Task: Innovation-then-Price Competition

Consider a two-stage game where firms invest in cost-reducing R&D in the first stage and set prices in the second stage. The firms' demand functions are D_i(p₁, p₂) = 1 - p_i + p_j, i, j = 1, 2, i ≠ j. The unit costs of both firms are c_i = 1 - y_i, where the reduction y_i causes R&D expenditures (y_i²).

(a) Solve for the symmetric open-loop equilibrium of the game where R&D activities are unobservable and show that the firms' optimal R&D expenditure is y² = 1/4. Solve for the equilibrium prices and profits.

(b) Now assume that R&D investment is not only observable but also leads to inter firm knowledge spillovers such that the firms' unit costs are c_i = 1 - y_i - sy_j where s ∈ [0,1] is the spillover parameter. Show that the firms' optimal R&D expenditure is y² = (1 - s)²/9. Solve for the equilibrium prices and profits in the limit cases of s = 0 and s = 1 and compare the results with those derived in (a).

(c) Classify and explain the R&D investment strategy in case of s = 0 according to the general taxonomy of business strategies.

(d) How do you expect the R&D expenditure to change qualitatively if there is quantity (capacity-price) competition instead of price competition? Explain.