A. Suppose that a person has an original amount a of a good. By sacrificing x of
it, they can produce y = 0g(x) of another good where 0 > 0 is a parameter (in
the sense discussed in the Readiness Check) that measures the productivity
of the production process. The person's utility depends on both goods, and
they solve the utility maximization problem maxz>ou(a-x, 0g(x)). Suppose
that: u(c, y) = log(c) + y; that g(x) = x, and that a > 1/0.
1. Give the FOCs for the maximization problem maxz2ou(a - x, g(x)).
2. Solve for x*(θ).
3. Show that x*(0) depends positively on @ by implicitly differentiating the
FOCs, and check your answer by differentiating your answer to the previ-
ous part of this problem.
4. How does this problem relate to Figures 1.1 and 1.4 in NS Chapter 1?
B. Person 1 has an original amount a₁ of a good while person 2 has an original
amount a₂ of a good. If person 1 sacrifices x1 of the good and person 2
sacrifices 22 of it, then the two of them can jointly produce y = 0g(x1+x2)
of another good where > 0 is as in the previous problem. Suppose that:
g(x) = x; that person 1's utility function of the two goods is 41 (C1, y) =
log(c₁) + y; that person 2's utility is 42 (C2, y) = log(c₂) + y; and that a₁ and
a2 are greater than 1/20.
1. Give the FOCs for the maximization problem
max [41(01-X1, 0(x1 + x2)) + U2(A2-X2, 0(x1+x2))].
21,22≥0
2. Solve for x*(0) = (x(0), x(0)).
3. Show that x*(0) depends positively on @ by implicitly differentiating the
FOCs, and check your answer by differentiating your answer to the previ-
ous part of this problem.
4. This is an example of public good production from voluntary contributions.
Give 3 examples of this kind of public goods in small groups.
C. Instead of finding the x* that maximizes the sum, we will now suppose that
both voluntarily contribute an amount 21 and 22, and that they do it to
solve the problem of making themselves as well off as they can given what
the other is choosing to do (this is known as a Nash equilibrium, and we
will study these simultaneous optimization problems extensively later on).
The equilibrium amounts Eq (2F9, F) have the property that solves
maxz1 U1 (21 - 11, 11+129) while 25 solves max2 42(A2 - X2, XE + X2).
1. Show that the simultaneous solutions to these two problems involve the
underprovision of the public good because it leaves one in a situation where
both people can be made better off.
Eq =
Eq
Eq
2. For the three examples that you gave in the previous problem, do we see
the underprovision?
3. How does this problem relate to Figure 1.4 in NS Chapter 1?
