Alpha Inc. and Beta Corp. compete in the market for smallgoods (processed meats). The smallgoods produced by the two firms are homogeneous, and the firms compete in Cournot (quantity) competition. Inverse demand in the market for smallgoods is given by the function,

𝑃
=
100
−
𝑄
𝐴
24
−
𝑄
𝐵
24
,

where
𝑄
𝐴
is the quantity of smallgoods produced by Alpha, and
𝑄
𝐵
is the quantity of smallgoods produced by Beta. Alpha Inc.'s marginal cost is
𝑀
𝐶
𝐴
=
$
20
, and Beta Corp.'s marginal cost is
𝑀
𝐶
𝐵
=
$
10
.

If Alpha Inc. were a monopolist, what quantity would it choose to produce?

Question

Alpha Inc. and Beta Corp. compete in the market for smallgoods (processed meats). The smallgoods produced by the two firms are homogeneous, and the firms compete in Cournot (quantity) competition. Inverse demand in the market for smallgoods is given by the function,

𝑃
=
100
−
𝑄
𝐴
24
−
𝑄
𝐵
24
,

where 
𝑄
𝐴
 is the quantity of smallgoods produced by Alpha, and 
𝑄
𝐵
 is the quantity of smallgoods produced by Beta. Alpha Inc.'s marginal cost is 
𝑀
𝐶
𝐴
=
$
20
, and Beta Corp.'s marginal cost is 
𝑀
𝐶
𝐵
=
$
10
.

If Alpha Inc. were a monopolist, what quantity would it choose to produce?

BlackTom AI · Accepted Answer

The prompt asks us to restate the question and analyze all answer options in detail before indicating the correct choice. Here, the provided data include a numerical answer of 192 but no answer options are given. I will therefore first restate the question clearly, then explore the standard ways one would approach a monopolist problem in this Cournot-style setting, and finally reconcile why the single numeric value provided (192) does not line up with the most straightforward derivations. This will help illuminate how the monopoly quantity could be determined under different interpretations, and why the given 192 might be a result of a specific assumption.

Restating the question:
- Market: smallgoods with homogeneous product.
- Competition framework: Alpha Inc. and Beta Corp. compete in Cournot (quantity setting). The inverse demand is P = 100 − Qa/24 − Qb/24, where Qa is Alpha’s output and Qb is Beta’s output.
- Costs: Alpha has marginal cost MC_A = 20; Beta has MC_B = 10.
- Question: If Alpha Inc. were a monopolist, what quantity would it choose to produce?
- Given answer field: ["192"], but there are no answer options provided.

Approach 1: Treat Alpha as a pure monopolist facing the market demand alone (Beta exits the market or is not producing for the monopolist’s problem).
- In this interpretation, with Beta effectively absent (Qb = 0), the demand faced by Alpha is P = 100 − Qa/24.
- Revenue for Alpha: R(Qa) = P·Qa = (100 − Qa/24)·Qa = 100 Qa − (Qa^2)/24.
- Profit for Alpha: π_A = R(Qa) − MC_A·Qa = (100 Qa − Qa^2/24) − 20 Qa = 80 Qa − (Qa^2)/24.
- FOC: dπ_A/dQa = 80 − (2 Qa)/24 = 80 − Qa/12 = 0 ⇒ Qa = 960.
- Check: At Qa = 960, price P = 100 − 960/24 = 100 − 40 = 60, and profit is (P − MC_A)·Qa = (60 − 20)·960 = 40·960 = 38,400.
- This yields a monopoly quantity of 960, not 192. If the problem's answer key lists 192, this interpretation would not match that key.

Approach 2: Alpha is monopolist but Beta remains in the market as a follower (i.e., the monopolist faces residual demand after anticipating Beta’s production).
- This is a less standard framing because a monopolist typically ignores a competitive follower, but one could imagine Alpha choosing Qa to maximize profit while Beta’s optimal response to Qa is Qb = BR_B(Qa).
- From the Cournot best-response conditions (two-firm Cournot), each firm maximizes profit given the other’s output.
- Alpha’s BR given Qb is: max_Qa (100 − (Qa+Qb)/24 − 20)·Qa ⇒ 80 Qa − (Qa^2)/24 − (Qb·Qa)/24; FOC: 80 − (2 Qa + Qb)/24 = 0 → 2 Qa + Qb = 1920.
- Beta’s BR given Qa is: max_Qb (100 − (Qa+Qb)/24 − 10)·Qb ⇒ 90 Qb − (Qa Qb)/24 − (Qb^2)/24; FOC: 90 − (Qa + 2 Qb)/24 = 0 → Qa + 2 Qb = 2160.
- Solving these simultaneously gives Qa = 560 and Qb = 800 (the classic Cournot equilibrium when both firms are active).
- In this interpretation, the monopolist quantity is not 192; the cooperative Cournot quantity for Alpha would be 560, with Beta producing 800.

Approach 3: Interpret 192 as a potential result under a different, perhaps misinterpreted, modeling assumption (e.g., a scaled or unit-corrected version of the problem).
- If the problem’s numbers come from a particular scaling or a variant of the demand function (for instance, if Qa and Qb were measured in tens or hundreds, or if the price function had a different coefficient), one could conceivably obtain Qa = 192 under some specific but nonstandard transformation.
- However, given the exact stated inverse demand P = 100 − Qa/24 − Qb/24 and MC_A = 20, the clean monopolist solution with Beta absent yields Qa = 960, not 192.

Reconciling the discrepancy:
- The most direct, standard interpretation of “Alpha as a monopolist” in a market that otherwise has Beta present would be either:
  - Alpha monopolist with Beta absent from the market (Qa = 960), or
  - Alpha monopolist while Beta is still in the system but Alpha ignores Beta (which is inconsistent with monopoly, since a monopolist would not account for a competitor in the objective function unless the setup specifies residual demand).
- The Cournot-equilibrium analysis with both firms active yields Qa = 560, not 192, indicating that if the problem intended a Cournot setting rather than a monopoly setting for Alpha, the monopoly framing is not the correct lens for Alpha.

Conclusion on the presented data:
- Under the standard monopoly framework with Beta removed, the quantity Alpha would choose is 960.
- Under a standard two-firm Cournot framework (Alpha as a firm, Beta as a firm), the Alpha output in equilibrium would be 560.
- The provided answer 192 does not align with these standard derivations either for the monopoly case (960) or for the two-firm Cournot case (560). This suggests either a difference in interpretation, a scaling factor not present in the stated data, or a mismatch between the problem’s setup and the listed answer.

In sum, with the given inverse demand and Alpha’s marginal cost, the conventional monopoly quantity is 960, and the conventional Cournot Alpha quantity is 560. The number 192 would require a nonstandard interpretation or a different set of assumptions than those stated in the problem.

ECON1056 Quiz 6: Capacity constrained industries

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