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```					Chiang                                     Exercise                             Chapter 12

Ch. 12 369 Optimization w/ equality constraints

12.1     370 Effects of a Constraint

12.2     372 Finding the Stationary Values
12.2     378 1a,1c|4,5
12.2-1(a)
Use the Lagrange-multiplier method to find the stationary values of z when
z = xy, subject to x + 2y = 2

12.2-1(c)
Use the Lagrange-multiplier method to find the stationary values of z when
z = x – 3y – xy, subject to x + y = 6

12.3     379 Second-Order Conditions                    386
12.3     1a,|1c
12.3-1(a)
Use the bordered Hessian to determine whether the stationary value of z obtained in each part of
Exercise 12.2-1(a) is a maximum or a minimum.
z = xy, subject to x + 2y = 2

12.3-1(c)
Use the bordered Hessian to determine whether the stationary value of z obtained in each part of
Exercise 12.2-1(c) is a maximum or a minimum.
z = x – 3y – xy, subject to x + y = 6

12.4     387 Quasi-concavity and Quasi-convexity 399
12.4     2a,2c,|4a,4c,5,7

598928ea-0191-4515-a889-bd82c5c111b3.doc   - 1 of 2 -                             8/29/2011
Chiang                                     Exercise                                Chapter 12

12.5      400 Utility Maximization and Consumer Demand 409
12.5      1b, 1c,|3,5,7
12.5-1
Given U = (x + 2)(y + 1) and Px = 4, Py = 6, and B = 130
a) Write the Lagrangian function
b) Find the optimal levels of purchase xbar and ybar.
c) Is the second-order sufficient condition for maximum satisfied?
d) Does the answer in b) give any comparative-staic information

12.5-2
Given U = (x + 2)(y + 1), but this time assign no specific numerical values to the price and income
parameters.
a) Write the Lagrangian function
b) Find xbar , ybar, and lambdabar in terms of Px, Py, and B.
d) By setting Px = 4 and Py = 6, and B=130, check the validity of your answer to 12.5-1.

12.5-3
Can your solution (xbar and ybar) in Exercise 12.5-2 yield any comparative-static information? Find the
comparative-static derivatives for (lambda, x, y) with respect to a change in B and Px, evaluate their
signs, and interpret their economic meanings.

12.6     410 Homogeneous Functions                         417
12.6     1a, 1c, 1e, 4,|7a, 8a, 8b, 8d

12.7     418 Least-Cost Combination of Inputs              430
12.7     1a,1b,2,4,|6a,6b,8a,8c

598928ea-0191-4515-a889-bd82c5c111b3.doc   - 2 of 2 -                                8/29/2011

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