Intermediate Micro
Homework Assignment 1
Due Thursday, July 10
1.) Find, and simplify, an equation for the level sets of the following functions (in the y=…
form).
a. 𝑘 = 2x + 3𝑦
𝑘
2
𝑦 = 3 − 3𝑥
b. 𝑘 = ln(𝑥) + 𝑏 ln(𝑦) , x>0, y>0
𝑘
−1
𝑦 = 𝑒𝑏𝑥 𝑏
c. 𝑘 = √𝑎𝑥 2 + 𝑏𝑦 2 , x>0, y>0, a>0, b>0
𝑘2
𝑎
𝑦 = √𝑏 − 𝑏 𝑥 2
𝑦
2.) Find the slope of the level set of 𝑓(𝑥, 𝑦) = ln(𝑥) + 3 𝑥 , x>0 .
3𝑦 − 𝑥
𝑥
3.) Find the x and y that maximize x + 𝑦 subject to the constraint 2 = √𝑥 2 + 𝑦 2 , for x>0,
y>0.
𝑥 = 𝑦 = √2
4.) Kelly’s dog kennel keeps small dogs (S), and large dogs (L). She faces 3 different
constraints to the number of dogs she can keep over the weekend.
For parts a-c, draw a graph of the pairs (S,L) that Kelly can keep under the constraint
given in that part. Each section should get its own graph. Label end points and any kink
points. When graphing, act as if it is possible to keep fractions of a dog (draw lines, not a
series of dots).
a. State law requires a kennel to have no more than 20 pets per employee watching
them. Kelly will have 3 workers watching the dogs at all times.
1
b. Kelly has 120 cups of dog food for each day. Small dogs need 1 2 cups of dog food
per day, while large dogs need 4 cups per day.
c. Kelly has 24 large cages, and 40 small cages. Each large dog needs its own large
cage, and each small dog needs its own cage of either size.
d. Draw a graph of all possible pairs (S,L) if Kelly obeys all of these restrictions.
Label any kink points
e.
i. If (S,L) = (58,2), what is the opportunity cost of adding 1 small dog?
1 large dog
ii. If (S,L) = (40,15), what is the opportunity cost of adding 1 small dog?
3/8 of a large dog
iii. If (S,L) = (12,24), what is the opportunity cost of adding 1 small dog?
0 large dogs (there are empty cages, extra dog food, and attention to
spare).
5.) There are two goods, essentials (s) and entertainment (n). As a rule, Adelaide spends
80% of her income on essentials, and 20% on entertainment. She has a monthly income
of $2000, and 𝑝𝑠 = $10, 𝑝𝑛 = $1.
a. Write down an equation for Adelaide’s budget line. What is her consumption of s
and n?
2000 = 10s + n
s = 160, n = 400
b. Suppose the government charges Adelaide $200 in tax. Write down an equation
for Adelaide’s budget line. What is her consumption of s and n?
1800 = 10s + n
s = 144, n = 360
c. Suppose that, instead of the lump-sum tax, the government charges a $0.50
quantity tax on entertainment goods. Write down an equation for Adelaide’s
budget line. What is her consumption of s and n?
2000 = 10s + 1.5n
s = 160, n = 266.7
d. Which tax scheme would Adelaide prefer?
There is not enough information to tell. 