Course Retrospective:
Review of Optimization
Christopher Makler
Stanford University Department of Economics
Econ 50
Resources
Technology
Stuff
Happiness
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Part I: The Real Economy
Demand
Supply
Equilibrium
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🏪
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Part II: Little Green
Pieces of Paper
Three Fundamental Tools of Analysis
Optimization
Given a fixed set of circumstances (prices, technology, preferences), how do economic agents (consumers, firms) make choices?
Comparative Statics
How do changes in circumstances (changing prices, shifting technology, preferences, etc.) translate into changes in behavior?
Equilibrium
How do economic systems converge toward certain outcomes?
Three Fundamental Tools of Analysis
Optimization
Constrained Optimization
Unconstrained Optimization
Tradeoffs between two good things
Goal: balance MB/MC ratio
Math characterization: tangency condition
Optimal amount of one thing
Goal: balance MB and MC
Math characterization: MB = MC
Constrained Optimization
Choice space:
all possible options
Feasible set:
all options available to you
Optimal choice:
Your best choice(s) of the ones available to you
Constrained Optimization
Canonical Constrained Optimization Problem
f(x_1,x_2)
\text{s.t. }
g(x_1,x_2) = k
k - g(x_1,x_2) = 0
\mathcal{L}(x_1,x_2,\lambda)=
\displaystyle{\max_{x_1,x_2}}
f(x_1,x_2)
k - g(x_1,x_2)
+ \lambda\ (
)
Suppose \(g(x_1,x_2)\) is monotonic (increasing in both \(x_1\) and \(x_2\)).
Then \(k - g(x_1,x_2)\) is negative if you're outside of the constraint,
positive if you're inside the constraint,
and zero if you're along the constraint.
OBJECTIVE
FUNCTION
CONSTRAINT
\mathcal{L}(x_1,x_2,\lambda)=
f(x_1,x_2)
k - g(x_1,x_2)
+ \lambda\ (
)
\displaystyle{\partial \mathcal{L} \over \partial x_1} =
FIRST ORDER CONDITIONS
\displaystyle{\partial \mathcal{L} \over \partial x_2} =
\displaystyle{\partial \mathcal{L} \over \partial \lambda} =
\displaystyle{\partial f \over \partial x_1}
\displaystyle{\partial f \over \partial x_2}
k - g(x_1,x_2)
=0
- \lambda\ \times
\displaystyle{\partial g \over \partial x_1}
\displaystyle{\partial g \over \partial x_2}
- \lambda\ \times
=0
=0
3 equations, 3 unknowns
Solve for \(x_1\), \(x_2\), and \(\lambda\)
How does the Lagrange method work?
It finds the point along the constraint where the
level set of the objective function passing through that point
is tangent to the constraint
Better to produce
more good 1
and less good 2.
MRS
>
MRT
Better to produce
less good 1
and more good 2.
MRS
<
MRT
Better to produce
more good 1
and less good 2.
MRS
>
MRT
MRS
<
MRT
“Gravitational Pull" Towards Optimality
Better to produce
more good 2
and less good 1.
These forces are always true.
In certain circumstances, optimality occurs where MRS = MRT.
We've just seen that, at least under certain circumstances, the optimal bundle is
"the point along the PPF where MRS = MRT."
CONDITION 1:
CONSTRAINT CONDITION
CONDITION 2:
TANGENCY
CONDITION
This is just an application of the Lagrange method!
Example: Linear PPF, Cobb-Douglas Utility
Chuck has 150 hours of labor, and can produce 3 coconuts per hour or 2 fish per hour.
His preferences may be represented by the utility function \(u(x_1,x_2) = x_1^2x_2\)
\text{s.t. }
\mathcal{L}(L,W,\lambda)=
\displaystyle{\max_{x_1,x_2}}
x_1^2x_2
150 - {1 \over 3}x_1 - {1 \over 2}x_2
+ \lambda\ (
)
OBJECTIVE
FUNCTION
CONSTRAINT
x_1^2x_2
150 - {1 \over 3}x_1 - {1 \over 2}x_2 = 0
{1 \over 3}x_1 + {1 \over 2}x_2 = 150
\mathcal{L}(L,W,\lambda)=
x_1^2x_2
150 - {1 \over 3}x_1 - {1 \over 2}x_2
+ \lambda\ (
)
\displaystyle{\partial \mathcal{L} \over \partial x_1} =
FIRST ORDER CONDITIONS
\displaystyle{\partial \mathcal{L} \over \partial x_2} =
\displaystyle{\partial \mathcal{L} \over \partial \lambda} =
2x_1x_2
x_1^2
150 - {1 \over 3}x_1 - {1 \over 2}x_2
=0
- \lambda\ \times
{1 \over 3}
{1 \over 2}
- \lambda\ \times
=0
=0
\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}
{1 \over 3}x_1 + {1 \over 2}x_2 = 150
\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}
2x_1x_2
x_1^2
3
2
MU_1
MU_2
MP_{L1}
MP_{L2}
\Rightarrow
Utility from last hour spent fishing
Utility from last hour spent collecting coconuts
Equation of PPF
\displaystyle{\lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}
{1 \over 3}x_1 + {1 \over 2}x_2 = 150
\displaystyle{ = \ \ \ \ \ \ \ \ \ \ \ \times}
2x_1x_2
x_1^2
3
2
MU_1
MU_2
MP_{L1}
MP_{L2}
Utility from last hour spent fishing
Utility from last hour spent collecting coconuts
Equation of PPF
{1 \over 3}x_1 + {1 \over 2}x_2 = 150
=
2x_1x_2
x_1^2
3
2
MU_1
MU_2
MP_{L1}
MP_{L2}
Equation of PPF
TANGENCY
CONDITION
MRS
MRT
CONSTRAINT
Optimization Subject to a Budget Constraint
\mathcal{L}(x_1,x_2,\lambda)=
u(x_1,x_2)+
(m - p_1x_1 - p_2x_2)
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1
First Order Conditions
\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial u}{\partial x_2} - \lambda p_2
\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1x_1 - p_2x_2
= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}
= 0 \Rightarrow \lambda = \frac{MU_2}{p_2}
"Bang for your buck" condition: marginal utility from last dollar spent on every good must be the same!
