## Statistical Equilibrium in a Simple Labor Market

Duncan K. Foley Labor Market Stat Eq 9/8/95 page 1 Statistical Equilibrium in a Simple Labor Market by Duncan K. Foley Department of Economics Barn...
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Duncan K. Foley

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idea of statistical equilibrium is to consider all feasible Pareto-improving market transactions, where a market transaction (x1,...,xn) specifies a trade xi in Rm for each agent, where m is the number of commodities traded, and negative components represent a supply of the commodity to the market. When there are many agents of each type, k = 1,...,K , characterized by a (finite) offer set Ak that specifies the trades an agent will accept voluntarily (her reservation price in the simplest Marshallian case), each Pareto-improving transaction implies a transaction distribution of the agents over their offer sets hk:Ak->[0,1] showing the proportion of agents of type k assigned to each point in Ak. The number of different market transactions that lead to a given transaction distribution (through permuting agents of the same type who receive different trades) is the multiplicity of the transaction distribution. The statistical market equilibrium is defined to be the transaction distribution that has maximum multiplicity. The logarithm of the multiplicity of a transaction distribution is closely approximated by the weighted informational entropy of the distributions, H = K

∑wk( ∑hk[x] lnhk[x]) ), the negative of the weighted sums of the expectation of k=1

xεAk

the logarithm of the probabilities, where the weights wk are the proportion of agents of type k in the market. Statistical market equilibria are Gibbsian canonical distributions. The maximization of the entropy subject to m resource constraints determines a vector of m Lagrange multipliers, or entropy prices, q = (q1 ,...,qm ), one for each traded commodity. The statistical market equilibrium trading distribution makes the probability of finding an agent at a particular trade in her offer set inversely proportional to the exponential of the value of the trade at the entropy prices, so that hk[x] ~ exp[-qx]. The entropy prices, q, must be chosen to clear the K

market, that is, make the average trade,

∑wk( ∑hk[x] x) ) = 0. k=1

xεAk

The normalizing constant in the distribution of each type's transaction distribution, considered as a function of the entropy prices, Zk[q] = ∑exp[-qx] , xεAk is called the partition function for type k, (which is also the Laplace transform of the indicator function of the offer set of type k). We see that hk[x|q] = exp[-qx]/Zk[q]. An easy calculation also shows that -dlnZk[q]/dq =

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∑exp[-qx]x/Zk[q] = xεAk

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_

∑hk[x]x = x k[q] is the average excess demand vector of xεAk

the agents of type k at the entropy prices q. The logarithm of the market partition function Z[q] is defined as the weighted sum of the logarithms of the partition K _ functions of each type, lnZ[q] = ∑wklnZk[q] , and -dlnZ[q]/dq = x [q] is the k=1

vector of excess market demand at the entropy prices q. Statistical market _ equilibrium occurs at the entropy prices q* at which x [q*] = 0. Statistical market equilibrium differs from Walrasian/Marshallian equilibrium in each of the three respects noted above. First, all feasible Pareto-improving multilateral trades are given some weight in the statistical equilibrium (although the weight will be very small for trades that have a high value at the equilibrium entropy prices). The statistical market equilibrium allows for trades between, for example, sellers with reservation prices above the Marshallian equilibrium price, and demanders with even higher reservation prices, although it renders those trades relatively improbable. Second, different agents of the same type get different market outcomes in statistical market equilibrium, due to their trading at different effective price ratios. The statistical market equilibrium induces endogenous horizontal inequality into market outcomes. Third, statistical market equilibrium moves toward, but does not fully achieve, Pareto-efficiency. In the statistical market equilibrium there will still be some mutually advantageous trades available. The relative rarity of these trades is measured by the equilibrium entropy prices. In particular, unemployment and excess capacity are equilibrium phenomena from the statistical point of view. In any given market, failure to trade may be large or small depending on the offer sets of the agents. The purpose of this paper is to analyze the statistical equilibrium of a highly simplified model of labor markets in order to make the application of the canonical method clearer in an economic context, and to contrast the statistical and Marshallian/Walrasian points of view. 2. A model of the labor market We consider a simplified model of the labor market, in which there are two types of agents, households and firms. Two commodities are traded in the July 22, 2008

