Question

- A consumer’s preferences over two goods
*(**x**1**,**x**2**)*are represented by the utility function*u**x**1**,**x**2**=5**x**1**+2**x**2*. The income he allocates for the consumption of these two goods is*m*. The prices of the two goods are*p**1*and*p**2*, respectively.

- Determine the monotonicity and convexity of these preferences and briefly define what they mean.

- Interpret the marginal rate of substitution
(
*MRS(**x**1**,**x**2**)*) between the two goods for this consumer.

- For any
*p**1**,**p**2*, and*m*, calculate the Marshallian demand functions of*x**1*and*x**2*including the corner solutions if they exist.

- Consider a price change in
*x**1*from*p**1**=£10*to*p**1**'**=£20*assuming*p**2**=£5*and*m=£50*. Calculate the substitution effect (SE) and income effect (IE) on*x**1*for the given price change. Considering this range of price change only, are these goods normal or inferior? Are they ordinary or Giffen? Explain using the SE and IE that you calculated.

- Is
*x**1*normal or inferior in general? Is it ordinary or Giffen in general? Briefly explain. Make sure to include its behaviour at the corner solutions (if they exist).

Answer #1

Alice’s preferences over two goods are described by the utility
function u(x1, x2) = 2x1+ 4x2. Her income is m= 100 and p1= 4, p2=
5. Assume now that the price of good 1 falls to p01= 2.
a) Find the substitution, income, and total effect for good
1.
b) Find the substitution, income, and total effect for good
2.
c) Verify that the Slutsky equation holds for both goods

Let Antonio and Kate’s preferences be represented by the utility
functions, uAntonio(x1, x2) = 9((x1)^2)(x2) and uKate(x1, x2) =
17(x1)((x2)^2), where good 1 is Starbursts and good 2 is M&M’s.
Antonio’s endowment is eA = (24, 0) and Kate’s endowment is eK =
(0, 200). Antonio and Kate will exchange candy with each other
using prices p1 and p2, where p1 is the price of one starburst and
p2 is the price of one M&M.
a) Determine Antonio’s and Kate’s...

2. A consumer has the utility function U ( X1,
X2 ) = X1 + X2 +
X1X2 and the budget constraint
P1X1 + P2X2 = M ,
where M is income, and P1 and P2 are the
prices of the two goods. .
a. Find the consumer’s marginal rate of substitution (MRS)
between the two goods.
b. Use the condition (MRS = price ratio) and the budget
constraint to find the demand functions for the two goods.
c. Are...

Bilbo can consume two goods, good 1 and good 2 where
X1 and X2 denote the quantity consumed of
each good. These goods sell at prices P1 and
P2, respectively. Bilbo’s preferences are represented by
the following utility function: U(X1, X2) =
3x1X2. Bilbo has an income of m.
a) Derive Bilbo’s Marshallian demand functions for the two
goods.
b) Given your answer in a), are the two goods normal goods?
Explain why and show this mathematically.
c) Calculate Bilbo’s...

Consider the utility function U(x1,x2) = ln(x1) +x2. Demand for
good 1 is: •x∗1=p2p1 if m≥p2 •x∗1=mp1 if m < p2 Demand for good
2 is: •x∗2=mp2−1 if m≥p2 •x∗2= 0 if m < p2 (a) Is good 1
Ordinary or Giffen? Draw the demand curve and solve for the inverse
demand curve. (b) Is good 2 Ordinary or Giffen? Draw the demand
curve and solve for the inverse demand curve. (c) Is good 1 Normal
or Inferior? Derive and...

Qin has the utility function U(x1, x2) = x1 + x1x2, where x1 is
her consumption of good 1 and x2 is her consumption of good 2. The
price of good 1 is p1, the price of good 2 is p2, and her income is
M.
Setting the marginal rate of substitution equal to the price
ratio yields this equation: p1/p2 = (1+x2)/(A+x1) where A is a
number. What is A?
Suppose p1 = 11, p2 = 3 and M...

11. a. Suppose David spends his income M on goods x1 and x2,
which are priced p1 and p2, respectively. David’s preference is
given by the utility function
?(?1, ?2) = √?1 + √?2.
(i) Derive the Marshallian (ordinary) demand functions for x1
and x2.
(ii) Show that the sum of all income and (own and cross) price
elasticity of demand
b.for x1 is equal to zero. b. For Jimmy both current
and future consumption are normal goods. He has...

Bundes preferences are given by the utility function
u(x1+x2)=x1+x2. Suppose p2=3 and m=24. Show all working and plot
this consumers PCC when p1 drops continuously from 6 to 2.

Consider a consumer who consumes two goods and has utility
function
u(x1,x2)=x2 +√x1.
The price of good 2 is 1, the price of good 1 is p, and income is
m.
(1) Show that a) both goods are normal, b) good 1 is an ordinary
good, c) good 2 is a gross substitute for good 1.

1. Suppose utility for a consumer over food(x) and clothing(y)
is represented by u(x,y) = 915xy. Find the optimal values of x and
y as a function of the prices px and py with an income level m. px
and py are the prices of good x and y respectively.
2. Consider a utility function that represents preferences:
u(x,y) = min{80x,40y} Find the optimal values of x and y as a
function of the prices px and py with an...

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