I assume there is a given industry (for example, electronics) in the global economy. The host country has several firms in this industry. The Home country also has several multinational firms selling in this industry. The firms and countries in the rest of the world have a given and fixed share of the global market, say, 40 percent of global output (sales in a universal currency). The host and Home firms therefore have the other 60 percent. It is this 60 percent that the host and Home bargain over.
I assume that the firms in the host country act as one firm and have a market share given by S1 and the Home country firms also act as one with a market share given by S2. Under these assumptions, the Herfindahl index of concentration for the global economy will equal 5200 for H=60sq + 40sq for rest of world. The Herfindahl index for the two countries then is given by the constraint equation, H1+2 = 3600 = (S1)2 + (S2)2. This is a concave function with a slope of dS2/dS1 = -S1/S2, which is decreasing algebraically. The constraint function is given by curve CC in the graph below. If the rest of the world increases its share of the global market, then the constraint function will shift inward to (say) C'C'.
The Nash bargaining problem (given the four assumptions of scale, Pareto optimum, irrelevant alternatives, and symmetry) has the solution given by maxU(S1)U(S2)=W, the product of each countries industry utility function, subject to the constraint function, CC. A typical equilibrium solution is given by point "a" in the graph. If the rest of the world's share increases, then the new (lower) equilibrium is given by point "b".
This model has direct application to international trade and intra-regional trade where the latter forms a trade zone or block vis-a-vis the rest of the world. (For an alternative model, see, M. Dutta, The Euro Revolution and the European Union: Monetary and Economic Cooperation in the Asia-Pacific Region, in Hooley and Yoo, Eds, The Post-Financial Crisis Challenges for Asian Industrialization, Research in Asian Economic Studies, Vol 10, 2002, pp. 271-300.)