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
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^*(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

\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}

[50Q only]

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?

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}

MRTS for Different Production Functions

aL + bK

Linear

Cobb-Douglas

AL^aK^b

CES

(aL^\rho + bK^\rho)^{1 \over \rho}
f(L,K)
MRTS(L,K)
\frac{a}{b}
\frac{a}{b}\times \frac{K}{L}
\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

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

\mathcal{L}(L,K,\lambda)=
wL + rK +
(q - f(L,K))
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = 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:

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) = 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.