Lecture 6: Practice

What does it cost to make a car? Scale and the cost of production

Work through each problem on paper before you open its Solution.

1. Denise’s bakery

Denise runs a small bakery. Rent and the ovens cost $300 a day whether the bakery bakes anything or not, and each loaf costs $2 in flour and labor.

(a) Write down the bakery’s cost function, \(C(Q)\), where \(Q\) is loaves per day.

Solution

The fixed cost is $300 and the variable cost is $2 per loaf, so \(C(Q) = 300 + 2Q\).

(b) Find the total cost, average cost, and marginal cost at 50, 100, and 300 loaves a day.

Solution
Loaves per day, \(Q\) 50 100 300
Total cost, \(C(Q)\) 300 + 100 = 400 300 + 200 = 500 300 + 600 = 900
Average cost, \(C(Q)/Q\) 400 / 50 = 8 500 / 100 = 5 900 / 300 = 3
Marginal cost 2 2 2

Marginal cost is the cost of one more loaf, $2, at every output. Average cost is \(2 + 300 / Q\): each loaf carries its own $2 plus a share of the $300, and that share shrinks as more loaves are baked.

(c) At what output does average cost fall to $3 a loaf?

Solution

Average cost is \(2 + 300 / Q\). Setting \(2 + 300 / Q = 3\) gives \(300 / Q = 1\), so \(Q = 300\) loaves a day. The table in part (b) confirms it.

(d) The landlord doubles the rent, so fixed costs rise to $600 a day. Which of the three cost measures change, and how?

Solution
  • Total cost rises by $300 at every output: \(C(Q) = 600 + 2Q\).
  • Average cost rises, by \(300 / Q\): at 100 loaves it goes from $5 to $8.
  • Marginal cost does not change. One more loaf still costs $2 in flour and labor. The rent is paid whether that loaf is baked or not.

2. Fixed or variable, and for how long

(a) Denise’s daily expenses are listed below. Which vary with the number of loaves baked, and which do not?

  • Rent for the shop.
  • Flour, yeast, and butter.
  • Staff paid by the hour, with more hours scheduled on busy days.
  • The yearly fee for the bakery’s website.
  • Electricity for the ovens.
Solution

Fixed: the rent and the website fee. Both are the same whether the bakery bakes one loaf or a thousand.

Variable: flour, yeast, and butter; hourly staff; and electricity for the ovens. Each of these rises with the number of loaves baked.

(b) Denise’s ovens can bake at most 300 loaves a day on normal shifts. To bake more, Denise has to pay staff overtime at a higher hourly wage. What happens to the marginal cost of a loaf beyond 300, and why does the book describe this as a short-run cost?

Solution

Beyond 300 loaves each extra loaf needs overtime labor, so marginal cost rises above $2. It is a short-run cost because it comes from holding the ovens fixed: in the short run the stock of equipment cannot be changed. In the long run Denise can buy another oven, and the extra loaves would again cost $2 each.

3. Lena’s screen-printing shop

Lena prints T-shirts. Her cost function, in dollars per day, is \[C(Q) = 90 + \frac{Q^2}{10}\] where \(Q\) is shirts per day.

(a) What is the marginal cost per shirt when Lena goes from 10 to 15 shirts a day? From 15 to 20?

Solution

First the total costs: \[C(10) = 90 + \frac{10^2}{10} = 100, \qquad C(15) = 90 + \frac{15^2}{10} = 112.5, \qquad C(20) = 90 + \frac{20^2}{10} = 130\]

From 10 to 15 shirts: \[\text{MC} = \frac{\Delta C}{\Delta Q} = \frac{C(15) - C(10)}{15 - 10} = \frac{112.5 - 100}{5} = \$2.50 \text{ per shirt}\]

From 15 to 20 shirts: \[\text{MC} = \frac{C(20) - C(15)}{20 - 15} = \frac{130 - 112.5}{5} = \$3.50 \text{ per shirt}\]

(b) Is Lena’s marginal cost constant, like Beautiful Cars’ was? What might make each extra shirt cost more than the last?

