EconS_305__Homework_1-13.pdf
EconS 305: Intermediate Microeconomics w/o Calculus
Homework 1:
Consumer Preference and Decision Making
Due: Friday, May 22nd, 2020 at 5:00pm via
Blackboard
- Please submit all homework solutions in the order the questions are presented and as one
.PDF.
- Please redo all the [EXAMPLE] solutions for Question 1, and submit them with you
homework.
- Please show all calculations as these exercises are meant to help refine your quantita-
tive abilities. If I can not follow your calculations or it seems as you just “copy and pasted”
answers from the internet, I will be deducting half the points from that solution.
1. Co
-Douglas with the preference restriction of ↵ + � = 1
Consider the following utility maximization problem of the consumer where the consume
prefers to consume some bundled amount of rice (q1) and beans (q2). We model the consumer’s
preference for these goods with the classic Co
-Douglas Utility function (U(q1, q2) = q↵1 q
�
2 ).
For more information on this utility function please navigate to to this website
(http:
www2.hawaii.edu/⇠fuleky/anatomy/anatomy.html). For simplicity, we assume that
↵+ � = 1 which means that the consumer prefers more of each good, and that each additional
amount of good yields the same utility as the the previous amount of good. We refer to this in
economics as a consumer having a constant returns to scale in terms of consumption. This also
means that we can represent the utility function as U(q1, q2) = q↵1 q
1�↵
2 since � = 1� ↵. Next,
we employ a linear budget constraint to model the consumer’s budget (p1q1 + p2q2 = M). p1
is the price the consumer pays to consume one unit of rice, and p2 is the price the consume
pays to consume one unit of beans. M is the consumer’s entire budget for rice and beans, and
we assume the consumer is willing to spend their entire budget for rice and beans, on rice and
eans.
We represent the consumer’s Utility Maximization Problem (UMP) as
max
q1,q2�0
U(q1, q2) = q
↵
1 q
1�↵
2
subject to the budget constraint of:
p1q1 + p2q2 = M
CALCULUS PART:
Using constrained optimization techniques from calculus, we can set the problem up with a
Lagrange multiplier s.t.
L(q1, q2) = q↵1 q1�↵2 � � [p1q1 + p2q2 �M ]
From here, we can take our derivatives and set them equal to zero
@L(q1, q2)
@q1
= ↵q(↵�1)1 q
1�↵
2 � �p1 = 0 (1)
@L(q1, q2)
@q2
= (1� ↵)q↵1 q
(�↵)
2 � �p2 = 0 (2)
@L(q1, q2)
@�
= p1q1 + p2q2 �M = 0 (3)
where we now have three equations ((1),(2), and (3)), and two choice variables (q1 and q2) to
solve for.
CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE.
(a) Find the consumer’s demand for q1 and q2 in equili
ium (i.e. find q⇤1 and q
⇤
2).
[EXAMPLE] - Please redo these calculations for practice and submit them with
your homework.
Solving for � in equations (1) and (2), and setting them equal to each other yield a result
of
↵q(↵�1)1 q
1�↵
2
p1
=
(1� ↵)q↵1 q
(�↵)
2
p2
Canceling on both sides
=) ↵q
�↵�1
1 q
1��↵
2
p1
=
(1� ↵)��q
↵
1⇢
⇢⇢q(�↵)2
p2
2
Where we get one choice variable in terms of the other choice variable
q2 =
(1� ↵)
↵
p1
p2
q1 (4)
Using this equation, we plug it into equation (3) and solve for the only choice variable we have
(q1) such that
p1q1 + p2
✓
(1� ↵)
↵
p1
p2
q1
◆
= M
p1q1 +⇢⇢p2
✓
(1� ↵)
↵
p1
⇢⇢p2
q1
◆
= M
p1q1
✓
1 +
(1� ↵)
↵
◆
= M
p1q1
✓
⇢↵� (1�⇢↵)
↵
◆
= M
p1q1
✓
1
↵
◆
= M
=) q⇤1(p1,M,↵) =
↵
p1
M (5)
which, intuitively, is the consumer’s demand function for rice. Notice that the the consumer’s
demand function for rice is a function in terms of variables in which the consumer is not choosing
(i.e. p1,M,↵). Plugging this function back into equation (4), we get
q⇤2 =
(1� ↵)
↵
p1
p2
q⇤1
q⇤2 =
(1� ↵)
⇢↵
⇢⇢p1
p2
✓
⇢↵
⇢⇢p1
M
◆
q⇤2(p2,M,↵) =
(1� ↵)
p2
M (6)
which is the consumers demand function for beans.
In summary, the quantity demanded for rice (q⇤1) and beans (q
⇤
2), respectively, is
(q⇤1(p1,M,↵), q
⇤
2(p2,M,↵)) =
✓
↵
p1
M ,
(1� ↵)
p2
M
◆
(7)
(b) What happens to each demand with an increase in it’s own price (pi), the other good’s price
(pj), the consumer’s budget (M), and the consumer’s preference for rice (↵)?
[EXAMPLE] - Please redo these calculations for practice and submit them with
your homework.
i) The consumer’s quantity demanded for rice and beans both decrease with respect to an
3
increase in their own price. This should be a property of your demand functions as it follows
the law of demand.
ii) Both demand functions are not e↵ected by the price of the other good.
iii) Both demand functions increase as the budget of the consumer increases. This makes
sense as an increase in wealth would mean that the consumer could consume more rice and
eans.
iv) Notice that as ↵ increase, the consumer’s demand for rice increase and their demand fo
eans decreases. This is because of the condition we set on ↵ such that ↵ + � = 1. This
implies that, from a preference stand-point, these goods can be substituted to some degree fo
the consumer. This is a neat concept because economists can then build models that estimate
these preference parameters to determine equili
ium conditions.
