# Take It Or Leave It Strategy: Social Optimum

Here is what I understood

Using Backward Induction, I inferred that buyer offers a price, say, $$P$$ and the seller will sell only if $$P \geq c(I)$$. Setting the lowest possible Price that will ensure that Seller sells the good would mean that buyer offers $$P=c(I)$$

The payoff to Seller would be $$U_S=P-C(I)-I$$ Putting $$P=C(I)$$ gives $$U_S=C(I)-C(I)-I$$ which would be negative. Therefore, seller would choose not to invest, that is, $$I=0$$

The subgame perfect strategy is Seller chooses to not invest, and Buyer offers $$P=C(I)$$

Is this the correct way to write the Subgame Strategy? Also, for the Social Optimum, I understand that we are supposed to find optimal $$U_B+U_S$$, i.e., $$P-C(I)-I+v-P$$, but I don't know how to proceed. Help would be appreciated!