Unit 7 The firm and its customers

7.6 Setting price and quantity to maximize profit

Like the producer of Cheerios, Beautiful Cars will choose its price, P, and quantity, Q, taking into account its demand curve and production costs. The demand curve determines the feasible set of combinations of P and Q. To find the profit-maximizing point, we can draw the isoprofit curves, and find the point of tangency as before.

The firm’s profit is the difference between its revenue (the price multiplied by quantity sold) and its total costs, C(Q):

\[\begin{align*} \text{profit} &= \text{total revenue} - \text{total costs} \\ &= PQ - C(Q) \end{align*}\]
profit, economic profit
A firm’s profit is its revenue minus its total costs. We often refer to profit as ‘economic profit’ to emphasise that costs include the opportunity cost of capital (which is not included in ‘accounting profit’).
normal profits
Normal profits are the returns on investment that the firm must pay to the shareholders to induce them to hold shares. The normal profit rate is equal to the opportunity cost of capital and is included in the firm’s costs. Any additional profit (revenue greater than costs) is called economic profit. A firm making normal profits is making zero economic profit.

This calculation gives us what is known as the economic profit. Remember that the return per dollar of investment that the firm must pay to shareholders to induce them to hold shares (which is equal to the opportunity cost of capital) is included in the firm’s cost function. These payments that must be made to shareholders are referred to as normal profits. Economic profit is additional profit above the minimum return required by shareholders.

Equivalently, profit (or more specifically, economic profit) is the number of units of output multiplied by the profit per unit, which is the difference between the price and the average cost:

\[\begin{align*} \text{profit} &= Q(P-\frac{C(Q)}{Q}) \\ &= Q(P- \text{AC}) \end{align*}\]

In general the shape of the isoprofit curves will depend on the shape of the average cost curve. For Beautiful Cars, with cost function \(C(Q) = F + cQ\), we can write profit as:

\[\text{profit} = Q(P-c)-F\]

The equation shows that the isoprofit curves for Beautiful Cars will have the same shape as the ones we drew in Figure 7.2b for Apple Cinnamon Cheerios. Both firms have constant (although different) marginal costs: $2 for a pound of Cheerios; $14,400 for a car. The main difference is that Beautiful Cars also has fixed costs, which affect the amount of profit on each isoprofit curve.

Figure 7.14 shows the isoprofit curves for Beautiful Cars. The lowest curve shown is the horizontal straight line where the price is equal to the marginal cost, P = $14,400. At this price the firm makes a loss equal to the fixed cost, $80,000. The next curve is the zero-economic-profit curve, which is also the average cost curve: the combinations of price and quantity for which economic profit is equal to zero, because the price is just equal to the average cost at each quantity. On the higher curves, economic profit is positive.

In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0. Another downward-sloping convex curve passing through points G (11, 35,309) and H is the isoprofit curve for profit 150,000. Point H has a higher quantity and a lower price than point G. A rectangle has a side of the size of the horizontal distance between the vertical axis and point G, and the other side of the size of the vertical distance between the two isoprofit curves. The area of the rectangle is total profit. Another downward-sloping convex curve lies above the curve passing through G and H, and passes through point K. Point K corresponds to the same quantity as point H, but a higher price.
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Figure 7.14 Isoprofit curves for Beautiful Cars.

When the price is equal to the marginal cost: In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000.
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When the price is equal to the marginal cost

The horizontal straight line at P = 14,400 (Isoprofit 1) is an isoprofit curve. The price is equal to the marginal cost of a car and the firm makes a loss equal to its fixed cost: profit = –$80,000.

When the price is equal to the average cost: In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0.
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When the price is equal to the average cost

The downward-sloping curve (Isoprofit 2) shown is the firm’s average cost curve. If P = AC, the firm’s economic profit is zero. So the AC curve is also the zero-profit curve: it shows all the combinations of P and Q that give zero economic profit.

The shape of the zero-economic-profit curve: In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0.
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The shape of the zero-economic-profit curve

Beautiful Cars has decreasing AC. When Q is low, it needs a high price to break even. As Q increases, the break-even price falls, but it is always higher than the marginal cost because the firm needs to cover its fixed cost.

A higher isoprofit curve: In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0. Another downward-sloping convex curve passing through points G (11, 35,309) and H is the isoprofit curve for profit 150,000. Point H has a higher quantity and a lower price than point G.
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A higher isoprofit curve

The downward-sloping curve (Isoprofit 3) shows the combinations of P and Q giving higher levels of profit. Profit is $150,000 at points G and H.

