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5. (a) Solve the problem max 1-rx² - y? subject to x+y m, with r > 0. (b) Find the value function f"(r, m), compute af"/ar and af"/am and verify (15). SM6. (a) Solve the problem max x + y? + z? subject to x + y² +4z? = 1 and x+3y+2z = 0

5. (a) Solve the problem max 1-rx² - y? subject to x+y m, with r > 0. (b) Find the value function f"(r, m), compute af"/ar and af"/am and verify (15). SM6. (a) Solve the problem max x + y? + z? subject to x + y² +4z? = 1 and x+3y+2z = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 31E
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Can you help with question 5? Item (15) is attached for reference. 

60 ト CHAPTER 3 I STATIC OPTIMIZATION 5. (a) Solve the problem max 1 – rx² – y² subject to x + y = m, with r > 0. :- (b) Find the value function ƒ*(r, m), compute af*/ar and af*/@m and verify (15). SM 6. (a) Solve the problem max x² + y² +z? subject to x² + y? + 4z² =1 and x+3y + 2z = 0 (b) Suppose we change the first constraint to x² + y? + 4z² = 1.05 and the second constraint to x+3y +2z = 0.05. Estimate the corresponding change in the value function. 7. (a) In Example 4 let U (x) = L=1 aj In(x; – aj), where aj, aj, Pj, and m are all positive constants with £-i ¤j = 1, and with m > Li=1 Pia¡. Show that if x* solves problem (16), then the expenditure on good j is the following linear function of prices and income UNJearnings.csv 2021S QF202 quiz w.R Removed prt sc トト 114 %24 4. 9. 08.
Transcribed Image Text:60 ト CHAPTER 3 I STATIC OPTIMIZATION 5. (a) Solve the problem max 1 – rx² – y² subject to x + y = m, with r > 0. :- (b) Find the value function ƒ*(r, m), compute af*/ar and af*/@m and verify (15). SM 6. (a) Solve the problem max x² + y² +z? subject to x² + y? + 4z² =1 and x+3y + 2z = 0 (b) Suppose we change the first constraint to x² + y? + 4z² = 1.05 and the second constraint to x+3y +2z = 0.05. Estimate the corresponding change in the value function. 7. (a) In Example 4 let U (x) = L=1 aj In(x; – aj), where aj, aj, Pj, and m are all positive constants with £-i ¤j = 1, and with m > Li=1 Pia¡. Show that if x* solves problem (16), then the expenditure on good j is the following linear function of prices and income UNJearnings.csv 2021S QF202 quiz w.R Removed prt sc トト 114 %24 4. 9. 08.
Envelope Result Consider the following version of the general Lagrange problem (1): max f(x, r) subject to gj(x, r) = 0, j= 1, ., m (13) where x = (x1,..., Xn) and r = (r1, ., rk) is a vector of parameters. Note that we can absorb each constant b; in (1) by including it as a component of the parameter vector r and by including a term -b; in the corresponding function g¡(x, r). If we put g = (g1, ..., gm), the m constraints can then be written as a vector equality, g(x, r) = 0. Note that in problem (13) we maximize w.r.t. x, with r held constant. The value function for problem (13) is f*(r) = max{ f(x, r) : g(x, r) = 0} (14) Let the Lagrangian be defined as L(x, r) = f(x, r) –E^18;(x, r) We want to find an expression for af*(r)/ar¡ at a given point F, assuming there is a unique optimal choice x*(F) for x. Let 1, . ., àm be the associated Lagrange multipliers. Under certain conditions (see Theorem 3.10.4), we also have the following relationship: ENVELOPE RESULT L (x, r)' (1).fe i = 1, ., k (15) (4),x==1/ JNJearnings.csv .R Removed
Transcribed Image Text:Envelope Result Consider the following version of the general Lagrange problem (1): max f(x, r) subject to gj(x, r) = 0, j= 1, ., m (13) where x = (x1,..., Xn) and r = (r1, ., rk) is a vector of parameters. Note that we can absorb each constant b; in (1) by including it as a component of the parameter vector r and by including a term -b; in the corresponding function g¡(x, r). If we put g = (g1, ..., gm), the m constraints can then be written as a vector equality, g(x, r) = 0. Note that in problem (13) we maximize w.r.t. x, with r held constant. The value function for problem (13) is f*(r) = max{ f(x, r) : g(x, r) = 0} (14) Let the Lagrangian be defined as L(x, r) = f(x, r) –E^18;(x, r) We want to find an expression for af*(r)/ar¡ at a given point F, assuming there is a unique optimal choice x*(F) for x. Let 1, . ., àm be the associated Lagrange multipliers. Under certain conditions (see Theorem 3.10.4), we also have the following relationship: ENVELOPE RESULT L (x, r)' (1).fe i = 1, ., k (15) (4),x==1/ JNJearnings.csv .R Removed
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