Econ 300 - Solutions to Problem Set 1 - Julien Bengui
10.1.8
First we write the firm's profit function using the factor and output price information as (K, L) = 4 9L1/3 K 1/3 - 12L - 6K = 36L1/3 K 1/3 - 12L - 6K. To find the optimal levels for each
Econ300 Spring 2008
Second Midterm Exam version W
This exam consists of 25 multiple choice questions. The maximum duration of the exam is 50 minutes. 1. In the spaces provided on the scantron, write your last name, then your first name, and also be
Econ 300, Problem Set 5, Suggested Answers
Professor Cramton
10.1.8.
First we write the rms prot function using the factor and output price information as
(K, L) = 4 9L1/3 K 1/3 12L 6K
= 36L1/3 K 1/3 12L 6K
To nd the optimal levels for each input, we writ
Econ 300 - Solutions to Problem Set 1 - Julien Bengui
2.1.2
(a)
Yes, this is a function. It maps each x in the domain into one and only one y in the range.
(b)
No, this is not a function. It maps each x in the domain into more than one y. For insta
1. y x 4 ln x . Then
A. y 4 x 3 ln x
1
B. y 4 x 3
x
C. y 4 x 3 ln x x 3
D. y 4 x 3 ln x x 5
E. None of the above
2. y ln( x 2 4 x 8) . Then
2
4
x
y e( 2 x 4)
2x 4
y
x
2x 4
y 2
x 4x 8
None of the above
A. y
B.
C.
D.
E.
3. y
4
. Then
x
A. y 2 x3/2
2
B. y
1
ECON300.1: Chapter 2
Introduction to Functions
Vocabulary
1. Consumption function: shows how consumption varies with income
2. Production function: shows how firms output varies with level of input
3. Qualitative variable: distinguishing characteristic
Econ 300: Problem Set 1
Due Wednesday, February 6, in class
1. Determine whether the following relationships are functions or not over the specied
domain for x values. Defend your answer in one sentence.
(a) y x ; x (, )
(b) y = + x ; x (0, ), where is a
Econ 300: Problem Set 4
Due Wednesday, April 17, in class
1. Find the partial derivatives, second partial derivatives and cross-partial derivatives
(a)
y = f (x1 , x2 ) = 5x5 2x2 x2 + x3
1
1
2
(b)
y = f (x1 , x2 ) = (x2 + 3x1 + 3)(5x2 + 1)
1
2. Consider t
Econ 300: Problem Set 4
7. (c)
dy
Fx
y 2 ey
y 2 ey
y
=
=
=
=
y + y 2 ey )
y (2 + y)
dx
Fy
x(2ye
xye
x(2 + y)
Not dened when Fy = 0. xyey (2 + y) = 0 x = 0,y = 0 or y = 2.
9. (a) gu = 20 4u 2v = 0
gv = 16 2v 2u = 0
Using elimination (subtract second equati
Econ 300: Problem Set 3
Due Monday, March 11, in class
1. Consider the function y = 4x2 2x + 7.
(a) Write down the expression for the dierence quotient and evaluate it at x0 = 3
and x = 3.
(b) Holding x0 constant, determine the impact on the dierence quot
8. (b)
y=
ln x
1
=
2x
2
ln x
x
To nd the stationary points set the derivative equal to zero:
dy
1
=
dx
2
1
xx
ln x
x2
=
1 1 ln x
=0
2 x2
1 ln x = 0 ln x = 1 x = e
To classify the point, check the second derivative:
1
d2 y
1 x x2 2x(1 ln x)
1 x 2x + 2x l
Econ 300: Problem Set 3
6. Find the stationary points of the following functions and determine whether they represent
local maxima or local minima.
dy
(a) dx
= 2x 12 = 0.
So, the stationary point is
Check second derivative to classify.
Then, the stationar
ECON 300 - WINTER 2012
MIDTERM EXAM SOLUTIONS
1.
(a) Not a function since ln x is not dened for x (1, 0]
(b) Not a function since the function maps a single value of x into multiple values of y for
x [2, 5]
(c) Not a function since there are two possible
ECON 300 - WINTER 2012
MIDTERM EXAM
1. (8) Which of the following expressions are functions and which are not functions in their
respective domains (domains are indicated inside the parentheses)? For those that are not
functions, provide the reason.
(a) y
Example 3: Find fl and f3, if fix , y} is given by
fix , y} = x eM 1"
Sclutipn tc Example 3:
Differentiate with respect Is 1: assuming 3; is censtant
3f 3
f=¢=lxe]=e+xyexf=c<y+tie
x a):
Differentiate with respect is 5r
3f 3
fy=[xe1=cnxen=x2e
w w
Selutien
ECON 300
Systems of Equations
Burak Trkgl
Systems of equations
Model many participants
Model many markets
Model many related variables
Solving systems of equations
Comparative statics
Find impact of change in exogenous variable on
endogenous variabl
1
ECON300.8: 9.1-9.2
Optimal Outcomes, Interest rates, etc
1. Maximum values identify
a. Highest profit, utility, tax rev
2. Minimum values identify
a. Lowest cost, price, risk
Identifying Extreme Values
1. Stationary Points
a. x* is a stationary point of
1
ECON300.6: Chapter 7.2-7.3
Second Derivative
1. First Derivative
a. Rate of change
b. Slope of function
c. Velocity
2. Second Derivative
a. Rate of change of the rate of change
b. Rate of change of the slope of function
c. Rate of change of orig functio
Chapter 6.1-6.4
Why differential calculus?
1. Economic models assume rational optimizers
a. Consumers maximize utility
b. Producers maximize profits
2. Optimization uses calculus to evaluate tradeoffs
a. How much to consume?
a.i. Consume until marginal ut
1
Chapter 3.1: Calculating Growth
Modeling Growth
1. Exponential functions
a. Constant percentage growth per unit time
2. Logarithmic functions
a. Inverse of expo functions
Rate of Growth of Income
1. High rates dramatic improvements standard of living ov
1
Chapter 4.1
Systems of Equations
1. Model many participants
2. Model many markets
3. Model many related variables
4. Solving systems of equations
5. Comparative statics
a. Find impact of change in exogenous variable on endogenous variables
b. Exogenous:
Econ 300, Problem Set 1 Professor Cramton
2.1.2.
Determine which of the following relationships represent functions. Assume that the interval is the set of real numbers unless otherwise indicated. (a) y = 5x (b) y x (c) y = a + (d) y = x2 (e) y 2 = x (f)
Problem Set 6
Econ 300
Fall 2013
Due Date: December 10th
Please return your answers in legible handwriting. Do dont forget to write your NAME and University
ID. Please show your work, you will be awarded partial credit for coherent attempts, but no credit
ECON 300
Univariate Calculus
Burak Trkgl
Overview
Rules of Dierentiation
Elasticities
Second Derivative and Curvature
Rules of Dierentiation
In the previous section, we saw calculating derivatives as the
dierence quotient as
x 0.
But this is cumbersome fo