.k*.sub.t] but flatter than between [P.sup.C*.sub.t]. and [P.sup.k*.sub.t]. Efficiency wage behavior clearly affects the shape of the aggregate supply curve, creating a second kink.

The aggregate demands portrayed in Figure 3 are the same as those in Figure 2. Notice that the intersection of AD([M.sup.L.sub.t]) with AS([W.sub.t], e[[w.sub.t]] < 1) is at a higher output price, [P.sup.L.sub.t], and at a lower level of output, than the intersection with AS([W.sub.t] e[[w.sub.t]] = 1) at [P.sup.L.sub.t]. Thus efficiency wage behavior may increase both output price and involuntary unemployment, similar to the conventional conclusion concerning efficiency wage behavior.

[FIGURE 3 OMITTED]

IV. GENERAL EQUILIBRIUM

Equilibrium occurs when aggregate demand (equation [7]) equals aggregate supply (equation [13]). This equality determines [P.sub.t] and [Y.sub.t], given [W.sub.t] and the realization of [M.sub.t]. This solution is employed to solve for the optimal wage (equation [15]). By solving for the equilibrium values of [P.sub.t] and [Y.sub.t] (and substituting [L.sub.t] derived using equation [8] for [Y.sub.t]) for each possible realization of [M.sub.t] and knowing all the possible equilibrium outcomes and their probabilities, equation (15) can be reduced to a complicated implicit equation in terms of [W.sub.t] and [M.sub.t]. (12) At each equilibrium value of [P.sub.t] and [Y.sub.t], each individual consumer, worker, and firm owner is fully optimizing subject to the relevant constraints. Each firm is maximizing profit subject to firms and workers choosing to voluntarily employ or be employed. Workers are involuntarily unemployed because firms do not voluntarily choose to hire them. However, they are optimizing subject to this constraint.

In our numerical analysis we considered three alternative specifications of [M.sup.L.sub.t] and [M.sup.H.sub.t] that generate employment determined on (1) the positively sloped section of the supply of labor, (2) the vertical section of the supply of labor, or (3) the demand for labor. (13) For each of the specifications we calculate the numerical value of the optimal wage. For the numerical example in which employment is determined on the demand for labor, involuntary unemployment equals 12.9% when [M.sup.L.sub.t] is realized. However, for this specification it is clearly not in the interest of the firm to set the competitive labor market clearing wage because the firm's expected profit is 80% higher under monopsony power in the labor market. There is no obvious mechanism to eliminate the resulting inefficiency in this model because such a mechanism would have to compensate the owners of firms for the consequent decrease in their expected profit.

As a formal matter, in a fully rational expectations model, changes in the quantity of money, [M.sup.L.sub.t] or [M.sup.H.sub.t], should affect [W.sub.t]. However, the numerical analysis revealed that E(II) is extremely flat with respect to [W.sub.t]--so flat that the computer analysis generates a large set of "optimal" wages and frequently could not converge to a unique value of [W.sub.t]. (We chose [W.sub.t], directly from a graph of this set.) This implies that the choice of a nonrational [W.sub.t] has a negligible effect on the expected profit of a firm in our example.

V. THE EFFECTIVENESS OF MONETARY POLICY

Aggregate supply, equation (13), and aggregate demand, equation (7), can be written as,

(16) [Y.sub.t] - AS([S.sup.j][w.sub.t]) = 0

(17) [Y.sub.t] - [(1 - [beta]).sup.-1] [M.sub.t]/[P.sub.t] = 0

Differentiating this system results in

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because employment will be determined either on the upward-sloped part of the labor supply curve, on the vertical part of the labor supply curve, or on the demand for labor, the aggregate supply function (equation [13]) incorporates (1) changes from a positive or vertical slope to a negative slope as the economy changes regimes from a state of recession to a state of expansion, (2) possible changes between inelastic and elastic labor supply in a state of expansion, and (3) possible changes between conventional and efficiency wage behavior in any regime. The different regimes in this model imply different specifications for [a.sub.12] and whether [a.sub.12] <,=, or > 0. We denote the Jacobian by, [[delta].sup.i], i = D or S when the demand or supply of labor is binding and assume that the upper bound of the possible parameter values of [[phi].sup.*] and [[psi].sup.*] ensure that these are both positive.

Given the nominal wage, if the Jacobian is nonzero, there is a monotonic (positive) relationship between the price of output and the quantity of money as we will demonstrate. Hence, there exists a unique value of the money supply, [M.sup.c.sub.t], and associated output price, [P.sup.c.sub.t], for which the aggregated marginal (efficiency) product of labor equals the aggregated supply of labor (refer to Figures 1, 2, or 3). The associated real wage and employment are [w.sup.c.sub.t] and [L.sup.c.sub.t].