\text{Also: }\frac{\partial \mathcal{L}}{\partial m} = \lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}
\text{Set two expressions }
\text{for }\lambda \text{ equal:}
\frac{MU_1}{p_1} = \frac{MU_2}{p_2}
The Tangency Condition
What happens when the price of a good increases or decreases?
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
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
Plug tangency condition back into constraint:
\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}
Same tangency condition, different constraints
The Social Planner's Problem
How do we maximize GDP subject to the PPF?
\text{s.t. }
\mathcal{L}(Y_1,Y_2,\lambda)=
\displaystyle{\max_{Y_1,Y_2}}
p_1Y_1 + p_2Y_2
\overline L - L_1(Y_1) - L_2(Y_2)
+ \lambda\ (
)
OBJECTIVE
FUNCTION
CONSTRAINT
p_1Y_1 + p_2Y_2
\overline L - L_1(Y_1) - L_2(Y_2) = 0
L_1(Y_1) + L_2(Y_2) = \overline L
DOLLARS
HOURS
What are the units?
DOLLARS
PER
HOUR
\mathcal{L}(Y_1,Y_2,\lambda)=
p_1Y_1 + p_2Y_2
\overline L - L_1(Y_1) - L_2(Y_2)
+ \lambda\ (
)
\displaystyle{\partial \mathcal{L} \over \partial Y_1} =
FIRST ORDER CONDITIONS
\displaystyle{\partial \mathcal{L} \over \partial Y_2} =
p_1
p_2
- \lambda\ \times
{dL_1 \over dY_1}
{dL_2 \over dY_2}
- \lambda\ \times
=0
=0
\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \times}
\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \times}
p_1
p_2
MP_{L1}
MP_{L2}
Tangency condition:
{p_1 \over p_2} = {MP_{L2} \over MP_{L1}}
Firms setting their own MRPL equal to the wage rate will choose the point along the PPF at which \({p_1 \over p_2} = MRT\)
Conditions for
GDP Maximization
{p_1 \over p_2} = MRT
TANGENCY CONDITION
CONSTRAINT CONDITION
L_1(Y_1) + L_2(Y_2) = \overline L
Firms in industry 1 set \(w = p_1 \times MP_{L1}\)
Firms in industry 2 set \(w = p_2 \times MP_{L2}\)
How does competition achieve this?
Wages adjust until the
labor market clears
Corner Solutions
Interior Solution:
Corner Solution:
Optimal bundle contains
strictly positive quantities of both goods
Optimal bundle contains zero of one good
(spend all resources on the other)
If only consume good 1: \(MRS \ge MRT\) at optimum
If only consume good 2: \(MRS \le MRT\) at optimum
What would Lagrange find...?
The Tangency Condition when Lagrange sometimes works
What happens when income decreases?
Kinked Constraints
Unconstrained Optimization
\pi(q) = r(q) - c(q)
Optimize by taking derivative and setting equal to zero:
\pi'(q) = r'(q) - c'(q) = 0
\Rightarrow r'(q) = c'(q)
Profit is total revenue minus total costs:
"Marginal revenue equals marginal cost"
Example:
p(q) = 20 - q \Rightarrow r(q) = 20q - q^2
c(q) = 64 + {1 \over 4}q^2
What is the profit-maximizing value of \(q\)?
Profit as a function of quantity
Profit as a function of labor
c(q) = wL(q) + r \overline K
1. Total costs = cost of required inputs
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
\pi(q) = pq - [wL(q) + r\overline K]
\pi^\prime(q) = p - w \times {dL \over dq} = 0
r(L) = p \times f(L)
1. Total revenue = value of output produced
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
\pi(L) = p\times f(L) - [wL + r\overline K]
\pi^\prime(L) = p \times {dq \over dL} - w = 0
p = w \times {dL \over dq}
p \times {dq \over dL} = w
\text{Profit }\pi = pq - [wL + rK]
Profit as a function of quantity
Profit as a function of labor
1. Total revenue = value of output produced
2. Profit = total revenues minus total costs
3. Take derivative of profit function, set =0
"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."
p = MC
r(L) = p \times f(L)
\pi(L) = p\times f(L) - [wL + r\overline K]
\pi^\prime(L) = p \times {dq \over dL} - w = 0
p \times {dq \over dL} = w
Profit as a function of quantity
Profit as a function of labor
"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."
"Keep hiring workers as long as the marginal revenue from the output of the last worker is at least as great as the cost of hiring them."
p = MC
w = MRP_L
L^*(w\ |\ p) = 8 \left({p \over w}\right)^2
LABOR DEMAND FUNCTION
Econ 50 | Lecture 27
By Chris Makler
Econ 50 | Lecture 27
Course Retrospective
