strategy set of player /, //., is nonempty, compact and convex subset of R, the real line, and further, the constraint correspondence <p, (7]_, ) is convex-valued and continuous. Proof. By defmition the strategy set of player i is Hi = [0,/£■]. The player can always choose not to lobby, implying that 0 e therefore, Hi * 0 . The set is obviously closed for it also includes boundary points. Compactness requires that the set be bounded.
Since Pt is bounded from above by P" [Assumption (A4)], and /?, is increasing in Pt, implies8 that/?,(/*") < +°°. It follows from equations (5.5), (5.6) and 0 < Ki < +«> that for all non-negative values of Tj_i we must have r\i < Kfi^P"). Therefore, Ht is compact. Convexity of //■ is automatically satisfied because it is a closed interval of a real line.
To show that the choice set is convex or the constraint correspondence is convex valued, let us consider the defmition of <p-( T7_4). Since (pi(r}_i) is a collection of
strategies (lobby levels) available to player i when player-/ has chosen to lobby rj_i amount of resources, <p4(77_.) can also be defined by
(5.11) <Pi(n-i) = [Vi e H; |n j(rj„r)_1)> 0 )
But this means (p^Tj_.) is a better set for the payoff function, which is quasiconcave by Lemma 5.2 implies that <p-(77_,) is a convex set.
Lastly, it remains to be shown that the constraint correspondence ( p t ( T ]_•) is continuous. Continuity of (p^rf_.) has been shown in Coggins, et al. (1991), which is restated here for the sake of completeness.
It suffices to show that (pi(TJ_i) is upper and lower hemicontinuous9. Since/* is continuous in //•, and Rt is continuous in /*, it follows from equation (5.5) that (77_t)
8 Note that the nature o f the rental functions depend on the economic system and the rental rates are defined for all prices, whereas the bounds o f the pricing function are in the domain o f the
government. For any Pi > Pi it follows from the monotonicity of the real rent that Ri(Pi)> Ri(Pi ). 9 A correspondence 9 from a subset S of a Euclidean space to a subset T o f a Euclidean space is upper hemicontinuous (u. h. c.) at a point x° of S if there is a neighbourhood o f x ° in which cp is
bounded, and few every sequence x4 in S converging to x ° in S, and every sequence yQ in T converging to y ° in T such that for every q, one has yQ in (p(x4), then y ° is in <p(x0). Upper hemicontinuity of <p on S is defined as upper hemicontinuity at every point of S.
is continuous in [0, 7}_.]. Thus the graph of < p . ( 7 i s closed. As (pfq'_.) is also compact valued by the above argument, it is upper hemicontinuous. To show lower hemicontinuity, consider a sequence {r/"(} converging to 77^, and take any arbitrary 7j] € <50(77!-). If r\° < 77.(77!.), then for N large we may set 77! = 77° for n > N. Then clearly 77* 77°, and the conditions for lower hemicontinuity are satisfied. If
77° = 77,(77!.), then let 77! = 77.(77!.). As 77.(77,.) is continuous, the conditions for lower hemicontinuity are again satisfied. It is concluded that <50,(77_.) is lower hemicontinuous.
Thus, it is continuous. Q. E. D.
T heo rem 5.2. (Existence o f an equilibrium in a tariff game). Given a Tariff Game in
T = (//.,! !,, <p.)i€/ as defined above, if the rental function Rt is real-valued, strictly increasing, concave and differentiable in P , and the pricing function Pl (77,, 77_.)
satisfies assumptions (Al) to (A4), then there exists at least one noncooperative Nash equilibrium.
Proof: Under the assumptions of the Theorem 5.2, Lemma 5.2 and Lemma 5.3 hold.
Since, by assumption, the pricing function is continuous in lobby expenditures 77. and 77_i. By the assumption of the theorem the rental rate Rt is continuous and real valued in f j . This implies that the payoff function of any arbitrary player i defmed by the
equation (5.3) is continuous and real valued in H and quasiconcave in its ith variable. Hence, the conditions of Theorem 5.1 are satisfied, and the theorem is proved.
