Question

3. Assume that the inverse demand function for a good is P = 40 – 2Q. A monopolist retailer has exclusive rights to sell this good. A monopolist manufacturer sells the good to the retailer at price R. The retailer has an additional marginal cost equal to $2 per unit. The manufacturer’s marginal cost is $4 per unit.

a) Assume that the two firms remain independent. Determine the value of R charged by the manufacturer.

b) Now assume that the two firms merge into a vertically integrated firm. Determine the change in aggregate profits (that is, combined profits) due to the merger.

c) Determine the change in consumer surplus social welfare due to a merger of the two firms.

Answer 1

a) for the manufacturer , he will sell when marginal cost = marginal revenue

revenue = p*q = (40-2q)q

marginal revenue=40-4q

hence,

4=40-4q=> 1=10-q => q=9

**therfore price at which manufacturer will sell to retailer(R) = 40-2(9) = 22

profit = revenue - total costs = 22*9 - 9*4 = 162

for the retailer , in the same way revenue=p*q=q(40-2q)

marginal revenue=40-4q

marginal cost = 4+2(additional cost as given) =6

hence marginal cost = marginal revenue

6=40-4q => 4q=34 => q=8.5

hence p =23

profit = revenue - total costs = 23*8.5 - 6*8.5=144.5

if they are integrated there would be no additional costs for the retailer.

hence combined profit be =2*profit of the manufacturer (this is because we need to consider profit of each individual unit .) = 2*162 = 324

****change in profits = profit of merger - (profit of manufacturer and retailer without merger)

=324-162-144.5=17.5

consumer surplus when price = 23 ( without merger consumers are getting products from retailer) is

=0.5*(40-23)*8.5 = 72.25

consumer surplus when price =22 (with merger the price will be set at R) is

= 0.5*(40-22)*9

=81

****change in consumer surplus = 81- 72.25 = 8.75

change in social welfare =(consumer surplus plus producer surplus after merger) -(consumer surplus plus producer surplus before merger)

= (81+324)-(72.25+162+144.5)

=8.75+17.5

= 26.25