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market, labor-power, v, and wage income y. Wage income may be thought of either as real income or as money income. In reality, of course, labor markets trade money for labor-power. Applying statistical equilibrium theory to a labor market requires accepting strong assumptions. The statistical equilibrium is a model of a single run of a market. In applying it to a labor market we must assume that the market has no elements of learning, experience, or memory, so that workers and firms enter the market with no information as to the likely level of the wage or employment. If the market takes place repeatedly under similar circumstances, the model would have to modified to take account of workers' and firms' learning from experience and modifying their offer sets accordingly. This lack of experience underlies the assumption of equal a priori probabilities on which the statistical equilibrium is based. If workers and firms entered the market with more information, we would have to modify the assumption of equal a priori probabilities in describing the offer set. As is true of all applications of statistical equilibrium reasoning, including those arising in the physical theory of statistical thermodynamics, the predictive relevance of statistical equilibrium is contingent on the statistical model incorporating all the constraints that produce observable regularities in the system under analysis. If the labor market shows regularities not reflected in the statistical equilibrium, we must trace these to further constraints on the states of the market and incorporate them explicitly into the model. Our confidence in the statistical equilibrium model might arise from dynamical considerations, if, for example, we believed that the market worked in such a way as to systematically explore all the possible mutually advantageous allocations. (This rationale would be analogous to the assumption of ergodicity in physical systems.) Observing the a particular run of the market we might then believe that we would be overwhelmingly likely to observe an outcome close to the statistical equilibrium because the huge multiplicity of such outcomes dominate all other possibilities. But an explicit dynamic rationale for the statistical method is not the only possible reason for adopting it. Even if the market does not systematically explore all the possible mutually advantageous reallocations, it might have no observable tendency to produce allocations different from the statistical equilibrium predictions because there is no information available to the participants that would allow them to reach a more organized outcome.

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3. The household partition function and excess demands For simplicity, assume that households either supply labor-power fulltime, corresponding to v = -1 (where we normalize the measure of labor-power to one full-time unit, or a job, and we measure the supply of labor power as a negative number) or are unemployed, corresponding to v = 0. Furthermore, assume that all households have a minimum reservation wage wm i n which they will accept for a job, but that they will work for any wage higher than this. Assume that unemployed households earn no wage. Then the household offer set consists of a point at the origin, (v,y) = (0,0), representing unemployment, and a vertical ray extending upward from (-1,wm i n ), as in Figure 3-1.

Figure 3-1 A simple household offer set. The household is either unemployed at (0,0), or offers 1 unit of labor-power (a job) at or above a reservation wage wm i n .

Despite the fact that this set is not finite, we could regard the ray as approximating a countable set of points spaced more and more widely as y increases, as in Figure 3-2.

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Figure 3-2 The household offer set containing a ray approximates a countable set.

Points with very high wage income will play a negligible role in the equilibrium as long as the entropy price of wage income is positive, because those points will be very improbable. As Figure 3-2 is drawn, the available points become sparser as wage income rises. We can represent this effect in the continuous ray set by assuming that there is a measure of density of points proportional to, say, exp[-fy] where f is a positive fixed number. The size of f also determines the practical importance of the single point of unemployment at the origin in relation to the ray. If we make the assumption of a declining density of points with wage income, we can find the partition function for employed households: exp[qv-(f+qy)wmin] Zeh[(qv,qy),f,wmin] = f+qy The average supply of labor and average wage of an employed worker can be found by differentiating the logarithm of the partition function with respect to the entropy prices: _ 1 x eh[(qv,qy),f,wmin] = {-1, f+qy + wmin} Since all employed workers by assumption supply a unit of labor-power, the average supply is -1, and the average wage depends on f, the density of wage income levels, and qy, the entropy price that determines what proportion of the employed workers will be at each wage income. As qy decreases, the average wage rises.

