Question

1. Two firms, one in Spain and the other in France, act as Cournot competitors in supplying mussels to Belgium. The inverse demand for mussels in Belgium is given by p = 100 − 2Q, where Q is the total quantity supplied. The marginal cost for each firm (including shipping) is 25.

(i) Calculate the Cournot equilibrium in Belgium and the profits of the two firms on their exports.

(ii) Now the French government agrees to subsidise French mussel producers at a level of s per unit. Recalculate the Cournot equilibrium and compare it to the equilibrium in part (i). Explain the intuition for this result.

Answer 1

i) The inverse demand for mussels in Belgium is

p =100-2Q

let quantity produced by firm in Spain = q₁ and quantity produced by firm in france = q₂ then

inverse demand function will be

p =100-2(q₁ +q₂ ) ............(i)

where Q= q₁ +q₂

The marginal revenue of firm in Spain (MR₁ ) = dpq₁/ dq₁ = 100-4q₁ -2q₂

& marginal revenue of firm in France (MR₂ ) =dpq₂ /dq₂ =100-4q₂ -2q₁

Also marginal cost (MC) of each firm = 25

therefore, Cournot Nash equilibrium conditions are

d$\pi$₁ /dq₁ = 100-4q₁ -2q₂ -25=0

or, 4q₁ +2q₂ =75 ..........(ii)

& d$\pi$₂ /dq₂ = 100-4q₂ -2q₁ -25=0

or, 4q₂ +2q₁ =75 .........(iii)

Solving (ii) and (iii) we get, q₁ = 12.5 and q₂ =12.5 Therefore, Cournot equilibrium output of each firm is 12.5

ii) If a subsidy of 5 per unit is provided by French government to firm in France, then

marginal cost of Firm in France = 25-5=20

Therefore, from part (i) we get,

d$\pi$₂ /dq₂ = 100-4q₂ -2q₁ -20=0

or, 4q₂ +2q₁ =80 ..........(iv)

Also from part (i), we get

4q₁ +2q₂ =75...........(v)

Solving (iv) and (v), we get q₂ = 14.16 and q₁ = 11.66

Here with government subsidy of 5/unit, firm in France produces more output than the firm in Spain. while with zero subsidy, each firm is producing the same output.

iii) With subsidy of 5 per unit, Firm in France reduces its cost of production and is able to produce higher output than firm in Spain. This violates the Cournot equilibrium output condition of both firms. With no government subsidy, both firms compete with each other in a Cournot Duopoly market ad produces equivalent level of output.