# WAGE AND UNEMPLOYMENT

Accordingly, we return to the relation of unemployment to the bargaining- determined wage. How much, indeed, would we expect the wage *z _{t}* to vary with changes in the rate of unemployment? Recall that

*z*has three components (equation A11.9k in the appendix to Chapter 11 and A14.3b in the appendix to this chapter):

_{t}

For an employee whose second-best alternative is to seek other employment, supposing the search for new employment is successful, the payoff of the new job will have the same components. Moreover, if the job search fails in the current period, the individual may obtain employment in a future period. Similarly, if he remains in his present employment, he can expect remuneration in future periods, which will comprise the same three components, along with some probability of disemployment in the future periods. These sequences of expected future remuneration could be quite complex, but we are concerned here with the response to changes in unemployment in the *current* period, so we let the expected discounted value of remuneration in future periods, conditional on separation from the current job, be a constant Z^{1}. We let the expected discounted value of remuneration in future periods conditional on continuation in the job for the current period be a constant Z^{2}. Let the probability of re-employment in the current period bep, y* the net share of the surplus to be expected in that case, and *g** the expected value of the compensation for effort in the new job. Suppose further that there is a fixed cost, g, of entering the labor market. This might be a resource cost such as the cost of travel to attend interviews. It might also be a subjective fixed cost. Surveys of self-assessed quality of life indicate that the unemployed are less happy than those who are employed *at the same income* (for example, Frey, 2008). Then this subjective fixed cost is a component of g. Thus, a rational agent then will leave the job if

That is, the rationality constraint *y _{t} $ w_{t}* is equivalent to
That is,

Assume for simplicity that Z^{1} = Z^{2}. Then (14.5) becomes
That is,

Thus,

With *V* the coalitional surplus in the present job net of the outside options of all members of the coalition. This will be illustrated by a numerical example. For simplicity and for comparability with received theory, assume that all employees have the same outside option y* = z* - *g** - w_{;}, *g(h) =* g*, and that there are *n* employees. Then

As *p* varies from zero to one *pip* varies from 0 to 0.5 and *ppp* varies from 1 to 0.5. Thus, the first term is a constant with respect top, the last varies directly with p, and the middle term varies inversely with p, provided *pz* > g.*

As a first approximation, let the overall average *p* be the ratio of the number of job openings, J, to the number unemployed, *U.* This may oversimplify - some job openings will be filled directly by transfers from other jobs, without a period of unemployment, and some openings may indeed not be available to the unemployed, for example. Further, Chapter 12 has proposed a different measure. Nevertheless, for simplicity and comparability with the received literature, and preliminarily, posit

*Figure 14.1 Job openings as a proportion of unemployment*

Taking data on the level of job openings in the United States for 2001-2014 from the JOLTS project of the Bureau of Labor Statistics and data on unemployment levels from the Bureau of Labor Statistics for the same period, we see the quotient as plotted in Figure 14.1. On the one hand, the quotient varies between 0 and 1 and so may, without contradiction (at least in this period) be interpreted as a probability. On the other hand, it varies over most of that range in a highly countercyclical way. Thus it would seem that, by identifying this quotient with the probability of quick re-employment, we err in the direction of exaggerating the impact of cyclical variations on the computed wage, rather than in the opposite direction.

For a numerical example, suppose that *1 _{i}n =* 1/2,

*z* =*42,500,

*g* =*35,000, and l (V*

*— w*= 1000 and g = 5000. Note that

_{A}— g i^bw)*In =*0.5, would imply that employees as a group obtain about half of the surplus: if l is the same for all employees, then they receive exactly half. Then we obtain the calculated values for

*z*in four contrasting periods as shown in Table 14.1.

