Theorem T.1: With Tax[y, q], minimum wage setting M = B + Tax[M], and balanced growth, then: if B is indexed on the net average wage and x and c on inflation only, then M rises faster than other wages, and unemployment rises.
Note: That M rises faster than other wages is not inconsistent with balanced growth. For M is only the selection of one of the proper wages that is taken to be the minimum wage.
Proof:
For all clarity, parameter r will not be indexed. Let the price level index again be P. Again W = P rwi W[0]. With real wage index rwi, the nominal index is wi = P rwi. For heterogeneous wages with wage density, we have w = wi w[0] along the balanced growth path.
For a dynamic path we have starting position B[0] giving M[0]. In the base year the minimum level is taxed at an average rate less than r, implying that B[0] > (1 - r) M[0].
We also use J as the index for the real minimum wage:
M = P J M[0] i.e. J = M / (P M[0])
(1) We first prove that J > rsi in the limit. There are two relations for B, with rsi given by the relation above:
B = P rsi[rwi] B[0]
B = M - Tax[M, (r, P x[0], P c[0])]