Exam Monday
- Non-OAE: Everyone in Dinkelspiel Auditorium
- OAE: starts at 10:30; if this doesn't work for you
because you have a class immediately after this one,
please respond to Paige Vasek's email right now. - Review session Sunday 6-8, requested Shriram104.
Production and Cost
Christopher Makler
Stanford University Department of Economics
Econ 50 : Lecture 12
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
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
Cost Minimization for Firms
Short-Run and Long-Run Costs; Cost Curves
Profit Maximization for a Firm
A Competitive Firm's
Supply Curve
Theory of the Firm
Theory of the Firm
Labor
Firm
🏭
Capital
⏳
⛏
Customers
🤓
p
p
w
w
r
r
q
q
L
L
K
K
Firms buy inputs
and produce some good,
which they sell to a customer.
PRICE
QUANTITY
Labor
w
w
L
L
Capital
r
r
K
K
Output
p
p
q
q
Firm
🏭
Costs
wL+rK
wL + rK
Revenue
pq
pq
Theory of the Firm
Firms buy inputs
and produce some good,
which they sell to a customer.
PRICE
QUANTITY
Labor
w
w
L
L
Capital
r
r
K
K
Output
p
p
q
q
Costs
wL+rK
wL + rK
Revenue
pq
pq
Profit
π=pq−(wL+rK)
\pi = pq - (wL + rK)
Next week: Solve the optimization problem
finding the profit-maximizing quantity q∗
The difference between their revenue and their cost is what we call profits, denoted by the Greek letter π.
Theory of the Firm
Costs
c(q)=wL(q)+rK(q)
c(q) = wL(q) + rK(q)
Revenue
r(q)=p(q)×q
r(q) = p(q) \times q
Profit
π(q)=r(q)−c(q)
\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.
Profit
π(q)=r(q)−c(q)
\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.
We will then just take the derivative with respect to q
and set it equal to zero to find the firm's profit-maximizing quantity.
π′(q)=r′(q)−c′(q)=0
\pi'(q) = r'(q) - c'(q) = 0
r′(q)=c′(q)
r'(q) = c'(q)
MR
MC
Output Supply
q∗(p)
q^*(p)
Theory of the Firm
Finally, we will solve for a competitive (price-taking) firm's optimal output as a function of the market price.
Today: Production and Costs
Production Functions
The Firm's Cost Function
The Firm's Cost-Minimimization Problem
(exactly like utility functions)
(exactly like the consumer's expenditure function from last time)
(exactly like the consumer's cost minimization problem from last time)
Really, no new math, just new context.
Production Functions
A mathematical form describing how much output is produced as a function of inputs.
Labor (L)
Capital (K)
Production Function f(L,K)
Output (q or x)
Isoquants
Economic definition: if you want to produce some amount q of output, what combinations of inputs could you use?
Mathematical definition:
level sets of the production function
Isoquant for q={(L,K) ∣ f(L,K)=q}
\text{Isoquant for }q = \{(L,K)\ |\ f(L,K) = q\}
Isoquant: combinations of inputs that produce a given level of output
Isoquant map: a contour map showing the isoquants for various levels of output
pollev.com/chrismakler
What happens to isoquants after an improvement in technology?
Marginal Products of Labor and Capital
Economic definition: how much more output is produced if you increase labor or capital?
Mathematical definition:
partial derivatives of the production function
MPL=∂L∂f(L,K)
\displaystyle{MP_L = {\partial f(L,K) \over \partial L}}
These are both rates: they are measured in terms of units of ouptut per unit of input.
=ΔL→0limΔLf(L+ΔL,K)−f(L,K)
\displaystyle{= \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}}
MPK=∂K∂f(L,K)
\displaystyle{MP_K = {\partial f(L,K) \over \partial K}}
=ΔK→0limΔKf(L,K+ΔK)−f(L,K)
\displaystyle{=\lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}}
pollev.com/chrismakler
Consider the production function
f(L,K)=4L21K
f(L,K) = 4L^{1 \over 2}K
What is the expression for the marginal product of labor?
Marginal Rate of Technical Substitution (MRTS)
Economic definition:
the rate at which a producer
can substitute one input for another
while keeping output at the same level
Visual definition:
slope of an isoquant
Mathematical definition:
we'll get to this in section and on Friday
Marginal Rate of Technical Substitution (MRTS)
Economic definition: the rate at which a producer can substitute one input for another while keeping output at the same level
Mathematical definition: slope of an isoquant
Recall: by implicit function theorem,
the slope of a level set is given by
MRTS=MPKMPL
\displaystyle{MRTS = {MP_L \over MP_K}}
dxdyf(x,y)=z=−∂f/∂y∂f/∂x
\displaystyle{\left.{dy \over dx} \right|_{f(x,y) = z} = -{\partial f/\partial x \over \partial f/\partial y}}
Therefore the formula for the MRTS is
(absolute value)
Functional Forms
Examples of Production Functions
f(L,K)=aL+bK
f(L,K) = aL + bK
Linear
f(L,K)=min{aL,bK}
f(L,K) = \min\{aL, bK\}
Leontief
(Fixed Proportions)
Cobb-Douglas
f(L,K)=ALaKb
f(L,K) = AL^aK^b
Constant Elasticity of Substitution (CES)
f(L,K)=(aLρ+bKρ)ρ1
f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho}
[50Q only]
Cobb-Douglas Production Function
q=f(L,K)=ALaKb
q=f(L,K)=AL^aK^b
MPL=aALa−1Kb
MP_L = aAL^{a-1}K^b
MPK=bALaKb−1
MP_K = bAL^aK^{b-1}
What story do these marginal products tell us?
