# Consumer Welfare vertical differentiation [SOLVED]

Suppose consumers have the standard Tirole's utility $$U_A = \theta q + \alpha X_A - P$$ if they buy product A and $$U_0 = 0$$ if they don't buy this product (where $$X_A$$ is the actual demand of product A). Profits for the firm come from prices and raising quality (q) has quadratic variable cost: $$\pi= X_A ( P - cq^2)$$

This model is described in https://onlinelibrary.wiley.com/doi/abs/10.1002/j.2325-8012.2001.tb00384.x .

After finding the partial market coverage eq., which I could replicate: $$\hat{\theta}=\frac{p-\alpha b}{s-\alpha};$$ $$p^*=\frac{(4 \alpha t+b+k) (4 \alpha t+7 b+k)}{72 t};$$ $$q^*=\frac{4 \alpha t+b+k}{6 t};$$ $$x^*=\frac{8 \alpha^2 t^2+k (2 \alpha t-b)-8 \alpha b t-b^2}{3 (2 \alpha t-b-k)};$$ At pag 973 they compute $$CS_M^{pmc}$$ supposedly as: $$CS =\int_{\hat{\theta}}^{b} U(\theta,s)f_\theta d\theta= \int_\hat{\theta}^b (\alpha x-p+s \theta) \, d\theta$$ $$= \int_\hat{\theta}^b \alpha x \, d\theta -\int_\hat{\theta}^b p \, d\theta+\int_\hat{\theta}^b q \theta \, d\theta$$ $$= (b-\hat{\theta})^2\alpha - (b-\hat{\theta}) p + q\left[\frac{\theta^2}{2}\right]_\hat{\theta}^b$$ $$= (b-\hat{\theta})((b-\hat{\theta})\alpha - p) + q\frac{b^2-\hat{\theta}^2}{2}$$

$$\frac{(-5 b + k + 4 t \alpha )^2 (b + k + 4 t \alpha )^3}{1728 t (b + k - 2 t \alpha)^2 } = \frac{(b+k+4t\alpha)}{(b+k-2t\alpha)}\frac12 \pi$$ SOLVED That is indeed the solution. Solved on my own and added solution as edits.