Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

The **profit** from \(q\) units of output

\pi(q) = r(q) - c(q)

PROFIT

REVENUE

COST

is the** revenue** from selling them

minus the** cost** of producing them.

We will assume that the firm sells **all units of the good** for the same price, \(p\). (No "price discrimination")

r(q) = p(q) \times q

The **revenue** from \(q\) units of output

REVENUE

PRICE

QUANTITY

is the** price** at which each unit it sold

times the **quantity** (# of units sold).

The price the firm can charge *may depend* on the number of units it wants to sell: **inverse demand** \(p(q)\)

- Usually
**downward-sloping**: to sell more output, they need to drop their price - Special case: a
**price taker**faces a**horizontal**inverse demand curve;

can sell as much output as they like at some constant price \(p(q) = p\)

- Derive the firm's
**revenue function** - Combine
**cost**and**revenue**to determine the firm's**profit** - Find the
**profit-maximizing**level of output, \(q^*\)

Demand curve:

quantity as a function of price

Inverse demand curve:

price as a function of quantity

QUANTITY

PRICE

Demand

Inverse Demand

q(p) = 20 - p

q(p) = 20 - 2p

p(q) =

p(q) =

\text{revenue} = r(q) = p(q) \times q

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p(q)

\displaystyle{\text{average revenue} = \frac{r(q)}{q} = p(q)}

If the firm wants to sell \(q\) units, it sells all units at the same price \(p(q)\)

Since all units are sold for \(p\), the average revenue per unit is just \(p\).

By the product rule...

let's delve into this...

Demand

Inverse Demand

q(p) = 20 - p

q(p) = 20 - 2p

p(q) = 20-q

p(q) = 10 - \tfrac{1}{2}q

Revenue

r(q) =

MR(q)=

AR(q)=

r(q) =

MR(q)=

AR(q)=

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

dr = dp \times q + dq \times p

p

p(q)

q

The total revenue is the price times quantity (area of the rectangle)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

dr = dp \times q + dq \times p

p

p(q)

q

dp

dq

Note: \(MR < 0\) if

dq \times p

{dq \over q} < {dp \over p}

\% \Delta q < \% \Delta p

|\epsilon| < 1

dp \times q

<

The total revenue is the price times quantity (area of the rectangle)

If the firm wants to sell \(dq\) more units, it needs to drop its price by \(dp\)

Revenue loss from lower price on existing sales of \(q\): \(dp \times q\)

Revenue gain from additional sales at \(p\): \(dq \times p\)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

= \left[{dp \over dq} \times {q \over p}\right] \times p + p

= {p \over {dq \over dp} \times {p \over q}} + p

= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by \(p/p\))

(simplify)

(since \(\epsilon < 0\))

Notes

Elastic demand: \(MR > 0\)

Inelastic demand: \(MR < 0\)

In general: the more elastic demand is, the less one needs to lower ones price to sell more goods, so the closer \(MR\) is to \(p\).

Demand curve:

quantity as a function of price

Inverse demand curve:

price as a function of quantity

QUANTITY

PRICE

Special case: perfect substitutes

Demand curve:

quantity as a function of price

Inverse demand curve:

price as a function of quantity

QUANTITY

PRICE

Special case: perfect substitutes

For a small firm, it probably looks like this...

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

= \left[{dp \over dq} \times {q \over p}\right] \times p + p

= {p \over {dq \over dp} \times {p \over q}} + p

= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by \(p/p\))

(simplify)

(since \(\epsilon < 0\))

Note

Perfectly elastic demand: \(MR = p\)

\pi(q) = r(q) - c(q)

Optimize by taking derivative and setting equal to zero:

\pi'(q) = r'(q) - c'(q) = 0

\Rightarrow r'(q) = c'(q)

Profit is total revenue minus total costs:

"Marginal revenue equals marginal cost"

Example:

p(q) = 20 - q \Rightarrow r(q) = 20q - q^2

c(q) = 64 + {1 \over 4}q^2

What is the profit-maximizing value of \(q\)?

p(q) = 20 - q \Rightarrow r(q) = 20q - q^2

c(q) = 64 + {1 \over 4}q^2

What is the profit-maximizing value of \(q\)?

\pi(q) = r(q) - c(q)

Multiply right-hand side by \(q/q\):

\pi(q) = \left[{r(q) \over q} - {c(q) \over q}\right] \times q

= (AR - AC) \times q

Profit is total revenue minus total costs:

"Profit per unit times number of units"

AVERAGE PROFIT

CHECK YOUR UNDERSTANDING

p(q)=20-2q

c(q)=10+5q+{1 \over 2}q^2

Find the profit-maximizing quantity.

MR = p\left[1 - \frac{1}{|\epsilon_{q,p}|}\right]

Recall our **elasticity** representation of marginal revenue:

MR = MC

Let's combine it with this

**profit maximization condition:**

p\left[1 - \frac{1}{|\epsilon_{q,p}|}\right] = MC

p = \frac{MC}{1 - \frac{1}{|\epsilon_{q,p}|}}

Really useful if MC and elasticity are both constant!

**Inverse elasticity pricing rule:**

p = \frac{MC}{1 - \frac{1}{|\epsilon_{q,p}|}}

Find the optimal price and quantity if a firm has the cost function $$c(q) = 200 + 4q$$ and faces the demand curve $$D(p) = 6400p^{-2}$$

**Inverse elasticity pricing rule:**

P = \frac{MC}{1 - \frac{1}{|\epsilon|}}

1 - \frac{1}{|\epsilon|} = \frac{MC}{P}

\frac{P - MC}{P} = \frac{1}{|\epsilon|}

Fraction of price that's markup over marginal cost

(Lerner Index)

What if \(|\epsilon| \rightarrow \infty\)?

- Lots of "small" firms selling basically the same thing

- One or a few "medium" or "large" firms selling differentiated products

- Firms face essentially
**horizontal**demand curve

- Firms face
**downward sloping**demand curve

\pi(q) = r(q) - c(q)

\pi(q) = p \times q - c(q)

Optimize by taking derivative and setting equal to zero:

\pi'(q) = p - c'(q) = 0

\Rightarrow p = c'(q)

Profit is total revenue minus total costs:

For a **price taker** (or** competitive firm**), revenue equals price times quantity:

"Price equals

marginal cost"

c(q) = 64 + {1 \over 4}q^2

r(q) = pq = 16q

\text{Let's assume }p = 16:

\text{Profit function is:}

\text{Total cost}

\pi(q) = 16q - [64 + {1 \over 4}q^2]

\text{Take derivative with respect to } q \text{ and set equal to zero:}

\pi'(q) = \ \ \ \ \ \ - \ \ \ \ \ \ = 0

16

{1 \over 2}q

Price

MC

\(q\)

$/unit

P = MR

16

MC = {1 \over 2}q

32

\Rightarrow q^* = 32

- All firms maximize profits by setting MR = MC
- If a firm faces a downward-sloping demand curve,

the marginal revenue is less than the price. - The more elastic a firm's demand curve,

the less it will optimally raise its price above marginal cost. - A competitive firm faces a perfectly elastic demand curve,

so its marginal revenue is equal to the price. - Next time: establish a competitive firm's
**output supply**and**labor demand**as functions of \(p\) and \(w\)