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The partition function for an unemployed household is just 1, since exp[-q0] = 1, no matter what q is: Zuh[(qv,qy),f,wmin] = 1 The full partition function for a worker is just the sum of the two partition functions: exp[qv-(f+qy)wmin] Zh[(qv,qy),f,wmin] = 1 + f+qy The average supply of labor per household, averaging out over both employed and unemployed workers, can also be calculated by differentiating the logarithm of the partition function: _ x h[(qv,qy),f,wmin] = exp[qv] {-exp[qv]+(f+qy)exp[(f+qy)wmin] , (1+(f+qy)wmin)exp[qv] (f+qy)(exp[qv]+(f+qy)exp[(f+qy)wmin]) } _ The first component of x h represents the employment rate, the expected labor supply of a household. The second is the average wage income per household, which is just the employment rate e multiplied by the average wage per employed worker. Figure 3-3 illustrates the interaction of the canonical distribution with the offer set. The straight lines are iso-entropy-value or iso-probability loci, with slope = -qw , where qw = qv /qy is the entropy wage. The density of the loci represents the probability, and falls as the value of the trade at the entropy prices rises.

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Figure 3-3 Entropy prices establish iso-probability lines over the offer set of the agent. In this case the probability of unemployment is the same as the probability of being employed at a wage income twice the reservation wage.

The higher is qv for a given qy , the steeper are the iso-probability lines, and the lower is the unemployment rate. We can see this by plotting the employment rate against qv , as in Figure 3-4.

Figure 3-4 The employment rate rises toward unity as the entropy price of labor increases, holding the entropy price of wage income constant.

As qy increases, on the other hand, employment declines, because a rise in qy corresponds to a fall in the entropy wage, qv /qy .

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Figure 3-5 As the entropy price of wage income rises, the employment rate falls monotonically, reflecting the decline in the entropy wage, qv/qy.

The units of the entropy prices are 1/commodity units. For example, the entropy price of labor-power, qv , has the units 1/units of labor-power and qy has the units 1/wage income, or 1/\$ more briefly, as does f. If we define the entropy wage qw = qv /(f+qy ), we can write the partition function in terms of the entropy wage and the inverse entropy price of wage income, t = 1/(f+qy ), which has the dimension of the wage income unit. temperature of a physical system. qv = qw/t qy = -f + (1/t)

t is analogous to the

In these variables, the partition function takes on a particularly simple form: Zh[(qw,t),f,wmin] = 1 + texp[(qw-wmin)/t] _ x h[(qw,t),f,wmin] = texp[qw/t] {-exp[qw/t]+texp[qw/t] , (t+wmin)texp[qw/t] exp[qw/t]+texp[qw/t] } We can plot the employment rate as a function of the entropy wage qw , holding t constant.

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Figure 3-6 This plot, identical to Figure 3-4, shows the employment rate as a function of the entropy wage.

We can also vary the temperature of the market, t, holding the entropy wage constant. This is a different experiment from varying qy , holding qv constant, which implicitly also changes the entropy wage qw = qv /qy .

Figure 3-7 A rise in the temperature of the labor market, holding the entropy wage constant, first reduces the employment rate, assuming that the entropy wage is higher than the reservation wage.

If the entropy wage is lower than the reservation wage, a rise in temperature has a different effect.

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Figure 3-8 When the entropy wage is lower than the reservation wage, a rise in temperature raises the employment rate.

The average wage of employed workers depends only on qy . In fact, the excess of the wages of employed workers over the reservation wage wm i n is distributed exponentially with the parameter 1/(f+qy ). We can also write this in terms of market temperature. _ x eh[(qw,t),f,wmin] = {-1,t+wmin} The excess of the mean wage of employed workers over the reservation wage is just t. Wages are distributed exponentially above wm i n with parameter t, and their standard deviation is also t, which could be a route for the empirical estimation of the temperature of the market, as well as an empirical test of this simple labor market model. There is a relation between the employment rate and the mean wage of employed workers, which we can see by varying t, holding the entropy wage, qw , constant.

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Figure 3-9 As the market temperature rises, holding constant the entropy wage, the mean wage of employed workers rises, while the employment rate first falls and then increases.

An alternative way to approach this relation is by varying qy , holding qv constant, so that both the entropy wage and the market temperature vary.

Figure 3-10 As qy rises, holding qv constant, a "tradeoff" between the employment rate and the average wage level appears, in which the average wage level rises sharply as the employment rate approaches 1.

