Lecture Notes 8: Monopolistic Competition
A monopolistically competitive market is a market in which there are a large number of firms and easy entry but, unlike in a perfectly competitive market, the products are differentiated. Think about restaurants – lots of choices and no huge entry barriers for starting a restaurant, but different restaurants serve different kinds of food. Most retail products are sold in a monopolistically competitive environment – clothes, cosmetics, etc…
There are a number of approaches to modeling monopolistic competition, and in this section we will go through the most important ones.
We will begin with two modifications of the Cournot model – one that allows free entry and another that allows product differentiation – to show the impact on the market outcome.
Cournot Model with Free Entry
In the standard Cournot model, the number of firms in the market is regarded to be fixed, presumably because of some kind of barriers to entry. But what if we take a Cournot model and modify it to allow free entry of firms? This model does not take product differentiation into account, but it does illustrate two key ideas in monopolistic competition – the impact of free entry, and especially the role of fixed costs in the market outcome.
Let’s consider the Cournot model from the previous section. The market demand curve is given by 𝑃𝑃 = 140 − 2𝑄𝑄 and production costs are $20 per unit. When there are 𝑛𝑛 firms in the market, equilibrium output by each firm is:
𝑞𝑞𝑖𝑖 =1 + 𝑛𝑛60
In turn, the total output in the market and the market price are:
𝑄𝑄 =1 + 𝑛𝑛60𝑛𝑛
The profit earned by each firm is:
Π𝑖𝑖 = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇
= 𝑃𝑃 ⋅ 𝑞𝑞 − 20𝑞𝑞
= �140 + 20𝑛𝑛1 + 𝑛𝑛 � �1 + 𝑛𝑛� − 20 �60 1 + 𝑛𝑛�60
=(1 + 𝑛𝑛)72002
Now, suppose that the market features free entry, but there is a fixed cost of 𝐹𝐹 = 5 required in order to enter the market. Firms will continue to enter as long as it is profitable – in other words, as long as the profit after entering is sufficient to cover the fixed cost. When 𝐹𝐹 = 5, the market will have 𝑛𝑛 = 36 active firms. To see why, if 𝑛𝑛 = 36 then each firm’s profit is Π = 7200
(1+36)2 = 5.26. But if 𝑛𝑛 = 37, then each firm’s profit is Π = 7200
(1+37)2 = 4.98. The 37th firm has no incentive to enter because it would lose money by doing so – the profits are not sufficient to cover the fixed cost of entering the market.
But what if the fixed cost rises to 𝐹𝐹 = 100? In this case, the market will have only 𝑛𝑛 = 7 firms. Each firm earns a profit of Π = 7200
(1+7)2= 112.5, which is enough to cover the fixed cost and still be profitable. But if one more firm enters, then Π = 7200
(1+8)2 = 88.89 for each firm, so there is no incentive for the 8th firm to enter the market.
How does this impact consumers and welfare in the market? Recall that the market price is:
𝑃𝑃 =140 + 20𝑛𝑛1 + 𝑛𝑛
When the fixed cost to enter is 𝐹𝐹 = 5 𝑛𝑛 = 36 firms enter in equilibrium 𝑃𝑃 = 23.24
When the fixed cost to enter is 𝐹𝐹 = 100 𝑛𝑛 = 7 firms enter in equilibrium 𝑃𝑃 = 35
The important lesson is this. Introductory economics students are sometimes taught that fixed costs have no impact on prices. But that’s wrong, at least in the long-run. Fixed costs impact entry. In turn, entry and the number of firms competing in the market impacts prices.
Cournot Model with non-Homogeneous Products
The standard Cournot model assumes that the products are homogeneous – the output of the various firms is perfectly substitutable. But what if the products offered by the firms are not
perfectly substitutable? In other words, the products are related – the firms impact each other, but the products are differentiated.
