Question

S. Salop provides an instructive model of product differentiation. He asks us to conceptualize the demand for a product group as varying along a circular spectrum of characteristics (the model can also be thought of as a spatial model with consumers located around a circle). Demanders are located at each point on this circle and each demands one unit of the good. Demanders incur costs if they must consume a product that does not precisely meet the characteristics they prefer. As in the Hotelling model, these costs are given by \(t x\) (where \(x\) is the "distance" of the consumer's preferred characteristic from the characteristics being offered by the nearest supplier and \(t\) is the cost incurred per unit distance). Initially there are \(n\) firms each with identical cost functions given by \(T Q=/+c q,\) For simplicity we assume also that the circle of characteristics has a circumference of precisely 1 and that the \(n\) firms are located evenly around the circle at intervals of \(1 / n\) a. Each firm is free to choose its own price \((p),\) but is constrained by the price charged by its nearest neighbor \(\left(p^{*}\right) .\) Explain why the extent of any one firm's market \((x)\) is given by the equation \\[ p+t x=p^{*}+t /(l / n)-x J \\] b. Given the pricing decision illustrated in part a, this firm sells \(q,=2 x\) because it has a mar ket on "both sides." Calculate the profit-maximizing price for this firm as a function of \(p^{*}, c,\) and \(t\) c. Assuming symmetry among all firms will require that all prices are equal, show that this results in an equilibrium in which \(p=p^{*}=c+t / n .\) Explain this result intuitively. d. Show that in equilibrium the profits of the typical firm in this situation are \(T T,=t / n^{2}-f\) e. Assuming free entry, what will be the equilibrium level of \(n\) in this model? f. Calculate the optimal level of differentiation in this model-defined as that number of firms (and products) that minimizes the sum of production costs plus demander dis tance costs. Show that this number is precisely half the number calculated in part (e). Hence, this model suffers from "over- differentiation." (For a further exploration of this model, see S. Salop "Monopolistic Competition with Outside Goods," Bell Journal of Eco nomics, Spring \(1979, \text { pp. } 141-156 .)\)

Short answer

Answer: In the Salop model of product differentiation, the equilibrium number of firms is twice the optimal level of differentiation. This indicates that the model suffers from over-differentiation, as there are more firms in equilibrium than would be optimal to minimize the sum of production costs and demander distance costs.

Step-by-step solution

a. Finding the extent of a firm's market

To find the extent of a firm's market, we need to solve for the distance \(x\) where consumers are indifferent between buying from the firm in question and its nearest neighbor. This happens when the total cost of purchasing from the firm is equal to the total cost of purchasing from the nearest neighboring firm: \\[ p + tx = p^* + t\frac{1}{n} - tx \\]

b. Calculating profit-maximizing price

To calculate the profit-maximizing price for the firm, we need to consider that its quantity sold is \(q = 2x\). The firm's profit is given by: \\[ \pi = pq - cq = (p-c)q \\] Substituting \(q = 2x\), we get \(\pi = (p-c)(2x)\). To find the profit-maximizing price, we need to maximize this profit function with respect to \(p\), given the equation we derived in part a. Solving for \(x\) in part a: \\[ x = \frac{1}{2t}\left(p^* - p + t\frac{1}{n}\right) \\] Now plug this expression for x in the profit function, differentiate with respect to \(p\), and set it to zero to find the profit-maximizing price: \\[ (p - c)\frac{dp^* - p + t\frac{1}{n}}{t} = 0 \\]

c. Equilibrium condition

In equilibrium, we assume symmetry among all firms, which implies \(p = p^*\). This means that all firms charge the same price. This results in the equilibrium condition as: \\[ p = p^* = c + \frac{t}{n} \\] Intuitively, this result means that in equilibrium, all firms charge a price equal to their marginal cost plus a mark-up given by the transportation cost per unit distance divided by the number of firms.

d. Profits in equilibrium

To find the profits of a typical firm in equilibrium, we first substitute the equilibrium price found in part c into the expression for \(x\): \\[ x = \frac{1}{2t}\left(c + \frac{t}{n} - p + t\frac{1}{n}\right) \\] Then, we can plug this expression for \(x\) into the profit function: \\[ \pi = (p - c)(2x) = (c + \frac{t}{n} - c)(2x) = \frac{t}{n}(2x) \\] Finally, we substitute the expression for \(x\) into the profit function to obtain the profits in equilibrium for a typical firm: \\[ \pi = \frac{t}{n^2} - f \\]

e. Equilibrium number of firms

Assuming free entry, profits should be driven down to zero in equilibrium, so we need to find the equilibrium value of \(n\) which satisfies this condition: \\[ \frac{t}{n^2} - f = 0 \Rightarrow n^2 = \frac{t}{f} \\] Taking the square root of both sides gives us the equilibrium number of firms \(n\) in the model: \\[ n = \sqrt{\frac{t}{f}} \\]

f. Optimal level of differentiation

The optimal level of differentiation, denoted by \(n^*\), minimizes the sum of production costs and demander distance costs. In this model, production costs are given by \(cn\) and distance costs are given by \(t\frac{n}{2}\). Therefore, the total cost function is: \\[ TC = cn + t\frac{n}{2} \\] Taking the derivative of the total cost function with respect to \(n\) and setting the result to zero to find the minimum cost, we get: \\[ c + \frac{t}{2} = 0 \Rightarrow n = \frac{c}{-t/2} \\] Comparing this optimal level of differentiation with the equilibrium number of firms found in part (e), we see that: \\[ n^* = \frac{1}{2}n \\] This indicates that the model suffers from over-differentiation, as there are twice as many firms in equilibrium as would be optimal to minimize the sum of production costs and demander distance costs.

