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.
