Finding the long-run equilibrium price:
Set the market demand equal to the supply of each individual firm at the equilibrium price (assuming identical firms): x(p) = q, where q is the quantity produced by each firm. Substitute the demand function: 100 - p = q. Solving for p: p = 100 - q.
Finding the minimum average cost (LAC):
Differentiate the long-run cost function clr with respect to y (output) to find the marginal cost (MC): MC = 20y.
In long-run equilibrium, MC = p (since each firm faces a perfectly elastic demand curve due to numerous competitors).
Substitute the equilibrium price expression from step 1: 20y = 100 - q.
Solve for y: y = 4 - 0.5q.
Equating LAC and market price:
Set the minimum average cost (LAC), which is just the long-run cost divided by output, equal to the equilibrium price: clr/y = p. Substitute the cost function and y expression from step 2: (10y^2 + 10) / (4 - 0.5q) = 10q.
Simplify and solve for q: q = 40.
Finding the number of firms:
Divide the market demand at the equilibrium price (from step 1) by the individual firm's output: Number of firms = x(p) / q = (100 - 100 + 40) / 4=1.
Therefore, in the long run, only one firm will be active in the market because only one firm can achieve the minimum average cost and earn normal profits at the equilibrium price. This is because the market demand cannot be efficiently divided among multiple firms with the given cost structure and demand function.
It's important to note that this assumes complete information and perfect competition in the market. In real-world scenarios, other factors like entry barriers and imperfect information can affect the number of firms in the long run.