Question

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

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

PA = 80 - 0.6qA

PB = 50 -  0.5qB

and that MC = \$40.

1. 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 (Pmarket = PA+ PB = 80 + 50 – 0.6 q – 0.5q)).

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

1. Draw the demands for persons A and B and the market demand in one graph.

Follow that the inverse market demand Pmarketfrom 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.

1. Calculate the efficient market allocation q*.  (Hint: set Pmarket = MC)

Set the equation of Pmarket= MC and then solve for efficient public good q*.

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

both individual demand equations).

Plug q* into PA* = 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 PB*.

e.   Calculate the free rider (private) allocation for A and B (Hint: Set PB =MC &

PA = MC and solve for their quantities). That is,

PA =80 - 0.6qA = MC and solve for qA

Then do the same for Person B by setting PB = 50 - 0.5qB = MC and solving for  qB.

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

stronger demand). That is,  (qA - qB) + qB

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

MC*qB + MC* (qA – qB)

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

MC * q* = 40*q*

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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