San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

An Economic Welfare
Analysis of Oligopoly

Background to the Problem

The economic welfare analysis of a protected monopoly or a protected monopsony is a straight forward bit of standard economic analysis. A protected monopoly which is unregulated makes profit by restricting production to raise the price of its product. It makes a profit but the gain in profit from monopolization of a market is less than the cost to consumers resulting from the higher price. Therefore there is a net social loss from a protected monopoly.

The loss to the consumers is the area of the pink-colored trapezoid. The loss in producers surplus is the area of the purplish-colored trapezoid. The profits of the monopolist is the green-hatched rectangle. Clearly the loss of consumer and producer surpluses is greater than the amount of the monopoly profits by the area of the triangles.

It is very important to remember that the above analysis applies only to the protected monopoly; i.e., a firm who is protected by the State from competition. If there is a small town which there is only a a single grocery store despite freedom of entry then that store has a monopoly but it is not a protected monopoly. Even if that store exploits its monopoly power there is no economic welfare loss due to monopoly. When the town grows enough it will get another store. The town will get another store when someone sees that the revenue it will generate exploiting all the opportunities for price setting and discrimination will be greater than the cost. For the people of the town it is not a choice between perfect competition and monopoly it is a choice between mon-opoly and zero-opoly. The towns people are clearly better off with the exploitative monopoly than they were with no store because they can if they ignore the existence of the store and continue to do whatever they did before the store located there.

The above point harkens back to the French engineer, Jules Dupuit, who essentially founded cost benefit analysis. He asked when should a bridge be built in a particular location. His answer was that it should be built when the revenues that would be generated if the bridge were operated as a monopoly taking advantage of price discrimination exceeds the costs of building it. In building and operating it the costs may be reduced taking advantage of monopsonistic advantages at the location. Dupuit did not assert that the bridge should actually be operated that way, he was only looking for a criterion about when or if the bridge should be built. However, any other mode of operation would require government subsidies financed out of taxes which would involve a transfer of welfare from one segment of the economy to another.

On this matter of possible welfare losses due to competition deemed not to be perfect the Austrian School of Economics deviates sharply from the Neoclassical School. Clearly in this matter the Austrian School is correct; freedom of entry and exit is the essential criterion of competitiveness.

The Analysis of Oligopoly

The economic analysis of oligopoly is always fraught with the dilemma of there being multiple models of the behavior of firms in an oligopolistic market. Additionally the welfare analysis of oligopoly has the problem of what is the condition for comparison. Perfect competition is usually the standard for comparison. The appropriate standard is usually the socially optimal condition.

In oligopoly theory there are two class of models

The conjectural variation models presume a fixed number of firms operating in the market and this corresponds to a situation of protected oligopoly. The limit pricing model of oligopoly corresponds to a condition of unprotected oligopoly.

The first task is to determine how a given level of output should be produced from a set of plants.

The Socially Optimal Number of Plants

Consider first the case in which there is an indivisibility constaint, such as the number of airplane flights on a given route. The social cost and benefit situation might be as shown below.

There is a small increase in cost for each additional passenger and a big discontinuous increase when an additional plane has to be put into service. An incorrect interpretation of the marginal cost-pricing principle would suggest that for economic efficiency the passengers should be charged the negligible cost of carrying one more passenger on a partially filled plane or the enormous cost of putting another plane into service. The correct interpretation of the marginal cost-pricing principle is that for economic efficiency the passengers should be charged the average cost per passenger of another planeload of passengers. As will be demonstrated, the relevant marginal cost for economic efficiency in general is the minimum average cost of the marginal plant rather than the intra-plant marginal cost. This is equivalent to the condition that the marginal plant is earning no economic profit, a condition that prevails when there is freedom of entry and exit to and from the industry.

In the preceding analysis there was a physical capacity which necessitated the introduction of additional plants, but in general the introduction of an additional plant and its operating level is a matter of choice. Suppose the cost functions for the different plants are known, say

ci = fi(xi)
for i=1,...,n.

Since the focus of interest is how many plants should be brought into existence and operation it will also be presumed that the cost function, particularly its fixed cost, for a plant does not apply until that plant is in operation. Otherwise the ficed costs of all the plants would be sunk costs and would not be relevant in determining the optimal plants and their outputs. The exclusion of the fixed costs of the plants if they are not in production can be achieved by requiring that for all i,

fi(0) = 0,
even though
limxi->0+ > 0.

Note also that this formulation which makes the cost of a plant a function only of its output presumes that each plant's costs are unaffected by the level of output of the industry.

The industry cost function is defined as:

C(X) = minxi Σin fi(xi)
subject to the constraint
Σinxi = X

The first order conditions are easily derived as:

fi'(xi) = λ
for all i such that
xi >0.

This means that the intra-plant marginal cost of all operating plants is the same; i.e., λ.

For purposes of finding the optimal operation of the plants the concept of the convex envelope of the industry cost function is relevant. The convex envelope of the cost function could be defined in terms of the boundary of the convex closure of the cost function; i.e., the intersection of all the convex sets which enclose the feasible cost combinations. Geometrically the envelope is easily constructed by drawing tangent lines to the industry cost function.

