1. Optimisasi dengan batasan persamaan
(Optimization with equality constraints)
Mengapa batasan relevan dalam kajian
ekonomi?
Masalah ekonomi timbul karena kelangkaan
(scarcity).
Kelangkaan menyebabkan keputusan ekonomi
(termasuk optimisasi) tidal dilakukan dalam
kondisi tidak terbatas.
Dengan kata lain, constrained optimization
merupakan pembahasan pokok dalam ekonomi
slide 0

2. Lagrange Multiplier
Merupakan suatu metode matematika yang
dapat menyatakan suatu persoalan nilai ekstrim
(maksimum atau minimum) yang mempunyai
batasan (constrained-extremum) dalam bentuk
yang bisa diselesaikan dengan menggunakan
First-Order condition (FOC)
slide 1

3. Iso-cost lines
Draw set of points where
z2 cost of input is c, a constant
Repeat for a higher value
of the constant
Imposes direction on the
diagram...
w1z1 + w2z2 = c"
w1z1 + w2z2 = c'
w 1z 1 + w 2z 2 =
c Use this to
z1 derive
optimum
slide 2

4. Cost-minimisation
The firm minimises cost...
z2
q Subject to output constraint
Defines the stage 1 problem.
Solution to the problem
minimise
m
Σ wizi
i=1
subject to φ(z) ≥ q
z* But the solution depends
on the shape of the input-
requirement set Z.
z1
What would happen in
other cases?
slide 3

5. Convex, but not strictly
convex Z
z2
Any z in this set is
cost-minimising
An interval of solutions
z1
slide 4

6. Convex Z, touching axis
z2
Here MRTS21 > w1 / w2
at the solution.
z1 Input 2 is “too
z*
expensive” and so isn’t
used: z2*=0. slide 5

7. Non-convex Z
z2
There could be multiple
z* solutions.
But note that there’s no
solution point between z*
z** and z**.
z1
slide 6

8. Aplikasi 1: Optimalisasi kepuasan
konsumen
The primal problem
objective function Tujuan konsumen adalah
x2 memaksimalkan utilitas
Batasannya adalah budget
max U(x) subject to
n
Constraint Σ pixi ≤ y
set i=1
x* Cara lain memandang
persoalan ini adalah...
x1
slide 7

9. The dual problem
Konsumen bertujuan
x2
z2
υ
q meminimalkan pengeluaran
Constraint Untuk mencapai utilitas
tertentu
set
minimise
n
Σ pixi
i=1
subject to U(x) ≥ υ
x*
z*
Contours of x1
z1
objective function
slide 8

10. The Primal and the Dual…
There’s an attractive
symmetry about the two n
approaches to the problem Σ pixi+ λ[υ – U(x)]
i=1
In both cases the ps are
given and you choose the xs. n
But… [
U(x) + µ y – Σ pi xi ]
…constraint in the primal i=1
becomes objective in the
dual…
…and vice versa.
slide 9

11. A neat connection
Compare the primal problem
of the consumer...
...with the dual problem
x2 x2υ The two are
equivalent
So we can link up
their solution
functions and
response functions
x*
x*
Run through
x1 x1 the primal
slide 10

12. Utilitas dan Pengeluaran
Maksimisasi utilitas dan minimisasi pengeluaran pada
dasarnya merupakan persoalan yang sama yang dilihat
dari sudut pandang berbeda
Dengan demikian, solusinya sangat terkait satu sama
lainnya
Primal Dual
n n
[
Problem: max U(x) + µ y – Σ pixi ] min Σ pixi + λ[υ – U(x)]
x i=1
x i=1
Solution
function:
V(p, y) C(p, υ)
Response x * = Di(p, y) xi* = Hi(p, υ)
function: i slide 11

13. Bentuk Umum
Objective Function
z = f ( x, y )
Constraint
c = g ( x, y )
Lagrangian
L = f ( x, y ) + λ [c − g ( x, y )]
slide 12

14. Penyelesaian (FOC)
Necessary Conditions
Lλ = c − g ( x, y ) = 0
Lx = f x − λg x = 0
L y = f y − λg y = 0
slide 13

15. Aplikasi 1: Maksimisasi utilitas dengan
pendapatan terbatas
Utility Function
U = x1 x2 + 2x1
Budget Constraint
4 x1 + 2 x2 = 60
Lagrangian
L = x1 x2 + 2 x1 + λ [60 − 4 x1 − 2 x2 ]
slide 14

16. Necessary Conditions
∂L
= 60 − 4 x1 − 2 x2 = 0
∂λ
∂L
= x2 + 2 − 4λ = 0
∂x1
∂L
= x1 − 2λ = 0
∂x1
Tentukan nilai x1 dan x2
slide 15

17. Teorema Envelope
Teorema yang membahas perubahan nilai
optimal suatu fungsi dengan berubahnya salah
satu parameter dalam fungsi tersebut
slide 16

18. The Envelope Theorem
Substituting into the original objective function
yields an expression for the optimal value of y (y*)
y* = f [x1*(a), x2*(a),…,xn*(a),a]
Differentiating yields
dy * ∂f dx1 ∂f dx 2 ∂f dxn ∂f
= ⋅ + ⋅ + ... + ⋅ +
da ∂x1 da ∂x 2 da ∂xn da ∂a
slide 17

19. Marshallian Demand
The derivation of
an ordinary
demand curve.
Budget lines B1,
B2 and B3 show
different prices of
apples but the
same income and
price of oranges.
DM is the ordinary
(Marshallian)
demand curve.
slide 18

20. Hicksian Demand
The derivation of an
income-adjusted demand
curve. Budget lines B1, B2
and B3 show different
combinations of prices
and income
corresponding to the
same real income. DH is
the resulting income-
adjusted (Hicksian)
demand curve.
slide 19