**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*):

- 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.

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:

- increase
*Q*by 1 - reduce
*P*by (*P*−*c*)/*Q*

then your profit will stay the same because the extra profit of (*P* − *c*) on car 12 will be offset by a fall in revenue of (*P* − *c*) 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.

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 (*P* − *c*), 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 |

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):

- 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.

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.

**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).

- 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).

- 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*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.