Here is one way to model this. Consider a market with two firms.1 The demand curves faced, respectively, by firm 1 and firm 2 are:
𝑃𝑃1 = 120 − 2𝑞𝑞1− 𝜆𝜆𝑞𝑞2
𝑃𝑃2 = 120 − 2𝑞𝑞2− 𝜆𝜆𝑞𝑞1
The idea is that 𝜆𝜆 measures the degree of substitutability of the products offered by the two firms. If 𝜆𝜆 = 0 then the products are not at all substitutable – firm 2’s output has no impact on the demand for firm 1’s product and vice versa. If 𝜆𝜆 = 2 then the products are perfectly substitutable – firm 2’s output impacts the demand for firm 1’s product in exactly the same way as firm 1’s own output. When 0 < 𝜆𝜆 < 2, the firms’ products are substitutable, but not perfectly so.
To keep things simple, assume that there are no production costs. Let’s find the Cournot Equilibrium. Firm 1’s profit function is given by:
Π1 = 𝑃𝑃1⋅ 𝑞𝑞1
= (120 − 2𝑞𝑞1− 𝜆𝜆𝑞𝑞2) ⋅ 𝑞𝑞1
= 120𝑞𝑞1− 2𝑞𝑞12− 𝜆𝜆𝑞𝑞1𝑞𝑞2
Firm 1 will choose its output to maximize profit.
𝜕𝜕Π1
𝜕𝜕𝑞𝑞1 = 120 − 4𝑞𝑞1− 𝜆𝜆𝑞𝑞2 = 0
4𝑞𝑞1 = 120 − 𝜆𝜆𝑞𝑞2
𝑞𝑞1 = 30 −14 𝜆𝜆𝑞𝑞2
This is firm 1’s reaction function. We could finish solving by finding firm 2’s reaction function and then substituting back, but let’s do it an easier way by using our trick. Since the problem is symmetric, we know that 𝑞𝑞1 = 𝑞𝑞2 in equilibrium. Making this substitution into the reaction function allows to solve easily:
𝑞𝑞1 = 30 −14 𝜆𝜆𝑞𝑞1
𝑞𝑞1+14 𝜆𝜆𝑞𝑞1 = 30
4𝑞𝑞1+ 𝜆𝜆𝑞𝑞1 = 120
𝑞𝑞1 =4 + 𝜆𝜆120
By symmetry, we know that the output of firm 2 is also 𝑞𝑞2 = 120
4+𝜆𝜆. We can find the equilibrium
prices by substituting these outputs into the demand functions.
𝑃𝑃1 = 120 − 2𝑞𝑞1− 𝜆𝜆𝑞𝑞2
= 120 − 2 �4 + 𝜆𝜆� − 𝜆𝜆 �120 4 + 𝜆𝜆�120
=4 + 𝜆𝜆240
The equilibrium price for firm 2 is:
𝑃𝑃2 = 120 − 2𝑞𝑞2− 𝜆𝜆𝑞𝑞1
= 120 − 2 �4 + 𝜆𝜆� − 𝜆𝜆 �120 4 + 𝜆𝜆�120
=4 + 𝜆𝜆240
Notice that the price in the market falls as 𝜆𝜆 rises. What is the economic intepretation? As the products become more and more substitutable for each other (higher value of 𝜆𝜆), there is more competition among the firms and the market price falls. For lower values of 𝜆𝜆, consumers do not regard the firms’ products to be substitutable, so there is less competition and the market price is higher.
• As products become less substitutable, market price rises.
Chamberlin Model
The Chamberlin model is the standard model of monopolistic competition taught in principles-level classes. It is easiest to understand the main ideas graphically.
The basic idea is that each firm is a price-maker with its own customers and faces a downward sloping demand curve. Because the products are differentiated, the demand for each firm is not totally elastic like a perfect competitor’s demand curve. But at the same time the demand for a typical firm will be more elastic than a monopoly’s demand. In other words, each firm has some limited ability to raise price and still retain some customers, but demand is still pretty elastic.
In the short-run, the firm’s demand curve can allow it to earn a profit or a loss. The diagram below shows a monopolistic competitor earning a short-run profit. Just like any other firm with market power, it sets its output where marginal revenue equals marginal cost and then sets the corresponding price on the demand curve.
