Optimisasi dengan batasan persamaan (Optimization with equality constraints) Mengapa batasan relevan dalam kajian ekonomi?

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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 z 2 w 1 z 1 + w 2 z 2 c" Draw set of points where cost of input is c, a constant Repeat for a higher value of the constant Imposes direction on the diagram... w 1 z 1 + w 2 z 2 c' w 1 z 1 + w 2 z 2 c z 1 Use this to derive optimum slide 2

4 Cost-minimisation z 2 q The firm minimises cost... Subject to output constraint Defines the stage 1 problem. Solution to the problem z* z 1 minimise m Σ w i z i i1 subject to φ(z) q But the solution depends on the shape of the inputrequirement set Z. What would happen in other cases? slide 3

5 Convex, but not strictly convex Z z 2 Any z in this set is cost-minimising An interval of solutions z 1 slide 4

6 Convex Z, touching axis z 2 z 1 Here MRTS 21 > w 1 / w 2 at the solution. z* Input 2 is too expensive and so isn t used: z 2 *0. slide 5

7 Non-convex Z z 2 z* There could be multiple solutions. z** But note that there s no solution point between z* and z**. z 1 slide 6

8 Aplikasi 1: Optimalisasi kepuasan konsumen The primal problem x 2 objective function Tujuan konsumen adalah memaksimalkan utilitas Batasannya adalah budget Constraint set x* max U(x) subject to n Σ p i x i y i1 Cara lain memandang persoalan ini adalah... x 1 slide 7

9 The dual problem x 2 z 2q υ Constraint set Konsumen bertujuan meminimalkan pengeluaran Untuk mencapai utilitas tertentu x* z* minimise n Σ p i x i i1 subject to U(x) υ Contours of objective function xz 1 slide 8

10 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 p i x i + λ[υ U(x)] i1 n U(x) + µ[ y Σ p i x i ] i1 slide 9

11 A neat connection Compare the primal problem of the consumer......with the dual problem x 2 x 2υ υ The two are equivalent x* x* So we can link up their solution functions and response functions x 1 x 1 Run through 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 Problem: Solution function: Response function: Primal n max U(x) + µ[y Σ p i x i ] x i1 V(p, y) x i * D i (p, y) Dual n min Σ p i x i x i1 C(p, υ) x i * H i (p, υ) + λ[υ U(x)] slide 11

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

14 Penyelesaian (FOC) Necessary Conditions L λ (, ) 0 c g x y L x f x λ g x 0 L y f y λ g y 0 slide 13

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

16 Necessary Conditions λ λ x L x x L slide 15 Tentukan nilai x1 dan x λ λ x x L x x

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 [x 1 *(a), x 2 *(a),,x n *(a),a] Differentiating yields dy da * 2 f dx1 f dx x da x da f x n dx da n + f a slide 17

19 Marshallian Demand The derivation of an ordinary demand curve. Budget lines B 1, B 2 and B 3 show different prices of apples but the same income and price of oranges. D M is the ordinary (Marshallian) demand curve. slide 18

20 Hicksian Demand The derivation of an income-adjusted demand curve. Budget lines B 1, B 2 and B 3 show different combinations of prices and income corresponding to the same real income. D H is the resulting incomeadjusted (Hicksian) demand curve. slide 19