This probably should have been split into three questions, one for each part. But I feel bad that it's been here for so long unsolved, and it's a good problem, so I will at last go through and solve all three. Next time please separate the questions appropriately.
For part A, we can get a Pareto-efficient solution by weighting the different citizens however we like, and then maximizing utility; but it sounds like what we really want is the utilitarian efficient solution, which weights each citizen uniquely.
Then basically what we do is imagine there's a central planner who takes all the money ($1,001,000) and divvies it up in such a way as to maximize overall utility for the whole town.
This total utility is given by adding up all the individual utilities, remembering that there are 1000 people of type 1:
U = 1000 (x_i - 100/G) + (x_w - 110/G)
We maximize this subject to the constraint that we need to pay $10 for each square meter of rink and $1 for each bottle of ale, using a Lagrangian; we also keep in mind that all the type 1 people will get the same amount of ale:
L = 1000 x_i - 100,000/G + x_w - 110/G + lambda (1000 x_i + x_w + 10 G - 1,001,000)
dL/dx_i = 0 = 1000 + 1000 lambda
lambda = -1
dL/dx_w = 0 = 1 + lambda
dL/dG = 0 = 100,000/G^2 + 110/G^2 + 10 lambda
100,100/G^2 - 10 = 0
100,110/G^2 = 10
G^2 = 10,011
G = 100.05
dL/dlambda = 0 = 1000 x_i + x_w + 10 G - 1,001,000
1000 x_i + x_w + 1000 = 1,001,000
1000 x_i + x_w = 1,000,000
Now, how do we decide how much ale the type 2 person gets compared to the type 1 person? Well, they all have the same constant marginal utility of ale, so it doesn't really matter---but we may as well assume it's split fairly so that x_i = x_w = x.
x = 1,000,000/1001 = 999
So, the utilitarian allocation is for everyone to buy 999 bottles of ale and contribute their last $1 to building 100 square meters of rink.
Part B is the majority-rule democratic solution. If everyone voted, how much rink would they want to be bought?
Clearly the type 1 people are going to win the vote, so we can basically ignore the type 2 person. The type 1 people will vote for the amount of spending that maximizes their utility:
U = 1000*x_i - 100,000/G
The budget constraint to use is still assuming that they have all the money to work with, because in a majority-rule system they can tax the type 2 person's money as well. Theoretically they could tax all of it, though this never happens in real life. In fact, it was stipulated that everyone must pay the same share.
L = 1000*x_i - 100,000/G + lambda(1000*x_i + 10*G - 1,001,000)
dL/dx_i = 0 = 1000 + 1000 lambda
lambda = -1
dL/dG = 0 = 100,000/G^2 + 10 lambda
100,000/G^2 = 10
G = 100
Now we said that everyone must pay the same share, so this means that the $1,000 to buy 100 square meters of rink must be shared equally through all 1001 citizens; so each one owes $1 (rounded from 0.999, but you can't pay a tenth of a cent), and then each will buy as much ale as possible with the rest, so that's 999 bottles of ale.
You'll note that this is the same (with some rounding) as the utilitarian Pareto-efficient allocation. It could actually be feasibly implemented using a 0.1% sales tax on ale.
But now, part C: What if there is no planner, no taxes, and everyone must decide how much rink to buy on their own?
Then, each person maximizes their own utility without regard to anyone else. They act as if they're paying for the whole thing. As noted in the problem, only one type of first-order condition can actually be satisfied; now the question is, whose will it be?
What we need is a condition where one type of person has the same marginal utility for ale and skating rink, while the other has strictly less marginal utility for the skating rink and will therefore not contribute.
So, whose marginal utility of skating rink is highest? Why, the type 2 person! They like skating rinks just a little bit more (marginal utility of 110/G^2 instead of 100/G^2) than everyone else--and as a result they're going to end up footing the bill.
So we maximize that person's utility subject to their budget constraint:
L = x_w - 110/G + lambda (x_w + 10 G - 1000)
dL/dx_w = 0 = 1 + lambda
lambda = -1
dL/dG = 0 = 110/G^2 + 10 lambda
110/G^2 - 10 = 0
G = 10
dL/dlambda = 0 = x_w + 10 G - 1000
x_w + 100 - 1000 = 0
x_w = 900
So, this person is going to spend $100 of their own money to buy a tiny little 10 square meter skating rink, only having $900 left to buy ale, while everyone else spends all their money on ale and free-rides on the skating rink---all because everyone was acting independently and nobody could force anyone else to pitch in. Yet even they are worse off than they would have been with a larger skating rink.
Notice how by being forced to pay a tax to provide for a public good, everyone can be made better off---this is the general lesson of public goods.
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