Consider an economy with only one producer, such that their production function is Y=4X{"version":"1.1","math":"Y=4X"} (Y and X are the output and input qualtities respectively) and the cost of the input per unit is equal to: w = 4$. The demand for the output is given by the individual demand function of four consumers, such as: q1=5-P; q2=7-P; q3=11-P; q4=13-P{"version":"1.1","math":"q1=5-P; q2=7-P; q3=11-P; q4=13-P"}. What would be the predicted equilibrium (aggregated quantity, price)?

Question

Consider an economy with only one producer, such that their production function is Y=4X{"version":"1.1","math":"Y=4X"} (Y and X are the output and input qualtities respectively) and the cost of the input per unit is equal to: w = 4$. The demand for the output is given by the individual demand function of four consumers, such as: q1=5-P; q2=7-P; q3=11-P; q4=13-P{"version":"1.1","math":"q1=5-P;&#xA0;q2=7-P;&#xA0;q3=11-P;&#xA0;q4=13-P"}. What would be the predicted equilibrium (aggregated quantity, price)?

BlackTom AI · Accepted Answer

We begin by restating the setup and the data in the question, then examine each answer option in turn before drawing a final interpretation.

First, the production side: Y = 4X, with input price w = 4. The firm’s profit is π = P·Y − w·X = P·(4X) − 4X = (4P − 4)X. The marginal profit with respect to X is dπ/dX = 4P − 4. This implies:
- If P > 1, the marginal profit is positive for any X, so the firm would like to push X (and hence Y) up without bound, as there is no stated production cap in the model. In other words, with P > 1, the profit-maximizing choice is to set X as large as possible (the model does not impose a finite upper limit on X). The associated supply is unbounded in this case.
- If P < 1, the marginal profit is negative for all X, so the firm would choose X = 0, producing nothing.
- If P = 1, the firm is indifferent across X; any X yields zero profit.
Thus, under the given cost structure, the competitive supply is not a finite, upward-sloping function of P. Instead, for P > 1 the firm would desire arbitrarily large output, while for P ≤ 1 it would produce zero (or be indifferent at P = 1).

Next, the demand side: The four consumers have q1 = 5 − P, q2 = 7 − P, q3 = 11 − P, q4 = 13 − P. Summing these gives aggregate demand Qd = (5 + 7 + 11 + 13) − 4P = 36 − 4P. This yields the inverse demand P = 9 − Q/4.

Now, evaluate each candidate (Q, P) pair against these observations:
- Option (12, 4.5): With P = 4.5, the aggregate demand would be Qd = 36 − 4·4.5 = 36 − 18 = 18. The proposed equilibrium quantity is Q = 12, which does not match the quantity implied by the price from the demand curve (the market would demand 18 units at P = 4.5). Moreover, since P > 1, the profit-maximizing behavior of the single firm would push output upward without bound, not settle at a finite Q like 12. This makes (12, 4.5) inconsistent with the model’s implications.
- Option (None of the other answers): This would be the choice if none of the finite (Q, P) pairs can be a market-clearing equilibrium given the above logic. Since P > 1 leads to unbounded supply, and the finite pairs fail to satisfy demand-consistent Q or the profit-maximization implication, this option remains a plausible candidate.
- Option (12, 6): With P = 6, Qd = 36 − 4·6 = 36 − 24 = 12. The market would demand exactly 12 units at P = 6, which matches the proposed Q = 12. However, with P = 6, since P > 1, the firm would profitably expand output without bound rather than producing a finite 12 units, so this pair cannot be an equilibrium under the stated production technology and cost structure.
- Option (9, 6.75): With P = 6.75, Qd = 36 − 4·6.75 = 36 − 27 = 9. The quantity aligns with Q = 9. Yet again, P > 1 implies the firm would want to increase X (and Y) without bound, so a finite Q = 9 is not consistent with profit-maximizing behavior.
- Option (9, 4.25): With P = 4.25, Qd = 36 − 4·4.25 = 36 − 17 = 19. This yields Qd = 19, which does not equal the proposed Q = 9. Also, since P > 1, the firm would prefer to raise output rather than settle at Q = 9, so this pair is not an equilibrium either.

In summary, for any finite price P > 1, the profit-maximizing production plan drives output to infinity in the absence of a cap, while P ≤ 1 would push output to zero. None of the finite (Q, P) combinations given can satisfy both market-clearing (Q = Qd) and profit-maximization under the described setup. Therefore, the set of plausible equilibria given the model does not include any of the finite pairs listed, and the option stating that lies outside the other finite choices.

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