Maniobra de Reves sobre Barinas
IV. EL RETO DEL FRACASO
the lobbying efforts o f the players which, in turn, are determined by the respective payoffs of the players (or the price function offered by the government) follows from the first-order condition (5A.3). See also Peltzman (1976).
Alternately, a government may induce no lobbying on the part of private players if it
A A
offers a constant pricing function, say Pl(ril,ri2) = Px for all rj{ and r]2, where Px could be a price that emerges at some Nash equilibrium and also maximizes its political support. In the later case, no lobbying certainly increases the payoffs of both players from what they would obtain at the corresponding Nash equilibrium with positive lobbying. In both cases no player is dissatisfied, and no player will spend in lobbying against the government or financing the political campaign of the opposition. Such a pricing mechanism would also raise the level of political support for the government. If such possibilities exist, then such a price corresponds to the (self-enforcing) cooperative outcome, which we shall study in the next chapter. However, if any one of the players aspires for a higher or lower price than Px, that is he is not satisfied with the ‘fixed- price’ offered by the government, then he may still spend resources in opposing the government, and supporting the opposition. The player can behave strategically to ‘educate’ the voters that will penalize the incumbent in the next election. To mitigate this opposition, the government has to offer a pricing function that induces a positive lobbying effort from the beneficiary. It is this situation that we analyze in this appendix.
The first order condition of the maximum to the Nash player i is: (5A.3)
f£L _ dRiiJx _ j _ o
dtli dPx <??7,
The condition (5A.3) implies that each player will spend on lobbying as long as a unit of output spent on lobbying yields one extra unit of output in rental income. This condition can be rewritten as
(5A.4) dRi ldPl = \l{ d P J d r \i)
The necessary condition for the maximization of the government's support function can be written as
(5A.5) cLS _ d S dRi d Tji
o
This condition states that the government will choose the level of relative price so that at the margin the gain in support is exactly balanced by the loss in support that arises due to the pricing policy. It follows that (5A.5) is satisfied whenever (5A.4) is satisfied and that the ‘lobbying derivatives’ (Baldwin, 1987) exist such that
The condition (5A.6) will follow automatically if the pricing function satisfies assumption (Al) of the text - that it is continuous in and rj2, and that d Px / d rfl * 0 for each i.
We know that dRx / dPx> 0 and dR2 / dPx < 0. Satisfaction of the first order condition of support maximization follows from the satisfaction of the first order condition (5A.3) if the government announces a pricing function satisfying (5A.6), and
r) P r) P
(5A.7) - ^ > 0 a n d - ^ - < 0 . 5tj, dr)2
at all nonzero values of and r\2. This means that the pricing function should be such that the relative price of commodity 1 increases with increased lobbying effort of player 1 and decreases with increased lobbying effort of player 2.
Second Order Conditions:
For each i, differentiating the first order condition (5A.3) with respect to T]iwe get
(5A.8) d2R;
is.
dP?
dR, d 2P, +
dP,drif
We know that (<?P, / c?7),)2 > 0, dPl / dPt >0, and we get the following conditions:
(i) If d 2Ri / dP2 < 0, then the sufficient (but not necessary) conditions for d 2Y\ J drj2 < 0 are that
d 2P J d r f < 0 and d2 Px / d T[22 > 0.
(ii) If d1Ri / dP2> 0, then the conditions that d2Px / d Tjf < 0 and <92 / d J]\ > 0
are necessary but not sufficient for d 2Yli / dr\2 < 0. A sufficient condition would require sufficiently large magnitude of d2P J d T]2.
This means that if the government supplies a pricing function such that the conditions
(5A.9) d2P J d r f < 0, and d2P^ / d T}\ > 0
hold with sufficiently large magnitudes of the second order derivatives of the pricing function at the point where the first order conditions are satisfied, then the second order
conditions of the maximization of the payoff functions of the two players are always satisfied.16 The pricing function, however, will met this requirement, by Lemma 5.2, if it is bounded, assumption (A4) in the text, and satisfies the condition (5A.6) and (5A.9). But the conditions (5A.6) and (5A.9) are the same as assumption (A3) of the text.
We now check whether the second order condition holds for the government's support maximization whenever it is satisfied for the private players. Differentiating equation (5A.5) with respect to P{ yields
(5A.10) 7 dR, ] d 2S
-j-
d s ( d*R,\ d P t dp,
J
d p , dn, ' dn,
1 dP,2 dP? )_ When the first order conditions are satisfied for the Nash players we have dRi / dPx = dr\i / dPx, therefore, it follows that(5A.11) d 2S
dP2< 0 if and only if ^
' dS d 1Ri d 2r11 1 dP,2 dP2 [ <0 Since —— > 0, a sufficient (but not necessary) condition for
" dS d 2R, Y
a n , < dP2 [ < o is that
(5A.12) f d % _ d \ '
dP2 dP2 < 0 for each player i.
Now it will be shown that the condition (5A.12) is satisfied whenever d 1Y\i / dTj2 < 0, and the first order condition (5A.5) is satisfied. In other words, the second order condition for the maximization of the government's support function is satisfied whenever the payoff functions of the two players are simultaneously maximized.
Clearly, for d2RX / dP2 < 0 the condition (5A. 12) is immediately satisfied, since by differentiating equation (5A.6) we can see that
(5A.13) d 2T]i = f d P x\ d 2Px