How to compute Social Welfare in this model?

The first part of this scientific article analyzes reward-based crowdfunding. The assumptions are:

• Seller is a monopolist
• Unit mass of consumers
• Unit demand
• Two periods
• Individuals are identified by $\theta$ and are uniformly distributed on [0,1].
• Individuals in period 1 derive surplus $U_1 = \theta_1(1+\sigma) - P_1$
• Individuals in period 2 derive surplus $U_2 = \theta_2 - P_2$
• Individuals in period 1 derive extra benefits, so-called community benefits denoted by $\sigma > 0$

The entrepreneur has to maximize profits in both periods while also inducing self-selection between the two consumer groups. The entrepreneur has to achieve profits in period 1 greater than some development cost, $K$.

Overall profits (in the unconstrained case) are derived as: $\frac{(1+2\sigma)^2}{4+16\sigma} - K$

My question is: How would you derive consumer surplus and thereby also social welfare in this model?

Assuming the consumer's have been successfully separated in the two periods, the demand for period 1 consumers is: $Q_1 = 1 - P_1$, and the demand for period 2 consumers is: $Q_2 = \theta_1 - P_2$.

Inserting the equilibrium prices yields the following quantities: $Q_1^* = (1+4\sigma-4\sigma^2)/(2+8\sigma)$ and $Q_2^* = (1+2\sigma)/(2+8\sigma)$.

The equilibrium prices are such that $P_1^* > P_2^*$. While $Q_1^* > Q_2^*$ if $\sigma < 1/2$ and $Q_1^* < Q_2^*$ if $\sigma > 1/2$.

Clearly, there is a mistake here. I believe that it might be due to the demand expressions. But I don't know what specifically there is wrong.

Any help is much appreciated.

• Any suggestions? – Toms.S Jul 26 '16 at 9:18