Question

Green Valley Water and Granite Slope Water are sellers of spring water in the Republic of Bogritania. Like mineral water sellers in many simple examples, they have no costs so their payoffs are the revenues they obtain from selling water. The price of mineral water in Bogritania is 1,000-q_v-q_s , where q_v is the quantity sold by Green Valley and q_s is the quantity sold by Granite Slope. As a result:

Proposition A: Each firm's best response is to sell exactly half the amount left over by the other firm: for example, the best response q_v is q_v= 1/2(1,000-q_s)

A. Determine the Nash Equilibrium for this duopoly.

B. Bones question. Using calculus: demonstrate why Proposition A must be true

Answer 1

Ans.

Assume the price P = 1000 - Q

Where Q = q_v + q_{s =} total quantity produced.

Also given that TC_v = TC_s = 0 (cost of both firms is zero)

Consider profit as π_v for green valley water and π_s for granite slope water.

A.

Profit for Green valley water is

π_v = P(q_v) - TC_V = (1000 - q_v - q_s)q_v

Similarly we get

π_s = (1000 - q_v - q_s)q_s

Nash equilibrium is a pair of quantities where one player doesn't want to make a move keeping the other player quantity constant because his profit will be decreased.

That means partial differentiation of profit of green valley water with the quantity of green valley water should be zero. (This rule also goes for granite slope water)

Partial differential of π_v with respect to q_v = 0

1000 - 2q_v - q_s = 0. -(1)

Similarly applying same condition on granite slope water we get

1000 - 2q_s - q_v = 0. -(2)

Solving (1) &(2) we get

q_v = q_s = (1000/3).

Therefore (q_v,q_s) = (1000/3,1000/3) is the Nash equilibrium.

B.

From answer A equation (1)

1000 - 2q_v - q_s = 0

q_v = (1000 - q_s) / 2. Will be the best response of the green valley water if we know q_s value.

*Note : equation (1) from answer A is only considered because best response function can only be achieved by keeping the other quantity constant and (1) is the only case where q_s kept as a constant and changing q_v.