Microsoft PowerPoint - 103Micro_Part1C1.pptxA. Microeconomics and Economic ModelsA. Microeconomics and Economic Models S l d d S l d dB. Supply and DemandB. Supply and Demand
C. The Mathematics of OptimizationC. The Mathematics of Optimization
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Part 1. Part 1. IntroductionIntroduction
C. C. The Mathematics of The Mathematics of The Mathematics of The Mathematics of Optimization Optimization Optimization Optimization
1.1. Unconstrained Optimization ProblemsUnconstrained Optimization Problems 2.2. Constrained Optimization ProblemsConstrained Optimization Problems
Perloff (2014, 3e, GE), Calculus Appendix, pp. 711-738 Nicholson and Snyder (2012, 11e), Ch. 2 2
The Optimization PrincipleThe Optimization Principle The Optimization PrincipleThe Optimization Principle People will try to choose what is best for them People will try to choose what is best for them (under their constraints)(under their constraints)(under their constraints).(under their constraints).
The Mathematics of Optimization The Mathematics of Optimization Methods used to solve the optimization optimization problemsproblems.pp
Max.Max. Objective FunctionObjective Function {choice variables}{choice variables}
s.ts.t.. ConstraintsConstraints
subject to
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1.1. Part 1C. Part 1C. The Mathematics of OptimizationThe Mathematics of Optimization
Unconstrained Unconstrained Optimization ProblemOptimization ProblemOptimization ProblemOptimization Problem
Unconstrained Optimization Problem: One Unconstrained Optimization Problem: One

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The Maximization ProblemThe Maximization Problem
Max. y = f (x) {x}
where f(x) objection function y objecty object x choice variable
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Necessary '( ) 0dy f x y condition0
( ) 0 dx
2
2
0dx dx dx dx
F.O.C. & S.O.C. are necessary and sufficient F.O.C. & S.O.C. are necessary and sufficient conditions for the optimization problems.conditions for the optimization problems.
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y d2y < 0
y
f(x)
f(x)y f(x)
* d2y = 0
Example: Example: Profit MaximizationProfit Maximizationpp Suppose the relationship b/w profit () and quantity produced (q) is given byproduced (q) is given by
Max. ( ) ( ) ( )q R q C q { }
( ) ( ) ( ) q
F.O.C.:F.O.C.: 0d dR dC d d d MR = MC q*
S.O.C.:S.O.C.:
2 2 2 0 dq dq dq
MaximumMaximum
dMR dMC i e the slop of the slop of MCMC is
10
dMR dMC dq dq
i.e., the slop of the slop of MCMC is bigger than that of MRMR.
Numerical Example:Numerical Example:pp
dx
11
dq
Analytical Example:Analytical Example:y py p
2
F.O.C.:F.O.C.: dy = 0
, 2 4
x y
Solutions:Solutions:
Solutions:Solutions:
( ) 2
2 *( )
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Comparative Static Effects:Comparative Static Effects: *( ) 1
2 dx a
The Maximization ProblemThe Maximization Problem
Max. y = f (x1, x2) {x1, x2}
where f(x) objection function y objecty object
x1, x2 choice variables
F.O.C.:F.O.C.: dy = f1dx1+f2dx2 = 0 y f1 1 f2 2
1 2 0f f dy = 0 holds if 1 2 0f fdy = 0 holds if
S.O.C.:S.O.C.: d2y < 0
d2y = (f dx +f dx )dx + (f dx +f dx )dx
2 2f dx f dx dx f dx dx f dx
d y = (f11dx1+f21dx2)dx1+ (f12dx1+f22dx2)dx2
11 1 21 2 1 12 1 2 22 2f dx f dx dx f dx dx f dx
Young’s Theorem: Young’s Theorem:
2 22 0f d f d d f d
gg if f is C2, then f12,= f21.
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2 2 11 1 12 1 2 22 22 0f dx f dx dx f dx
Theorem: Theorem: MaximumMaximum
2 20 holds if 0 ( 0 ) and 0d y f f f f f
11 12f f
11 22 11 22 120 holds if 0, ( 0, ) and 0d y f f f f f
11 12
21 22

f f f
Negative Negative DefiniteDefinite
11 21 22
, ( )f f f
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ProofProof: : (only for your understanding)( y y g) 2 2 2
11 1 12 1 2 22 22 0d y f dx f dx dx f dx
2 2 12 12
11 1 1 2 22 f ff dx dx dx dx
11 1 1 2 2 11 11
f f f

