Question

A consumer has a demand function for good 2, ?2, that depends on the price of good 1, ?1, the price of good 2, ?2, and income, ?, given by ?2 = 2 + 240/(??2) + 2?1. Initially, assume ? =40, ?2 = 1, and ?1 = 2. Then the price of good 2 increases to ?2′ = 3.

a) What is the total change in demand for good 2?

b) Calculate the amount of good 1 consumed at the optimal bundle before and after the price change.

c) What is the amount of income needed to make the initial optimal bundle just affordable at the new prices? [5 marks]

d) Find the optimal consumption of good 2 at the income level you found in part c) after the price change. [1 mark]

e) Calculate the amount of change in good 2 due to the Slutsky substitution effect of the price change. [3 marks]

f) Calculate the amount of change in good 1 due to the Slutsky substitution effect of the price change. [2 marks]

g) Calculate the amount of change in each good due to the income effect of the price change. [3 marks]

h) For good 2, check that the Slutsky identity does indeed hold in this example. [1 mark]

i) Assume well-behaved preferences and draw a diagram, with ?1 on the horizontal axis and ?2 on the vertical axis, showing the Slutsky substitution effect and income effect you calculated in parts e) through

g). Label the initial budget line, the final budget line, and the line that passes through the initial bundle and is parallel to the final budget line, and show the axis intercepts for each. Show the coordinates of the optimal bundles on each budget line and draw the indifference curves through these optimal bundles. Show the substitution and income effects on both goods with arrows. [6 marks]

j) Using your diagram, for each good, determine whether it is an inferior good or normal good. Justify your answer. If there is not sufficient information to determine the type of good, explain why. [3 marks]

k) Using your diagram, for each good, determine whether it is an ordinary good or Giffen good. Justify your answer. If there is not sufficient information to determine the type of good, explain why.

Answer #1

Solution :

A) total change in demand for x2= -4 (demand decreased by 4 units)

B) before price change optimal x1= 14

After price change, optimal x1=8

C) Amount of income needed to make the initial bundle just affordable at new prices is 64

D) optimal consumption of good 2 at income in part (c) and new prices is 7.25

Problem 1 [32 marks] A consumer has a demand function
for good 2, ?2, that depends on the price of good 1, ?1, the price
of good 2, ?2, and income, ?, given by ?2 = 2 + 240 ??2 + 2?1.
Initially, assume ? = 40, ?2 = 1, and ?1 = 2. Then the price of
good 2 increases to ?2 ′ = 3.
a) What is the total change in demand for good 2? [2 marks]
b)...

Please solve all the parts.Thank you.
A consumer can spend her income on two products, good X and good
Y . The consumer’s tastes are represented by the utility function
U(x, y) = xy.
a. Suppose that Px = 4 and Py = 1, and I = 16.
Draw the budget line and mark it as BL1. Initial optimum is at A.
Find the optimal amounts, xA and yA and locate A on the graph. Find
the initial level of...

3. Nora enjoys fish (F) and chips(C). Her utility function is
U(C, F) = 2CF. Her income is B per month. The price of fish is
PF and the price of chips is PC. Place fish
on the horizontal axis and chips on the vertical axis in the
diagrams involving indifference curves and budget lines.
(a) What is the equation for Nora’s budget line?
(b) The marginal utility of fish is MUF = 2C and the
Marginal utility of chips...

A consumer has utility for protein bars and vitamin water
summarized by the Cobb-Douglas utility function U(qB,qW) =
qBqW.
e. Find the consumer’s Engel curve for vitamin water when PB =
PW = 1.
f. What is the consumer’s optimal bundle when M = 100 and PB =
PW = 1?
Suppose the price of protein bars increases to P’B = 2.
g. Find the new optimal bundle.
h. Find the substitution effect of the price increase on
purchases of...

Amy has income of $M and consumes only two goods: composite good
y with price $1 and chocolate (good x) that costs $px per unit. Her
util- ity function is U(x,y) = 2xy; and marginal utilities of
composite good y and chocolate are: MUy = 2x and MUx = 2y.
(a) State Amy’s optimization problem. What is the objective
function? What is a constraint?
(b) Draw the Amy’s budget constraint. Place chocolate on the
horizontal axis, and ”expenditure all other...

3. Suppose that a consumer has a utility function given by
U(X,Y) = X^.5Y^.5 . Consider the following bundles of goods: A =
(9, 4), B = (16, 16), C = (1, 36).
a. Calculate the consumer’s utility level for each bundle of
goods.
b. Specify the preference ordering for the bundles using the
“strictly preferred to” symbol and the “indifferent to” symbol.
c. Now, take the natural log of the utility function. Calculate
the new utility level provided by...

Utility cobb douglas function = 2X.5Y2
MUx =Y2/X.5 MUy
=4X.5Y Px=1 PY=2 and M=100
1.Graph the consumer optimization problem in(X,Y) space. Clealy
label the precise location of the optimal bundle, the budget
constraintm, and the shape of the furthest obtainable indifference
curve.
2.Assume Px increase to 2. What is the total effect of the price
change in terms of X and Y.
3. What is the precise location of the bundle used to decompose
the substitution and income effect?
4.What...

4. Suppose a consumer has perfect substitutes preference such
that good x1 is twice as valuable as to the consumer as good
x2.
(a) Find a utility function that represents this consumer’s
preference.
(b) Does this consumer’s preference satisfy the convexity and
the strong convex- ity?
(c) The initial prices of x1 and x2 are given as (p1, p2) = (1,
1), and the consumer’s income is m > 0. The prices are changed,
and the new prices are (p1,p2)...

Suppose a consumer only consumes two goods. There is a price
drop of good 1 and the quantity demanded of good 1 increased from 5
unit to 20 unit, the substitution effect is 10 unit. Use a graph to
show the income effect and substitution effect for these two goods.
Label the direction of substitution effect and income effect and
calculate income effect.

Consider utility function u(x1,x2)
=1/4x12
+1/9x22. Suppose the prices of good
1 and
good 2 are p1 andp2, and income is
m.
Do bundles (2, 9) and (4, radical54) lie on the same
indifference curve?
Evaluate the marginal rate of substitution at
(x1,x2) = (8, 9).
Does this utility function represent
convexpreferences?
Would bundle (x1,x2) satisfying (1)
MU1/MU2 =p1/p2 and (2)
p1x1 + p2x2 =m be an
optimal choice? (hint: what does an indifference curve look
like?)

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