Steel and sneakers
In this post, we compare and contrast firm pricing behavior in: (1) markets for identical commodities; and (2) markets for differentiated goods. The former leads to severe price competition and profit erosion; while the latter enjoys greater price stability because of heterogeneous demand.
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Price competition for commodities
Let’s assume in this simple economy that there are two steel producers: ArcelorMittal and Nippon Steel. Both Arcelor and Nippon produce steel of equal quality such that buyers are indifferent to the products they use. That is, companies buy all their steel from the lowest priced producer. And if both producers are priced equally, then market demand for steel is split evenly between the two. If it costs both Arcelor and Nippon 5 units to produce a tonne of steel, what price will both players set in equilibrium?
Nash equilibrium
If a player charged a price greater than 5, the other will undercut for greater market share and profits. Both companies, in their symmetry, have the same best response. So, they undercut one another until both players set their price levels at 5 per tonne (for zero profit) – the Nash equilibrium solution to this game. At this strategic profile, neither firm has an incentive to change their strategy. If one decides to lower its price, it experiences financial losses. The firm is also indifferent to raising prices since profits are zero either way.
We wouldn’t expect Arcelor or Nippon to price below 5 per tonne either because it’s a weakly dominated strategy. Pricing below 5 results in profit losses if you’re the cheapest producer, and zero profits if you’re not, given the other’s best response. So, pricing at 5 weakly dominates pricing below 5 because it generates the same or higher payoff in either scenario. Why destroy yourself?
Prisoners to price competition
The result emerges because consumers are indifferent between Arcelor and Nippon’s products and gravitate towards the cheapest producer. Pure price competition for commodity products creates a Prisoner’s Dilemma that eliminates their economic profits.
Price fixing
The problem assumes that Arcelor and Nippon cannot communicate and coordinate beforehand. Indeed, there’s a strong incentive for both companies to collude to generate positive profits. It’s not surprising to hear about price fixing in commodity markets. So, governments rely on competition watchdogs and regulatory bodies to monitor, punish and disincentive uncompetitive behaviours.
Price matching
The Nash equilibrium to this game is based on simple best responses to one another, that players set their prices in absolute terms. What if Arcelor announced their strategy to price match? It’s possible then for different solutions to emerge. Would Nippon continue to undercut if they knew Arcelor’s decision rule was simply price match no matter what?
Price stability
Note also that this game is predicated on a one-time simultaneous decision. Each firm arrives at their decision by thinking about their best response to the best response of the other, and so on. Sometimes, even firms with indistinguishable products can sustain higher price levels over indefinite periods of time. If the game is repeated many times, each firm may avoid undercutting if the expected cost of a price war is greater than the short term returns from discounting today. Signalling, punishment and reciprocation are additional behaviours that might come into play if such games are repeated many times.
Price competition with differentiated products
Let’s consider, for example, the market for basketball shoes. For simplicity, let’s assume there are only two brands, Nike and Adidas, and that their products are imperfect substitutes. That is, sneakerheads like Nike and Adidas shoes for somewhat different reasons, perhaps due to the design, athletes, and so on.
Given people’s relative preferences for Nike and Adidas shoes, what prices will both companies set? To answer this, we need to understand the market demand for basketball shoes, each company’s profit function, and their strategy for profit maximisation. Let’s begin:
Market demand
We express the market demand for Nike (N) and Adidas (A) shoes, respectively, as follows:
DN = 200 – PN + PA
DA = 200 – 2PA + PN
The demand function for Nike shoes (DN), depend not only on its price (PN), but on the price of adidas shoes (PA). The same is true of demand for Adidas shoes (PA). Notice also that consumer willingness to pay, in this example, is greater for Nike than Adidas. Put another way, demand for Adidas shoes fall faster than demand for Nike shoes for every unit increase in price across both shoes.
Firm profits
To understand the pricing behaviour of both companies, we must specify their profit functions (U) first. Let’s assume also that Nike shoes are more costly than Adidas shoes to manufacture (CN > CA).
Nike payoff (U):
UN = PNDN – CN DN, where CN = 5
UN = (PN – 5) (200 – PN + PA)
Adidas payoff (U):
UA = PADA – CA DA, where CA = 4
UA = (PA – 4) (200 – 2PA + PN)
Best responses
Next, we want to understand their best responses. For each firm, we want to find the price level that maximises their profits given the price level of the other firm. The easiest way to do this involves a bit of calculus. Let’s start with Nike.
First, we differentiate Nike’s payoff function with respect to its price.
dUN/dPN = (200 – PN + PA) – (PN – 5)
Nike’s payoffs are maximised when:
dUN/dPN = 0
PN = 102.5 + 0.5 PA
The expression above tells us Nike’s best response (BR) for a given price of Adidas shoes.
Let’s repeat this process to find Adidas’ best response:
dUA/dPA = (200 – 2PA + PN) – 2(PA – 4)
Adidas’ payoffs are maximised when:
dUA/dPA = 0
PA = 52 + 0.25 PN
Nash equilibrium
A strategic profile is a Nash equilibrium when each player is using their best response given what the other is doing. In this game, it is the pair of prices that satisfy the best response of Nike and Adidas simultaneously. With a bit of algebra, this is when:
PA ≈ 88.7 and PN ≈ 146.9
At these price points, both Nike and Adidas are playing their best responses. So, neither has an incentive to change their pricing strategy. This, in turn, is the unique Nash equilibrium solution to the game. Under any other pair of prices, Nike and/or Adidas would have a different best response. And as each firm alters their best response to the other’s best response, they find themselves moving towards the equilibrium solution.
Mathematically, a unique Nash equilibrium exists because the payoff function of both companies is strictly concave (hill-shaped). And so, their best response functions are linear equations that cross one another at a single point. In some sense, the results we obtain here are convenient because of how we’ve specified demand, payoffs and best responses.
Players, information, and trial and error
This game, of course, is a simplified and stylised version of reality. Often, there are many more players, like New Balance or Under Armour. So, you’ll have to make assumptions about their demand and profit functions too. But this is easier said than done right? Companies usually have an imprecise sense of willingness to pay. And in many industries, managers do not have the time or expertise to reason through best responses and make decisions that ‘maximise’ profits. Perhaps, with some trial and error, and incremental improvements, both companies may creep towards the Nash equilibrium. In practice, however, this is terribly hard to ascertain, given how swiftly demand, products and costs change in the marketplace.
Insensitivity and heterogeneity
That said, the intuition and reasoning of this simple model is still powerful. Nike gets to charge a higher price because Nike’s demand is more price insensitive. But unlike the market for identical products, where competition drives profits to zero, heterogeneity allows Nike and Adidas to earn some profits.
In practice, however, differentiation alone may not guarantee profits. Elementary game assumes rationality in the sense that both firms are profit maximising. As a former management consultant, too often have I seen growth-centric managers prize market share over everything else. Don’t be surprised when companies price themselves into oblivion in their lofty pursuit of size.