Monetary policy is considered effective if an increase in money increases output.

Recessions

A recession or contraction is defined by deficient aggregate demand. This occurs whenever [M.sub.t] < [M.sup.c.sub.t]. In this event employment is determined by firms' demand for labor, and aggregate supply, denoted with the superscript D, depends on the sum of all firms' marginal efficiency products of labor,

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[H.sub.e] is a heavy-side variable. If the real wage is sufficiently high and there is no shirking, [w.sup.*.sub.t] [less than or equal to] [w.sub.t], then [H.sub.e] = 1. In this event [a.sub.12] = [-K[theta]/(1-[theta])][([w.sub.t]).sup.-[theta]]/ (1-[theta])][[theta].sup.[theta]/(1-[theta])] [([P.sub.t]).sup.-1] 0, and the Jacobian, [[delta].sup.D] > 0. If [H.sub.e], = 0 (as [w.sup.*.sub.t] > [w.sub.t]), then e([w.sub.t]) < 1 and [a.sub.12] = - [K[[theta].sup.1/(1-[theta])] (1 - [phi][theta])/ (1 - [theta])] x [[[([w.sub.t]).sup.1 - [phi][theta]] [([w.sup.*.sub.t]).sup.[phi][theta]]].sup.-[theta]/(1 -[theta])] x [([P.sub.t]).sup.-1] [less than or equal to] 0, if [phi] [less than or equal to] 1.

A positive monetary shock increases output when [M.sub.t] < [[M.sup.c.sub.t], if [H.sub.e], = 1 or if [H.sub.e] = 0 and [phi] < 1 (the elasticity of effort is less than one).

(20) d[Y.sub.t]/d[M.sub.t] = -[([[DELTA].sup.D])sup.-1] [a.sub.12]/(1 - [beta])[P.sub.t] > 0.

A positive monetary shock always increases the price level,

(21) d[P.sub.t]/d[M.sub.t] = -[([[DELTA].sup.D])sup.-1]/ (1 - [beta])[P.sub.t] > 0.

If [H.sub.e] = 0 and [phi] = 1, then this behavior is consistent with the simple quantity theory of money (equation [7]), d ln [P.sub.t]/d ln [M.sub.t], = 1, and d [Y.sub.t]/d[M.sub.t] = 0.

Because employment is determined by firms' demand for labor, a monetary shock that decreases the price level increases the real wage and decreases employment and output. This is consistent with the empirical observations of cyclical involuntary unemployment, a positive covariation between money and output (and price), and a negative covariation between real wages and employment in recessions.

Expansions

An expansion is defined by [M.sub.t] > [[M.sup.c.sub.t]. For [M.sub.t] > [[M.sup.c.sub.t], the realized price will be above [[M.sup.c.sub.t] and the real wage will be below [w.sup.c.sub.t]. The amount of labor that firms will want to hire will be determined by their marginal (efficiency) product of labor (equation [12]), but the desires of firms will be thwarted because the number of potential employees willing to work at this wage, determined by the supply of labor, is less than what firms desire.

With employment determined by the supply of labor, aggregate supply, denoted with the superscript S, can be expressed as

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[H.sub.L] is a heavy-side variable that has the value [H.sub.e] = 1, if [w.sub.t] [greater than or equal to] [w.sup.k.sub.t] and [S.sub.j]([w.sub.t]) = [[N.sup.j.sub.t] Otherwise, [H.sub.L] = 0 and S(w) = [w.sup.[PSI]]. Increases in the quantity of money may either not affect output or actually decrease it. If [H.sub.e] = 1 and [H.sub.L] = 1 (the real wage is sufficiently high that there is no shirking and labor is fully participating), then [a.sub.12] = 0, the Jacobian is [[DELTA].sup.S] > 0 and monetary policy is ineffectual,

(23) d[Y.sub.t]/d[M.sub.t] = 0.

There are three possibilities for [a.sub.12] > 0 and an increase in money to decrease output, d[Y.sub.t]/d[M.sub.t] < 0 The decrease in output can be entirely the result of a decrease in the quantity of labor supplied, entirely the result of efficiency wage behavior, or a combination of the two.

