Let the political support function (M) be described by:
M = M(p, Π)
Where p is price established for the regulated service (e.g., electricity) by the regulatory authority (e.g., the New York Public Service Commission) and Π is the level of profit earned by the regulated firm (e.g., New York Edison) at that level.
M falls with p but rises with Π:
dM/dp = M_p< 0
dM/dΠ =M_Π > 0
An example of a function M of the form described above is: M = e^(αΠ-βp)
In this case,
- α = sensitivity of the special interest’s support for the regulator given changes to their profits
- β = sensitivity of consumers’ support for the regulator given changes to prices
Profit, Π, is itself a function of production costs “c” and the price “p” .
Π = f(p,c)
(See footnote at the end of this post to mix this model with normal production models of perfect and monopolistic competition. )
Such that profits fall as production costs rise…
dΠ/dc = Π_c < 0
… and profits rise with prices but only up to a level, above which the fall in demand would undermine profits
dΠ/dp = Π_p > 0
dΠ_p/dp = Π_pp < 0
The regulator sets the legal price by constrained optimisation of the following lagrangian:
L = M(p, Π) + λ( Π – f(p,c))
which yields the solution
– (M_p/f_p) = M_Π = -λ
Notice that the equilibrium will depend on the steepness of the two curves:
- M(p,Π) will be steeper the more sensitive to price consumers’ support is.
- M(p,Π) will be flatter the less sensitive to price consumers’ support is.
- Π = f(p,c) will be steeper the more sensitive to profits the firms’ support is.
- Π = f(p,c) will be flatter the less sensitive to profits the firms’ support is.
Given a combination of these 4 alternatives, there are 4 solutions involving extreme positions:
From the above considerations it is possible to preliminarily conclude that
- The highest price is achieved when the firm’s profits are potentially very high and the voters are not very sensitive to prices. The lowest price is achieved when the voters sensitivity to prices is high but the firms profits’ are potentially quite small.
p(Cl,Fh) > P(Cl,Fl) > p(Ch,Fh) > p(Ch,Fl)
- As regards the isovote curve, clearly, if potential profits are quite small, the regulator could in principle gather the same level of support with a lower price for that industry as he would for an industry with higher potential profits with a higher price (compare “Ch,Fl” with “Ch,Fh”).
- Also, if consumers/voters don’t care, the level of support to be gathered by the regulator will be are lower if profits are potentially low and higher if they are potentially high. This suggests it is best to regulate high profit industries than low profit ones (of course this ignores the number of firms making this profit or assumes that the profit curve describes industries, not firms).
(Unfortunately, I have not had the time to finish this, but the discussion below is the beginning of an introduction to the model described in this post with some functional forms)
* Briefly returning to the function Π = f(p,c), we could express it in the normal way it is described by basic production theory, where traditionally
Π = TR-TC = p(Q)*Q -c(Q)
- c(Q) = (wL+rK), which assuming that it takes 1 unit of Labour to get 1 unit of output, L=Q, then
- c(Q) = (wQ+rK)
- p^ = market rule + RegDev
- Demand Function: pD(Q) = a – bQ
- Supply Function: pS(Q) = c + dQ
- RegDev is the deviation imposed by the Regulator. The value of RegDev will depend on the level of capture suffered by the regulator.
- RegDev =0, the regulator is not captured
- RegDev >0, the regulator is captured by the industry
- RegDev < 0, the regulator is captured by consumers and there’s profit
The market rule will depend on the nature of the market:
- p(Q) = MC(Q) = w -> for perfect competition
- Notice that in this case Π = 0
- p(Q^): MR(Q^) = MC(Q^) <=> a-3bQ^ = w <=> Q^ = (a-w)/3b
- Notice that in this case Π > 0