Profit = Q(P − AC): In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0. Another downward-sloping convex curve passing through points G (11, 35,309) and H is the isoprofit curve for profit 150,000. Point H has a higher quantity and a lower price than point G. A rectangle has a side of the size of the horizontal distance between the vertical axis and point G, and the other side of the size of the vertical distance between the two isoprofit curves. The area of the rectangle is total profit.
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Profit = Q(P − AC)

At G, where the firm makes 11 cars, the price is $35,309 and the average cost is $21,673. The firm makes a profit of $13,636 on each car, and its total profit is $150,000, shown by the area of the shaded rectangle.

Higher prices, higher profits: In this diagram, the horizontal axis shows quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A horizontal line at price 14,400 is the isoprofit curve for profit negative 80,000. A downward-sloping, convex curve which lies above the horizontal line at all points is the isoprofit curve for profit 0. Another downward-sloping convex curve passing through points G (11, 35,309) and H is the isoprofit curve for profit 150,000. Point H has a higher quantity and a lower price than point G. A rectangle has a side of the size of the horizontal distance between the vertical axis and point G, and the other side of the size of the vertical distance between the two isoprofit curves. The area of the rectangle is total profit. Another downward-sloping convex curve lies above the curve passing through G and H, and passes through point K. Point K corresponds to the same quantity as point H, but a higher price.
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Higher prices, higher profits

Profit is higher on the curves closer to the top-right corner in the diagram. Point H has the same quantity as K, so the average cost is the same, but the price is higher at K.

Isoprofit curves are steep when price is high, and flatter when price is close to marginal cost. At any point on an isoprofit curve the slope is given by:

\[\text{slope of isoprofit curve} = -\frac{(P- \text{MC})}{Q}\]

To understand why, think again about point G in Figure 7.14 where Q = 11, and the price is much higher than the marginal cost. If you:

  1. increase Q by 1
  2. reduce P by (Pc)/Q

then your profit will stay the same because the extra profit of (Pc) on car 12 will be offset by a fall in revenue of (Pc) on the other 11 cars.

Figure 7.15 shows the profit-maximizing choice of price and quantity for Beautiful Cars. Its feasible set is all points on or below the demand curve. It obtains the highest profit at E, where the demand curve is tangent to an isoprofit curve.

In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. A horizontal line at price 14,400 is the marginal cost. A downward-sloping straight line connecting points (0, 40,000), E (32, 27,200) and (100, 0). Three parallel, downward-sloping, convex curves are shown. The lowest two lie above the marginal cost line and intersect the straight line in two points, while the highest one is tangential to the straight line at point E. The area enclosed by points (0, 16,900), (0, 27,200), E and (32, 16,900) is profit.
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Figure 7.15 Maximizing the profit for Beautiful Cars.

The profit-maximizing price and quantity are P* = $27,200 and Q* = 32. The average cost per car is $16,900, giving a profit of $10,300 on each car. Its total profit is 32 × $10,300 = $329,600, which is equal to the area of the shaded rectangle.

The firm maximizes profit at the tangency point, where the slope of the demand curve is equal to the slope of the isoprofit curve, so that the two trade-offs are in balance.

marginal rate of substitution (MRS)
The trade-off that a person is willing to make between two goods. At any point, the MRS is the absolute value of the slope of the indifference curve. See also: marginal rate of transformation.
marginal rate of transformation (MRT)
The quantity of a good that must be sacrificed to acquire one additional unit of another good. At any point, it is the absolute value of the slope of the feasible frontier. See also: marginal rate of substitution.
price markup
The price minus the marginal cost divided by the price. In other words, the profit margin as a proportion of the price. If the firm sets the price to maximize its profits, the markup is inversely proportional to the elasticity of demand for the good at that price.
  • The isoprofit curve is the indifference curve, and its slope represents the marginal rate of substitution (MRS) in profit creation, between selling more and charging more.
  • The demand curve is the feasible frontier, and its slope represents the marginal rate of transformation (MRT) of lower prices into greater quantity sold.

At E, the profit-maximizing point, MRS = MRT.

profit margin
The difference between the price of a product and its marginal production cost.

Remember that the slope of the isoprofit curve depends on (Pc), the difference between price and marginal cost, which we call the profit margin. At point E, the profit margin is the additional profit the firm makes by producing and selling the 32nd car. Remember also that the slope of the demand curve is related to the price elasticity of demand, \(\varepsilon\): \(\varepsilon=-\frac{P}{Q } \times \text{slope}\), or equivalently \(\text{slope}=-\frac{P}{\varepsilon Q}\).