1. If [H.sub.e] = 1 and [H.sub.L] = 0 (the real wage is sufficiently high that there is no shirking but the upward-sloped section of the labor supply curve is operational), then [a.sub.12] > 0 and the Jacobian, [[DELTA].sup.s] > 0, restricts [PSI] to not be too large defined by, 0 [less than or equal to] [PSI] < [[PSI].sup.*] = ([M.sub.t]/[P.sub.t])([w.sub.t).sup.-[PSI][theta]/(1 - [beta]) [theta]K.

2. If [H.sub.e] = 0 and [H.sub.L] = 1 (the real wage is sufficiently high that all potential workers are participating in the labor force but there is shirking), then [a.sub.12] > 0 and the Jacobian, [[DELTA].sup.S] > 0, restricts [[theta].sup.*] to be ([M.sub.t/[P.sub.t])[([w.sub.t])sup.-[phi][theta] [([w.sup.*]).sup.[phi][theta]]/(1 -[beta])[theta][N.sub.t] > [[psi].sup.*].

3. If [H.sub.e] = 0 and [H.sub.L] = 0 (there is both shirking and not all potential workers are participating in the labor force), then [a.sub.12] > 0 and the Jacobian, [[DELTA].sup.S] > 0, restricts ([phi] + [psi]) < ([[phi].sup.*] + [[psi].sup.*]) = ([M.sub.t/[P.sub.t])[([w.sub.t])sup.-[theta] ([phi]+[psi])]([w.sup.*]).sup.[phi][psi]]/(1 - [beta])[theta]K. [[phi].sup.*] and [[psi].sup.*] are the least of the restrictions necessary for a positive Jacobian.

Note that an increase in money that decreases output can decrease employment (and perhaps effort) but does not cause cyclical involuntary unemployment in any of the three cases.

A positive monetary shock always increases the price level,

(24) d[P.sub.t]/d[M.sub.t] = [([[DELTA].sup.S])sup.-1]/ (1 - [beta])[P.sub.t] > 0.

The aggregate demand equation (7) implies. d ln [Y.sub.t] + d ln [P.sub.t] = d ln [M.sub.t]. If d [Y.sub.t]/d[M.sub.t] = 0, then d ln [P.sub.t]/d[M.sub.t], = 1, as argued by monetary economists who believe in the quantity theory of money. However, when d[Y.sub.t]/d[M.sub.t] < 0 , this implies d ln [P.sub.t]/d[M.sub.t] > 1 or that a monetary expansion will create a larger percentage increase in inflation during an expansion than monetary economists have previously argued. This is due to the decrease in output associated with either or both decreased effort or the labor force participation rate by workers accompanying the decrease in real wages in an expansion.

VI. CONCLUSIONS

We demonstrated that when firms have monopsony power to set the optimal nominal wage before the stochastic output price is realized and are faced with an upward-sloped or bounded supply of labor, the potential for disequilibrium in the labor market arises even if all agents have fully rational expectations and there is market clearing in the goods market. This model generates two distinct regimes and results consistent with three key empirical characteristics of the business cycle. (1) Unemployment fluctuates with recessions and expansions and at least some unemployment seems best characterized as involuntary. (2) Monetary shocks have output and employment effects at least some of the time, and such effects are asymmetric with respect to the state of the economy. (3) The relationship between real wages and employment appears to be negative in contractions and either positive or zero during expansions.

Optimal behavior in the labor market by both firms and laborers, with or without efficiency wage behavior, produces two regimes. These are characterized by a kink in the aggregate supply of output curve at the price associated with the competitive labor market-clearing real wage. Below the kink the demand for labor is binding, excess supply of labor and involuntary unemployment occurs, income and employment vary inversely with the real wage rate, and monetary policy is effective. Above the kink in the aggregate supply of output the labor supply function is binding; there is excess demand for labor and no involuntary unemployment; income and employment either vary directly or not at all with the real wage rate; and monetary policy is ineffective. In this model as real wages fall below their competitive market clearing value, the supply of labor, effort, and output may remain constant in a range, but ultimately the supply of labor and effort decreases. Thus the aggregate supply of output may be vertical for a range of prices but eventually is negative or backward-bending at values of the model corresponding to (portions of) expansions or cyclical full employment. (14)

The microfoundation for these attributes of the business cycle proposed here is attractive for several reasons. The results do not depend on imperfections in the goods market, errors in expectations, ad hoc rules of thumb, bounded rationality, nonoptimizing, or less than fully rational expectations behavior by some agents. The assumption of a market clearing goods market is consistent with the consensus of opinion that goods markets are predominantly competitive, whereas the assumption of monopsony in the labor market is consistent with recent conclusions that labor markets are better characterized as monopsonistic or monopsonisticly competitive, rather than either competitive or by monopoly power wielded by unions. (15) Inefficiency occurs and persists in this model because the interests of the owners of the firms are distinct from those of their employees, and firms have a higher expected profit by not setting the competitive market clearing wage.