4. A firm partition function and excess demands In order to close the model and calculate a full equilibrium, we need to represent the buyers of labor-power, firms. In order to keep the firm model as simple as possible, we will assume that each firm has a given amount of money

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capital (a "wage fund"), k, and also has a maximum reservation wage, wm a x , which could represent the productivity of labor. In order to simplify the model, we will assume that every firm actually spends its whole wage fund, so that there is no excess capacity on the firm side of the market similar to the unemployment we have allowed for on the household side. The firm could hire k/wm a x workers, if it spent its whole wage fund at its maximum wage. We will allow firms to benefit from bargains in the labor market, and to employ more workers than this. The firm offer set is shown in Figure 4-1:

Figure 4-1 The firm is assumed to spend its entire wage fund k at an effective wage no larger than its reservation level, wm a x .

Assuming an exponential density of points for the firm with the parameter g, the firm's partition function can be found by integrating exp[-(qv v + qy y)exp[-g v ] over the set Af = {(v,y)|y = -k, v ≥ k/wm a x }: exp[kqy-(k(g+qv))/wmax] Zf[(qv,qy),g,wmax,k] = g+qv The mean demand for labor-power and supply of wage income can be calculated by differentiating the logarithm of the firm partition function. _ x f[(qv,qy),g,wmax,k] = 1 k {g+qv + wmax ,-k} The mean demand for labor by the firm depends only on qv . The firm's supply of wage income to the market has to be, under the assumptions, -k. The mean wage is the ratio:

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k 1 k g+qv + wmax Since both the mean demand for labor and the mean wage for a firm in this simple model depend only on qv, we can plot a statistical demand schedule for labor by varying qv parametrically.

Figure 4-2 The statistical demand for labor in this simple model is a hyperbola.

5. Market clearing The labor market clears at entropy prices for labor and wage income, qv and qy, where the total supply of labor on the part of households equals the total demand on the part of firms, and the total household demand for wage income equals the total firm supply of wage income. Another way to think of this equilibrium is to require the supply and demand for labor to be equal and the mean wage from the point of view of firms and households to be equal. If there are H households, and J firms, we can write these market clearing conditions: H exp[qv] 1 k exp[qv]+(f+qy)exp[(f+qy)wmin] = J(g+qv + wmax ) (1+(f+qy)wmin)exp[qv] (f+qy)(exp[qv]+(f+qy)exp[(f+qy)wmin])

= J k

An alternative way to express the market clearing conditions is to equate the supply and demand for labor-power and the mean wage:

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H exp[qv] 1 exp[qv]+(f+qy)exp[(f+qy)wmin] = J(g+qv 1 k wmin + f+qy = 1 k g+qv + wmax

k + wmax )

It is easy to find market equilibrium in this model because the firm demand has such a transparent structure. In fact, the mean employment and mean wage of the firm are monotonic functions of qv alone. We can solve for qv in terms of w, the mean wage, on the demand side: gk(wmax-w)-w wmax qv = k(wmax-w) With this substitution, we can find a statistical demand curve for labor in terms of the mean market wage. Since each firm is required to spend its entire wage fund, this statistical demand curve is just a hyperbola. The curves plotted below correspond to J = 1000 firms, each with a capital k = 10 and a reservation wage wm a x = 2, and H = 12000 households, each with a reservation wage wm i n = 1. The parameters f and g are both assumed to be .1. vD[w,J,g,k,wmax,H,f,wmin] =

J k w

Figure 5-1 The statistical demand for labor is a section of a hyperbola.

In a similar fashion, the mean wage for households is a monotonic function of qy alone, so that we can solve for qy in terms of the mean wage on the supply side:

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qy =

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1-f(w - wmin) , or w-wmin 1 f+qy = w-wmin

It is not possible to establish a one-to-one relation between the mean wage and the supply of labor-power on the household side, because the supply of labor-power depends on both qv and qy . The mean wage determines qy, but the effective supply of labor-power also depends on qv . We can, however, find a pseudo-supply curve for labor-power by substituting using the relation between the mean wage and qv from the demand side. Given a mean wage w, we can find the qv corresponding to that wage from the demand side, and the qy corresponding to that wage from the supply side, and use these values to find the effective supply of labor-power: vS[w,J,g,k,wmax,H,f,wmin] = gk(wmax-w)-w wmax H exp[] k(wmax-w) gk(wmax-w)-w wmax exp[wmin/(w-wmin)] exp[]+ k(wmax-w) w-wmin

Figure 5-2 The statistical supply of labor rises smoothly from the reservation wage to full employment.

