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Goal of the firm
More generally, when a firm has [[market power]], it has some leeway in setting prices but, in return, has to explicitly consider the [[market demand curve]] it faces. When a firm has monopoly, the leeway in setting prices as well as the importance of the market demand curve are maximal.
 
The overall conclusion will be the following:
 
{{quotation|The absolute (i.e., global) maximum for profit must occur either at one of these three points: zero quantity of production, the maximum quantity of production, or some local maximum at an intermediate quantity of production. In the last case, that ''must'' be a point where <math>MR = MC</math> and, more specifically, the value <math>MR - MC</math> is changing sign from positive to negative. Here, <math>MR</math> and <math>MC</math> denote respectively the marginal revenue and marginal cost.}}
 
To jump directly to the meat of the analysis, see the [[#Formal mathematical analysis|formal mathematical analysis section]].
==Similar determinations==
As before, we assume that the firm serves an entire market and has no competing firms. The firm has two pieces of information:
{| class="sortable" border="1"
! Piece of information !! Nature !! Equivalent pieces of information!! How it's decided!! How the firm gets to know it
|-
| the marginal cost curve of the firm || This is a curve describing the marginal cost as a function of quantity produced. Traditionally, the quantity is plotted along the horizontal axis and the cost function along the vertical axis. It is not a single number but rather a functional relationship. || Knowledge of the marginal cost curve is mathematically equivalent to knowledge of the average variable cost curve, the variable cost curve, and (subject to knowledge of the fixed cost),the average fixed cost curve, the average total cost curve, and the total cost curve. This means that knowledge of any one of these curves can be used to deduce the others. || Determined by the firm's production technology and [[factor of production|factors of production]]. The curve itself is not subject to changein the short run. || The firm may know this curve through explicit theoretical knowledge of its production process, or it may collect the data through experimentation with different production quantities.
|-
| the market demand curve facing the firm || This is the [[market demand curve]] that plots the quantity demanded by the market the firm serves as a function of the price. || || See [[determinants of demand]] for details. || See [[demand curve estimation]] for details.
|}
* The price at which it sells whatever it produces. Call this <math>P</math>.
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 <math>Q(P - AVC)</math> where <math>Q</math> is the quantity produced, <math>P</math> is the price ''chosen'' by the firm, and <math>AVC</math> is the average variable cost (dependent on <math>Q</math>).
===Note on conventions for profit: accounting and not accounting for fixed costs===
In particular, the net profit (not accounting for fixed costs) is zero at <math>Q = 0</math>.
The net profit accounting for fixed costs is <math>R - TC</math>, which becomes <math>(P - ATCAC) \cdot Q</math>, where <math>AC</math> is the average total cost, including both average fixed costs and average variable costs.
This is smaller than the profit calculated by not accounting for fixed costs. Also, for the short-run choices, this profit value may well be negative, because the firm treats its fixed costs as [[sunk cost]]s.
===Definition of marginal revenue===
Now that we have determined the firm's price as a function of the quantity produced, the firm's total revenue (<math>PQ</math>) also becomes a function of the quantity produced. Thus, we can take the derivative of total revenue with respect to quantity produced. This derivative is called '''marginal revenue''' and is abbreviated '''MR'''. Geometrically, the marginal revenue is the derivative with respect to the quantity produced of the area of the rectangle between the axes and the (quantity, price) point on the demand curve.
Note that marginal revenue depends, ''not'' on the firm's production technologies, but rather, ''completely'' on the firm's market demand curve. Moreover, in analogy terms:
{{quotation|Marginal revenue curve : Market demand curve :: Marginal cost curve : Average variable cost curve}}
 
===Relationship between marginal revenue curve and market demand curve===
 
The marginal revenue curve ''lies below the market demand curve'' everywhere. This follows from the [[law of demand]]: the demand curve is downward-sloping. In particular, this means that increasing the quantity demanded requires reducing the price somewhat. The derivative of total revenue with respect to quantity is therefore less than the price.
 
The extent to which the marginal revenue curve diverges from the market demand curve is directly related to the extent of [[deadweight loss due to market power of sellers|deadweight loss]] created in the situation.
===Relationship between marginal revenue and price-elasticity of demand===
It is still relevant, for a monopolistic firm, to consider the question of how shifts (expansions or contractions) in the demand curve affect the price and quantity produced. However, the analysis must revert directly to the calculation problem for the monopolistic firm and cannot use the short run supply curve as a visual aid.
==Formal mathematical analysis under differentiability assumptions==
===Determining the profit-maximizing quantity of production===
| 2 || The derivative of the profit function with respect to <math>Q</math> is <math>MR - MC</math> where <math>MR</math> 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 <math>MC</math> is the marginal cost (dependent on <math>Q</math>). || <toggledisplay>By the [[calculus:differentiation is linear|linearity of differentiation]], the derivative of the profit function is <math>Q \mapsto \frac{d}{dQ}(QP) - \frac{d}{dQ}(VC)</math>. The derivatives are <math>MR</math> and <math>MC</math> respectively.</toggledisplay>
|-
| 3 || For those quantities of production where <math>MR > MC</math>, the profit function is increasing with respect to <math>Q</math>. For those quantities of production where <math>MR < MC</math>, the profit function is decreasing with respect to <math>Q</math>. || See [[calculus:positive derivative implies increasing|positive derivative implies increasing]], [[calculus:negative derivative implies decreasing|negative derivative implies increasingdecreasing]].
|-
| 4 || The local maxima for the profit function must occur at points where <math>MR = MC</math>. Further, under reasonable assumptions of the function not being heavily oscillatory, we must have <math>MR > MC</math> for <math>Q</math> slightly less than the point of local maximum and <math>MR < MC</math> for <math>Q</math> slightly more than the point of local maximum. In other words, the <math>MC</math> curve is overtaking the <math>MR</math> 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 <math>Q</math>.
| 5 || The absolute (i.e., global) maximum for profit must occur either at one of these three points: zero quantity of production, the maximum quantity of production, or some local maximum at an intermediate quantity of production. In the last case, that ''must'' be a point where <math>MR = MC</math> and the <math>MC</math> curve is overtaking the <math>MR</math> curve. || Follows directly from Step (4).
|}
 
==Graphical interpretation of revenue and profit==
 
We have the following:
 
* The revenue is given by the product of the price and quantity produced. It is the area of a rectangle with two sides along the axes and the optimal (quantity,price) vertex as the one vertex not on the axes.
* The profit is given by the signed area between the marginal cost curve and the horizontal line for the price, up till the optimal quantity produced. Explicitly, it is:
 
<math>\int_0^{Q_{\mbox{opt}}} (P_{\mbox{opt}} - MC) \, dQ</math>
 
where <math>Q_{\mbox{opt}}</math> is the optimal quantity and <math>P_{\mbox{opt}}</math> is the corresponding price.
 
Note that, unlike the [[Determination of quantity supplied by firm in perfectly competitive market in the short run|case of perfect competition]], we do not have <math>P = MC</math> at the optimum quantity of production. In other words, at the firm's profit-maximizing choice of price and quantity, its marginal cost is not equal to its price. Rather, the marginal cost is equal to the ''marginal'' revenue. This is somewhat less than the price because the marginal revenue curve is below the market demand curve, or more concretely, because increasing the quantity sold requires reducing the price, and this has a compensating downward effect on revenue.
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