# Management/ Price Discrimination is a Remarkably Widespread Business Practice. term paper 41875

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How Microeconomic Theory can be used to capture this Phenomenon.

Price discrimination, i.e. when buyers are charged different prices for the same commodity, is a widespread business practice. This essay will detail how microeconomic theory can explain such phenomena. Price discrimination in monopoly in the first and second degrees will first be discussed. Monopolistic third degree price discrimination will then be introduced, and thus leading onto the idea of spatial price discrimination under oligopoly, in particular Cournot- Nash and Bertrand- Nash price discrimination.

First and Second Degree Price Discrimination

In first degree price discrimination, the monopolist is able to identify each consumerï¿½s demand functions, and also preventing arbitrage. This is the re- selling of goods from those offered lower prices to those offered higher prices. Therefore the monopolist extracts all consumer surpluses amongst all buyers. A Pareto efficient outcome is achieved as each buyer pays a price that is equal to its marginal cost (Gravelle and Rees 196).

To examine this, assume that there are two types of buyers, n1 and n2, in the market, and the preferences of each type can be represented by their reservation constraints ūi:

(1.1)

where i=1, 2. qi is the monopolized good, Ui the utility of consuming it, and yi is a composite commodity, the price of which is the same for both types of buyers. It is also assumed that type 2 buyers have a stronger preference for qi, i.e.:

(1.2)

Both buyers have identical incomes M, and the monopolist produces at a constant marginal cost, c. Each buyer typesï¿½ budget constraints are vertically parallel. They buyerï¿½s choice problem:

(1.3)

where p is the price of good qi and F is a fixed charge. The following first order conditions result:

(1.4)

(1.5)

Hence we yield the following demand functions for xi and yi:

(1.6)

(1.7)

Thus the following indirect utility function is obtained:

(1.8)

First degree price discrimination can thus be represented graphically in figure 1. Each buyer maximizes their utility with respect to their preferences, and type 2ï¿½s reservation constraint is steeper. Thus, q1c and q2c are each buyerï¿½s demand of the monopolized good, at the point where p= c (Price= Marginal Cost). The area under the derived demand curves D1 and D2, the consumer surpluses, are denoted by S1 and S2 for each buyer type, which is also the distance denoted on the yi axis (Gravelle and Rees 198). The sum of these two surpluses is thus the monopolistï¿½s profit.

Fig. 1. First Degree Price Discrimination

Source: Gravelle, Hugh, and Ray Rees. Microeconomics. Edinburgh: FT Prentice Hall, 1992: 198

It may now be seen that the monopolistï¿½s optimal policy is to set a price of each type equal to marginal cost, which equate to different quantities of the commodity, and to set a fixed charge Fi= Si (i= 1,2) (Gravelle and Rees 199). This is also known as ï¿½perfectï¿½ price discrimination as not only entire consumer surpluses are extracted, each buyer reveals the value of the product by paying the highest price he or she is willing to pay. These two- part tariffs are carried out frequently in amusement parks, where the highest possible entrance fee is charged to the customer, as the price of each ride within the park is set equal to marginal cost, and hence the Pareto efficient outcome (Phlips 137).

So far it has been assumed that consumer demand is distinguishable by the monopolist. Assume now that the defining criterion between buyers is unobservable. Therefore the pricing policies described insofar would be undesirable as type 1 buyers, representing themselves as type 1s, would exhibit a positive consumer surplus and therefore the monopolist would ideally use a pricing policy that encourages consumers to sort themselves into appropriate buying types (Norman 116). This is known as second degree price discrimination, and the two part tariff is used as before, except that the buyer is now offered a quantity and a fixed charge instead of a price and a fixed charge. Hence the monopolistï¿½s profit:

(1.9)

and the buyerï¿½s reservation constraints now expressed as direct utilities:

(1.10)

where yi= M- Fi. In addition, self- selection constraints exist to ensure each buyer chooses the appropriate deal:

(1.11)

(1.12)

Type i will accept these deals if qi and Fi are satisfied by them. Solving for xi and Fi by maximizing profit with respect to equations 1.11 and 1.12, it may be seen that a type 2ï¿½s reservation constraint and a type 1ï¿½s self selection constraint are self- binding (Gravelle and Rees 200). Type 2 buyers must be offered (q2, F2) such that u2>ū2, thereby satisfying both reservation and self selection constraints for both types (Gravelle and Rees 200- 201). Thus, as before, type 2 buyers are offered a higher price than that of type 1s.

