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A year ago
Consider a competitive firm with the short-run cost function
      C(q) = 20 + 6q + 5q2
The firm faces a market price of p for its output.
a.   Derive the firm's profit maximizing condition. Is the sufficient second order condition satisfied?
b.   Suppose a specific tax of t (t < p) is levied on only this firm in the industry. What is the profit maximizing level of output as a function of p and t? (Assume the price is high enough that the firm does not shut down)
c.   How does the output change as the tax increases? Use calculus to determine the relevant comparative static.
d.   How does the firm's profit chance as the tax increases? Again, use calculus to determine the relevant comparative static. Show that profit decreases as t increases.
Source  Download
Microeconomics: Theory and Applications with Calculus
Edition: 4th
Author:
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Answer verified by a subject expert
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A year ago
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a.   Profit maximization occurs where p = MC:
      p = 6 + 10q
dMC/dq = 10 > 0, so the sufficient second order condition is satisfied.
b.   A tax will lower the price to p - t, so
      p - t = 6 + 10q
Solve for q* = (p - t - 6)/10
Note that we are assuming p - t > 6 (or the firm will shut down).
c.   dq*/dt = -1/10.
d.   First compute the profit function
      π = (p - t)q* - C(q*) = (p - t)(p - t - 6)/10 - 20 - 3(p - t - 6)/5 - (p - t - 6)2/20
Then compute dπ/dt = -(p - t - 6)/10 - (p - t)/10 + 3/5 + (p - t - 6)/10
This simplifies to
      dπ/dt = -(p - t - 6)/10 < 0
This verified answer contains over 160 words.
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* Majoring in business & math
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