A Competitive Firm's Supply Curve

Christopher Makler

Stanford University Department of Economics

Econ 50 | Lecture 20

Optimization

What is an agent's optimal behavior for a fixed set of circumstances? 

x_1^*
x_2^*
q^*
L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

Comparative Statics

How does an agent's optimal behavior change when circumstances change? 

x_1^*
x_2^*
q^*
L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

x_1^*(p_1\ |\ p_2,m)
x_2^*(p_2\ |\ p_2,m)
q^*(p\ |\ w)
L^*(w\ |\ p)

Assumptions

We will be analyzing a
competitive (price-taking) firm 
in the short run

  • output price \(p\)
  • wage rate \(w\)

This Week's Agenda

Review of firm profit maximization, with and without market power

Edge cases

Deriving output supply
as a function of \(p\),
holding \(w\) constant

Deriving labor demand
as a function of \(w\),
holding \(p\) constant

Analyzing how a change in \(w\)
shifts the supply curve

Analyzing how a change in \(p\)
shifts the labor demand curve

[ MOVEMENT
ALONG CURVES]

[ SHIFTS OF
 CURVES]

Wednesday

Friday

Monopsony

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

Review: Profit Maximization

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

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

r(q) = 20q - q^2
\pi(q) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -

Optimize by taking derivative and setting equal to zero:

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

Profit is total revenue minus total costs:

"Marginal revenue equals marginal cost"

(64 + {1 \over 4}q^2)
(20q - q^2)
r'(q)
c'(q)
{1 \over 2}q
20 - 2q
r'(q)
c'(q)
{1 \over 2}q
20 - 2q
40 - 4q = q
40 = 5q
q^* = 8

Average Profit Analysis

\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

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:

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

Competitive (Price-Taking) Firms

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...

c(q) = 64 + {1 \over 4}q^2
r(q) = pq = 12q
\text{Let's assume }p = 12:
\text{Profit function is:}
\text{Total cost}
\pi(q) = 12q - [64 + {1 \over 4}q^2]
\text{Take derivative with respect to } q \text{ and set equal to zero:}
\pi'(q) = \ \ \ \ \ \ - \ \ \ \ \ \ = 0
12
{1 \over 2}q

Price

MC

\(q\)

$/unit

P = MR

12

MC = {1 \over 2}q

24

\Rightarrow q^* = 24

Output Supply as a Function of \(p\) with Fixed \(w\)

When price is fixed at 12

For a general price

1. Costs and Revenues

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = 12q - (64 + \tfrac{1}{4}q^2)
\pi^\prime(q) = 12 - {1 \over 2}q = 0
r(q) = pq
\pi(q) = pq - (64 + \tfrac{1}{4}q^2)
\pi^\prime(q) = p - {1 \over 2}q = 0
q^* = 24
q^*(p) = 2p

Profit-Maximizing Output Choice when \(w = 8\), \(r = 2\), and \(\overline K = 32\)

c(q) = wL(q) + r\overline K = {1 \over 4}q^2 + 64
r(q) = pq = 12q
c(q) = wL(q) + r\overline K = {1 \over 4}q^2 + 64

NUMBER

FUNCTION

f(L,K) = \sqrt{LK}, \overline K = 32
64 + {1 \over 4}q^2
{1 \over 2}q
pq
\pi(q) = \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\pi^\prime(q) = \ \ \ - \ \ \ \ \ \ \ \ = 0
p

TR

TC

MR

MC

Take derivative and set = 0:

Solve for \(q^*\):

q^*(p) = 2p

SUPPLY FUNCTION

When \(p = 4\), this function says that the firm should produce \(q = 8\).
If it does this...

When \(p = 12 , w = 8, \overline K = 32\) 

For a general \(p, w\) and \(\overline K\)

1. Costs and Revenues

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = 12q - (\tfrac{1}{4}q^2 + 64)
\pi^\prime(q) = 12 - {1 \over 2}q = 0
r(q) = pq
\pi(q) = pq - (\tfrac{w}{\overline K}q^2 + r\overline K)
\pi^\prime(q) = p - {2w \over \overline K}q = 0
q^* = 24
q^*(p\ | w, \overline K) = {\overline K p \over 2w}

Now's let's also let wage and capital be a variable

c(q) = {1 \over 4}q^2 + 64
r(q) = pq = 12q
c(q) = w \times {q^2 \over \overline K} + r\overline K

NUMBER

FUNCTION

c(q) = wL(q) + r \overline K

1. Costs for general \(w\) and revenue for general \(p\)

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = pq - [wL(q) + r\overline K]
\pi^\prime(q) = p - w \times {dL \over dq} = 0
p = w \times {1 \over MP_L}
\text{Profit }\pi = pq - [wL + rK]
r(q) = p \times q

MARGINAL COST (MC)

"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it." 

p = MC

Edge Cases

Edge Case 1:

Multiple quantities where P = MC

Edge Case 2:

Corner solution at \(q = 0\)

"The supply curve is the portion of the MC curve above minimum average variable cost"

Other Edge Cases to
Watch Out for On Exams

  • Discontinuities
  • Capacity constraints
  • Quantities produced with capital
  • Don't just trust formulas —
    perform a gut check!

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.
  • A firm's supply curve shows its optimal quantity to produce as a function of the price at which it can sell the good