Consider a homogenous good industry with four firms. Total demand is given by D(p)=100-p. The variable (=marginal) cost of each of the firms is c1=20, c2=40, c3=50 and c4=55. Firms compete in prices. Suppose firms 1 and 2 merge into one entity and produce with a marginal cost of k>0. What is the threshold value for k below which the merger increases total welfare? [if applicable, round up to the second decimal place.]
Question
Consider a homogenous good industry with four firms. Total demand is given by D(p)=100-p.
The variable (=marginal) cost of each of the firms is c1=20, c2=40, c3=50 and c4=55. Firms compete in prices.
Suppose firms 1 and 2 merge into one entity and produce with a marginal cost of k>0.
What is the threshold value for k below which the merger increases total welfare? [if applicable, round up to the second decimal place.]
Solution
To solve this problem, we need to understand that total welfare in this context is the sum of consumer surplus and producer surplus. A merger will increase total welfare if it leads to a decrease in price, which increases consumer surplus, and/or an increase in total profit, which increases producer surplus.
Step 1: Determine the pre-merger equilibrium price and quantity.
In a Bertrand competition with differentiated products, each firm sets its price equal to its marginal cost. Therefore, before the merger, the prices are p1=20, p2=40, p3=50, and p4=55. The total quantity demanded is D(p)=100-p. The firm with the lowest price will serve the entire market, so the pre-merger equilibrium price is p=20 and the quantity is Q=100-20=80.
Step 2: Determine the post-merger equilibrium price and quantity.
After the merger, the new entity will set its price equal to its marginal cost, which is k. The other firms will not change their prices. Therefore, the post-merger equilibrium price is the minimum of k, 50, and 55. The quantity is Q=100-p.
Step 3: Determine the threshold value for k.
The merger will increase total welfare if the post-merger price is lower than the pre-merger price, i.e., if k<20. However, this is not possible because k>0 and the marginal cost of the merged entity cannot be lower than the marginal cost of the cheapest firm before the merger.
Therefore, the threshold value for k below which the merger increases total welfare does not exist.
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