PART A: Answer any 5 of the questions A.1 – A.7. Do not answer additional questions. This part is worth 15% of the exam.
Briefly define and give a specific example of:
A.1. Perfect competition
A.2. Monopoly
A.3. Monopolistic competition
A.4. NAICS Code
A.5. Location Quotients
A.6. An inverse matrix
A.7. Price discrimination
PART B: Answer each of the questions B.1 – B.3. This part is worth 30% of the exam.
B.1. Why is “1” a critical value in terms of Location Quotients? If the Location Quotient of some industry is greater than one, what does this mean for the area whose Location Quotient this is?
B.2. Why do all suppliers want to price discriminate?
B.3. Why don’t all suppliers price discriminate?
PART C: Answer only 1 of the questions C.1 – C.2. Do not answer additional questions. This part is worth 55% of the exam.
C.1. Outline the structure of an input-output model (including assumptions about supply and demand). What is an inverse matrix? Why is inverting a matrix significant in terms of input-output analysis?
C.2. Describe a Linear Programming (LP) Problem. Specifically, describe (you can use an example):
- Primal Linear Programming Problem
- Dual Linear Programming Problem
- Interpretation of the Primal Linear Programming Problem
- Interpretation of the Dual Linear Programming Problem
NAME:___________________________________
PART A: Answer any 5 of the questions A.1 – A.7. Do not answer additional questions. This part is worth 15% of the exam.
Briefly define and give a specific example of:
A.1. Perfect competition
A.2. Monopoly
A.3. Monopolistic competition
A.4. NAICS Code
A.5. Location Quotients
A.6. An inverse matrix
A.7. Price discrimination
PART B: Answer each of the questions B.1 – B.3. This part is worth 30% of the exam.
B.1. Why is “1” a critical value in terms of Location Quotients? If the Location Quotient of some industry is greater than one, what does this mean for the area whose Location Quotient this is?
B.2. Why do all suppliers want to price discriminate?
B.3. Why don’t all suppliers price discriminate?
PART C: Answer only 1 of the questions C.1 – C.2. Do not answer additional questions. This part is worth 55% of the exam.
C.1. Outline the structure of an input-output model (including assumptions about supply and demand). What is an inverse matrix? Why is inverting a matrix significant in terms of input-output analysis?
C.2. Describe a Linear Programming (LP) Problem. Specifically, describe (you can use an example):
· Primal Linear Programming Problem
· Dual Linear Programming Problem
· Interpretation of the Primal Linear Programming Problem
· Interpretation of the Dual Linear Programming Problem
Regional Economics
Lecture 3
J. M. Pogodzinski
carol
Agenda
• Governance structures
• Sources of
• Evidence for
• Effects of
• Market structures
• Perfect Competition
• Monopolistic Competition
• Review of simple Keynesian macroeconomic model (basis for the
economic base model)
Keynesian Macroeconomic Model
• Equilibrium condition
• Consumption function
• Marginal propensity to consume
• Investment function
• Solving the model
• The spending multiplier
Governance Structures
• Governance structures, incentives, and efficiency (e.g., SB50)
• ABAG
• Tax sharing incentives
Governance Structures
• Spillovers from municipal investment projects
• Zoning and building codes – spillover effects
• Efficiency implications
• NIMBY-ism
Market Structures
• Perfect competition
• Assumptions
• Efficiency implications
• Mathematical/graphic representation
• Imperfect competition
• Non-price discriminating monopolist
• Assumptions
• Efficiency implications
• Mathematical/graphic representation
Market Structures
• Imperfect competition
• Perfectly price discriminating monopolist
• Assumptions
• Efficiency implications
• Mathematical/graphic representation
• Monopolistic competition
• Assumptions
• Efficiency implications
• Mathematical/graphic representation
• Why not just assume perfect competition?
Monopolistic Competition
• Many varieties of goods
• Can determine equilibrium number of firms/number of varieties
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Linear Programming
Part 1
J
.