The table in Figure 7.16 shows that the tangency condition MRS = MRT tells us something important: when the firm maximizes profit, it sets its price so that the price markup (the profit margin as a proportion of the price) is equal to the inverse of the elasticity of its demand curve.

Slope of isoprofit curve Slope of demand curve
MRS MRT
$$ - \frac{(P-c)}{Q} $$ $$ - \frac{P}{\varepsilon Q} $$
MRS = MRT
$$ \frac{(P-c)}{Q} = \frac{P}{\varepsilon Q} $$
$$ \frac{(P-c)}{P} = \frac{1}{\varepsilon} $$
The price markup is equal to the inverse of the demand elasticity

Figure 7.16 Interpreting the tangency condition.

When the intensity of competition from other firms is low, \(\varepsilon\) will be low too. This result tells us that the firm will then set a higher price markup than it would if it faced more competition.

Profit maximization and fixed costs

How do the firm’s fixed costs affect its choice of price and quantity? The answer may surprise you: if the fixed costs changed, the profit-maximizing choice would not.

Suppose Beautiful Cars’ fixed costs increase by $1,000, while the marginal cost remains the same. Remember that \(\text{profit} = (P - c)Q - F\). Then if two different (P, Q) combinations were equally profitable before, they still give the same amount of profit as each other, although it is $1,000 lower.

So all the isoprofit curves in Figure 7.15 remain in exactly the same places. The only difference is that we need to relabel them to reduce the profit on each one by $1,000. The firm makes the same choice of P and Q, but receives $1,000 less profit.

Using marginal revenue and marginal cost to find the profit-maximizing quantity

In Figure 7.15 we worked out how the firm would maximize profit by finding the values of P and Q that would achieve the highest profit within the feasible set. An alternative approach is to work out how profit varies with Q, allowing for the effect of changing Q on the price at which cars can be sold.

Remember that profit is the difference between revenue and costs, so for any value of Q, the change in profit if Q is increased by one unit (marginal profit) will be the difference between the change in revenue (marginal revenue, MR), and the change in costs (marginal cost, MC):

\[\begin{align*} \text{profit} &= \text{total revenue} - \text{total costs} \\ \text{marginal profit} &= \text{MR} - \text{MC} \end{align*}\]
  • If MR > MC, the firm could increase profit by raising Q.
  • If MR < MC, the marginal profit is negative. It would be better to decrease Q.
  • So at the profit-maximizing Q, MR = MC.

Figure 7.17 shows you how to calculate the marginal revenue for each value of Q along the demand curve, and use it to find the point of maximum profit for Beautiful Cars. Remember that Beautiful Cars has constant marginal cost; the horizontal line at $14,400 represents MC.

In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and E (32, 27,200). This is the demand curve. Another downward-sloping line passes through points (0, 40,000) and E-prime (32, 14,400). This is the marginal revenue curve.
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Figure 7.17 Marginal revenue and marginal cost.

Demand and marginal cost curves: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and B (20, 32,000). This is the demand curve. The area enclosed by points (0, 0), (0, 32,000), (20, 32,000) and (20, 0) is the revenue.
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Demand and marginal cost curves

The diagram shows the demand curve for Beautiful Cars, and its marginal cost. At point B on the demand curve, Q = 20, P = $32,000, and revenue is $640,000 (the area of the rectangle).

Calculating the marginal revenue, MR: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000), B (20, 32,000) and (21, 31,600). This is the demand curve. The area enclosed by points (0, 0), (0, 32,000), (20, 32,000) and (20, 0) is the revenue at quantity 20. The area enclosed by points (0, 0),  (0, 31,600), (21, 31,600) and (21, 0) is the revenue at quantity 21.
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Calculating MR when Q = 20
Revenue, R = P × Q
Q = 20 P = $32,000 R = $640,000
Q = 21 P = $31,600 R = $663,600
ΔQ = 1 ΔP = –$400 MR = $23,600

Calculating the marginal revenue, MR

The marginal revenue is the change in revenue when Q increases by 1 unit. If Q increases from 20 to 21, P falls by $400. The table shows you how to calculate MR.

