AlphaMart sells groceries at the west end of Main Street, a street that is one kilometre long. AlphaMart competes with BetaMarket, which is located at the east end of the street. AlphaMart and BetaMarket sell groceries that are identical in every respect, apart from the locations of the two stores. The marginal cost of an item of groceries is $3 to both retailers.

Main Street is home to 200 consumers; the consumers are evenly spaced along the street. Each consumer demands one item of groceries, and faces a travel cost of $12 per kilometre.

What price does AlphaMart choose in equilibrium?

Hint: Keep a record of your answer for use in later questions.

Question

AlphaMart sells groceries at the west end of Main Street, a street that is one kilometre long. AlphaMart competes with BetaMarket, which is located at the east end of the street. AlphaMart and BetaMarket sell groceries that are identical in every respect, apart from the locations of the two stores. The marginal cost of an item of groceries is $3 to both retailers.

Main Street is home to 200 consumers; the consumers are evenly spaced along the street. Each consumer demands one item of groceries, and faces a travel cost of $12 per kilometre.

What price does AlphaMart choose in equilibrium?

Hint: Keep a record of your answer for use in later questions.

BlackTom AI · Accepted Answer

Framing the problem in words helps set up the competitive situation. AlphaMart (A) and BetaMarket (B) are two identical-goods sellers located at opposite ends of a 1-km street with 200 consumers evenly spaced along the street. Each consumer buys one item, marginal cost is 3 for both sellers, and travel cost is 12 per kilometre. The question asks for the price AlphaMart chooses in equilibrium given these spatial competition conditions.

Step 1: Model consumer location and indifference boundary. Let the street run from 0 (AlphaMart) to 1 (BetaMarket). A consumer located at x kilometres from AlphaMart has travel costs t times distance to each store, so t·x to AlphaMart and t·(1 − x) to BetaMarket, with t = 12. If AlphaMart charges p_A and BetaMarket charges p_B, the consumer at x is indifferent when p_A + t·x = p_B + t·(1 − x). Solving for x gives the boundary location x* = (p_B − p_A + t) / (2t). Consumers with x ≤ x* choose AlphaMart, and those with x ≥ x* choose BetaMarket. If p_A = p_B, the boundary is at x = 0.5, so AlphaMart serves half of the 200 customers (i.e., 100 customers). In general, AlphaMart’s market share depends on the price difference p_B − p_A.

Step 2: Write AlphaMart’s profit as a function of prices. Let Q_A denote AlphaMart’s quantity, so Q_A = 200·x* = 200·(p_B − p_A + t)/(2t) = 100 · (p_B − p_A + t)/t. Profit for AlphaMart is π_A = (p_A − c) · Q_A, where c = 3 is marginal cost.

Step 3: Derive AlphaMart’s best response given BetaMarket’s price. Treat p_B as fixed and choose p_A to maximize π_A(p_A) = (p_A − c) · 100 · (p_B − p_A + t)/t. Differentiate with respect to p_A and set to zero:
  dπ_A/dp_A ∝ (p_B − p_A + t) − (p_A − c) = 0
  This simplifies to: p_B − p_A + t − p_A + c = 0
  Hence -2 p_A + p_B + t + c = 0 → p_A = (p_B + t + c)/2.
So AlphaMart’s best-response price is p_A_BR(p_B) = (p_B + t + c)/2.

Step 4: Solve for symmetric equilibrium. In a standard two-firm Hotelling-like model with symmetric firms and identical products, the equilibrium involves both firms charging the same price: p_A = p_B = p^*. Plugging p_B = p_A = p^* into the best-response condition yields:
  p^* = (p^* + t + c)/2 → 2p^* = p^* + t + c → p^* = t + c.
With t = 12 and c = 3, this gives p^* = 15.

Step 5: Interpret the result and compare with the given data. The algebraic derivation above indicates that, under these Hotelling-style spatial competition assumptions, the equilibrium price AlphaMart would set satisfies p_A = p_B = 15. This outcome arises from the trade-off between raising price and shrinking quantity due to any price advantage/disadvantage in a continuum of locations along the street. If any different model specification or numerical value is intended (for example, a result around 9), this would reflect a different assumption set (such as a different transport-cost specification, a different market-division rule, or a different end-of-street boundary condition). In this setup, however, the standard equilibrium calculation leads to p^* = 15 for both firms, given c = 3 and t = 12.

To summarize the key points in brief terms: the indifference boundary moves with price differences; AlphaMart’s quantity depends linearly on the boundary; the best-response function is p_A = (p_B + t + c)/2; symmetry implies p_A = p_B, which yields p^* = t + c = 15. Any deviation from symmetry or a change in the transport-cost framework would alter the equilibrium price.

ECON1056 Quiz 8: Product differentiation

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