M. Pogodzinski
carol
carol
carol
carol
Agenda
•
Mathematical Programming
Problems
• Economic Theory and Mathematical Programming Problems
•
Linear Programming Problems
• The Objective Function
• The Inequality
Constraints
• The Non-Negativity Constraints (which are inequality constraints)
• Equality Constraints?
• The Feasible Set
• Does a Solution Exist to a Linear Programming Problem? (the existence question)
• Applications (Uses) of Linear Programming
• Solving Linear Programming Problems
• Theorems About Linear Programming
Mathematical Programming Problems
• A Mathematical Programming Problem consists of:
• An objective function
• Constraints defined somehow – equations, inequalities,…
• Little can be said about such a general problem – we need to make
assumptions about the objective function and/or about the
constraints before we can say anything about the existence of
solutions, algorithms for finding solutions (if they exist), properties of
solutions
About Objective Functions
• Very common to assume there is only one objective function
• Objective functions are either maximized or minimized – the generic term is
optimized. The specific problem determines whether maximization or
minimization is appropriate. There are deeper connections between
maximization and minimization. Maximization problems can be restated as
minimization problems. More importantly, specific maximization problems are
associated with specific minimization problems through duality
.
• It is possible to consider multi-objective mathematical programming problems
(there is a legitimate topic called multi-objective linear programming)
• What do you get out of multi-objective linear programming (if there is a
solution)?
• The Pareto Frontier
• We will not consider multi-objective linear programming because it is
computationally difficult
About Objective Functions
• Example (from microeconomics): Consumers maximize utility subject
to a budget constraint
• 𝑚𝑎𝑥𝑥,𝑦 𝑈 𝑥,𝑦 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We assume that 𝑈 𝑥,𝑦 is a quasi-concave continuous function
(Note: famous paper “Quasi-Concave Programming” by Kenneth J.
Arrow and Alain C. Enthoven, Econometrica, Vol. 29, No. 4 (Oct.,
1961), pp. 779-800)
• A function 𝑈 𝑥,𝑦 is quasi-concave if its upper level sets are convex
sets
Constraints
• Most common to define constraints by one or more equations or
inequalities
• Note on finite constraint sets – existence of optimum
• For example, in the consumer choice problem mentioned in the
previous slide, an equation called the budget equation defined the
constraint set – 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 (and 𝑥 ≥ 0 and y ≥ 0)
• We might also have defined the constraint set with several
inequalities: 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑥 ≥ 0 and y ≥ 0
• We can write the equation 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀 as two inequalities:
𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≤ 𝑀 and 𝑝𝑥𝑥 + 𝑝𝑦𝑦 ≥ 𝑀
The Consumer Choice Problem
𝑚𝑎𝑥𝑥,𝑦 𝑈 𝑥,𝑦 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝𝑥𝑥 + 𝑝𝑦𝑦 = 𝑀
(and 𝑥 ≥ 0 and y ≥ 0)
Draw the constraint set (the feasible set)
Draw some upper level sets of the objective function
Are these sets convex sets?
Convex Sets – Yes or No?
• Examples
Linear Programming Problems
• An LP Problem has:
• A linear objective function
• Linear inequality constraints
• Non-negativity constraints
LP Problem – General Form
• Decision variables: 𝑥𝑗 (𝑗 = 1,…,𝑛)
• (Linear) Objective function: Π = 𝑐1𝑥1 + 𝑐2𝑥2 + ⋯+ 𝑐𝑛𝑥𝑛
• (Linear) Inequality constraints:
𝑎11𝑥1 + ⋯+ 𝑎1𝑛𝑥𝑛 ≤ 𝑏1
𝑎21𝑥1 + ⋯+ 𝑎2𝑛𝑥𝑛 ≤ 𝑏2
𝑎𝑚1𝑥1 + ⋯+ 𝑎𝑚𝑛𝑥𝑛 ≤ 𝑏𝑚
• Non-negativity constraints: 𝑥1 ≥ 0, 𝑥2 ≥ 0,…, 𝑥𝑛 ≥ 0
Write this in
matrix
notation
Write this in
matrix
notation
x – variables
a, b, c – parameters (constants)
Write this
in matrix
notation
LP Problem – An Example
• A Production Problem
• 𝑥𝑗 amount of good j to be produced (j=1,…,n)
• 𝑎𝑖𝑗 amount of resource i required to produce one unit of good j*
• 𝑏𝑖 amount of resource i available
• 𝑐𝑗 profit per unit of good j
*CONVENTION: (i,j) = (row, column)
Graph it!
