Profit Maximization
Christopher Makler
Stanford University Department of Economics
Econ 50 : Lecture 14
pollev.com/chrismakler
When does the production function
f(L) = 10L^a
exhibit diminishing marginal product of labor?
Diminishing marginal product:
as you use more of a unit, holding other inputs constant,
each additional unit produces less additional output.
\frac{\partial MP_L}{\partial L} < 0
\text{Check: when does }f(L,K) = L^aK^b \text{ have diminishing }MP_L?
MP_L = \frac{\partial f(L,K)}{\partial L} = aL^{(a-1)}K^b
\frac{\partial MP_L}{\partial L} = a(a-1)L^{(a - 2)}K^b
This has the same sign as \((a-1)\); so it's negative when \(a < 1\).
Today's Agenda
- Wrap up discussion of long-run vs. short-run costs from last time
- Revenue as a function of quantity
- Profit as a function of quantity
- Punting elasticity discussion to next time...
Friday
Monday
Wednesday
Cost Minimization for Consumers
Cost Minimization for Firms
Short-Run and Long-Run Costs; Cost Curves
Profit Maximization for a Firm
A Competitive Firm's
Supply Curve
Market Supply and Demand
Midterm II
Market Equilibrium in the Long Run
Market Equilibrium in the Short Run
Unit II: Firms and Markets
Decomposing a Price Change into Income & Substitution Effects
Week 4
Week 5
Week 6
Week 7
Efficiency of Markets: Consumer & Producer Surplus
Midterm I
Theory of the Firm
Labor
Firm
🏭
Capital
⏳
⛏
Customers
🤓
p
w
r
q
L
K
Firms buy inputs
and produce some good,
which they sell to a customer.
PRICE
QUANTITY
Labor
w
L
Capital
r
K
Output
p
q
Firm
🏭
Costs
wL + rK
Revenue
pq
Theory of the Firm
Firms buy inputs
and produce some good,
which they sell to a customer.
PRICE
QUANTITY
Labor
w
L
Capital
r
K
Output
p
q
Costs
wL + rK
Revenue
pq
Profit
\pi = pq - (wL + rK)
The difference between their revenue and their cost is what we call profits, denoted by the Greek letter \(\pi\).
Theory of the Firm
Costs
c(q) = wL(q) + rK(q)
Revenue
r(q) = p(q) \times q
Profit
\pi(q) = r(q) - c(q)
Theory of the Firm
Our approach will be to write costs, revenues, and profits all as functions of the ouput \(q\).
Long-Run vs. Short-Run
Highly Precise and Technical Definitions
Long Run
Short Run
Something is fixed
That thing isn't fixed
In the context we're talking about today:
Capital is fixed at some amount \(\overline K\),
labor is variable
All inputs are variable
c^{LR}(q) = wL^c(w,r,q) + rK^c(w,r,q)
Conditional demand for labor
Conditional demand for capital
TC(q\ |\ \overline{K}) = wL(q\ |\ \overline{K}) + r\overline{K}
Variable cost
Fixed cost
Cost of the lowest-cost way of producing \(q\) units if you can vary both labor and capital
Cost of producing \(q\) units if your capital is fixed, so you can only scale production by adding labor
Costs of Production
Long Run (can vary both labor and capital)
Short Run with Capital Fixed at \(\overline K \)
- Increasing marginal product:
MPL is increasing in L - Constant marginal product:
MPL is constant in L - Diminishing marginal product:
MPL is decreasing in L
Scaling Production
- Increasing returns to scale (IRS):
doubling all inputs more than doubles output. - Constant returns to scale (CRS):
doubling all inputs exactly doubles output. - Decreasing returns to scale (DRS):
doubling all inputs less than doubles output.