Third Degree Price Discrimination under Monopoly

Third degree price discrimination, also known as spatial price discrimination, is where the monopolist may divide the market for the output into subgroups (not within groups), often based on exogenous criteria such as distance from the seller. It is assumed that the cost of supplying the two groups are identical, and the demand (and therefore marginal revenue) of the subgroups are known by the monopolist. A fixed level of output, q0, is divides between the subgroups 1 and 2 as q1 and q2 respectively. The monopolist will divide this quantity between subgroups in a way to maximize profit, and therefore to maximize revenue. Marginal revenue in each submarket must be equal; otherwise more profit can be made from taking output from one submarket and selling it to another. Hence, profit maximization may be expressed as:

(2.1)

where MRi (i=1, 2) are the marginal revenues for markets 1 and 2. In addition, price discrimination will only exist if the price elasticity of demand in each subgroup is unequal. If ε1 and ε2 are the elasticities of demand in each submarket, then

(2.2)

or

(2.3)

i.e. if ε1< ε2, then p1

Fig. 2. Third Degree Price Discrimination

Source: Norman, George. ï¿½Price Discrimination.ï¿½ The New Industrial Economics: Recent developments in industrial organization, oligopoly and game theory. Ed. Manfredi La Manna and George Norman. Aldershot: Edward Elgar, 1995: 122.

The quantity level q0*, determined by MC= ΣMR (the aggregate marginal revenue), gives rise to the division of q0* according to the p0* levels of MR1 and MR2, which gives rise to levels of p1* and p2*, and thus the higher price p1* arises from the more inelastic demand of group 1 buyers.

Third Degree Price Discrimination under Oligopoly

1. Cournot-Nash

Spatial price discrimination is often an occurrence under oligopolistic conditions, and is a result of distance from the firms to the consumers. According to Greenhut, ï¿½Each seller is cognizant of his closest rivals, each is affected by the prices of said rivals, and from time to time each is aggressively concerned over how the rival (down the street) will react to his own actions. Costly distance limits consumers in choosing suppliers, and in the process establishes a network of spatial oligopolists.ï¿½ (229). Considering a spatial market, k, with nk competing sellers, kï¿½s aggregate inverse demand function is as follows:

(3.1)

where Qk is the aggregate supply to market k. Assuming marginal cost is constant at ci and transport costs to market k are linear at tik per unit, and the quantity supplied by firm i to market k is qik, then the profit π of firm i supplying market k:

(3.2)

The Cournot- Nash equilibrium price in market k, which is the ï¿½reaction functionsï¿½ (system of nk first order conditions), and after summing over nk firms, and rearranging for pk is as follows (Norman 124- 125):

(3.3)

where Â¬ k and k are the mean marginal production and transportation costs respectively. Equation (C.3) gives rise to a range of cases of price discrimination. Assume 2 markets (k= 1, 2), which contain a local seller able to sell in both markets, i.e. nk= 2. Assuming ci= 0, and 1= 2t/3 and 2= t. Thus Cournot- Nash equilibrium prices will be higher in market 2 than in market 1 as, with reference the equation C.3, the higher the average transportation costs the lower the equilibrium price (Norman 125- 126).

2. Bertrand- Nash

Bertrand- Nash price discrimination assumes buyers are evenly distributed over a given market space. Assume a closed line, marked from 0 to 2, and three sellers, each occupying points 0, 1, and 2, as in figure 3 below.

Fig. 3. Third Degree Price Discrimination: Bertrand-Nash

Source: Norman, George. ï¿½Price Discrimination.ï¿½ The New Industrial Economics: Recent developments in industrial organization, oligopoly and game theory. Ed. Manfredi La Manna and George Norman. Aldershot: Edward Elgar, 1995: 129.

All buyers have the identical demand functions:

(3.4)

where p(r) and q(r) is the price and quantity for the buyer at a given distance r respectively. Consider a linear transport function tr. In the absence of oligopoly, the optimal price in monopolistic third degree price discrimination with 50% freight absorption for firm i (i= 0,1,2) would be as follows:

(3.5)

The Bertrand- Nash is the competitive interaction between sellers at every point (Norman 127-128), and is marked as the thick line in figure 4. Sellers will either exploit monopoly power P0M, or will undercut a competitor by selling at marginal cost Ci+tr. (Seller 1 has no monopoly power in this case). It may be seen that price falls with distance from the seller in regions where there is competition for buyers.

Conclusion

Price discrimination, a prevalent business practice, takes a wide variety of forms, whether it occurring through monopolistic or competitive forms. The varieties that have been covered, however, are only a small fraction of the different explanations of price discrimination. In order to discuss the microeconomic explanations in full, a much larger debate is needed.

Works Cited

Gravelle, Hugh. and Ray Rees. Microeconomics. Edinburgh: FT Prentice Hall, 1992.

Norman, George. ï¿½Price Discrimination.ï¿½ The New Industrial Economics: Recent developments in industrial organization, oligopoly and game theory. Ed. Manfredi La Manna and George Norman. Aldershot: Edward Elgar, 1995, 123-69.

Greenhut, Michael. ï¿½The Impact of Distance on Micro- Economic Theory.ï¿½ Manchester School 46.4(1978): 17- 40.

Phlips, Louis. ï¿½Price Discrimination: A Survey of the Theory.ï¿½ Journal of Economic Surveys. 2.2(1998): 135- 67.

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