**Ramsey Pricing**

Ramsey pricing is a linear pricing scheme designed for the multiproduct natural monopolist
[see Frank Ramsey. "A Contribution to the Theory of Taxation," *Economic Journal*, March 1927].

**Basic idea:** Set prices (or rates) on the various services provided by the regulated firm such as to
*maximize social welfare (TS) subject to a profit constraint:*

**
TR - TC = M**

Where M is some fixed amount. In fact, one possibility is to set M = 0, in which case we have a break-even constraint.

**Assumption:** Constant marginal cost (MC) so that producer surplus (PS) is equal to zero. Thus:

**TS = TR + CS - TC
[1]**

Maximize [1] subject to:

**TR - TC = M
[2]**

We use calculus (specifically, the Lagrangean method. Note to students: This presentation will explain the math, but you do not need to know the Lagrangean method for testing purposes. ) to derive the Ramsey Pricing Rule, which can be stated as follows:

**P _{a} - MC_{a}/ P_{a} =
l /h **

where:

Pa is the price of service a;

MC_{a} is the marginal cost of service a;

is a constant; and

is elasticity of demand for service a;

The price that maximizes social welfare (TS) subject to a profit constraint will exceed marginal cost by an amount that is inversely proportional to elasticity of demand.

**Example:**

Let X be rail transport (measured in ton miles) of a high value article such as computers. | |

Let Y be rail transport (measured in ton miles) of a low value article such as oranges. | |

MC |

Provision of the services at a price equal to marginal cost would produce revenues sufficient to cover variable costs but not fixed costs. Marginal cost pricing would produce a loss of $1800. The cost function is given by

The cost function is given by:

**C = 1800 + 20X + 20Y**

The demand functions are given by:

**X = 100 - P _{X
}Y = 120 - P**

Option 1: Proportionate price increase.
Raise prices proportionately above MC until revenues are just equal to total costs.

The single price that equates TR and
TC is $36.30.

Note that with a proportionate price_{ }increase above MC:

**
DWL = DFH + JKH**

Where DFH is the loss of TS attributable to increasing the rate of shipment for the low value article (oranges) and JKH is the loss of TS due to the rate increase for the high value article (computers).

Note that the dead weight loss is greater for the commodity with higher elasticity of demand, i.e.

DFH > JKH.

**Option 2:**
The Ramsey Pricing Rule.
Raise price above marginal cost in inverse proportion to elasticity of demand.

The prices derived from use of the Ramsey pricing formula are:

**
P _{X }= $30 P_{Y} = $40**

The change in TS is:

where NTV is the DWL attributable to the rate increase on the low value article (oranges) and MTV is the DWL attributable to the rate increase for the high value article (computers).

Final notes:

The differential rate structure that emerges under the Ramsey pricing rule minimizes the welfare loss given the legal imperative of allowing a "fair" rate of return to the regulated firm. | |

The ICC has long used a variant of Ramsey pricing to establish interstate rail rates. Viscusi, Vernon, and Harrington write that "It has been common for rail rates for shipping gravel, sand, potatoes, oranges, and grapefruits to be lower relative to shipping costs for liquor, cigarettes, electronic equipment, and the like"[345]. |

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