Suppose the envelope cost function is designated as C*(X). The algorithm proposed is to find the set of plants that should be in operation by finding the value of X which maximizes SB(X) - C*(X), say X*. The value of X* is then the basis for finding the value of X which maximizes SB(X) - C(X).

On the basis of a consideration of production runs the industry cost function can be reformulated as:

C(X) = minzi,ti Σin tifi(zi)
subject to the constraints
Σintizi = X
ti≥0, zi≥0,

where ti is the fraction of the time the i-th plant is running, zi is the rate of production of the i-th plant when it is running and the annual rate of production xi is equal to tizi.

The first order conditions with respect to the production levels in the plants are:

tif'i(zi) = λti
for all i such that
zi > 0.

This reduces to the condition previously found; i.e.,

fi'(zi) = λ
for all i such that
zi > 0.

The interesting new element in the problem is the first order conditions with respect to the ti's:

f(zi) = λzi

Since λ = fi'(zi) from the first order conditions for zi and λ = fi(zi)/zi it follows that an operating plant is operating at its economic capacity if it is operating less than full or equal to full time; i.e., the rate of production at which average cost is a minimum. If the value of ti is limited by the constraint that ti≤1 then the above first order condition does not apply.

At this point it is convenient to introduce what might be loosely described as the supply function for a plant. Supply function is the wrong term because the relationship constructed may involve more than one quantity supplied for a particular price. This means the proper terminology would be supply correspondence. But a better term would be the inverse supply curve; i.e., the price require to elicit the different quantities supplied. Strictly speaking this would be the inverse of the supply correspondence.

Consider a plant not yet built so the cost for zero production is zero. For prices below the minimum average cost the quantity supplied would be zero. An example of the marginal and average cost functions are as shown.

At a price equal to the minimum average cost the quantity supplied could be either zero or the quantity equal to the economic capacity. If it is presumed that a plant's operation can be switched on or off at no cost then the plant can achieve an average production of any quantity from zero up to its economic capacity simply by varying the proportion of the time that it is in operation. For outputs above the economic capacity the inverse supply curve is the same as the marginal cost curve for the plant.

Each potential plant would have its inverse supply curve and thus its supply correspondence. The industry supply correspondence is simply the horizontal sum of the plant correspondences. This industry supply correspondence corresponds to the social marginal cost curve.

The above analysis indicates that the industry cost function is constructed by building and putting into operation the plants in the order of their minimum average cost values. When a plant is put into operation the value for t increases from 0 to 1. At the levels of industry output that has the marginal plant operating full time marginal cost has to increases and the marginal plant operates full time at a production level above its economic capacity. When the level of industry output increases to the level such that the industry marginal cost reaches the level of the next plant; i.e., the plant with the lowest minimum average costs of those not yet in production, then that next plant is brought into production. The value of t for that marginal plant increases from 0 to 1. This process is repeated indefinitely or until all possible plants are in operation and the industry is producing at the highest level it can. This latter would apply if there were some restricting factor such as land which would prevent building an unlimited number of plants.

This method of construction of the industry cost function is equivalent to finding the convex hull of the industry cost function constructed without allowance for partial operation of plants. The relevant marginal cost is slope of the convex hull of the industry cost function and this is closely related to the minimum average cost of the marginal plant.

Unprotected Oligopoly (Undifferentiated product)

The industry marginal cost function consists of flat portions whose levels are equal to the minimum average cost of the plant that is brought into operation at that level of industry production. There are also transition sections of the marginal cost for the transition between the levels of minimum average cost of the plants. For the moment ignore these transition sections. The condition for a social optimum is that the marginal social benefit (which is the same as market price) be equal to the marginal cost. But the marginal cost is equal to the minimum average cost of the marginal plant. This means that the marginal plant is operating at zero economic profit. This is just the condition for industry equalibrium when there is freedom of entry and exit.

The end result is that if the marginal social benefit function intersects the industry marginal cost function then the marginal plant is just breaking even; i.e., the market price is equal to the minimum average cost. Such a condition would prevail if there is freedom of entry and exit for the industry. The plants in operation with minimum average costs below the market price would be earning economic profits.

If the marginal social benefit function intersects the industry marginal cost at a point in between the flat sections then market price would be sufficient for the plants in operation to earn an economic profit but not sufficient for the incipient plant to break even. This too is a condition that is consistent with industry equilibrium with freedom of entry and exit.

Under the limit-pricing model of oligopoly a dominant firm would establish a price just under the entry price of a potential entrant. This might involve a price greater than the socially optimal price. Such a price would involve the existing firms with their existing plants wanting to supply more than the market wants to purchase at that price. The market could not be in equilibrium until the market price dropped to the level that would balance quantity demanded with the quantity supplied. An alternative resolution would be for the existing firms to establish different prices for their product.

Thus the existence of the transition sections of the industry marginal cost function allows for the existing firms to be making an economic profit because another firm entering would not be able to break even. This is exactly the condition that would prevail in an industry with freedom of entry and exit. Thus there is no difference between the social optimum and the condition that would prevail under freedom of entry and exit. Thus there is no social welfare loss for an unprotected oligopoly.

(To be continued.)

Protected Oligopoly

If the number of firms and plants in operation is legally limited then there could be a social welfare loss.

(To be continued.)

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