Profit Maximization

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

Profit

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.

Revenue

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 \(p(q) = p\)

Today's Agenda

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

Revenue

Demand and Inverse Demand

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...

Total, Average, and Marginal Revenue

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

Marginal Revenue and Elasticity

= \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 and Inverse Demand

Demand curve:

quantity as a function of price

Inverse demand curve:
price as a function of quantity

QUANTITY

PRICE

Special case: perfect substitutes

Demand and Inverse Demand

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

Marginal Revenue for Perfectly Elastic Demand

= \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: \(MR = p\)

Profit

\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.

Elasticity and Profit Maximization

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:

One more way of slicing it...

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

Special Case: Price Takers

Competition

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

Market Power

  • 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

Summary

  • 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 \(w\)

Econ 50 | Winter 2022 | 12 | Profit Maximization

By Chris Makler

Econ 50 | Winter 2022 | 12 | Profit Maximization

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