While the firm is able to earn a short-run profit, the profit will not continue in the long-run. Because the market features easy entry, new firms will enter. This entry of new firms reduces demand among existing firms. And the entry will continue all the way until the profits are driven to zero.
can never earn excess profits. No matter how great your product is and how differentiated it is, if entry is easy then someone else will come along to compete and chip away at your profits.
To examine the implications for long-run equilibrium it is useful to contrast with a perfectly competitive firm. The competitive firm earns zero profit in the long run. The market price adjusts until it reaches the zero-profit level where P=ATC.
A firm in a monopolistically competitive industry also earns zero profit in the long-run equilibrium, but its demand is downward sloping. Competition and entry implies that demand adjusts until P=ATC at the firm’s output level, and the firm earns no profits.
There are a number of consequences.
First, notice that the long-run equilibrium (zero-profit) price P* is above marginal cost in the long-run equilibrium. Thus, monopolistic competition leads to deadweight loss.
• Monopolistically competitive markets do not achieve allocative efficiency. The long run equilibrium price is higher than marginal cost, leading to deadweight loss.
words, each firm’s cost per unit of output produced is higher than what it really needs to be. This problem is called excess capacity – each firm produces less than the output level that would minimize average costs.
Think about what’s going on here. The market features too many firms, with each firm producing too little output to be efficient. It’s not hard to think of examples. A shopping mall has many clothing stores, each serving relatively few customers. We could still serve all the customers, even with fewer clothing stores, and think about the savings for society. We could sell the same number of clothes, but we wouldn’t have to pay rent at all those shops. In other words, we wouldn’t have to keep paying the fixed cost so many times.
• There is excessive entry in monopolistically competitive markets in the long-run equilibrium. The number of firms is higher than the efficient level and each firm’s output is lower than the efficient level (excess capacity).
This excess capacity represents a waste of society’s resources because it is more expensive to deliver clothes to consumers than it would be if there were fewer firms, each firm with more sales. On the other hand, consumers value variety. If we explicitly add in the value that consumers place on product variety, then the number of brands in equilibrium might be too high or too low.
Indeed, the question of optimal product diversity is a complicated one if we add in the fact that consumers place a value on product diversity. Even if consumers want a new brand, firms might not be willing to produce it if they cannot capture sufficient profits to overcome the fixed cost of starting the brand up. The basic problem is that the firm doesn’t capture the full social benefit of producing the new brand. It only captures the profit, not the consumer surplus. So a brand might be efficient to add (CS + Profit exceeds fixed cost of starting brand), but the firm’s profit alone is not sufficient to cover the fixed cost.
• If products are homogeneous, then the number of brands is certainly too high (excess capacity). If products are differentiated and consumers value diversity, the number of brands might be too high or too low.
Linear City (Hotelling) Model
The linear city model was developed by Harold Hotelling (1929). In this model, products are differentiated on one dimension. For a concrete example, suppose there is a boardwalk a mile long. There are two ice cream vendors, and each vendor can choose where to locate along the boardwalk. The consumers are uniformly distributed across the boardwalk, and will visit the closest firm.2 Again, you could think about the “distance” more broadly as product variety.
From the perspective of social efficiency, the optimal location for the firms is at 1/4 and 3/4. All the customers to the left of the 1/2 mark will visit Firm 1. All the customers to the right of the 1/2 mark will visit Firm 2. Nobody has to travel too far. Everyone can find a firm reasonably close to his own location / preferences.
Now, what happens when firms choose their own locations? The firm located at 1/4 will start moving towards the right – it can capture more customers in the center, and it won’t lose any of its customers to its left. Similarly, the firm located at 3/4 will start moving towards the left – it can capture customers in the center, and won’t lose any customers to its right.
In fact, the Nash Equilibrium in this game is for both firms to operate exactly at the 1/2 point. At any other locations, firms can move towards the center to capture more customers. Both firms operating at the 1/2 point is the Nash Equilibrium. Neither firm has an incentive to move left or right, because doing so will only cause it to lose customers.