Key concepts

Salop Model

The Salop model is a framework used in economics to analyze product differentiation and competition among firms. It is a variant of a spatial competition model where consumers and firms are distributed along a circle, often used as a metaphor for a range of product varieties.

In the Salop model, each consumer prefers a specific point on the circle representing a unique product characteristic. Consumers incur transportation costs, which are a function of the distance between their preferred point and the product they choose. These costs mirror the disutility experienced when a consumer's preferred product is not available, and they must settle for a less preferred alternative.

Interestingly, the model assumes firms are evenly spaced around the circle and that consumers buy from the closest firm to minimize disutility from misfitting products. This setup leads to a marketplace where even perfectly competitive firms with identical costs can maintain market power due to product differentiation—each firm has a local monopoly over consumers located nearest to them on the product characteristic circle.

A key point in the Salop model is understanding that firms set their prices considering both their production costs and the costs consumers face when substituting away from their ideal product. This dual consideration helps explain why firms may not always compete purely on price, as differentiation creates a buffer that allows for higher pricing strategies.

As firms differentiate their products to capture a portion of the market, the model also illustrates potential for over-differentiation, which occurs when too many firms enter the market. This leads to inefficiencies, as some firms will be unable to cover their fixed costs and the total social costs (production plus transportation costs) exceed the minimum possible.

Hotelling Model

The Hotelling model is another celebrated example of spatial competition in microeconomics, focusing on how firms position themselves to maximize market share. This model can be imagined as a line segment (or 'Main Street') where consumers are uniformly spread along its length and have distinct preferences based on their location.

Firms, according to the Hotelling model, choose locations and prices for their products in order to attract the maximum number of customers. Here, like in the Salop model, transportation or disutility costs are incurred by customers when products purchased are not exactly what they preferred—the closer the product is to their ideal, the less cost they bear.

A critical aspect of the Hotelling model is that it predicts firms tend to cluster in the center of the market to minimize transportation costs on average for all consumers, known as the 'principle of minimum differentiation' or 'Hotelling's law.' This tendency towards convergence in product properties can be counterintuitive, as one might expect firms to spread out to cover different market segments. However, the desire to capture the most significant portion of the market leads firms to choose positions that are centrally located.

This concept overlaps with the common real-world observation where similar businesses, like gas stations and fast-food restaurants, are often found close together, despite them ostensibly being in competition. The model underscores that while differentiation is significant, there's a strategic positioning element that plays a pivotal role in competitive markets.

Equilibrium Pricing

Equilibrium pricing is a fundamental concept in economics that occurs where the supply and demand for a product are equal. This balance dictates that the price level is stable and no incentives exist for buyers or sellers to alter their current behaviors.

In the context of the Salop model, equilibrium pricing happens when all firms, assuming they are symmetrical, set the same price. This price includes the marginal cost plus a markup that covers the 'transportation' or disutility costs divided by the number of firms. The firms each have a section of the market to themselves, and thus, they face a trade-off between setting a high price and attracting fewer customers due to increased transportation costs, or setting a lower price and attracting more customers but earning less per unit sold.

Equilibrium pricing ensures that in a competitive differentiated market, no single firm can unilaterally change its price without losing customers to competitors. This equilibrium is delicate as changes in external variables like transportation costs, the number of firms, or the introduction of new product varieties can disrupt the current pricing balance and demand a new equilibrium.

Moreover, equilibrium pricing isn't just about the balance between cost and consumer preference; it also represents the strategic interactions between firms, where the actions of one firm directly influence the pricing decisions of others. These factors combined highlight the complex nature of price-setting in a differentiated market, which cannot be understood by considering supply and demand alone but requires a deeper analysis of competition and strategy.

Market Differentiation

Market differentiation refers to the process by which firms make their products stand out from competitors' offerings. This can be achieved through a variety of means including, but not limited to, differences in product quality, features, branding, customer service, and pricing strategies.

In differentiated markets, each firm's product is slightly different from the next, which can create a 'monopolistic competition' environment. This allows firms to have some degree of market power despite the presence of competitors because each firm's product is seen as unique by consumers. This uniqueness reduces the direct price competition as consumers do not view the products as perfect substitutes.

Market differentiation is central to the Salop and Hotelling models. The models showcase how firms balance locating close to competitors to capture more market share with the need to be distinct enough to attract their own loyal customer base. Differentiation can be beneficial to consumers by providing variety and catering to specific preferences, but as the Salop model warns, over-differentiation can lead to inefficiencies where too many options increase the overall cost without proportional benefits.

It's essential for students of economics to understand market differentiation because it not only informs pricing and product positioning strategies but also aligns with innovation and customer satisfaction. The balance achieved through differentiation can lead to a thriving market with a rich diversity of products, ultimately meeting various consumer needs and preferences.