When Q = 20, MR is positive: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000), B (20, 32,000) and (21, 31,600). This is the demand curve. The area enclosed by points (0, 0), (0, 32,000), (20, 32,000) and (20, 0) is the revenue at quantity 20. The area enclosed by points (0, 0),  (0, 31,600), (21, 31,600) and (21, 0) is the revenue at quantity 21. The gain in revenue outweighs the loss in revenue.
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When Q = 20, MR is positive

The gain in revenue on the 21st car outweighs the loss due to the fall in price for the other 20 cars. MR > 0.

Plotting MR on the diagram: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and B (20, 32,000). This is the demand curve. At quantity 20, marginal revenue is 23,600. This point is labelled B-prime.
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Plotting MR on the diagram

The MR when Q = 20 is $23,600. We have plotted it as point B′ on the same diagram. The marginal revenue is always less than the price. The firm gains P when it sells an extra car, but it loses revenue on the other cars because their price is lower than before.

MR at other points: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and B (20, 32,000). This is the demand curve. At quantity 20, marginal revenue is 23,600. This point is labelled B-prime. Points A-prime, C-prime and D-prime show the marginal revenue at other quantities. Coordinates are the following: A-prime (10, 32,000), C-prime (40, 8,000) and D-prime (55, -4,000).
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MR at other points

We have calculated MR at other points on the demand curve in the same way, and plotted them on the diagram. Moving down the demand curve, P falls and MR falls by more. The gain on the extra car gets smaller, and the loss on the other cars is bigger. At point D MR is negative: the loss outweighs the gain.

The marginal revenue curve: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and B (20, 32,000). This is the demand curve. At quantity 20, marginal revenue is 23,600. This point is labelled B-prime. Points A-prime, C-prime and D-prime show the marginal revenue at other quantities. Coordinates are the following: A-prime (10, 32,000), C-prime (40, 8,000) and D-prime (55, -4,000). The straight line passing through points A-prime, B-prime, C-prime and D-prime is the marginal revenue curve.
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The marginal revenue curve

Joining the MR points gives the MR curve. It is below the demand curve (because MR is always less than P) and slopes downward.

The point of maximum profit: In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A horizontal line starting at (0, 14,400) is the marginal cost. A downward-sloping straight line passes through points (0, 40,000) and E (32, 27,200). This is the demand curve. Another downward-sloping line passes through points (0, 40,000) and E-prime (32, 14,400). This is the marginal revenue curve.
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The point of maximum profit

From the diagram we find that MR = MC at point E′, when Q = 32. At this point, marginal profit is zero, and profit is maximized. The firm should make 32 cars, and sell them at the price on the demand curve, at point E.

The marginal revenue curve is usually (although not necessarily) a downward-sloping line. Figure 7.17 demonstrates that MR = MC at point E′, where Q = 32. The diagram shows the following:

  • When Q < 32, MR > MC: so marginal profit is positive; profit increases with Q.
  • When Q > 32, MR < MC: marginal profit is negative; profit decreases with Q.

So the firm would not want to choose any Q below 32, because profit could be increased by choosing a higher value. And it would not want any Q above 32 because profit is decreasing: it would be better to choose a lower value.

The profit-maximizing quantity is Q = 32. What should the price be? To maximize profit, the firm should set the highest price at which 32 cars can be sold, according to the demand curve. So profit is maximized at point E: Q = 32, and P = $27,200.

Figure 7.18 shows that point E′, where MR = MC, leads us to the same profit-maximizing point that we found before by finding a tangency point.

In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 80. The vertical axis shows the price, and ranges from negative 10,000 to 40,000. Coordinates are (quantity, price). A straight line passes through points (0, 40,000) and E (32, 27,200). This is the demand curve. A downward-sloping, convex curve is tangential to the demand curve at point E. Another straight line passes through points (0, 40,000) and E-prime (32, 14,400 0). This is the marginal revenue curve. A horizontal line at price 14,400 is the marginal cost line and intersects the marginal revenue curve at point E-prime.
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Figure 7.18 The profit-maximizing point can be found from MR and MC, or from isoprofit curves.

Question 7.9 Choose the correct answer(s)

Figure 7.15 depicts the demand curve for Beautiful Cars, together with the marginal cost and isoprofit curves. At point E, the quantity–price combination is (Q*, P*) = (32, 27,200) and the profit is $329,600.

Suppose that the firm chooses instead to produce Q = 32 cars and sets the price at P = $27,000. Using this information, read the following statements and choose the correct option(s).

