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Unformatted text preview: UCLA
Economics 11 – Fall 2010
Professor Mazzocco
Problem Set 6
Due by November 12 before 9:00am in the box located outside room 2221E, Bunche Hall
Question 1)
Ms Ramirez has the following utility function: U(x,y)= ln(x)+ln(y)
Her total income is equal to I and she faces prices Px for good x and Py for good y.
a) Calculate the Marshallian demands and Hicksian demands.
Marshallian demands: compensated demands: b) Find the expenditure function. c) Suppose Px=10, Py=1 and I=22. If Px increases by 1%, what is the percentage change in the
Marshallian demand functions for the goods x and y? What is the percentage change in the
Hicksian demand functions for the goods x and y? (Hint: compute the elasticities)
New price
0.010891089 If you used the formula for infinitesimal changes: d) Are x and y gross substitutes? Are they net substitutes?
X and Y are not gross substitutes since marshallian demands do not depend in the other good’s
price.
They are Net substitutes since the compensated demand are increasing functions of the other
good’s price.
e) How much additional income is necessary in order to compensate Ms Ramirez for the price
change? (Hint: use the compensating variation) f) Suppose that the price didn’t change. What is amount of income that Ms Ramirez has to give
up to have the same level of utility as if the price had changed? Question 2) (Clarification of Problem) – This problem will not be considered for grading Mr. Macleod likes to drink wine. He has the option to buy wine in two different packages.
Package X has 1 liter of wine and Package Y has 1.6 liters of wine (Y). X and Y are perfect
substitutes. Each liter of wine gives Mr. Macleod 1 unit of utility.
a) Write Mr. Macleod utility function.
U(x, y) = x + 1.6 y represent Mr. Macleod preferences. This representation is not unique.
Someone could multiply the representation above by a positive constant and still would be a
utility function that represents his preferences. Note that all representations have the same MRS. b) Suppose that Px=5. Calculate the Marshallian demand for Y. Draw a graph for this demand
(Y as function of Py).
Note that 1.6 equals 8/5.
If MRS = 5/8 > 5/Py then y=0: If Py > 8 then y = 0
If MRS = 5/8 < 5/Py (or Py < 8) then y = I / Py
If Py=8 then
DEMAND: c) Now suppose that initially Px=5 and Py=10. But then the price of y changes to $7 . Calculate
the total change in demand for y. Explain your result.
MRS = 5/8 > 5/10 then y = 0. When Py=7, MRS = 5/8 < 5/7 so y=I/7. Then the change is I/7. Question 3)
The preferences of an individual are represented by the following utility function:
U(x,y)= ln(x)+ 2y
a) Determine if x and y are gross substitutes or gross complements. If (CASE 1) Then
And
If (CASE 2 – CORNER SOLUTION) Then
And
In case 1 a small increase in the price of y increases demand of x. A small increase in px doesn’t change
the demand for y.
In case 2 a small increase in the price of y doesn’t change the demand of y and an increase in px doesn’t
change the demand for x. b) Determine if x and y are net substitutes or net complements.
Hicksian Demands: if Note that and that the marshallian demand for x is equal to the hickisian demand (there is no wealth effect on good x) . Also
. so there is substitution for y when px increases and ...
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