Midterm Exam for Practice
Instructions. You have 90 minutes to solve the questions. To get full credit you need to provide a
detailed justication of your claims (a picture is not a proof!).
1. A consumer with income y buys x1 units of good 1 at price p1 a
Solution to Problem Set 1
1. For each function below, determine whether it is concave, quasiconcave,
or neither. (Assume x; y 0:)
p
a. f (x; y) = x y
Concavity
1 p1
2 y
3
1
4x
4y
0
H (f (x) =
1 p1
2 y
11
4y
Since jH (f (x)j =
!
< 0 and f is C2 , then f is
Southern Economic Journal 2005, 71(3), 636660
Supermodularity and Complementarity in
Economics: An Elementary Survey
Rabah Amir*
The literature on supermodular optimization and games is surveyed from the perspective of potential
users in economics. This m
Theory of the Firm
Slides 6
UC Santa Cruz
2016
(UC Santa Cruz)
2016
1 / 13
Readings
Chapter 3 of J - R.
Chapter 5 of M - W - G
(UC Santa Cruz)
2016
2 / 13
Primitives of the Model
The main ingredient of the model is a production function f : Rn+ ! R+
with
Natalia Lazzati
ECON 204A: Microeconomic Theory
Mathematical Preliminaries
Note 2: Quasiconcave and Pseudoconcave Functions
Note 2 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21).
Monotonic Transformations: Cardinal Versus Ordinal
Natalia Lazzati
ECON 204A: Microeconomic Theory
Mathematical Preliminaries
Note 1: Convex Sets and Convex Functions
Note 1 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and
21).
Concave functions play a key role in optimiz
Choice-Based Approach to Consumer Theory
Slides 5
UC Santa Cruz
2016
(UC Santa Cruz)
2016
1/9
Readings
Chapter 2.3 of J - R.
Chapter 2 of M - W - G
(UC Santa Cruz)
2016
2/9
Choice-Based Approach
Denition
A Walrasian demand function maps price-income pairs
Donald Wittman
APPENDIX A1
Mathematical Background
Constrained Calculus: Example:
Maximizing utility subject to a budget
Greater Utility
constraint. The triangle bounded by the
y
x and y axes and the budget constraint
.
is the feasible set. Choose the hig
1
Concavity, Convexity, Quasi-concavity and Quasi-convexity
There are two concepts relating to the curvature of a function that are important in economic theory. These are concavity (and convexity) and quasi-concavity (quasiconvexity). Concavity (convexit
Economics 204A Midterm 2 Fall 2011
1. Given the following indirect utility function, determine the demand curve:
3,?
P1+P2
V(p|=pzay):
. . 9%
2. Show that the foklowing is a cost function. Also determine K. ([8)
r
L
c* = 6 31w + 323 + 2133(wi)2J B[,B2,
HOMEWORK 2
The following questions are formulated so that you will gain an intuitive feel for the KT
conditions.
For the following, first establish (1) whether the conditions on the objective function are
met and (2) whether the constraint qualifications
,0 = t
)1+ k(
t
.0 1+ Tk
, tc
f+1
and
) k( f +
t
) 1+tc(
0=
tk )
)1+ k(
t
.0 =
t
t
1+ T
> 0k
k
1+ Tk
L
1+t
k
1+tk
L
0 1+tk
0=
t
tk )
u = ) tc( u
1+ Tk ) Tc(
0=
u T
t T
t T
t
0=
0=
1+tk
L
0
t
t
t T
0 t 1 T
f + 1 1+t =
1+tk
L
0
c
,0 ) c( u =
L
c 0 1
Q97
l. The individual maximizes the sum of discounted expected utilities over two times periods.
3140- [/0014 (LA0
Max 2 = U(W[1s]) + [ld][l/2][U(sW[l+r+e]) + U(sW[l+reJ)]
Where W is original wealth; s is the savings rate; d < l is the discounted value of
Economics 204A MTl ' 2011
1. Suppose that there is a perfectly competitive firm and that Q(K, L) is negative definite and
that QKL < O.
