Homeowners 1, 2, and 3 live at the end of a badly deteriorated road. Fixing the road would cost $C. The value to Homeowner 1 of fixing the road is $3,000, the value to Homeowner 2 is $5,000, and the value to Homeowner 3 is $8,000. Each homeowner claims that fixing the road is not worth much to him, because each wants the others to pay the cost. The local government suspects that the total value to these homeowners of fixing the road is greater than $C and has decided to require the three homeowners to use the VCG mechanism to determine whether to fix the road. Since the government had no idea of the individual values for fixing the road, it decided to allocate the costs equally among the three homeowners. Each homeowner is asked to report his value for fixing the road. If the sum of the reported values is greater than C, the road will be fixed and each homeowner will have to pay $C/3 and also will have to pay an additional tax as calculated by the VCG mechanism.
(a) Suppose that C = $13, 500, so that each homeowner has to pay $4,500 as his share of the cost. If homeowners report their values accurately, the sum of the reported values will be ____________. Since the sum of reported values is greater than $C, the government will choose to build the road. To calculate Homeowner 3’s VCG tax, we reason as follows. If the road is repaired, Homeowners 1 and 2 will have to pay a total of ____________ while the sum of their values for the repairs is ___________ so their net change in utility is _______________ Since Homeowner 1’s response changes the decision about whether the road is built, she would be assessed a VCG tax of ___________ in addition to her $4,000 share of the costs. What would be Homeowner 1’s VCG tax? _________ Homeowner 2’s VCG tax? _____________ Would Homeowner 1 be better off or worse off with the outcome of the VCG mechanism than if the road is left unrepaired? _____________ Would Homeowner 2 be better off or worse off than if the road is left unrepaired? ____________ Would Homeowner 3 be better off or worse off with the VCG outcome than if the road is left unrepaired? ____________ When the VCG tax is used, would the sum of the utilities of the three homeowners be higher or lower than if the road were left unrepaired? __________
(b) Suppose that C = $18, 000 and so each homeowner will have to pay a $6,000 share of the cost. If homeowners report their values accurately, the sum of the reported values will be __________ Since the sum of reported values is less than $C, the government will choose not to repair the road. Let us calculate Homeowner 1’s VCG tax. If the road is repaired, the total amount that Homeowners 2 and 3 have to pay is _________ and the sum of the values to Homeowners 2 and 3 of fixing the road will be ___________, so their net gain from the project is ___________ Since Homeowner 3’s response changes the outcome of whether the road is repaired, she would be assessed a VCG tax of _________
c) Suppose that instead of 3 homeowners at the end of the road, there were 30 homeowners, 10 of type 1 who valued repairing the road at $3,000, 10 of type 2 who valued repairing the road at $5,000 and ten of type 3 who valued repairing the road at $8,000. Suppose that the cost of repairing the road is C = $135, 500. If the road is repaired, each homeowner would have to pay a tax of $135, 000/30 = $4, 500. If the VCG mechanism is used and each individual reports his true valuation, the road will be repaired, since the sum of reported valuations will be $160,000. To calculate the VCG tax of a type 3 consumer, we observe that the sum of the values of the project to all other consumers is ___________. The total amount that other consumers would have to pay if the project is undertaken is __________. Would a type 3 homeowner have to pay a positive amount of VCG tax? _________ How about type 1 and 2 homeowners? _______________________________