An industry is a natural monopoly if the production of a particular good or service by a single firm minimizes cost.

Classical regulatory problem: economic efficiency (production at minimum cost) v. market power.

Note: Technology changes or growth in demand can transform what had been a natural monopoly into an oligopoly or even an effectively competitive market structure. Examples: Natural gas transmission, electricity, telephony.

Subadditive Cost Functions and Natural Monopoly: It used to be that natural monopoly was simply defined as existing when the AC curve is everywhere downward-sloping relative to market demand (economies of scale for for a single firm to produce the market-clearing Q). As it ends up, this is a bit too restrictive-natural monopoly can occur when there are diseconomies of scale. Let's see why.

Baumol (1977) and others pioneered the notion of subadditive costs, and that we can define natural monopoly (both for single- and multi-product lines) as existing when the cost function is subadditive.

So what does subadditivity mean? A cost function is subadditive when the total cost of producing industry output is lowest when a single firm produces it.

It is easiest to see subadditivity for a single-product case. Start with the usual U-shaped average cost curve for a single firm. Assume that other firms have access to this same technology (they have identical costs). We can then construct an "industry" average cost curve through horizontal summation (for example, the minimum point in the "industry AC" curve is N times the minimum point in the "single-firm AC" curve, where N is the number of firms). DRAW…. Note that the industry total cost of producing a given "Q" is (industry AC)*Q, while the individual-firm total cost of producing a given "Q" is given by (single-firm AC)*Q.

Find the point where the single-firm AC and the industry AC curves intersect. The single-firm AC curve features subadditivity for all outputs below the point of intersection…. WHY?

NOTE: As figure 11.4 in the Viscusi et al. text illustrates, natural monopoly based on the subadditivity definition can occur even in a range of outputs for which there are diseconomies of scale in production (i.e., on the upward-sloping portion of the individual-firm AC curve). Thus the presence of economies of scale is sufficient (in the single-product case) but not necessary for the existence of natural monopoly. In the case of multiple products, the existence of economies of scale in the production of any one product is neither necessary nor sufficient (because of economies of scope in the production of multiple products).

NOTE: Subadditivity does not imply that a natural monopoly is sustainable. If a natural monopoly produces on the upward-sloping portion of its AC curve (and is assumed to continue doing so for some time after entry) and sets a zero-profit price P = AC, then an opportunistic entrant could potentially produce the "Q" at the minimum point of the AC curve, undercut the natural monopolist's price and displace the natural monopolist-implying that the natural monopoly is unsustainable. A natural monopoly is assured of being sustainable if it is producing on the downward-sloping portion of its AC curve.

Efficient Pricing and Natural Monopoly

Suppose that we have a sustainable natural monopoly, that its costs were observable, and that demand is stationary. What would be the best way to set price?

1. Linear MC-pricing: Allocative efficiency satisfied. What about profit for the firm? Negative. Solutions? Use tax revenues to meet shortfall? This generates additional distortions, and it could be that TB < TC; weakens incentive to dynamically minimize cost and implement cost-reducing innovation; involves a cross-subsidy from non-consumers. In the absence of external benefits or costs, most economists argue that pricing systems rather than taxes should be used to recover costs.

2. Linear AC-pricing: Fails allocative efficiency. Firm earns normal economic profits. Weak incentives to dynamically minimize cost and implement cost-reducing innovation;

3. Non-Linear Tariffs: Most basic form would be a two-part tariff: Tariff = a + bQ, where "b" is set equal to MC, and "a" contributes toward covering the negative economic profits that derive from MC-pricing. If "a" is set so that when summed across all consumers it just equals the profit shortfall in (1) above, then this system assures allocative efficiency AND normal economic profits-socially ideal from a static perspective. It still fails to provide incentive to dynamically minimize cost and implement cost-reducing innovation. Problem: In the real world consumers have differing demand curves, and the uniform ("nondiscriminatory) fixed fee "a" may exceed consumer surplus for some low-demand consumers who would have otherwise been willing to pay MC prices, which is a failure of allocative efficiency itself. Thus the zero-profit constraint implies a tradeoff between balancing the inefficiency due to excluding low-demand consumers with "a" against the inefficiency due to "b" > MC. The socially optimal (and discriminatory) two-part tariff will usually have an "a" that excludes some consumers (failure of universal service) and a "b" > MC (failure of allocative efficiency).

Real-World Example of a Nonlinear Price Schedule: "Declining-Block Tariffs"

Public utilities (power, telephone) have used declining-block tariffs. Here's an example:

Fixed fee = $10

+ 10 cents per call for the first 100 calls; + 5 cents per call for between 100 and 200 calls; + 2 cents per call for more than 200 calls

Consumers then self-select their tariff schedule (and thus reveal their demand) by the number of calls they make. Note that the first segment of the tariff schedule has the formula Tariff = $10 + $0.1Q; the second segment of the schedule has the formula Tariff = $15 + $0.05Q; the third segment has the formula Tariff = $21 + $0.02Q (WHY?). DRAW DIAGRAM…

Thus declining-block and similar pricing systems can be thought of as allowing consumers to select their own preferred two-part tariff based on their own demand characteristics.

Ramsey Pricing

Frank Ramsey developed a method for establishing efficient pricing and taxing schemes. In the context of a multi-product monopolist, each product would have a linear price, and the set of prices would minimize deadweight social losses subject to the zero profit constraint. Ramsey pricing can be complex when the products are interdependent.

When the demand for the products are independent, Ramsey pricing can be done at the assumed level of technical expertise for this course.

Example:

Suppose that a two-product natural monopolist has the following cost structure:

TC(Q1,Q2) = $1000 + 50Q1 + 50Q2

Demands:

Q1 = 150 - P1

Q2 = 200 - 2P2

Note that MC-pricing fails to meet the zero-profit constraint. Thus each product's price must exceed MC. But by how much?

Ramsey Pricing Rule:

(Pi - MCi)/Pi = X/Ei, i = 1, 2, …, N products,

Where X is a constant and Ei is the (own) price elasticity of demand. Thus the Ramsey-optimal pricing scheme features an inverse relationship between the Price-MC margin and the elasticity of demand. For this independent demand case, the Ramsey-optimal pricing scheme implies cutting the marginal-cost output for each product by the same proportion until zero profit is realized.

In our example above, the marginal-cost outputs are Q1 = 100 = Q2. If the two marginal-cost outputs are to be reduced by the smallest proportion such that profit is zero, then let "z" = (1-smallest proportion) be the largest proportion of the marginal-cost Q that solves the Ramsey-optimal pricing problem:

Profit = 0 = P1(Q1=100z)*100z + P2(Q2=100z)*100z - 1000 - 50*100z - 50*100z,

where inverse demands P1(Q1=100z) = 150-100z, P2(Q2=100z) = 100-50z.

Solving for "z" gives the following: 15,000z - 15,000z**2 - 1000 = 0; dividing across by -1000 gives

15z**2 - 15z + 1 = 0.

Quadratic equation: z = (15 +/- sqrt(225 - 4*15*1))/30 = (15 +/- 12.845)/30 = 0.9282, 0.072. Solution should be 0.9282.

Check: Profit = 13,923 - 12,923 - 1000 = 0!!!

What are the implied Ramsey prices? P1 = 150 - 92.82 = $57.18; P2 = 100 - 46.41 = $53.59.

Note: Using the elasticities at P = $50 as an approximation, the constant "X" in the Ramsey equation above is approximately 0.064.