Question

Suppose the individual inverse demand curves for person A and person B, respectively, are given by:

** ***P _{A}
= 80 - 0.6q_{A}*

* P _{B}
= 50 - 0.5q_{B}*

and that MC = $40.

*Derive the inverse market demand curve? (Hint: sum the two demand curves vertically). What’s the price and the quantity at the kink point?*

First draw the inverse individual demands for persons A and B in the same graph by connecting their horizontal and vertical intercepts.

(Hint: Sum up vertically the two
individual demands and call it the inverse market demand
(*P _{market}* =

Plug the horizontal intercept of the
weaker demand (for Person B which 50*2 = 100) into the inverse
market demand *P _{market}*and find the kink
point).

- D
*raw the demands for persons A and B and the market demand in one graph.*

Follow that the inverse market demand
*P _{market}*from its vertical intercept as it goes
down and ends at the kink point on the demand for person A which is
the stronger demand, thus showing the kink.

*Calculate the efficient market allocation q*. (Hint: set P*_{market}= MC)

Set the equation of
*P _{market}*=

*Estimate the efficient pricing system for persons A and B. (Hint: Plug q* in*

*both individual demand
equations).*

Plug
*q** into ** P_{A}* = 80 -
0.6q*_{.}** This is the efficient price for
person A. Do the

same for person B’s inverse
demand and calculate B’s efficient price
** P_{B}***.

e. *Calculate
the free rider (private) allocation for A and B (Hint: Set
P _{B} =MC &*

* P _{A} = MC and
solve for their quantities)*. That is,

*PA
=80 - 0.6q _{A} = MC* and solve for

_{ }

_{ }Then
do the same for Person B by setting *P _{B} = 50 -
0.5q_{B} = MC* and solving
for

Determine the allocation for the free rider (which is the person with the

stronger demand). That
is*, (q ^{A} - q^{B}) +
q^{B}*

f. *Calculate the cost of free
rider and efficient allocations*.

*MC*q ^{B} +
MC_{*} (q^{A} – q^{B})*

g. *Calculate the cost of efficient allocation?
(MC _{*}q*).*

*MC _{*} q* =
40*q**

Answer #1

A) Market demand of public good is vertical sum of individual demand.( For private good ,it is horizontal sum )

Inverse market demand:p=Pa+ Pb=80-0.6q+50-0.5q=130-1.1q

Kink point will be at where,slope of demand changes.

Slope of market demand will change at, Q=100

And p=130-1.1*100=20

B)

C) Efficient allocation of public good at,

Inverse market demand=MC

130-1.1q=40

Q=90/1.1=81.81

D) efficient Price charge to person A:P=80-0.6*90/1.1=30.9

Efficient Price charge to person B:p=50-0.5*90/1.1=9.09

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