_{i}We see that the projected value of z varies only slightly as the unemployment rate fluctuates from its lows to its highs over “the great recession.” The variation is less than 2 percent. The quotient *J/U* and the predicted

*Table 14.1 Computed wage payments*

Period |
Average unemployment rate |
Average quotient |
Computed wage |

2002-2005 |
5.6 |
0.439459449 |
46144.85541 |

2006-2008 |
5.008 |
0.561981044 |
46349.20262 |

2009-2011 |
9.28 |
0.200062587 |
45625.16298 |

2010-2014 |
7.6 |
0.331186084 |
45932.96334 |

*Figure 14.2 Thepredicted wage and the quotient J/U*

wage are plotted together in Figure 14.2 with the predicted wage expressed as a proportion of 100,000 for comparability.

A reason for this result is that g*, the compensation for the disutility of effort, is relatively large by comparison with z*, the expected wage conditional on re-employment, and *z.* Since the individual’s bargaining share, *y ,,* is *net* of this compensation, and the compensation recurs in the case of re-employment, it reduces the impact of *J/U* on the predicted wage. To illustrate this, consider a second example with z* = 40,000, g* = 0, g = 0, and *1 _{i} (V** —

*w*[ gw,) =40,000. Then we have Table 14.2 and Figure 14.3 by contrast with Table 14.1 and Figure 14.2. While the variation in the wage from the low to high points in 2002-2014 does not appear large by comparison with that of

_{A}— g ,*J/U,*it is in the range of 9 percent, about five times as great as the previous example.

*Table 14.2 Computed wage payments in an alternative example*

Period |
Average unemployment rate |
Average quotient |
Computed wage |

2002-2005 |
5.6 |
0.439459449 |
46105.89551 |

2006-2008 |
5.008 |
0.561981044 |
47195.7473 |

2009-2011 |
9.28 |
0.200062587 |
43334.20255 |

2010-2014 |
7.6 |
0.331186084 |
44975.80448 |

*Figure 14.3 The predicted wage and the quotient J/U in the alternative example*

It appears that the application of bargaining theory to the determination of wages in search-and-matching models has overlooked a key question: what proportion of the wage is compensation for the disutility of effort? If that proportion is large, then the model does not predict any considerable countercyclical variation of wages with unemployment. We note that the compensation for effort depends on the structure of production, and not on the employee’s outside option. In a more complex model, there might be other components of employee compensation that depend on the persistent structure of production rather than on the employees’ outside option, and this result probably is more valuable as an instance of that dependence than as a complete theory of wages. The question should be, to what extent are wages determined by the structure of production

*Figure 14.4 Two measures of the probability of re-employment*

rather than by the employee’s outside option? The evidence that real wages are sticky suggests that wages are primarily determined by the persistent structure of production.

All in all, the prediction that the wage would fluctuate countercyclically is weakly supported if at all. If the bargaining model is somewhat more realistically complex, and in particular includes a component of the wage that compensates for efficient effort, then the predicted countercyclical fluctuation in wages could be very slight, if not indeed negligible.

Thus far we have used the ratio J/U as the probability of prompt reemployment in determining the individual employee’s outside option and thus wage. Suppose instead we adopt the index p = [1 — (1 — U*) ^{J}* ] suggested by the search model of section 12.2.1. A preliminary examination of the two probability expressions suggests that there will be little difference, but that this substitution would further reduce the responsiveness of the wage to unemployment. This is suggested by Figure 14.4, which compares the two estimates of the probability of re-employment.

Then, replacing *J/U* with the index, we have Table 14.3 in place of Table 14.1, and in place of Figure 14.2 we have Figure 14.5.

It seems that the bargaining hypothesis cannot be dismissed on the grounds that the wage does not vary proportionately with the rate of unemployment.

*Table 14.3 Computed wage payments*

Period |
Average unemployment rate |
Average index |
Computed wage |

2002-2005 |
5.6 |
0.354234575 |
45980.90809 |

2006-2008 |
5.008 |
0.425803662 |
46119.90436 |

2009-2011 |
9.28 |
0.180904544 |
45574.46814 |

2010-2014 |
7.6 |
0.281289003 |
45823.25982 |

*Figure 14.5 The probability of re-employment and the predicted wage in a third example*