  • The profit remains the same at $329,600.
  • The profit is reduced to $323,200.
  • The average cost of production is $17,000.
  • The firm is unable to sell all the cars.
  • Since Q is still 32, production costs remain the same, but revenue falls, so profit falls.
  • Since Q is still 32, production costs remain the same. Revenue falls by $200 on each car, so by $6,400 in total. So profit is $329,600 – $6,400 = $323,200.
  • At E, where Q* = 32 and P* = $27,200, the profit is $329,600. So the profit per car is $329,600/32 = $10,300. Since $27,200 – AC = $10,300, AC must be $16,900.
  • At the lower price the demand is higher than 32, so the firm will have no problem selling all 32 cars at the new price.

Question 7.10 Choose the correct answer(s)

Figure 7.15 depicts the demand curve for Beautiful Cars, together with the marginal cost and isoprofit curves.

Suppose that the firm decides to switch from P* = $27,200 and Q* = 32 to a higher price, and chooses the profit-maximizing level of output at the new price. Using this information, read the following statements and choose the correct option(s).

  • The firm reduces the quantity of cars produced.
  • The marginal cost of producing an extra car is higher.
  • The total cost of production is higher.
  • The profit increases due to the new higher price.
  • At a higher price than P*, the maximum number of cars that can be sold is less than 32, and the firm will not produce more cars than it can sell.
  • The marginal cost of production is constant regardless of the number of cars produced.
  • The firm will produce fewer than 32 cars, so its total costs will be lower.
  • Any feasible point other than E is on a lower isoprofit curve.

Question 7.11 Choose the correct answer(s)

Figure 7.17 shows marginal cost, demand and marginal revenue for Beautiful Cars. Using the figure, read the following statements and choose the correct option(s).

  • When Q = 40, the marginal cost is greater than the marginal revenue so the firm’s total profit must be negative.
  • Total revenue is greater when Q = 10 than if Q = 20.
  • The firm would not choose to produce at the point where the marginal cost and marginal revenue curves intersect, because marginal profit at that point is zero.
  • Total profit is greater when Q = 20 than when Q = 10.
  • When Q = 40, the marginal cost is greater than the marginal revenue so the marginal profit is negative, but the total profit (summed across all units sold) is not necessarily negative.
  • The marginal revenue is greater at Q = 10 than Q = 20. But because the marginal revenue is positive as output increases from 10 to 20, revenue is increasing: it is higher at Q = 20.
  • Marginal profit is zero at the point where the marginal revenue and marginal cost curves intersect. But this is the profit-maximizing point, so the firm will choose it. (Remember that the firm still makes a positive profit on all previous units sold, so total profit is positive.)
  • At all levels of output up to point E, marginal revenue is greater than marginal cost. So profit increases as output increases—it is higher at Q = 20 than Q = 10.

Extension 7.6 Profit maximization

In the main part of this section, we used diagrams to show how a firm (Beautiful Cars) would set its profit-maximizing price and quantity. We took two different approaches: one involving isoprofit curves and another involving marginal revenue and marginal costs. In this extension, we show how to do this mathematically, using calculus (differentiation). In particular, we apply the method for solving constrained choice problems, which is explained in Extension 3.5; you may need to re-read it before reading this extension.

We have found the profit-maximizing price and quantity diagrammatically, in two ways: first by drawing isoprofit curves and finding the point of tangency with the demand curve, and secondly by drawing the MR and MC curves. In this extension, we demonstrate how to do this mathematically, using calculus.

Beautiful Cars has linear demand and cost functions; marginal cost is constant ($14,400) and fixed cost is $80,000. Figure E7.2 (taken from Figure 7.15) shows the isoprofit curves and profit-maximizing point (E) for this case.

In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. A horizontal line at price 14,400 is the marginal cost. A downward-sloping straight line connecting points (0, 40,000), E (32, 27,200) and (100, 0). Three parallel, downward-sloping, convex curves are shown. The lowest two lie above the marginal cost line and intersect the straight line in two points, while the highest one is tangential to the straight line at point E. The area enclosed by points (0, 16,900), (0, 27,200), E and (32, 16,900) is profit.
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Figure E7.2 Isoprofit curves and profit-maximisation for Beautiful Cars.

As in Extensions 7.4 and 7.5, we will examine the more general case of a firm with cost function, \(C(Q)\), and inverse demand function, \(P=f(Q)\). Remember that the average and marginal cost functions are:

\[\text{AC}(Q)=\frac{C(Q)}{Q} \text{ and } \text{MC}(Q)=C'(Q)\]

The symbol Π is the Greek capital letter ‘pi’, and is often used in economics to represent profit.

