NOTES AND PROBLEMS
IN MICROECONOMIC THEORY
PETER B. DIXON
La Trobe University
SAMUEL BOWLES
University of Massachusetts at Amherst
DAVID KENDRICK
University of Texas at Austin
in collaboration with
LANCE TAYLOR
Massachusetts Institute of Technology
MARC ROBERTS
Harvard University
m
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM • NEW YORK • OXFORD
vn
CONTENTS
Introduction
XI
Introduction to the Markham edition
XV
Chapter 1. Notes and problems in the theory of mathematical
programming
1.1.
1.2.
1.3.
1.4
1.5.
1.6
1.7.
1.8.
Introduction
Goals, reading guide and references
Formal statement of the problem
The gradient vector and contour diagrams
The necessary conditions for a constrained maximum
The role of convexity
The Lagrangian function
Displacement analysis and the interpretation of the Lagrangian
multipliers
Problem set 1
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
Exercise
1
2
4
6
8
15
18
21
24
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
1.10.
1.11.
1.12.
1.13.
1.14.
1.15.
1.16.
1.17.
1.18.
Standard form
Contour-gradient diagram
Necessary conditions for a constrained maximum
An easy optimization problem
A more difficult optimization problem
An exceptional case
Direction of steepest ascent
Sign constraints
Equality constraints
Linear constraints
Linear objective function
A nonquasiconcave objective function
Nonuniqueness of the problem solutions
Linear programming, primal and dual
On the interpretation of the Lagrangian multipliers
An example of nonunique Lagrangian multipliers
A problem with multiple unconnected solutions
The interpretation of the Lagrangian multipliers when
there are inequality constraints
Exercise 1.19. Restrictions on the domain of/and g
24
24
25
28
29
34
37
38
40
43
44
44
45
47
51
52
55
59
62
VIII
Contents
Chapter 2. Theory of the consumer: introduction
2.1.
2.2.
2.3.
2.4
Goals, reading guide and references
Notes on utility maximizing
Systems of demand equations
The implications of utility maximizing for demand systems
Problem set 2
Exercise 2.1.
Exercise 2.2.
Exercise 2.3.
Exercise 2.4.
Exercise 2.5.
Exercise 2.6.
Exercise 2.7.
Exercise 2.8.
Exercise 2.9.
Exercise 2.10.
Exercise 2.11.
Exercise 2.12.
Exercise 2.13.
Exercise 2.14.
Exercise 2.15.
Exercise 2.16.
Exercise 2.17.
65
69
73
75
77
The Engel aggregation
Homogeneity restriction
The linear expenditure system
The marginal utility of income
Displacement analysis, an example
Displacement analysis and the symmetry restriction
The triad
The Cournot aggregation
The Hicks-Slutsky partition
The negativity of the own-price substitution effect
The inferiority of Giffen goods
The sign of cross-elasticities of demand
Substitutes, the two-good case
Consumer behavior under rationing
The allocation of time
Pure exchange and Pareto optimality
Additive utility functions
Chapter 3. Theory of the consumer: extensions
3.1.
65
Goals, reading guide and references
Problem set 3
A. Problems on revealed preference
Exercise 3.1. Revealed preference for an individual
Exercise 3.2. Community welfare decisions (a revealed preference
problem)
Exercise 3.3. International welfare comparisons
Exercise 3.4. Revealed preference versus utility maximization
B. Problems using the concept of economic surplus
Exercise 3.5. A consumer surplus approach to the theater problem
Exercise 3.6. The application of the consumer surplus concept to social
decisions through cost-benefit analysis
(For discussion)
Exercise 3.7. Compensated demand functions, consumer surplus and
excess burden
Exercise 3.8. The costs of protection
Exercise 3.9. The costs of average cost pricing
77
78
79
84
86
90
93
95
96
97
101
102
103
104
107
109
114
123
123
Contents
Exercise 3.10. Economies of Scale, intraindustry specialization and the
costs of protection
IX
157
C. Some difficulties concerning the use of micro restrictions at the
macro level
165
Exercise 3.11. Aggregation across households
Exercise 3.12. Homogeneity and aggregate demand functions
165
168
Exercise 3.13. The additivity assumption and aggregation across households
169
D. An integrability proposition
173
Exercise 3.14. The triad and integrability
173
E. Obstacles to Pareto optimal market solutions
181
Exercise 3.15. Interdependent utilities
181
Exercise 3.16. Alliance military expenditures as a public good
183
F. Intertemporal consumer behavior
185
Exercise 3.17. The elimination of stock variables from demand equations
for durables
Exercise 3.18. Savings as a function of the rate of interest
Exercise 3.19. Intertemporal consumption and investment
Exercise 3.20. Vicious circles
Exercise 3.21. Terminal conditions for multiperiod consumption models
Exercise 3.22. Intertemporal inconsistency
G. Decisions under uncertainty
Exercise 3.23. The determination of utility functions for risky decisions
Exercise 3.24. The St. Petersberg paradox
Exercise 3.25. The boundedness of the utility function
Exercise 3.26. Measurable utility (a review question)
186
188
190
193
195
202
207
207
212
213
214
Chapter 4. Production theory
4.1.
4.2.
Goals, reading guide and references
Background notes on some recent developments in production
function theory
Problem set 4
A. Some properties of production functions
Properties of two-factor, linearly homogeneous production
functions
Exercise 4.2. Factor price frontier
Exercise 4.3. Properties of the Cobb-Douglas production function
Exercise 4.4. A'Generalized'Cobb-Douglas production function
215
215
221
224
224
Exercise 4.1.
224
230
231
234
Contents
B. Static optimization
Exercise 4.5. Optimization subject to neoclassical technology
Exercise 4.6. Optimization in a two-sector-model
Exercise 4.7. A planning problem
Exercise 4.8. The construction of the production possibilities frontier
from a set of production functions and resource endowments
Exercise 4.9. The construction of a production function
Exercise 4.10. Class interests and induced technical change
Exercise 4.11. Technological change, factor supplies and income distribution
C. An introduction to linear models
Exercise 4.12.
Exercise 4.13.
Exercise 4.14.
Exercise 4.15.
Optimization with linear technologies
An illustration of Samuelson's (non)substitution theorem
Income distribution in Ricardia
A factor-price frontier for an economy with linear technologies
D. Theoretical developments underlying modern production function
econometrics
Exercise 4.16.
Exercise 4.17.
Exercise 4.18.
Exercise 4.19.
Exercise 4.20.
Exercise 4.21.
Exercise 4.22.
Some formal analysis of the standard cost minimizing model
Some properties of cost functions
The Allen definition of the elasticity of substitution
Alternative definitions of the elasticity of substitution
The CES production function
The CRESH production function
An example of a flexible functional form: the generalized
Leontief function
Exercise 4.23. Direct specifications of the input demand functions
Index
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