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.

            

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 (P_market = P^A+ P^B = 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 P_marketand find the kink point).

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

Follow that the inverse market demand P_marketfrom 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.

3. Calculate the efficient market allocation q*.  (Hint: set P_market = MC)

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

4. 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 qA

         

          Then do the same for Person B by setting P_B = 50 - 0.5q_B = MC and solving for  q^B.

      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