Solve it!
Write in matrix
notation!
Products Variables
socks x1
shirts x2
Resources Parameters
Looms b1 10
Sewing Machines b2 15
Labor b3 12
Coefficient Matrix
1 1
3 1
2 1
Product net revenue Pi-1 Pi-2 Pi-3
socks c1 1 2 1
shirts c2 1 1 2
Linear Programming
Part 2
J. M. Pogodzinski
Agenda
• Some LP Theorems
• Excel skills: sumproduct, matrix multiplication, matrix inversion
• Using Excel to Solve LP Problems (demo)
• Binding and non-binding constraints
•
The Dual LP Problem
• The Dual Solution Variables
• The Value of the Program
• The
Duality Theorem
• Interpretation of the Dual Solution Variables
Some Linear Programming Theorems
• If a Linear Programming (LP) Problem has a solution, the solution
occurs at a vertex of the feasible set.
• If the feasible set is bounded, there are a finite number of vertices.
• This means there is an “obvious” algorithm for solving an LP Problem
with a bounded feasible set: compute the value of the objective
function at every vertex of the feasible set – the largest value is the
maximum; the smallest value is the minimum.
• Not every LP Problem that has a solution has a bounded feasible set
Excel Skills (demos and downloadable file)
• Dot product
• Transpose Matrix
• Inverse Matrix
• Solve an LP Problem
The Dual LP Problem
• For every LP Problem, there is a related LP Problem called the “dual”
of the first problem
• The first problem is called the Primal LP Problem, the related problem
is called the Dual LP Problem
• The dual of the Dual LP Problem is the Primal LP Problem
Going from the Primal LP Problem to the Dual
LP Problem
The Primal LP Problem
𝑎11𝑄1 + ⋯+ 𝑎1𝑛𝑄𝑛 ≤ 𝐶1
.
.
.
𝑎𝑚1𝑄1 + ⋯+ 𝑎𝑚𝑛𝑄𝑛 ≤ 𝐶𝑚
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 Π = 𝑃1𝑄1 + ⋯+ 𝑃𝑛𝑄𝑛
𝑄1 ≥ 0,𝑄2 ≥ 0,…,𝑄𝑛 ≥ 0
Baumol’s “mad gremlin”
The Dual LP Problem
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 Α = 𝐶1𝑉1 + ⋯+ 𝐶𝑚𝑉𝑚
𝑎11𝑉1 + ⋯+ 𝑎𝑚1𝑉𝑚 ≥ 𝑃1
𝑉1 ≥ 0,𝑉2 ≥ 0,…,𝑉𝑚 ≥ 0
𝑎1𝑛𝑉1 + ⋯+ 𝑎𝑚𝑛𝑉𝑚 ≥ 𝑃𝑛
.
.
.
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 Π = 𝑃1𝑄1 + ⋯+ 𝑃𝑛𝑄𝑛
𝑎11𝑄1 + ⋯+ 𝑎1𝑛𝑄𝑛 ≤ 𝐶1
.
.
.
𝑎𝑚1𝑄1 + ⋯+ 𝑎𝑚𝑛𝑄𝑛 ≤ 𝐶𝑚
𝑄1 ≥ 0,𝑄2 ≥ 0,…, 𝑄𝑛 ≥ 0
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 Α = 𝐶1𝑉1 + ⋯+ 𝐶𝑚𝑉𝑚
𝑎11𝑉1 + ⋯+ 𝑎𝑚1𝑉𝑚 ≥ 𝑃1
.
.
.
