Industrial Economics

...

The specular case when pA > pB is analogous.

Notice then that an indifferent consumer will consequently exist only in the first case.

- The strategy combination composed of pA = pB = 1 – t1 determines an equal division of the market and profits for both firms equal to πA = πB = (1 – t1)/2.

If one of the two firms increases the price, she will lose the whole demand (and profits will become zero). The deviation is then not profitable.

If one of the two firms ‘slightly’ decreases the price (i.e., the price remains greater than 1 – t2) she will not obtain any new market share, and the profits will consequently decrease. Also in this case the deviation is then not profitable.

If one of the two firms ‘significantly’ decreases the price (i.e., the price is not greater than 1 – t2 → exactly 1 – t2 would be the optimal strategy), that firm will conquer the entire market and the profit consequently becomes (1 – t2) instead of (1 – t1)/2. If (1 – t2) t1)/2, i.e., if t1 t2 – 1, that is if t1 is sufficiently small w.r.t. t2, the deviation is not profitable and then pA = pB = 1 – t1 is a Nash equilibrium. Otherwise it is not an equilibrium.

Exercise 3

A natural monopoly produces two different products (X and Y) facing a cost function equal to:

C = 1.050 + 20X + 20Y

Market demands for goods X and Y are respectively:

Px = 100 – X ; PY = 60 – 0.5 Y

Answer to the following questions:

- Assume that the regulator sets the prices for both product using the “fully distributed cost methodology”, given that fixed costs are allocated at 75% to product X. Determine the price for both products, the quantity sold and the deadweight loss of such solutions;

- Assume now that the regulator would like to set prices in order to obtain the second best solution in a multi-product setting. Let the shadow cost of public funds λ = 1/6. Determine the optimal prices and the dead weight loss for both products X and Y.

- Give the above results, compare the results in 1) and 2) and explain which solution is socially preferable and why (Note: use economic arguments).

- Finally, assume that the regulator can adopt a two part tariff, with a fixed fee T and a per-unit price equal to p. At which level would the regulator set the two components of the tariff T and p? What would it be the deadweight loss in this case?

Solution outline

- The fully distributed cost methodology implies that the prices of each product would be set equal to the average cost of production of each product. This average cost incorporates a share of the total fixed costs whose allocation is given by the text (75% to product X and consequently 25% to product Y). In symbols:

Px = ACx = [pic 1]

PY = ACY = [pic 2]

Solving the above expressions (by approximating to the first decimal), we obtain:

Px = 31.5 ; X = 68.5

PY = 23.6 ; Y = 72.8

To determine the deadweight loss, note that the first best solution is Px = Py = 20 and Xfirst = Yfirst = 20.

Hence it results DWLX = 66.125, DWLY = 11.52, for total welfare loss equal to 77.645.

- In a multi-product setting, the optimal second best regulation is when the prices follow the so called Ramsey-Boiteaux formula, i.e. [pic 3] where εd is the demand elasticity. Hence, in this case the optimal solutions are:

[pic 4]

[pic 5]

Hence it results DWLX = 50, DWLY = 25, for a total deadweight loss equal to 75.

- It is straightforward to note that in case 2) the social loss is lower than in case 1). This is because in presence of multiple goods with different demand the regulator needs to take into account of the different demand sensitivity when fixing the final price.

- In case of a fixed fee, the regulator would like to set a tariff in order to cover firm’s cost and maximize consumer surplus. This happens when the regulator sets p = 20 and the fixed fee T at the level that covers the fixed costs, i.e. T = 1.050, so that the firm just breaks even. This solution dominates (from a social perspective) the previous ones, since the deadweight loss in this case is null as in the perfect competition case.

Exercise 4

From