For public goods such as bus service, sewage services, power grids, water systems, roads, vaccines, education, law and order, reliable knowledge, and many others, the marginal cost of providing an amount q, MC(q), is positive but decreasing. Suppose that the demand curve for the good is D(q), i.e. D(q) is societal willingness to pay for the marginal unit of the public good at level q, D(q) 0 and D0(q) < 0. This problem asks you to compare several methods of deciding how much q to provide. (a) Produce using the competitive rule, p = MC(q). The solution is denoted qcomp ∗ . (b) Produce as a Monopolist, i.e. solve max q≥0 qD(q) - C(q): The solution is denoted qMon ∗ . (c) Produce so as to maximize social surplus from the industry, i.e. solve the problem max q≥0 [CS(q) + qD(q)] - C(q); where CS(q) = R0q[D(x) - D(q)] dx is consumer surplus. The solution is denoted qISS ∗ . (d) Produce so as to maximize the total social surplus, i.e. solve the problem max q≥0 [B(q) + CS(q) + qD(q)] - C(q);
For public goods such as bus service, sewage services, power grids,
water systems, roads, vaccines, education, law and order, reliable
knowledge, and many others, the marginal cost of providing an
amount q, MC(q), is positive but decreasing. Suppose that the
demand curve for the good is D(q), i.e. D(q) is societal willingness
to pay for the marginal unit of the public good at level q, D(q) 0
and D0(q) < 0. This problem asks you to compare several methods
of deciding how much q to provide.
(a) Produce using the competitive rule, p = MC(q). The solution
is denoted qcomp ∗ .
(b) Produce as a Monopolist, i.e. solve
max
q≥0 qD(q) - C(q):
The solution is denoted qMon ∗ .
(c) Produce so as to maximize social surplus from the industry, i.e.
solve the problem
max
q≥0 [CS(q) + qD(q)] - C(q);
where CS(q) = R0q[D(x) - D(q)] dx is
solution is denoted qISS ∗ .
(d) Produce so as to maximize the total social surplus, i.e. solve the
problem
max
q≥0 [B(q) + CS(q) + qD(q)] - C(q);
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