logo
How it worksCoursesResearch CommunitiesBenefitsAbout Us
Schedule Demo
Learn Before
Example

Verification of the Tangency Condition at the Profit-Maximizing Point for P=44-0.5Q

For a firm with cost function C(Q)=320+2Q+0.2Q2C(Q) = 320 + 2Q + 0.2Q^2 and inverse demand P=44−0.5QP = 44 - 0.5Q, the profit-maximizing point is Q∗=30Q^* = 30 and P∗=29P^* = 29. At this point:

  1. Slope of the Demand Curve: The derivative of the inverse demand function P=44−0.5QP = 44 - 0.5Q is a constant −0.5-0.5.
  2. Slope of the Isoprofit Curve: The slope of an isoprofit curve is given by the formula: dPdQ=MC−PQ\frac{dP}{dQ} = \frac{MC - P}{Q}
  3. Calculation: Since marginal cost is MC=C′(Q)=2+0.4QMC = C'(Q) = 2 + 0.4Q, at Q∗=30Q^* = 30 we have MC=2+0.4(30)=14MC = 2 + 0.4(30) = 14. Plugging in the values: dPdQ=14−2930=−1530=−0.5\frac{dP}{dQ} = \frac{14 - 29}{30} = \frac{-15}{30} = -0.5

Since both slopes are equal to −0.5-0.5, the isoprofit curve is tangent to the demand curve, satisfying the profit-maximization condition.

0

1

Updated 2026-07-01

Contributors are:

Gemini AI
Gemini AI
🏆 3

Who are from:

Google
Google
🏆 3

References

Tags

Social Science

Empirical Science

Science

Economy

CORE Econ

Economics

Introduction to Microeconomics Course

The Economy 2.0 Microeconomics @ CORE Econ

Ch.7 The firm and its customers - The Economy 2.0 Microeconomics @ CORE Econ

The Economy 2.0 Macroeconomics @ CORE Econ

Related
Learn After