A. Derive the KuhniTucker Conditions for a profitwmaximizing firm. (4) f :1! [5;
B. Prove that K and L are homogeneous
Economics 205A
Fall 2012
K. Kletzer
Problem Set 3 Sample Answers
1. a) Ans: The households problem is given by
sd sc gol )ts(e
and
a
t t = t
vd vr st
,0
ar
sa, sccfw_
r
t
s
e sa mil
. For the household,
t
+
t
a
t
given initial financial wealth,
t
= U x
[ii-Mill. W i. Joe Ho hit (imi
1. Assume the following: it NW "Vi/i Ti
A) Completenesshe can rank order all the feasible contingent outcomes (La, Lb, Lc).
B) Preferences are transitive and reexive.
C) The individual chooses the most preferred feasible con
1
Prot Maximization
The theory of the rm is rst presented in terms of general functional forms (Lectures 1-4)
and then in Lecture 6 we consider the Cobb-Douglas production function. For Lectures
1-4 the homework is to redo the previous lecture under the a
Econ 204A: Fall 2016
Producer Theory, Consumer Theory, and Decision Theory
Problem Set 3
1. Suppose that a consumer (with a standard utility function) buys clothing for the warmth, w,
it provides and that each unit of clothing provides k units of warmth.
Natalia Lazzati
ECON 204A: Microeconomic Theory
Mathematical Preliminaries
Note 4: Implicit Function Theorem
Note 4 is based on Apostol (1975, Ch. 13), de la Fuente (2000, Ch.5) and Simon and Blume
(1994, Ch. 15).
This note discusses the Implicit Function
Natalia Lazzati
ECON 204A: Microeconomic Theory
Mathematical Preliminaries
Note 3: The Envelope Theorem
The Envelope theorem plays a key role in microeconomics as it describes whether consumers
or rms are better-o or worse-o after a change in their enviro
Preference-Based Approach: Hicksian Demand
Slides 4
UC Santa Cruz
2016
(UC Santa Cruz)
2016
1 / 10
Expenditure Minimization Problem
The expenditure minimization problem is given by
minx fp x : x 2 Rn+ , u (x )
where p 2 Rn+ , u
ug
u (0) and p x = ni=1 pi
Consumer Theory
Slides 2
UC Santa Cruz
2016
(UC Santa Cruz)
2016
1 / 25
Readings
Chapters 1.1 and 1.2 of J - R.
Chapters 1 and 3A - C of M - W - G
(UC Santa Cruz)
2016
2 / 25
Aim of Consumer Theory
We want a model (or a theory) of how a person chooses wha
Econ 204A: Fall 2016
Producer Theory, Consumer Theory, and Decision Theory
Problem Set 4
1. Calculate the cost function and the conditional demand functions for y = min f x1 ; x2 g :
2. Under what assumptions do we have @x1 (w; y) =@w2 > 0 for a two-input
Econ 204A: Fall 2016
Producer Theory, Consumer Theory, and Decision Theory
Problem Set 1
1. For each function below, determine whether it is concave, quasiconcave, or neither. (Assume
x; y
0:)
p
(a) f (x; y) = x y
(b) f (x; y) = u (x) + v (y) ; with u (x)
Econ 204A: Fall 2016
Producer Theory, Consumer Theory, and Decision Theory
Problem Set 2
1. Solve for the Marshallian demands in (with a > 0):
max(x1 ;x2 )2R2+ fU (x1 ; x2 ) = minfx1 ; ax2 g : p1 x1 + p2 x2
yg :
2. J-R, 1.28, p. 62.
3. Given the indirect
Mathematical Preliminaries
Slides 1
UC Santa Cruz
2016
(UC Santa Cruz)
2016
1 / 10
Outline of the Course
List of topics:
1
Consumer theory
2
Theory of the rm
3
Monotone comparative statics
4
Decision under uncertainty and risk aversion
Methodological dime
Economics 205A
Fall 2012
K. Kletzer
Problem Set 4 Sample Answers
1. a) Ans: The household solves the problem,
0=t +
1xam cfw_
ta, tc
, tc tT tw + ta ) tr + 1( = 1+ta
0 1+ Ta
and the solvency condition,
sr + 1 1=s T
1 T mil
0a
given initial assets,
, ) tc(