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
Econ 50 | Lecture 20
By Chris Makler
Econ 50 | Lecture 20
Output Supply and Labor Demand for a Competitive Firm
