Cost Minimization

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 11

Today's Agenda

  • Cost Minimization
  • Compensated Demand Functions
  • Indirect Utility and Expenditure Functions

The Tangency Condition

Cost Minimization

Utility Maximization

Cost Minimization

\max \ u(x_1,x_2)
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U)
x_2^c(p_1,p_2,U)

Utility Maximization

\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m

Utility Maximization

\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m

Plug tangency condition
into constraint:

\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
p_1x_1 + p_2 \left[ {p_1 \over p_2} x_1 \right] = m
p_1x_1 + p_1x_1 = m
x_1^*(p_1,p_2,m) = {m \over 2p_1}

Plug \(x_1^*\) back into tangency condition:

x_2^*(p_1,p_2,m) = {p_1 \over p_2} \left [{m \over 2p_1}\right] = {m \over 2p_2}

Marshallian (ordinary) demand functions

Cost Minimization

\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }x_1x_2 = U

Cost Minimization

\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }x_1x_2 = U

Plug tangency condition
into constraint:

\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
x_1 \left[ {p_1 \over p_2} x_1 \right] = U
{p_1 \over p_2} \times x_1^2 = U
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}

Plug \(x_1^*\) back into tangency condition:

x_2^*(p_1,p_2,U) = {p_1 \over p_2} \left [\sqrt{p_2U \over p_1}\right] = \sqrt{p_1U \over p_2}

Hicksian (compensated) demand functions

Same tangency condition, different constraints

Utility Maximization

Cost Minimization

\text{Objective function: } x_1x_2
\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Constraint: }x_1x_2 = U
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

Utility Maximization

Cost Minimization

\text{Objective function: } u(x_1,x_2) = x_1x_2
\text{Objective function: } c(x_1,x_2) = p_1x_1 + p_2x_2

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

What is the optimized value of the objective function?

V(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m))
E(p_1,p_2,U)=c(x_1^c(p_1,p_2,U),x_2^c(p_1,p_2,U))

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

Utility from utility-maximizing choice,
given prices and income

Cost of cost-minimizing choice,
given prices and a target utility

Utility Maximization

Cost Minimization

\text{Objective function: } u(x_1,x_2) = x_1x_2
\text{Objective function: } c(x_1,x_2) = p_1x_1 + p_2x_2
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

What is the optimized value of the objective function?

V(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m))
E(p_1,p_2,U)=c(x_1^c(p_1,p_2,U),x_2^c(p_1,p_2,U))
= x_1^*(p_1,p_2,m) \times x_2^*(p_1,p_2,m)
= \displaystyle{{m \over 2p_1} \times {m \over 2p_2}}
= \displaystyle{{m^2 \over 4p_1p_2}}
=\displaystyle{p_1\sqrt{p_2U \over p_1} + p_2\sqrt{p_1U \over p_2}}
=p_1x_1^c(p_1,p_2,U) + p_2x_2^c(p_1,p_2,U)
=2\sqrt{p_1p_2U}

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

Utility Maximization

Cost Minimization

V(p_1,p_2,m) = \displaystyle{{m^2 \over 4p_1p_2}}
E(p_1,p_2,U)=2\sqrt{p_1p_2U}

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

Set \(V(p_1,p_2,m)=U\) and solve for \(m\).

Set \(E(p_1,p_2,U)=m\) and solve for \(U\).

{m^2 \over 4p_1p_2}=U
m^2=4p_1p_2U
m=2\sqrt{p_1p_2U}
2\sqrt{p_1p_2U} = m
\sqrt{U} = {m^2 \over 2\sqrt{p_1p_2}}
U = {m^2 \over 4p_1p_2}

These functions are inverses of one another!

Utility Maximization

Cost Minimization

V(p_1,p_2,m) = \displaystyle{{m^2 \over 4p_1p_2}}
E(p_1,p_2,U)=2\sqrt{p_1p_2U}

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

What is the marginal utility of another dollar?

What is the  marginal cost  of another util?

\displaystyle{{\partial V(p_1,p_2,m) \over \partial m} = {2m \over 4p_1p_2}}
\displaystyle{= {m \over 2p_1p_2}}
\displaystyle{{\partial E(p_1,p_2,U) \over \partial U} = {\sqrt{p_1p_2 \over U}}}

Write these down, and let's look at Lagrange one last time.

\max

The Lagrange Method: Utility Maximization

x_1,x_2
\text{s.t.}
m - p_1x_1 - p_2x_2

Income left over

x_1x_2

Utility

\ge 0

The Lagrange Method: Utility Maximization

m - p_1x_1 - p_2x_2

Income left over

\mathcal{L}(x_1,x_2,\lambda)=
\lambda
x_1x_2
+
(
)

Utility

(utils)

(dollars)

utils/dollar

\frac{\partial \mathcal{L}}{\partial x_1} = x_2 - \lambda p_1

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = x_1 - \lambda p_2
\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1x_1 - p_2x_2
= 0 \Rightarrow \lambda = \frac{x_2}{p_1}
= 0 \Rightarrow \lambda = \frac{x_1}{p_2}

The Lagrange Method: Utility Maximization

m - p_1x_1 - p_2x_2
\mathcal{L}(x_1,x_2,\lambda)=
\lambda
+
(
)
x_1x_2
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}

Solutions:

What's the value of \(\lambda\) at the optimal bundle?

= \frac{m/2p_2}{p_1}
= \frac{m/2p_1}{p_2}
= \frac{m}{2p_1p_2}
\frac{\partial \mathcal{L}}{\partial x_1} = p_1 - \lambda x_2

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = p_1 - \lambda x_2
\frac{\partial \mathcal{L}}{\partial \lambda} = U - x_1x_2
= 0 \Rightarrow \lambda = \frac{p_1}{x_2}
= 0 \Rightarrow \lambda = \frac{p_2}{x_1}

The Lagrange Method: Cost Minimization

p_1x_1 + p_2x_2
\mathcal{L}(x_1,x_2,\lambda)=
\lambda
+
(
)
U - x_1x_2

Solutions:

What's the value of \(\lambda\) at the optimal bundle?

= \frac{p_1}{\sqrt{p_1U \over p_2}}
= \frac{p_2}{\sqrt{p_2U \over p_1}}
= \sqrt{p_1p_2 \over U}
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

Last (!) Multivariate Concept: Cost Minimization, Envelope Theorem / Value Functions

  • Cost minimization is the inverse problem of utility maximization. We'll use it a lot for firms!
  • Indirect utility function = utility of utility-maximizing bundle
  • Expenditure function = cost of cost-minimizing bundle
  • Both of these are "value functions" - and their derivative is the Lagrange multiplier, which shows you how much the optimized value changes when the constraint changes by one unit.