• (Minimal differentiation) When firms choose only location (not prices), the equilibrium involves both firms choosing identical locations – those that match the median customer.
In this model, the outcome is too little product differentiation relative to the efficient level. Both firms compete for customers in the middle, and the customers located closer towards the end have
2 In particular, we assume that travel costs are quadratic. Basically, the aggravation of walking an additional distance
to travel a long way and don’t have a product close to their location / preferences.3 It would be better for society overall to have the firms locate at 1/4 and 3/4, but that’s not an equilibrium.
For a practical example, many businesses of similar type often cluster together. Think about groups of restaurants or gas stations.
We have been assuming so far that the firms are choosing only their locations. But what if firms can choose their locations and their prices? Now it gets more complicated. I won’t go into the details, but the basic idea is that consumers place a value on location and are willing to pay more to purchase from a firm that is located close by (i.e. with a product more similar to their own preferences). In this case, firms will start to move farther apart from each other in order to make their core customers happy and willing to pay more. It turns out that the equilibrium in this case is exactly the opposite of the previous section – the firms locate at the two ends.
• (Maximal differentiation) When firms choose both location and price, the equilibrium involves firms operating at the extreme ends of the product space.
In this case, firm profits increase with differentiation and society actually ends up with too much
product differentiation in equilibrium. This is consistent with the result in the previous section where the long-run equilibrium involved an over-proliferation of branding and differentiation. In a humorous observation, Hotelling once noted that businesses should stop supporting improvements to roads and infrastructure. Making transportation between locations more costly creates more differentiation between firms from the perspective of consumers and, in this model, increases the profits of the firms.
The basic source of the inefficiency here is that the welfare gains of moving more towards the center (lower transportation costs for consumers) are not captured by the firm. Thus, the firms aggressively pursue product differentiation in order to maximize the price they can charge to consumers.
3 This is the “median voter theorem” in political science. In a two-candidate election, both candidates will stake out a
Circular (Salop) Model
The circular approach was pioneered by Steven Salop (1979).
Customers are evenly distributed along a circle. There are 𝑛𝑛 firms, and of course the firms will locate equidistantly from each other in order to capture the most customers. We normalize the circumference of the circle to be equal to 1. Thus, since there are 𝑛𝑛 firms, the distance between any two firms is given by 1
