Assume that budgets and preferences are identical and that each person chooses how much to contribute to a public good independently of everyone else's choice.
Theorem: If ui = s(wi - gi) + o(n,G), then there are functions s and o such that G is unbounded as n approaches infinity,
where wi is person i's budget, gi is i's contribution to the public good, s is the benefit to i of privately consuming wi - gi resources, n is the population, G is the total provision of a public good, (thus, if G-i is the total provision minus gi, then G = G-i + gi = ;g-i +gi = ngi), o is the benefit to i of n people consuming public good G, and ui is i's utility function for contributing gi to a public good and spending the rest on private goods.
Proof. Let s = log f, f = (w-g), o = n(log(G)). Each person chooses gi so as to maximize (u). Each utility maximizer treats G-i (what others contribute to the public good) as a constant, which, under Nash equilibrium, we can assume is equal to (n-1)gi. We find max (u) by setting u' = 0. Thus:
1) u' = 0 = s'+o' = 0 sum rule
2) d (log(f(g)))/dg * (f'(g) + o' = 0 chain rule
3) (1/(w-g))f'(g) + o' = 0 derivative of a log
4) (f'(g)/w-g) + o' = 0 multiplying
5) (-1/w-g) + o' = 0 derivative of f(g)
6) (-1/w-g) + n (d log(G)/dg) = 0 derivative of n(.)
7) (-1/w-g) + n (d log G/dg * dG/dg) = 0 chain rule
8) (-1/w-g) + n (d log G/dg * 1) = 0 derivative of G
9) (-1/w-g) + n ((1/G) * dG/dg) = 0 chain rule
10) (-1/w-g) + n (1/G) * 1 = 0 derivative of G
11) (-1/w-g) + n (1/ng) = 0 Nash conjecture
12) (-1/w-g) + 1/g = 0 multiplying
13) -g/w-g + 1 = 0 multiplying
14) -g/w-g = -1 adding
15) -g = -w + g multiplying
16) g = w - g multiplying
17) 2g = w adding
18) g = w/2 multiplying
Since G = ngi, G approaches infinity as n approaches infinity.
Efficient Provision Without Matching
Without a matching contract, the contributions of a father and mother to their child will be efficient, under certain conditions.
Let Uf equal the father's utility, w equal the father's (or mother's) income, x equal the father's expenditure on his clothing, and z equal the mother's expenditure on her clothing. The father's utility will be a function of what he spends on his clothing and what is spent on toys for his child. More particularly, we shall assume that his overall utility is the sum of the utility he derives from his clothing and his child's toys. If the mother's income equals the father's, if she spends z on her clothing and the rest of her income on toys for the child, then the total amount spent on the child will be (w-x) + (w-z). Hence, the father's utility will be:
Uf = f(x) + g([w-x]+[w-z])
= f(x) + g(2w-x-z)
To figure out what the father would spend on himself if he were to maximize his utility, we set the derivative of Uf equal to zero and solve for x.
0 = d(f(x))/dx + d(g(2w-x-z))/dx
If the parents do not match each other's gift to the child, then as far as the father is concerned, z acts like a constant. So, for example, if f(x) = g(x) = log (x), then
0 = d(log(x))/dx + d(log(2w-x-z))/dx
By the chain rule:
0 = 1/x + (1/(2w-x-z))*(-1)
0 = 1/x - (1/(2w-x-z))
1/(2w-x-z) = 1/x
x = 2w-x-z
For simplicity we assume that the father and mother are similar in several respects. We have already mentioned that their incomes are the same. In addition, we assume they have the same utility functions and make similar evaluations of the relative value of giving to the child versus spending on herself. Both are rational, expect Nash behavior from the other and have mutual knowledge of each other's rationality, preferences and expectations. So the father can expect that the mother will be solving the same equation. If there is to be a Nash equilibrium, z must equal x. Hence,
x = 2w - 2x
x = 2w/3
What distribution would maximize everyone's welfare? Let us suppose that the parents are not only alike in their preferences regarding their child, but actually correct in how they value spending on clothes versus spending on toys. If that is so and if a fair representation of total welfare gives equal consideration to each parent, then we may model total welfare as the sum of three functions:
Ut = f(x) + g(y) + h(z)
Where Ut = the total welfare of all, x = the expenditure on the father's clothes, y = the total expenditure on the child's toys and z = the expenditure on the mother's clothes.
Because of a budget constraint, the combined income of the father and mother, 2w, equals x + y + z. So y = 2w-x-z.
Ut = f(x) + g(2w-x-z) + f(z)
To find the maximum, we simply set the partial derivatives for x and z equal to zero.
0 = d(fx)/dx + d(g(2w-x-z))/dx
0 = d(hz)/dz + d(g(2w-x-z))/dx
Hence the value of x which maximizes the welfare of the group will also maximize the father's utility. (The exact same reasoning applies to the mother. Note that if the parents are similar in their preferences, h(.) will be identical with f(.).)
For example, again suppose that f(x) = g(x) = log(x). Then from our first partial derivative we have:
0 = d(log(x))/dx + d(log(2w-x-z))/dx
0 = 1/x + (1/(2w-x-z)) (-1)
0 = 1/x - 1/(2w-x-z)
1/(2w-x-z) = 1/x
x = 2w-x-z
Since x = z:
x = 2w/3
Over-provision With Matching
The father will give too much to the child if there is matching.
In such an instance, z no longer behaves like a constant. If the father and mother agree to match each other's contribution, dollar for dollar, the father's utility function becomes:
Uf = f(x) + g(2w-2x)
Taking the derivative and applying the chain rule, we get:
0 = d(f(x)/dx + (d(g(2w-2x))/dx)(-2)
The solution in this instance will differ from the solution when z acts like a constant. For example, if f(x) = g(x) = log(x), then:
0 = d(log(x))/dx + d(log(2w-2x))dx (-2)
0 = 1/x + [1/(2w-2x)] [-2]
0 = 1/x - 2/(2w-2x)
2/(2w-2x) = 1/x
2x = 2w-2x
x = w/2
Thus, the father will spend less on himself as the "price" of the toys declines with matching contributions from the mother, as we would expect.
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