The firm’s profit, which we call \(\Pi\), is a function of \(P\) and \(Q\):

\[\Pi (P, Q) = PQ - C(Q)\]

Drawing isoprofit curves and calculating their slopes

The isoprofit curves are a family of curves in the \(Q–P\) plane, each of which corresponds to a given level of profit. The equation of a typical isoprofit curve is:

\[PQ - C(Q) = \Pi_0\]

where \(\Pi_0\) is a constant that represents the level of profit. There is a different curve for each value of \(\Pi_0\).

We can work out the shape of the isoprofit curves by examining the algebraic properties of this equation. To represent them in a diagram with \(P\) on the vertical axis, it is helpful to rearrange this equation to express \(P\) as a function of \(Q\):

\[P = \frac{C(Q)+\Pi_0}{Q}\]

This equation implies that if \(\Pi_0\) increases, then \(P\) also increases for any given \(Q\). So in a diagram depicting the family of isoprofit curves, higher curves correspond to higher levels of profit. Also, the zero-profit curve (\(\Pi_0=0\)) corresponds to the average cost curve, \(C(Q)/Q\): the firm makes a profit of zero when its price is equal to the average cost of a unit of output.

To find the slope of an isoprofit curve at any point, we can differentiate using the rule for a quotient:

\[\frac{dP}{dQ}= -\frac{QC'(Q)-(C(Q)+\Pi_0)}{Q^2}\]

And since \(C(Q)+\Pi_0=PQ\), we have:

\[\frac{dP}{dQ} = \frac{C'(Q)-P}{Q}=\frac{\text{MC}-P}{Q}\]

This is the result we obtained in the main part of this section for Beautiful Cars. In that case, \(\text{MC}=c\), a constant. Figure E7.2 shows that the isoprofit curve where \(P=c\) is flat, and that all the curves above it, where \(P>c\), slope downward.

For cost functions where MC is not constant, isoprofit curves may slope upward for some values of \(P\) and \(Q\), and downward for others—as the following example illustrates.

Example

Figure E7.3 shows the isoprofit curves for a quadratic cost curve, \(C(Q)=320+2Q+0.2Q^2\). The equations of the marginal cost and isoprofit curves are as follows:

  • Marginal cost: \(\text{MC}(Q)=C'(Q)=2+0.4Q\)
  • Isoprofit curve: \(P=\frac{320+\Pi_0}{Q}+2+0.2Q\)

This firm has fixed costs of 320, and its marginal cost increases with output—MC is an upward-sloping line. We have plotted MC, together with the isoprofit curve for \(\Pi_0=0\) (the AC curve), and the curves for \(\Pi_0=310\) and \(640\).

In this diagram, the horizontal axis shows the quantity, Q, and ranges from 0 to 100. The vertical axis shows the price, P, and ranges from 0 to 60. Coordinates are (quantity, price). The marginal cost curve is an upward-sloping straight line with the equation MC = 2 + 0.4Q. The average cost curve is a U-shaped function with the equation 320/Q + 2 + 0.2Q. Two U-shaped curves lie above the average cost curve but do not intersect it. The lower curve is the isoprofit curve corresponding to a profit of 310, and intersects the marginal cost curve at (56, 24). The upper curve is the isoprofit curve corresponding to a profit of 640, and intersects the marginal cost curve at (68, 29).
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Figure E7.3 MC, AC, and isoprofit curves for \(C(Q)=320+2Q+0.2Q^2\).

Consider again the expression we obtained for the slope of an isoprofit curve:

\[\frac{dP}{dQ} = \frac{\text{MC}-P}{Q}\]

Figure E7.3 shows that the isoprofit curves slope downward in the areas of the diagram where \(P > \text{MC}\), and upward where \(P < \text{MC}\). We have already shown in Extension 7.4 that if the AC curve is U-shaped, the MC curve crosses it at the point where AC is minimized. Now we know that on any isoprofit curve, MC crosses at the point with lowest value of \(P\).

Maximizing profit

The owners of the firm want to choose their price and quantity from the feasible set to make the highest possible profit, \(\Pi=PQ-C(Q)\). The feasible combinations of \(P\) and \(Q\) lie on or below the demand curve, \(P=f(Q)\). A point above the demand curve would not be feasible because the price would be too high for it to sell the corresponding quantity of output. In other words, when choosing \(P\) and \(Q\), the firm faces a constraint \(P\leq f(Q)\).

