Changes

Determination of price and quantity supplied by monopolistic firm in the short run

, 22:15, 9 February 2014
Determining the profit-maximizing quantity of production
| 2 || The derivative of the profit function with respect to $Q$ is $MR - MC$ where $MR$ is the marginal revenue (defined as the derivative of total revenue with respect to quantity, using price as a function of quantity via the market demand curve) and $MC$ is the marginal cost (dependent on $Q$). || <toggledisplay>By the [[calculus:differentiation is linear|linearity of differentiation]], the derivative of the profit function is $Q \mapsto \frac{d}{dQ}(QP) - \frac{d}{dQ}(VC)$. The derivatives are $MR$ and $MC$ respectively.</toggledisplay>
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| 3 || For those quantities of production where $MR > MC$, the profit function is increasing with respect to $Q$. For those quantities of production where $MR < MC$, then the profit function is decreasing with respect to $Q$. || See [[calculus:positive derivative implies increasing|positive derivative implies increasing]], [[calculus:negative derivative implies decreasing|negative derivative implies increasing]].
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| 4 || The local maxima for the profit function must occur at points where $MR = MC$. Further, under reasonable assumptions of the function not being heavily oscillatory, we must have $MR > MC$ for $Q$ slightly less than the point of local maximum and $MR < MC$ for $Q$ slightly more than the point of local maximum. In other words, the $MC$ curve is overtaking the $MR$ curve. || Follows from (3). We can also interpret this in terms of the [[calculus:first derivative test|first derivative test]] applied to the profit function as a function of $Q$.