This imaginary free market for coupons is an effective tool for the analysis of the processes leading to changes in the optimal value of relative wages.
First, suppose that, starting from some equilibrium situation depicted by the solid line in figure V, with demand for labour J, the average job preference goes up, that is: the average critical value m shifts down to m` as it would in case of a shift from the solid line to the interrupted line in figure V.
Figure V: Displacements of equilibrium
Supply and demand then no longer intersect at point 1 in the lower half of fig. V but at point 2. The optimal relative wage has gone down. In a coupon market, the upward shift of average job preference would raise the coupon price and thus lower w/t, until pc would be such that, for the new (interrupted line) distribution Nm `,s
which is the ordinate of point 2. This means that average job preference (critical value) m is inversely related to the optimal relative wage.
Now suppose that from this new distribution Nm `,s , J, the distribution of the population shifts to a denser one (with s `< s ). The dotted lines in fig. V exemplify such a Nm `,s `. Graphically, it is easily seen how this affects the supply curve: CNm `,s ` will be steeper than CNm `,s in the middle, and less steep in both extremes.
What would cause a population's distribution of critical values to become denser? We may imagine cultural reasons, like increasing conformism. But a decreasing gross wage sum has the same effect. A decreasing wage sum W causes a downward shift of the budget line of fig. II. In fig. VI we have at once drawn a "minimum budget line" in the following way. We shaded the area in which either wage is below the bare subsistence level (wmin ) or transfer income is below the bare subsistence level (tmin ). The minimum budget line is defined as the line through the point with coordinates (wmin , tmin )
Figure VI
If W would sink to the level belonging to the minimum budget line, then (wmin,tmin) would be the only point at which the relative wage is such that neither the employed, nor the unemployed are below subsistence. No person would have a critical value (w/t)p � wmin/tmin. Opting for employment in case of w/t < wmin/tmin would reduce his income below subsistence, but choosing unemployment in case of w/t > wmin/tmin would also reduce his income below subsistence. So, shifting downward to the minimum budget line, we arrive at a distribution where m = wmin/tmin and s = 0. In our coupon market system, any increase of the relative wage above wmin/tmin would result in a run to the market of the unemployed in order to buy coupons, which would raise pc and reverse the direction taken by the value of relative wages. Any decrease of the relative wage below wmin/tmin would result in a run to the market by the employed in order to sell coupons, hereby lowering the coupon price and raising the relative wage again. In this predicament, job preference would not have any chance to determine the relative wage. It would however still decide who would be the employed and who would be the unemployed.
Concluding: we may adopt the view that a decrease of the total gross wage sum is associated with a denser distribution of critical values in a population, like the dotted distribution in figure V (for a sketch of the strategy for a formal proof see the next section). Again, this implies a leftward shift of the equilibrium point (from 2 to 3). Therefore a decrease of the total gross wage sum indeed lowers the relative wage. In the coupon market system it would raise the coupon price. It is of importance to observe that a curious reversal of the effect arises once we have more than 50% unemployment: in the lower half of the traject, the dotted line lies below the interrupted line, causing a reverse effect.
The effect of decreasing employment J`< J is quite unambiguously to decrease the relative wage. In the coupon system any downward shift of J to J` given P, would raise the coupon price, which is needed in order to have the newly unemployed J`-J to transgress their critical value downward. With distribution Nm ,s [(w/t)p], the equilibrium shifts from 1 to 4, with Nm ,s `[(w/t)p] it shifts from 2 to 5, and with Nm `,s [(w/t)p] it shifts from 3 to 6. Only with zero standard deviation of N do we have the extreme case of a vertical supply curve, where J has no effect on the equilibrium value of w/t, which is then either solely determined by job preference, if the reason for s = 0 would be extreme conformism; or equal to wmin/tmin if its reason would be a gross wage per head at bare subsistence level.