(c) Find the consumers utility in terms of their optimal demand (i.e. find U(q⇤1, q
⇤
2)). In economics,
we refer to this as the indirect utility function as it is in terms of exogenous parameters (factors
other than the choice variables).
[EXAMPLE] - Please redo these calculations for practice and submit them with
your homework.
U(q⇤1, q
⇤
2) =
✓
↵
p1
M
◆↵ ✓
(1� ↵)
p2
M
◆1�↵
U(q⇤1, q
⇤
2) =
✓
↵
p1
◆↵ ✓
(1� ↵)
p2
◆1�↵
M
(d) Find the own price elasticity of demand for rice using
@q⇤1
@p1
= � ↵
p21
M .
[EXAMPLE] - Please redo these calculations for practice and submit them with
your homework.
Using the definition of the own price elasticity of demand s.t.
"q⇤1 ,p1 =
�q⇤1
�p1
p1
q⇤1
=
@q⇤1
@p1
p1
q⇤1
And substituting in our results
"q⇤1 ,p1 = �
↵M
p21
p1⇣
↵M
p1
⌘
"q⇤1 ,p1 = �
↵M
p21
p21
↵M
= �1
4
Where we can interpret this as a “unit elastic” demand where a 1% increase in good 1’s price,
yields a 1% decrease in the quantity demanded for good 1 (rice). Also note that it is the
standard in economics to take the absolute value of the elasticity so that we can characterize
the type of elasticity it is (i.e. it is unit elastic since |"q⇤1 ,p1 | = 1).
(e) Find the cross price elasticity of demand for rice using
@q⇤1
@p2
= 0.
[EXAMPLE] - Please redo these calculations for practice and submit them with
your homework.
Using the definition of the cross price elasticity of demand s.t.
"q⇤1 ,p2 =
�q⇤1
�p2
p2
q⇤1
=
@q⇤1
@p2
p2
q⇤1
And substituting in our results
"q⇤1 ,p1 = 0
p2⇣
↵M
p2
⌘
"q⇤1 ,p1 = 0
Where we can interpret this as a “perfectly inelastic” demand with respect to the other good’s
price where a 1% increase in good 2’s price, yields a 0% change in the quantity demanded fo
good 1 (rice). Essentially, a price change in beans has no e↵ect on the consumer’s quantity
demanded for rice. Also note that it is the standard in economics to take the absolute value of
the elasticity so that we can characterize the type of elasticity it is (i.e. it is perfectly inelastic
since |"q⇤1 ,p2 | = 0).
5
2. Co
-Douglas with General Preferences
Consider the same setting we were operating in in Question 1, but now lets consider that
0 < ↵ + � 1. Notice that we cannot simply the problem as we did before (by replacing
� = 1� ↵).
We set up the consumer’s Utility Maximization Problem (UMP) as
max
q1,q2�0
U(q1, q2) = q
↵
1 q
�
2
subject to the budget constraint of:
p1q1 + p2q2 = M
CALCULUS PART:
Using constrained optimization techniques from calculus, we can set the problem up with a
Lagrange multiplier s.t.
L(q1, q2 : �) = q↵1 q
�
2 � � [p1q1 + p2q2 �M ]
From here, we can take our derivatives and set them equal to zero
@L(q1, q2;�)
@q1
= ↵q(↵�1)1 q
�
2 � �p1 = 0 (8)
@L(q1, q2;�)
@q2
= �q↵1 q
(��1)
2 � �p2 = 0 (9)
@L(q1, q2;�)
@�
= p1q1 + p2q2 �M = 0 (10)
where we now have three equations ((8),(9), and (10)), and two choice variables (q1 and q2) to
solve for.
CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE.
(a) Find the consumer’s demand for q1 and q2 in equili
ium (i.e. find q⇤1 and q
⇤
2).
(b) What are the similarities and di↵erences between the demands you found in Question 1?
(c) What happens to each demand with an increase in it’s own price (pi), the other good’s price
(pj), the consumer’s budget (M), and the consumer’s preference for rice (↵)?
(d) Find the consumers utility in terms of their optimal demands (i.e. find U(q⇤1, q
⇤
2)).
(e) Find the own price elasticity of demand for rice using
@q⇤1
@p1
= � ↵
(↵+�)p21
M .
(f) Find the cross price elasticity of demand for rice using
@q⇤1
@p2
= 0.
(g) Find the income elasticity of demand for rice using
@q⇤1
@M =
↵
(↵+�)p1
.
6
3. Log Utility with an Ad Valorem Tax
Consider a similar setting to Question 2, but now we have a new utility function. This utility
function is called the “Log” utility function, and is one of the more simpler “production” func-
tions to work with. We are still considering that 0 < ↵ + � 1, and now we are introducing
an “Ad Valorem Tax” (1+ ⌧) to the price of rice (good 1). Notice that the tax is some “added
value” to the price of rice, and that this it is applied to the price of each and every unit sold.
Also notice how it enters into the budget constraint of the consumer. This is the price the
consumer pays, and this is also the price the theoretical producer will get.
We modify the Co
-Douglas and set up the consumer’s Utility Maximization Problem (UMP)
as
U(q1,