C.2: Contracting with and without commitment

C2 (a)

According to John, a fully specified algebraic solution is not necessary. We simply need to note that there will be efficiency for the low type (high cost) and the high type (low cost) will earn an information rent. The "key tradeoff" driving the structure of the contract is: Efficiency at the low end vs paying information rent at the high end.

Note that this is an "indirect" mechanism.

The solution is given by $\max_{t_{H},t_{L},q_{H},q_{L}}[\sum_{i=0}^{\infty}[\delta^{i}(0.5(u(q_{H})-t_{H})+0.5(u(q_{L})-t_{L}))]]$

C2 (b)

This is a direct mechanism, which is equivalent to the indirect mechanism above. This is a trick question because they're equivalent.

C2 (c)

We can use the revelation principle to limit attention to direction mechanisms in which the agent reveals his type in the first period. As a result, the principle knows his type going into the second period and can therefore pin the agent at its IR for the second period (ie, the agent has no profits during its second period). Because of this, the high type agent will demand more profits/information rent in the first period and the principal cannot earn as much in this setup as he did in (a) and (b).