Economics, Organization, and Management
Examination Set (June7th, 2013)
(Paolo Bertoletti)
Time: 90 minutes.
I)Consider the problem of providing an input to a firm division, and suppose that only the slope of the (linear) marginal benefit of such an input is known to the planning division, which overestimates it. Assuming that the (linear) marginal cost is perfectly known to be almost vertical , show graphically that the best instrument to decentralize the provision is to use a quantity constraint (rather than a price signal).
The situation is described in the following Figure.
The welfare loss induce by the use of the price signal P is equal to the area EFAD, while on the contrary the use of Q as a quantity constraint creates a welfare loss given by the smaller area ABC.
II)A potential buyer can have either a willingness to pay of 4 (with probability 1/2) or of 2 (with probability 1/2). A potential seller has either a cost of 1 (with probability 1/2) or a cost of 3 (with probability 1/2). Will they be able to exchange efficiently? Why or why not?
The situation is described in the following matrix:
B\S / 1 (50%) / 3(50%)2 (50%) / 2 / -
4(50%) / p / 3
Taking the perspective of a neutral middleman, consideration of the participation constraints suggests p(2, 1) = 2 and p(1, 3) = 3, while efficiently the exchange should not take place in the case in which evaluations are given by (2, 3).
The incentive compatibility constraint for a buyer with evaluation 4 requires:
(4 - p)50% + (4 – 3)50% = 2,5 – 0,5p ≥ (4 - 2)50% = 1,
that is
3 ≥ p.
The incentive compatibility constraint for a seller with cost 1 requires:
(2 - 1)50% + (p – 1)50% = 0,5p ≥ (3 - 1)50% = 1,
that is
p ≥ 2.
Accordingly, to guarantee an efficient exchange it is in principle possible, and a price satisfying 3 ≥ p ≥ 2 would do the job.
III)Explain what an “allocation problem with design attributes” is, and how it should be dealt with.
In problems with design attributes:
- A great deal of a priori information about the form of the optimal solution is available;
- Failing to achieve the right relationship among the variables is generally more costly than other kind of errors.
Design variables need to be determined in a centralized fashion and communicated to the lower levels of the organization and to the individual members of the organization.