𝑎1𝑛𝑉1 + ⋯+ 𝑎𝑚𝑛𝑉𝑚 ≤ 𝑃𝑛
𝑉1 ≥ 0, 𝑉2 ≥ 0, …,𝑉𝑚 ≥ 0
Primal and Dual: Head-to-Head
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 Π = 𝑃1𝑄1 + ⋯+ 𝑃𝑛𝑄𝑛
𝑎11𝑄1 + ⋯+ 𝑎1𝑛𝑄𝑛 ≤ 𝐶1
.
.
.
𝑎𝑚1𝑄1 + ⋯+ 𝑎𝑚𝑛𝑄𝑛 ≤ 𝐶𝑚
𝑄1 ≥ 0,𝑄2 ≥ 0,…, 𝑄𝑛 ≥ 0
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 Α = 𝐶1𝑉1 + ⋯+ 𝐶𝑚𝑉𝑚
𝑎11𝑉1 + ⋯+ 𝑎𝑚1𝑉𝑚 ≥ 𝑃1
.
.
.
𝑎1𝑛𝑉1 + ⋯+ 𝑎𝑚𝑛𝑉𝑚 ≥ 𝑃𝑛
𝑉1 ≥ 0, 𝑉2 ≥ 0, …,𝑉𝑚 ≥ 0
Objective Function
P – parameters
Q – choice variables
n
Objective Function
C – parameters
V – choice variables
m
Transposed Coefficient
Matrices
Primal (mxn)
Dual (nxm)
Constraints: m inequalities in n unknowns Constraints: n inequalities in m unknowns
Analysis of Primal
and Dual LP
Problems
Primal LP
Problem
Dual LP
Problem
Duality Theorem
• Definition: the value of the program for an LP Problem is the value of
the objective function at the optimal solution.
• THEOREM: If the Primal LP Problem has a solution, then the Dual LP
Problem has a solution, and Value of the Primal Program equals the
Value of the Dual Program.
• Note: the values of the programs are equal, not the solutions of the
programs
Demo: The Dual LP Problem
• Formulation of the Dual LP Problem in Excel (demo)
• Solution of the Dual LP Problem using Excel (demo)
Interpreting the Dual Solution Values
Linear Programming
Part 3
J. M. Pogodzinski
Agenda
• Applications of LP
•
The Diet Problem
• Transportation Problems
The Diet Problem
• How to feed an army in the most economical way while meeting nutritional
requirements (also, according to Hadley, feeding livestock, provisioning
submarine or spacecraft)
• Dietician must select amounts of n foods (F1, F2, …, Fn) that provide
certain amounts of m nutrients (N1, N2, …, Nm)
• Each person must consume (G1, G2, …, Gm) of the corresponding nutrient
• aij denotes the amount of the i-th nutrient contained in one unit of the j-th
food, i.e., aij is the amount of Ni contained in one unit of Fj
• The choice variable is the amounts of foods to be eaten (e1, e2,…,en)
• The prices of each food are (p1, p2, …,pn)
Coefficient Matrix for the Diet Problem
F1 F2 . . . Fn
N1 a11 a12 a1n
N2 a21 a22 a2n
.
.
.
Nm am1 am2 amn
In-Class Activity
• Assume
• (G1, G2, …, Gm)
• (e1, e2,…,en)
• (p1, p2, …,pn)
are column vectors, i.e., G is mx1, and e and p are nx1
• Using matrix notation, write down the LP Problem
• Using matrix notation, write down the Dual LP Problem
Diet Problem – In-Class Example
Transportation Problem
• Let a commodity be produced at each of m plants (P1,P2,…,Pm) [m origins]
• The commodity is needed in each of n markets (M1,M2,…,Mn) [n
destinations]
• Denote by Ci the capacity of the i-th plant [availability at origins]
• Denote by dj the demand in the j-th market [needs at destinations]
• Denote by cij the cost of shipping one unit of the commodity from plant i
to market j
• Determine a shipping schedule xij (the amount to be shipped from plant i
to market j)
• Assume σ𝑖=1
𝑚 𝐶𝑖 = σ𝑗=1
𝑛 𝑑𝑗 [What does this mean?]
