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Microeconomics Assignment: Utility Functions and Elasticities, Assignments of Economics

Marshall UniversityEconomics

Solutions to assignment questions related to utility functions, cross-elasticity of demand, income elasticity, and price elasticity of demand. Students will learn how to calculate these elasticities and interpret their meaning.

Typology: Assignments

Pre 2010

Uploaded on 07/30/2009

koofers-user-ylf
koofers-user-ylf 🇺🇸

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Assignment 6
Intermediate Micro
1. Ella’s utility function for two goods, ‘x’ and ‘y’ is given by min {3x, y}.
a. If the price of x is 9 and the price of y is 15, and her income is $500 how much of each
will she purchase?
b. What is the cross-elasticity of demand between goods ‘x’ and ‘y’? Interpret your
answer.
c. What is the cheapest bundle that Ella could purchase that she would like as well as the
bundle (x, y) = (25, 12)?
d. Find the income elasticity of good x. Interpret your answer.
2. Janice has a utility function of the form :
U = ( xy) 10501
6503
a. Can you write Jan’s utility function in a simpler form? If Jan’s income is $1000 and
px = 5 and py = 10, how much x and y will Jan purchase?
b. Using the information from part a.) above, if the government imposed a lump-sum
tax of $200, how much x and y will Jan purchase?
c. Using the information from part a.) above, what is Jan’s price elasticity of demand
for x? How does it change if px=10?
d. Using the information from part a.) above, what is her income elasticity of demand
for y? How does it change if income goes up to $2000?
3. Brian only cares about pencils. He gets all his satisfaction from them and none from
any thing else. He cares about the color of his pencils, but he will always be indifference
between 3 blue and 5 red pencils, and multiples of these amounts.
a. Provide three mathematical expressions that represent his utility.
b. Let pb be the price of blue pencils and pr be the price of red pencils. Give
expression of Brian’s demand function for the blue pencil and red pencils (Be
sure to include the range of prices the demand is relevant).
c. Brian has $1.00 to spend. If the price of blue pencils is $0.10 and the price of red
pencils is $.05, how many of each color will he buy?
d. Using the information from part c.) above, if the government imposes a quantity
tax of $.02 on red pencils, how many red pencils will he buy? How many blue
pencils?
e. Using the information from part c.) above, if the price of red pencils falls to $.04,
how many red pencils will he buy? How many blue pencils?
pf3
pf4
pf5

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Assignment 6 Intermediate Micro

  1. Ella’s utility function for two goods, ‘x’ and ‘y’ is given by min {3x, y}. a. If the price of x is 9 and the price of y is 15, and her income is $500 how much of each will she purchase? b. What is the cross-elasticity of demand between goods ‘x’ and ‘y’? Interpret your answer. c. What is the cheapest bundle that Ella could purchase that she would like as well as the bundle (x, y) = (25, 12)? d. Find the income elasticity of good x. Interpret your answer.
    1. Janice has a utility function of the form : U = ( xy) 10501 6503 a. Can you write Jan’s utility function in a simpler form? If Jan’s income is $1000 and px = 5 and py = 10, how much x and y will Jan purchase? b. Using the information from part a.) above, if the government imposed a lump-sum tax of $200, how much x and y will Jan purchase? c. Using the information from part a.) above, what is Jan’s price elasticity of demand for x? How does it change if px=10? d. Using the information from part a.) above, what is her income elasticity of demand for y? How does it change if income goes up to $2000?
    2. Brian only cares about pencils. He gets all his satisfaction from them and none from any thing else. He cares about the color of his pencils, but he will always be indifference between 3 blue and 5 red pencils, and multiples of these amounts. a. Provide three mathematical expressions that represent his utility. b. Let pb be the price of blue pencils and pr be the price of red pencils. Give expression of Brian’s demand function for the blue pencil and red pencils (Be sure to include the range of prices the demand is relevant). c. Brian has $1.00 to spend. If the price of blue pencils is $0.10 and the price of red pencils is $.05, how many of each color will he buy? d. Using the information from part c.) above, if the government imposes a quantity tax of $.02 on red pencils, how many red pencils will he buy? How many blue pencils? e. Using the information from part c.) above, if the price of red pencils falls to $.04, how many red pencils will he buy? How many blue pencils?
  1. U = min {3x, y} = min {x, ⅓y} a) For x: m = pxx + pyy m = pxx + py (3x) m = x (px+3py) x* = m = 500 = 500 = 9. px+3py 9+15(3) 54 For y: m = pxx + pyy m = px (⅓y) + pyy m = y (⅓px + py) y= m = 500 = 500 = 27. ⅓px + py ⅓ (9) +15 18 b) є = d x. py d py x x = m = m(px+3py)‾¹ px +3py d x*= -1(m)(px+3py) ‾ ² ∙ d py d x = -3m = -3(500) = -1500 = -0. d py (px+3py)² [9+3(15)]² 2916 є = -.514 * (15/9.26) = -.833 complements c) U=min{3x,y} For bundle (x, y) = (25, 12) → U = min {3(25), 12} U= min {75, 12} =12 of Y Note: Ella requires 3 of Y for every 1 of X Therefore, if she has 12 of Y → she requires 4 of X Cheapest bundle (4, 12)

b) Lump sum tax of $200 → m= 1000-200 = 800 X= 800. 1/2 = 160 = 80 5 2 Y= 800. 1/2 = 40 10 c) Ex = dx .px dpx x x* = m = px-1^ m 2px 2 dx = -½ m px- dpx dx = -m dpx 2 (px)^2 є = -m. px 2px^2 x m=1000 px=5 py=10 x= є = -1000. 5 = -5000 = - 2(25) 100 5000 Take the absolute value of є → 1, unitary elastic If px = 10? x = m = 1000 = 50 2px 2 (10) Є = -1000. 10 = - 2(100) 50 absolute value → 1 → unitary elastic d) η = dy. m dm y y* = m 2py dy = 1 dm 2py

η = 1. m 2py y m=1000 px=5 py=10 y* = 50 = 1. 1000 = 1 → normal good (border between luxury 2(10) 50 and necessity) if m = 2000 y* = m = 2000 = 100 2py 2(10) η = 1. m 2py y = 1. 2000 = 1 → normal good 2(10) 100 3.) 3 blue & 5 red a. U = 5b + 3r, U = 10b + 6r, U = 5/2 b + 3/2 r b. b* = m and r= 0 if 3pb < 5pr Pb r = m and b=0 if 3pb > 5pr Pr Any combination of blue and red pencils that spends all of m if 3pb = 5pr c. m = $1 pb = .10 pr =. 3pb 5pr 3(.10) 5(.05) .30 > .25 → buy red pencils only r = 1 = 20, b* = 0 .