When do these functions have diminishing marginal products?
Elasticity of Substitution
- Measures the substitutability of one input for another
- Key to answering the question: "will my job be automated?"
- Formal definition: the inverse of the percentage change in the MRTS
per percentage change in the ratio of capital to labor, K/L - Intuitively: how "curved" are the isoquants for a production function?
CES Production Function
q=f(L,K)=(aLρ+bKρ)ρ1
q=f(L,K)=(aL^\rho + bK^\rho)^{1 \over \rho}
MRTS for Different Production Functions
aL+bK
aL + bK
Linear
Cobb-Douglas
ALaKb
AL^aK^b
CES
(aLρ+bKρ)ρ1
(aL^\rho + bK^\rho)^{1 \over \rho}
f(L,K)
f(L,K)
MRTS(L,K)
MRTS(L,K)
ba
\frac{a}{b}
ba×LK
\frac{a}{b}\times \frac{K}{L}
ba×(LK)1−ρ
\frac{a}{b}\times \left(\frac{K}{L}\right)^{1-\rho}
Production Functions and Utility Functions
Partial Derivative
Marginal Utility
Level Set
Isoquant
Slope of a
Level Set
Marginal Rate of Technical Substitution
MATH
UTILITY
PRODUCTION
Marginal Product
Indifference Curve
Marginal Rate of Substitution
Cost Minimization
Cost Minimization Subject to a Utility Constraint
Cost Minimization Subject to an Output Constraint
minp1x1+p2x2 s.t. u(x1,x2)=U
\min p_1x_1 + p_2x_2 \text{ s.t. } u(x_1,x_2) = U
minwL+rK s.t. f(L,K)=q
\min wL + rK \text{ s.t. } f(L,K) = q
solutions:x1c(p1,p2,U),x2c(p1,p2,U)
\text{solutions}: x_1^c(p_1,p_2,U), x_2^c(p_1,p_2,U)
solutions:Lc(w,r,q),Kc(w,r,q)
\text{solutions}: L^c(w,r,q), K^c(w,r,q)
Hicksian Demand
Conditional Demand
pollev.com/chrismakler
If labor is shown on the horizontal axis and capital is shown on the vertical axis, what is the magnitude of the slope of the isocost line, and what are its units?
Cost Minimization: Lagrange Method
L(L,K,λ)=
\mathcal{L}(L,K,\lambda)=
wL+rK+
wL + rK +
(q−f(L,K))
(q - f(L,K))
λ
\lambda
∂x1∂L=w−λMPL
\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L
First Order Conditions
∂x2∂L=r−λMPK
\frac{\partial \mathcal{L}}{\partial x_2} = r - \lambda MP_K
∂λ∂L=q−f(L,K)=0⇒q=f(L,K)
\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)
=0⇒λ=w×MPL1
= 0 \Rightarrow \lambda = w \times {1 \over MP_L}
=0⇒λ=r×MPK1
= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}
MRTS (slope of isoquant) is equal to the price ratio
Also: ∂q∂L=λ=Marginal cost of producing last unit using either input
\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}
Set two expressions
\text{Set two expressions }
for λ equal:
\text{for }\lambda \text{ equal:}
MPKMPL=rw
\frac{MP_L}{MP_K} = \frac{w}{r}
f(L,K)=LK
f(L,K) = \sqrt{LK}
Tangency condition: MRTS=w/r
Constraint: q=f(L,K)
MRTS=MPKMPL=
MRTS = {MP_L \over MP_K} =
21L21K−2121L−21K21
{{1 \over 2}L^{-{1 \over 2}}K^{1 \over 2} \over {1 \over 2}L^{1 \over 2}K^{-{1 \over 2}}}
=LK
= {K \over L}
Conditional demands for labor and capital:
Total cost of producing q units of output:
Expansion Path
A graph connecting the input combinations a firm would use as it expands production: i.e., the solution to the cost minimization problem for various levels of output
Exactly the same as the income offer curve (IOC) in consumer theory.
(And, if the optimum is found via a tangency condition, exactly the same as the tangency condition.)
Total Cost of q Units
c(q)=wLc(w,r,q)+rKc(w,r,q)
c(q) = wL^c(w,r,q) + rK^c(w,r,q)
Conditional demand for labor
Conditional demand for capital
"The total cost of producing q units
is the cost of the cost-minimizing combination of inputs
that can produce q units of output."
Exactly the same as the expenditure function in consumer theory.
Next Time
- Scaling production in the short run and long run
- Short-run vs. long run costs
- Cost curves
Exam Monday
- Non-OAE: Everyone in Dinkelspiel Auditorium
- OAE: starts at 10:30; if this doesn't work for you
because you have a class immediately after this one,
please respond to Paige Vasek's email right now. - Review session Sunday 6-8, requested Shriram104.
Exam Monday Non-OAE: Everyone in Dinkelspiel Auditorium OAE: starts at 10:30; if this doesn't work for you because you have a class immediately after this one, please respond to Paige Vasek's email right now . Review session Sunday 6-8, requested Shriram104.
Econ 50 | Spring 25 | Lecture 12
By Chris Makler
Econ 50 | Spring 25 | Lecture 12
Production and Cost
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