Inequality and Non-Negativity Constraints of the
Transportation Problem
The Objective Function of the Transportation Problem
• σ𝑗=1
𝑛 𝑥𝑖𝑗 = 𝐶𝑖 𝑖 = 1,…,𝑚
• σ𝑖=1
𝑚 𝑥𝑖𝑗 = 𝑑𝑗 𝑗 = 1,…,𝑛
• 𝑥𝑖𝑗 ≥ 0
• 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = σ𝑖,𝑗 𝑐𝑖𝑗𝑥𝑖𝑗
The Transportation Problem in Matrix Notation
• 𝒙 = (𝑥11,…,𝑥1𝑛,𝑥21,…,𝑥2𝑛,…,𝑥𝑚𝑛)
• 𝒃 = (𝐶1,…,𝐶𝑚,𝑑1,…,𝑑𝑛)
• 𝑨 is an 𝑚 + 𝑛 x 𝑚𝑛 matrix
• Transportation Problem
• 𝑨𝒙 = 𝒃,𝒙 ≥ 0max𝑜𝑟 min𝑧 = 𝒄𝒙
Linear Programming
Part 4
J. M. Pogodzinski
Agenda
• Dual of the Diet Problem
• More about the Transportation Problem
• Variants of the Transportation Problem
• Dual of the Transportation Problem
Recall: The Diet Problem
• How to feed an army in the most economical way while meeting nutritional
requirements (also, according to Hadley, feeding livestock, provisioning
submarine or spacecraft)
• Dietician must select amounts of n foods (F1, F2, …, Fn) that provide
certain amounts of m nutrients (N1, N2, …, Nm)
• Each person must consume (G1, G2, …, Gm) of the corresponding nutrient
• aij denotes the amount of the i-th nutrient contained in one unit of the j-th
food, i.e., aij is the amount of Ni contained in one unit of Fj
• The choice variable is the amounts of foods to be eaten (e1, e2,…,en)
• The prices of each food are (p1, p2, …,pn)
The Diet Problem in Matrix Notation
(Example)
F1 F2 F3 F4 F5
N1 1 0 1 1 2
N2 0 1 0 1 1
G1 700
G2 400
e1
e2
e3
e4
e5
≥ 0
≥ 0
≥ 0
≥ 0
≥ 0
p1 2
p2 20
p3 3
p4 11
p5 12
What is the objective
function and do you
maximize it or minimize
it?
Explain the logic of the
Diet Problem
The Dual to the Diet Problem (Example)
G1 700
G2 400
p1 2
p2 20
p3 3
p4 11
p5 12
What is the objective
function and do you
maximize it or minimize
it?
Explain the logic of the
Dual to the Diet
Problem
N1 N2
F1 1 0
F2 0 1
F3 1 0
F4 1 1
F5 2 1
v1
v2
≥ 0
≥ 0
Recall: The Transportation Problem
• Let a commodity be produced at each of m plants (P1,P2,…,Pm) [m origins]
• The commodity is needed in each of n markets (M1,M2,…,Mn) [n
destinations]
• Denote by Ci the capacity of the i-th plant [availability at origins]
• Denote by dj the demand in the j-th market [needs at destinations]
• Denote by cij the cost of shipping one unit of the commodity from plant i
to market j
• Determine a shipping schedule xij (the amount to be shipped from plant i
to market j)
• Assume σ𝑖=1
𝑚 𝐶𝑖 = σ𝑗=1
𝑛 𝑑𝑗 [What does this mean?]
Transportation Problem (Example)
m 2 origins
n 4 destinations Ax = b
Matrix A (m+n)x(mn) rows 6
columns 8
1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
Vector b a1 origin 10
a2 origin 15
b1 destination 2
b2 destination 6
b3 destination 8
b4 destination 9
Vector c c11 3
c12 4
c13 2
c14 5
c21 3
c22 6
c23 7
c24 4
Vector x x11
x12
x13
x14
x21
x22
x23
x24
What is the objective function and do you maximize
it or minimize it?
Explain the logic of the Transportation Problem
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