This is a constrained choice problem, like the problem facing a worker wishing to maximize utility (Extensions 3.5 and 5.5). We have expressed the constraint as an inequality, but for any given value of \(Q\), the firm makes the most profit if it chooses the highest feasible price. So we know it will choose a combination of \(P\) and \(Q\) that lies on the demand curve, and we can write the constraint as \(P=f(Q)\). Then the constrained choice problem has the same form as the worker’s problem, and also the employer’s problem studied in Extension 6.10.

The firm’s constrained choice problem

Choose \(P\) and \(Q\) to maximize \(\Pi(P,\ Q)\), subject to the constraint \(P=f(Q)\).

The simplest mathematical approach to this problem is the method of substitution. We use the constraint to substitute for \(P\), giving profit as a function of \(Q\) alone:

\[\Pi= Qf(Q) - C(Q)\]

This is the profit function that we drew for the example of Cheerios in Figure 7.4b, reproduced below as Figure E7.4. It tells us the amount of profit at each point on the demand function.

There are two diagrams.
   In diagram 1, the horizontal axis displays the quantity of Cheerios in pounds, and ranges from 0 to 80,000. The vertical axis displays the price of Cheerios in dollars per pound, and ranges from 0 to 10. Coordinates are (quantity, price). A downward-sloping, convex curve  which connects points (0, 7) and (80,000, 1) is the demand curve. Three, downward-sloping, convex curves which do not intersect one another are the isoprofit curves for Cheerios. From the upper to the lower curve, the isoprofit curves correspond to profits of $60,000, $34,000 and $10,000. The isoprofit curve for $60,000 lies above the demand curve. The isoprofit curve for $34,000 is tangent to the demand curve at point E (14,000, 34,000), which corresponds to point (14,000, 34,000) in diagram 2. The isoprofit curve for $10,000 intersects the demand curve in two points: one is high price and low quantity, the other is low price and high quantity. A horizontal line at 2 dollars per pound is the isoprofit curve corresponding to $0 profits and lies below all the other isoprofit curves.
   In diagram 2, the horizontal axis displays the quantity of Cheerios in pounds, and ranges from 0 to 80,000. The vertical axis displays profits in dollars, and ranges from -15,000 to 45,000. Coordinates are (quantity, price). A concave curve starts from (0, 0), peaks at (14,000, 34,000), and crosses the horizontal axis at (38,000, 0). The point (14,000, 34,000) corresponds to point E in diagram 1. Profit  falls to $0 when the price is equal to the unit cost, $2. This point corresponds to the point in diagram 1 where the isoprofit curve for $0 profit intersects the demand curve.
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Figure E7.4 Profit at each point on the demand function.

To find the value of \(Q\) that maximizes this function, we differentiate with respect to \(Q\), using the product rule:

\[\frac{d\Pi}{dQ} = f(Q) + Qf'(Q) - C'(Q)\]

The profit-maximizing quantity, \(Q^*\), satisfies the first-order condition \(d\Pi/dQ = 0\) (where the slope of the profit function in Figure E7.4 is zero):

\[f(Q)+Qf '(Q)= C'(Q)\]

If we knew the specific form of the functions, \(f(Q)\) and \(C(Q)\), we could try to solve the equation to find \(Q^*\) explicitly. The profit-maximizing price could then be calculated: \(P^*=f(Q^*)\).

But without knowing the functions, we can still consider what the first-order condition tells us. Since the profit-maximizing value of \(Q\) is on the demand curve, \(f(Q) = P\) and the first-order condition can be written as:

\[f'(Q)= \frac{C'(Q) - P}{Q}\]

The left-hand side of this equation is the slope of the demand curve, and (from above) the right-hand side is the slope of the isoprofit curve. Therefore, the first-order condition tells us precisely that the profit-maximizing choice lies at a point of tangency between the demand and isoprofit curves.

The first-order condition also gives us the relationship between the price markup and the demand elasticity at the profit-maximizing point. Rearranging the first-order condition:

\[P-C'(Q)=-Qf'(Q)\]

and using the formula for the elasticity in terms of the inverse demand function, \(\varepsilon=-\dfrac{f(Q)}{Qf'(Q)}=-\dfrac{P}{Qf'(Q)}\), we get:

\[\frac{P-C'(Q)}{P}=\frac{1}{\varepsilon}\]

That is, the markup of price over marginal cost is equal to the inverse of the price elasticity of demand.