We can visualize the statistical equilibrium of this simple labor market by plotting the statistical supply and demand curves:

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Figure 5-3 The statistical equilibrium lies at the intersection of the statistical demand and supply curves for labor.

The statistical equilibrium mean wage lies above the household reservation wage, giving rise to Keynesian involuntary unemployment. 6. A comparison of Marshallian and statistical equilibrium The Marshallian demand curve for this market is just the union of the section of the hyperbola w = Jk/v to the right of 5000 units of labor-power (corresponding to the minimum employment at the reservation wage wm a x ) with a horizontal line at the reservation wage. In this simple model the statistical and Marshallian demand curves actually coincide on the hyperbolic portion.

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Figure 6-1 The Marshallian Demand curve is a section of a hyperbola, cut off by the reservation wage level.

The Marshallian supply of labor for this simple model is a vertical section of the axis up to the household reservation wage, wm i n , a horizontal line at the household reservation wage, and a vertical segment at the maximum supply of labor-power, H.

Figure 6-2 The Marshallian supply curve is infinitely elastic at the household reservation wage up to the maximum supply of labor-power.

The Marshallian equilibrium of this market lies at the intersection of the Marshallian supply and demand curves. The equilibrium wage is equal to the household reservation wage, so that there is no involuntary unemployment.

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Figure 6-3 The Marshallian equilibrium of the labor market has no involuntary unemployment.

If we juxtapose the Marshallian and statistical equilibria, we see how the statistical equilibrium corresponds to involuntary unemployment, with the wage above the reservation wage of the households, but the level of employment below the maximum supply of labor-power.

Figure 6-4 The Marshallian and statistical equilibria of the market differ because the statistical supply curve of labor lies above the Marshallian supply curve.

In Marshallian equilibrium, the existence of even a single unemployed worker creates enough competition to drive the rents of employed workers down to zero. As a result, the market can come into equilibrium only when the wage is equal to the reservation wage, so that this competition does not occur. July 22, 2008

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In statistical equilibrium, the existence of unemployed workers is not completely effective in competing away the rents of employed workers. As employment rises, so to speak, the competitive pressure of the remaining unemployed gradually diminishes, allowing the wage of the employed to rise. As a result of this process the statistical equilibrium is reached with a mean wage above the reservation wage of households, and is thus compatible with involuntary unemployment. The magnitude of the involuntary unemployment, of course, depends on the exact parameters of the model. An increase in the reservation wage of the firms, or in their number, will increase employment and the equilibrium wage. A decrease in the reservation wage of households or an increase in their numbers will increase equilibrium employment with a fall in the wage.

Figure 6-5 An increase in the number of firms raises the statistical demand for labor-power, leading to an increase in the equilibrium wage and employment levels.

7. Statistical Labor Market Equilibrium and the Efficiency Wage Theory of Unemployment The theory of efficiency wages put forward by Samuel Bowles (1985), and Carl Shapiro and Joseph Stiglitz (1984) (for a more complete review of the relevant literature, see Michael Woodford (1994) and Edmund S. Phelps (1994)) argues that firms will voluntarily pay workers rents above their reservation wages as part of a second-best strategy of discipline aimed at maximizing worker effort. A worker earning a wage equal to her reservation wage has no fear of being fired, and hence the threat of dismissal is ineffective in motivating her to