Q. E. D. Theorem 5.2 has guaranteed that at least a Nash equilibrium will exist in an economy where tariff policy responds to the lobby efforts of the interest groups. Since 0 e H , no lobbying is always a feasible strategy to both the players, it is a candidate for an equilibrium. It is natural to ask, at this point, whether (77,., 77,.) = (0,0) can be a Nash equilibrium of the game. The following Lemma answers this question.
L em m a 5.4. In addition to the conditions of Theorem 5.2, if the pricing function is such that the partial elasticities of price with respect to the lobbying expenditures are locally constant, then (77., 77..) = (0,0) cannot be a Nash equilibrium.
Similarly, A correspondence 9 from a subset S of a Euclidean space to a subset T of a Euclidean space is lower hemicontinuous (1. h. c.) at a point x° of S if, for every sequence (x9) in S converging to x° in S, and every y° converging to <p(x°), there is a sequence (y9) in T converging to y° such that for every q, one has y9 in (p(x^). Lower hemicontinuity on S is defined as lower hemicontinuity at every point of S.
Continuity of a Correspondence at a point or on a set is defined as the conjunction of upper and lower hemicontinuity. (Debreu, 1982: 698-701).
Proof: Let us consider the behaviour of player i when player -i chooses not to spend on lobbying, that is r\_i = 0. The payoff to player i is given by
n i(T)„0) = «,(/> (TJ^.O))- rjf, 77i £ <p,(0).
The first order condition for the maximum requires that player i choose lobby expenditure satisfying
0dRi / dPiKdPi / d Vi) ^1, and 77, (d n , / 9 rtf = 0.
The conditions will yield a solution of 77, = 0 if and only if (dRt / dPi){dPl / d77.) < 1 for all 7ji e <p,(0). But
^ ^ A <9 77,
g d R j Y V
y /?, dPi <9 rji J K77, y
Since, under the conditions of the Lemma, the first two terms on the right (elasticities) are always positive, and locally constant around the origin of the lobbying space; and the real rental income is positive10 at all prices therefore, in the neighbour of the origin, we have lim Vi-*0 f d R , Y A i / V 1 ( \ f lim Vi~*0 > L i5 . ^ 3 Vi V Ä.N a 1 .; = +0 0.
That is, (^/?, / dPi)(dPi / <9 77.) > 1 in the neighbourhood of 77, = 0, and the first order necessary condition for a maximum is not satisfied. Therefore, 77, = 0 cannot be a best reply of player i to the strategy 77_. = 0 of player Since choice of i is arbitrary, we find that (77,., 77_,) = (0,0) cannot be a Nash equilibrium (see Definition 5.2) in the tariff
game. Q. E. D.
Thus, under the assumed behaviour of the pricing function and the behaviour of the rental functions, Lemma 5.4 guaranteed that in any Nash equilibrium we will
observe that the players will be spending positive amounts on lobbying. It is tempting to conclude that all prices corresponding to Nash equilibrium are necessarily different from free trade prices. This is not always the case, however. In fact, as Coggins, et al. (1991) also have shown, if the lobby expenditure of one player is matched appropriately by another player in equilibrium then the free trade price may result. The arguments are as follows:
10 If there are increasing returns to scale, then at very low price the rental income may be negative even if the slope of the rental function with respect to price is positive. In that case, not spending on lobbying might be the best reply for the player and hence Lemma 5.4 may not hold. However, this study is based on the assumption of constant returns to scale in both sectors.
Let and ry_. be any feasible level of lobby expenditures of the two players. Then by the assumption of productive lobbying (A3) we have
pi(rii, o ) >p im ) = p ; > p l(o,Ti_i).
By the mean value theorem, there exist feasible expenditures fji e (0,77.) and f)-i e (0, tj.,. ) such that Pt (77., ) = P*.
Thus, the result that non-zero lobby levels may generate a free trade price ratio is still possible provided the government has no interest of its own in deviating from free trade. It is not yet known whether such combination of lobby expenditures can constitute an equilibrium of the game. What has been shown so far is insufficient to rule out the possibility that at least one Nash equilibrium outcome of the game will yield the free trade price ratio.
Definition 5.3. Given a description of a small open Walrasian economy a