Compared to the previous Cournot model, the cost functions of the two-stage model are given by C(q1, K1) + F(K1) and C(q2, K2) + F(K2), C(.) is, in effect, variable cost and K1 and K2 act like parameters. F(K1) is the cost of capital and it includes the cost of doing business in a foreign country. The games are solved backwards.
In the first game, for any K1 and K2 (using the previous topic model), the first-order conditions give the reaction curves conditional on K1, K2, and A (for firm 1), namely, q1 = R1(q2, K1, A) and q2 = R2(q1, K2), Fig. 2. The first-stage Nash solution is q*1(K1, K2, A) and q*2(K1, K2, A). Substituting these output functions into the respective profit functions gives
P1 = R1(K1, K2, A) - C(K1, K2, A) - F(K1)
P1 = R2(K1, K2, A) - C(K1, K2, A) - F(K2).
The first-order marginal profit functions (conditions) are
MPK1 = MRPK1(K1, K2, A) - TMCK1(K1, K2, A) = 0
MPK2 = MRPK2(K1, K2, A) - TMCK2(K1, K2, A) = 0,
where TMCK1 and TMCK2 are the total or full marginal cost of capital. These conditions give the second-stage reaction functions, K1 = S1(K2, A) and K2=S2(K1, A). The Nash solution is K*1(A) and K*2(A), in Fig. 3. The equilibrium capital structure is K* = K*1 + K*2 and foreign capital intensity is (K*1/K*) = f(A), ignoring other parameters. An increase in A results in K*1 increasing and K*2 decreasing.
The collusive solution and the Nash bargaining solution are similar to those given in the previous Cournot topic. (Ref. Lee and Martin also apply.)