11 11

f f f
Figure: Figure: Maximum Maximum gg
d 0 d2y < 0 Con cavedy = 0, d2y < 0, Con cave x2
y
x2*
xx *
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x1x1*
{x1 x2}{x1, x2}
S.O.C.:S.O.C.: d2y > 0 holds if
2 11 22 11 22 12 0, ( 0, ) and 0f f f f f
11 12
21 22

f f f
2 + 4x2 + 5
F.O.C.:F.O.C.: dy = 0
xxf dx dy 2042 *
S.O.C.:S.O.C.: d2y < 0S.O.C.:S.O.C.: d y 0
02 2
MaximumMaximum2 11 22 12 4 0 4 0f f f
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Example: Example:
E li it FE li it FExplicit FormExplicit Form y = mx +b or y = f(x)
y – mx – b = 0
Implicit FormImplicit FormImplicit FormImplicit Form F(x, y, m, b) = 0
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Explicit vs. Implicit FunctionsExplicit vs. Implicit Functionsp pp p It may not always be possible to solve implicit implicit functionsfunctions of the form g(x,y) = 0 for unique explicitexplicitfunctionsfunctions of the form g(x,y) 0 for unique explicit explicit functionfunctions of the form y = f(x).
Explicit Function:Explicit Function: x = x (x ) (1)Explicit Function:Explicit Function: x2 = x2 (x1) (1)
2 2 1( )
dx dx x the trade off b/w x1 and x2
1 1
dx dx
Implicit Function: Implicit Function: F(x1, x2) = 0 (2)1 2
Totally differentiating Eq. (2) gives dF F d F d
1 21 2
F Fdx dx



The Implicit Function TheoremThe Implicit Function Theorempp The trade off between x1 and x2, dx2/dx1, can be directly derived by implicit functions of the formdirectly derived by implicit functions of the form g(x,y) = 0.
2 1 1

12 dx dx
1dx
Implicit Function: Implicit Function: F(x1, x2) = x1 + x2 – 6 = 0 (2) Totally differentiating Eq. (2) gives
dF = dx1 + dx2 = 0
1dx 1 1 1
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Numerical Example:Numerical Example:pp Implicit Function: Implicit Function: f(x, y) = x2 +0.25 y2 – 200 = 0 T t ll diff ti ti b E iTotally differentiating above Eq. gives
df = 2xdx + 0.5ydy = 0
2 4 0 5
dy x x d
Calculate fx = 2x and fy = 0.5y, by theorem
2 4 0.5

Nicholson and Snyder (2012, 11e), Example 2.4, pp. 32-33.
The Envelope TheoremThe Envelope TheoremThe Envelope TheoremThe Envelope Theorem
QQ: Suppose that we have derived the maximum : Suppose that we have derived the maximum value function, value function, yy = = ff((xx; ; aa)), how can we obtain the , how can we obtain the comparative static property of comparative static property of aa on on yy, , dydy*/*/dada? ?
Method 1: Derive it directlyMethod 1: Derive it directly Derive y* = f(x(a); a) = y*(a) as a function of a,Derive y f(x(a); a) y (a) as a function of a, and then calculate dy*/da directly. Method 2: Use the Envelope Theorem Method 2: Use the Envelope Theorem Method 2: Use the Envelope Theorem Method 2: Use the Envelope Theorem Treat x* as a constant, and then calculate dy/da di tl h f( *)directly where y = f(a; x*).
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* ( )x x ada da

It h th ti l lth ti l l f f tiIt concerns how the optimal valuethe optimal value for a function changes when a parameter of the function changeschanges. i.e., the comparative static property of a on y*.
Proof:Proof: * *( ) ( ( ); ))dy a df x a a d d
da da
* *( ) ( )x x a x x a dx da da da
= 0= 0
2Max. y ax x where a parameter { }x
y where a parameter
x a
…… Choice (Solution) functionChoice (Solution) function
(M i ) V l F ti(M i ) V l F ti( ) 4
y a …… (Maximum) Value Function(Maximum) Value Function
C ti St ti Eff t f C ti St ti Eff t f * *( )dy a aComparative Static Effect of Comparative Static Effect of a on on y*:: ( )
2 dy a a
* *( ) ( )dy a df a a
29 * ( )

Solutions:Solutions:
( ) 2
2 *( )
4
Comparative Static Effects:Comparative Static Effects: *( ) 1
2 dx a
30
2da
Table: Table: Optimal Values of Optimal Values of xx and and yy for Alternative Values of for Alternative Values of aapp yy in in yy = = axax –– xx22
Value of Value of aa Value of Value of xx** Value of Value of yy**
00 00 0000 00 00
11 1/21/2 1/41/4
22 11 11
33 3/23/2 9/49/4
44 22 44
66 33 99
Figure: Figure: Illustration of the Envelope TheoremIllustration of the Envelope Theoremgg pp
As a increases, the maximal value for y10
y*
9
10
*( ) ay a 6
4
5 The envelope theorem states that the slope of the relationship between y
2
3
slope of the relationship between y (the maximum value of y) and the parameter a can be found by calculating the slope of the auxiliary
0
1
a
calculating the slope of the auxiliary relationship found by substituting the respective optimal values for x i t th bj ti f ti d
32Nicholson and Snyder (2012, 11e), Figure 2.3, p. 36.