Suppose a farmer is a price taker for soybean sales with cost functions given by the following:

TC = 0.1q2 + 2q + 30

MC = 0.2q + 2

The firm's supply curve is given by

Question

Suppose a farmer is a price taker for soybean sales with cost functions given by the following:

TC = 0.1q2 + 2q + 30

MC = 0.2q + 2

The firm's supply curve is given by

BlackTom AI · Accepted Answer

A price-taking firm chooses output where P = MC (provided price covers average variable cost).

1. Relevant knowledge points and concepts
- For a competitive (price-taking) firm, the short-run supply curve is the portion of the marginal cost (MC) curve above the shutdown point (minimum average variable cost, AVC).
- Given total cost TC = 0.1q^2 + 2q + 30, variable cost VC = 0.1q^2 + 2q and AVC = VC/q = 0.1q + 2.
- Marginal cost is given: MC = 0.2q + 2.

2. Complete reasoning process step by step
- Set price equal to marginal cost for positive output: P = MC.
- Substitute MC: P = 0.2q + 2.
- Solve for q in terms of P.

3. Calculations with intermediate steps
- P = 0.2q + 2
- P - 2 = 0.2q
- q = (P - 2) / 0.2
- Since 1/0.2 = 5, q = 5(P - 2) = 5P - 10.

4. Why the correct answer is correct
- This algebra shows the firm’s supply relation (inverse MC) is q = 5P - 10. However, the firm will only produce if price covers AVC. AVC = 0.1q + 2 is minimized at q = 0 (it's linear increasing), so the shutdown price is AVC_min = 2. At P = 2, the formula gives q = 0; for P < 2 the firm shuts down (q = 0). Thus the supply curve is q = 5P - 10 for P ≥ 2 (and q = 0 for P < 2). The multiple-choice option gives the algebraic supply function q = 5P - 10, which matches the derived relation.

5. Why the other options are incorrect
- "q = 0.2P + 2" incorrectly treats MC as q = 0.2P + 2 instead of solving P = 0.2q + 2; it confuses coefficients.
- "q = 10P – 2" has the wrong slope and intercept (factor 10 rather than 5 and wrong constant).
- "q = 2P – 5" also has incorrect slope/intercept; none of these follow from P = 0.2q + 2.

6. Conclusion
The firm’s supply function is q = 5P – 10 (with the practical constraint q = 0 for P < 2). This matches the provided answer q = 5P – 10.

Suppose a farmer is a price taker for soybean sales with cost functions given by the following: TC = 0.1q2 + 2q + 30 MC = 0.2q + 2 The firm's supply curve is given by单项选择题

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Suppose a farmer is a price taker for soybean sales with cost functions given by the following: TC = 0.1q2 + 2q + 30 MC = 0.2q + 2 The firm's supply curve is given by单项选择题

类似问题

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