𝑛𝑛. Consumers incur a cost 𝑡𝑡 for each unit of distance traveled.
Suppose that your neighbor charges a price 𝑝𝑝. Your price is 𝑝𝑝𝑖𝑖 and you are deciding what price to set. Consider a consumer who is located at distance 𝑥𝑥 from your firm. His total cost to purchase from you, including the price and the travel cost, is given by:
𝑝𝑝𝑖𝑖+ 𝑡𝑡𝑥𝑥
On the other hand, if the consumer goes to your neighbor instead the distance to your neighbor’s firm is 1
𝑛𝑛− 𝑥𝑥. So his total cost to purchase from your neighbor is given by:
𝑝𝑝 + 𝑡𝑡 �1𝑛𝑛 − 𝑥𝑥�
Thus, the distance of the consumer right on the margin between choosing you and choosing your neighbor is given by:
𝑝𝑝𝑖𝑖+ 𝑡𝑡𝑥𝑥 = 𝑝𝑝 + 𝑡𝑡 �1𝑛𝑛 − 𝑥𝑥�
𝑝𝑝𝑖𝑖+ 𝑡𝑡𝑥𝑥 = 𝑝𝑝 +𝑛𝑛 − 𝑡𝑡𝑥𝑥𝑡𝑡
2𝑡𝑡𝑥𝑥 = 𝑝𝑝 − 𝑝𝑝𝑖𝑖+𝑛𝑛𝑡𝑡
Now, your firm gets demand from the consumers on the right and on the left, up to the marginal consumer who is going to visit your neighbors instead. Thus, your total demand is:
𝑞𝑞 = 2𝑥𝑥 =𝑝𝑝 − 𝑝𝑝𝑡𝑡𝑖𝑖 + 𝑡𝑡𝑛𝑛
Now, each firm incurs a cost 𝑐𝑐 to produce each unit. Thus, your firm’s profit is given by:
Π𝑖𝑖 = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇
= 𝑝𝑝𝑖𝑖𝑞𝑞 − 𝑐𝑐𝑞𝑞
= 𝑝𝑝𝑖𝑖�
𝑝𝑝 − 𝑝𝑝𝑖𝑖+ 𝑡𝑡𝑛𝑛
𝑡𝑡 � − 𝑐𝑐 �
𝑝𝑝 − 𝑝𝑝𝑖𝑖+ 𝑡𝑡𝑛𝑛
𝑡𝑡 �
= �𝑝𝑝𝑡𝑡� 𝑝𝑝𝑖𝑖− �1𝑡𝑡� 𝑝𝑝𝑖𝑖2+ �𝑛𝑛� 𝑝𝑝1 𝑖𝑖 −𝑐𝑐𝑝𝑝𝑡𝑡 + �𝑐𝑐𝑡𝑡� 𝑝𝑝𝑖𝑖 −𝑛𝑛𝑐𝑐
The firm’s objective is to choose its price 𝑝𝑝𝑖𝑖 to maximize profit.
𝑑𝑑Π𝑖𝑖
𝑑𝑑𝑝𝑝𝑖𝑖 =
𝑝𝑝 𝑡𝑡 −
2 𝑡𝑡 𝑝𝑝𝑖𝑖 +
1 𝑛𝑛 +
𝑐𝑐 𝑡𝑡 = 0 �2𝑡𝑡� 𝑝𝑝𝑖𝑖 = 𝑝𝑝𝑡𝑡 +𝑐𝑐𝑡𝑡 +1𝑛𝑛
𝑝𝑝𝑖𝑖 = 𝑝𝑝2 +𝑐𝑐2 +2𝑛𝑛𝑡𝑡
Now we apply the trick that we learned for symmetric models with many identical firms. In equilibrium, all firms have to set the same price (since firms are identical), and thus in equilibrium 𝑝𝑝𝑖𝑖 = 𝑝𝑝. We can make this substitution to solve for the equilibrium price.
𝑝𝑝 =𝑝𝑝2 +𝑐𝑐2 +2𝑛𝑛𝑡𝑡 𝑝𝑝 2 = 𝑐𝑐 2 + 𝑡𝑡 2𝑛𝑛 𝑝𝑝 = 𝑐𝑐 + 𝑡𝑡
𝑛𝑛
Each unit sold results in a profit of:
One thing to notice is that the equilibrium price and profit falls as the number of firms rises.
Now, each firm has proportion 1
𝑛𝑛 of the total customers, so the firm’s total profits are:
Π𝑖𝑖 =𝑛𝑛 �1 𝑛𝑛� =𝑡𝑡 𝑛𝑛𝑡𝑡2
Now, suppose that there is a fixed cost of 𝐹𝐹 to enter the market. Firms will enter as long as they can capture enough profit to recover their fixed costs. So the equilibrium number of firms is:
𝑡𝑡 𝑛𝑛2 = 𝐹𝐹
𝑛𝑛2 = 𝑡𝑡
𝐹𝐹 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒= �𝑡𝑡 𝐹𝐹⁄
The expression makes sense. As the travel cost for consumers rises, the number of active firms increases. But as the fixed cost to enter the market rises, the number of active firms falls.
By contrast, let’s now ask what the efficient number of firms is. There are two components of the total cost to society of having 𝑛𝑛 firms in operation.
• Fixed costs – The fixed cost has to be paid for each firm that enters. In total, this is 𝑛𝑛𝐹𝐹.
• Travel costs – The distance between firms is 1
𝑛𝑛. Thus, all customers who are located
between 0 and 1
2𝑛𝑛 will visit your firm (the others will go to the neighboring firm). Thus, the
average travel distance a customer has to travel is 1
4𝑛𝑛. Each unit of travel costs 𝑡𝑡, so travel
costs are given by 𝑡𝑡
4𝑛𝑛.
