$.$ Consider two countries denoted by i $=1,2,$ each of which has one firm producing a homogenous product only for export, to be sold in the world market. The world$$s demand for the product is given by where $.$ The pre$-$innovation unit cost of each firm is $,$ where $.$ Let denote the amount of R&D sponsored by the government in country i$.$ When government i undertakes R&D$,$ the unit production cost for the firm producing in country i is reduced to $.$ The total cost to government i of engaging in R&D at level is : $($a$)$ Assuming that the firms play a Cournot quantity game in the world market for given R&D levels, calculate the equilibrium profit level for firm i$.[50\%$ of marks$]($b$)$ Assuming that country i$$s welfare is given by the sum of the profit earned by firm i minus the cost of R&D$,$ calculate the unique Nash equilibrium levels of R&D in country i$.[50\%$ of marks$]$