PROBLEM (2) (Signaling and screening) The government is trying to contract the construction of a government building. A contractor can be either a (H)igh or (L)ow quality type. A (H)igh quality contractor builds a high quality building that is worth $30m to the government and it costs $26m to the contractor to construct. On the other hand, a (L)ow quality contractor builds a low quality building that is worth $20m to the government and it costs $16m to the contractor to construct. Assume the contractor knows its type (quality), but the government does not know the contractor's type. The government offers a bid (price) for the construction of the building, and the contractor can accept or reject the offer (when there is no accepted offer, both get 0; it is the outside option for both players, that is). Both players (government and contractor) are risk neutral. (a) What should be the minimum probability that the contractor is the H type, for a pooling equilibrium to exist where both types of contractors accept the bid? (b) If the probability of contractor being H type is %70, what is the possible range of equilibrium government bids (prices) in a pooling equilibrium (that attracts both types)? (c) In pursuit of better utilization of public resources, the government tries a signaling and screening scheme to separate the types. The government forces the contractors to take an industry excellence test. This test has scores from 0-10 and it costs nothing ($0m) for either type of firm to get 0 on this test. However, to increase the test score by 1 point costs $1m in firm structuring costs for the H type firm, and $7m for the L type firm, for each point (x can be any REAL number). For example, it costs $21m for a L firm to get 3 points on this test. The government offers two contracts to separate the types: If you get 0 points from the test I will pay you $16m and have you construct the building. If you get x points from the test, I will pay you $30m and have you construct the building. Writing down the incentive constraints and participation constraints for each type, calculate what values x can take, so that the two contracts above form a separating equilibrium? (d) Forget about (c), return to (b). It is now uncovered that the H types are actually composed, in equal probability (%35 each, comprising the %70 probability for the H type), of two sub-types: (R)eal-high type, and (N)ot-so-high 1 type contractors. R types make real high-quality buildings worth $32m to the government and it costs them $28m to construct. N types make not-so-high quality buildings worth $28m to the government and it costs them $24m to construct. Notice that, as they (R and N) are equally likely, the numbers average out to the numbers for the \"H type\" as an umbrella type; %2 32+ %2 28 = 30m the worth of an average H building, 2 28 + % 24 = 26m, the average cost of an H building. A contractor knows whether it is R, N or L type as before, but the government doesn't. (i) Is there an equilibrium where R, N and L types all accept the bid? (That is, does the pooling equilibrium in (b) survive in this expanded types scenario?) (ii) Is there an equilibrium where only N and L types accept the bid? (iii) Is there an equilibrium where only L types accept the bid? If yes, give the possible range of prices (bids) for that equilibrium in each case above