(1.) In the United States, unemployment reached 18% in 1894 and 25% in 1932 (McCallum 1989, p. 8) and exceeded 30"/0 in 1933 in Germany.

(2.) Benassy (2002) uses the same terminology.

(3.) For example, see Blanchard and Fisher (1989, sec. 8.1) and Romer (2001, sec. 6.4).

(4.) It is common to specify an upward-sloped, firm-specific labor supply in macroeconomic modeling, albeit without the assumption of monopsony behavior (Blanchard and Fisher 1989, section 8.1; Romer 2001, section 6.4; Woodford 2003, pp. 147-49).

(5.) The assumption that firms do not contract away their uncertainty concerning output price either by futures contracts for output or cost of living adjustment is consistent with the fact that less that 16% of the employees of firms in the United States are covered by such contracts.

(6.) Conventionally, a stationary monetary policy will generate an unstable stationary equilibrium in an OG model when population and technology are constant. See Blanchard and Fisher (1989)or Jullien and Picard (1998).

(7.) The supply behavior of households, in general, should be based on a vector of consumer goods prices or an index of their price level. This would differ from the price of the output for particular firm except in the case of a composite commodity model.

(8.) This assumption, commonly made in macroeconomics (see note 2), is similar to that made, usually implicitly, in the analysis of the welfare effects to a distortion to a competitive market.

(9.) See the Reader's Guide for the derivation available at www.economics.buffalo.edu/facultystaff/ jholmes/readersguide.pdf.

(10.) As workers are paid lower real wages, [w.sub.t] < [w.sup.*.sub.t] the average productivity at firm j decreases as more employees shirk. Efficiency wage behavior, while making the firm's choice more complex, presents no conceptual difficulties.

(11.) Entry of new firms is limited by the structure assumed.

(12.) Equation (15) involves terms not defined for probability distributions that give a positive weight to [P.sub.t] = 0. Hence, it is necessary to either restrict the class of probability distributions of money such that a price of zero cannot occur or to consider nonrational expectations solutions to the monopsonist's problem, that is, rules of thumb. If one recognizes that the lowest positive price that can be paid is 1 cent, then restricting [P.sub.t] away from 0 should not be a problem for discrete distributions, such as the Bernoulli. For continuous probability density functions, the distribution of price could be truncated at $0.01 and the probability density function adjusted appropriately.

(13.) The Reader's Guide available at our Web site presents the details of how the labor and goods markets are simultaneously solved and present the optimal values of nominal wage and the general equilibrium values of prices, real wages, and employment when the model is parameterized as ([theta],[psi], [phi], [w.sup.*.sub.t], k, [N.sup.j.sub.t], [a.sup.j.sub.t], K) = (0.6, 0.9, 0.0, 1, 0.5, 0.5. 1,1) and X = 0.20. [theta], labor's share of output, is set at 0.6 to approximate average data in the United States, [PSI] = 0.9 satisfies the sufficient condition for a solution, and the remainder of the parameter values are arbitrary, simplifying values. X = 0.20 approximates the relative frequency of recessions.

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(14.) It is tempting to speculate whether this portion of this model might explain the behavior of employment and real wages during parts of the German hyperinflation when a fall in real wages was apparently accompanied by low unemployment rates and decreases in output.

(15.) Woodford (2003, pp. 219-20) has argued for the direct opposite on the basis that the rigorous theoretical foundations presented here and the empirical evidence presented elsewhere did not exist. It is conceptually impossible for involuntary unemployment to occur in his framework.

PATRICIA A. HUTTON, We wish to thank Ronald Bodkin, an anonymous referee, an associate editor, and especially Thomas Cosimano for helpful comments of earlier drafts.

Holmes: Professor of Economics, State University of New York at Buffalo, Buffalo, NY 14260. Phone 1-716-688-2461, Fax 1-716-689-6072, E-mail ecoholme@buffalo.edu

Hutton: Professor of Economics, Canisius College, Buffalo, NY 14208. Phone 1-716-888-2673, Fax 1-718-689-6072, E-mail hutton@canisius.edu

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Author: | Holmes, James M.; Hutton, Patricia A. |
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Publication: | Economic Inquiry |

Geographic Code: | 1USA |

Date: | Jan 1, 2005 |

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