# Firms and markets I

Materials for class on Thursday, October 10, 2019

## Contents

## Slides

Download the slides from today’s class.

## Stone cold sober chocolate milk

### Finding the cheapest cost of production

#### Cost function

Quantity | Milk | Sugar | Chocolate powder | Spoon | Pot | Fridge |
---|---|---|---|---|---|---|

$0 | 0 | 0 | 0 | 1 | 4 | 15 |

$1 | 2 | 0.5 | 0.1 | 1 | 4 | 15 |

$2 | 6 | 1.5 | 0.3 | 1 | 4 | 15 |

$3 | 12 | 3 | 0.6 | 1 | 4 | 15 |

$4 | 20 | 5 | 1 | 1 | 4 | 15 |

$5 | 30 | 7.5 | 1.5 | 1 | 4 | 15 |

$6 | 42 | 10.5 | 2.1 | 1 | 4 | 15 |

$7 | 56 | 14 | 2.8 | 1 | 4 | 15 |

$8 | 72 | 18 | 3.6 | 1 | 4 | 15 |

$9 | 90 | 22.5 | 4.5 | 1 | 4 | 15 |

$10 | 110 | 27.5 | 5.5 | 1 | 4 | 15 |

#### Total costs

Quantity | TFC | TVC | TC |
---|---|---|---|

$0 | 20 | 0 | 20 |

$1 | 20 | 2.6 | 22.6 |

$2 | 20 | 7.8 | 27.8 |

$3 | 20 | 15.6 | 35.6 |

$4 | 20 | 26 | 46 |

$5 | 20 | 39 | 59 |

$6 | 20 | 54.6 | 74.6 |

$7 | 20 | 72.8 | 92.8 |

$8 | 20 | 93.6 | 113.6 |

$9 | 20 | 117 | 137 |

$10 | 20 | 143 | 163 |

If we decompose total costs into fixed and variable costs, we see that the rise in costs is driven almost entirely by increasing variable costs.

#### Average costs

Quantity | TC | AFC | AVC | ATC | MC_chunk | MC_instant |
---|---|---|---|---|---|---|

0 | $20.00 | — | — | — | — | $1.30 |

1 | $22.60 | $20 | $2.60 | $22.60 | $2.60 | $3.90 |

2 | $27.80 | $10 | $3.90 | $13.90 | $5.20 | $6.50 |

3 | $35.60 | $7 | $5.20 | $11.87 | $7.80 | $9.10 |

4 | $46.00 | $5 | $6.50 | $11.50 | $10.40 | $11.70 |

5 | $59.00 | $4 | $7.80 | $11.80 | $13.00 | $14.30 |

6 | $74.60 | $3 | $9.10 | $12.43 | $15.60 | $16.90 |

7 | $92.80 | $3 | $10.40 | $13.26 | $18.20 | $19.50 |

8 | $113.60 | $2 | $11.70 | $14.20 | $20.80 | $22.10 |

9 | $137.00 | $2 | $13.00 | $15.22 | $23.40 | $24.70 |

10 | $163.00 | $2 | $14.30 | $16.30 | $26.00 | $27.30 |

The optimal point on the ATC curve occurs when Q = 3.93. This is also not coincidentally where the MC curve intersects the ATC curve. The optimal price at this quantity is $11.52 per gallon of milk, but the firm won’t necessarily be able to set the price at that point on its own (unless it’s a monopoly; and even then, they’ll set it higher).

**IMPORTANT NOTE**: Because we’re dealing with curves and not lines, calculating marginal values with Excel by subtracting the previous value from the current value *will not be 100% accurate*. The only way to get perfectly accurate marginal values is to use calculus to find the instantaneous derivative at exactly that point.

### Finding the optimal price and quantity

The firm’s quantity decision depends on the market demand for chocolate milk. The demand curve for this market looks like this:

With this demand curve, we can find the price and quantity that would produce the maximum revenue, assuming there were no costs to production.

Quantity | Price | TR |
---|---|---|

0 | $55 | $0 |

1 | $50 | $50 |

2 | $45 | $90 |

3 | $40 | $120 |

4 | $35 | $140 |

5 | $30 | $150 |

6 | $25 | $150 |

7 | $20 | $140 |

8 | $15 | $120 |

9 | $10 | $90 |

10 | $5 | $50 |

The firm can maximize its revenue by producing 5.5 gallons of milk, which would create $151.25 in revenue.

However, this doesn’t take into account the firm’s costs. The firm’s profit maximizing point is defined as \(MC = MR\), so we need compare marginal costs and marginal revenues and calculate total profit (π) across all quantities of output.

**Again**, note that using chunky marginal values by subtracting previous values *won’t be as accurate* as calculus-based instant marginal values.

To get correct marginal values we can find the formula for the total cost function by using Wolfram Alpha. If you make a list of all the points in the quantity and total cost columns (i.e. `{{0, 20}, {1, 22.6}, {2, 27.8}}`

), you can fit a polynomial formula to those points with Wolfram Alpha.

For example, search for “quadratic fit {0, 20}, {1, 22.6}, {2, 27.8}, {3, 35.6}, {4, 46}, {5, 59}, {6, 74.6}, {7, 92.8}, {8, 113.6}, {9, 137}, {10, 163}” and you’ll see that the best formula for total cost is \(y = 1.3x^2 + 1.3x + 20\).

The marignal cost is the first derivative of the total cost, or \(y = 2.6x + 1.3\).