Solution

No. Marginal cost rises with output, from $2.50 a shirt to $3.50. With a fixed press and a fixed number of hours in the day, printing more means squeezing more out of the same equipment: overtime pay, a tired crew, more misprints. That is the short-run case from the lecture, where marginal cost rises while the equipment is fixed.

4. Returns to scale

(a) Each firm below doubles all of its inputs. Classify its technology as increasing, constant, or decreasing returns to scale.

  • A brewery’s output rises from 100 barrels to 250.
  • A print shop’s output rises from 100 posters to 200.
  • A consulting firm’s output rises from 100 reports to 150.
Solution
  • The brewery more than doubles its output: increasing returns to scale, also called economies of scale.
  • The print shop exactly doubles it: constant returns to scale.
  • The consulting firm less than doubles it: decreasing returns to scale, also called diseconomies of scale.

(b) The print shop has constant returns to scale in production. Can its average cost still fall as it prints more posters? Explain.

Solution

Yes, if it has fixed costs. Constant returns mean the variable cost per poster stays the same, but a fixed cost such as the lease on the press is spread over more posters as output rises. With \(C(Q) = F + cQ\), average cost is \(c + F / Q\), which falls as \(Q\) rises even though \(c\) does not change.

5. Multiple choice

1. A firm’s cost function is \(C(Q) = 300 + 2Q\). Its marginal cost is:

  1. $300.
  2. $2.
  3. \(300 / Q + 2\).
  4. $302.
Solution

Marginal cost is the increase in total cost from one more unit, which is the slope of the cost function: $2. Option (c) is the average cost.

2. A cereal producer’s cost function is \(C(Q) = 2Q\), where \(Q\) is pounds of cereal. Select all the statements that are correct.

  1. There are no fixed costs of production.
  2. The marginal cost of production is 2.
  3. The producer’s average cost falls with output.
  4. For any quantity \(Q\), average cost and marginal cost are the same.
Solution

At \(Q = 0\) total cost is 0, so there are no fixed costs. The slope of the cost function is 2, so marginal cost is 2. Average cost is \(2Q / Q = 2\) at every output, so it does not fall, and it equals marginal cost everywhere. (a), (b), and (d) are correct.

3. Select all the statements that are correct.

  1. With constant returns to scale, doubling all inputs doubles output.
  2. With decreasing returns to scale, doubling all inputs more than doubles output.
  3. With economies of scale, cost per unit falls as the firm expands its production.
  4. With increasing returns to scale, tripling all inputs more than triples output.
Solution

Decreasing returns mean doubling inputs less than doubles output, so (b) is wrong. The other three follow from the definitions: with increasing returns the firm needs a less than proportional increase in inputs to raise output, so its cost per unit falls. (a), (c), and (d) are correct.

4. Which of the following is an example of network economies of scale?

  1. A bakery buys flour in bulk at a discount.
  2. A messaging app becomes more useful to each user as more of their friends join.
  3. A carmaker spreads the cost of designing a model over more cars.
  4. Doubling the material in a brewery’s pipe more than doubles the liquid it can carry.
Solution

Network economies of scale are on the demand side: the product is worth more to each user because other users are connected to it. That is (b). Option (a) is bargaining power, (c) is a fixed cost spread over more units, and (d) is an economy of scale in production.

5. A carmaker has fixed costs and a constant cost per car. Which statement about its average cost (AC) and marginal cost (MC) is correct?

  1. AC is below MC at every output.
  2. AC equals MC at every output.
  3. AC is above MC at every output, and gets closer to MC as output rises.
  4. AC rises with output while MC stays constant.
Solution

Average cost is the constant cost per car plus a share of the fixed cost, so it is always above marginal cost. The share shrinks as more cars are produced, so average cost falls toward marginal cost without reaching it. (c) is correct.