Book Iii. Another Approach Is More Formally, and a Commonly Used Route

here is the assumption of a constant macro-economic progression factor. This factor is the elasticity of tax revenue with respect to income (Koopmans (1975:103)), thus mepf = (Y / TAX) ( TAX / Y). The factor is determined by tax parameters, their indexation, the income distribution and its change. In this case, without a deficit, the progression factor applies to expenditure too, which may be taken to mean, effectively, that taxes are indexed such that tax revenue follows expenditure.

We shall take the progression factor for the average wage, which is exclusive of profits and the growth of employment. Thus our = (W / g) ( g / W). We assume a nominal position, thus include price developments in government expenditure relative to the average wage. We set gn = 0 now, since it can be included mathematically with gp. We also assume that is equal for gs and gp, so that gs = gs W / W = gs0 W and gp = gp W / W = gp0 W . Thus g = g W / W with properly g = gp + gs.

Then g / W g / W = NG / WT. This has the specific property that = 1 implies that the quote g / W = g / W is constant, and thus NG /WT is constant too. We will use this property below.

Taking W separate:

and hence

(27.3)

Inclusion of the progression factor does not cause special observations yet. If < 1 then in the limit of W the indexation can be rather simple, especially if Pb qr / W goes to zero too. If > 1, then there could be a point where the markup on W is zero, or subsistence would have to be zero - which would suggest an unrealistic tax function. The progression factor becomes more useful if we regard special cases.

Special cases

Definition: A (democratic) state is “Madisonian”, iff gs = 0. James Madison remarked that a proper democracy with a majority rule actually safeguards the interests of the minorities.

Definition: A “real welfare state” aspires at a constant RIR and takes q = 0. The idea on the latter is that breathing air is prerequisite to utility and no source of it. The berries in the field are owned by someone, and no longer free. (If they were free, then Coase’s Theorem shows that they could be counted as part of income, and hence they would no longer be free for all practical purposes.)

Definition: A “pragmatic” real welfare state sets u = 0 in the determination of the benefit level and RIR. The factor B u really does not amount to much.

Definition: “Uniform prices” means P = Pb = Pgs = Pgb = Pgn = Pq. If this happens then one price index P suffices.

Theorem B1: In a pragmatic Madisonian real welfare state with Ricardian equivalence and uniform prices, (i)

RIR = (B + g) / ((1 + Z) W) (base year)

and

B = W ((1 + Z) RIR - NG/WT) (henceforth)

(ii) If RIR is constant, then: (1) A constant quote for government layouts (or progression factor = 1) only allows for some variation in B/W by variation in the average tax rate difference Z. (2) If Z is constant, then B is fully indexed on W.

Proof:

(i) For the base year: substitute the results of the definitions in the RIR (vide (27.2)), note that the prices cancel and that g = gp. Then find the base year result as stated, and then use (NG /WT) W = g to get the annual expression.

(ii) For (1), we use = 1 NG /WT = g / W from above. Then simply rework the equation for a constant.

For (2), if NG/WT and Z are constant, write B = c W. Then B / W = c = B / W. Hence Log[B] = Log[W].

Q.E.D.

Theorem B2: In a pragmatic Madisonian real welfare state with Ricardian equivalence and uniform prices, net income indexation is only feasible for special tax functions.

Proof: To see what happens if B is indexed on Net[W], write n = Net[W] / W. Note that 1- n is the marginal tax rate for W, and that B / W = B / Net[W] n.

With B = W (1 + Z) RIR - g (theorem B1) use W (1 + Z) = (Net[W] + g + b) and get:

B = RIR Net[W] - (1 - RIR) g + RIR b

Note that b 0, since we have set u = 0 only in the determination of the RIR. Then:

B / W = (RIR Net[W] - (1 - RIR) g + RIR b) / W

= RIR n - (1 - RIR) g / W + RIR u B / W / / / / B / W= (RIR n - (1 - RIR) g / W) / (1 - RIR u) / / / / / / / / We again find a small multiplier. Dividing by n gives the transform to Net[W]:

B / Net[W] = (RIR - (1 - RIR) NG / WT / n) / (1 - RIR u)

LogB / Log[Net[W]] = Net[W] / B (RIR - (1 - RIR) g / W / n) / (1 - RIR u)

Indexation on Net[W] means that the left hand side is 1, and that Net[W] / B is some constant. Setting net income ratio B / Net[W] = NIR:

NIR = (RIR - (1 - RIR) g / W / n) / (1 - RIR u) / / / We want to find the conditions under which RIR is a constant (for the ‘real welfare state’). Solving above expression for RIR gives:

A special case has = 1 and thus NG/WT = g / W constant, and n constant, i.e. for the Bentham tax function n = 1 - r. This is only feasible if u is constant too. There is a more general class when g / W / n is some constant, but u must be constant here too. In other cases the RIR is implicitly adjusted to make B / Net[W] constant. But nonconstancy of the RIR conflicts with above definition of the welfare state (that must have constant RIR).

Q.E.D.

28. Phillipscurve

This chapter deals with the confrontation of labour supply with labour demand, and the equilibrating dynamics. With high unemployment, wage growth may be reduced. With low unemployment there may be ample room for wage demands, and wage inflation can rise.