I'm reading this paper on credit lending cycles and trying to understand how it works for the case of one bank. This is the set-up of the model: There are two players, a bank and the market. There are also two states of the world: adverse and normal.
The banks lends across time t=0,1 and 2. The bank can be of two types: High type, which generates profitable loans with a high probability $\theta$ in normal times and generates profitable loans with probability 0 in adverse times. Low type, which generates profitable loans with a probability lesser than $\theta$ in normal times and probability 0 in adverse times.
The bank lends loans in period 0 and becomes aware of the state of the world in period 1 when the loans turn out good or bad. If the loan is bad the bank can either terminate the loan with a cost of -1 dollars or can decide on a credit policy ( for eg extending the term of the bad loan). The credit policy varies between [0,a] and comes with a cost c in period 2 for the bank. After the credit policy is implemented the market observes the bank's earnings and updates its reputation, which is given by p+ = Prob( H/positive earnings in period1) and p- = Prob (H/negative earnings in period 1). Having implemented the credit policy, the bank's earnings can remain unaffected with probability a, and can become negative with probability 1-a.
Now in such a scenario, given the utility function of the manager as shown in equation (1) of the image attached, how do I formulate the incentive compatibility problem of the manager. I should reach equation(3) from the ICC, but I'm not able to at the moment on my own.
image of equ (1) and (3) : https://i.stack.imgur.com/6hlyi.jpg