Firm Production Functions and Cost Minimization

Christopher Makler

Stanford University Department of Economics

Econ 50 : Lecture 14

From Lecture I: Two Kinds of Optimization

Tradeoffs between two goods

Optimal quantity of one good

🍎

(not feasible)

(feasible)

🍌

Optimal choice

🙂

😀

😁

😢

🙁

🍎

benefit and cost per unit

Marginal Cost

Marginal Benefit

Optimal choice

Tradeoffs between two goods

Optimal quantity of one good

Checkpoint 1: October 13

Model 1: Consumer Choice

Model 2: Theory of the Firm

Checkpoint 2: October 27

WEEK 1

WEEK 2

WEEK 3

Modeling preferences with multivariable calculus

Constrained optimization when calculus works

Constrained optimization when calculus doesn't work

WEEK 4

WEEK 5

Consumer Demand

Application: Financial Economics

Checkpoint 3: November 10

Final Exam: December 12 (cumulative)

WEEK 6

WEEK 7

WEEK 8

Production and Costs for a Firm

Profit Maximization

Short-Run Equilibrium

WEEK 9

WEEK 10

Long-Run Equilibrium

Applications: Public Economics

Model 3: Market Equilibrium

Theory of the Firm

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)

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 \(\pi\).

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 \(q\).

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 \(q\).

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.

\pi'(q) = r'(q) - c'(q) = 0
r'(q) = c'(q)

MR

MC

Output Supply

q^*(w,r,p)

Theory of the Firm

Finally, we will solve for a competitive (price-taking) firm's optimal output and inputs as functions of the price of output \((p)\) and inputs (\(w\) and \(r\)).

Profit-Maximizing Input Demands

L^*(w,r,p)
K^*(w,r,p)

Theory of the Firm

L^*(w,r,q)
K^*(w,r,q)

Exogenous Variables

Endogenous Variables

technology, \(f(L,K)\)

level of output, \(q\)

input prices \(w, r\)

Cost Minimization

Profit Maximization

cost function, \(c(w,r,q)\)

revenue function \(r(q)\)

\text{Optimal Output }q^*

Special Case: Competitive Firm

L^*(w,r,q)
K^*(w,r,q)

Exogenous Variables

Endogenous Variables

technology, \(f(L,K)\)

level of output, \(q\)

input prices \(w, r\)

Cost Minimization

Profit Maximization

cost function, \(c(w,r,q)\)

market price \(p\)

\text{Supply }q^*(w,r,p)
L^*(w,r,p)
K^*(w,r,p)

Friday

Monday

Wednesday

Firm Production Functions and Cost Minimization

Profit Maximization

Input and Output Decisions of a Competitive Firm

Unit III: Theory of the Firm

Week 6

Week 7

Checkpoint II

Cost Functions
and Cost Curves

Elasticity and Market Power:
From Monopoly to Competition

Monday

Checkpoint III

Week 8

RECORDED - NO LIVE CLASS

Today: Production Functions and Cost Minimization

Production Functions

The Firm's Cost-Minimimization Problem

(exactly like utility functions,
but the level matters)

(flipping the script on utility maximization)

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 (\(q\))

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

\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

\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.

\displaystyle{= \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}}
\displaystyle{MP_K = {\partial f(L,K) \over \partial 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) = 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

\displaystyle{MRTS = {MP_L \over MP_K}}
\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

Linear

f(L,K) = \min\{aL, bK\}

Leontief
(Fixed Proportions)

Cobb-Douglas

f(L,K) = AL^aK^b

Constant Elasticity of Substitution (CES)

f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho}

Examples of Production Functions

f(L,K) = aL + bK

Linear

f(L,K) = \min\{aL, bK\}

Leontief
(Fixed Proportions)

Cobb-Douglas

f(L,K) = AL^aK^b

Constant Elasticity of Substitution (CES)

f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho}

We will only be using Cobb-Douglas this quarter.

Cobb-Douglas Production Function

q=f(L,K)=AL^aK^b
MP_L = aAL^{a-1}K^b
MP_K = bAL^aK^{b-1}

What story do these marginal products tell us?

When do these functions have diminishing marginal products?

Analyzing Production Functions Two Ways

Along an isoquant:
elasticity of substitution

Between isoquants:
returns to scale

Why do economists use different functional forms?

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)=(aL^\rho + bK^\rho)^{1 \over \rho}

Returns to Scale

What happens when we increase all inputs proportionally?

For example, what happens if we double both labor and capital?

Does doubling inputs -- i.e., getting \(f(2L,2K)\) -- double output?

f(2L,2K) > 2f(L,K)
f(2L,2K) = 2f(L,K)
f(2L,2K) < 2f(L,K)

Decreasing Returns to Scale

Constant Returns to Scale

Increasing Returns to Scale

f(L,K) = 4L^{1 \over 2}K

Does this exhibit decreasing, constant or increasing returns to scale?

f(2L,2K) = 4(2L)^{1 \over 2}(2K)
= 2^{3 \over 2} \times 4L^{1 \over 2}K
= 2^{3 \over 2} \times f(L,K)
> 2 \times f(L,K)

Increasing returns to scale

pollev.com/chrismakler

When does the production function

f(L,K) = AL^aK^b

exhibit constant returns to scale?

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

Utility Maximization Subject to a Budget Constraint

Cost Minimization Subject to an Output Constraint

\max u(x_1,x_2) \text{ s.t. } p_1x_1 + p_2x_2 = m
\min wL + rK \text{ s.t. } f(L,K) = q
\text{solutions}: x_1^*(p_1,p_2,m), x_2^*(p_1,p_2,m)
\text{solutions}: L^c(w,r,q), K^c(w,r,q)

Conditional Demand Functions

Demand Functions

on \(q\)

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

\mathcal{L}(L,K,\lambda)=
wL + rK +
(q - f(L,K))
\lambda
\frac{\partial \mathcal{L}}{\partial L} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial K} = r - \lambda MP_K
\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)
= 0 \Rightarrow \lambda = w \times {1 \over MP_L}
= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}

MRTS (slope of isoquant) is equal to the price ratio

\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}
\text{Set two expressions }
\text{for }\lambda \text{ equal:}
\frac{MP_L}{MP_K} = \frac{w}{r}
f(L,K) = \sqrt{LK}

Tangency condition: \(MRTS = w/r\)

Constraint: \(q = f(L,K)\)

MRTS = {MP_L \over MP_K} =
{{1 \over 2}L^{-{1 \over 2}}K^{1 \over 2} \over {1 \over 2}L^{1 \over 2}K^{-{1 \over 2}}}
= {K \over L}

Conditional demands for labor and capital:

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

Total Cost of \(q\) Units

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