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supply effort on the job. The efficiency wage theory argues that premia of actual wages over reservation wages are a pervasive phenomenon in actual labor markets. The theory of efficiency wages undoubtedly corresponds to a real practice in labor markets in advanced capitalist economies. It also explains why we commonly observe queues for desirable jobs in that setting. It is not so clear that it actually constitutes a theory of equilibrium labor market unemployment, since there could be a low productivity sector of the economy in which the efficiency premium would be zero, and no job queue could exist (perhaps largely involving self-employment). The statistical labor market equilibrium also implies the existence of a gap between actual wages and reservation wages that could allow the threat of dismissal to motivate worker effort. In the statistical labor market equilibrium the average wage of employed workers exceeds their reservation wage. This gap establishes the involuntary character of the resulting equilibrium unemployment. But it also, viewed from the point of view of employed workers, makes the loss of a job costly, because being thrown into the pool of unemployed workers carries with it the finite probability of continuing unemployment. This gap occurs from the statistical equilibrium point of view whether or not firms actually seek to establish it as a matter of policy, because the clearance of the labor market is only statistical, and not complete. 8. Wage Subsidies and Employment in a Statistical Labor Market Equilibrium As a first step toward understanding the implications of the statistical labor market equilibrium for employment policy, consider the impact of a wage subsidy on the statistical equilibrium in this simple model. In a Marshallian market it doesn't make any difference whether workers or employers receive a subsidy, which is viewed as lowering the supply price of labor, or raising the demand price for labor by the amount of the subsidy at every level of employment. The situation in a statistical equilibrium is somewhat different. A subsidy paid to workers will change the workers' offer sets, while a subsidy paid to employers would change the employers' offer sets. For example in the present simple model, a subsidy to workers will lower wm i n , while a subsidy to employers will raise wm a x . Because these two parameters enter into the market

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equilibrium condition asymmetrically, the equilibrium entropy wage and temperature in the market may not be the same when the subsidy is paid to the workers as when it is paid to the employers. In order to simplify, we will model the wage subsidy when it is offered to employers as changing only wm a x , not k. In this interpretation the firm adds the wage subsidy to its profits, leaving its capital budget for production unchanged. The subsidy raises the profitability of each employed worker to the firm, thus increasing the firm's reservation wage, but does not supply more capital to the firm. With this assumption, we can see the impact of a subsidy paid to employers on the statistical labor market equilibrium:

Figure 8-1 A subsidy paid to employers shifts the pseudo-labor supply curve upward, lowering employment and raising the average wage.

The effect of the wage subsidy to employers, given their capital constraints, is to make them compete harder for labor, and thus to raise the average wage. Given the limited capital available in the market, the higher average wage corresponds to a lower level of employment. If we make a similar subsidy, paid to workers, the market looks like this:

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Figure 8-2 A subsidy paid to workers lowers the pseudo-supply curve of labor and raises employment while lowering the wage.

Subsidizing workers, on the other hand, lowers their reservation wage and this leads to a lower overall wage and a higher level of employment. These results are in striking contrast to the Marshallian analysis. In part this is due to the assumption of a fixed capital constraint underlying the demand for labor in the statistical equilibrium model. In general the results of applying a subsidy or tax to employers and workers will be different. We can compare the two different subsidies by plotting the equilibria on the same graph, together with the original, unsubsidized equilibrium:

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Figure 8-3 The labor market equilibria with the subsidy paid to employers (upper pseudo-supply curve), no subsidy (middle pseudo-supply curve), and to workers (lower pseudo-supply curve).

An advantage of the statistical approach to the analysis of the labor market is that some equilibrium level of unemployment is inherent in the statistical conception of market equilibrium. Models in the WalrasianMarshallian tradition start by predicting no real unemployment in market equilibrium, so that unemployment appears as a kind of anomaly in these models. The statistical approach, however, also raises more delicate questions of modeling. Because of the combinatorial nature of statistical equilibrium, parameters such as the number of individual firms and the capital budget of each firm, which play no role in the Marshallian demand schedule for labor, make a big difference from the statistical point of view, as the present very tentative exploration of the impact of wage subsidies makes clear. 9. References Bowles, Samuel. 1985. The production process in a competitive economy: Walrasian, neo-Hobbesian, and Marxian models. American Economic Review 75(1) (March), 16-36 Foley, Duncan K. 1994. A statistical equilibrium theory of markets. Journal of Economic Theory 62(2) (April), 321-345 Phelps, Edmund S. 1994. Structural Slumps: the Modern Equilibrium Theory of Unemployment, Interest and Assets. Cambridge, MA: Harvard University Press Shapiro, Carl and Joseph Stiglitz. 1984. Equilibrium unemployment as a worker discipline device. American Economic Review 74(3) (June), 433-44 July 22, 2008

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Woodford, Michael. 1994. Structural slumps. Journal of Economic Literature XXXII (December) pp 1784-1815

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