Example

Consider again the firm with cost curve is \(C(Q)=320+2Q+0.2Q^2\), and suppose its inverse demand function is \(P=44-0.5Q\). Substituting for \(P\), we can write profit in terms of \(Q\) only and differentiate to obtain the first-order condition:

\[\begin{align*} \Pi&=Q(44-0.5Q)-(320+2Q+0.2Q^2)\\\frac{d\Pi}{dQ} &= 42-1.4Q=0 \\ \text{and hence } Q^*&=30 \end{align*}\]

So the profit-maximizing quantity is \(Q^*=30\) and the corresponding price from the demand curve is \(P^*= 44-0.5Q^*=29\). Figure E7.5 shows the solution graphically.

In this diagram, the horizontal axis shows the quantity, Q, and ranges from 0 to 100. The vertical axis shows the price, P, and ranges from 0 to 60. Coordinates are (quantity, price). The marginal cost curve is an upward-sloping straight line with the equation MC = 2 + 0.4Q. The average cost curve is a U-shaped function with the equation 320/Q + 2 + 0.2Q. Two U-shaped curves lie above the average cost curve but do not intersect it. The lower curve is the isoprofit curve corresponding to a profit of 310, and intersects the marginal cost curve at (56, 24). The upper curve is the isoprofit curve corresponding to a profit of 640, and intersects the marginal cost curve at (68, 29). The inverse demand curve has the equation P = 44 - 0.5Q and is tangent to the isoprofit for 310 at the point (30, 29).
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Figure E7.5 Maximizing profit with costs, \(C(Q)=320+2Q+0.2Q^2\), and inverse demand, \(P=44-0.5Q\).

The slope of the inverse demand function is −0.5, and we can verify that the slope of the isoprofit curve at \((Q^*, P^*)\) is also −0.5:

\[\begin {align*} \text{slope of isoprofit} &= \frac{C'(Q)-P}{Q}\\&=\frac{2+0.4Q^* - P^*}{Q^*}\\&=\frac{2+0.4\times 30-29}{30}=-0.5 \end {align*}\]

Since demand curves always slope downward, the profit-maximizing point will always be at a point where the isoprofit curve slopes downward. Hence, the firm sets a price that exceeds marginal cost.

The condition \(MR = MC\)

The main part of the section shows that the profit-maximizing quantity can be found diagrammatically at the point where marginal revenue is equal to marginal cost.

To derive this algebraically, we write the firm’s revenue, \(R=PQ\), in terms of \(Q\) only using the inverse demand function, \(P=f(Q)\). The revenue function is:

\[R(Q)=Qf(Q)\]

and the marginal revenue, MR, is given by:

\[MR=R'(Q)=f(Q)+Qf'(Q)\]

So the first-order condition for profit maximization is exactly equivalent to the condition, MR = MC:

\[\begin{align*} f(Q)+Qf'(Q)&=C'(Q)\\ \text{MR}&=\text{MC} \end{align*}\]

Since profit is equal to revenue minus cost (\(\Pi(Q)= R(Q)-C(Q)\)) you can solve a profit maximization problem either by differentiating the profit function and solving the equation \(\Pi'(Q)=0\), or by finding MR and MC by differentiating the revenue and cost functions and solving the equation, MR = MC. Both approaches give you the same answer.

Exercise E7.4 Profit maximization

A firm has the cost function, \(C(Q) = 50 + 4Q + Q^2\), and faces the inverse demand function, \(P = 100-2Q\).

  1. Write the equation for the isoprofit curve corresponding to an amount of profit, \(\Pi_0\).
  2. Draw a diagram like Figure E7.3, that shows the inverse demand function and the isoprofits corresponding to profits of 200, 500, and 1,000. (Make sure to label each isoprofit curve.)
  3. Write an expression for profits, \(\Pi\), and use the constrained choice method to find the profit-maximizing choice of \(Q\) and the corresponding price, \(P\). Indicate this point in your diagram from Question 2.
  4. Draw and label the isoprofit curve passing through the profit-maximizing point. Verify that the slope of the isoprofit curve at this point equals the slope of the inverse demand function. What level of profits do the points on this isoprofit curve correspond to?
  5. Write an expression for marginal revenue, and an expression for marginal cost. Plot both of these functions in your diagram from Question 2, and verify that the profit-maximizing point occurs at the quantity where the MR curve intersects the MC curve.

Read more: Chapter 8 (on curve-sketching, and finding maxima and minima) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.