# Changes

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

, 05:26, 21 January 2017
Goal of the firm
* The price at which it sells whatever it produces. Call this $P$.
The goal of the firm is to maximize its profit. Assuming that the firm is able to sell everything it makes, the profit (not accounting for fixed costs) is given by $Q(P - AVC)$ where $Q$ is the quantity produced, $P$ is the price ''chosen'' by the firm, and $AVC$ is the average variable cost (dependent on $Q$).
===Note on conventions for profit: accounting and not accounting for fixed costs===
| 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 $MR < MC$, 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 increasingdecreasing]].
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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 $Q$.