To find the equation for total revenue we need to multiply the demand function by Q (or x), since the formula for total revenue is \(TR = PQ\). By looking at the demand curve above, we can see that the y-intercept is 55 and the slope is −5, giving us \(y = -5x + 55\). If we multiple that by x, we get a total revenue formula of \(y = -5x^2 + 55x\) (or \(P = -5Q^2 + 55Q\)).

The marginal revenue is the first derivative of the total revenue, or \(y = -10x + 55\).

We can set MR and MC equal to each other to find the optimal x (or Q) (\(2.6x + 1.3 = -10x + 55\)), solve for x, and then plug that x into one of the total equations to find the optimal price.

The point where \(MC = MR\) can’t be seen in the table, since it happens between 4 and 5 gallons. If the firm produces 4.26 gallons of milk at a price of $12.38 per gallon, it will achieve its maximum profit of $94.43.

Quantity | Price | TR | TC | MR_chunk | MR_instant | MC_chunk | MC_instant | π |
---|---|---|---|---|---|---|---|---|

0 | $55.00 | $0.00 | $20.00 | — | $55.00 | — | $1.30 | $-20.00 |

1 | $50.00 | $50.00 | $22.60 | $50.00 | $45.00 | $2.60 | $3.90 | $27.40 |

2 | $45.00 | $90.00 | $27.80 | $40.00 | $35.00 | $5.20 | $6.50 | $62.20 |

3 | $40.00 | $120.00 | $35.60 | $30.00 | $25.00 | $7.80 | $9.10 | $84.40 |

4 | $35.00 | $140.00 | $46.00 | $20.00 | $15.00 | $10.40 | $11.70 | $94.00 |

4.262 |
$33.69 |
$143.59 |
$49.15 |
$17.38 |
$12.38 |
$11.08 |
$12.38 |
$94.43 |

5 | $30.00 | $150.00 | $59.00 | $10.00 | $5.00 | $13.00 | $14.30 | $91.00 |

6 | $25.00 | $150.00 | $74.60 | $0.00 | $-5.00 | $15.60 | $16.90 | $75.40 |

7 | $20.00 | $140.00 | $92.80 | $-10.00 | $-15.00 | $18.20 | $19.50 | $47.20 |

8 | $15.00 | $120.00 | $113.60 | $-20.00 | $-25.00 | $20.80 | $22.10 | $6.40 |

9 | $10.00 | $90.00 | $137.00 | $-30.00 | $-35.00 | $23.40 | $24.70 | $-47.00 |

10 | $5.00 | $50.00 | $163.00 | $-40.00 | $-45.00 | $26.00 | $27.30 | $-113.00 |

### Elasticity of demand

Finally, we calculated the elasticity of demand of chocolate milk. Recall the formula for elasticity:

\[ \begin{align} \varepsilon &= -\frac{\% \text{ change in demand}}{\% \text{ change in price}} \\ &= - \frac{\% \Delta Q}{\% \Delta P} \end{align} \]

Remember that \(\% \Delta Q = \frac{Q_{\text{new}} - Q}{Q}\) and that \(\% \Delta P = \frac{P_{\text{new}} - P}{P}\) (or just \(\frac{\text{new} - \text{old}}{\text{old}}\)). We can also write \(Q_{\text{new}} - Q\) as \(\Delta Q\), or just the change in \(Q\) (and also \(\Delta P\)) This means we can rewrite the equation like so:

\[ \begin{align} \varepsilon &= - \frac{\% \Delta Q}{\% \Delta P} \\ &= - \frac{\frac{Q_{\text{new}} - Q}{Q}}{\frac{P_{\text{new}} - P}{P}} \\ &= - \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} \end{align} \]

We can then simplify this huge hairy fraction by multiplying both the numerator and denominator by the inverse of the denominator, \(\frac{P}{\Delta P}\):

\[ \begin{align} \varepsilon &= - \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} \times \frac{\frac{P}{\Delta P}}{\frac{P}{\Delta P}} \\ &= - \frac{\Delta Q}{Q} \times \frac{P}{\Delta P} \\ &= - \frac{\Delta Q}{\Delta P} \times \frac{P}{Q} \end{align} \]

That’s the final version of the price elasticity of demand formula: \(\varepsilon = - \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}\). Conveniently, \(\frac{\Delta Q}{\Delta P}\) is also the slope of the demand curve.

Quantity | Price | \(\frac{\Delta Q}{\Delta P}\) | \(\frac{P}{Q}\) | ε |
---|---|---|---|---|

1 | $50 | -0.2 | 50 | 10 |

2 | $45 | -0.2 | 22.5 | 4.5 |

3 | $40 | -0.2 | 13.33 | 2.667 |

4 | $35 | -0.2 | 8.75 | 1.75 |

5 | $30 | -0.2 | 6 | 1.2 |

6 | $25 | -0.2 | 4.167 | 0.8333 |

7 | $20 | -0.2 | 2.857 | 0.5714 |

8 | $15 | -0.2 | 1.875 | 0.375 |

9 | $10 | -0.2 | 1.111 | 0.2222 |

10 | $5 | -0.2 | 0.5 | 0.1 |

Demand is elastic as long as the slope of the revenue function is positive, and demand is inelastic when the slope of revenue is negative, as seen here:

## Clearest and muddiest things

Go to this form and answer these three questions:

- What was the muddiest thing from class today? What are you still wondering about?
- What was the clearest thing from class today?
- What was the most exciting thing you learned?

I’ll compile the questions and send out answers after class.