Long Run (can vary both labor and capital)
Short Run with Capital Fixed at \(\overline K \)
f(L,K) = \sqrt{LK}
L^c(q) = \sqrt{\frac{r}{w}}q
K^c(q) = \sqrt{\frac{w}{r}}q
TC^{LR}(q) = wL^c(q) + rK^c(q)
=2\sqrt{wr}q
L^c(q | \overline K) = {q^2 \over \overline K}
TC(q) = wL^c(q | \overline K) + r \overline K
={wq^2 \over \overline K} + r \overline K
CONSTANT RETURNS TO SCALE => CONSTANT MC (LINEAR FUNCTION)
DIMINISHING MPL =>
INCREASING MC (CONVEX FUNCTION)
Long Run (can vary both labor and capital)
Short Run with Capital Fixed at \(\overline K \)
f(L,K) = \sqrt{LK}
Long Run (can vary both labor and capital)
TC^{LR}(q)=2\sqrt{wr}q
Short Run with Capital Fixed at \(\overline K \)
TC(q)={wq^2 \over \overline K} + r \overline K
Let's fix \(w= 8\), \(r = 2\), and \(\overline K =32\)
TC^{LR}(q)=8q
TC(q)=64 + {1 \over 4}q^2
Relationship between
Short-Run and Long-Run Costs
TC(q | \overline{K}) = \text{Cost if capital is fixed at }\overline{K}
TC(q,K^c(q)) = \text{Cost if capital is fixed at optimal }K \text{ for producing }q
What conclusions can we draw from this?
TC(q,\overline{K}) \ge TC^{LR}(q)
= TC^{LR}(q)
TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ curves}
Total, Fixed and Variable Costs
c(q\ |\ \overline{K}) =
Fixed Costs \((F)\): All economic costs
that don't vary with output.
Variable Costs \((VC(q))\): All economic costs
that vary with output
explicit costs (\(r \overline K\)) plus
implicit costs like opportunity costs
r\overline{K}
wL(q,\overline{K})
+
TC(q) =
+
F
VC(q)
e.g. cost of labor required to produce
\(q\) units of output given \(\overline K\) units of capital
TC(q) = {wq^2 \over \overline K} + r\overline K
VC(q) = {wq^2 \over \overline K}
F = r\overline K
pollev.com/chrismakler
Generally speaking, if capital is fixed in the short run, then higher levels of capital are associated with _______ fixed costs and _______ variable costs for any particular target output.
\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{q}
Fixed Costs
Variable Costs
Average Fixed Costs (AFC)
Average Variable Costs (AVC)
Average Costs
\text{Total Cost}: TC(q) = F + VC(q)
\text{Average Cost}: ATC(q) = \frac{F}{q} + \frac{VC(q)}{q}
Average Costs
TC(q) = 64 + {1 \over 4}q^2
ATC(q) = {64 \over q} + {1 \over 4}q
\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q) = 0 + VC'(q)
Fixed Costs
Variable Costs
Marginal Cost
(marginal cost is the marginal variable cost)
\text{Total Cost}: TC(q) = F + VC(q)
\text{Marginal Cost}: MC(q) = TC'(q)
Marginal Cost
TC(q) = 64 + {1 \over 4}q^2
MC(q) = {1 \over 2}q
Relationship between Marginal Cost and Marginal Product of Labor
TC(q) = wL^c(q | \overline K) + r \overline K
{dTC(q) \over dq} = w \times {dL^c(q) \over dq}
= w \times {1 \over MP_L}
Revenue
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 \(p(q) = p\)
Demand and Inverse Demand
Demand curve:
quantity as a function of price
Inverse demand curve:
price as a function of quantity
QUANTITY
PRICE
\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
\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
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 \(p\): \(dq \times p\)
Demand
Inverse Demand
q(p) = 20 - p
p(q) = 20-q
Revenue
r(q) =
MR(q)=
AR(q)=
20q - q^2
20 - 2q
20 - q
pollev.com/chrismakler
Suppose instead that the firm faced the demand function
(not inverse demand!)
\(q(p) = 20 - 2p\).
What would their marginal revenue function \(MR(q)\) be?
Correctness matters on this one...
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)=
20q - q^2
20 - 2q
20 - q
10q - \tfrac{1}{2}q^2
10 - q
10 - \tfrac{1}{2}q
Profit
\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
CHECK YOUR UNDERSTANDING
p(q)=20-2q
c(q)=10+5q+{1 \over 2}q^2
Find the profit-maximizing quantity.
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
Next Time
- How does demand elasticity affect a firm's ability to mark up its price above marginal cost?
- Extreme example: a firm facing perfectly elastic demand (i.e., a "price taker" or "competitive firm")
- Derive supply curve
Econ 50 | Spring 25 | Lecture 14
By Chris Makler
Econ 50 | Spring 25 | Lecture 14
Production and Cost for a Firm
