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
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
Iso-cost lines Draw set of points where cost of input is c, a constant
z2
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
z1
Use this to derive optimum
slide 2
Cost-minimisation The firm minimises cost... Subject to output constraint
z2
q
Defines the stage 1 problem. Solution to the problem
minimise m
Σ wizi i=1
subject to φ(z) ≥ q
z*
z1
But the solution depends on the shape of the inputrequirement set Z. What would happen in other cases?
slide 3
Convex, but not strictly convex Z z2
Any z in this set is cost-minimising
An interval of solutions
z1 slide 4
Convex Z, touching axis z2
Here MRTS21 > w1 / w2 at the solution.
z*
z1
Input 2 is “too expensive” and so isn’t used: z2*=0. slide 5
Non-convex Z z2
z*
There could be multiple solutions.
z**
But note that there’s no solution point between z* and z**.
z1 slide 6
Aplikasi 1: Optimalisasi kepuasan konsumen The primal problem x2
Tujuan konsumen adalah memaksimalkan utilitas Batasannya adalah budget
objective function
max U(x) subject to n
Σ pixi ≤ y
Constraint set
i=1
x*
Cara lain memandang persoalan ini adalah...
x1 slide 7
The dual problem Konsumen bertujuan meminimalkan pengeluaran Untuk mencapai utilitas tertentu
xz22
υq Constraint set
minimise n
Σ pixi i=1
subject to U(x) ≥ υ
x* z*
Contours of objective function
xz11 slide 8
The Primal and the Dual… There’s an attractive symmetry about the two approaches to the problem
In both cases the ps are given and you choose the xs. But… …constraint in the primal becomes objective in the dual… …and vice versa.
n Σ pixi+ i=1
λ[υ – U(x)]
[
n
U(x) + µ y – Σ pi xi i=1
]
slide 9
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* x1
x1
Run through the primal
slide 10
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
Problem: max U(x) + µ y – Σ pixi x
Solution function:
V(p, y)
Response x * = Di(p, y) function: i
i=1
]
n
min Σ pixi + λ[υ – U(x)] x
i=1
C(p, υ) xi* = Hi(p, υ) slide 11
Bentuk Umum
Objective Function Constraint
z = f ( x, y )
c = g ( x, y )
Lagrangian L = f ( x, y ) + λ [c − g ( x, y )]
slide 12
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
Aplikasi 1: Maksimisasi utilitas dengan pendapatan terbatas
Utility Function Budget Constraint
U = x1 x2 + 2x1 4 x1 + 2 x2 = 60
Lagrangian L = x1 x2 + 2 x1 + λ [60 − 4 x1 − 2 x2 ]
slide 14
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
Teorema Envelope
Teorema yang membahas perubahan nilai optimal suatu fungsi dengan berubahnya salah satu parameter dalam fungsi tersebut
slide 16
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 = ⋅ + ⋅ + ... + ⋅ + ∂x1 da ∂x 2 da ∂xn da ∂a da slide 17
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
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 incomeadjusted (Hicksian) demand curve. slide 19
