Duopoly and Product Differentiation Notes

Duopoly and Product Differentiation

Learning Objectives

• Understand product differentiation.
• Understand the Hotelling model.
• Understand location equilibrium.
• Understand minimum product differentiation.
• Understand maximum product differentiation.
• Understand product differentiation and consumer preference.

What is Product Differentiation

• (Supply) condition for product differentiation
• Monopolistic competition and oligopoly (focus).
• Oligopoly: homogenous or differentiated products

Horizontal Product Differentiation

• If two products $A$ and $B$ are sold at the same price, consumers have no consensus on which one is preferred.
• Relative: some prefer $A$, while others prefer $B$.

Vertical Product Differentiation

• If two products $A$ and $B$ are sold at the same price, all consumers prefer one over the other.
• Absolute: all consumers prefer $A$ to $B$, or $B$ to $A$.

Horizontal vs Vertical Product Differentiation

• Example: Business class vs. economy class for airlines

Product Differentiation and the Hotelling Model

• How to describe a product
  • A set of utility-bearing attributes
  • Differentiation: one or more of the attributes are different
• How to analyze product differentiation
  • $A$ and $B$ are identical on all attributes but one ($a_1$)
  • $A$ and $B$ are differentiated on $a_1$
  • Products to be differentiated on one attribute at a time
• Example: Two restaurants selling the same burger
  • McDonald’s
  • Burger King

Basics of the Hotelling Model

• A market with two firms
  • Duopoly: $F<em>1$ and $F</em>2$ compete
  • $L$-mile-long street (L > 0)
• Supply side
  • $F<em>1$ and $F</em>2$ sell the same product but their locations could be different
  • No constraints on production capacity
  • Same marginal cost: $c<em>1 = c</em>2 = c$
  • Same price given: $p<em>1 = p</em>2 = p$
• Demand side
  • Consumers uniformly distributed
  • Each consumer incurs transportation cost: $t$ per mile (t > 0)
  • Each consumer has the same willingness to pay

How $F<em>1$ and $F</em>2$ Compete

• $F<em>1$’s profit: $\pi</em>1 = (p<em>1 - c</em>1)q<em>1 = (p - c)q</em>1$
• $F<em>2$’s profit: $\pi</em>2 = (p<em>2 - c</em>2)q<em>2 = (p - c)q</em>2$

What Determines Firm Profit

• Quantity demanded
• Location of firms

Product Differentiation in the Hotelling Model

• Product differentiation and location choice
  • $F<em>1$ and $F</em>2$ choose same location: no product differentiation
  • $F<em>1$ and $F</em>2$ choose different locations: product differentiation
• Consumer choice
  • Same location: consumers have no preference
  • Different locations: consumers have preference
• Hotelling model explains product differentiation
  • Location choice (long run): product differentiation on location
  • Pricing (short run): given locations, what about price?

Deriving the Location Equilibrium

• Location equilibrium and the minimum product differentiation (unique)

The Minimum Product Differentiation

• Location equilibrium is unique
  • $F<em>1$ and $F</em>2$ locate back to back in the center
  • The minimum product differentiation

Key Implications

• Location equilibrium is realized in the long run: products eventually look very similar
• Minimum product differentiation is not unique
• Price competition with minimum product differentiation
  • No market power: $p<em>1 = p</em>2 = c$
  • Zero economic profit: $\pi<em>1 = \pi</em>2 = 0$

Pricing in Maximum Product Differentiation

• Degree of product differentiation
  • Product differentiation: $F<em>1$ and $F</em>2$ locate differently
  • Minimum product differentiation: same location

Maximum Product Differentiation

• $F<em>1$ and $F</em>2$ must locate at two ends
• Maximum product differentiation is unique, but not stable

Pricing in Maximum Product Differentiation

• Assume $F<em>1$ and $F</em>2$ cannot change their locations
  • In the short run: firms cannot change locations
  • Only possible behavior: pricing through undercutting
• In equilibrium, consumer $i$ is indifferent
  • Quantity demanded for $F<em>1$: $q</em>1 = x$
  • Quantity demanded for $F<em>2$: $q</em>2 = L - x$

Derive the Demand Functions

• Consumer surplus

• Condition of equilibrium
  • $r - p<em>1 + tx = r - p</em>2 + t(L - x)$
• We obtain the demand functions
  • $q<em>1 = x = \frac{p</em>2 - p_1}{2t} + \frac{L}{2}$
  • $q<em>2 = L - x = \frac{p</em>1 - p_2}{2t} + \frac{L}{2}$

Profit Maximization for $F<em>1$ and $F</em>2$

• $F<em>1$’s and $F</em>2$’s profit functions
  • $\pi<em>1 = (p</em>1 - c)q<em>1 = (p</em>1 - c)(\frac{p<em>2 - p</em>1}{2t} + \frac{L}{2})$
  • $\pi<em>2 = (p</em>2 - c)q<em>2 = (p</em>2 - c)(\frac{p<em>1 - p</em>2}{2t} + \frac{L}{2})$
• Let marginal profit equal zero ($F_1$)
  • $\frac{\partial \pi<em>1}{\partial p</em>1} = \frac{p<em>2 - p</em>1}{2t} + \frac{L}{2} - \frac{p_1 - c}{2t} = 0$
  • $p<em>1 = \frac{p</em>2 + Lt + c}{2}$
• Let marginal profit equal zero ($F_2$)
  • $\frac{\partial \pi<em>2}{\partial p</em>2} = \frac{p<em>1 - p</em>2}{2t} + \frac{L}{2} - \frac{p_2 - c}{2t} = 0$
  • $p<em>2 = \frac{p</em>1 + Lt + c}{2}$

Maximum Product Differentiation

• In equilibrium:
  • $p<em>1 = p</em>2 = Lt + c$
  • $\pi<em>1 = \pi</em>2 = \frac{1}{2}L^2t$

Implications for Maximum Product Differentiation

• Market power: p1 = p2 > c
• Positive economic profit: \pi1 = \pi2 > 0
• Price and profit depend entirely on $L$ and $t$

Consumer Preference and Product Differentiation

• $L$: Dispersion of consumer preference
• Two extremes: $L = 0$ and $L \rightarrow \infty$

Intensity of Consumer Preference

• $t$: Intensity of consumer preference
• Two extremes: : $t = 0$ and $t \rightarrow \infty$

Consumer Preference and Product Differentiation

• Consumer preference
  • No consumer preference: $L = 0$, or $t = 0$, given the other, then products cannot be differentiated
  • Extremely strong preference: $L \rightarrow \infty$, or $t \rightarrow \infty$, then products can be differentiated anyway
• Consumer preference product differentiation
  • Consumer preference is the precondition for product differentiation
  • Product differentiation is the manifestation of consumer preference

List of Concepts/Terms Covered in Sequence

• horizontal product differentiation
• vertical product differentiation
• Hotelling model
• duopoly
• location choice
• location equilibrium
• minimum product differentiation
• maximum product differentiation
• consumer preference
• dispersion of consumer preference
• intensity of consumer preference