The total cost to society of having 𝑛𝑛 firms in operation includes both pieces.
𝑇𝑇 = 𝑛𝑛𝐹𝐹 + 𝑡𝑡
4𝑛𝑛 = 𝑛𝑛𝐹𝐹 + 1 4 𝑡𝑡𝑛𝑛−1
𝑑𝑑𝑇𝑇
𝑑𝑑𝑛𝑛 = 𝐹𝐹 − 1
4 𝑡𝑡𝑛𝑛−2= 0 𝐹𝐹 =4𝑛𝑛𝑡𝑡2
4𝐹𝐹𝑛𝑛2 = 𝑡𝑡
𝑛𝑛2 = 𝑡𝑡
4𝐹𝐹 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒 =1
2�𝑡𝑡 𝐹𝐹⁄
Comparing this with the free market equilibrium gives us a familiar result. The equilibrium number of entrants in a free market is �𝑡𝑡 𝐹𝐹⁄ , but the efficient number of firms is only 1
2�𝑡𝑡 𝐹𝐹⁄ . Thus, like
Exercises
Problem 1
Consider a market where demand is given by 𝑃𝑃 = 1000 − 𝑄𝑄. The market features 𝑛𝑛 firms, so the market output is 𝑄𝑄 = 𝑞𝑞1+ 𝑞𝑞2+ ⋯ + 𝑞𝑞𝑛𝑛. The cost for any firm to produce 𝑞𝑞𝑖𝑖 units of output is given by 𝑇𝑇𝑇𝑇 = 𝑞𝑞𝑖𝑖2.
a. Calculate the profit of each entrant. Your answer will depend on 𝑛𝑛.
b. Now suppose that there is a fixed cost of 𝐹𝐹 to enter the market. Give an expression for the equilibrium number of entrants to the market if there is free entry. Ignore integer issues. c. (Hard) What would be the efficient number of entrants. How does this compare to the
equilibrium number of entrants?
Problem 2
There are ten firms that simultaneously choose their outputs. The prices that firms can fetch on the market are given by:
𝑃𝑃1 = 1 − 𝑞𝑞1− 𝜆𝜆𝑞𝑞2− 𝜆𝜆𝑞𝑞3− ⋯ − 𝜆𝜆𝑞𝑞10
𝑃𝑃2 = 1 − 𝑞𝑞2− 𝜆𝜆𝑞𝑞1− 𝜆𝜆𝑞𝑞3− ⋯ − 𝜆𝜆𝑞𝑞10
⋮
𝑃𝑃10= 1 − 𝑞𝑞10− 𝜆𝜆𝑞𝑞1 − 𝜆𝜆𝑞𝑞2− ⋯ − 𝜆𝜆𝑞𝑞9
As in the lecture notes, the idea is that 𝜆𝜆 measures the degree of substitutability of the products offered by the different firms. There are no production costs.
a. Write out the profit function for firm 1.
b. How much output does each firm produce in Cournot equilibrium? (Hint: Use symmetry) c. What is the market price?
d. How does the market price depend on 𝜆𝜆? Give an economic interpretation.
Problem 3
Consider a firm operating in a monopolistically competitive market that faces demand for its product 𝑃𝑃 = 10 −1
2𝑄𝑄, and with production costs 𝑇𝑇𝑇𝑇 = 86 − 14𝑄𝑄 + 𝑄𝑄2
a. How much output will the firm produce in the short-run? b. What price will the firm charge in the short-run?
Problem 4
In the Chamberlin model, suppose that fixed costs fall.
a. In the short-run, what happens to economic profits of operating firms? b. Describe the adjustment to long-run equilibrium.
Problem 5
Consider a Hotelling model where firms choose only location (not price) along a 1-mile long boardwalk. Suppose now there are three firms, instead of just two. Explain why there is no Nash Equilibrium.
Problem 6
