Consider a manufacturer that sells his product to a dealer who resells it to final consumers. The market demand is given by P = 100 - Q. The manufacturer's constant marginal cost is $20 and the dealer's constant marginal cost is $10. The dealer's alternative profit (i.e., his profit if he does not operate in the market) is 0. The manufacturer offers the dealer a contract (w,T) where w is the wholesale price per unit and T is the lump sum that the dealer will pay to the manufacturer.

1. What is the contract (w,T) that maximizes the manufacturer's profit?
2. Suppose that before the manufacturer offers the contract to the dealer, the dealer can invest $2000 and increase demand to P = 200 - Q. Should the dealer make the investment? If your answer is "yes", how will the optimal contract look in this case?
3. Instead of the cases described above, suppose that after the manufacturer and the dealer sign the contract, the dealer can invest $2000 and increase demand to P = 200 - Q. Suppose, however, that the dealer's investment is not verifiable. That is, the contract can only be of the form (w,T) and cannot be conditioned on whether or not an investment has been made. How will the manufacturer's profit-maximizing contract look in this case?